Thierry Jaekel

Simona Cocco

Ulisse Ferrari

Gaia Tavoni

Tomoyuki Obuchi

Francesco 'équipe Du Pbpcd

Sébastien Zamponi

Federico Balibar

Alessandro Stella

Treves

J'ai

Benjamin Assel

Fabien Nugier

Konstantina Kontoudi

Evgueny Sobko

Pawel Laskós-Grabowski

Bruno Le Floch

Jean Taris

Alice Coucke

Antoine Bourget

R Monasson

S Rosay

A Statistical Mechanics approach to the modelling and analysis of place-cell activity

Keywords: hippocampus, place cells, attractor neural network, decoding, disordered systems, memory PACS number(s): 84.35.+i, 05.20.-y, 87.85.dq, 87.19.lv

Place cells in the hippocampus are neurons with interesting properties such as the correlation between their activity and the animal's position in space. It is believed that these properties can be for the most part understood by collective behaviours of models of interacting simplified neurons. Statistical mechanics provides tools permitting to study these collective behaviours, both analytically and numerically.

Here, we address how these tools can be used to understand place-cell activity within the attractor neural network paradigm, a theory for memory. We first propose a model for place cells in which the formation of a localized bump of activity is accounted for by attractor dynamics. Several aspects of the collective properties of this model are studied. Thanks to the simplicity of the model, they can be understood in great detail. The phase diagram of the model is computed and discussed in relation with previous works on attractor neural networks. The dynamical evolution of the system displays particularly rich patterns. The second part of this thesis deals with decoding place-cell activity, and the implications of the attractor hypothesis on this problem. We compare several decoding methods and their results on the processing of experimental recordings of place cells in a freely behaving rat.

Les cellules de lieu de l'hippocampe sont des neurones aux propriétés intrigantes, comme le fait que leur activité soit corrélée à la position spatiale de l'animal. Il est généralement considéré que ces propriétés peuvent être expliquées en grande partie par les comportements collectifs de modèles schématiques de neurones en interaction. La physique statistique fournit des outils permettant l'étude analytique et numérique de ces comportements collectifs.

Nous abordons ici le problème de l'utilisation de ces outils dans le cadre du paradigme du "réseau attracteur", une hypothèse théorique sur la nature de la mémoire. La question est de savoir comment ces méthodes et ce cadre théorique peuvent aider à comprendre l'activité des cellules de lieu. Dans un premier temps, nous proposons un modèle de cellules de lieu dans lequel la localisation spatiale de l'activité neuronale est le résultat d'une dynamique d'attracteur. Plusieurs aspects des propriétés collectives de ce modèle sont étudiés. La simplicité du modèle permet de les comprendre en profondeur. Le diagramme de phase du modèle est calculé et discuté en comparaison avec des travaux précedents. Du point de vue dynamique, l'évolution du système présente des motifs particulièrement riches. La seconde partie de cette thèse est à propos du décodage de l'activité des cellules de lieu. Nous nous demandons quelle est l'implication de l'hypothèse des attracteurs sur ce problème. Nous comparons plusieurs méthodes de décodage et leurs résultats sur le traitement de données expérimentales.

Mots-clés : hippocampe, cellules de lieu, réseau attracteur, décodage, systèmes désordonnés, mémoire.

Introduction

The work presented here is both about physics and neuroscience. More precisely, it tackles issues of computational neuroscience with tools from statistical mechanics. It deals with a neural system -the hippocampus -viewed as an attractor neural network -a paradigm for memory -from two different perspectives: the issue of modelling and the one of decoding. Modelling means that we propose a schematic picture of how the experimentally observed activity could be explained. Under the assumption that neuronal activities reflect an aspect of the external world, decoding consists in retrieving this aspect from the observation of the neurons' activity itself. The general aim is to show how the attractor neural network assumption can help better understand the observed properties of neurons in the hippocampus called place cells. This work is in line with the efforts in the physics community to apply its methods to problems across the boundaries of its discipline. It shares the view according to which many observed phenomena can be accounted for by collective behaviours of many agents at a smaller scale. This includes mental events, as we shall explain.

All these terms need a more precise definition. This will be the object of Chapter 1. The ideas underlying our approach will be clarified, introducing the attractor neural network theory (Section 1.3). The biological facts about the hippocampus and its place cells will be exposed in Section 1. 4.

In Chapter 2, we will present our work on the modelling issue. This chapter is based on the results presented in references [1,2], provided in the appendix. A model for place cells is introduced and its properties are investigated. The comparison with previous works on the same system is discussed in Section 2.6. A tentative link with experimental data is made in Section 2.7.

Chapter 3 deals with the decoding problem. After having discussed the definition of the problem (Section 3.1), we introduce and compare several methods to decode place-cell activity. These methods are applied to experimental data described in Section 3.2.

This thesis summarises the work done during the three years of my PhD. It would be pointless to attempt to present it as an accomplished and completed venture. In Chapter 4, we overview what is still to be done, what would be interesting to do, and the questions that remain. By "how the brain works", we mean two things. The first refers to its observed activity (electrical, metabolical activity. . . ). The second refers to its presumed functions (perception, motion, memory, consciousness. . . ). Indeed, nervous systems are living systems, and because they have been shaped by evolution they can often be described as fulfilling a certain function. Of course, the activity and the function are linked, and one would like to make the connection between them.

Nervous systems are made of cells called neurons, connected one to the other by synapses (see Figure 1.1). There is a great deal of evidence that neurons are the fundamental units of the brain's working. Neurons have an electrical activity whose main feature is the action potential (or spike), a very stereotyped phenomenon that is similar across species, neuron types, and brain areas. To describe it briefly, each neuron receives electrical currents from the other neurons (in its dendrites) and integrates them in its membrane potential. If the total input exceeds a certain threshold, the neuron "fires": the membrane potential abruptly increases before returning to its initial value, within about one milisecond. This action potential propagates along the axon and is in turn received by other neurons, via synapses. The synaptic connection relies on chemical substances called neurotransmitters, that differ from one synaptic type to the other. Depending on the neurotransmitter, the synapse is either excitatory or inhibitory -that is, an action potential coming from the presynaptic neuron can either favor or prevent a spike in the postsynaptic neuron.

Because the activity of a given neuron depends on the activity of its neighbours through the strength of the synaptic couplings, then ultimately the activity of the whole network is determined by the synapses. More precisely, many configurations of neural activities can be observed in a given network, but not any configuration (by configuration we mean which neurons spike and which are silent at a given time). Which configurations are possible is dictated by the synaptic structure, in the same way as the rules of soccer or rugby dictate which game configurations are allowed in each game. Furthermore, this synaptic structure is not fixed in time, but is influenced in turn by the neural activityalbeit the timescale over which this change takes place is far larger than the timescale of the action potential. This phenomenon, called synaptic plasticity, has been shown to be the basis of learning. Indeed, since the strength of the synapses governs the activity, it is intuitive that changing the rules will change the game.

Neurons are many. To give an order of magnitude, the human brain contains of the order of 10 11 neurons, connected by 10 15 synapses. Therefore, the brain's activity can be described at very different scales. Experimental techniques also cover a wide range of scales. At short time and length scales it is possible to record the action potentials of individual neurons, by micro-electrodes inserted in the neural tissue or by fluorescence (calcium imaging). These recording techniques have been greatly improved over the past decades, making it possible to observe simultaneously an increasing number of neurons with an increasing time resolution. At larger scales, the simultaneous activity of large numbers of neurons give rise to oscillatory electrical fields that can be detected on the scalp (the electro-encephalogram or EEG). At the scale of a brain area (a few centimeters), the activation of a whole region is visible, for instance, in functional magnetic resonance imaging (fMRI).

With the development of recording techniques, massive neural activity data become available. What should we do with it? What do we want to know exactly? When do we consider that we have understood what is going on?

Just as experimental techniques permit to observe the brain at various scales, a theorist willing to understand it can try to do it at different levels. The relevant scale at which we should place ourselves is still a matter of debate. Incidentally, we should keep in mind that we are dealing with a relatively young discipline: the idea that neurons are the basic functional units of nervous systems is but one century old (Ramón y Cajal, early 20 th century).

Computational neuroscience is a theoretical subfield of neuroscience. Its approach consists in proposing mathematical models for neural systems. These models aim at reproducing certain observed features of brain activity or a possible way a given function could be implemented [3]. Whether they do is checked by analytical and numerical tools. Whether a given model allows us to reach a satisfactory degree of understanding is however a subjective question. We will come back to it in the discussion (see Section 4.4).

Inside computational neuroscience, connectionism is a prominent theory [4]. It postu-lates that cognitive events (perception, memory. . . ) are collective states that emerge from the microscopic activity of large numbers of interconnected neurons. Since the 1970's, this approach has been very popular and has permitted some progress in many issues. We will place ourselves in this framework.

The Statistical Mechanics Approach

At this point, the reader may wonder why on earth this thesis is labeled "theoretical physics". Statistical mechanics is indeed a branch of physics, but the reason for this is more historical than anything else. Historically, this field was born in the context of thermodynamics at the end of the 19 th century and developed during the subsequent decades in the framework of condensed matter, with the objective to deduce macroscopic quantities as the result of microscopic laws, and to explain collective behaviours such as phase transitions. It relies on the constatation that, when moving from the individual to the collective level of description, novel, non-trivial phenomena appear -"more is different", as P.A. Anderson put it [5]. In the case of gases or condensed matter, the motivation for resorting to statistical mechanics is two-fold: (1) what we observe experimentally is the macroscopic scale, so we are mostly interested in macroscopic quantities and (2) even if we wanted, we would not be able to compute microscopic quantities, because the number of parameters is too big. A statistical approach thus encompasses both what we want to do and what we can do. In a sense, statistical mechanics is more a mathematical framework grown on physics grounds than true-blue physics. Thus, it is not limited to problems of physics: situations with multiple levels of description are ubiquitous. During the past decades, tools from statistical mechanics have been applied in many fields such as sociology, economy, finance, biology. . . and neuroscience, as we will explain here.

Coming back to the brain, it should now appear clearly that the aforementioned connectionist hypothesis makes mental states a statistical mechanics problem. There are microscopic units -the neurons -in interaction on a network -via the synapsesand each of them follows its own local, microscopic dynamics. From this we would like to deduce the global states. The relationship with statistical mechanics is thus straightforward. More precisely, this problem has been found to be particularly reminiscent of magnetic systems in which spins (magnetic moments) interact with each other: spins are binary units that can be either in a up or a down state, like as a neuron can be either firing a spike or quiescent. There is thus a direct kinship between the celebrated Ising model, which is the seminal model for magnetic systems, and schematic representations of neural networks [6]. To complete the analogy, in neural networks there may also be randomness, both in the structure of the network ("quenched noise") and in the response of each neuron ("fast noise", equivalent to a temperature parameter). Therefore, all the tools developed in the former case are also usable in the latter.

A lot of work has thus been done to tackle computational neuroscience problems with statistical mechanics approaches. Neural network models have been proposed that account for observations of the brain activity at macroscopic scales, or that perform certain functions. Moreover, another source of useful tools for computational neuroscience has been information theory, a field closely tied to statistical mechanics. It is helpful both in addressing issues such as perception or neural representation and massive data analysis.

Therefore, interdisciplinarity in general has proved a fruitful approach. Still, the communication between two different scientific communities is not always easy: for a physicist, "temperature" and "energy" refer to noise and probability, while for a neurophysiologist they evoke body warmth and metabolism -just to give an example.

The Attractor Neural Network Theory

Let us now turn to the particular issue of memory. Before asking how memory can be supported by nervous systems, we have to define what memory is. Here comes the attractor neural network hypothesis, proposed by Donald Hebb in his Organization of Behavior (1949) [7]. It can be expressed as follows:

What is memorised are configurations of activity of the neurons (i.e. firing patterns). A configuration is said to be memorised when it is a stable state of the network. In other words, a memory item is an activity configuration that is an attractor of the network's dynamics -hence the name of attractor neural network.

This assumption may seem surprising and artificial at first glance. Yet it is quite well accepted among neuroscientists and has received some experimental support. Very schematically, the idea is that, for something to be memorised, one has to be able to maintain for some time the neural activity that corresponds to this memory. The next question is: can we propose a neural model that stores memories? We have said before that the states of a neural network are driven by the synapses. We can thus rephrase the question: for given activity configurations, can we find synaptic couplings for which those configurations are attractors?

In 1982, John Hopfield [8] proposed such a network that became famous as the Hopfield model. It consists in a number N of binary (i.e. either active either silent) neurons {σ i } i=1...N and stores a number p of randomly selected binary configurations {ξ µ i } i=1...N,µ=1...p . ξ µ i is equal to +1 if neuron i is active in configuration µ, and -1 otherwise. The synaptic couplings that allow these configurations to be attractors are given by the so-called Hebb rule:

J ij = 1 N µ p=1 ξ µ i ξ µ j ∀ i, j . (1.1)
The idea underlying this rule is that, during a previous hypothetical learning phase, the p patterns have been presented to the network (i.e. the network has been forced to adopt these p configurations successively) and that neurons that were then active together reinforced the connection between them. The reinforcement of couplings between coactivated neurons, already postulated by Hebb in 1949, has been observed experimentally since then [9]. It is summarised by the proverb: "neurons that fire together wire together". The rule 1.1 also assumes that these successive modifications of the couplings are summed up additively. Note that each neuron is a priori coupled to all the others J ij = 0 ∀ i, j: the network is said to be recurrent.

The last thing to define is the dynamics of the network. In the original 1982 paper [8], time was discretized and at each time step neurons responded deterministically to their local fields. Later studies (see ref. [10]) incorporated the possibility of stochasticity in the response, through a noise (or temperature) parameter T , so that the system obeys detailed balance at temperature T under the Hamiltonian E = -i<j J ij σ i σ j .

(1.

2)

The introduction of this model aroused much excitement in the statistical mechanics community. Indeed, the Hopfield model has many common points with spin glasses (frustrated1 disordered magnetic systems), a very active domain of statistical mechanics. The properties of the Hopfield model have been intensively studied. The first question is of course to check whether the patterns {ξ µ i } i=1...N,µ=1...p are indeed attractors of the dynamics: it turns out that, for p and T not too large (i.e. not too strong noise and memory load), they are. From any initial state, the system will converge to the closest attractor, i.e. the stored pattern that most resembles this initial state. For this reason we speak about "autoassociative memory": feeding the network with a partial cue leads to the retrival of the whole pattern. The maximal value for p is proportional to the number N of neurons.

Many other aspects have been investigated: the dynamics of the system, its response to an external field. . . Moreover, lots of variants have been proposed to the model, in order to take into account this or that property of real neurons, or to give the network this or that function. For instance, the observation of low activity in real cortical networks can be incorporated [11]. . . It would be too long to review here all the work and variations that have been made on the Hopfield model (the reader can refer to ref. [12]). We will discuss at length, in the case of the hippocampus, how a basic attractor model can be thoroughly explored and refined.

To finish on a general remark, in all this work that has been done on attractor neural networks during the past thirty years, it is not always very clear whether the properties exhibited in this or that model are supposed to happen "for real" in biological systems. Statistical physicists, when they propose a model, like to explore every inch of it -even the parts that are beyond the scope of the system as a model for reality -just of the fun of it, if I may say so. Thus, the boundary between what is for real and what is for art's sake is not always neat. This is not necessarily bad: this taste for exhaustivity gives rise to very elegant investigations, and sometimes the properties thus exhibited turn out later to happen for real in real life. But one should always keep in mind that all the reality is not contained in the model, nor are all the model's properties present in reality.

The Hippocampus

The Hopfield model and its variants are theoretical views on what memory could be. Now, there are experimental observations indicating that the attractor mechanism is indeed the basis of memory, at least some sorts of memory in some parts of the brain, of which the hippocampus is one.

This section is dedicated to presenting this brain area and some of the empirical knowledge on it. Theoretical models will be presented and discussed in Chapter 2.

Overview

Types of memory Memory is the ability to retrieve after a delay an information or a skill that has been learnt before. Learning and retrieval are the two fundamental components of this process. Several types of memory are distinguished based on their contents, timescales and retrieval modes.

First, memory can be either short-or long-term: the former decays after some seconds to minutes (e.g. remembering a phone number while dialling it), while the latter can last a lifetime. Then, long-term memory can be either declarative or procedural. Declarative memory refers to explicit knowledge, that can be consciously recalled; procedural memory refers to non-conscious memory such as skills (e.g. riding a bike) and conditioning (e.g. salivating at the microwave bell). Within declarative memory itself, we distinguish episodic memory and semantic memory. The former is the memory of autobiographical events (e.g. one's childhood in Combray) and the latter is the memory of general facts (Marcel Proust was a writer). The two can contain similar facts, but in episodic memory these facts are associated to the context of their learning while in semantic memory they constitute a knowledge detached from one's personal experience. This classification of memory types is summarised in Figure 1.2. When discussing the role of the hippocampus, we will be mainly concerned with declarative memory.

Anatomy

The hippocampus is a region of mammalian brains; it has an homologue in most vertebrate species. In humans it is located ventrally, under the cortex in the medial temporal lobe, and its aspect makes think of a seahorse (hence its name). In rodents, in contrast, it lies caudally (see Fig. 1.3). Despite these differences of aspect from one species to the other, the general connectivity scheme of the hippocampus is well preserved across mammals. It is sketched in Figure 1.4. The hippocampal formation is composed of the CA fields (CA1 and CA3, from cornu ammoni, "Ammon's horn"2 ), the dentate gyrus (DG) and the subicular complex. Synaptic outputs from many cortical areas converge to the hippocampal formation through the entorhinal cortex (EC). The synaptic input from the entorhinal cortex to the hippocampal formation is called the perforant pathway (PP). The dentate gyrus projects to CA3 through the mossy fibers (MF) and CA3 projects to CA1 through the Schaffer collaterals (SC). Within CA3 itself there is a dense recurrent connectivity. CA1 sends axons back to the entorhinal cortex both directly and through the subicular complex.

Research on the hippocampus

The hippocampus is one of the most studied parts of the brain. Historically, it has been the object of two parallel, almost separate lines of research. The first concerns the human hippocampus, studied mostly through a behavioural approach in lesioned patients. The second line of research, making use of microelectrode recordings, focusses on the rodent hippocampus. As we will see, these two independent approaches led to apparently very different visions of the hippocampus.

In humans, the research on the memory function of the hippocampus really started with the seminal case of patient H.M. reported in 1957 by Scoville and Milner [13]. H.M. had undergone a bilateral medial temporal lobectomy in an attempt to relieve his epilepsy -in a nutshell, he had no hippocampus anymore. After the surgery, he suffered from memory disorders: he was unable to form new memories (anterograde amnesia) and also to remember some of his memories anterior to the lobectomy (retrograde amnesia). His other intellectual and perceptual skills, including short-term and procedural memories, were intact. The case of H.M. inspired the theory of an episodic memory function for the hippocampus and oriented the subsequent research carried out on humans, mostly patients with a lesioned hippocampus (see [14] for a review). Marr introduced an influential model for the role of the hippocampus as a temporary memory centre, where memories of new events are encoded before being gradually transferred to the neocortex, where they are permanently stored [15]. More recently, single-units recording data in humans became available thanks to the pre-surgical implantation of some epileptic patients. This is how Quian Quiroga et al discovered neurons in the hippocampus that are activated specifically by the presentation of pictures or the name of a person or an object, e.g. Jennifer Aniston, Halle Berry or the Sidney Opera House [16]. This discovery supported the assumption that the hippocampus forms episodic memories by binding together concepts, objects and people.

In rodents, on the other hand, everything started in 1971 when O'Keefe and Dostrovsky recorded single units activity in freely behaving rats and discovered that some cells in the hippocampus have the astonishing property to be active if and only if the rat is located at a given position [17]. These cells were named "place cells" and the location in space where a given place cell is active is called its "place field" (see Figure 1.5). They are pyramidal cells -a type of excitatory neurons, named after the shape of their soma -found in CA1 and CA3. This original finding led to the postulate that the hippocampus supports spatial memory and navigation [18]. Since then, a lot of research has been carried out on the rodent hippocampus, focussing on its presumed spatial function. Left: trajectory of the rat (black line) and positions where spikes were emitted by the cell (each red dot represents a spike). Right: corresponding smoothed rate map (i.e. average number of spikes per time unit as a function of position), displaying the characteristic bump of activity. The recording is taken from the dataset described in Section 3.2.

The most widely used experimental technique is the micro-electrode recording of indi-vidual cells in the behaving rat. With this technique, rats trained for a given spatial task are surgically implanted micro-electrode arrays in the hippocampus. After recovery, the position of the electrode is adjusted and the activity is recorded while the animal performs the task. The recording is numerically treated to sort out the spikes and allocate them to individual neurons. This way, the properties of place cells have been intensely investigated. They will be detailed in the next paragraph. In addition, other types of cells with spatial correlates were discovered over the years in regions neighbouring the hippocampus (Figure 1.6): there were first the "head-direction cells" found by Ranck and colleagues in 1984 in the presubiculum [19] and later reported in several other areas. These cells are active when the animal is facing a certain preferred direction, irrespectively of its position and speed. Then, "boundary cells" (or "border cells", responsive to walls of an environment) were reported in several regions of the hippocampal formation [20]. Last but not least, the "grid cells" were discovered by the Moser lab in 2004 in the medial entorhinal cortex [21,22]. These, like place cells, fire specifically at certain positions of space, but instead of being limited to a single spot their firing fields exhibit a nice spatial periodicity, with active positions located at the vertices of a triangular grid, the orientation and mesh of which depend on the recorded cell. All these discoveries, besides generating the excitement of the community, supported the trend to look for a spatial function of the hippocampal formation. In this respect, the research on the rodent hippocampus seems at odds with the more episodic flavour of its human counterpart. We shall see nevertheless how these two views are eventually reconciled.

Apart from humans and rodents, hippocampal activity has been studied in many species. Equivalents of place and grid cells have been found in sometimes phylogenetically distant species including primates [23] and bats [24].

Properties of place cells

Let us now turn to the experimental properties of place cells. In order to avoid any speculative interpretation about what the neurons "represent" or "compute", we will limit ourselves to what is observed, that is their firing correlates.

Some numbers

The estimated number of pyramidal cells in the rat CA3 is 2 × 250, 000; in CA1 it is around 2 × 390, 000 and in the dentate gyrus, 2 × 1, 000, 000 [25] -the factor 2 comes from the two brain hemispheres. Among them, a great majority are place cells (92% according to ref. [26]). The recurrent connectivity in CA3 is very dense, around 4%: each pyramidal cell hence receives around 12,000 axons from other CA3 cells, against 4000 perforant pathway axons and a few tens of mossy fibers, on average [27]. In addition to (excitatory) pyramidal cells, CA3 and CA1 contain inhibitory interneurons, which do not display spatial selectivity. Their number is less important -around 10% of the total number of neurons, but each interneuron is connected to thousands of pyramidal cells.

Place fields have variable sizes, shapes and peak firing rates. Moreover, field sizes depend on the size of the enclosure and on the position of the recorded cell [28]. To give an order of magnitude, the diameter of a place field ranges from a few centimeters to a few meters and the firing rate at its centre ranges from a few Hertz to tens of Hertz [29]. This peak firing rate can be modulated by running velocity [30]. Place cells have place fields in both two-dimensional and one-dimensional environments, though in the latter case (linear tracks) the place fields are most of the time directional, that is the place cell is active only when the field is crossed in a certain direction [30]. It has been also observed that the activity of a place cell within its place field is very variable from one trial to the other, a property called overdispersion [31].

Reference frame

When saying that a place cell is active at a fixed position, one needs to specify relative to what this position is fixed. Indeed, in the absence of an absolute origin, the reference frame must be stated. In the case of experiments with rodents, there are basically three reasonable alternatives (Figure 1.7): the position can be defined either relative to the ground (lab frame) or relative to the platform where the rat is evolving (platform frame) or relative to any object the rat has visual access to (and there are as many such cue frames as there are cues). A set of experimental studies have thus been carried out in order to determine to which frame the place fields are attached.

Place fields are stable in the dark [32] and under substitution of part of the visual cues [33]; different place cells are active in different environments sharing a common object [34]: these facts rule out the possibility of place fields tied to visual cues only. Nevertheless, visual cues do exert some influence on place fields as geometrical deformation of a box leads to a corresponding stretching of place fields [20,34]. Actually, all three alternatives mentioned above have found some experimental indications. Place fields linked respectively to the three frames have been found in an experiment where a platform was rotated in the dark and in the light [35]. In experiments where cues are moved, some place fields are bound to the cues while others remain fixed [36][37][38]. When a linear track is elongated, the place fields are fixed relative to the ground during the first part of the journey and switch to a cue frame near the end of the track [34]. However, modifying the visual aspect of the enclosure does not affect the position of the place fields as soon as the rat knows its position relative to the ground has not changed [39,40]. Consequently, when there is no disorientation, the lab frame seems to dominate.

The issue of the reference frame has been posed in the different terms of navigation [18]. If we assume that the hippocampus computes the animal's position, an eloquent parallel can be drawn with marine or aeronautic navigation. A ship or an aircraft can compute her position either by triangulation relative to daymarks, beacons, stars and satellites (visual, radar, celestial and GPS navigation respectively), or by integrating its trajectory by estimating the elapsed time, speed and heading from a previously calculated position ("dead reckoning" or "path integration"). Of course, all methods should in principle lead to the same result as the landmarks are supposed to be fixed with respect to the ground Figure 1.7: Illustration of the reference frame problem: Effect of cue displacement and platform shifting on the position of a place cell's place field centre (p.f.c.), in the cases of a p.f.c. fixed relative to the lab frame (left), to the platform frame (centre) and to the cue frame (right). and to each other. The parallel with place fields is straightforward: if place fields are fixed relative to the visual cues, then the rat will be assumed to perform visual navigation. If place fields are fixed relative to the platform, then the navigation is rather based on path integration. If finally place fields are fixed relative to the ground, then the navigation is also based on path integration, but with an additional inertial integration of the platform's motion, much alike the taking into account of currents in marine dead reckoning (or wind in aircraft).

According to the experimental results reviewed above, place fields can be tied to several frames. It seems nevertheless that the lab frame is the preferential one. In navigational terms, just as the sailor compares and combines different methods to check and refine its position, the rat uses both visual navigation and path integration, more precisely it corrects the error of path integration thanks to visual cues. Here we should emphasise that trying to disentangle the reference frames is after all quite artificial: in natural situations, these frames are most of the time equivalent and their dissociation requires drastic experimental manipulations where the animal is likely to be quite disoriented. More importantly, we do not know whether navigation is the ultimate function of the hippocampus.

Allocation of place fields

Besides their spatial firing correlate, another striking property of place cells is the apparently random allocation of their place fields. First, the relative position of the place fields of two given cells is totally uncorrelated with these cells' anatomical location in the neural tissue [41]. Moreover, one given cell can have several place fields and their positions do not display any regularity, so that if one considers two different "environments" (see definition below) then the populations of neurons having a place field respectively in each environment will be uncorrelated [42,[START_REF] Kubie | Multiple representations in the hippocampus[END_REF], as illustrated in Figure 1.8 . This property called remapping has been studied by various experiments, reviewed hereafter. Note the absence of systematic spatial relationship from one environment to the other. Also, place cells do not necessarily have a place field in all environments (there are "silent cells"), and place fields may overlap.

The notion of environment Many papers about the hippocampus make use of the notion of "environment" or "map" as a discrete unit of space, similar to rooms in a house for example [START_REF] Kubie | Multiple representations in the hippocampus[END_REF]. Indeed, this chunking of space makes sense in the experimental context, where the rat evolves in enclosed boxes of size around 50cm-1m.

Thus, in a given environment, only a fraction of place cells has place fields, ranging from 20 to 70% depending on the environments [26,[START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF]. These cells are called active cells in this environment, the others are called silent cells.

This notion of environment is convenient (though not necessary) to study the remapping phenomenon. In natural conditions there is no such obvious division of space, yet the place fields seem to be randomly allocated all the same [START_REF] Park | Ensemble place codes in hippocampus: Ca1, ca3, and dentate gyrus place cells have multiple place fields in large environments[END_REF]. Fragmenting space into discrete pieces raises the issue of their connection [18,[START_REF] Derdikman | A manifold of spatial maps in the brain[END_REF].

Conditions of remapping

Experiments aimed at determining the conditions under which a set of active cells was replaced by another. First, by definition of remapping, there is of course the case when the animal is taken from one environment to an other, different one [42]. Then, remapping also occurs between two environments that have similar appearances, as soon as the rat understands that they are located at difference places [39]. In similar conditions, partial remapping (some place cells remapping while the others maintaining their place fields) has also been observed in CA1 [START_REF] Skaggs | Spatial firing properties of hippocampal ca1 populations in an environment containing two visually identical regions[END_REF]. The reverse case, i.e. two boxes of different appearances located at the same place, does not lead to remapping per se but to a change of firing rate in the place fields (a property called "rate remapping" [39]). Some differences have been observed in this respect between CA1 and CA3, see Ref [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF] for more details.

Mechanism

The random allocation of place fields seems to be due to the dentate gyrus input to CA3 [START_REF] Treves | Computational constraints suggest the need for two distinct input systems to the hippocampal ca3 network[END_REF][START_REF] Leutgeb | Pattern separation in the dentate gyrus and ca3 of the hippocampus[END_REF]. Moreover, the remapping of place fields between two environments is accompanied by a simultaneous rotation and a shift of the "grid fields" of the grid cells in entorhinal cortex [START_REF] Fyhn | Hippocampal remapping and grid realignment in entorhinal cortex[END_REF]. So the randomness and variability observed in place fields is generated in upstream regions.

Learning and formation of place fields Place fields form in the few first minutes of exploration of a novel environment [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF][START_REF] Wilson | Dynamics of the hippocampal ensemble code for space[END_REF]. This mechanism seems to be innate: place, head-direction and grid cells are already present in the third week after birth (as soon as the pups start to navigate) [START_REF] Langston | Development of the spatial representation system in the rat[END_REF]. There is debate regarding whether there exist or not pre-wired spatial maps to which other, non-spatial stimuli would be associated [18,40].

Theta rhythm

Up to now our description of place-cell activity has been limited to whether a cell was active or not at a given position. But the observed activity is more complex than a binary on/off variable: the firing rate and the timing of the spikes are not trivial. When the rat is moving around, the global activity of hippocampal pyramidal cells is modulated by an oscillatory pattern of frequency 5-10 Hz clearly visible on the EEG signal, called theta rhythm. An interesting property of place cells is the so-called phase precession: when the animal is in the place field, the spikes are emitted relative to the theta rhythm at a preferential phase which advances as the animal moves forward [START_REF] O'keefe | Phase relationship between hippocampal place units and the eeg theta rhythm[END_REF]. A lot of experimental work has been done on the properties of the theta rhythm and phase precession (see [START_REF] Buzsáki | Theta oscillations in the hippocampus[END_REF] for a review).

Anticipative or delayed firing

Place cells do not only fire when the animal is in its place fields. They have also been observed to be sequentially activated when the animal has been or will be in the place field. The former, called replay, occurs when the animal is sleeping after having explored an environment [START_REF] Wilson | Reactivation of hippocampal ensemble memories during sleep[END_REF][START_REF] Skaggs | Replay of neuronal firing sequences in rat hippocampus during sleep following spatial experience[END_REF]. The latter consists in place cells firing a few seconds in advance forward of the rat's actual position [START_REF] Johnson | Neural ensembles in ca3 transiently encode paths forward of the animal at a decision point[END_REF].

Towards a reconciliation between human and rat theories

As space is a component of any animal's experience, episodic and spatial theories for the hippocampus are not mutually exclusive. Rather, the episodic view is more general than the spatial one and for this reason includes it. Even if this idea is not new -it dates back to Edward Tolman's visionary concept of "cognitive map" (1948) [START_REF] Tolman | Cognitive maps in rats and men[END_REF] and has been applied to the hippocampus by O'Keefe & Nadel [START_REF] O'keefe | The hippocampus as a cognitive map[END_REF] -the spatial and episodic aspects have been studied essentialy independently. Recent research on the hippocampus tends to unify these two approaches.

On the rodent side, although much attention has been given to the spatial aspect of place cells, substantial evidence has also been brought that their correlates are not purely spatial. Some cells in the hippocampus have both spatial and non-spatial or only nonspatial correlates (e.g. odour) [START_REF] Wood | The global record of memory in hippocampal neuronal activity[END_REF]. Some place cells' activity depends on the context of the task [START_REF] Markus | Interactions between location and task affect the spatial and directional firing of hippocampal neurons[END_REF][START_REF] Smith | Hippocampal place cells, context, and episodic memory[END_REF]. These additional dimensions in the correlates phase space have been proposed as a cause for the observed "overdispersion" mentioned above [START_REF] Jackson | Network dynamics of hippocampal cell-assemblies resemble multiple spatial maps within single tasks[END_REF]. Nevertheless, the definition and study of episodic memory in nonverbal animals are not straightforward, nor is the comparison with its equivalent in humans. On the human side, on the other hand, the spatial aspect is not absent. Patients with hippocampal lesion display spatial memory impairment. Above all, place cells and grid cells have been observed in the human hippocampus [START_REF] Ekstrom | Cellular networks underlying human spatial navigation[END_REF][START_REF] Doeller | Evidence for grid cells in a human memory network[END_REF], which completes the circle.

Hence a broader vision of the role of the hippocampus has emerged. In agreement with the temporary memory buffer hypothesis (initially made in humans, see above), some research has focussed on memory consolidation and transfer to the neocortex [START_REF] Mcclelland | Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory[END_REF][START_REF] Káli | Off-line replay maintains declarative memories in a model of hippocampal-neocortical interactions[END_REF][START_REF] Battaglia | The construction of semantic memory: grammar-based representations learned from relational episodic information[END_REF]. With the fast improvement of recording methods, many more discoveries on the hippocampal region are to be expected.

A candidate Attractor Neural Network

One of the reasons that have brought so much attention on the hippocampus is that it could be an example of the memory principle imagined by Hebb two decades before the first experimental results on place cells. Let us briefly review the facts that support this idea.

The first property of an attractor network is that it supports memory. And indeed, the place cells' firing fields, which do not exist at the first exploration of an environment, are then preserved for a very long time, at least several weeks [START_REF] Lever | Long-term plasticity in hippocampal place-cell representation of environmental geometry[END_REF]. Moreover, sequences of place cell activity are replayed during sleep. Hence, the firing fields are learnt and retrieved, they are not the direct output of some sensory processing. Besides, long term synaptic plasticity occurs in the hippocampus [START_REF] Shapiro | Hippocampus as a memory map: synaptic plasticity and memory encoding by hippocampal neurons[END_REF].

Then, an attractor neural network is characterised by its extensive recurrent connectivity, a feature also displayed by the CA3 field (see paragraph 1.4.1). This, together with experimental indications of long term potentiation in the hippocampus, led McNaughton and Morris to the attractor hypothesis [START_REF] Mcnaughton | Hippocampal synaptic enhancement and information storage within a distributed memory system[END_REF].

When two distinct configurations are attractors of the same network, then the activity can jump abruptly from one to the other, but not interpolate smoothly between them. Experimenters have tested this non-linear response property by inducing the formation of two separate "attractors" (i.e. two independent place fields allocations in two distinct environments) and then continuously morphing the input from one attractor into the other [40,[START_REF] Wills | Attractor dynamics in the hippocampal representation of the local environment[END_REF][START_REF] Leutgeb | Progressive transformation of hippocampal neuronal representations in 'morphed' environments[END_REF] 3 . They observe that the place cells' activity corresponds to one of the environments in an all-or-none manner, without intermediary configuration (see also [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF]), and that the transition between the two occurs suddenly and globally at some point of the morphing path, as would be expected in an attractor neural network.

Finally, the activity of an attractor neural network is dominated by the recurrent connectivity: the external input, if any, can modify which attractor will be retrieved but not the fact that the network converges to an attractor. As a consequence, the activity of the network can be characterised with great precision by the datum of the co-activated units. In the hippocampus, Harris et al have shown that the detection of cell assemblies improves the prediction of which neuron will fire at a given time, compared to to a prediction based on independent cells [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF]. We will come back to this idea in Chapter 3.

To sum up, the picture that emerges goes as follows: CA3 is an attractor neural network where each attractor corresponds to an environment, i.e. a continuous set of stable states corresponding to the different positions within this environment. In other words, it is a discrete set of continuous attractors. Each time the rat explores a new environment, it associates random place cells with different locations and the non-spatial features of these locations. This process forms a new continuous attractor manifold in the space of neural configurations that is added to the other previously learnt maps. When the rat enters a familiar environment, even a weak input is able to make the network retrieve the right attractor.

The work presented here is based on this assumption.

Chapter 2

A model for place cells

As CA3 is believed to perform as an attractor neural network, we have proposed a model that is an extension of the Hopfield network to the case of spatial memory. Instead of storing memory items that are discrete configurations of the activity, the network memorizes "maps" or "environments" that are continuous sets of configurations and that are called "quasi-continuous attractors" for this reason. Apart from this complexification, both the ingredients of the model (binary units, incorporation of noise, randomness of the patterns, Hebbian couplings...) and the methods of analysis are much similar to those already used in [8] and in the subsequent works on the Hopfield model. We have investigated the static and dynamical properties of this model. Thanks to its simplicity, it can be studied in great analytical details. Yet, because of the quasi-continous nature of the attractors, its dynamics is particularly rich compared to the original Hopfield model. In this Chapter, we will present the model and our main results on it. The technical details have been developed in [1] and [2]: they are reported in appendix.

On models

"All models are wrong, but some are useful", George E.P. Box said. A model is a representation of reality used for mental comprehension. Therefore, it contains approximations and simplifications, as the mind cannot grasp all the complexity of reality. When we say that a model accounts for something, the verb can be understood both in the sense of to represent as well as to explain.

A model starts from ingredients taken among experimental facts and/or assumptions. With them it deduces the necessity of other experimental facts as well as non-observed facts that are called predictions1 . There may also be experimental facts that are not used by the model at all, neither as ingredients nor as outcomes. In summary, a model proposes a causality between some experimental observations, together with assumptions.

That being said, we can classify neural network models according to several criteria:

• qualitative models (description with words) vs computational models (with mathematical equations),

• whether the aim is to account for an observed property (e.g. phase precession) or for a hypothetical function (e.g. navigation),

• the level of detail (from microscopic to macroscopic),

• the scope (CA3 only, the whole hippocampal region, the hippocampus plus the neocortex...).

Our model is a computational model accounting for the formation of stable place fields in CA3, at the single-unit level, assuming attractor dynamics. We will compare it to other models of this family in Section 2.6. Here, we just give an overview of other types of models that have been proposed for the hippocampus.

Input models These models neglect the recurrent connectivity of CA3 and account for the localized bump of activity from the assumed expression of external inputs. They are for example the models of [START_REF] Zipser | A computational model of hippocampal place fields[END_REF] [START_REF] Zipser | A computational model of hippocampal place fields[END_REF] and [START_REF] Sharp | Computer simulation of hippocampal place cells[END_REF] [START_REF] Sharp | Computer simulation of hippocampal place cells[END_REF]. After the discovery of grid cells, certain such models have shown how the summation of grid inputs can lead to localized place fields, in a Fourier transform-like fashion [START_REF] Solstad | From grid cells to place cells: a mathematical model[END_REF]. An even more extreme class of models takes the shape of the bump as a starting point and deduces from it the place fields response to cues manipulations [20].

Models for phase precession or replay Some models have investigated how the theta rhythm combined with a certain structure of connectivity could lead to the observed phenomenon of phase precession. The famous study by Tsodyks et al (1996) [79] is one of them. Another feature studied by computational models is the sleep replay of place cells sequences. See for instance [START_REF] Shen | Modeling the spontaneous reactivation of experience-specific hippocampal cell assembles during sleep[END_REF] [START_REF] Shen | Modeling the spontaneous reactivation of experience-specific hippocampal cell assembles during sleep[END_REF].

Models for hippocampal functions

It is commonly assumed that the hippocampus subserves spatial navigation (see Section 1.4.1). Many models have proposed possible ways how this function could be implemented. It can consist in a path integration system to compute the animal's position [START_REF] Touretzky | Theory of rodent navigation based on interacting representations of space[END_REF][START_REF] Samsonovich | Path integration and cognitive mapping in a continuous attractor neural network model[END_REF] or in an algorithm to find a trajectory to a goal.

In the latter case, some models assume that the animal learns trajectories [START_REF] Blum | A model of spatial map formation in the hippocampus of the rat[END_REF] while others propose mechanisms for mental exploration [START_REF] Hopfield | Neurodynamics of mental exploration[END_REF][START_REF] Ponulak | Rapid, parallel path planning by propagating wavefronts of spiking neural activity[END_REF].

The hippocampus is also believed to support episodic memories that are later transferred to the neocortex. There is a whole zoo of models for this function, derived from the seminal work by [START_REF] Marr | Simple memory: a theory for archicortex[END_REF] [15]. See also [START_REF] Mcclelland | Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory[END_REF] [START_REF] Mcclelland | Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory[END_REF].

Finally, if one assumes that CA3 works as an autoassociative memory network, then the question arises of the competition between learning and retrieval, or, equivalently, between pattern separation and pattern completion. With quantitative estimates based on a schematic network, [START_REF] Treves | Computational constraints suggest the need for two distinct input systems to the hippocampal ca3 network[END_REF] [START_REF] Treves | Computational constraints suggest the need for two distinct input systems to the hippocampal ca3 network[END_REF] have demonstrated that two distinct input pathways with different synaptic properties are necessary for the associative network to be able to do both. The mossy fiber and perforant pathway inputs, respectively, do have these properties, suggesting that retrieval is elicited by the perforant pathway while learning is triggered by the mossy fiber input. Though these questions are important in any attractor neural network theory of the hippocampus, our study focusses more on the recurrent network itself.

Description

We introduce a model for place cells in one-or two-dimensional space (the extension to higher dimensions would be straightforward, though more difficult to treat analytically).

The N place cells are modeled by binary units σ i equal to 0 (silent state) or 1 (active state)2 . These neurons interact together through excitatory couplings J ij . Moreover, they interact with inhibitory interneurons whose effect is assumed to fix the total activity of the place cells to a fraction f of active cells (global inhibition). We also assume that there is some stochasticity in the response of the neurons, measured by a parameter of noise T . All these assumptions come down to considering that the network's states are governed by a partition function

Z J (T ) = σ such that i σ i =f N exp(-E J (σ)/T ) , (2.1)
where the "energy" (in the thermodynamic sense) of a configuration σ reads

E J (σ) = - i<j J ij σ i σ j . (2.2)
We want to store L + 1 environments in the coupling matrix. These environments are 1D or 2D manifolds of size 1 (i.e. segments of length 1 or squares of area 1). We call place field a region of space where a place cell preferentially fires. An environment is defined as a random permutation of the N neurons' place field centres (assuming that these points are regularly arranged on a grid). This models the experimentally observed remapping of place fields from one map to the other3 (see 1.4.2). With this definition, an environment is said to be stored when activity patterns localized in this environment are stable states of the dynamics. In other words, an environment is said to be stored when the configurations where active neurons have neighbouring place fields in this environment are equilibrium states. To make this possible, we assume a Hebbian prescription for the couplings J ij that is a straightforward extension of the Hopfield synaptic matrix to the case of quasi-continous attractors:

• additivity:

J ij = L =0
J ij where the sum runs over all the environments.

• potentiation of excitatory couplings between units that are active together:

J ij = 1 N if d ij ≤ d c 0 if d ij > d c , (2.3)
where d ij is the distance between the centres of the place fields of i and j in the environment . d c represents the distance over which place fields overlap; it is chosen so that, in each environment, each cell is coupled to wN neighbours. The 1/N factor in 2.3 ensures that the total input received by a cell remains finite as N goes to infinity, a limit case ("thermodynamic limit") in which we will place ourselves because exact calculations then become possible.

This rule is illustrated in Figure 2.1. Like in the study of the Hopfield model, the next step is to check whether, with this prescription for the couplings, our environments are indeed attractors.

Phase diagram 2.3.1 Analytics

The aim of this calculation is to study the stable states of the network, and to find under which conditions these stable states correspond to activity patterns localized in one of the environments. In other words, we want to know for which parameter values the Hebbian synaptic matrix (2.3) ensures the retrieval of the stored maps. The system under study has disorder (from the random allocation of place fields in each map) and frustration (from the competition between excitatory synapses and the global inhibition).

It is analytically tractable in the large N limit (thermodynamic limit). In this limit, where we will carry our calculations, each unit has an infinite number of neighbours so the mean field approximation is exact.

The stable states are the ones that minimize the free energy of the system

F J = -T log(Z J (T )) . (2.4)
We therefore need to compute this quantity, which depends on the realization of the random permutations in each map. We assume that in the large N limit, the free energy is self-averaging: its particular value for a given realization of the disorder is typically close to its average F over all possible realizations of the disorder, which is thus a good approximation of the typical F J . The randomness of the remapping process is therefore a key hypothesis for the model to be tractable. To compute the average of the logarithm of Z J (T ) we use the replica method [START_REF] Kirkpatrick | Infinite-ranged models of spin-glasses[END_REF][START_REF] Mézard | Spin glass theory and beyond[END_REF]: we first compute the n th moment of Z J (T ), and then send n → 0. Since we are interested in configurations where active cells' place fields in one of the environments are spatially concentrated, we arbitrarily select one of the environments (called "reference environment") and do the averaging over the L other permutations. This choice is totally arbitrary because the difference between environments is eventually averaged out. In the reference environment the neurons are indexed in the same order as their place fields, which allows us to move from a microscopic activity configuration σ to a macroscopic activity density over continuous space

ρ(x) ≡ lim →0 lim N →∞ 1 N (x-2 )N ≤i<(x+ 2 )N σ i J , (2.5)
where the overbar denotes the average over the random remappings (quenched noise) while the brackets correspond to the average over the thermal noise (fast noise).

The n th moment of Z J (T ) is given by equation (20) in [1]. The averaged term depends on the configurations σ 1 , • • • σ n only through the overlap matrix with entries

q ab ≡ 1 N j σ a j σ b j . (2.6)
Then, to perform the n → 0 limit we make use of the replica symmetric Ansatz, which assumes that the overlaps q ab take a single value q (the Edwards-Anderson parameter [START_REF] Edwards | Theory of spin glasses[END_REF], measuring the fluctuations of the local spin magnetizations from site to site) for replica indices a = b. This Ansatz allows us to compute the free energy as a function of the order parameters ρ(x), µ(x) (chemical potential conjugated to ρ(x)), q and r (conjugated to q): see [1] in Appendix A.

Finally, extremization of the free-energy functional leads to the saddle-point equations

r = 2(q -f 2 ) k≥1 kπ sin(kπw) -β(f -q) -2 , q = ˆdx ˆDz 1 + e -βz √ αr-βµ(x) -2 , ρ(x) = ˆDz 1 + e -βz √ αr-βµ(x) -1 , (2.7
)

µ(x) = ˆdy J w (x -y) ρ(y) + λ ,
where β ≡ 1/T , α ≡ L/N , Dz ≡ dz exp(-z 2 /2)/ √ 2π is the Gaussian measure, and λ is determined to enforce the fixed activity level constraint ´dx ρ(x) = f . We find three solutions to these coupled equations:

• a paramagnetic phase (PM), in which the average local activity is uniform over space, ρ(x) = f , and neurons are essentially uncorrelated, q = f 2 .

• a glassy phase (SG), in which the local activity σ i varies from neuron to neuron (q > f 2 ), but does not cluster around any specific location in space in any of the environments (ρ(x) = f after averaging over remappings). In this SG phase the crosstalk between environments is so large that none of them is actually stored in the network activity.

• a 'clump' phase (CL), where the activity depends on space, i.e. the density ρ(x) varies with x and is localized in the reference environment. ρ(x) then displays a characteristic bump of activity as the result of local excitation and global inhibition [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF] (see Fig. 5 and6 in [1], Appendix A). Thus the symmetry by translation is broken. This phase corresponds to the 'retrieval phase' where the environment is actually memorized. In fact, all the L + 1 environments are memorized since any of them could be chosen as the reference environment, while in the PM and CL phases none of them is memorized. This is the 'black-out catastrophe' [12] already described in the Hopfield model where retrieval also takes place in an all-or-nothing fashion.

We now need to know which one of these solutions is the thermodynamic phase for given α, T , that is the phase of lowest free energy, that will be thermodynamically favoured. This first requires to compute each solution's domain of stability against longitudinal and replicon modes and the transition lines between them. To study the stability, we write the Hessian of the free energy and study its eigenvalues in the longitudinal and replicon sectors. Then, the transition between two phases is the line where the free energy in both phases equalizes. We have done these calculations in the one-dimensional case, as detailed in [1] (Appendix A). The outcome is the phase diagram shown in Fig. 2.2, displaying the three phases domains in the (α, T ) plane:

• the paramagnetic solution exists for all α, T and is stable for T > T PM (α) displayed in dotted-dashed line in Figure 2.2. This phase therefore corresponds to high levels of noise T .

• the glassy phase exists for T < T PM (α) and is always replica-symmetry broken. This phase corresponds to large loads α. • the clump phase's longitudinal stability region is computed numerically and shown in thin dashed line in Figure 2.2. The clump is stable against replicon modes except in a little low-T high-α region (dotted line). An interesting feature of the clump phase stability domain is the reentrance of the high-α boundary.

We will comment further on this phase diagram in paragraph 2.3.4.

Simulations

We have done Monte Carlo simulations of the model for different N to verify our theoretical results. The algorithm is the following: at each time step, we select at random one spin up and one spin down. We compute the change in energy ∆E that flipping those spins would provoke. If ∆E < 0 then the flip is accepted. If ∆E ≥ 0 then the flip is accepted with probability exp(-β∆E) and rejected otherwise. This procedure ensures that the total activity remains fixed to f N while the system converges to a thermodynamic equilibrium at temperature 1/β. First, we have looked at equilibrium quantities such as the correlation between spins, the chemical potential or the local field on a spin, and we have found an excellent agreement with the theory (see Figures 12,13,14 in [1], Appendix A).

Then, we have corroborated numerically the phase diagram by varying α and T and looking for transitions: see Figures 11,15,16,17 in [1].

Finally, we have studied out-of-equilibrium effects at the PM-CL transition (unpublished). For simplicity, we consider the case α = 0. In the region T PM < T < T CL both paramagnetic and clump phases exist. At equilibrium, T c is the true thermodynamical critical temperature. Nevertheless, depending on the initial conditions, a hysteresis can take place and the system can be trapped in one phase or the other. The trapping time grows exponentially with N . However we expect the barriers to scale as f w at small f, w, and to be easy to cross unless N 1/(f w). As long as N = 1600 (for f = .1, w = .05) thermalization is achieved with a small number of Monte Carlo steps per spins, see Fig. 12 in [1]. Out-of-equilibrium effects can be seen at larger sizes. We show in Fig. 2.3 how the system gets trapped in a metastable phase if the simulation time is not sufficient to ensure proper thermalization. For instance, a system with N = 6400 spins can remain stuck in the clump phase when T c < T < T clump , or in the paramagnetic phase when T para < T < T c . Another consequence of this metastability is that spontaneous fluctuations from one phase to the other will occur on long time scales in the coexistence region.

Quantitative study

Effect of dimension Concerning the replica calculation, the two-dimensional case does not differ much from the one-dimensional one. It is detailed in [1], Appendix A. Qualitatively, the phase diagram is not expected to differ, though we have not computed it in details, except the clump phase stability region. Quantitatively, for equal f and w, the clump phase is slightly reduced in 2D as compared to 1D (see Figure 23 in [1]). We expect the three phases to exist in any dimension of space, although the calculation would be more difficult.

Effect of parameters

We have also looked at the influence of the model's parameters f and w on the phase diagram, especially on the clump phase domain, which is the phase of interest. For this we consider critical quantities such as T CL , α CL , T c and α g (see Figure 2.2's caption). We compute these quantities for varying f or w. The result is shown in Figures 18,19,20 and 21 in [1], Appendix A. Regarding the critical temperatures T c and T CL , the dependence on f and w is monotonic. A more interesting effect appears with the critical loads α g and α CL which exhibit a maximum for values around roughly f ∼ w. This optimum can be understood as a tradeoff between sparsity and limiting the cross-talk with other stored environments.

Silent cells Finally, we have extended our calculation to take into account "silent cells" (Section 1.4.2). We incorporate in the averaging procedure the random selection of a given fraction c of cells having a place field in a given environment (0 < c < 1), the others cells being silent. The calculation, detailed in [1], Appendix A, leads to a simple modification of the free-energy functional. It turns out that this additional hypothesis alters the phase diagram only quantitatively, with T c and α g increasing monotonically with c (see Fig. 22 in [1]).

Parallel with the Hopfield model phase diagram

It is instructive to compare the phase diagram we have obtained (Fig. 2.2) with the Hopfield model's [10,[START_REF] Amit | Statistical mechanics of neural networks near saturation[END_REF] (Fig. 2

.4).

Common points The Hopfield model also exhibit three phases -paramagnetic, spin glass and ferromagnetic ('retrieval', where the stored patterns are attractors) -with an obvious parallel with our paramagnetic, spin glass and clump phases respectively. The retrieval phase is also replica-symmetry broken in a low-T , high-α region, though this region is much smaller than in the present case. As in our model, there is a second order transition between the paramagnetic and spin glass phases, along a line scaling as √ α. Another common point is that in both models the spin glass phase is always unstable against replicon modes.

As regards the derivation of this phase diagram, the replica calculation performed by Amit, Gutfreund and Sompolinsky [START_REF] Amit | Statistical mechanics of neural networks near saturation[END_REF] follows basically the same steps as ours -average over disorder of the partition function replicated n times, introduction of conjugated parameters, saddle-point equations, replica-symmetric Ansatz. Our isolating of the 'reference environment' by the indexing choice is replaced, in the case of discrete patterns, by an external symmetry-breaking field that selects the pattern(s) to retrieve and that is finally sent to zero.

Differences

The main difference concerns the shape of the ferromagnetic/clump domain. In the Hopfield model, it does not overlap with the paramagnetic domain, so these two phases do not coexist. Moreover, in the Hopfield model it has a triangular shape (the storage capacity decreases monotonically with the temperature) so there is no reentrance as in our case. A very weak reentrance was nevertheless found by subsequent studies [START_REF] Naef | Reetrant spin glass behaviour in the replica symmetric solution of the hopfield neural network model[END_REF], but in a much lesser extent. This can be interpreted in terms of robustness to noise. Intuitively, noise can much more easily disrupt a point attractor than a continuous one. Our phase diagram shows that a moderate level of noise can even increase the storage capacity.

One major feature of the Hopfield model is the stability of the so-called 'spurious states' which are linear sums of several stored patterns 4 . In our model, sums of patterns are not 4 These spurious states, considered as prejudicial to retrieval, have been shown to be suppressed above stable states because the 0 and 1 states are not symmetric (fixed-activity constraint) as are the -1 and +1 states in the Hopfield model. The equivalent of spurious states here would consist in the "double bumps" that appear briefly during a transition between environments (see Section 2.4). But, in the parameter range we have explored, these states have a higher free energy than the clump phase, hence they do not disturb the retrieval.

The patterns stored in those two models have fundamentally different structures. A consequence is that the parallel between both models is not always straightforward. Take for instance the notion of sparsity. In the Hopfield model (or rather an extension of it), a sparse representation refers to patterns with a small fraction of up spins. It has been shown that sparsity could increase the storage capacity of the network (see refs [11,[START_REF] Tsodyks | The enhanced storage capacity in neural networks with low activity level[END_REF]): in the case of 0-1 units, the storage capacity has been shown to scale as -1/(a log a), where a is the fraction of active units in each pattern. By naive extrapolation, one could expect that in our model the storage capacity follows

α CL ∼ -1 f log f , (2.8)
given that f denotes the constant fraction of active units. We observe that it is not the case: α CL vanishes when f goes to 0. (see Figure 20 in [1]) because what we call a pattern is not one configuration of sparsity f but a continuous set of configurations, each with average activity f . However, in Figures 18 and20 in [1] it seems that, by decreasing w and f to zero while keeping them equal f ∼ w (which is optimal, as discussed above), the storage capacity increases and diverges close to zero. It would be interesting to check if this increase follows Eq. 2.8. Battaglia & Treves (1998) [START_REF] Battaglia | Attractor neural networks storing multiple space representations: a model for hippocampal place fields[END_REF] gave the argument that a continuous map with connection range w could be roughly approximated, in terms of storage capacity, by 1/w configurations of average activity w. By analogy with the Hopfield model, they deduced that the storage capacity should scale as -1/ log w (that goes to 0 as w becomes small) instead of -1/(w log w). They observed this scaling, but in their model the inhibition does not fix the total activity so they do not have necessarily f ∼ w. More work would be required to understand these scalings, and the role of correlations between configurations5 in the storage capacity. The conclusion here is that one should be cautious when drawing comparisons between discrete and continuous attractor networks.

We now turn to the dynamical aspects of the model in its clump phase. It is a general property of attractor networks with noise that once the system has reached an attractor, it does not remain in it indefinitely. In the case of quasi-continous attractors, there are two possible evolutions:

• the noise can either make the network jump to another attractor, i.e. form a clump in another map (we will call it 'transition'),

• or it can make the activity evolve quasi-continously within this attractor, i.e. form a clump in the same map, but at different contiguous positions along time (we will call it 'diffusion').

These two phenomena observed in a simulation are illustrated in Figure 2.5. Thus the a certain level of noise. This beneficent effect of noise in the case of the Hopfield model is reminiscent to the the aforementioned advantage of noise in our case. Between time 0 and time ≈ 1000 the activity is localized in map 2 and delocalized in map 1. Then it undergoes several transitions between 1 and 2. Between times ≈ 2000 and ≈ 3700, the network is in attractor 1 and the bump diffuses within this attractor. Finally, it ends up and diffuses in map 2. Note the abruptness of the transitions between maps.

dynamics of quasi-continuous attractors is much richer than in the case of more basic models as the Hopfield network, where the only possible evolution is to transit from one attractor to the other. This complex behaviour and the competition between both phenomena had not been analytically studied and were little understood. Here I will report the results we obtained in the case of our model.

Transitions between maps 2.4.1 Occurrence

At finite temperature, the system evolves in time stochastically (see 2.3.2). The Metropolis algorithm ensures that the sequential states of the system sample the Boltzmann distribution with energy E and temperature T . At the macroscopic level, the density function samples the Boltzmann distribution with free energy F (T ). Hence, in the clump phase, the system will explore each of the maps (i.e. the activity will stay localized for some time in each environment), and the relative time spent in each configuration tends to its Boltzmann weight. Just as a Hopfield network -in the retrieval phase at finite temperature and without external input -necessarily jumps between attractors from time to time [12], we observe transitions between maps in our simulations, as sketched in Figure 2.6. The frequency of these transitions increases with the temperature, as would be expected from an Arrhenius law. It is also an increasing function of the cross-talk between maps, hence of α. Finally, the barriers to transitions scale linearly with the number of units N , so the frequency of transitions decreases exponentially with N (see Fig. 14 in [2]). These effects are illustrated in Figures 2.7 (one-dimensional case) and 2.8 (two-dimensional case).

Mechanism

The initial and final states of a transition correspond to a bump of activity concentrated around a certain position in the initial environment and in the final environment, respectively. What happens in between? What do the intermediary states look like? There are basically two possibilities: either the activity is not localized in any of the environments, either it is localized in both.

Analytics: instanton calculation

Here again, the replica method allows us to answer this question. In the replica calculation, we can isolate two environments (the starting environment and the arrival environment) and average over the disorder coming from all the other environments. We end up with saddle-point equations with a density order parameter for each environment, ρ 1 (x) and ρ 2 (x), with a constraint enforcing that both environments are linked by a fixed permutation. These saddle-point equations have several solutions at a given (α, T ): first, as expected, we retrieve the solutions where the activity is localized in one of the two environments and not in the other (clump phase). These are the global minima of the free energy. But we also find local minima corresponding to a "double clump", that is the activity is localized in both environments, albeit the bump in each map is smaller than the bump in the retrieved map in the clump phase. This solution correspond to the activated state through which the transition takes place. There is a small high-T domain where this solution is not stable anymore, and the transition occurs through a totally delocalized state.

The details of this calculation will be reported in a forthcoming publication. The approach taken here also allows us to estimate the free energy barrier to transitions.

Simulations: preferential positions of transitions

Such a "double clump" intermediary state is observable in simulations: see for instance Fig. 12 in Ref. [2] in Appendix A. We then conjectured that the transition between two positions in two maps would be all the more frequent as these two positions "look alike" in terms of remapping. Indeed, the free energy of an intermediary state localized in both positions is all the lower as the two positions share many common neurons. In the extreme case where these two positions are "identical", i.e. in the eventuality that the permutation of place fields from one map to the other is locally equal to the identity, the double clump state is equal to a full clump in both maps, hence its free energy is very low and this state is very stable. McNaughton et al (1996) [18] had already conjectured these preferential positions for transitions where the environments most look alike (they called them "worms holes"), even if in their paper they reckoned that the phenomenon of transitions would be much lesser than the one of diffusion.

We have verified in simulations that the frequency of transitions between two given positions respectively in two maps is indeed directly correlated with their similarity. We define the local resemblance between, say, map 1 and map 2 in position x 1 and position x 2 as

Res 12 (x 1 , x 2 ) ≡ N i=1 ρ 1 (π 1 (i))ρ 2 (π 2 (i)) , (2.9)
where ρ 1 and ρ 2 denote the density profiles in the clump phase centered respectively in x 1 and in x 2 ; π is the permutation giving the position of the neurons' place field centres in map . Note that here ρ denotes the thermal average σ i ; it is defined on the microscopic units i. Res 12 (x 1 , x 2 ) measures a sort of dot product between the two clump phases: it can be seen as a projection of the density profile in (x 1 , map 1) on the density profile in (x 2 , map 2), or vice versa. If Res 12 (x 1 , x 2 ) measures some kind of energy (log-likelihood), then the frequency of transitions between the two positions Φ 12 (x 1 , x 2 ) would follow a law of the form

Φ 12 (x 1 , x 2 ) ∝ exp(A • Res 12 (x 1 , x 2 )) , (2.10)
where A is a constant. It turns out that it is well verified in simulations, as illustrated by Fig. 2.9.

The resemblance between environments is of course symmetric: Res 12 (x 1 , x 2 ) = Res 21 (x 2 , x 1 ) As expected, the frequency of transitions observed in the simulations is also symmetric: Φ 12 (x 1 , x 2 ) = Φ 21 (x 2 , x 1 ). We therefore have a good understanding of how, where and when the transitions between environments occur.

Retrieval

To carry on with the parallel between our model and the Hopfield model, we have addressed the question of pattern retrieval. The case of the Hopfield model is quite simple as the stored patterns are point configurations [12]: retrieval can be obtained with an external input colinear to the pattern to retrieve, for instance. In our case, because of our quasi-continuous attractor structure, there are more possibilities. Indeed, "retrieval" can mean two things: retrieval of an environment or retrieval of a position in an environment. In the former case, the system is said to have retrieved the environment when it reaches the clump phase in this environment, regardless of the position of the bump within it. In the latter case, the system is said to have retrieved the right position when it stabilizes into a clump centered on this position in the right environment.

We have studied the retrieval properties in both cases (see [2] in Appendix A). The problem of retrieving a position is closest to pattern retrieval in the Hopfield model because it is about reaching a point activity configuration. It can be achieved with an external excitatory field that is localized around this position. As in the Hopfield model, the retrieval is all the faster as the external field is strong (see Figure 22 in [2]). Retrieving a map without focussing on a position is a more exotic issue as it cannot be achieved with a constant external input. We have tried to increase the weight of the map to retrieve by adding a coefficient in the couplings Hebbian additivity rule. We observe that, here again, the retrieval time is a decreasing function of this coefficient (see Figure 23 in [2]).

Our study of retrieval remains however very schematic and we do not claim to mimic the biological systems. More work would be required to model the various synaptic inputs to CA3 and to evaluate their effects on the attractor dynamics.

2.5 Diffusion within one map 2.5.1 Emergence of a 'quasi-particle': demonstration in the α = 0 case By translational invariance of the saddle-point equations, in the clump phase the bump of activity can concentrate equivalently around any position of the environment. It is then intuitive to consider the bump as a solid and to describe its motion in this map with a single coordinate located at its centre. In other words, it seems natural to extend the macroscopic description of the activity established at equilibrium to the dynamics. This kind of description has already been adopted before (see for instance [START_REF] Samsonovich | Path integration and cognitive mapping in a continuous attractor neural network model[END_REF]). Yet, it was then postulated without demonstration: the fact that noise makes the clump move globally with little deformation had not been proven so far.

In the case of our model without disorder (single-environment case, α = 0) and in the large N limit, we have analytically derived this quasi-particle behaviour 6 . More precisely, we have shown that the microscopic stochastic evolutions of the units result in a collective pure diffusion motion of the clump; while the shape of the clump only shows small fluctuations around its equilibrium. The position of the centre of the clump, x c (t) (collective coordinate), obeys the diffusion equation

ẋc (t) = D 0 ξ(t) , (2.11)
where ξ(t) is a Gaussian noise and D 0 is an effective diffusion constant. In the thermodynamic limit, we have obtained an analytical expression for D 0 given by Equation ( 31) in [2]. D 0 only depends on the order parameters of the activity profile and the microscopic dynamics, and scales as 1/N . The fluctuations of the activity profile ρ(x, t) are of the order 1/ √ N . The calculation is detailed in [2]. Here we will briefly summarize its main steps, the reader is invited to refer to Appendix A for more technical details. We start from the microscopic dynamics of the system, defined as in the Monte Carlo simulations (see 2.3.2). Then we move to a macroscopic description by expressing the spins flip in terms of density increment ∆ρ(x) and converting the energy change in free-energy change. We then write the Fokker-Planck equation describing the stochastic evolution of the probability distribution of the density profile. To achieve this we introduce the diffusion tensor D(x, y) ≡ ∆ρ(x)∆ρ(y) , where the average is taken over the microscopic dynamics. We are then able to move from the Fokker-Planck equation to the equivalent Langevin equation for ρ(x). We linearize it around small fluctuations, which allows us to derive both the diffusion coefficient and the vanishing of the shape deformations.

We have run Monte-Carlo simulations in one dimension in order to measure the diffusion coefficient of the centre of the clump and compare it with the theoretical prediction D 0 . Indeed, we do observe trajectories that look like Brownian motion, both in one and two dimensions (see Figure 2.10). Nevertheless, measuring the diffusion constant is not trivial: because the position of the centre of the clump is not well-defined in the case of periodic boundary conditions, we have to resort to a binning of the space, which in turn strongly distorts the measure of the diffusion coefficient. These effects have to be corrected a posteriori: for this purpose, we numerically invert the distortion function (see details in [2]). After correction, the value of the diffusion coefficient in simulations is in very good agreement with the theory (see Figure 6 in [2]). The agreement is all the better as N is large, as expected for a prediction valid in the thermodynamic limit.

At first view, one may think that this emergence of an effective quasi-particle from the individual microscopic dynamics is a nice statistical mechanics result, but without much interest as concerns neuroscience. Indeed, the pure diffusion here is a consequence of the absence of any external input to the network, a situation highly unrealistic in the case of CA3. However, two objections can be raised to this view.

The first is that diffusion is closely linked to response: the behaviour of the quasiparticle under the sole effect of noise also gives information about how it responds to a force. This is resumed by the celebrated Einstein relation

µ = β 2 D (2.12)
linking the mobility of a particle (i.e. its velocity response to an applied force) to its diffusion coefficient. The equivalent of this relation in our quasi-particle case could be derived, giving a theretical prediction for its mobility. Here again, we checked it with Monte-Carlo simulations with excellent agreement for large N (typically N 1000).

The second objection is that, if pure diffusion is not likely to occur in CA3, there are other similar neural systems where it could be the case. In particular, spatial working memory in prefontal cortex is characterized by a persistent bump of activity localized at the position of the stimulus after extinction of this stimulus [START_REF] Funahashi | Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex[END_REF]. This persistent activity is believed to be underlied by continuous attractor dynamics, but this assumption remains to be proven. In a recent publication, Wimmer et al (2014) [START_REF] Wimmer | Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory[END_REF] study the diffusion of this bump during the delay following stimulus extinction in an oculomotor delayed response task (a spatial working memory task) in awake monkeys. An example of this diffusion is shown in Figure 2.11 (note the similarity with Fig. 2.5). By looking at the correlation between bump drifting and error in the subsequent behavioural response, they show that the observed activity is consistent with predictions from a continuous attractor bump diffusing in the absence of external input, and also invalidates other possible underlying mechanisms. Our model (in the α = 0 case) could therefore be applied to spatial working memory; it would be interesting to confront our results on diffusion with experiments on prefrontal cortex.

Effects of disorder (α > 0)

Storing several environments in the network complicates matters. Its effect is two-fold. First, there is now disorder: the effective free-energy landscape experienced by the quasiparticle is not flat anymore, so its motion is not a pure diffusion (Figure 2.12). The resulting free-energy barriers hinder the motion and slow it down. Then, there is now the possibility of transitions between environments that enters into competition against diffusion.

Competition between transition and diffusion

This point is a very robust feature of the model: an increase in the diffusion constant is always accompanied by an increase in the rate of transitions to other environments. We have tested several ways to improve the motility of the clump: besides moving to the edge of the clump's stability domain (by increasing α or T or by lowering c), we have tried additional, out-of-equilibrium mechanisms such as adaptation, theta rhythm or asymmetric dilution of the synapses. They are described in [2]. We observe that, whatever the mechanism, as soon as diffusion is enhanced, so are transitions.

Two simple explanations can be proposed for that. First, facilitating diffusion basically means, one way or another, increasing the noise level. So the clump is destabilized and more likely to transit to another map. Then, the more the bump diffuses, the more it is susceptible to find a favourable position for transition (see Section 2.4). These two scenarii are not mutually exclusive, and it seems reasonable to say that they both contribute to the ubiquitous competition between the two dynamics.

Position x c

Free energy F l b ∆F Figure 2.12: Sketch of the effective free energy landscape probed by the bump of neural activity (dashed bump) moving through space. Fluctuations of the free energy are of the order of ∆F , and are correlated over a space scale equal to b .

Barriers to diffusion

The replica method allows us to study the statistics of the effective free-energy landscape (Figure 2.12). Here are the main results of these calculations detailed in [2], see Appendix A. The typical height ∆F of the barriers corresponds to the width of the distribution of the free energy accross realizations of the disorder. This width can be estimated using the replica method, by a cumulants expansion of the n th moment of the partition function. Indeed,

Z n = exp -n β F (2.13) = exp -n β F + 1 2 n 2 β 2 (F 2 -F 2 ) + • • • ,
where the overbar still denotes the average over quenched disorder (remappings) and β is the inverse temperature. Therefore, the standard deviation of F can be accessed through the second derivative of Z n with respect to n in n = 0. We find that ∆F scales as √ N with a multiplicative factor that increases with α (see Figure 7 in [2]).

A very similar calculation allows us to access to the average width of the barriers. Supposing that a bump is moved across space, this latter quantity is defined as the typical distance in space over which its free energy will vary. Equivalently, if I take two bumps of activity centered on different positions, up to which distance will their free energies be correlated? This question can be answered again by using the replica method. If we split the replicas in two groups, with the n 2 replicas in the first group having an activity profile centered in x and the n 2 replicas in the second group having an activity profile centered in y, then, as before, the second derivative of the replicated partition function taken in n = 0 is directly related to the correlation of the free energies in each replica subgroup (see Equations ( 52) and (53) in [2]). Then, this correlation is calculated for varying distances between x and y. The width of the barrier corresponds to the distance at which this correlation vanishes (see Figure 9 in [2]). As expected, this width is comparable with the size of the bump.

These calculations of the statistics of the free energy landscape allow us to draw several conclusions. First, they allow us to estimate the depinning force to apply on a bump stuck in a free-energy minimum, with a good agreement with simulations (see paragraph VI A. in [2]). More importantly, the fact that the free-energy barriers scale as √ N results in a drastic limitation of diffusion. Quantitatively, we estimate that cross-over size as the size N c at which β∆F = 1. If we consider that the crossing of barriers follows an Arrhenius law, N c gives us an idea of network sizes at which the barriers are not too high so that they can be crossed in a 'reasonable' time. The result is without call: as shown in Figures 8,11 and 15 in [2], as soon as N exceeds a few hundreds or even tens, the barriers cannot be crossed. Consequently, the disorder severely hinders the diffusion. This effect is a very robust feature of the model. Nevertheless, this conclusion must be qualified with a discussion on dimensions.

Effect of dimension

An interesting point, this hindering of diffusion is much less pronounced in two dimensions than in one dimension. We hypothetized that this could be explained by the fact that in 2D, unlike in 1D, the barriers can be bypassed. We have verified it in simulations where we apply a force along the x-axis: we then observe that, instead of moving at constant y and increasing x, the bump exhibits a tortuous motion along the y-axis, with preferred y positions. The statistics of this trajectory in the (x, y) plane gives access to an estimate of the free-energy landscape (see Figure 20 in [2], Appendix A). This is a direct illustration of the by-passing of barriers.

Thus, increasing the dimension alleviates the hindering effect of disorder on diffusion. This is reminiscent of the effect of dimension on Anderson localization. As, in natural environments, place fields are likely to be most of the time two-dimensional, the smooth motion of the bump within one map may still be possible. Yet, the slowness of motion under the sole influence of noise has already been noted in 2D by J. Hopfield [START_REF] Hopfield | Neurodynamics of mental exploration[END_REF]. At this level of simplification, it is difficult to tell to what extent this effect is relevant for biological systems.

Comparison with previous models

Several models have addressed place cell activity as quasi-continuous attractors in a recurrent network with distance-dependent excitatory couplings. The simplest case of a single one-dimensional environment has been treated by Tsodyks & Sejnowski (1995) [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF]. They use rate units and Gaussian couplings to show numerically the formation of localized bumps and their tendency to cluster together due to inhomogeneities in the place field centres distribution. This latter effect is reminiscent of our trapping in local minima of the free energy. In this simple single-environment form, the problem is equivalent to other continuous attractor models proposed for other parts of the brain, e.g. visual cortex [START_REF] Ben-Yishai | Theory of orientation tuning in visual cortex[END_REF] or head-direction cells in the subiculum [START_REF] Zhang | Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory[END_REF].

Subsequent models have incorporated the remapping phenomenon and multiple-environment storage. This is how the hypothesis of a discrete set of quasicontinuous attractors has been made. Samsonovich and McNaughton (1997) [START_REF] Samsonovich | Path integration and cognitive mapping in a continuous attractor neural network model[END_REF] thus proposed a pre-wired attractor network of leaky integrate-and-fire units with a path integration system. The activity is modulated by theta oscillations. By numerical simulations, they showed that the bump of activity drifted as a quasi-particle and also exhibited phase precession. They then introduced an effective motion equation for the centre of the bump and studied its response to geometrical manipulations of the environment. This is a quite complete model, covering several areas and incorporating many phenomena. Seven years before the discovery of grid cells, its predicts the existence of an absolute metric system upstream of CA3. Nevertheless, it is not consistent with partial remapping (observed e.g. in [START_REF] Skaggs | Spatial firing properties of hippocampal ca1 populations in an environment containing two visually identical regions[END_REF]) or relative motion of place fields (observed e.g. in [38]).

Battaglia & Treves (1998) [START_REF] Battaglia | Attractor neural networks storing multiple space representations: a model for hippocampal place fields[END_REF] have carried out a study quite similar to ours, with this difference that they use threshold linear units instead of binary units (see discussion below), and a generic kernel for the connections. The computation is based on the replica method, adapted to the case of rate units by Treves (1990) [START_REF] Treves | Graded-response neurons and information encodings in autoassociative memories[END_REF]; it is performed at zero temperature. They thus estimate the storage capacity of the network, both in one and two dimensions, as a function of the range of interactions -the other parameters being taken at their optimal value. Interestingly, their quantitative estimate of the storage capacity is in very good agreement with ours. In addition to the storage capacity, the information capacity is computed. Another point we have not addressed, the case of diluted synapses is investigated. The notion of sparsity in continuous attractors is tackled, as discussed above (paragraph 2.3.4).

Other models have investigated the crosstalk between environments. If the maps are correlated or if the subpopulations active in each map do not overlap, multiple bumps can appear [START_REF] Romani | Continuous attractors with morphed/correlated maps[END_REF][START_REF] Stringer | Self-organising continuous attractor networks with multiple activity packets, and the representation of space[END_REF]. Stella and Treves [START_REF] Stella | Associative memory storage and retrieval: involvement of theta oscillations in hippocampal information processing[END_REF] have simulated a network with two orthogonal subpopulations encoding for two maps, and shown that the transitions between the two maps can be synchronized with the theta rhythm (as observed by [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF]) as soon as the network is fed with a conflicting input oscillating in antiphase.

All these models, as well as ours, take the connectivity matrix as given. Some other models, in contrast, start one step before by assuming only a learning rule and an exploratory trajectory. They then explore the consequences of the learning procedure on the structure of the couplings (Muller 1991 [103]) or on the shape, directionality and geometry of place fields (Brunel & Trullier 1998 [104], Káli & Dayan 2000 [START_REF] Káli | The involvement of recurrent connections in area ca3 in establishing the properties of place fields: a model[END_REF]). In general, the learning procedure leads to couplings that decrease with the distance between place fields, so at the end of the day all models end up with basically the same structure for couplings, apart possibly from the degree of disorder.

How does our model position itself relative to these models? There are several common points. The coupling structure is one of them. We use a step function of distance for J 0 ij instead of, for instance, a Gaussian, but we do not expect this choice to be critical. Another similarity is the assumption of a global inhibition. On the other hand, the main pecularities of our model are • the use of binary units instead of rate or spiking neurons,

• the high degree of simplification (no theta rhythm, inputs, learning rule...),

• the incorporation of noise at the microscopic level (and not on average, as is the case when using sigmoidal transfer functions).

These features make our approach quite extreme in schematization. Its main drawback is its limited scope: the model, at least in its present form, does not account neither for response to cue manipulations, nor for navigational abilities, nor for phase precession. . . On the other hand, its key point is the fact that it is analytically tractable in depth. We think that analytics offers more insight, deeper undestanding of a phenomenon than any numerical simulation. Therefore, there is a tradeoff to find between the complexity of a model and the comprehension we can get on it. In our case, we could of course imagine to add more details and ingredients to the model (and we have done it to some extent in [2]), but this inevitably means a loss in analytical tractability, hence in insight.

Our hope is that the collective behaviours captured by the model and understood in great detail are robust to further complexifications. So, to repeat the words of George E.P. Box, we know that our model is wrong but we hope it is useful.

Remark on binary units A lingering ambiguity remains in the literature as regards binary units. Indeed, as discussed in Amit's book [12], binary units can represent two very different things:

• either individual spikes: σ i (t) = 1 if neuron i has emitted a spike between t and t + ∆t 1 , 0 otherwise. Firing rates are the averages over ∆t > ∆t 1 of the σ i . These units are therefore more microscopic than continuous rates.

• or a binary description of rates, useful in the case of neurons working at saturation: σ i (t) = 1 if neuron i has fired at maximal rate between t and t + ∆t 2 , 0 if it has been silent. In this case, binary units are an approximation of continuous rate units with saturation. This approximation is not suitable for neurons working close to the threshold, because they do not saturate.

Both describe the same reality but on different timescales: ∆t 2 ∆t 1 . Why the ambiguity persists is because there is no explicit timescale in most attractor neural network models: the dynamics is only defined through transition rates with respect to an arbitrary time. The confusion is such that the same model is sometimes discussed from the two incompatible points of view depending on the publications (in the case of the Hopfield model, see for instance [8] vs [10]).

We argue that, as soon as the units follow stochastic dynamics, they have to be considered as microscopic spike units. Indeed, rate models average the activity over time or over a population [START_REF] Ginzburg | Theory of correlations in stochastic neural networks[END_REF], allowing to move from a microscopic, stochastic description to a macroscopic, deterministic one [12] 7 . Varying the level of noise in the microscopic binary model is equivalent to varying the slope of the (deterministic) response function in the macroscopic rate model. Hence, in our model the level of description is more microscopic than in rate models. Besides, the rate description is retrieved in the macroscopic order parameter ρ(x). Of course, it remains a simplified model of the neural responses, notably because we do not have refractory periods.

There are many possible ways of describing neurons. The correspondence between them and their differences is a crucial point to understand the scope of each model [START_REF] Shriki | Rate models for conductance-based cortical neuronal networks[END_REF].

Estimation of the parameters from experimental data 2.7.1 The problem

A natural question is now: what would be the values of our parameters (namely w, f , c, T and α) in real life? In fact, this question is ambiguous. Given that our model is but a model, there is of course no one-to-one correspondence between these effective quantities and biological ones. Therefore, there are several ways to make the model stick to reality. Regarding the parameters f (average level of activity) and c (proportion of active cells in each environment), the only problem is the definition of environment and of the unit of time. We will come back to it. More work is required for the parameters w, T and α which do not have direct biological correlates.

The first idea that comes to mind is to start from the microscopic details of neurons (the strength of synapses, the average connectivity, the level of noise in the synaptic release. . . ) and to try to link them to the model's parameter. This approach is not usable here: because the strength of synapses is already imposed by the Hebb rule (Eq. 2.3), all the other quantities are defined relative to it. In conclusion, at the microscopic level, it is impossible to fit our parameters to biological quantities.

A more macroscopic approach, that we will take here, is to compare our density profile ρ(x) with the shape of the experimentally observed bumps as the one shown in Figure 1.5. The profile indeed depends on the parameters, so we can try to find the parameters for which ρ(x) most resembles real bumps. But of course, real bumps do not all have the same shape, and there are many features that we could try to fit. The next issue is thus to decide which quantities we want to reproduce.

Method

We choose to make the following assumptions:

• The timescale is given by the maximal observed firing rate: ρ(x) = 1 in the model corresponds to the maximal observed peak firing rate ρ max in the data. As a consequence, the estimated value for f writes (we will hereafter denote the estimated values with an asterisk)

f * = r(t) i,t ρ max , (2.14)
where r(t) is the measured firing rate; it is averaged across neurons i and time t.

• We consider the height and the width of a bump as a measure of its shape. The height h is defined as the peak firing rate in the centre of the place field. The width s is defined as the fraction of the total space area where the firing rate is above h 2 .

• We look at average quantities across the cells we have at hand and their standard deviations.

• The definition of environments is natural in the dataset we have (see below), so this point does not pose any problem here.

There is some degree of arbitrariness in these assumptions, in the sense that many other choices would have been possible. Nevertheless, they seemed to us the most simple ones. Moreover, there are also features that our model is unable to reproduce, for instance the great variability of bump shapes across cells, or the fact that silent cells (that do not have a place field in a given environment) have an average activity much lower than the active cells, while in our model both silent and active cells have an average activity over space that is equal to f . The global level of activity itself is far to be constant over time in the data: its standard deviation is of the order of its mean (this is why we average over time in Eq. 2.14). Finally, the choice itself of fitting bump shapes is questionable. We will come back to these issues in the discussion in Chapter 4.

Experimental data

We have used the recordings from the Moser lab on which our work on decoding will be based (Chapter 3). For a detailed description of the data, see Section 3.2; here we just need to say that we have the rate maps (i.e. the average firing rate as a function of position, see paragraph 3.3.1) of 34 place cells in 2 different environments A and B.

We consider that a given neuron has a place field in a given environment if the maximum of its average firing rate r max in this environment exceeds a threshold r 0 = 2.5 Hz: i has a place field ⇐⇒ r max i > r 0 .

(2.15)

This threshold value is motivated by the observation of the rate maps (maps below r 0 in general do not have localized fields) and by the shape of the graph of the percentage of cells having a place field as a function of r 0 (Figure 2.13). With this criterion, and with the definitions made above, we can compute the statistics of the place fields. They are recapitulated in Table 2.1. Some cells (1 in environment A, 4 in environment B) have 2 place fields in the same environment, a property that our model does not account for. Nevertheless, in general one of these two place fields is much lesser than the other. We will therefore neglect these second place fields here.

Results

Estimated average activity f *

We have ρ max = 38.0 Hz. By averaging over all cells in both environments, from Eq. 2.14 we get f * 0.02 ± 0.005 .

(2.16)

Note that this average activity is quite low: this is a characteristic feature of place cells, we will come back to it in Chapter 3. The error bar is estimated from the variation of the level of activity from one environment to the other.

A

Estimated fraction of active cells c *

To infer the fraction of silent cells in each environment, we must not only consider the population of 34 cells in the data set, but also take into account all the cells that did not displayed any localized activity in both environments and that have been excluded from the dataset for this reason. According to K. Jezek (personal communication), 60 to 80% of cells are active in neither environment. These cells are nevertheless detected because they become active during sleep. So, let us consider that our 34 cells represent 70% of a sample. We therefore have 113 cells in total. On average, in a given environment, 20 of them have a place field (see Table 2.1). Therefore, the average fraction of cells having a place field in a given environment is 20 divided by 113, hence

c * 0.18 ± 0.06 , (2.17)
where the error bar reflects the range of uncertainty on the fraction of inactive cells. This quantity is quite low compared to other experimental estimates, ranging rather between 20 and 70% [26,[START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF], maybe due to differences in the experimental procedure, or merely to fluctuations from sample to sample.

Estimated range of connections w *

The estimation of w is facilitated by the observation that, for fixed c = c * and f = f * , the size s of our model's bump depends practically only on w, with very low dependence on T and α (Fig. 2.14). We thus estimate w * 0.17 ± 0.05 .

(2.18)

Here the error bar combines the error due to the uncertainty on f * and c * and the error on s(w).

Estimated level of noise T * and load α * Finally, we estimate T and α by fitting as best as possible the average of the peak firing rate h and its standard deviation across cells, with c = c * , f = f * and w = w * . In the model, these quantities correspond, respectively, to the density at the centre of the bump ρ(0) and its standard deviation δρ(0). This latter quantity is the average over disorder of the thermal fluctuations of individual magnetizations:

δρ(x) = σ 2 i -σ i 2 (2.19) = q(x) -ρ 2 (x) .
(2.20)

An example of these fluctuations in the 1D case is given in Figure 2.15.

-0,4 -0,2 0 0,2 0,4 Having a precise estimate is quite arduous, first because for these values of c, f, w the numerical resolution of the saddle-point equations 2.7 is quite tedious, second because the height of the bump ρ(0) and its standard deviation over disorder δρ(0) = q(0)ρ(0) 2 do not vary much across the clump stability domain.

x 0 0,2 0,4 0,6 0,8 1 ρ(x)+δρ(x) -0,4 -0,2 0 0,2 0,4 x 0 0,2 0,4 0,6 0,8 1 ρ(x)+δρ(x)
Nevertheless, we find that the best fit is in the low-T , high-α region of this domain, with the rough estimate

T * 10 -4 ± 10 -4 ; α * 0.001 ± 0.0005 , (2.21)
for which we get the values ρ(0) = 0.33 and δρ(0) = 0.25, that are of the same order as h i ρ max and SD i (h). The error bars are large, for the reasons mentioned above.

Decoding

We now turn to another problem: after having proposed a mechanism how a given form of neural activity could be caused, we wonder what we can infer about the external world from the observed activity. This question will be made more precise in the first section, and we will review how it has been treated in the case of the hippocampus. Then we will present the experimental data from the Moser lab we have been working on, our approach to analyse them and our results.

Stating the decoding problem

What does decoding mean? The term of "neural code" contains the idea of semantics, of a language to be translated. The underlying picture is the one of a representation of the organism's environment by its neural activity. We can thus see that the kind of questions it arises is quite different from the ones we have been dealing with in the previous chapter: while the latter were about deducing collective activity strictly "mechanistically" from microscopic rules, here we try to assign them a meaning1 .

To illustrate the situation, let us imagine a Martian who has found a text in latin alphabet and who knows that it means something. In order to decipher the text, he needs first to have some idea about what it could be about, and second to find out how the meaning is supported in the text. Thus, stating the decoding problem leads to asking two (almost independent) questions, namely what is encoded, and how it can be decoded.

What do we decode?

It is impossible to translate an unknown language without any cue about its content. As concerns the neural code, the decoding issue has been tackled mostly for sensory and motor areas [START_REF] Rieke | Spikes: Exploring the neural code[END_REF]. In this case, things are a bit simpler because we a priori have some idea about what could be encoded; at least we can reasonably eliminate a lot of possibilities. Maybe due to this predominance of sensory systems, most work on decoding has been based on a stimulus-response paradigm, so that the decoding issue has essentially reduced to inverting an input-output transfer function. This simple scheme calls for discussion.

The problem of stimulus description Even in the simpler case of a primary sensory area, describing the stimulus is not obvious at all. One needs to find the relevant quantities among a great lot of possibilities. Take for instance the visual system: is it sensitive to colors, shapes, contrast, movement, orientation, a combination of them? The property of neurons to be responsive to a given external quantity is called feature selectivity. It is not a straightforward task to experimentally determine which are these features, all the more so as most experiments are performed on animals, so the experimenter has to guess somehow what perception could be for non-human species.

Noise and hidden variables Another problem with the stimulus-response paradigm is that, for a given same stimulus, the response is variable across trials. Whether this variability is noise or the result of hidden variables is a difficult question. Hidden variables can be external (taking us back to the previous question of stimulus description) or internal (e.g. mood, expectations, attention, chaotic cortical dynamics...) [START_REF] Renart | Variability in neural activity and behavior[END_REF]. What is more, history may also matter, so that these multiple variables have to be integrated over the animal's past experience. There are therefore many "unknown unknowns". Thus, the "constant stimulus" paradigm is much more difficult to control experimentally than it seems at first glance, here again because we do not know what variables are relevant.

Non-sensorymotor areas Things turn even more complicated in the case of deep brain areas (e.g. the hippocampus) which receive inputs from many cortical areas and for which the relationship with external variables is all the more remote. It is alluring -but hazardous -to make assumptions about the functions of these areas and to imagine the meaning of neural activities with respect of these functions. For instance, if neurons have a robust firing correlate (e.g. position for place cells), one is tempted to state that their activity "represents" the value of the correlate. This approach is quite speculative, so one needs to be cautious when resorting to it. We will come back to this when discussing the issue of decoding in the case of the hippocampus.

How can it be decoded?

Asking how to decode the information contained in the activity often boils down to asking how it has been encoded [START_REF] Bialek | Reading a neural code[END_REF][START_REF] Paninski | Statistical models for neural encoding, decoding, and optimal stimulus design[END_REF] 2 . A striking feature of neural activity is the universality of the spike: the action potential is a phenomenon shared with high similarity across all neurons of all species, making it likely that the language of the brain is written in spike alphabet [START_REF] Rieke | Spikes: Exploring the neural code[END_REF]. Let us turn back to our Martian. He sees letters of different shapes separated by spaces and arranged in parallel lines. He can try to decode at the letter level, that is to assign a meaning to individual letters. Maybe he will think that "e" and "c" have close significations because they look alike, and he would be mistaking. Alternatively, he can try to decode at the paragraph level, looking for a meaning in the global shape of a paragraph, and he would be wrong again. In summary, there is nothing obvious in the fact that he should consider words and phrases as semantic units. The same kind of difficulties arises when trying to decode spike trains. What we englobe in the much-touted "rate or spike" dilemma has several components [START_REF] Rubin | Neural coding and decoding with spike times[END_REF]:

• The problem of spike train description: what are the relevant parameters (spike times, interspike intervals, spike rates, cofiring. . . )? This is the corollary on the neural side of the problem of stimulus description on the external world's side.

• The problem of sampling: should I look at single-cell activity, at a population of cells (and which one), at the whole brain? What can I decode from the activity of a few tens of randomly selected neurons, as in micro-electrode recording? How to deal with noise correlations across a population [START_REF] Averbeck | Neural correlations, population coding and computation[END_REF]?

• The problem(s) of timescales: first, what is the relevant biological timescale? Second, which timescale gives us enough statistics so that we are able to decode? These two questions may not have the same answer. The latter is bound to the problem of sampling.

• The problem of biological implementation: the brain itself should somehow decode neural activities, otherwise there would be no cognition. How can a given decoding algorithm be implemented in neural circuits? See for instance references [START_REF] Rubin | Neural coding and decoding with spike times[END_REF][START_REF] Seung | Simple models for reading neuronal population codes[END_REF].

We will not address this question on the read-out mechanism, though it is important from the point of view of neural computation.

Various answers can be tentatively made to these questions, leading to a whole jungle of decoding procedures (see for instance [START_REF] Rubin | Neural coding and decoding with spike times[END_REF][START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF]) leading to different outcomes. To make a choice between them, one can select the one with the best decoding performance (assuming that the decoded quantity was the right one, i.e. the encoded one), or the most "biological" one (but it is sometimes very hard to decide).

To complicate things a bit more, nothing tells us that there is a neural code: it could well be that there are as much neural codes as there are brain areas and/or animal classes, and that for each of them the questions enumerated above get a different answer. The unfortunate Martian is not even sure that his text is written in a single language. . .

Previous work on hippocampal activity decoding

Let us now review how these decoding issues have been applied to the hippocampus. To the question of what is encoded, the consensus is quite broad: the majority of these "code" studies answer by considering that the activity represents the animal's position. Some of these works do not tackle the decoding problem itself, but rather try to characterize the strength of the positional firing correlate. They make use of information theory to estimate how much information about position is contained in hippocampal place cell activity. See for instance references [START_REF] Skaggs | An information-theoretic approach to deciphering the hippocampal code[END_REF][START_REF] Rolls | The neuronal encoding of information in the brain[END_REF][START_REF] Cerasti | How informative are spatial ca3 representations established by the dentate gyrus?[END_REF].

A second group of studies deals with building decoding algorithms, that is reconstructing position from neural activity. The methods employed follow the same basic ideas, with only small variations across studies:

1. Assume that the information is encoded in firing rates ("rate coding").

2. Build activity templates (rate maps) corresponding to average rate population vectors for each position.

3. Compare the instantaneous population activity vector (on a given timescale ∆t) to the template at each position.

4. The decoded position is either the one where population vectors are most similar ("template-matching") or the most likely one ("Bayesian" methods). The latter case demands to make assumptions on the cells firing statistics, most of the time Poisson distribution and independent neurons are assumed [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF]. Wilson & McNaughton (1993) [START_REF] Wilson | Dynamics of the hippocampal ensemble code for space[END_REF] measured the precision of the template-matching method as a function of ∆t and of the number N c of recorded cells: the decoding error (distance between the decoded position and the actual position) decreases with increasing N c and decreases then saturates (because of the rat's velocity) when increasing ∆t. By extrapolation, they estimated that, with this method and with ∆t = 1s, the activity of about 130 cells would be necessary to reach an accuracy of 1 cm. Zhang et al (1998) [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF], followed by Brown et al (1998) [START_REF] Brown | A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells[END_REF], made a comparative study of these rate-based methods. They found Bayesian approaches to be more accurate than template-matching, and even more when a continuity constraint was added. The performances were compared to the estimate of the theoretical maximum [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF]. The issue of the read-out mechanism's biological implementation was addressed [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF].

Jensen & Lisman (2000) [START_REF] Jensen | Position reconstruction from an ensemble of hippocampal place cells: contribution of theta phase coding[END_REF] added phase coding on top of rate coding, by incorporating phase precession to their decoding paradigm. They found this additional parameter to greatly improve the decoding precision, probably because in their experiment the rat runs in constant direction on a linear track. Indeed, phase precession is much more salient in 1D than in 2D: consequently, using firing phase to decode position in a 2D open field would probably have a much lesser effect. They also discuss the problem of intrinsic tracking error: the "true" position of the rat has itself error bars that complicate the measure of the decoding precision. Finally, they address the issue of low activity that is a feature of the hippocampus: on timescales smaller than one second, most recorded cells are silent, so a trade-off has to be found between the "real" relevant timescale and the one that gives enough statistics. This is a problem we also had to face, we will come back to it.

However, some works used approaches differing from this rate-based paradigm. Recently, Agarwal et al [START_REF] Agarwal | Spatially distributed local fields in the hippocampus encode rat position[END_REF] took a completely different path and showed that the position could be reconstructed not from spikes but from local field potentials (LFP) with a comparable precision.

In general, all the aforementioned studies are based on single-environment experiments and focus on decoding the position in this environment. We will see how the multipleenvironment case can be addressed.

The cell assembly hypothesis

An alternative view on the decoding issue starts from a conjecture about brain processes. In 1949, Donald Hebb postulated that cognitive events were the result of the activation of "cell assemblies" -subpopulations of neurons distributed across a large population [7]. A given neuron can be part of several cell assemblies, but what is important is the simultaneous firing of a whole assembly. This is the so-called cell assembly hypothesis. To relate it with what we have said in Chapter 1, the cell assembly hypothesis is more precise than the connectionist hypothesis and more general than the attractor paradigm. (1) Connectionism states that cognitive events are collective behaviours. (2) Hebb then made more precise the notion of collective behaviour in terms of cell assembly activation. (3) Finally the attractor paradigm focusses on memory (which is a particular case of cognitive event) and postulates that not only does a cell assembly coactivate, but it remains active for a while.

What are the consequences of this paradigm on the decoding problem? To the "how" question (paragraph 3.1.2), it answers that, more than individual firing rates or spike trains, it is the knowledge of which neurons are active together that matters. Here again, what "together" means depends on the timescale. We will discuss it below. To the "what" question (paragraph 3.1.1), it does not give an absolute answer, but it claims that, whatever the external quantities that define a stimulus, internal processes do also come into play, because the connectivity itself contributes to shape the cell assemblies. In other words, neural activities do not merely mirror the animal's external environment, and the input-output scheme does not hold [START_REF] Harris | Neural signatures of cell assembly organization[END_REF].

To our knowledge, this cell assembly approach to decoding has not been applied yet to the case of hippocampal activity. Nevertheless, it has been adopted by Harris et al [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF] to the problem of spike times prediction. Given the rat's position, it consists in predicting which cells will fire and when -the inverse problem to decoding, as it were. Harris et al have shown that taking into account correlations between cells greatly enhanced the prediction performance as compared to considering cells as independent. Interestingly, they have also shown that using peer activity only (without taking position into account) gave a better prediction than the knowledge of position and the assumption that cells fire independently (see Figure 3b in [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF]). This constitutes an important, non-trivial result in favor of the cell assembly hypothesis.

To finish, the very notion of cell assembly coactivation requires to specify a timescale. Harris et al have noticed that the spike times were best predicted from peer activity on a time window of 10-30 ms. They argue that this timescale corresponds to physiological events such as membrane time constants and synaptic plasticity time windows [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF].

We will see how we can apply those ideas on decoding.

Experimental data

In this section I will present the data on which the subsequent analysis is based. This was made possible thanks to Karel Jezek and Edvard Moser, from NTNU Trondheim, who kindly gave us access to their recording data. The data come from an experiment they had carried out a few years before and whose results have been reported in a 2011 paper [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF].

The teleportation experiment

The experiment was originally designed to study abrupt transitions between two representations. To this end, Jezek and his colleagues used two square boxes (environments) A and B that differed only by the lighting conditions so that it was possible, by suddenly switching the lights, to instantly move from one environment to the other. They called this procedure "teleportation", because the rat has been trained in the two boxes as distinct environments, so when the lights are switched the animal is like teleportated from one place to the other. Here we briefly sum up the experimental procedure, see ref. [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF] for more details. 6 rats were used in the experiment. They were implanted in CA3 with 14-tetrode bundles and trained to forage for randomly scattered food pellets in each box. The training procedure was specifically designed to form orthogonal representations between the two environments, that is to force the remapping from one box to the other, a procedure already described in [40]. Moreover, recording from CA3 instead of CA1 minimizes the overlap between the populations of cells that are active -i.e. have place fields -in each environment [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF].

After the training phase was completed, a 10-minute 'reference session' was recorded in each environment that would be used for establishing reference population vectors (i.e. rate maps, see 3.3.1 below). Then, they proceeded to a 10-minute 'test session': the rat was put in one of the environments, and the lights were suddenly switched to the other environment. The teleportation was thus repeated every 40-60 seconds.

The results were striking in several respects. First, the neural activity was almost always either very 'A-like' or 'B-like' (a more rigorous definition will be given below), but not a combination -or 'mixed state' -of the two. This is consistant with what would be expected from an attractor neural network, and with what had already been observed before (see paragraph 1.4.4). But this is not the end of the story. Right after the teleportation, the neural activity abruptly jumps from the old environment to the new one, but then it goes back and forth for a few seconds between the two representations. Jezek et al have named this phenomenon "flickering". What is more, this flickering seems to be synchronized by the theta rhythm, an effect to which a great part of their analysis is devoted.

Description of the dataset

The dataset we have studied comes from a recording in one rat's dorsal CA3, during one reference session in A, one reference session in B and one test session (with 15 teleportations). The filtered and sorted signal came from a total of N c = 34 place cells. For each session, we thus have the following data:

• the spike times {t s k } of each neuron k = 1 • • • N c ,
• the trajectory (x(t), y(t)) of the rat,

• the EEG signal giving the local field potential LFP(t),

• in the Test Session, the "teleportation times", that is the times when the lighting were switched from one environment to the other.

An example of a neuron's activity together with the trajectory of the rat is given in Figure 1.5. We define the correlation between two neurons i and j at time-shift τ with resolution ∆ t as the quantity

C ij (τ ) ≡ T tot N i N j ∆t t N i t i =t i1 t N j t j =t j1 θ(∆t -|t i + τ -t j |) , (3.1)
where θ is the Heaviside step function, N i (resp. N j ) is the total number of spikes of neuron i (resp. j) and the t i , t j denote the spike times of i and j respectively. If i = j, the graph of C ij as a function of τ (cross-correlogram) informs about the co-activation of i and j. If i = j we speak about neuron i's auto-correlogram.

Motivations

What we want to do now is to decode the spike trains {t s k } k=1•••Nc . As we have seen before, this can be done in many ways depending on what we assume to be encoded and how we assume it to be encoded. Suppose that, after binning the time, we have the activity vectors

n(t) = (n 1 (t), • • • , n Nc (t)) (3.2)
where n k (t) is the number of spikes emitted by neuron k between times t and t + ∆t (the timescale ∆t remains to be defined).

Ultimately, with the teleportation data, we want to decode an environment rather than a position. To do so, we will introduce an approach (called "Ising-based") based on the effective network underlying the neurons' spiking activity in each environment. This approach is set out in Section 3.4, and the properties of the inferred network are studied in Section 3.5. The Ising-based approach only allows us to decode an environment, but not a position. For the sake of comparison with more classic methods and with previous studies, we also make use of rate-based methods (both template-matching and Bayesian) described in paragraph 3.3. We will test the rate-based methods on the standard problem of position-decoding (paragraph 3.3.6). We will compare these methods, from the point of view of their performances in decoding an environment (Section 3.6) and from the point of view of their predictions of spontaneous transitions (Section 3.7). Finally, we will try to draw the link between both classes of methods (Section 3.8).

The low activity issue and its relationship with time binning

As mentioned before, one of the issues in decoding hippocampal activity is the low average spiking rate of place cells. In our dataset, the mean activity

a ≡ 1 N c 1 T tot k t n k (t) (3.3)
is equal to 0.87 Hz in reference session A, 0.65 Hz in reference session B and 0.99 Hz in the test session.

In order to decode the activity, one has to choose a timescale for the description of this activity. Several conflicting points come into play:

1. because the overall activity is low, time bins of less than a second contain only a few spikes, or even no spike at all (Fig. 3.2). That is why the position-decoding precision generally increases when increasing the bin up to one second [START_REF] Wilson | Dynamics of the hippocampal ensemble code for space[END_REF].

2. because of the motion of the rat, the longer the time bin, the larger the distance in space over which the activity is averaged out. So if the time bin is too long, the activity is blurred.

3. we do not know what is the "biological" timescale, that is the timescale used by the brain for decoding. The flickering reported by Jezek et al occur on the timescale of the theta rhythm (around 120 ms). Harris et al [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF] and Tavoni et al [123] estimate the coactivation of cell assemblies to take place over a few tens of ms. To conclude, as concerns position decoding, points 1 and 2 lead to a trade-off around 1 s where the position is decoded with maximal precision, consistently with the results of Zhang et al [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF] but more biological considerations indicate that the relevant phenomena happen at smaller timescales. To circumvent the problem of point 1 and shorten the timescale, people have found ways to get rid of low activity time bins, either by merely excluding them from the dataset [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF][START_REF] Jensen | Position reconstruction from an ensemble of hippocampal place cells: contribution of theta phase coding[END_REF] or by limiting their effect with a continuity constraint on the decoded trajectory [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF][START_REF] Brown | A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells[END_REF] 3 . We have tried to decode the activity on several timescales (see below). We have decided not to exclude bins of low activity, first because they do contain information, and also because we think that a decoder that would be extremely performant but only on a small fraction of the bins would not be very useful.

We will now introduce 7 different decoding methods and discuss the ideas underlying their approaches. We will compare their outcomes in decoding the present dataset, in terms of performance and in terms of detection of spontaneous transitions.

Rate-based decoding methods

We have implemented standard decoding methods based on the comparison of population activity vectors with reference rate maps (see paragraph 3.1.3). There are many ways to perform this comparison. As mentioned above, they fall into two main classes described in detail by Zhang et al [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF]: the "template-matching methods" looking for the position that resembles most the current activity vector, and the "Bayesian methods" computing the most probable position given assumptions on the cells firing probability distribution. We have tried both types of methods. The particularity here is that we have two maps: the methods therefore have to be extended to the multiple-environment case, which is quite straightforward.

Rate maps

In either class of methods, one has first to build each neuron's rate map in each environment. To do this, we follow the same procedure as Jezek et al (see [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF]). Space is discretized into 2 cm × 2 cm bins. Rate maps are based on reference sessions: for each neuron k, in each environment and at each position x, the average firing rate is defined as the total number of spikes emitted by k when the animal was located in the bin x devided by the total time spent in x. The outcomes are "raw rate-maps". They are then smoothed by convolution with a Gaussian kernel. We end up with smooth rate maps {r M k ( x)} k, x,M , where r is the averaged, smoothed firing rate, k is the cell index, x is the position and M is the map (A or B).

Template-matching method: dot-product (DP)

The simplest way to compare the population activity vector n(t) with the reference vectors {r M k ( x)} k, x,M is to project the former on the latter and to look for the position that maximizes this dot-product. The decoded position is thus:

( x DP , M DP ) ≡ arg max x,M n(t) • r M ( x) .
(3.4)

Here the method has been applied to the multiple-environment case simply by taking the maximum over both positions x and maps M. The decoded map is merely M DP .

Incorporating knowledge of position: local dot-product (loc-DP)

In the analysis of their teleportation experiment, Jezek et al [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF] make use of two different template-matching methods to decode the environment (A or B): a dot-product comparison as above, and a Pearson correlation comparison. But in both cases, instead of looking for the most resembling reference vector among all positions in both environments, they restrict the comparison to two reference vectors only: the one at the rat's current position in A and the one at the rat's current position in B, namely r A ( x rat (t)) and r B ( x rat (t)), where x rat (t) denotes the position of the rat in the square box. This approximation simplifies the procedure for decoding a map by assuming that the only unknown is the map (A or B), while the position in this map is known. So this is only an environment-decoding method, not a position-decoding method.

For comparison with their study, and to estimate the effect of assuming the knowledge of position on the decoding performance, we will hereafter add their "local dot-product" method in our set:

M loc-DP ( x rat ) ≡ arg max M n(t) • r M ( x rat ) . (3.5)
We have chosen the dot-product similarity criterion rather than Pearson correlations because this latter method depends strongly on the sampling, and is not defined on the (numerous) bins with no spike.

A difference between our "local dot-product" method and the one of [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF] lies in the time binning procedure: Jezek et al aimed at studying the role of the theta rhythm in the flickering phenomenon. To this end, they binned time into successive theta cycles, defined with respect to the filtered EEG signal. Because we want to vary the timescale, and because we do not intend to look at the theta rhythm effects, we have chosen the simpler option of constant-length bins.

Bayesian method based on Poisson statistics: max. posterior (MP)

Instead of wondering which reference vector most "looks like" the current activity vector n(t) (with all the arbitrariness in the choice of a resemblance criterion), one may ask what is the probability for n(t) to have been emitted at each position of each environment, and thus to find the most likely position through the Bayes' formula

P ( x, M| n(t))P ( n(t)) = P ( n(t)| x, M)P 0 ( x, M) , (3.6)
where P 0 ( x, M) represents the a priori knowledge we have on the rat's position (certain positions are more often visited than others, increasing their weight in the decoding procedure). The choice of a resemblance criterion is replaced by the (less arbitrary) choice of a firing probability distribution P ( n(t)| x, M). The standard assumption is that cells fire independently following Poisson statistics [START_REF] Sanger | Probability density estimation for the interpretation of neural population codes[END_REF] (each cell spikes with a Poisson distribution of mean equal to the cell's average firing rate in x, M):

P ( n(t)| x, M) = k exp(-r M k ( x)∆t) • (r M k ( x)∆t) n k n k ! . (3.7)
Both the Poisson statistics and independence assumptions have shown to be inexact (see, respectively, refs. [31] and [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF]), and this is one of the motivations for our Ising-based approach. Yet they have been considered as good approximations in previous studies on decoding [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF].

The Bayesian approach incorporates an a priori on the decoded quantity through the prior distribution P 0 ( x, M). Here, we take this probability as proportional to the time spent in position x in map M. The term P ( n(t)) plays the role of a normalization factor, so it does not need to be directly evaluated: one just has to make sure that the posterior distribution P ( x, M| n(t)) is correctly normalized to one.

In order to decode a position, one has to maximize the posterior likelihood across positions and maps:

( x MP , M MP ) ≡ arg max x,M P ( x, M| n(t)) .
(3.8)

In order to decode a map now, one has to sum over all positions. Indeed, the likelihood of map M is given by the posterior

P (M| n(t))P ( n(t)) = P ( n(t)|M)P 0 (M) , (3.9)
where

P ( n(t)|M) = x P 0 ( x|M) k exp(-r M k ( x)∆t) • (r M k ( x)∆t) n k n k ! . (3.10)
Finally, the decoded map is the one that maximizes the posterior:

M MP ≡ arg max M P (M| n(t)) . (3.11)
Note that, contrary to Eq. 3.7, the probability 3.10 does not factorizes as a product over neuron indices k, hence the environment-decoding procedure does not consider the cells as independent. To retrieve the independence of cells, one can average the rate maps over positions: this is the "rate -independent procedure", described hereafter.

Averaged rate maps: independent cells (IND)

This method consists in assuming that an environment M is represented by a reference vector r moy,M that is the average over all positions of the reference vectors at each position, namely:

∀k = 1 . . . N c , r moy,M k ≡ r M k ( x) x . (3.12)
Assuming a Poisson activity as before, the probability to emit a population vector n(t) in map M now reads

P ind ( n(t)|M) = k exp(-r moy,M k ∆t) • (r moy,M k ∆t) n k n k ! . (3.13)
Note that this probability does not depend on the position anymore, since the dependence on x has been averaged out.

Assuming an uniform prior on the environment (that is, no environment M is a priori more likely than the other), we end up with

P ind (M| n(t)) = 1 P ind ( n(t)) k exp(-r moy,M k ∆t) • (r moy,M k ∆t) n k n k ! , (3.14)
where, as before, P ind ( n(t)) is just a normalization factor. And as before, the decoded map is the one that maximizes the posterior:

M IND ≡ arg max M P ind (M| n(t)) . (3.15)
Now the probability 3.14 is indeed a product over cells, so the model is actually an independent-neurons model. For this reason, it is simpler than the previous model (Eq. 3.10) which takes into account correlations between cells through the sum over all positions.

Decoding a position

In order to make comparisons with previous studies, we have tried to decode the position of the rat with the 2 aforementioned position-decoding methods, namely the "dot-product" method and the "max. posterior" method. Figure 3.4 shows the precision of these methods as a function of the decoding timescale. As expected, the error decreases as ∆t increases and then saturates. For timescales larger than a few seconds, the error increases again because of the motion of the rat (not shown). We also notice that the "max. posterior" method performs better than the "dot product", consistently with Zhang et al 's study [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF].

Quantitatively, the performances are in good agreement with previous studies. For the same number of cells, Wilson & McNaughton [51] found error values that closely match ours. Zhang et al [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF] and Brown et al [START_REF] Brown | A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells[END_REF] obtain better performance with like-for-like timescale and number of cells (the error is approximatively divided by two). We expect it to be mainly due to details in the decoding algorithm, albeit other differences could be invoked (decoding of CA1 instead of CA3, 1D environment in [START_REF] Brown | A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells[END_REF]. . . ). Their processing of bins without spikes (see paragraph 3.2.3) and their use of continuity constraints have the consequence that each bin is not decoded alone but as a function of the previous bins, explaining a better performance.

We could try here to optimize our algorithms in order to improve the performance in the task of decoding a position. Nevertheless, this has already been done and our point is not to find how to lower the decoding error as much as possible. Rather, we are more interested in the task of decoding an environment; in this purpose we make use of rate-based methods only for comparison.

The inverse-Ising approach to decoding

The rate maps introduced above provide a description of the neurons activity as a function of position. Since we aim at decoding an environment rather than a position, it seems natural to describe the typical neural activity globally in each environment. A quite precise description would consist in providing the whole distribution of population vectors, namely for each possible activity vector n giving the probability P ( n). However, in practice this method is impossible to implement because there is an exponential number of such possible n (even an infinity if we do not bound the n k ). One way to circumvent this difficulty is to assume that the distribution P ( n) is the outcome of the dynamics of an underlying system and to try to characterize this hidden system as best as possible. Why this is relevant is because the underlying model has in general much less parameters than the huge number of possible n.

The point is now to choose this underlying model. It has been shown that neural activity could be well described by one-and two-point statistics [START_REF] Schneidman | Weak pairwise correlations imply strongly correlated network states in a neural population[END_REF]. In other terms, as soon as we know the averages σ i t and σ i σ j t for all neurons i, j (where the σ i describe the activity with binary units on a given timescale), then we can estimate with good accuracy the higher order statistics σ i 1 • • • σ i k t . Therefore, a model that reproduces the mean activity of the neurons and their pairwise correlations should be a good approximation. Among all the models that could do that, the Maximum Entropy principle leads us to choose the "most neutral" one, the one that makes no additional assumptions on the activity statistics 4 . This model is the Ising model. It has been widely used to describe biological or neural systems [123,[START_REF] Schneidman | Weak pairwise correlations imply strongly correlated network states in a neural population[END_REF][START_REF] Cocco | Neuronal couplings between retinal ganglion cells inferred by efficient inverse statistical physics methods[END_REF]. We have already mentioned it, in a different context, when introducing the analogy between neural and magnetic systems. Though the motivations were not the same -we wanted then to represent real neurons and synapses as magnetic spins and couplings, while now we only want to reproduce the firing statistics by an underlying, effective model -the ubiquity of the Ising model is not surprising: it comes from the fact that neural networks are mostly formed by two-point synapses (and not, for instance, synapses between 3, 4. . . neurons).

In the Ising model, binary units σ i receive external fields h i and are connected to each other by (symmetric) couplings J ij = J ji . The probability of a configuration σ = {σ i } is given by

P ( σ) = 1 Z exp i h i σ i + i<j J ij σ i σ j , (3.16)
where Z enforces the normalization of the probability distribution. We now have to retrieve the parameters of the Ising network (namely the fields {h i } and the couplings {J ij }) from the observed statistics (namely the magnetizations m i ≡ σ i t and the two-point correlations p ij ≡ σ i σ j t ). This is the famous inverse Ising problem; it is hard to solve exactly when the number of units exceeds 10-20.

Here we use an algorithm proposed by Cocco & Monasson [START_REF] Cocco | Adaptive cluster expansion for inferring boltzmann machines with noisy data[END_REF] to estimate these parameters respectively in each of the three sessions A, B and T. We do the inference on a timescale of 10 ms: here again, this choice is the result of a compromise due to the low activity. It is a trade-off between observing pairwise correlations and having a sufficient number of timebins for the statistics to be reliable. After running the inference algorithm on this timescale, we end up with a set of couplings and fields that will hereafter be denoted

{h M i , J M ij } M=A, B, T i,j=1•••Nc .
How can we use these quantities to decode an environment? The first idea is in the "template-matching" spirit: since we have a characterization for neural activities in A and in B respectively, then for a given population vector n we can tell whether it looks more "A-like" or "B-like". This approach will be made more precise in paragraph 3.4.1. An alternative approach, based on the cell assembly hypothesis, is to look at which neurons coactivate and to use the inferred coupling matrices to see whether these coactivations are typical of A or B. Instead of looking for an absolute resemblance, the idea here is to try to detect the formation of cell assemblies relative to a background of activity, regardless of the absolute value of this background. We will define this method in paragraph 3.4.2.

Decoding with the coupled model

The first way to decode with the inferred Ising networks is simply to compute the likelihood of n -the population vector to be decoded -in the Ising model A and in the Ising model B respectively, and to consider that the decoded environment is the one with the largest likelihood.

The log-likelihood L of the (10 ms) configuration σ in map M is given by

L M 10 ( σ) ≡ i<j J M ij σ i σ j + i h M i σ i -log Z M . (3.17)
The only subtlety here concerns timescales: the Ising models have been inferred on 10 ms bins while we want a method for any decoding bin length (say greater or equal than 10 ms). So to decode the activity n(t) on a bin ∆t we just average the log-likelihood L over all the 10 ms sub-bins in the bin t:

L M ( n(t)) ≡ L M 10 ( σ(τ )) ∆t . (3.18)
For a given population vector n(t), the decoding algorithm is then simple:

• evaluate L A ( n(t)) and L B ( n(t)),

• the decoded environment M is the one of maximal L M .

Decoding by co-activation detection: rescaled model

The second idea is that the inferred effective networks give us access to cell assemblies. The {J M ij } are supposed to capture the tendency of pairs of neurons to coactivate regardless of an external modulation of the activity. In other terms, the representation by an effective Ising model disentangles the correlation due to recurrent connectivity from the one due to common external inputs.

We thus define a new measure of the likelihood of a coactivation pattern in each map M. It is done in two steps: (1) defining the instantaneous fluctuations of activity and (2) moving from a 10 ms timescale (for which the couplings have been inferred) to a decoding timescale ∆t.

Definition of instantaneous fluctuations of the activity

We want to measure the degree of coactivation of two cells with respect to their instantaneous firing rates. As before we start from the observed number of spikes n i (t) measured on a time window ∆t. The idea, proposed by Renart et al [START_REF] Renart | The asynchronous state in cortical circuits[END_REF] , is to substract from n i (t) its average on a larger time window ∆t 2 . In their study they take ∆t 2 = 4∆t. Here, we find a trade-off between what we think would be the "real" timescale (a common external input is supposed to last for the duration of place-field crossing -typically a few seconds) and the need to consider long enough time windows in order to cope with low statistics. We have observed that by increasing ∆t 2 the performance of our decoder (to be defined below) increases and saturates around ∆t 2 ≈ 30 s. So we take ∆t 2 = 30 s, keeping in mind that this value is an ad-hoc compromise. For each time bin t of length ∆t we thus consider the quantities

δn i (t) ≡ n i (t) -n i ∆t 2 , (3.19)
where n i ∆t 2 is obtained by convoluting the spike trains with a Gaussian of width n i ∆t 2 .

Rescaling procedure

The next step is to "convert" the couplings inferred on 10 ms to a timescale ∆t. Here we just rescale the couplings by the mean activities. Writing

J ij σ i σ j = J ij m i m j σ i m i σ j m j , (3.20)
(where we have dropped the superscript M and where the m i ≡ σ i t ), we define the

rescaled couplings Ĵij ≡ J ij m i m j (3.21)
that are supposed to be now time-dimensionless. This mean that we also need to rescale the activity variables with respect to their average (as in Eq. 3.20), so we introduce

δi (t) ≡ n i (t) -n i ∆t 2 n i ∆t 2 . (3.22)
The δi thus measure how much neuron i is activated on the timescale ∆t relatively to its mean activity on the larger timescale ∆t 2 .

Measure of the likelihood of a co-activation pattern

Finally, we define the likelihood of a co-activation pattern δ in map M as the interaction part of the log-likelihood 3.18 with rescaled couplings and activity variables:

I M ( δ) ≡ i<j ĴM ij δi δj . (3.23)
To decode an activity pattern n(t) with the measure I, we just follow the steps below:

• rescale the couplings according to Eq. 3.21,

• compute δ(t) given by Eq. 3.22,

• evaluate I A and I B given by Eq. 3.23,

• the decoded environment M is the one of maximal I M .

Independent model

Lastly, we want to estimate how the couplings in the inferred Ising network improve the decoding with respect to an Ising model with independent units (only the external fields intervene). Such a model only reproduces the average activities σ i t and not higher order statistics. This case of independent units is an easy one. The effective fields h ind i then have a straightforward expression as a function of the averages:

h ind i = log m i 1 -m i . (3.24)
The log-likelihood L of the configuration σ in map M reads

L ind,M 10 = i h ind,M i σ i -log Z ind,M , (3.25)
where the partition function reads

Z ind,M = i 1 + exp(h ind,M i ) . (3.26)
As with the log-likelihood with couplings 3.18, we then average over the ∆t time bins the log-likelihoods computed on 10 ms bins:

L ind,M ( n(t)) ≡ L ind,M 10 ( σ(τ )) ∆t , (3.27)
and the decoding procedure is the same:

• evaluate L ind,A ( n(t)) and L ind,B ( n(t)),

• the decoded environment M is the one of maximal L ind,M .

Summary of methods

In total, we have introduced 7 decoding methods: 4 rate-based and 3 Ising-based. A recapitulative summary of these methods is given in Table 3

Structure of the inferred effective networks

The inference algorithm gives the effective couplings between neurons. We have asked ourselves how this effective network was related to the place field allocation in each map. Another interesting question is how the A and B networks are reflected in the test session network. We report here our main results on the structure of the inferred connectivity matrices. Because the couplings are computed with some error bar ∆J ij , we need to introduce a reliability criterion. Hereafter, a coupling will be considered as reliable when it satisfies

|J ij | ∆J ij > 3 .
(3.28)

The couplings correlate with the overlap between place fields

At the core of all the attractor models for place cells, there is the Hebbian idea that neurons with overlapping place fields form excitatory couplings during learning (see Section 2.6). We wondered whether some connectivity rule of this kind could be visible in the effective network. We have plotted the value of the inferred couplings as a function of the normalized overlap between place fields, defined in map M (=A or B) as

O M (i, j) ≡ x (r M i ( x) -r M i x )(r M j ( x) -r M j x ) x (r M i ( x) -r M i x ) 2 x (r M j ( x) -r M i x ) 2 . ( 3.29)
The results are shown in Figure 3.5. A positive correlation clearly appears both in the raw couplings J ij and in the rescaled couplings Ĵij . Some remarks are necessary here. First, we notice that the correlation is not perfect: there are some strong couplings between neurons with low overlap and vice-versa. This effect is not problematic, rather it was expected: according to the model, the couplings embed the structure of the current environment but also some disorder coming from the other stored memories. Of course, one must be careful not to confuse the "real" synapses and the inferred couplings which are only effective quantities. Still, it is not surprising that the co-activations captured by the latter are strongly dependent on the arrangement of place fields, but not totally determined by it.

Then, we have used normalized centered overlaps as a measure of the similarity between two neurons' rate maps. Indeed, we have tried several other measures (distance between place field centres, un-normalized or un-centered overlaps) and it turns out that O M (i, j) defined by (3.29) is the quantity that correlates best with the couplings. This is an interesting point: it means that the couplings capture the similarity between rate maps with respect to their respective background levels of activity rather than in absolute values.

The couplings satisfy an additivity rule

The test session consists in an alternation between maps A and B. One may then wonder how the effective coupling structure computed in this session is related to its respective counterparts in A and B. Interestingly, as shown in Figure 3.6, it seems that the J T ij are a linear superposition of the J A ij and J B ij , with a coefficient equal to the relative time spent in each map during the test session:

J T ij ≈ γJ A ij + (1 -γ)J B ij , (3.30)
with The reliable couplings satisfy quite well the additivity law. Interestingly, the agreement is even better if we consider the rescaled couplings (Fig. 3.6, right), supporting the idea that our rescaling procedure is pertinent to the attempt to compare different regimes of activity.

γ ≡ τ T (A) τ T tot ≈ 0.41 . (3.31)
We can exclude at least three artefacts that could have possibly led to this additivity feature. (1) First, the additivity does not come from the inference algorithm itself: suppose you have data from a network with couplings J A ij and other independent data from another network with couplings J B ij . Then, if you artificially concatenate both datasets and run the inference algorithm on the resulting big dataset (provided both networks have the same number of units), the inferred couplings will a priori not be the average of the couplings inferred separately in A and B. To convince ourselves, we can for instance consider the simple case of two spins coupled by J A 12 , J B 12 and J T 12 in A, B and T respectively. Suppose that there are no external fields. Then, we can immediately show that

J M 12 = log 3 σ 1 σ 2 M 1 -σ 1 σ 2 M , for M = A, B, T . (3.32)
We have on the other hand

σ 1 σ 2 T = γ σ 1 σ 2 A + (1 -γ) σ 1 σ 2 B . Eq 3.32 is obviously not linear in σ 1 σ 2 M .
Hence in general we do not have the relationship 3.30.

(2) Another situation that would trivially lead to the observed additivity law would be the case where the J A ij , J B ij and J T ij are equal or have very close values. We have checked that it is not the case (see Table 3.2).

(3) Last, if some of these pairs of neurons were decorrelated in one of the two maps, say A, then the additivity would be trivially explained. In this case, J A ij would be close to zero for these pairs: this does not happen (see Table 3 3.2: Values of the inferred couplings in each map for the 11 pairs of neurons with reliable couplings in all three sessions (magenta points in Fig. 3.6). We can see that the additivity rule (Eq. 3.30) is not the trivial result of

J A ij ≈ J B ij ≈ J T ij nor (J A ij ≈ 0 or J B ij ≈ 0).
In the end, the inferred network in the test session seems to capture some non-trivial additivity property.

To conclude, the two fundamental elements of Hebb's rule are found in our effective couplings, namely the reinforcement of couplings between co-activating neurons ("neurons that fire together wire together"), and the additivity of patterns. The former is visible in the correlation between couplings and maps overlap in A and B respectively. The latter appears in the couplings inferred in the test session. Nevertheless, these results concern only the effective networks: we are of course not able to state whether they reflect a learning rule at the synaptic level.

Decoding an environment

We now turn to our initial goal: retrieving the rat's current environment from the observed neural activity. We will compare the different decoding methods introduced above on different timescales. To do this, we introduce a measure of the performance Π(X) of a given decoding procedure X on a timescale ∆t as the percentage of correctly decoded time bins of length ∆t:

Π(X) ≡ N bin (correct X) N tot bin . (3.33)
This requires to specify what we consider as a "correct" decoding. As pointed out before, we do not know what the hippocampal activity ultimately "means" for the brain, hence we do not know what is the right answer a decoder should give: we have to resort to assumptions. Here, we will assume that the hippocampal activity encodes the environment in which the animal currently is. Under this hypothesis, a perfect decoding method should decode A in all time bins in reference session A and in time bins in the test session when the lights are in the A configuration, and B in all time bins in reference session B and in time bins in the test session when the lights are in the B configuration. This means considering spontaneous transitions and flickers as decoding errors, since neural activity is assumed to always reflect the current external configuration. It is a convenient approximation when it comes to comparing decoding methods, given that the spontaneous transitions and the flickers, if they do occur, only represent a tiny fraction of the total number of bins: the measured quality of the decoder will not be much affected by them. Nevertheless, we will reconsider this assumption in the next section, as we believe that there are spontaneous transitions and flickers that have a reality in terms of neural dynamics and are not mere decoding artefacts. With this performance measure in hand, we are now able to quantitatively compare our rate-based and Ising-based methods. We will do it first when all neurons are included in the dataset, second with a subset of them, with the aim of addressing the issue of non-orthogonal representations. What we will present here are preliminary results; there is still ongoing work on this part.

Performance of the decoding procedures -all neurons included

We have run our decoding algorithms on several timescales on all three recording sessions (A,B,T). Here, we report the results in the test session (T). As mentioned above, we include all the time bins in the decoding, even those with no spike at all. Figure 3.7 illustrates the Ising-based "coupled" decoder on the test session. As expected, the performance decreases at short timescales due to the large number of bins with zero spikes. This is particularly true for the dot-product methods because they are unable to decode those bins. The "local -dot product " performs better than the "dot product", meaning that the knowledge of the rat's position helps retrieving in which environment the rat is.

A striking feature is that 4 methods ("rate -max. posterior", "rate -independent", "Ising -coupled" and "Ising -independent") have almost equal performances. There are two points in this result. First, the models based on rate maps and Poisson firing give results comparable to the Ising-based models. Though there is still ongoing work to fully understand this similarity, a tentative link between both classes of models is proposed hereafter (Section 3.8). Second, there is the fact that the independent models give similar results to their coupled counterparts -"rate -independent" vs "rate -max. posterior" on the one hand, "Ising -independent" vs "Ising -coupled" on the other hand. This means Red: Ising-based methods. All N c = 34 neurons were included in the decoding. 4 graphs ("rate -max. posterior", "rate -independent", "Ising -coupled" and "Ising -independent") are almost merged together.

that assuming that cells fire independently is a reasonable approximation here, in the sense that incorporating two-point couplings does not improve significantly the decoding. It can be understood by the high degree of orthogonality between firing patterns in both maps (see below), and also by the low activity level. Indeed, bins with at least two spikes are rare. At certain timescales, the "Ising-independent" method performs even slightly better than the coupled one, a result that can only be explained by the predominance of bins with 0 or 1 spike. Lastly, we can see that, relative to those 4 methods, the Ising-rescaled method has rather poor performances at large timescales. This is in agreement with the good performance of independent-cell methods: the Ising-rescaled method only takes couplings into account, descarding a lot of information contained in the fields. However, its relative performance is better at short timescales, consistently with the idea that cell assemblies co-activate on the timescale of a few tens of ms [START_REF] Harris | Organization of cell assemblies in the hippocampus[END_REF]123]. Finally, the fact that its performance is far above chance level indicates that the sole co-activation of cells does capture some aspect of the neural activity typical of each environment.

Case of overlapping representations

An important feature of our dataset is the orthogonality of the representations of A and B: most cells have a place field either in A or in B but the overlap between the two is low. Only 5 neurons among the N c = 34 have place fields in both maps, but even in their case there remains a strong asymmetry between A and B (the average firing rate is much higher in one than in the other). This orthogonalization of representations is a characteristic feature of CA3 [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF][START_REF] Leutgeb | Pattern separation in the dentate gyrus and ca3 of the hippocampus[END_REF]. In most experiments looking for attractor dynamics, this phenomenon is welcome because it makes a neat distinction between patterns of activity. Training protocols have been designed to favor the formation of orthogonal representations [40].

As far as decoding is concerned, orthogonal representations are a major advantage for independent-cell decoding procedures because the mere knowledge of which cells spiked and which were silent -regardless of the exact number of spikes, not to mention the correlations between cells -generally suffices to decide in which environment the animal was. This is why we have observed that the independent-cell methods perform as well as their coupled counterparts. In the case where the subpopulations active in each map are overlapping, however, independent-cells methods could turn out to be less efficient and the interest of taking correlations into account appear. This situation could for instance occur with CA1 data [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF] or any other case where the cell assemblies are not turning on in the middle of silence but instead co-activate within a background of activity [123].

In order to test this possibility with the CA3 data we had, we have tried to artificially "un-orthogonalize" the representations by limiting the sampling to the subset of 5 cells with place fields in both environments. We have thus run our decoding procedures with this subset only. As concerns the rate-based approaches, it simply consists in taking only these 5 neurons in the dot-products (3.4,3.5) and in the product over cells in the Poisson distribution (3.7). Regarding the Ising-based methods, we ran the inference algorithm on the dataset limited to this subpopulation. The results are shown in Figure 3.9.

Quantitatively, the overall performances are obviously worse than in the previous case because we have used less data for the decoding. Qualitatively, there is no major change compared to the previous case. Of course in the N c = 5 case the low-activity problem is all the more marked. Because the bins with no spike are very numerous, we have looked at the performances if those silent bins were excluded. The outcome is shown in Figure 3.10.

It turns out that, compared to Fig. 3.8, the rank of the methods is changed. The "local -dot product" becomes very good, indicating that its bad performance was mostly due to the bins with no spikes. The "Ising -coupled " becomes better than the "rate -max. posterior", "rate -independent" and "Ising -independent". This is encouraging because it could mean that this method is more robust than the others to non-orthogonality. Nevertheless, this result remains to be confirmed: the fact that the "Ising -rescaled" method does not perform better than chance level is consistent with the fact that bins with more than two spikes are very rare in this dataset, as could be expected from 5 cells with low activity. This is corroborated by the fact that the performance does not increase any more with the timescale.

Conclusion on decoding performances

Do these performance results allow us to draw conclusions about the "real" encoding of representations in the brain? In other words, can we conclude, based on which method is better and on which timescale, about what is encoded and how? Our considerations on low activity and orthogonal representations show that the relative performances of those approaches are plagued by the lack of statistics (due to both the low number of cells and their low activity), and strongly depend on the sampling and on the structure of the representations. Hence, more work on larger datasets, maybe from other brain area (CA1, prefrontal cortex. . . ) would be needed to rigorously conclude on this point. We insist on the fact that these results are preliminary; in any case they demonstrate that one should be cautious about the effects of the dataset on the conclusions about decoding. Nevertheless, the strikingly close performances of the methods based, respectively, on the log-likelihood of configurations in underlying Ising networks and on the probability of configurations from Poisson units, is a non-trivial result that deserves better understanding. A first step in this regard will be presented in Section 3.8.

Decoded transitions between maps

All the decoding methods studied here give a substantial amount of "wrong" answers -decoding A when the animal is in B and vice versa. Yet, maybe these presumed uncorrectly decoded bins are not all errors coming from the decoding method, maybe some of them actually reflect some "real" transitions to the alternative environment in the rat's mind. This is besides the assumption on which is based the study by Jezek et al [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF]: the "flickers" are assumed to take place in the rat's neural activity, not to be decoding artefacts.

So we have looked at the transitions -i.e. "erroneous" bins -decoded by each method in the test session. Since, nevertheless, a lot of them are supposed to be decoding artefacts all the same, only the "strongest" transitions are considered. This requires to set a criterion on which transitions are significant and which are just decoding noise. We have followed a similar approach than Jezek et al [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF] for selecting significant transitions:

• we choose a timebin of 120 ms,

• only bins with at least two active cells are included,

• among those bins, the ones that are considered to reflect a transition from A to B (resp. from B to A) are the ones that are more similar to B than 80% of bins in the reference session in A (resp. more similar to A than 80% of bins in the reference session in B) and less similar to A than 20% of bins in the reference session in A (resp. less similar to B than 20% of bins in the reference session in B). In other words, we choose as thresholds the 80 th percentiles of the distributions of X B in reference session in A and of X A in reference session in B, and the 20 th percentiles of the distributions of X A in reference session in A and of X B in reference session in B.

The choice of this threshold is quite arbitrary, but it allows us to compare the outcomes of the different methods. These outcomes are shown in Figure 3.11. It appears first that the number of transitions considered as significant greatly varies from one method to the other, even between methods that have similar decoding performances. This means that the methods with the lowest number of significant transitions have many non-significant transitions, i.e. many "mixed states", resembling both A and B. Moreover, we have arbitrarily chosen the criterion for significant transitions, but for some methods the number of these transitions depends also strongly on the criterion. For other methods, it does not. For instance, moving from a 80 th -20 th percentiles criterion to a 90 th -10 th percentiles criterion devides the number of transitions by 15 in the case of the "Ising -rescaled" method without altering the number of transitions in the "ratemax. posterior" method. More work is needed to understand the causes of these differences. Nevertheless, it indicates that the conclusions of Jezek et al on the low occurrence of mixed states may be dependent on the decoding method used to detect them. Second, we can see in Figure 3.11 that transitions are more frequent just after a switch, and this feature is robust across methods (see Figure . 3.12). Thus, we find again the flickering phenomenon exhibited by Jezek et al, with quite different decoding methods: hence, at least after some of the switches, the occurrence of flickering seems to be robust to the decoding procedure. More generally, we note that there is a certain amount of consistency between the transitions found by each method, even if their number varies. This supports the assumption that these transitions are not artefacts coming from the decoding process, but do reflect some event at the neuronal level.

Finally, there are also transitions that are not flickers, that is transitions ocurring far from the switches. We are interested by them because they could be the equivalent of the spontaneous transitions between maps predicted by our model for place cells. The study of these transitions is the object of ongoing work.

Correspondence between the "Ising -coupled"

and the "rate -max. posterior" methods

Let us consider a timescale ∆t equal to 10 ms and an environment M. The "Isingcoupled" and the "rate -max. posterior" methods both give the probability to observe a certain configuration σ given the map M, respectively by the formulae 3.10 and 3.16. We would like to better understand the relationship between the two. Because we compare these methods in a same environment, we will hereafter drop the superscript M in order to lighten notations.

As we have already explained, the sum over positions in the "rate -max. posterior" method (see Eq. 3.7) introduces effective couplings between neurons because the probability does not factorizes over neurons any more. These couplings are not limited to two-point couplings as in the Ising model, there can be higher-order couplings: 3-point, 4-point. . . Let {h i } i , {J ij } i,j , {K ijk } i,j,k respectively the fields, 2-point and 3-point couplings in the "rate -max. posterior" method. Our aim here is to draw the correspondence between these quantities and the fields {h i } i and couplings {J ij } i,j of the Ising model.

In the Ising model the neurons are assumed to be binary units. Since for ∆t = 10ms, neurons emit at most one spike per bin (bins with two spikes are very rare), we have n(t) = σ(t) for all t. Note that σ k ! is then always equal to one. Eq. 3.10 thus writes:

P Poisson ( σ(t)|M) ∝ x P 0 ( x|M) k exp(-r M k ( x)∆t) • (r M k ( x)∆t) σ k , (3.34)
with ∆ t = 10ms. By definition of the effective fields and couplings, this can be rewritten in general

P Poisson ( σ|M) = 1 Z exp i h i σ i + i<j J ij σ i σ j + i<j<k K ijk σ i σ j σ k + . . . , (3.35)
where the sum runs over all orders of couplings. Thanks to the use of 0-1 units, σ 2 i = σ i so we can discard auto-couplings J ii , K iii . . . that are absorbed in the h i . We remind that in the Ising model the probability of the same configuration σ is given by

P Ising ( σ|M) = 1 Z exp i h i σ i + ij J ij σ i σ j . (3.36)
In order to establish the link between P Poisson and P Ising , we consider successively configu-rations where one, two, three. . . neurons are active and the others are inactive 5 . Denoting S a ≡ (0, . . . , 0, 1, 0, . . . , 0)

S ab ≡ (0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0) . . .

where the superscript indicates which spins are up, we notice that

P Poisson ( 0|M) = 1 Z , P Poisson ( S a |M) = 1 Z exp(h a ) , P Poisson ( S ab |M) = 1 Z exp(h a + h b + J ab ) , P Poisson ( S abc |M) = 1 Z exp(h a + h b + h c + J ab + J bc + J ac + K abc ) , . . .
Hence, we can express the fields and couplings as a function of the probabilities P Poisson ( S a ) . . ., which in turn have simple expressions as a function of the rate maps, given by Eq. 3.34: (3.39) This way we are able to compute the fields and couplings in the "rate -max. posterior" model and to compare them to the fields and couplings in the "Ising -coupled" model. The outcome is shown in Figure 3.13. The fields in the rate-based model are very close to the ones in the Ising-based model. The two-point couplings in the former model correlate well with the ones in the latter model, but they are weaker in absolute value and they seem to saturate for strong couplings. This can be explained partly by the existence of nonzero three-point couplings (shown in Fig. 3.14) that do not exist in the Ising approach. Plus, a perfect correspondence between both models is not expected because they do not describe the same probability distribution.

h a = log P Poisson ( S a |M) P Poisson ( 0|M) = x P 0 ( x|M)(r M a ( x)∆t) k exp(-r M k ( x)∆t) x P 0 ( x|M) k exp(-r M k ( x)∆t) ; ( 3
As a conclusion, we have established the link between the models underlying the "rate -max. posterior" method on the one hand and the "Ising -coupled" method on the other. The effective two-point couplings and the fields in both models are in good match, except for strong couplings in the Ising-based method, a discrepancy that can be explained from the higher-order couplings in the Poisson-based method. Note that the latter model is based on an assumption on the firing statistics of neurons and only reproduces their average activity. Besides, the activity of hippocampal neurons has been shown to significantly deviate from a Poisson distribution [31]. In contrast, the Ising model reproduces both the first and second-order statistics of the activity. In conclusion, it is normal that the match between both models should not be perfect. The relationship between both models is currently being further investigated.

Chapter 4

Conclusions & Extensions

Summary of results

We have tackled several aspects of the hippocampus viewed as an attractor neural network storing spatial maps. This work is in line with a thirty-year-old tradition of physicists aiming at understanding neural systems by collective behaviours of interconnected neurons. In the present case, the attractor neural network hypothesis is motivated by the properties of place cells observed in experiments.

We have first proposed an attractor neural network model for 1D and 2D place cells. It assumes that space is divided into maps, each of which consists in a random pairing between positions (place fields) and neurons (place cells). A Hebbian rule is assumed for the couplings, so that neurons with neighbouring place fields tend to coactivate together. The neurons are modelled by binary units whose activity is a stochastic function of their total input. This model, reminiscent of disordered magnetic systems, is simple enough to be studied analytically. It is a direct offspring of the Hopfield model for discrete attractors, and the methods and calculations are very similar. Yet, the more complex structure of the stored patterns sometimes leads to novel features, especially on the time evolution of the system.

Finding under which range of parameters the system works as an autoassociative memory requires to study its equilibrium states. We thus have computed the model's phase diagram as a function of the level of noise T and the memory load α. A "clump phase" is the stable phase in the low-T , low-α region. It corresponds to an activity configuration in which active neurons have neighbouring place fields in one of the stored maps -which is supposed to correspond to the experimentally observed localized patterns of activity. The other two phases are a uniform (paramagnetic) phase dominated by noise and a glassy phase dominated by the cross-talk between stored maps. The phase diagram is corroborated by numerical simulations, and its dependence on the parameters of the model -namely, w, f and c -is investigated. We have discussed the parallel with the Hopfield model's diagram.

We have then explored the dynamics of the system in the clump phase, an important aspect in many experimental situations. Two phenomena happen to be in competition: transitions between maps and collective motion of the bump of activity within one map. Transitions between maps occur more often as α and T increase. This can be understood by the calculation of the free-energy barrier to transitions, through an activated state where the activity is localized simultaneously in two environments. This barrier is lower in positions of local similarity between environments, because the formation of such a double bump is less costly energetically. Consistently, we find that transitions from one map to the other occur more often in these favourable positions; more precisely the relative frequency of transitions between two given positions in two maps is proportional to the exponential of their local resemblance. The other possible evolution for the bump is to smoothly move across one map. In the case without disorder (single-environment case, α = 0), we have demonstrated that the bump behaves as a quasi-particle emerging from the microscopic dynamics of the individual units, undergoing a purely diffusing dynamics. We have calculated its effective diffusion constant and its effective motility, in good agreement with numerical simulations. We have also studied the case with disorder, in which the bump experiences a rough free-energy landscape.

The model's parameters w, f , c, T and α have been estimated from activity patterns observed experimentally in CA3. We have used the average characteristics of the bumps of activity to fit the model's clump shape.

In the second part of this work, we have addressed the issue of decoding. We had access to the activity of 34 place cells recorded while the animal was put in two distinct environments. Decoding this activity requires making assumptions on what is encoded and how it is encoded: a first step of the work thus consists in defining such a framework.

This being done, we have introduced and compared seven decoding methods, four of them being based on rate encoding, and the other three based on a picture of the activity as the outcome of an Ising model whose parameters have been inferred thanks to an algorithm for the inverse-Ising problem. The four first rate-based methods, very widespread in hippocampal activity decoding, serve as a reference for comparison.

The idea here is that if CA3 does work as an attractor neural network, then its activity must be driven in great part by the recurrent collaterals. Observing the co-activation of cells would then inform on the regime of activity. An encouraging result, the inferred couplings reflect well the tendency of pairs of cells to co-activate with respect to their average activity. Moreover, the couplings inferred during a an alternation of the two environments turn out to be a linear combination of the couplings inferred separately in each environment -a relationship analogous to a Hebbian rule. This indicates that the inferred couplings capture the memory structure in a non trivial way. All these observations support the view of an attractor neural network underlying CA3 place cells activity.

We have compared the performances of these seven methods in terms of their ability to retrieve the animal's current environment. The so-called 'Ising -coupled' and 'Ising -independent' methods -consisting in calculating the log-likelihood of the observed activity configuration in each effective network -turns out to perform roughly as good as the rate Bayesian methods -which calculates the posterior probability of this same configuration in a framework where cells fire with Poisson statistics. At short timescales, the decoding task is plagued by low activity bins, nevertheless the Ising -coupled method becomes relatively better, consistently with the view that coactivations take place on timescales of a few tens of ms. These results, based on a small set of neurons with rather low activity and with two orthogonal representations, need to be compared to results on other datasets in order to draw general conclusions.

We have also seen that the transitions between environments detected by the different methods and considered as significant are more or less numerous depending on the methods. Nevertheless, some of them are robust across methods, indicating transition events taking place at the neural level. Those of these events that occur in the few seconds after a switch of maps are candidate flickers as exhibited by Jezek et al [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF]. The other are possible spontaneous transitions as the ones predicted by the model.

Lastly, we have shown that a direct link can be made between the "rate -max. posterior" method and the "Ising -coupled" one in terms of effective fields and couplings. There is a good match between these quantities in both models, even if the two-point couplings in the former are on average weaker than in the latter. In spite of a huge literature on the hippocampus and the presumed attractor dynamics it undergoes, a satisfactory understanding of this system has still not been reached. At the microscopic level of this thesis all the more, the work presented here is far to be completed: several interesting aspects have been left aside that would deserve further study. I will review them here, some of them are currently under study.

Refining the model 4.2.1 Continuation of the analysis

First, the estimate of the model's parameters would deserve to be refined. We could for instance infer w, T and α separately in environments A and B and see if the values match. Or even infer them separately on each bump of activity instead of averaging over all the bumps. The statistics of the quantities thus fitted would be interesting to study. A totally different approach would consist in starting from the observation that neurons work near their threshold, far from saturation. This means that inhibition balances excitation. We could imagine fitting the noise and inhibition parameters T and f from the observed neuronal response, though some work would be needed to define this problem rigorously. From the effective coupling matrices inferred in 3.4, we could try to estimate the quenched noise parameter α.

Most of the analytics have been done in the one-dimensional model (transition lines in the phase diagram, stability regions, calculation of the diffusion coefficient. . . ) . The twodimensional case is important because it has more experimental counterparts. It would be especially useful to study it with the parameter values estimated from experimental data (Section 2.7), all the more as these values turned out to be 'exotic' as compared to the ones used in the simulations. The 2D phase diagram with parameters w, f , c computed in 2.7 should be computed in order to estimate the working regime of the model. However, the 2D case is more tedious to study analytically and we do not expect major qualitative differences between the phase diagrams in both cases.

More importantly, it would be interesting to address the issue of information storage. Indeed, the number of patterns the network can store does not tell us what is the quantity of retrievable information, all the more so as in continuous maps the retrievable configurations are correlated: in the extreme case of totally correlated patterns, the storage capacity tends to infinity while the information capacity vanishes [11,[START_REF] Tsodyks | The enhanced storage capacity in neural networks with low activity level[END_REF]! Following the approach of Amit, Gutfreund and Sompolinsky [START_REF] Amit | Statistical mechanics of neural networks near saturation[END_REF] on the Hopfield model and of Battaglia & Treves [START_REF] Battaglia | Attractor neural networks storing multiple space representations: a model for hippocampal place fields[END_REF] on continuous maps, we could estimate the information contained in the network as a function of the model's parameters w, f , c, T and α. There are several interesting questions. The role of noise is one of them: does a moderate level of noise T improve the information capacity as it improves the storage capacity? Is there a general relationship between storage capacity and information capacity? According to previous studies, it seems that the answer is no. How does the information content scales with f in the limit of sparse representations (to be linked to the discussion of paragraph 2.3.4)? Is the choice f ∼ w optimal also in terms of information capacity? Do we find quantitative estimates of the information capacity that match the values found by Battaglia & Treves, as was the case for storage capacities?

Then, further work is required to better understand the link between networks storing discrete patterns (as the Hopfield model) and networks storing continuous patterns (as ours). A continuous map is equivalent to a set of N configurations, with a high degree of correlation among them. In one dimension for instance, we can rewrite the couplings

J ij = 1 N L =0 θ(wN -d ij ) (4.1)
(θ is the Heaviside step function and d ij is the distance between the centres of the place fields of i and j in the environment ), as a sum over (L + 1) × N patterns indexed by , x 0 :

J ij = 1 N L =0 N x 0 =0 ξ ,x 0 i ξ ,x 0 j /(d ij -wN ) , (4.2)
where the x 0 index indicates the centre of a square bump of width w in environment :

ξ ,x 0 i = θ(wN -d ix 0 ) . (4.
3)

The N patterns of same are correlated through the constraint

x 0 ξ ,x 0 i = wN . (4.4)
It would be interesting to make the link with the Hopfield model with correlated attractors. For instance, some works have extended the Hopfield model to include correlations between patterns in the Hebb rule [START_REF] Griniasty | Conversion of temporal correlations between stimuli to spatial correlations between attractors[END_REF][START_REF] Cugliandolo | Correlated attractors from uncorrelated stimuli[END_REF]:

J ij = 1 N µ,ν ξ µ i X µν ξ ν j . (4.5)
A more rigorous parallel between both cases would be welcome.

A more general remark, we can note that the duplicity between discrete and continuous representations has been ubiquitous throughout this thesis, being binary vs rate units, discrete maps vs continuous attractors, discretization of time for decoding. . . even in simulations, where the measure of the quasi-particle's position required us to discretize the space and to correct for the discretization error thus introduced. The richness of the model's dynamics, with the competition between transitions and diffusion, is a consequence of the half-discrete half-continuous structure of the stored memories. These very disparate issues, and the discussions they led, had in common to deal with the discretecontinuous dilemma. It may seem an anecdotal point, but this notion is important in any task of modelling or data analysis.

Towards more realism

One of the characteristics of our model is its extreme simplification -see the comparison with other models in paragraph 2.6. It has allowed us to go quite far in the analytical study, but it could be relaxed in several ways (discussed in [1,2], see Appendix).

The direst simplification is the absence of external inputs. Even if we do have introduced inputs in our study of retrieval (see 2.4.3) and motion under an external force (see 2.5.2), further work would be needed to fully understand how inputs affect the properties of the recurrent dynamics. First, we need to justify the choice of the form of these inputs (intensity, connectivity, dynamics. . . ), if possible on experimental grounds. Modelling the mossy fibers and the perforant pathway would require assumptions on their content and on their relative strength [START_REF] Treves | Computational constraints suggest the need for two distinct input systems to the hippocampal ca3 network[END_REF], and a comprehensive picture of the separation between retrieval and learning. Then, from the point of view of the equilibrium properties of the network, we would have to determine how the phase diagram 2.2 is modified in the presence of external fields. Amit, Gutfreund and Sompolinsky [START_REF] Amit | Statistical mechanics of neural networks near saturation[END_REF] have addressed this question in the case of the Hopfield network. In this case, however, things are simpler because there is a natural form for the external field to apply to retrieve a pattern. In the case of continuous maps, we don't have such an obvious way to proceed and have to choose between several alternatives. Finally, how external inputs affect the dynamics is a very vast question, of which we have only begun to scratch the surface.

Our assumption for the coupling matrix (2.3) is also quite schematic. Possible extensions include the addition of disorder in the learning process (place fields randomly scattered instead of regularly arranged on a grid), asymmetric couplings, synaptic dynamics [START_REF] Fung | Dynamical synapses enhance neural information processing: gracefulness, accuracy, and mobility[END_REF], non-linearities in the Hebbian prescription [START_REF] Sompolinsky | Neural networks with nonlinear synapses and a static noise[END_REF][START_REF] Van Hemmen | Nonlinear neural networks: efficient storage and retrieval of information[END_REF]. . . One more fundamental modification would consist in storing only a finite number of environments (independent of N ), and replacing the quenched noise coming from all the other environments by a noise term -say, Gaussian -of width √ αr. This background of synaptic noise could model other, unknown, memories stored in the network (e.g. nonspatial components, in an episodic memory framework), or the innate randomness in the connectivity structure that preexists the learnt patterns. In this latter view, learning occurs on top of a disordered, spin-glass-like network, instead of an initial tabula rasa [START_REF] Changeux | Selective stabilisation of developing synapses as a mechanism for the specification of neuronal networks[END_REF][START_REF] Toulouse | Spin glass model of learning by selection[END_REF]. It would be interesting to draw a parallel between our model thus modified and previous works on the so-called learning by selection [START_REF] Toulouse | Spin glass model of learning by selection[END_REF]. In the same spirit, the "palimpsest" hypothesis (that is, learning new memories by erasing the oldest ones) [START_REF] Nadal | Networks of formal neurons and memory palimpsests[END_REF] could be applied to continuous attractors as ours.

Finally, as discussed in Chapter 2, the use of binary neurons is a severe simplification. Nevertheless, we believe that the equilibrium and dynamic properties we have been able to exhibit -thanks to this simplification that made the analytics tractable -are fundamental features of the interplay between excitatory couplings between neurons of neighbouring place fields, fast noise, slow noise and global inhibition. Hence, we therefore believe that those results are very robust and would also be true with more realistic units as, for instance, integrate-and-fire neurons. It would be very interesting to run numerical simulations with such realistic units in order to check this prediction.

Towards more generality

The model presented here is an extension of the Hopfield model to more structured memories, in this particular case continuous stored patterns. But it is only a starting point to further possible extensions. One could imagine introducing, for instance, a hierarchy between the patterns, as has been done in the case of discrete memories [START_REF] Parga | The ultrametric organization of memories in a neural network[END_REF].

Another modification would consist in dropping the chunking of space into discrete environments by merging the maps together into a single big continuous attractor. The motivation for this latter idea is that, in natural (non-experimental) conditions, space is not discretized into maps [START_REF] Park | Ensemble place codes in hippocampus: Ca1, ca3, and dentate gyrus place cells have multiple place fields in large environments[END_REF]. For instance, it could be implemented the following way: assume that the learnt space has size L (linear length or area depending on the dimension) and that N neurons are randomly allocated place fields in this space. More precisely, each neuron has a probability f to have a place field in each region of unit size (therefore, a given neuron will in general have several place fields in total). Then, for each pair of neighbouring place fields, add up an excitatory connection between the corresponding neurons. A uniform inhibition is also assumed. The problem would consist in finding the maximal size L up to which the stable states are localized in space. It is quite close to the initial case, even if the average over disorder is expected to be more tricky.

Extending the model to higher dimensions would be useful. First, the 3D case is interesting as regards spatial memory for flying or swimming animals (for example bats [24]). But more generally, in the effort to unify theories for spatial and episodic memories, additional dimensions could incorporate the possibility of non-spatial correlates of place cell firing. Another extension, in the same spirit, would be the introduction of distorted metrics in the coupling matrix: this would model the effect of non-spatial quantities as modulating the relative weight of different positions, instead of acting as additional dimensions. The predictions that would be made in each case could help to resolve whether the hippocampus forms episodic memories within a spatial framework ("cognitive map"), or if space is but one of the components of these memories [START_REF] Tolman | Cognitive maps in rats and men[END_REF][START_REF] O'keefe | The hippocampus as a cognitive map[END_REF][START_REF] Eichenbaum | Hippocampus: Mapping or memory?[END_REF].

Future work on the decoding issue

The conclusions on the decoding methods studied in Chapter 3 need to be backed up by further results. Indeed, we have seen that their relative performance depends strongly on the dataset (sampling, orthogonality between representations, level of global activity. . . ). It is therefore necessary to test these decoding paradigms on other data. For instance, CA1 recordings would allow us to test the effect of overlapping subpopulations, as spatial representations in CA1 appear to be less orthogonal than in CA3 [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF]. The low activity issue would be resolved by recording more cells simultaneously. We could also resort to other brain areas (e.g. the prefrontal cortex [123]), even if the whole approach would have to be rethought in this very different context. Moreover, the very close performances of the "rate -max. posterior" and "Ising -coupled" methods need to be better understood; as the kinship between both underlying models.

Another way to address the problem would consist in using synthetic data. This means simulating numerically a model for the system to decode (e.g. the model we have proposed in Ch. 2), recording its activity as a function of time, and then performing the decoding procedure on the recordings thus obtained. See for instance ref. [START_REF] Zhang | Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells[END_REF]. The advantage here is that we know the "good answer", i.e. what the decoder should find.

This approach thus allows one to test directly the effects of the dataset on the decoding performance, because one can tune the parameters of interest (relative strength of external input vs recurrent collaterals, sampling, orthogonality between representations, level of global activity. . . ). Of course, it relies on a model for the hippocampus and nothing tells us that what is encoded in the model is also what is encoded in real place cells. But at least, with this approach one can test the decoding methods in a well defined framework and make predictions as for real neural activity's encoding and decoding.

In all our Chapter 3 on decoding, we have limited ourselves to the assumption that place cell activity encoded either a position of space, either a whole environment. Here there are a lot of open questions, and a lot of things to do. As we have mentioned before, we do not know what is ultimately encoded in hippocampal activity, and there are experimental results indicating that space is not the only feature (see paragraph 1.4.3).

The possibility of additional dimensions could be investigated through the inferred coupling matrix. Indeed, one should note that in our inference procedure, what we call an environment could englobe non spatial components in a broader notion of "context" (all we do is to lump together firing statistics coming from the same session, but one session could represent more than just the place in which it has been recorded). We could for instance use a multi-dimensional scaling algorithm to sort out the couplings between all pairs and estimate the dimensionality of the space they live in. The dataset we have used would probably not suffice for this task -the error bars on the inferred couplings are too large because the recording sessions are quite short. Therefore, longer recordings with more cells would be required. An alternative effect of non-spatial correlates of place cell activity would be to distort the 2D metric. Here again, the inferred couplings could be used to test this hypothesis, with methods that are still to be set up.

A related feature is the so-called "overdispersion" property of place cells, first shown by [START_REF] Fenton | Place cell discharge is extremely variable during individual passes of the rat through the firing field[END_REF] [31]. It denotes the fact that a place cell's firing at a given position is very variable from trial to trial -more variable than if its activity followed a Poisson law with mean r M i ( x) (the average firing rate at this position). Jackson & Redish (2007) [START_REF] Jackson | Network dynamics of hippocampal cell-assemblies resemble multiple spatial maps within single tasks[END_REF] have interpreted this phenomenon as the signature of rapid (≈ 3 Hz) transitions between submaps, reminiscent of Jezek et al 's flickering [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF], but taking place all the time. Addressing these issues with the analysis tools we have in hand would help to better understand the relationship between firing variability, dimension of the correlates space and spontaneous transitions between maps. In our dataset, indeed, we do observe overdispersion (not shown here). Further work would consist in determining whether this variability can be accounted for by rapid jumps between submaps or instead by additional dimensions in the firing correlates.

As it appeared throughout this study, a lingering problem in the decoding issue is that we do not know ultimately what the activity encodes. A smart way to circumvent this difficulty in experiments is to couple neural activity recordings together with behavioural outcomes. The rat cannot tell us what it thinks, but we can design tasks where its behaviours reflects its internal state. Behavioural correlates are stronger than external ones because they are the result of cognition. A method of this kind has been employed by Wimmer et al [START_REF] Wimmer | Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory[END_REF] in the case of spatial working memory in prefrontal cortex. In our case, for instance, imagine we decode the rat's position from its hippocampal activity while it performs a spatial task. If a shift between the decoded and the actual positions is systematically accompanied by an error of the same distance in the rat's behaviour, then the hypothesis that place cells encode position is supported. Conversely, if such a correlation is not observed, then the hypothesis is ruled out. Behavioural tests are thus strong ways to test assumptions on neural computation. Novel experimental paradigms could be of great help in addressing the decoding issue.

Concluding remarks

Being when it comes to modelling or to decoding, we have seen that a crucial point is the question of relevant quantities. This issue is of course central in any scientific work, but it is particularly difficult to tackle in neuroscience. At the beginning of this thesis, we have made the connectionist assumption which, if it does not tell us what are the relevant quantities, at least allows us to consider that they are macroscopic. Nevertheless, one should keep in mind that it is but an assumption.

We have said that, at its origin in the context of thermodynamics, the recourse to statistical mechanics was driven by a double motivation: the fact that macroscopic quantities were the relevant quantities, and the fact that in any case there was no other choice because microscopic quantities were too complex to tackle. When addressing problems of neuroscience, connectionism has the same two motivations. But what if the first one failed? What if we could not do without the tremendous quantity of details at the level of individual neurons and synapses?

We would then have to embrace the dreadful complexity of neural systems, with little hope to be ever able to do anything analytical any more. Numerical simulations would be the only way to tackle such complex systems. With the fast improvement of computing powers, there is no doubt that it will be soon possible to simulate very realistic brain systems [START_REF] Gerstner | Theory and simulation in neuroscience[END_REF]. Mark Buchanan said numerical simulations were "a kind of telescope for the mind": they would allow us to go further, where the analytics cannot go, like the invention of the telescope allowed scientists to see what the eye could not see.

This brings us back to the question of what understanding means. Suppose we were able to fully simulate a whole brain with all its details at the molecular scale, can we said that we have understood it? I would tend to say no. The ability to reproduce a biological entity in a big labyrinthine system is not comparable to the insight one gets thanks to schematic pictures and analytical comprehension. Besides, numerical simulations are not considered as equivalent to a theory; rather they are kind of in between the experiment and the theory (we speak about "in-silico experiment"). So the possibility exists that we are not able to fully understand the brain because of its complexity. In the meantime, a great deal of understanding has been reached so far thanks to connectionism and its understandable models.

I. INTRODUCTION

Understanding the representation of space by the brain is a long-lasting question, which has been addressed using many varied methods. This includes memory of places, localization of one's position, mental exploration, and planning of forecoming trajectories. During the last decades, the use of microelectrodes allowing single cell recordings has revolutionized our knowledge of neural networks. In 1971, O'Keefe and Dostrovsky [1] recorded neural activity in the hippocampus of rats and discovered the existence of place cells, which fire only when the animal is located in a certain position in space (called place field). This discovery suggested that the hippocampus could be the support for space representation or a "cognitive map." Since then, many experimental and theoretical studies have been carried out on the hippocampus, making it one of the most, if not the most, studied parts of the brain [2].

The properties of place cells, their conditions of formation, and the sensory and behavioral correlates of place fields have been investigated experimentally [3][4][5]. Place fields have the striking property to appear as randomly distributed, independently of the neurons' locations in the neural tissue: two neighboring neurons can have very distant place fields. Furthermore, several "environments" or "maps" can be learned, and a given neuron can have place fields in several environments, which are apparently randomly assigned, a property called remapping [6]. Place fields are controlled primarily by visual cues but the activity of place cells persists in the dark [7] and is also driven by self-motion signals, that is, "path integration" [8]. More recently, the discovery of grid cells [9,10] in the entorhinal cortex (that feeds input into the hippocampus) opened a new way in the comprehension of a complex system of interacting brain regions [11]. Many theoretical models have been proposed to account for these experimental results. Beyond the comprehension of the hippocampus itself, the motivation is to reach more insights about the functional principles of the brain [2].

Experiments show that the hippocampus is able to learn, memorize, and retrieve spatial maps. The massive intrinsic connectivity in hippocampal CA3 led to the hypothesis of an attractor neural network [12][13][14] where memorized activity patterns are the attractors of the dynamics, such as in the celebrated Hopfield model [15]. In the Hopfield model it is assumed that the patterns are additively stored in the synapses, through a Hebbian learning mechanism. A deep and quantitative understanding of the Hopfield model was made possible by the use of the statistical physics theory of mean-field spin glasses [16,17]. In the case of the rodent hippocampus, the memorized items are space manifolds called environments [6]. Neural network models for place cells have been proposed, in particular by Battaglia and Treves, who carried out a mean-field calculation of the storage performance of a network with linear thershold units [18]. Recently Hopfield proposed a similar model for mental exploration in a network with adaptation [19]. However, the crosstalk between the different environments encoded in the network, and the transitions that can occur between them as observed experimentally [20] remain poorly understood.

Here, we propose a model of interacting binary units and study the different regimes of activity in the presence of neural noise. The model is defined in Sec. II. We study the case where multiple environments are memorized in Sec. III, and derive the different regimes of activity of the network under given conditions of neural noise and memory load in Sec. IV. The phase diagram of the system is computed in Sec. V and compared to numerical simulations. We show that an activity of the network that is locally spatialized in one of the stored maps, as observed experimentally, is the stable state of the network provided that both the neural noise and the memory load are small enough. For high noise and/or loads the the activity is delocalized in all environments, either uniformly over space or with spatial heterogeneities controlled by the crosstalk between environments (glassy phase). We finally discuss the value of the parameters (Sec. V C) and the hypothesis of the model (Sec. VI) compared to previous works. The study of the free energy landscape and of the dynamics of the model will be addressed in a companion publication [21].

II. MODEL

A. Definition

The N place cells are modeled by interacting binary units σ i equal to 0 or 1 corresponding to, respectively, silent and active states. We suppose that, after learning of the environment and random allocation of place fields, each place cell preferentially fires when the animal is located in an environment-specific location in the D-dimensional space, defining its place field. For simplicity, space is assumed to be a segment of length N for D = 1, and a square of edge length √ N in D = 2, with periodic boundary conditions. The N centers of the place fields are assumed to be perfectly located on a D-dimensional regular grid: two contiguous centers are at unit distance from each other. This simplification allows us to concentrate on the interference between the stored spatial maps as the only source of structural noise.

Let d c be the extension of a place field, that is, the maximal distance between locations in space recognized by the same place cell. Place cells whose place fields overlap, and, therefore, spike simultaneously as the animal wanders in the environment, are assumed to strengthen their synaptic connections. Calling d ij the distance between the place field centers of cells i,j in the environment we assume that the reinforcement process results in the production of excitatory synaptic couplings given by

J 0 ij = 1 N if d ij d c , 0 if d ij > d c . ( 1
)
The fact that all environments are equivalent (there is no privileged permutation) is basic to our theory. We choose the place extension d c such that each cell i is connected to the same number of other cells j , independently of the space dimension D. Let w N be this number: w( 1) is the fraction of the neural population any neuron is coupled to. Hence, andd

d c = w 2 N in dimension D = 1,
c = √ w N π in dimension D = 2. The 1 N
scale factor is such that the total contribution to the local field received by a place cell is finite when the number of cells N is sent to infinity. Note that we assume here that the environment is perfectly explored: couplings depend on the distance d ij only, and not on the particular sequence of positions occupied by the animal during the time spent in the environment. The case of partial, nonhomogeneous explorations was studied in [14]. Couplings defined by prescription (1) are symmetric, and only reflect the local structure of the environment.

Each time the rodent explores a new environment a remapping of the place fields takes place. Let L be the number of explored environments, in addition to the environment above (hereafter called reference environment). We assume that the remapping is represented by a random permutation of the N place-cell indices associated to the place fields in the reference environment, denoted by = 0. Let π be the permutation corresponding to remapping number , where = 1, . . . ,L is the index of the environment. In environment cells i,j interact if the distance d π (i)π (j ) is smaller than d c , and do not interact at larger distances. An obvious modification of (1) defines the coupling matrix J corresponding to environment . We finally assume that all environments contribute equally and additively to the total synaptic matrix,

J ij = L =0 J ij = J 0 ij + L =1 J 0 π (i)π (j ) . ( 2
)
For the sake of a better understanding, we consider an example of a matrix J in the very simple case N = 6, w = 2 6 , L + 1 = 2, and D = 1, illustrated in Fig. 1. For the reference environment the coupling matrix is

J 0 = 1 6 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 3
)
For another environment obtained through the random permutation π = (3,6,1,5,2,4) we obtain the coupling matrix

J 1 = 1 6 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 4
)
The total coupling matrix is therefore

J = J 0 + J 1 = 1 6 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 0 1 2 1 0 2 1 0 0 0 2 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 2 0 0 1 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 5
)
In addition to pyramidal cells, the network contains longrange, inhibitory interneurons whose activity is modeled by a global inhibition on place cells. We assume that the main effect of inhibition is to fix the total neural activity. We introduce the parameter f to denote the fraction of active cells:

N i=1 σ i = f N. ( 6
)
Once the coupling matrix J ij (2) and the constraint over the global activity ( 6) are defined the probability of a neural activity configuration σ = (σ 1 ,σ 2 , . . . ,σ N ) is assumed to be

P J (σ ) = 1 Z J (T ) exp(-E J [σ ]/T ), (7)
where the "energy" of the configuration reads

E J [σ ] = - i<j J ij σ i σ j , (8)
and the partition function is

Z J (T ) = σ with constraint (6) exp(-E J [σ ]/T ). (9)
Parameter T , which plays the role of temperature in statistical mechanics, fixes the amount of noise in the model. Large values of T corresponds to essentially flat distributions over the neural configuration space. Low T concentrate the probability distribution P J around the configurations with lowest energies E J . In all numerical computations hereafter we will take the parameters values w = 0.05 and f = 0.1, except in Sec. V C where the effect of those values on the results will be discussed.

B. Case of a single environment

Our model is an extension of the Hopfield model [15] to the case of space-dependent interactions. Despite this additional complexity in the model it remains exactly solvable in the infinite N limit, due to the long-range nature of the interactions [22].

We start by considering the case of a single environment, for which the coupling matrix is given by (1). To lighten notations we consider the D = 1 case. In the large N limit, a continuous approach can be introduced by defining the locally coarsegrained activity

ρ(x) ≡ lim →0 lim N→∞ 1 N (x-/2)N i<(x+ /2)N σ i J , ( 10
)
where . J denotes the average over distribution P J (7). Note that the order of limits is important for the local average to be correctly defined. Due to the presence of periodic boundary conditions we choose x ∈ [-1 2 ; 1 2 ]. The density of activity ρ(x) is found upon minimization of the free energy functional

F({ρ(x)}) = - 1 2 dx dy ρ(x)J w (x -y)ρ(y) + T dx{ρ(x)lnρ(x) + [1 -ρ(x)]ln[1 -ρ(x)]}, ( 11
)
where J w (u) = 1 if |u| < w 2 , and 0 otherwise. The minimum is taken over the activity densities fulfilling

dx ρ(x) = f. ( 12
)
All integrals run over the [-1 2 ; 1 2 ] interval. The minimization equation for ρ(x) can be written as

ρ(x) = 1 1 + e -μ(x)/T , ( 13
)
μ(x) = dy J w (x -y)ρ(y) + λ, ( 14
)
where μ(x) plays the role of a chemical potential, and the constant λ is chosen to satisfy (12). We discuss the solutions of these equations in the following sections. Note that the free energy per site,

F (T ) = lim N→∞ - T N lnZ J (T ), (15)
is simply given by the value of the free-energy functional F in its minimum ρ(x), solution of ( 13) and (14).

C. Relationship with rate models

Neurons are often described by their firing rate, i.e., the short-term average of the number of spikes they emit. A straightforward relationship can be drawn with binary models [23]. The current incoming onto neuron i evolves according to

τ dI i dt = -I i + j J ij g(I j ). (16)
Here, g(x) is the characteristic function expressing the firing rate of the neuron as a function of the current. It is a sigmoidal function, running between 0 and 1 (saturation of the postsynaptic neuron at high currents), and J ij includes both the positive coupling J 0 (1) between neighboring cells, and a constant, global inhibition contribution J I , whose value is chosen to enforce (6). The dynamical equation admits a stationary state, implicitly defined through

I i = j J ij g(I j ). (17)
Identifying

I i → μ i , g(I i ) → ρ i , (18)
and choosing

g(I ) = 1 1 + exp(-I/T ) , ( 19
)
we observe that Eq. ( 17) for the stationary currents is identical to Eq. ( 14) for the chemical potential in the single-environment case. The constant term λ in ( 14) is related to the constant inhibitory contribution to J through λ = J I f . The parameter T fixes the slope of g at the origin.

III. STATISTICAL MECHANICS OF THE MULTIPLE ENVIRONMENT CASE

A. Average over random remappings

In the presence of multiple environments the partition function Z J becomes a stochastic variable, which depends on the L remappings, or, equivalently, on the L random permutations π , with = 1 . . . L. We assume that, in the large N limit, the free energy of the system is self-averaging, i.e., concentrated around the average. To compute the average free energy we need to average the logarithm of Z J (T ) over the random permutations. To do so we use the replica method: we first compute the nth moment of Z J (T ), and then send n → 0. The neural configuration is now a set σ = (σ 1 , . . . ,σ n ) of n × N spins σ a i , where i = 1 . . . N is the spin index and a = 1 . . . n is the replica index. The nth moment of the partition 062813-3 function reads

Z J (T ) n = σ exp ⎡ ⎣ β n a=1 i<j J 0 ij + L =1 J ij σ a i σ a j ⎤ ⎦ = σ exp ⎡ ⎣ β n a=1 i<j J 0 ij σ a i σ a j ⎤ ⎦ ( σ ) L , ( 20
)
where β = 1/T and the overbar denotes the average over the random remappings. The sum over σ is restricted to configurations with average activity equal to f (within each replica), and

( σ ) = 1 N! π exp ⎡ ⎣ β i<j J 0 ij n a=1 σ a π (i) σ a π (j ) ⎤ ⎦ . ( 21
)
Here the equivalence between permutations is explicitly exploited. The calculation of the average over the random permutation π is not immediate, but can be done exactly in the large N limit. Details are reported in Appendix B. The result is

ln σ = N β 2 nwf 2 - λ =0 Tr ln[Id n -βλ(q -f 2 1 n )], (22)
where Id n denotes the n-dimensional identity matrix, q is the overlap matrix with entries

q ab ≡ 1 N j σ a j σ b j , ( 23
)
and 1 n is the n × n matrix whose all entries are equal to 1. The sum in (22) runs over all the nonzero eigenvalues of the matrix J 0 . Explicit expressions for those eigenvalues will be given in the next section for the D = 1 case, while the two-dimensional case is treated in Appendix A.

A key feature of ( 22) is that depends on the spin configuration σ through the overlaps q ab only. Those overlaps thus play the role of order parameters for the activity in the environment 1, as does ρ(x) for the environment 0. Calculation of the nth moment of the partition function therefore amounts to estimating the entropy of neural activity configuration σ at fixed {q ab ,ρ(x)}, which can be done exactly in the N → ∞ limit.

B. Replica-symmetric theory

To perform the n → 0 limit we make use of the replica symmetric ansatz, which assumes that the overlaps q ab take a single value q for replica indices a = b. The validity of the ansatz will be discussed in Sec. IV. The Edwards-Anderson order parameter q, defined through

q ≡ 1 N N i=1 σ i 2 J , ( 24
)
measures the fluctuations of the local spin magnetizations from site to site. Values for q range from f 2 to f . We expect q to be equal to f 2 when the local activity σ i J (averaged over the configurations with distribution P J ) is uniform over space, and to be larger otherwise.

As in the single environment case we define the order parameter ρ(x) as the density of activity around point x in space, see (10),

ρ(x) ≡ lim →0 lim N→∞ 1 N (x-/2)N i<(x+ /2)N σ i J . ( 25
)
The difference is that, in the multiple environment case, the density ρ(x) appearing in the replica theory is averaged over the environments. Local fluctuations of the density from environment to environment can be calculated [21], but will not be considered here; only global fluctuations, averaged over space, are considered through the order parameter q.

As in the single environment case a chemical potential μ(x), conjugated to ρ(x), is introduced. In addition, a new order parameter r plays the role of the conjugated force to q, and controls the fluctuations of the spin magnetizations. All order parameters are determined through the optimization of the free energy functional F(q,r,{ρ(x)},{μ(x)}), see Appendix C, whose expression for the D = 1 case is given by

F = αβ 2 r(f -q) - α β ψ(q,β) + dx μ(x) ρ(x) - 1 2 dx dy ρ(x) J w (x -y) ρ(y) - 1 β dx Dzln(1 + e βz √ αr+βμ(x) ), (26)
where Dz = exp(-z 2 /2)/ √ 2π is the Gaussian measure, and

ψ(q,β) ≡ k 1 β(q -f 2 ) sin(kπw) kπ -β(f -q) sin(kπw) -ln 1 - β(f -q) sin(kπw) kπ . ( 27
)
Parameter α ≡ L/N, hereafter called load, denotes the ratio of the numbers of environments and of cells. Extremization of the free energy functional leads to the saddle-point equations

r = 2(q -f 2 ) k 1 kπ sin(kπw) -β(f -q) -2 , q = dx Dz [1 + e -βz √ αr-βμ(x) ] -2 , ( 28
) ρ(x) = Dz [1 + e -βz √ αr-βμ(x) ] -1 , μ(x) = dy J w (x -y) ρ(y) + λ,
where λ is determined to enforce (12). The expression of F and of the saddle-point equations for the D = 2 case can be found in Appendix A.

IV. THE PHASES AND THEIR STABILITY

In both D = 1 and 2 dimensions three qualitatively different solutions are found for the extremization equations of F, corresponding to three distinct phases of activity: a paramagnetic phase in which the activity is uniform over space, a "clumplike" phase in which the activity is localized in one of the stored spatial maps, and a glassy phase where the activity is neither uniform nor coherent with any map. We now discuss the domains of existence and stability of each phase. We are chiefly interested in the clump phase domain, which corresponds to the experimentally observed regime where memorized maps can be retrieved. As usual all expressions given below correspond to the D = 1 case, while the case D = 2 is treated in Appendix A; all numerical results were obtained for f = 0.1, w = 0.05.

A. High noise: Paramagnetic phase

At high temperature we expect the activity to be dominated by the noise in the neural dynamics, and to show no spatial localization. The corresponding order parameters are

ρ(x) = f, q = f 2 (paramagnetic phase: PM).
The activity profile is shown in Fig. 2(a). The paramagnetic phase (PM) exists for all values of the control parameters, with corresponding potentials:

μ(x) = T ln f 1 -f , r = 0 (PM).
We now discuss its stability.

Case of a single environment (α = 0)

In the single environment case the stability of the paramagnetic solution is determined by computing the Hessian of the free-energy functional F (11). We find that

δ 2 F δρ(x)δρ(y) = T f (1 -f ) δ(x -y) -J w (x -y). ( 29
)
The solution is stable as long as the Hessian is definite positive.

In the one-dimensional case the most unstable mode corresponds to a spin wave δρ(x) ∝ sin(2π k x), with wave number k = 1; note that the k = 0 mode is forbidden according to condition (12). The instability develops under the spinodal temperature

T PM = f (1 -f ) sin πw π ≈ 0.0045. ( 30
)
T PM and, more generally, all thermodynamic quantities are invariant under the changes f → 1f or/and w → 1w, which simply amount to reverse σ i → 1σ i , i.e., to change active spins into holes and vice versa.

In dimension D = 2 a similar calculation shows that the first unstable mode is a one-dimensional spin wave along one arbitrary direction in the plane. The corresponding spinodal temperature is

T 2D PM = f (1 -f ) √ w sin(π √ w) π . ( 31
)

Case of multiple environments (α > 0)

The study of the stability of the PM phase in the multiple environments case is reported in Appendix E 1. As in the single environment case the PM solution is unstable at all temperatures T < T PM against perturbation of the activity of the type δρ(x) ∝ sin(2π k x). In addition coupled fluctuations of λ,r,q may lead to instabilities if T is smaller than T PM (α), implicitly defined through

k 1 T PM (α) kπ f (1 -f ) sin(kπw) -1 -2 = 1 2α . ( 32
)
The instabilities correspond to the transition to the glassy phase; see Sec. IV C. Note that T PM defined in (30) corresponds with T PM (α = 0). As a conclusion, in the (α,T ) plane, the PM phase is stable in the region T > T PM (α). This region is sketched in Fig. 3.

B. Moderate noise and load: The clump phase

In experiments place cells exhibit patterns of localized activity where neurons with neighboring place fields are active together. Our modeling reproduces such localized-in-space activity patterns (called "bumps" or "clumps" of activity) at sufficiently low (α, T ). The corresponding phase, hereafter referred to as the "clump phase" (CL), is characterized by the order parameters ρ(x) = f, q > f 2 (clump phase: CL).

Correspondingly, the chemical potential μ(x) will vary over space, and the conjugated force r is strictly positive. FIG. 3. The paramagnetic (PM) phase is stable in the upper part of the (α,T ) plane, defined by T > T PM (α). The spin glass (SG) phase exists below this line; replica symmetry is broken everywhere in the T < T PM (α) region.

Single environment (α = 0)

When the temperature T is sent to 0 + , assuming that f > w, we find a solution to ( 13), ( 14) that is localized in space:

μ(x) = ⎧ ⎨ ⎩ w 2 if |x| < 1 2 (f -w) f 2 -|x| if 1 2 (f -w) |x| < 1 2 (f + w) -w 2 if |x| 1 2 (f + w) , ( 33
)
and

ρ(x) → 1 if |x| f/2, 0 if |x| > f/2. ( 34
)
Any translation x → x + x 0 defines another ground state with the same free energy. The activity profile is shown in Fig. 2

(b).

At small but finite temperature we have solved Eqs. ( 13) and ( 14) numerically by discretizing space with a large number M of bins of width 1/M, such that Mw and Mf are both much larger than unity. The activity profile is now rounded off by the thermal noise; see Fig. 2(c) for a representative example. Cells far away from the center of the clump are active with some positive probability < f . This clump is reminiscent of a liquid phase, surrounded by its vapor. The clump persists up to some critical temperature T CL , e.g., T CL 0.008 for f = 0.1,w = 0.05, at which it disappears. The dependence of T CL on f and w will be studied in Sec. VI. Notice that T CL also slightly depends on the number of bins of discretization M as shown in Fig. 4.

The clump phase is also present for D = 2. An example of a two-dimensional clump is shown in Fig. 5.

Crosstalk between different environments (α > 0)

We now look for a solution with localized activity in the first environment, and nonlocalized activity in the other environments. Keeping the temperature T fixed and increasing the load α has the effect of squeezing and lowering the clump (Fig. 6). Once the disorder (random remappings) is averaged out, the clump solution is translationally invariant as in the single environment case. Here we assume that no external input (which would be important for retrieval properties of the network, and would break translation invariance) is present. We have studied the stability of the clump solution against longitudinal and replicon modes. The longitudinal stability domain is found by determining the boundary in the (α,T ) plane along which the clump abruptly collapses. This boundary, shown in Fig. 7, can be described as follows:

(i) at small α the clump phase is longitudinally stable for T < T CL (α), a slowly decreasing function of α, which coincides with the temperature T CL found for a single environment when α → 0;

(ii) at small temperature, the clump phase is longitudinally stable if α < α CL (T ), an increasing function of T . We denote α CL its value when T → 0;

(iii) at intermediate temperatures a weak reentrance is present. The curves T CL (α) and α CL (T ) merge at a point where the tangent is vertical and the reentrance begins.

Along the boundary of the clump phase the value of the Edwards-Anderson parameter increases from q = f 2 in (α = 0,T = T CL ) to q = f in (α = α CL ,T = 0).

Calculation of the stability against replicon modes is detailed in Appendix E 3. We find that the replica-symmetric solution is stable, except in a small region confined to small T and α close to α CL . This result is shown by the dashed line in Fig. 7. It is reminiscent of the results for the "retrieval phase" in the Hopfield model [16].

C. High load: The glassy phase

At large α the disorder in the interactions is strong enough to magnetize the spins locally, without any coherence with any spatial map. Again, the average of the activity σ i J will depend on the realization of the environments, while the average over the environment, σ i J , will be uniform in space and equal to f . In this glassy (SG) phase the order parameters will take values ρ(x) = f, q > f 2 (glassy phase: SG).

Correspondingly the chemical potential μ(x) does not depend on x, and r > 0.

As reported in Appendix E 2 a glassy solution is found when T < T PM (α), corresponding to the paramagnetic stability line calculated above. Within this region, the SG phase is always stable against clumpiness (localization of the activity). The spin glass phase is unstable against the replicon mode, indicating that replica symmetry is broken, similarly to the spin glass phase in the Hopfield model [16]. Results are summarized in Fig. 3.

V. PHASE DIAGRAM

A. Transitions between phases

Transition lines between the phases described above are determined in the (α,T ) plane from the comparison of their free energies:

(i) The clump-paramagnetic transition at high temperature is located slightly below the clump instability line. We denote T c (α) the corresponding temperature for a given α.

(ii) The clump-glass transition occurs at a load denoted α g (T ) for a given temperature T . Here again, we find a slight reentrance at moderate temperature: α g (T ) is maximal for T ≈ 0.004. As replica symmetry is broken in the SG phase the true free energy is expected to be higher than the RS value, and the true transition line should be slightly shifted to higher values of α.

(iii) At high α, T there is a second-order phase transition between the PM and the SG phases. The phase diagram in dimension D = 1 is summarized in Fig. 8.

It is interesting to emphasize the differences between this phase diagram and the one of the Hopfield model computed in [16]. In the Hopfield model, the "retrieval" or "ferromagnetic" (FM) phase (which corresponds to our clump phase) has a triangular shape in the (α,T ) plane. The temperature at which the FM phase becomes unstable at a given α is smaller than T PM (α). There is no coexistence between the PM and FM phases, and both are separated by the glassy phase. Moreover, for the Hopfield model, T FM (α) is monotonously decreasing so the capacity is maximal at zero temperature [24]. Consequently, it seems that our model of attractor neural network is much more robust to noise than the standard Hopfield model. This can be understood considering the structure of the coupling matrix. In the Hopfield model one pattern defines a single direction in the configuration space; interference with other patterns and dynamical noise may push the activity configuration in the high-dimensional orthogonal subspace, and the memory of the pattern is easily lost. In the present case, on the contrary, one map defines a whole collection of configurations (bumps) centered on different locations, thus the synaptic matrix will make the network converge to one of the attractors, even in the presence of a high level of noise.

When a first-order transition line is crossed the order parameter q is discontinuous. We have computed numerically the value of the Edwards-Anderson parameter at different points and plotted its evolution at the clump-paramagnetic transition at fixed α (Fig. 9) and at the clump-glass transition at fixed T (Fig. 10).

B. Numerical simulations

We have performed Monte Carlo simulations to check our theoretical predictions. The system is initialized with two types of conditions (respectively, uniform and clump configurations). At each time step, two neuron indices i,j are chosen such that σ i = 1σ j . We then calculate the change in the energy when the two spins are flipped, and accept the flip or not according to Metropolis' rule. As a consequence the activity is kept constant (and equal to f N over the neural 062813-7 FIG. 9. q as a function of T for fixed α: α = 0 (solid line), α = 0.01 (dashed line), and α = 0.015 (dots), computed with M = 1000. A discontinuity is observed at the clump-paramagnetic transition.

population), and the system is guaranteed to reach equilibrium for sufficiently long simulation times.

Single environment case

Figure 11 shows the average energy E(T ) vs the temperature T , for various sizes N . At high temperature, E(T ) = -1 2 f 2 w as expected in the paramagnetic phase. At low temperature, the shape of the activity clump varies with T , and so does E(T ). We find a clear signature of the first order transition as N grows. The critical temperature is in excellent agreement with the theoretical value for T c .

We plot in Fig. 12 the spin-spin correlation σ i σ j as a function of the normalized distance, d = |i-j | N :

C(d) = σ i σ i+d N . ( 35
)
At low temperature, finite size effects are negligible and C(d) is a nontrivial decreasing function of d in the large N limit. At small d, C(d) is of the order of f , and then decreases to a much smaller value over a distance of the order of f . As the location of the clump is arbitrary, we expect its center x 0 to be uniformly distributed over the [-1 2 ; 1 2 ] interval. The FIG. 10. q as a function of α for fixed temperature: T = 0.002 (solid line) and T = 0.004 (dashed line), computed with M = 1000. A discontinuity is observed at the clump-glass transition. correlation is therefore given, in the thermodynamic limit, by

C(d) = dx 0 ρ(x 0 ) ρ(x 0 + d). ( 36
)
At zero temperature, this formula gives

C(d) = f -d for d < f , C(d) = 0 for d f .
At finite temperature, we compute ρ from the extremization equation ( 13), and plug the value into the right-hand side of (36). The agreement with the correlation C(d) obtained from MC simulations is perfect (Fig. 12). At high temperature and for finite N, C(d) decreases over a distance d w 2 to the paramagnetic value f 2 . When N → ∞, C(d) is uniformly equal to f 2 at all distances d > 0. As an additional check of the value of T c we find that the spin-spin correlation decays quickly with increasing N for T = 0.0074, and saturates to a d-dependent value larger than f 2 for T = 0.0072 (not shown).

Multiple environments

We now report the outcome of Monte Carlo simulations with L + 1 environments (L > 0), obtained through random permutations of the sites. We have verified numerically the theoretical predictions for μ(x) (Fig. 13) and r (Fig. 14 noise on the field comes from the contribution of environments 1: z √ αr is a Gaussian random variable of mean 0 and standard deviation √ αr independent of x. In our simulations we have measured the contributions h i ≡ 1 N L =1 j J ij σ j of the environments 1 to the local fields at different locations. The distribution of the h i 's perfectly agrees with the theoretically predicted Gaussian (inset in Fig. 14).

We have also investigated the behavior of the system for varying levels of noise and load, and compared it to the phase diagram found theoretically. In simulations we have considered the environment of lowest energy (in which the activity acquires a clumplike shape) and measured its contribution to the energy density, E [{σ i }] = -1 N i<j J ij σ i σ j . This quantity is compared with the theoretical value - The agreement with theoretical predictions is very good in the case of the clump-paramagnetic transition (Fig. 15). Concerning the clump-glass transition (Fig. 16), as we mentioned above we expect the transition to occur at larger load, α g (T ) < α observed g < α CL (T ), due to the replica-symmetry broken nature of the glass phase. This expectation is corroborated by Fig. 17, which represents the fraction of simulations ending in the glassy phase as a function of α for T = 0.004. We have checked that this fraction does not depend on the initial conditions of the simulation. The transition occurs around α 0.018 ± 0.001 (uncertainty due to long thermalization times in the simulations), while α g 0.0173 for T = 0.004 used in the simulation.

C. Dependence on parameter values

All the numerical computations above were performed with parameter values w = 0.05 and f = 0.1. To gain insight on the influence of the parameter values on the behavior of the clump phase, we focus on two quantities representing its stability domain, namely α CL and T CL , respectively the load at which the clump phase becomes unstable at T = 0 and the temperature at which the clump phase is unstable when α = 0. We also study the influence of w and f on first-order transitions, through α g and T c , respectively the load of transition to the glassy phase at T = 0 and the temperature of transition to the PM phase at α = 0.

Reduced-distance parameter w

Parameter w controls the maximal distance d c between the place field centers of interacting place cells; see (1) and Sec. II A. It fixes the width of the clump in the phase of localized activity. Experiments on rats have shown that the size of place fields depends on the size and complexity of the environment and on the behavioral context. A value w = 0.05, i.e., place fields occupying a few percent of the total space, is reasonable [25]. We have varied w for different values of f , and have found that T CL is a monotonously increasing function of w (Fig. 18). This result agrees with the intuition that increasing w makes the clump phase more favorable energetically. It also appears that α CL (w) has a maximum around w ∼ f . In terms of storage capacity, this result suggests that there exists an optimal choice for the parameters: for a given level of inhibition hence a given number f N of active neurons, choosing w ∼ f maximizes the proportion of these active neurons that are located in the place field. Given that the quenched noise coming from other environments is statistically uniform over space (Fig. 14), w ∼ f represents a tradeoff between limiting the crosstalk and using the active neurons in the area covered by the place field.

As far as thermodynamic transitions to the PM and glassy phases are concerned we find that T c and α g behave similarly to, respectively, T CL and α CL when w varies, as shown in Fig. 19. Consequently, the qualitative aspect of the phase diagram remains the same when w varies.

Total activity f

Parameter f is the activity level of the network fixed by global inhibition. As expected, T CL is a monotonously increasing function of f (Fig. 20). We find again a maximum of α CL when f is of the order of w, consistently with the previous results. We also find that the boundary of the transition lines in phase diagram, α g and T c , behave similarly to α CL and T CL (Fig. 21).

VI. EXTENSIONS AND DISCUSSION

A. Taking silent cells into account

Thompson and Best [3] report that not all pyramidal cells have place fields in a given environment, a significative fraction of them (63% in their recording in CA1) being silent, i.e., with no place field, in this particular environment. To take this effect into account, our model can be further refined to incorporate partial activity of the cell ensemble. We assume that a fraction c < 1 of cells are active in any environment:

(i) In the reference environment (environment 0), cN given spins σ i among the N are assigned regularly spaced place field centers p(i).

(ii) For each one of the other environments, say, 1, each spin σ i (among all N spins) is selected with probability c, and the place field centers are reshuffled by a random permutation π . The set of chosen spins is encoded in the dilution variables The resulting expression for the coupling matrix is

J ij = J 0 p(i)p(j ) + L =1 J 0 π (i)π (j ) τ i τ j . ( 38
)
We incorporate this new hypothesis in the calculation of the average over disorder of the replicated partition function. The average is now over two types of disorder: the permutations π and the selection of involved cells τ i . Neural configurations σ still satisfy (6). Moreover, we expect that the global inhibition is homogeneously distributed over the different subpopulations of neurons, and, for each realization of the τ i , we restrict the sum to configurations such that

1 cN N i=1 τ i σ i = f. ( 39
)
A detailed calculation, reported in Appendix D, shows that ( 6) implies (39) up to corrections of the order of 1 √ N . In addition we give in Appendix D the expression for the free energy in dimension D = 1. The corresponding extremization equations are

r = 2c 2 (q -f 2 ) k 1 kπ sin(kπw) -βc(f -q) -2 , q = c dx Dz[1 + e -βz √ αr-βμ(x) ] -2 + (1 -c) Dz[1 + e -βz √ αr-βμ 2 ] -2 , ρ(x) = Dz[1 + e -βz √ αr-βμ(x) ] -1 , μ(x) = c dy J w (x -y)ρ(y) + λ, f = dxρ(x), f = Dz[1 + e -βz √ αr-βμ 2 ] -1 . ( 40
)
In the partial activity model, the active spins (with activity ρ(x)) obey equations that are very similar to the previous case, with a dilution factor coming from the silent spins which are in a paramagnetic phase. From a qualitative point of view the behavior of the system does not differ significantly from the system with all spins active (c = 1). We have computed the effect of varying c on the value of T c and α g : T c is found to be a linear function of c, while α g is a monotonously increasing function of c. Results are shown in Fig. 22.

B. Relationship with linear threshold models and previous studies

Several attractor neural network models for the hippocampus have been proposed in previous works. Tsodyks and Sejnowski [14] proposed a rate model with semilinear threshold neurons, uniform inhibition, and excitatory synapses between neurons with neighboring place fields, with a strength decaying exponentially with distance. Their study was limited to the single environment, one-dimensional case. They showed the formation of localized activity. Moreover, they studied the effect of inhomogeneities in the synaptic matrix due to irregularities in the learning process, an interesting effect that we do not address here.

Battaglia and Treves [18] introduced the multiple environment storage in additive synapses. They studied the case of linear threshold neurons with generic form of kernel of connection weights. The free energy is calculated implementing the threshold linear transfer function and averaging over disorder in the replica-symmetric approximation, along the lines developed in [26]. The clump phase is studied at zero temperature, and the storage capacity is found as the maximal value of α for which localized solutions exist. Different forms of couplings and varying sparsity of the representation are considered, and an enlightening parallel with episodic memory is proposed. The issue of information storage is addressed.

Our method is in the same spirit as [18], but the model differs as we consider binary units instead of threshold linear units (i.e., without saturation) for a simple coupling matrix and an explicit form of inhibition. Nevertheless, a parallel can be drawn between the range of interaction w in our model and the "map sparsity" 1 |M| in [18]. In spite of the differences between the models, the order of magnitude of the maximal storage capacity is the same in both models: ∼3.10 -2 in one dimension, ∼8.10 -3 in two dimensions (see Figs. 1 and2 in [18]). The "chart sparsity" α c in [18] corresponds to our parameter c. The main difference between both models lies in the way noise is taken into account. In [18], the level of noise is embedded in the rate model, in the gain g of the units, and is not taken into account in the thermodynamics since the study is carried out at zero temperature. Our model considers binary units with a level of noise T corresponding to the thermodynamic temperature. On average, binary neurons behave as rate neurons with sigmoidal transfer function of gain 1 T (see Sec. II C). From this point of view our model is more microscopic than the one in [18], as we have a description of noise at the neuron level. Furthermore, we have looked at the stability of the clump phase against replicon modes. Our study also includes the other regimes of activity of the model (i.e., the PM and SG phases) and their thermodynamic stability compared to the clump phase, summarized in the phase diagram.

C. Conclusion

In this paper we have introduced an attractor neural network model for the storage of multiple spatial maps in the hippocampus. Although very simplified, the model accounts for experimentally observed properties of place cells, such as the remapping of place fields from one environment to the other. We showed that multiple maps can be simultaneously learned in the same network, i.e., with the same synaptic coupling coefficients, even in the presence of noise in the neural response. Remarkably, moderate levels of noise can even slightly increase the capacity storage with respect to the noiseless case. Notice that the qualitative behavior of the model is robust to changes in the value of the parameters; for instance we do not expect that changing the couplings from a square-box function into an exponentially decreasing function over the distance wN in D = 1 or √ wN in D = 2 would have much effect on the phase diagram.

The storage of a map manifests itself through the fact that the neural activity is localized, and acquires a clumplike shape in the corresponding environment. When the load (number of environments) or the noise are too high the neural activity cannot be localized any longer in any one of the environments. For high noise, the activity, averaged over time, simply becomes uniform over the space. For high loads the activity is not uniform, but is delocalized with spatial heterogeneities controlled by the crosstalks between the (too many) maps. The prevalence of the glassy phase at high load and of the uniform (paramagnetic in the physics language) phase at high noise moderately limits the extension of the clump phase. Moreover, we have found that in the glassy phase the replica symmetric assumption is not correct, and we may expect from general consideration about replica symmetry breaking that the first-order transition from the clump phase to the glassy phase occurs at higher loads α. Remarkably the clump phase is therefore the thermodynamically dominant phase in nearly all of its stability domain.

The present work is a direct offspring of spin-glass models of attractor neural networks [16], with the difference that here one pattern corresponds to one map, i.e., a whole set of coherent neural configurations, instead of a single configuration of activity. This explains the robustness of the patterns against neural noise in our case compared to the Hopfield model case, as discussed in Sec. V. This generalization of the notion of "stored pattern" is interesting and would deserve further consideration. It appears that the concept of attractor neural network can embed memory items with much richer structure than the ones originally considered; it is quite encouraging that the theoretical framework built for the original Hopfield model can be extended to deal with those structured items. In the case of the hippocampus, it is widely believed that CA3 is the support of episodic memory, that is, the memory of autobiographical events and contextual experiences. According to this view, the hippocampus could learn not only spatial relations between locations but also associate them to events, times, and emotions. In our model the coupling matrix associates nearby places together. We could imagine a generalization of it to a network which makes associations between units coding for other, more abstract nonspatial features, although characterizing the "metric" properties of general feature space is much harder than for the usual Euclidean space.

Our work would deserve to be extended along other directions. First the assumption that synaptic couplings additively sum up the contributions coming from all the environments could be lifted. We could replace the synapses J ij with a nonlinear function G(J ij ). The additive case corresponds to G(x) = x, while a strongly nonadditive synapse is obtained with the choice G(x) = min(x, 1 N ): synapses can be written only once, and contributions from different environments do not add up but saturate the synaptic coupling. It would be worth extending the study of nonlinear synapses done for the Hopfield model [27,28] to the present model.

Second, we have considered that the only source of quenched noise was the interference between the multiple environments. In other words, in the single-environment case, our synaptic matrix is translationally invariant and the center of the activity clump can be moved at no energy cost in space. This idealizing assumption was done to study the effect of multiple-environment crosstalk only. However, even in the single environment case, place fields do not define a perfectly regular covering of space. We expect that such heterogeneities in the couplings will further destabilize the clump phase, and decrease the storage capacity [27]. Quantifying those effects would be interesting.

However, the most important extension seems to us to be the study of the dynamics. The richness of the phase diagram we have unveiled here and the multiplicity of phases for the system raise the question of if and how the network activity makes transitions between those phases. Multiple environments stored in the same network not only influence the shape of the clump and lead to transitions to a glassy phase, but they can as well provoke transitions between attractors. The study of these transitions and of the corresponding reaction paths will be reported in a companion paper [21]. It could prove useful to interpret recent experiments, where changes of the hippocampal activity resulting from the "teleportation" of the rat have been recorded [20]. In addition it would be interesting to understand in a more quantitative way the activated diffusion process of the clump in an environment. In the presence of other maps, the invariance by translation is lost and the clump does not freely diffuse. Quantifying the barriers opposing motion, as well as understanding the qualitative difference between motions in 1D and 2D spaces, would be very useful.

where

ψ 2D (q,β) ≡ 2 (k 1 ,k 2 ) =(0,0) β(q -f 2 ) φ(k 1 ,k 2 ) -β(f -q) -ln 1 - β(f -q) φ(k 1 ,k 2 ) , (A2) with φ(k 1 ,k 2 ) ≡ k 1 k 2 π 2 sin(k 1 π √ w) sin(k 2 π √ w) . ( A3
)
Hence the saddle point equations write

r = 4(q -f 2 ) (k 1 ,k 2 ) =(0,0) [φ(k 1 ,k 2 ) -β(f -q)] -2 , q = d x Dz[1 + e -βz √ αr-βμ( x) ] -2 , (A4) ρ( x) = Dz[1 + e -βz √ αr-βμ( x) ] -1 , μ( x) = d y J w ( x -y)ρ( y) + λ,
where λ is determined to enforce constraint (12).

In We thus computed ρ(r) in the clump phase and found the region in the (α,T ) plane where this solution is stable against longitudinal modes. We find that this region is reduced compared to the D = 1 case, but its shape is qualitatively similar. The result is shown in Fig. 23.

APPENDIX B: AVERAGE OF THE BOLTZMANN FACTOR OVER A RANDOM ENVIRONMENT

The purpose of this appendix is to calculate

( σ ) = 1 N! π exp ⎡ ⎣ β i<j J 0 ij n a=1 σ a π(i) σ a π(j ) ⎤ ⎦ = Cξ ( σ ) ( B 1 )
with

C ≡ exp N β 2 nwf 2 ξ ( σ ) ≡ 1 N! π exp ⎡ ⎣ β 2 i,j J 0 ij n a=1 σ a π(i) -f σ a π(j ) -f ⎤ ⎦ , (B2)
where the sum is carried out over all permutations of N elements.

The eigenvectors of the matrix J 0 are plane waves. Let v q,j denote the j th (real-valued) component of the qth normalized eigenvector, and λ q the associated eigenvalue. Then,

i,j J 0 ij σ a π(i) -f σ a π(j ) -f = N-1 q=1 λ q ⎛ ⎝ j v q,j σ a π(j ) -f ⎞ ⎠ 2 . ( B3
)
Due to condition (6) we have discarded the homogeneous mode q = 0 from the sum in (B3). Introducing a set of n(N -1) independent Gaussian variables with zero mean and variance unity, denoted by a q , we can write (all odd powers of √ β vanish after integration over the Gaussian measure)

ξ ( σ ) = exp β q,a,j λ q v q,j a q (σ a π(j ) -f ) π, = 1 + k 1 β k (2k)! q i ,a i ,j i i=1•••2k v q 1 ,j 1 v q 2 ,j 2 . . . v q 2k ,j 2k × λ q 1 λ q 2 . . . λ q 2k T a 1 ,a 2 ...a 2k j 1 ,j 2 ...j 2k a 1 q 1 a 2 q 2 . . . a 2k q 2k , (B4)
where T a 1 ,a 2 ,...,a 2k j 1 ,j 2 ,...,j 2k ≡ σ a 1 π(j 1 )f σ a 2 π(j 2 )f . . . σ a 2k π(j 2k )f π .

(B5) Using Wick's theorem the 2k-point correlation function of the variables is easy to calculate. The outcome is a multiplicative factor (2k -1)!!, and the replacement of the 2k sums over the indices q m ,a m by only k independent sums over q m ,a m . The value of T (B5) depends only on the number of distinct indices, i m , and of their associated multiplicities. Power counting shows that T a 1 ,a 2 ,...,a 2k i 1 ,i 2 ,...,i 2k vanishes in the infinite N limit unless the set {i 1 ,i 2 , . . . ,i 2k } includes exactly k distinct indices, each one with multiplicity 2. When this condition holds we write (a m ,a m ), the replica indices attached to the mth distinct index i, with m = 1,2, . . . ,k. Then, in the large N limit,

T a 1 ,a 1 ,a 2 ,a 2 ,...,a k ,a k i 1 ,i 1 ,i 2 ,i 2 ...,i k ,i k = k m=1 (q a m a m -f 2 ). (B6)
We assume that the quantities βλ q j v q,j (σ a π(j )f ) do not diverge when the limit N → ∞ is taken for each one of the terms in the series over k in (B4). This hypothesis breaks down if the permutation π is "close" to the identity permutation, or, equivalently, if the configuration σ π = {σ π(i) } is coherent with the environment 0. As π is randomly chosen the probability that this is the case vanishes for large N.

We are left with the summation over the j m indices. Using the identities j v q,j v q ,j = δ q,q , (B7)

we obtain from (B4) the following expression:

ξ ( σ ) = 1 + k 1 (β/2) k k! P w(P), ( B8
)
where the last sum runs over all weighted pairings among 2k points, described as follows:

(i) We define 2k points. The first k points carry the pair indices (q m ,a m ), with m running from 1 to k. The second k points carry the same pair indices. Hence, each pair index (q m ,a m ) is shared by two points.

(ii) A pairing P is a set of k bonds b ≡ {(q m ,a m ),(q m ,a m )}, = 1,2, . . . ,k, each joining one pair of points (dimer coverage).

(iii) The weight of the pairing is

w(P) ≡ a 1 ,...,a k q 1 ,...,q k k m=1 λ q m k =1 δ q m ,q m q a m a m -f 2 . (B9)
We denote q the overlap matrix with entries q ab and 1 n the n × n matrix whose all entries are equal to 1. Let us introduce a notation for the moments of the eigenvalues:

h ≡ q 1 λ h q = 2 q 1 sin(qπw) qπ h . ( B10
)
Two examples of pairings are shown in Fig. 24. The weight associated to the pairing P A is simply

w(P A ) = k m=1 q m λ q m a m q a m a m -f 2 = ( 1 Tr[q -f 2 ]) k = [ 1 n f (1 -f )] k , (B11)
as all Kronecker δ in (B9) are equal to 1 by construction. The weight associated to the second pairing in Fig. 24 is

w(P B ) = { 3 Tr[(q -f 2 ) 3 ]} ( 1 Tr[q -f 2 ]) k-3 . (B12)
For a given pairing, (i) the horizontal bonds represent independent replicas: point number m leads to a factor q m λ q m a m (q a m a mf 2 ) in the weight of the pairing;

(ii) the vertical and diagonal bonds couple replicas together. We then have to calculate the combinatorial multiplicity of the weights, i.e., how many pairings have the same weight in the sum (B8). For a given k, a pairing associates points by groups of j coupled replicas indices (i.e., 2j points). Let m j be the number of such groups in a given pairing. We have j jm j = k. Pairings P with the same (j,m j ) have equal weights

w(P) = w[{(j,m j )}] = j ( j T j ) m j , ( B13
)
where we set T j ≡ Tr[(qf 2 1) j ].

Combinatorial study shows that the number of such pairings is

N [{(j,m j )}] = k! j 1 m j ! 2 j -1 j m j . (B14)
Finally, using j jm j = k and (B10), we can rewrite

ξ ( σ ) = 1 + k 1 β 2 k j m j 0 1 m j ! 2 j -1 j m j ( j T j ) m j = exp j 1 2
β j j j T j = exp - λ =0 Tr ln[Id n -βλ(q -f 2 1 n )] . (B15)

APPENDIX C: REPLICA SYMMETRIC CALCULATION OF THE FREE ENERGY

We introduce parameters r ab conjugated to the overlaps q ab . With this notation, we have (up to a multiplicative irrelevant constant): where

Z n J = σ
G({q ab ,r ab }, σ ) = Nαβ 2 a<b r ab 1 N i σ a i σ b i -q ab + β 2N a |i-j |<wN/2 σ a i σ a j -αN λ =0 Tr ln[Id n -βλ(q -f 2 1 n )]. (C2)
We rewrite the sum over spin configurations as a path integral over continuous-space average densities and over the conjugated potentials,

σ exp αβ 2 a<b r ab i σ a i σ b i + β 2N a |i-j |<wN/2 σ a i σ a j = a Dρ a (x)Dμ a (x)dλ a exp N dx lnZ({μ a (x),r ab }) + N a βλ a dx ρ a (x) -f ) -β dx ρ a (x)μ a (x) + β 2 dx dy ρ a (x)J w (x -y)ρ a (y) , ( C3
)
where we have defined

Z({μ a (x),r ab }) ≡ {σ a } exp αβ 2 a<b σ a σ b r ab + β a μ a (x)σ a . (C4)
In the replica symmetric (RS) ansatz, we assume

∀ a = b, ∀ x, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r ab = r, q ab = q, ρ a (x) = ρ(x), μ a (x) = μ(x), λ a = λ. (C5)
We obtain

T j = (n -1)(f -q) j + [f -f 2 + (n -1)(q -f 2 )] j , ( C6
)
and

Z(μ(x),r) = 1 + n Dzln(1 + e βz √ αr+βμ(x)-αβ 2 r/2 ) (C7)
up to O(n 2 ) corrections. We now make the change of variable μ(x) → μ(x) -αβr 2 . The averaged partition function is, for small n, Z n ∼ dq dr dλ Dμ(x)Dρ(x)e -nNβF[μ(x),ρ(x),q,r,λ] , (C8) where

F = αβ 2 r(f -q) - α β ψ(q,β) -λ dxρ(x) -f + dxρ(x)μ(x) - 1 2 dxdy ρ(x)J w (x -y)ρ(y) - 1 β dxDzln(1 + e β √ αrz+βμ(x) ) ( C 9 ) with ψ(q,β) ≡ j 1 2
β j j j [j (q -f ) 2 (f -q) j -1 + (f -q) j ]. (C10)
When N → ∞ the integral is calculated through the saddlepoint method, and we look for the extremum of F over its arguments. We now give the expression of ψ and of the order parameter r = -2

β 2
∂ψ ∂q in dimensions D = 1 and D = 2. Case D = 1. We define A k ≡ πk sin(πkw) and write

1D j = 2 k 1 (A k ) -j . (C11)
We immediately obtain

ψ 1D (q,β) = k 1 β(q -f 2 ) A k -β(f -q) -ln 1 - β(f -q) A k (C12)
and

r 1D = 2(q -f 2 ) k 1 [A k -β(f -q)] -2 . ( C13
)
Case D = 2: see Appendix A.

APPENDIX D: SILENT CELLS CASE: CALCULATION OF THE FREE ENERGY

We now consider the hypothesis that a fraction c < 1 of the cells are involved in each environment's representation. Two types of disorder are present: the random permutations of the place field centers as before, and the choices of the subsets of cells taking part to each map , i.e., the dilution variables τ i . The nth moment of the partition function reads

Z n J = σ exp ⎡ ⎣ β a i<j J 0 ij τ 0 i τ 0 j σ a i σ a j ⎤ ⎦ × exp β L =1 a i<j J 0 ij τ π (i) τ π (j ) σ a π (i) σ a π (j ) π ,τ , (D1)
where τ denotes one realization of the L × N random variables τ i , and the τ 0 i are 1 if i is a multiple of 1/c, 0 otherwise. The sum is now taken over spin configurations satisfying (6) and (39) for each replica index a.

Using the function 1(x) = 1 if x = 0 and 0 elsewhere, we write

Z n J = C all σ 1 1 N i σ a i -f e β a i<j J 0 ij τ 0 i τ 0 j σ a i σ a j χ ( σ ) L , ( D2
)
where C is a constant and

χ ( σ ) ≡ 1 1 cN i τ i σ a i -f × e β a i<j J 0 ij τ π(i) τ π(j ) (σ a π(i) -f )(σ a π(j ) -f ) π τ . ( D3
)
The average over π follows exactly the steps described in Appendix B. Defining

( q) ≡ - λ =0 Tr ln[Id n -βλ q -cf 2 1 n ], ( D4
)
where q is now the n × n matrix of elements

qab ≡ 1 N i τ i σ a i σ b i , (D5)
we end up with χ ( σ ) = d qab exp[ ( q)] χ ( q, σ ) with

χ( q, σ ) = C iR dλ a dR ab e √ N(cf a λ a + a b R ab qab) × i exp -τ i a λ a √ N σ a i + R ab √ N σ a i σ b i τ i , ( D6
)
where we have introduced parameters R ab conjugated to qab and Lagrange multipliers λ a to enforce the constraint on σ . We now perform the average over the decoupled variables τ i . Introducing the order parameters

T abc ≡ 1 N i σ a i σ b i σ c i , S abcd ≡ 1 N i σ a i σ b i σ c i σ d i , (D7)
and carrying out the Gaussian integration over the leading terms (when N 1) in λ a and R ab , we have (up to a multiplicative constant)

χ( q, σ ) = C e -(N/2) a,b,c,d [A -1 ] abcd ( qab -cq ab )( qcd -cq cd ) , ( D8
)
where

A abcd ≡ c(1 -c) ⎛ ⎝ S abcd + 2 e,f [Q -1 ] ef T abe T cdf ⎞ ⎠ . (D9)
Hence, in the large N limit, the integral over qab is dominated by qab = c q ab .

The replica symmetric calculation of the free energy proceeds along the steps of Appendix C. The only difference is that, here, μ(x) and ρ(x) describe the activity of the cN cells involved in the reference environment, while the (1c)N remaining cells have uniform activities = f . We obtain the expression of the energy functional

F c = αβ 2 r(f -q) - α β ψ c (q,β) + c dx μ(x)ρ(x) + (1 -c)μ 2 f - c 2 2 dx dy ρ(x)J w (x -y)ρ(y) -λc dx ρ(x) -f - c β dx Dzln 1 + e βz √ αr+βμ(x) - (1 -c) β Dzln 1 + e βz √ αr+βμ 2 , ( D10
)
where q is defined as before and

ψ c (q,β) = k 1 βc(q -f 2 ) sin(kπw) kπ -βc(f -q) sin(kπw) -ln 1 - βc(f -q) sin(kπw) kπ . ( D11
)

APPENDIX E: STABILITY OF THE REPLICA SYMMETRIC SOLUTION

The extremization of the free energy functional under the fixed-activity constraint and under the replica symmetric assumption leads to three solutions corresponding to three different phases. We want to study the stability of those solutions in the (α,T ) space. We will limit ourselves to the one-dimensional case. A small perturbation of the solution

ρ a (x) → ρ a (x) + δρ a (x), μ a (x) → μ a (x) + δμ a (x),
(E1) r ab → r ab + δr ab , q ab → q ab + δq ab results in F → F + 1 2 δ 2 F, where

δ 2 F = dxdy ⎡ ⎢ ⎢ ⎢ ⎣ δρ a (x) δμ a (x) δr ab δq ab ⎤ ⎥ ⎥ ⎥ ⎦ † M(x,y) ⎡ ⎢ ⎢ ⎢ ⎣ δρ c (y) δμ c (y) δr cd δq cd ⎤ ⎥ ⎥ ⎥ ⎦ . (E2)
The Hessian matrix M(x,y) reads, in the {δρ a (x),δμ a (x),δr ab ,δq ab } basis,

M = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂ 2 F ∂ρ a (x)∂ρ c (y) ∂ 2 F ∂ρ a (x)∂μ c (y) 0 0 ∂ 2 F ∂μ a (x)∂ρ c (y) ∂ 2 F ∂μ a (x)∂μ c (y) ∂ 2 F ∂μ a (x)∂r cd 0 0 ∂ 2 F ∂r ab ∂μ c (y) ∂ 2 F ∂r ab ∂r cd ∂ 2 F ∂r ab ∂q cd 0 0 ∂ 2 F ∂q ab ∂r cd ∂ 2 F ∂q ab ∂q cd ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . ( E3
)
Using the notations

t(x) ≡ σ 3 (x) = Dz[1 + e -βz √ αr-βμ(x) ] -3 , s(x) ≡ σ 4 (x) = Dz[1 + e -βz √ αr-βμ(x) ] -4 , (E4) t ≡ dx t(x); s ≡ dx s(x); q 2 ≡ dx q 2 (x), we have ∂ 2 F ∂ρ a (x)∂ρ c (y) = -J w (x -y)δ ac , (E5) ∂ 2 F ∂ρ a (x)∂μ c (y) = δ(x -y)δ ac , (E6) ∂ 2 F ∂μ a (x)∂μ c (y) = δ(x -y)β[ρ 2 (x) -ρ(x)] if a = c, δ(x -y)β[ρ 2 (x) -q(x)] otherwise, (E7) ∂ 2 F ∂μ a (x)∂r cd = αβ 2 [q(x)ρ(x) -t(x)] if a = c = d, αβ 2 [q(x)ρ(x) -q(x)] otherwise, (E8) ∂ 2 F ∂r ab ∂r cd = ⎧ ⎪ ⎨ ⎪ ⎩ α 2 β 3 q 2 -q if a = c and b = d, α 2 β 3 q 2 -s if a = b = c = d, α 2 β 3 q 2 -t otherwise, (E9)
and, letting

B k ≡ kπ sin(kπw) -β(f -q), C 1 ≡ k 1 β B 2 k , C 2 ≡ k 1 β 2 (q -f 2 ) B 3 k , C 3 ≡ k 1 β 3 (q -f 2 ) 2 B 4 k , ∂ 2 F ∂q ab ∂q cb = ⎧ ⎪ ⎨ ⎪ ⎩ -2α(C 1 + 2C 2 + 2C 3 ) if a = c and b = d, -4αC 3 if a = b = c = d, -2α(C 2 + 2C 3 )
otherwise.

(E10)

The eigenvector equation writes

M • v = λ • v, (E11)
where v is the vector of fluctuations around the saddle point:

v(x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ δρ a (x) . . . δμ a (x) . . . δr ab . . . δq ab . . . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . ( E12
)
According to [29] the symmetry of the matrix elements under permutation of the indices imposes to look for an orthogonal set of eigenvectors with the particular forms:

v 1 (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ δρ a (x) = δρ(x) ∀a, δμ a (x) = δμ(x) ∀a, δr ab = δr ∀a,b, δq ab = δq ∀a,b, (E13)
v 2 (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ δρ a (x) = δ ρ(x) if a = θ, = δ ρ(x) otherwise, δμ a (x) = δ μ(x) if a = θ, = δ μ(x) otherwise, δr ab = δ r if a or b = θ, = δ ř if a and b = θ, δq ab = δ q if a or b = θ, = δ q if a and b = θ, (E14) v 3 (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ δρ a (x) = δ ρ(x) if a = θ or θ , = δρ * (x) otherwise, δμ a (x) = δ μ(x) if a = θ or θ , = δμ * (x) otherwise, δr ab = δ r if a = θ and b = θ , = δ r if a or b = θ or θ , = δr * if a and b = θ,θ , δq ab = δ q if a = θ and b = θ , = δ q if a or b = θ or θ , = δq * if a and b = θ,θ , (E15)
where θ and θ are two fixed replica indices. v 1 (x) and v 2 (x) are called longitudinal modes; v 3 (x) are called transverse or "replicon" modes. Imposing the orthogonality conditions

v 1 (x) • v 2 (x) = v 1 (x) • v 3 (x) = v 2 (x) • v 3 (x) = 0, (E16)
and taking the n → 0 limit in Eq. (E11), we end up with the eigensystem in the longitudinal sector,

-dy J w (x -y)δρ(y) + δμ(x) = λ δρ(x), δρ(x) + β(q -ρ)(x)δμ(x) + αβ 2 (q -t)(x)δr = λ δμ(x), 2αβ 2 (t -q)δμ + α 2 β 3 (-q + 4t -3s)δr + αβδq = λ δr, αβδr -2α(C 1 -2C 2 )δq = λ δq, (E17)
and in the replicon sector,

α 2 β 3 [-q + 2t -s]δr * + αβδq * = λ δr * , (E18) αβδr * -2αC 1 δq * = λ δq * .
For each one of the three phases (PM, CL, SG) the stability region in the (α,T ) plane is delimited by lines where one of the eigenvalues vanishes.

Note that the matrix of system (E17) is not symmetric while the hessian matrix M(x,y) is: a -1 2 factor appears when taking the n → 0 limit since there are n(n-1) 2 two-replica-indice components. This multiplicative factor does not change the points where a given eigenvalue changes signs. Nevertheless, it has the effect of giving nonreal eigenvalues.

Paramagnetic phase stability region

Taking ρ(x) = f , q(x) = f 2 , t(x) = f 3 , and s(x) = f 4 for all x in (E17) leads to a very simple system, invariant under translation in the x space. The eigenmodes in the (δρ(x),δμ(x)) sector are plane waves, e 2iπkx , with integer wave vectors k. The eigensystem (E17) decomposed on each Fourier mode gives the following:

(i) k > 0 components of the longitudinal matrix. The corresponding determinant is

-sin(πkw) πk 1 1 β(f 2 -f ) . (E19) It vanishes for β(k) = πk sin(πkw)(f -f 2 )
which is minimal for k = 1. For f = 0.1 and w = 0.05, T 1 ≈ 0.0045.

(ii) k = 0 component of the longitudinal matrix. We get a system with determinant

f 2 -f β (f 2 -f 3 ) 0 2αβ(f 3 -f 2 ) αβ 2 (-f 2 + 4f 3 -3f 4 ) 1 0 1 -2 C 1 β . ( E20
)
These modes appear for (α,T 0 (α)) at which this determinant vanishes, i.e.,

k 1 T 0 (α) kπ f (1 -f ) sin(kπw) -1 -2 = 1 2α . ( E21)
(iii) Replicon modes. These modes solve a system with determinant

αβ 2 (-f 2 + 2f 3 -f 4 ) 1 1 -2 C 1 β . (E22)
This defines the same stability line (E21) as found above.

To sum up, the paramagnetic phase is stable at high temperatures; when T decreases at fixed α, it becomes instable at T PM (α) = max {T 0 (α),T 1 } as depicted in Fig. 3.

Glassy phase stability region

We find a uniform solution to the saddle point equations (28) with q > f 2 only for T < T PM (α): the region of existence of the glassy phase hence corresponds to the region where the PM solution is unstable. In this region we find that the RS solution is always stable against longitudinal modes (E17) and always unstable against transverse modes (E18). The replica symmetric ansatz is therefore not correct in the case of the glassy phase.

Clump phase stability region

(i) Longitudinal modes. Due to the x dependence in this phase, we must use a numerical approach in a discretized space to study the eigenvalues of the longitudinal matrix. Since computation time for the matrix diagonalization limits dramatically the number of points in the discretization, we chose to study the longitudinal stability with a different method. Scanning the (α,T ) plane, ρ(x) is computed by solving iteratively the saddle-point equations (28) starting from an initial clump; the line of stability corresponds to the points where the clump collapses, i.e., the iteration converges to a uniform activity ρ(x) = f ∀x. The result is shown in Fig. 7 in the main text.

(ii) Replicon modes. For all α,T , we compute numerically q, t, s by solving iteratively the saddle-point equations as before, allowing us to calculate the determinant of system (E18). We looked for the line where this determinant vanishes. We found that replica symmetry breaking is limited to a small region in the low T -high α edge of the region of longitudinal stability; see Fig. 7. The dynamics of a neural model for hippocampal place cells storing spatial maps is studied. In the absence of external input, depending on the number of cells and on the values of control parameters (number of environments stored, level of neural noise, average level of activity, connectivity of place cells), a "clump" of spatially localized activity can diffuse or remains pinned due to crosstalk between the environments. In the single-environment case, the macroscopic coefficient of diffusion of the clump and its effective mobility are calculated analytically from first principles and corroborated by numerical simulations. In the multienvironment case the heights and the widths of the pinning barriers are analytically characterized with the replica method; diffusion within one map is then in competition with transitions between different maps.

Crosstalk and transitions between multiple spatial maps in an attractor

I. INTRODUCTION

Since the discovery of place cells in the hippocampus of rodents [1], the hippocampus is believed to support spatial memory and representation. Place cells are neurons that fire specifically when the animal is located at certain positions of space called place fields. Their properties have been extensively studied, revealing striking features. In particular, the memorized places appear to be organized in several discrete "maps" or "environments" [2]. A given neuron can have place fields in different environments, and these place fields appear randomly allocated, independently of the place cell's location in the neural tissue [3]. This random reallocation of place fields in each new environment is called "remapping" [4]. Place fields are also stable in the dark [5] and after alteration of visual cues [6], suggesting that their firing is driven in part by self-motion information ("path integration" [7]).

Many theoretical models have been proposed to account for the formation and the firing properties of place cells. An important class of them is formed by attractor neural network models [8][9][10][11][12][13]. These models postulate that an environment is memorized when the corresponding neural activities are stable states of the network [14], such as in the celebrated Hopfied model [15], an assumption motivated here by the high degree of recurrent connectivity in the CA3 area of the hippocampus [16]. In a majority, these studies focus mainly on the static properties of the models, that is, the stable states of the network. The conditions of formation of spatially localized attractors, their robustness to noise, and the storage capacity of such networks have been investigated in great detail. How the network dynamically evolves within one map and between maps remains, however, poorly understood in this framework, at least analytically. Yet this dynamical aspect plays a crucial role in most experiments, whether they involve physical motion of the animal [17][18][19], mental trajectory planning [20], "sleep replay" [21], or modification of visual cues [18,[22][23][24].

Attractor neural networks are an important paradigm in the attempt to understand and model the principles of memory. Following their introduction by Hopfield 30 years ago [15], the properties of attractor neural networks have been investigated in detail using tools from statistical mechanics of disordered systems [14]. In the "basic," most common version, a memorized pattern corresponds to an activity configuration of the network. In the present case of spatial memory, in contrast, a memory item corresponds to a space manifold (a spatial map), i.e., the whole collection of neural activity configurations obtained when the animal is located in various points of this map. As a consequence attractors are more complex than in the original Hopfield model. As far as dynamics is concerned, again, the present case displays much richer behaviors. Indeed, in the presence of noise in the neural response, the network activity can either jump between maps (as is the case between attractors in the Hopfield model) or evolve continuously within one attractor. In the latter case, the pattern of activity corresponds successively to positions along a continuous trajectory in one of the maps, as if the neural activity configuration "moved" in this map. As a result such an extension of the Hopfield model paves the way for refinements and complexification of the structure of the modeled memory. In this context, the comprehension of its complex dynamics has a theoretical interest in itself.

Furthermore, from the point of view of statistical mechanics, the study of a spatially localized phase as a bump of activity in hippocampal neurons is of great interest. How a "quasiparticle" emerges from the interactions of microscopic units, and how the dynamics of its location (being considered here as a collective coordinate for the neural activity) can be characterized, are nontrivial questions, which highlight the rich connection between statistical mechanics and computational neuroscience.

In a previous article [25], we proposed an attractor neural network model for hippocampal place cells encoding onedimensional (1D) and two-dimensional (2D) spatial maps. We studied the stable states and the phase diagram for varying levels of noise and of memory load. We showed that, under certain conditions, the stable states are "clumps" (bumps) of activity localized in one of the stored environments, similar to the activity patterns observed in microelectrode single-unit recordings. In the present work, we address the issue of the evolution of the network within one attractor, that is, within one map, when 1539-3755/2014/89(3)/032803 (23) 032803-1 ©2014 American Physical Society the network is in this clump phase. Its dynamics is studied both analytically and numerically. It appears that the crosstalk between environments has the effect of hindering the motion of the clump and virtually suppresses motion for a wide range of control parameters. This phenomenon is particularly salient in the 1D case. Neural noise, by itself, may therefore not be sufficient to make the clump move, and additional mechanisms have to be proposed to retrieve this motion [26]. We show that diffusion within one map is in competition with transitions between maps, corresponding to the sudden disappearance of the localization of the activity at one specific position in the map under consideration, followed by its localization at another position in another map. The detailed study of those transitions and of the distribution of the tunneling positions within the maps will be addressed in a companion publication.

In Sec. II we briefly recall the model and summarize the results of Ref. [25] on its stable phases. The main results of the present paper on the dynamics of the activity in one map are reviewed in Sec. III. In Sec. IV we study the singleenvironment case and analytically show that the dynamics can be described by an effective diffusion for the center of the clump; we also characterize the mobility of the clump in response to an external force. In Sec. V we show that the presence of disorder limits drastically the motion of the clump within one environment and propose additional mechanisms to enhance motion. In Sec. VI we address the retrieval process of the attractor neural network in the presence of input. Finally, in Sec. VII we study the effect of other, out-of-equilibrium mechanisms on the motion of the clump.

II. REMINDER ON THE MODEL AND ITS PHASES

The N place cells are modeled by interacting binary units σ i equal to 0 or 1, and corresponding to, respectively, silent and active neurons. Let us first consider a first environment (that can be either 1D or 2D). We suppose that, after learning of the environment and random allocation of place fields, each place cell preferentially fires when the animal is located in an environment-specific location in the 1D or 2D space, defining its place field. For simplicity space is assumed to be a segment of length N in dimension 1, and a square of edge length √ N in dimension 2, with periodic boundary conditions. The N centers of the place fields are located on the nodes of a 1D or 2D regular grid: two contiguous centers are at unit distance from each other.

Pairs of cells whose place field centers lie within some distance d c from each other are coupled with an excitatory coupling J 0 ij = 1 N . We choose the cutoff distance d c such that each cell i is connected to the same number w N of other cells j , independently of the space dimension: w( 1) is the fraction of the neural population any neuron is coupled to. The 1 N scale factor in the coupling J 0 ij is such that the total input received by a place cell is finite when the number of cells, N, is sent to infinity.

Then we consider other additional environments. Each time the rodent explores a new environment a remapping of the place fields takes place. We assume that the remapping is represented by a random permutation of the N place-cell indices associated to the place fields on the regular grid. Let π be the permutation corresponding to remapping (environment) number , where = 1, . . . ,L is the index of the new environments. We assume that all environments contribute equally and additively to the total synaptic matrix, with the result

J ij = J 0 ij + L =1 J 0 π (i)π (j ) . (1)
Note that all environments are statistically equivalent. We will look hereafter for the presence of localized activity in the environment 0 (hereafter called reference environment), but this choice is arbitrary.

In addition to pyramidal cells, the network contains longrange, inhibitory interneurons, which maintain the fraction of active place cells at a fixed level, f . The probability of a neural activity configuration σ = (σ 1 ,σ 2 , . . . ,σ N ) is then assumed to be

P J (σ ) = 1 Z J (T ) exp ⎛ ⎝ i<j J ij σ i σ j /T ⎞ ⎠ , (2)
where the partition function Z J (T ) is such that the sum of P J (σ ) over all activity configurations with exactly f N active neurons is normalized to unity. Parameter T , which plays the role of temperature in statistical mechanics, is indicative of the level of noise in the response of neurons to their inputs (local fields). In Ref. [25] we have analytically characterized the possible regimes, or phases, of the model in the limit of large size, N → ∞, and at a fixed ratio of the number of environments per neuron, α ≡ L/N, hereafter called load. The phases are defined in terms of the behaviors of the local average of the activity,

ρ(x) = lim →0 lim N→∞ 1 N i:|x-i N |< 2 σ i , (3)
and of the Edwards-Anderson overlap describing the fluctuations of the local activities:

q = 1 N N i=1 σ i 2 . ( 4
)
The overbar above denotes the average over the random remappings (permutations π ), while the brackets • correspond to the average over distribution P J (2). The outcome of the analysis is the phase diagram shown in Fig. 1. Three stable phases are found (see Ref. [25] for details):

(1) The paramagnetic phase (PM), corresponding to high levels of noise T , in which the average local activity is uniform over space, ρ(x) = f , and neurons are essentially uncorrelated, q = f 2 .

(2) A glassy phase (SG), corresponding to large loads α, in which the local activity σ i varies from neuron to neuron (q > f 2 ) but does not cluster around any specific location in space in any of the environments (ρ(x) = f after averaging over remappings). In this SG phase the crosstalk between environments is so large that none of them is actually stored in the network activity.

(3) A "clump" phase (CL), for small enough load and noise, where activity depends on space; i.e., ρ(x) varies with Thick solid lines: transitions between phases. Thin dashed lines: stability region of each phase against fluctuations. Insets show the corresponding activity profiles in the 2D model (averaged over one round of Monte Carlo simulations after thermalization). In the clump phase we represent the same activity profile in the retrieved environment (top) and in another stored environment (bottom). See Ref. [25] (Fig. 8) for more quantitative details.

x. In the present case, the activity is localized in the first environment (reference environment). This is the consequence of our choice for the reference environment, but in practice the activity could be localized in any environment. Which environment is retrieved may depend on external factors (initial configuration of activity, specific inputs, etc.) and may vary with time due to thermal fluctuations.

Unless stated otherwise, we take the parameter values w = 0.05 and f = 0.1 in the numerical simulations throughout this work.

III. OVERVIEW OF RESULTS

While the system is in the clump phase, the bump of activity can either move over space in the coherent environment (hence, stay in the same attractor) or switch between environments (transition to another attractor). Transitions from one environment to another have been observed experimentally [22,24] and will be addressed in a forthcoming publication. In this paper we focus on the dynamics of the neural activity "within" one map only. We now briefly review our main results.

The dynamics we consider defines an evolution for the microscopic configuration of neural activity, that is, the set of all neuron states (silent or active). As we know from the study of equilibrium properties [25], the statistics of the activity can be characterized through the average density profile, ρ * (x) (the asterisk superscript refers to the equilibrium value). It is a natural question whether such a macroscopic characterization of configurations also exists for dynamics. We show, through a careful study of the single-environment case for which the dynamics can be studied in great analytical details, that the answer is positive. Two main features emerge in the large system size limit, summarized below and in Fig. 2: (1) The position of the center of the clump (center of mass of the activity), x c (t), plays the role of a collective coordinate for the neural configurations. It undergoes a pure diffusion motion, whose diffusion coefficient is of the order of 1 N . The clump velocity under an external force satisfies the Einstein relation, with a mobility of the order of 1 N . The diffusion coefficient and the mobility depend on the exact shape of the equilibrium density profile, as well as some specific details of the microscopic neural evolution.

(2) In addition to the motion of the center of the clump, the activity profile ρ(x,t) shows fluctuations around the equilibrium profile ρ * [xx c (t)]. Those fluctuations are small, of the order of N -1/2 .

Informally speaking the clump has the status of a quasiparticle. It behaves like a quasirigid body, moving in space, and the only time-dependent and relevant variable to consider is the position of its center, as was already observed in simulations of previous models [9]. The properties above and the calculation of the diffusion and mobility coefficients are presented in Sec. IV.

How does this result extend to the case of multiple environments? We assume that the load and the level of noise are such that the clump is the stable phase of the system. The crosstalk between the environment in which the activity is localized and the other maps encoded in the couplings now hinders the motion of the clump center x c . This effect can be intuitively modeled by the presence of an effective free-energy potential acting on the clump, varying with the center position, x c . We expect that this potential will be random and quenched (independent of time). This phenomenon is illustrated in Fig. 3, which sketches the free energy of the clump as a function of x c . Two important features of this free-energy landscape are the typical height of free-energy fluctuations, F , and the typical space scale over which fluctuations are correlated, b . Those two quantities will be computed in Sec. V. The barrier height F is found to increase as the square root of the number N of cells, which makes the diffusion coefficient vanish as the exponential of minus the square root of N. Hence, diffusion is strongly activated and the clump may remain trapped for 032803-3 a long time at specific space locations when the size of the neural population exceeds a few tens or hundreds, depending on the values of the control parameters. In practice, therefore, diffusion is possible in a small part of the stability region of the clump phase (close to the small α and large T border) only. As expected the maximal size N for which diffusion is possible increases with the fraction of silent cells in each environment (this fraction ranges from 50% to 80% according to experiments [27]).

Diffusion of the clump within one environment coexists with the presence of abrupt transitions from one map to another. In such transitions, the clump of localized activity in the first environment disappears and reforms in another environment, where the activity is now localized, and diffusion can resume. We show some examples of transitions in Sec. V B 2. Disappearance and reformation take place at environment-specific place positions, corresponding to local resemblances of the environments [28]. Small values of N, which favor diffusion, make transitions more likely to occur too. Diffusion within maps and transitions between maps are therefore two competing phenomena, both very important for the mobility of the clump.

The results above were obtained in the absence of any external input. In the presence of an external force the clump may, however, easily move, with a finite velocity. We have investigated the dependence of the velocity on the force value and on the dimension of the space (1 or 2). However, the force cannot exceed a critical value above which the clump disintegrates, and the neural activity ceases to be localized. We estimate the upper bound on the force in Sec. VI. A force can also be used to move the clump towards a specific position in space, to retrieve a particular location. We show in Sec. VI that this mechanism can efficiently drive the clump to the desired position, in a time essentially independent of its initial position in the environment. Larger forces make the retrieval time smaller.

Finally we study several biologically inspired mechanisms, including adaptation and theta-related variations of the activity, with numerical simulations in Sec. VII and show how those mechanisms affect the diffusion properties of the neural clump. Adaptation seems to be particularly effective to avoid trapping in local minima of the free-energy potential.

IV. PURE DIFFUSION: SINGLE-ENVIRONMENT CASE

We start with a detailed study of the single-environment case. Since we have considered regularly spaced place fields, neglecting any noise coming from the learning process, there is no disorder in the connections in this case. We first define the dynamics undergone by the microscopic configurations σ = {σ 1 , . . . ,σ N }, in terms of transition probabilities between nearby configurations. We then show how the center of the clump emerges as a collective coordinate of the neural population. The dynamics can be described as a diffusion for the clump center, accompanied by low-amplitude fluctuations of the clump shape around its equilibrium profile. We then report the results of Monte Carlo simulations, in excellent agreement with the analytical findings.

A. Transition rates for the dynamics of the neural activity configuration σ

The dynamics is defined as follows. We start from a configuration σ of the neural activity, whose corresponding "energy" is defined as

E = - i<j J ij σ i σ j .
(

We then choose (1) a neuron i uniformly at random among the N(1f ) neurons which are silent, i.e., such that σ i = 0, and

a neuron j uniformly at random among the Nf neurons which are active, i.e., such that σ j = 1. Let us define the change in energy, E, when the states of both neurons are flipped, that is, σ i and σ j become, respectively, equal to 1 and 0. A short calculation leads to

E = - k( =i,j ) (J ik -J jk )σ k . ( 6
)
The joint flip of the two spins is accepted with rate (probability per unit of time) ω( E), with detailed balance:

ω( E) ω(-E) = exp(-β E). (7)
A possible choice for the rate function is ω( E) = N exp(-β E/2), or the Metropolis prescription:

ω( E) = N if E < 0, and ω( E) = N exp(-β E) if E 0.
The multiplicative N factor in the rate function ω ensures that the typical time for a round of the dynamical procedure (N joint flip attempts) is independent of the system size, and equal to unity in the infinite size limit. Note that the joint flip allows us to keep the global activity unchanged. The procedure is then iterated (choice of a new couple of spins, acceptance or rejection of the joint flip, and so on). As a consequence of detailed balance and of the obvious irreducibility of the Markov chain the system reaches equilibrium at long times.

B. The clump is an emergent collective "coordinate" of the neural activity

Transition rates for the dynamics of the density ρ

The previous dynamics over neurons defines an effective dynamics for the average density profile over space, ρ = {ρ(x)}. Let us denote by a = i/N and b = j/N the reduced positions of the two spins we attempt to flip. Let also J w (u) = 1 if |u| < w 2 and 0 otherwise. Observe first that the change in energy resulting from a joint flip is, according to ( 6),

E = -dx[J w (a -x) -J w (b -x)]ρ(x), (8)
up to corrections of the order of N -1/2 [the contributions coming from the spins i and j , which are discarded in Eq. ( 6), are of the order of N -1 ]. In the formula above ρ denotes the activity density associated to the configuration σ . A rigorous procedure would require us to bin the activity into boxes of width W , with 1 W N, and send N → ∞ first, W → ∞ next. To lighten notations we omit this binning procedure here.

The joint flip results in a change ρ of the activity density equal to

ρ(x) = 1 N δ(x -a) - 1 N δ(x -b), (9)
and in a change of the free energy [see Eq. ( 11) in Ref. [25]] given by

F = N F[ρ + ρ] -N F[ρ] = δF δρ(a) - δF δρ(b) = -dx [J w (a -x) -J w (b -x)] ρ(x) + T log ρ(a) [1 -ρ(a)] -T log ρ(b) [1 -ρ(b)]
, (10) when N is sent to infinity.

As the probability of choosing a silent spin at reduced position a and an active spin at reduced position b is equal to [1-ρ(a)]ρ (b) f (1-f ) , we may write the rate for the small change ρ → ρ + ρ,

ω(ρ; a,b) = [1 -ρ(a)]ρ(b) f (1 -f ) ω( E) = [1 -ρ(a)]ρ(b) f (1 -f ) × ω -dx [J w (a -x) -J w (b -x)] ρ(x) . ( 11
)
It is a simple check from Eq. ( 10) that the ratio of the forward and backward rates is equal to

ω(ρ; a,b) ω(ρ + ρ; b,a) = exp(-β F [ρ]). ( 12
)
Hence detailed balance is obeyed at the level of activity density profiles ρ.

Fokker-Planck equation for the activity density ρ

Let us call P[ρ,t] the probability density that the average density profile is equal to ρ at time t. Detailed balance condition (12) ensures that, at long times, equilibrium is reached and the activity density converges to its equilibrium value ρ * , as the infinite-size limit suppresses fluctuations. We now propose a heuristic derivation of the Fokker-Planck equation satisfied by P at finite times t. For simplicity we will restrict to a simplified version of this equation, describing the evolution around the equilibrium profile ρ * only.

The essential components of the Fokker-Planck equation are the diffusion tensor, the effective force as a function of the activity density, and the mobility tensor. The diffusion tensor is given by

D(x,y) = ρ(x) ρ(y) = δ(x -y) N f (1 -f ) [1 -ρ * (x)] db ρ * (b) ω * (x,b) + ρ * (x) da [1 -ρ * (a)] ω * (a,x) - 1 N f (1 -f ) [1 -ρ * (x)]ρ * (y) ω * (x,y) + ρ * (x)[1 -ρ * (y)] ω * (y,x) , ( 13
)
where the average is taken over the joint flips a,b with rate ω (11), and

ω * (x,y) ≡ 1 N ω -dz [J w (x -z) -J w (y -z)] ρ * (z) . ( 14
)
Note that ω * is of the order of 1 as ω is of the order of N . We have ρ(x) = 0 for all positions x since fluctuations cancel on average around the equilibrium density ρ * . It is easy to check that D is a real-valued, symmetric, and semidefinite positive operator: N dx dy (x)D(x,y) (y)

= dx dy [1 -ρ * (x)]ρ * (y) f (1 -f ) ω * (x,y)[ (x) -(y)] 2 0. (15)
The only zero mode of D is uniform over space: (x) = 0 . Under the action of diffusion a current of probability J dif [ρ,t] is produced, proportional to the gradient of P[ρ,t] over the density space, and to the diffusion tensor. This current is an infinite-dimensional vector whose component x is given by

J dif [ρ,t](x) = - 1 2 dy D(x,y) δP[ρ,t] δρ(y) . ( 16
)
We now turn to the force acting on the activity density, which we denote by A. The force includes thermodynamic contributions, proportional to minus the gradient of the freeenergy function N F, and external input contributions (to be made more precise in Sec. VI). Under the action of this effective force a velocity v in the activity density space is produced, whose component x at "point" ρ is v[ρ,t](x) = dy μ(x,y) A[ρ,t](y), (17) where μ is the mobility tensor and A[ρ,t] is the force at "point" ρ and time t. The components of the current of probability 032803-5

J force [ρ,t] resulting from the action of the force are

J force [ρ,t](x) = P[ρ,t] v[ρ,t](x). ( 18
)
The corresponding Fokker-Planck equation for P

[ρ,t] reads ∂P[ρ,t] ∂t = -dx δ δρ(x) [J dif [ρ,t](x) + J force [ρ,t](x)]. (19)
We see that

P[ρ] ∝ exp(-NβF[ρ]
) is a stationary solution of the Fokker-Planck equation above with the force given by A(y) = δ(-N F[ρ])/δρ(y), if the mobility tensor is chosen to be

μ(x,y) = β 2 D(x,y), ( 20
)
which is the celebrated Einstein identity.

Quasiparticle description around the equilibrium density ρ * and effective diffusion coefficient

We are now able to write the Langevin equation for the activity density equivalent to the previous Fokker-Planck equation, with the result ∂ρ(x,t) ∂t =dy μ(x,y) δNF[ρ] δρ(y)

+ dy D 1/2 (x,y) η(y,t), ( 21
)
where η is a white noise process, uncorrelated in space and in time, η(y,t) = 0, η(y,t) η(y ,t ) = δ(yy ) δ(tt ), (22) and D 1/2 is the square root of D (in operator terms):

D(x,y) = dz D 1/2 (x,z) D 1/2 (z,y). ( 23
)
Note that the drift term in Eq. ( 21) is of the order of 1 as N 1, while the effective noise term is of the order of N -1/2 . We stress that the Langevin equation ( 21) is expected to be valid for ρ close to ρ * ; far away from ρ * the diffusion tensor would have a different value, as one would need to compute the connected two-point correlation of the activity density fluctuations.

Let us write now ρ = ρ * + , with "small." Then

δβF[ρ] δρ(y) = dy H (y,y ) (y ), (24)
where

H (x,y) = δ 2 βF δρ(x)δρ(y) ρ * = -β J w (x -y) + δ(x -y) ρ * (x)[1 -ρ * (x)] . ( 25
)
The Langevin equation ( 21) reduces to a Ornstein-Uhlenbeck process for , described by

∂ (x,t) ∂t = - N 2 dy dz D(x,y) H (y,z) (z,t) + dy D 1/2 (x,y) η(y,t). ( 26
)
The integral of the right-hand side member above over x vanishes since the constant function 1 is an eigenmode of D and D 1/2 with zero eigenvalue. So dx (x,t) is independent of time, and equal to zero according to the initial condition at time t = 0: the activity is constant, as was expected from the use of joint flips for the elementary moves of the dynamics.

Let us denote by u m (x) and λ m the eigenmodes and the (real-valued) eigenvalues of the operator

N 2 D • H. Then d m dt (t) = -λ m m (t) + ξ m (t), (27)
where ξ m (t) and m (t) denote the components on u m of D 1/2 η(t) and (t), respectively. Note that all eigenvalues are positive as the equilibrium profile of the clump is a minimum of the free energy. We find the following:

(1) For the modes m with λ m > 0:

m (t) = m (0)e -λ m t + t 0 ds ξ m (s)e -λ m (t-s) . ( 28
)
These modes reach equilibrium at long times. More precisely the equilibrium distribution of the coefficient m is asymptotically Gaussian with a variance proportional to the variance of the noise term and to the inverse of λ m . Loosely speaking, those modes are thermalized at very low temperature (of the order of 1/N ) and describe very weak fluctuations around the equilibrium clump shape ρ * .

(2) For the zero mode (associated to λ 0 = 0):

0 (t) = 0 (0) + t 0 ds ξ 0 (s). (29)
This mode freely diffuses with a small diffusion coefficient of the order of 1/N. It is easy to convince oneself that the only zero mode of H, denoted by u 0 , is proportional to the derivative of the equilibrium clump shape:

u 0 (x) = 1
dy dρ * (y) dy 2 dρ * (x) dx . ( 30
)
Indeed, a global translation of the clump by δx does not affect the free energy. As ρ * (x + δx) ρ * (x) + δx dρ * (x) dx we conclude that (30) is the normalized zero mode of H. Note that, in more than one dimension, the derivative of ρ * (x) in Eq. ( 30) must be replaced by the gradient vector with respect to the space coordinates.

Hence, the effective diffusion coefficient characterizing the diffusive motion of the center of the clump is given by

D 0 = u 0 |D|u 0 = 1 N dx dy [1 -ρ * (x)]ρ * (y) f (1 -f ) ω * (x,y)[u 0 (x) -u 0 (y)] 2 . ( 31
)
This prediction is in very good agreement with simulations, as detailed in Sec. IV C.

Effective mobility of the quasiparticle

The velocity v of the density profile in the ρ space in response to an external force A is controlled by the mobility tensor μ; see (17) and (20). Here we derive an explicit expression for the effective velocity of the center of the clump, hereafter denoted by V 0 , as a function of the applied force. We assume that the clump behaves as a quasiparticle, i.e., that the temperature and the applied force are not too large.

The velocity v(x) in Eq. ( 17) can be decomposed as a linear combination of the different eigenmodes u m (x); see Sec. IV B 3. According to the results above all projections on the modes m = 0 will decay exponentially fast to zero. The projection along u 0 (x) is simply related to the velocity V 0 of the center of the clump. Indeed, consider the displacement of the clump during the time δt, from the activity profile ρ(x,0) = ρ * (x) to ρ(x,δt) = ρ * (x -V 0 δt). The velocity of the profile in the ρ space is

v(x) = ρ * (x -V 0 δt) -ρ * (x) δt = -V 0 dρ * (x) dx = -V 0 dy dρ * (y) dy 2 u 0 (x). ( 32
)
Comparing expressions (17), (20), and (32) we deduce the following expression for the effective velocity of the center of the clump:

V 0 = dx μ 0 (x) A(x), (33)
where A(x) is the force acting on position x of the clump, and the component μ 0 (x) of the effective mobility is

μ 0 (x) = -β dy D(x,y) u 0 (y) 2 dy dρ * (y) dy 2 . ( 34
)
Note that the effective mobility is, as the effective diffusion coefficient, of the order of 1/N. This theoretical prediction will be shown to be in very good agreement with simulations in Sec. VI.

C. Numerical simulations

We now report Monte Carlo simulations done with the Metropolis prescription above and in the region of stability of the clump phase. In this section we consider only the motion in the absence of an external force; the case of an input is considered in Sec. VI.

We observe that the stochastic evolution of neural units at the microscopic level results in a macroscopic erratic motion of the clump, in both one and two dimensions. To characterize this motion we compute the position of the clump center from the coarse-grained activity of the network. Space is binned into boxes of size approximatively equal to the clump width. We look for the box where the activity is maximal at time t and compare it to the box of maximal activity at time t -1, taking into account periodic boundary conditions. This provides us with the displacement of the clump between times t -1 and t. The position of the clump is obtained by adding those displacements over time. Two examples of trajectories are shown in Figs. 4 and5.

Method for estimating the diffusion coefficient

We assume that the trajectories of the clump correspond to realizations of a diffusion process with diffusion constant D. We want to infer D from the t M measured displacements { x t } t=1,...,t M . Bayes's formula gives the posterior distribution for D:

P (D|{ x t }) = P ({ x t }|D) P 0 (D) P ({ x t }) . ( 35
)
We choose a flat prior over the diffusion coefficients: P 0 (D) = (D) (Heaviside step function). The likelihood of the trajectories given D is

P ({ x t }|D) = t M t=1 1 √ 2πD exp - x 2 t 2D , ( 36
)
where we have fixed the time interval between two successive measured positions to unity. The denominator in Eq. ( 35) is a normalization factor. Maximization of P (D|{ x t }) with respect to D in Eq. ( 35) gives the most likely value for D, here denoted D * :

D * = 1 t M t M t=1 x 2 t , (37)
and the standard deviation of D with posterior distribution (35) is about δ = D * / √ t M .

Corrections of systematic errors due to binning

The exact position of the center of the clump of activity is not well defined in simulations. As explained above, we therefore bin space into boxes of length a roughly equal to the width of the clump, and estimate the diffusion coefficient through

D mes ≡ 1 t M t M t=1 (a t ) 2 , ( 38
)
with t = 0,±1,±2, . . . denoting the change in the box number between times t -1 and t.

We now want to estimate the error on the estimate of the diffusion coefficient due to binning. Let us consider a pure diffusion process with coefficient D in 1D continuous space x. The trajectory is observed during t M steps, and D mes is estimated according to (38). During a unit time interval t → t + 1 the continuous walker has moved by a quantity z t , which is a Gaussian random variable with zero mean, and standard deviation equal to √ D. We generically note by k the integer part of the ratio of z t over a, and u the remainder of the division, i.e., z t = k a + u. We need to relate t to z t , that is, to k and u.

For simplicity, we consider that, up to time t = t 1 ≡ a 2

4D

(diffusion time in a box), the displacement is counted from the middle of a box, while, for larger times t, the clump position is uniform at random in the box. (This approximation is not valid when D is too small, typically D 10 -5 : in simulations, we therefore have to adapt the length of one round in order to avoid low D effects when applying the correction.) It is then easy to show that, for t > t 1 , t = k with probability 1 -u a and t = k + 1 with probability u a . We conclude that the estimate of the diffusion coefficient is on average

D mes = a 2 t M t 1 +∞ k=-∞ a 2 -a 2 du √ 2πD e -(ka+u) 2 /(2D) k 2 + (t M -t 1 ) +∞ k=-∞ a 0 du √ 2πD e -(ka+u) 2 /(2D) × k 2 1 - u a + (k + 1) 2 u a . ( 39
)
The formula above gives the estimated D mes as a function of the "true" diffusion coefficient D. In practice, for each D * estimated according to (38) we numerically solve D mes (D) = D * . The same reasoning in two dimensions leads to a similar result (with a multiplicative factor 2 because we bin both the x and the y axes).

Statistical error bars

Once the individual values {D * n } n=1...N sim measured in N sim simulations have thus been corrected, we estimate the diffusion coefficient D as their average: FIG. 6. Diffusion of the clump in the single-environment case (α = 0) and 1D space. The theoretical prediction for the diffusion constant, D 0 , given by Eq. (31), is plotted as a function of 1 N for T = 0.005 (full lines) and T = 0.006 (dashed lines) and compared to the results of Monte Carlo simulations D sim (after correction of the binning effect). The agreement with the analytical prediction (done in the N → ∞ limit) improves as N increases. This also explains why the discrepancy is larger than error bars for smaller N . Therefore, simulations corroborate well the theoretical analysis, and the diffusion properties of the clump can be understood analytically in the single-environment case. Simulation time: 1000 rounds of 100N steps. Depending on the computational cost, each point is averaged over a number of simulations ranging from 5 (for large N ) to 100.

D = 1 N sim N sim n=1 D * n . ( 40
)
The error bars on the inferred D must take into account two sources of uncertainty: the width δ n = D * n / √ (t M ) n of the distribution of each D * n due to the randomness in the Monte Carlo process, and the standard deviation δ of the diffusion coefficients due to the random realization of the maps in each simulation. In practice, for the long MC runs, we consider that the former error is negligible compared to the latter. We therefore estimate the error bar on D through

δ = 1 N sim N sim n=1 ((D * n ) 2 -D 2 ). (41)
We compare the value of D to the theoretical prediction D 0 given by (31). The results in dimension 1 are plotted in Fig. 6 and show that the agreement is very good. The prediction gets better and better as N increases: indeed, it is valid in the large N limit.

V. ACTIVATED DIFFUSION: MULTIPLE-ENVIRONMENT CASE

In the presence of multiple environments the motion of the clump within the retrieved environment is not purely diffusive any longer. The crosstalk between the stored maps indeed creates an effective (free-energy) potential for the clump, which is not uniform over the space, as sketched in Fig. 3. In this section we first compute the typical height F of the barriers in this potential and their typical width b . We then show results of simulations and address the issue of partial activity of place cells.

A. Characterization of free-energy barriers

Barrier heights

In the presence of disorder, the distribution of the free energy F J = -T log Z J (T ) is centered around its typical value, with a nonzero width for finite size N. To compute this width, we use the replica method. Expanding the nth moment of the partition function, Z J (T ) n , in cumulants of F J we write

Z J (T ) n = exp(-n β F J ) = exp -n β F J + 1 2 n 2 β 2 F 2 J -F J 2 + • • • . (42)
Hence, the variance of F J can be computed from the knowledge of the second derivative of Z J (T ) n in n = 0:

F 2 J -F J 2 = ∂ 2 ∂n 2 n→0 T 2 log Z J (T ) n . ( 43
)
The calculation of this second derivative is reported in Appendix B, with the result

F 2 J -F J 2 = V (α,T ) N, ( 44
)
where

V (α,T ) = -α r q + αT 2 (q -f 2 ) 2 ϕ(q,T ) + T 2 dx Dz log 2 (1 + e βz √ αr+βμ(x) ) -T 2 dx Dz log(1 + e βz √ αr+βμ(x) ) 2 . ( 45
)
In the formula above, μ(x) is the field conjugated to the average density ρ(x) (not to be confused with the mobility tensor μ introduced above), and r is the conjugated force to q (see Appendix A); Dz = dz √ 2π exp(-z 2 /2) denotes the Gaussian measure. The function ϕ(q,T ) is given by

ϕ 1D = k 1 T πk sin(πkw) + q -f -2 (46)
in dimension 1, and by

ϕ 2D = 2 (k 1 ,k 2 ) =(0,0) T π 2 k 1 k 2 sin(π k 1 √ w) sin(π k 2 √ w) + q -f -2 (47)
in dimension 2. The typical barrier height, F , is given by the standard deviation of the free energy: F = √ N √ V from Eq. ( 44). We have computed V for different values of α,T and verified that it is a definite positive quantity. We plot in Fig. 7 the barrier height F , after division by √ N, as a function of the load α. We see that F increases very quickly with the load for small α and reaches a maximal value close to the stability boundary of the clump phase.

To gain some intuition on the barriers heights we look for a simple estimate of the standard deviation E of the energy E = -i<j J ij σ i σ j . To do so, we keep the spin configuration fixed and compute the variations due to the stochastic coupling matrix J , with the result

E ∼ f (1 -f ) α w(1 -w) 2 √ N, ( 48
)
to dominant order in N . Numerically, we find that E in the formula above takes values close to F . Hence, the much simpler formula for E offers some insight on the order of magnitude of the barriers, as well as on their dependence on the model parameters.

As the barrier heights against diffusion scale as √ N we can plot in the phase diagram the contour lines of different crossover sizes N c , corresponding to barrier heights such that β F = 1. The crossover size N c is thus defined through

N c = 1 β 2 V (α,T ) . ( 49
)
The outcome is shown in Fig. 8. In dimension 1 we can estimate that diffusion will be approximatively free for N < N c . For N > N c barriers cannot be neglected, and diffusion is activated. We see that, except in a narrow region of the phase diagram, the clump cannot freely diffuse for realistic values of N (of the order of thousands). In dimension 2, this argument is not true anymore because barriers can be bypassed. Nevertheless, simulations show that diffusion is quite limited also in that case, albeit to a lesser extent (see Sec. VI). Furthermore, in both one and two dimensions, in the low α-high T region where diffusion can occur, we observe in simulations that this process is in competition with transitions between environments (see Sec. V B 2).

Barrier widths

In order to estimate the typical width b of the barriers depicted in Fig. 3, we calculate the correlation between the free energies [denoted F J (x) and F J (y)] of the clump centered, respectively, on two positions x and y of space, that is, cov(F J (x),F J (y)) ≡ F J (x)F J (y) -F J (x) F J (y). [START_REF] Fyhn | Hippocampal remapping and grid realignment in entorhinal cortex[END_REF] This quantity can be derived using the replica method. We split the n replicas in two groups: the first n 2 replicas have an activity profile centered in x, while the remaining n 2 replicas have an activity profile centered in y. All n replicas share the same interaction matrix J and are coupled once these quenched couplings are averaged out. The resulting partition function for the n-replica system reads

Z(n,x,y) = exp - n 2 β [F J (x) + F J (y)] . ( 51
)
Similarly to the calculation above, by expanding in cumulants and taking the second derivative of Z(n,x,y) in n = 0,

∂ 2 ∂n 2 n→0 T 2 log Z(n,x,y) = N 2 [V + W (x,y)] , (52)
where

W (x,y) ≡ 1 N cov[F J (x),F J (y)]. ( 53
)
V was defined in Eqs. ( 44) and ( 45), and we use that, by translational invariance, the average of F J (x) over J does not depend on the position x. By translational invariance again, W (x,y) depends only on the distance xy and is equal to W (xy).

The calculation of Z(n,x,y) is detailed in Appendix B. We denote q 12 the overlap between two replicas, respectively, belonging to the group with a clump in x and the group with a clump in y. The outcome is

W (x -y) = -α r 12 q 12 + αT 2 (q 12 -f 2 ) 2 ϕ(q,T ) -T 2 dx Du log(1 + e β √ αru+βμ(x ) ) × Dv log(1 + e β √ αrv+βμ(x -x+y) ) -DuDv κ(u,v) log(1 + e β √ α(r-r 12 )u+βμ(x ) ) × log(1 + e β √ α(r+r 12 )v+βμ(x -x+y) ) (54)
where

κ(u,v) = exp ⎡ ⎣ r 12 2 ⎛ ⎝ u 2 r + r 12 - v 2 r -r 12 + 2uv r 2 -r 2 12 ⎞ ⎠ ⎤ ⎦ (55)
and

q 12 = dx DuDv κ(u,v)/[1 + e -βu √ α(r-r 12 )-βμ(x ) ] /[1 + e -βv √ α(r+r 12 )-βμ(x -x+y) ]. ( 56
)
The conjugated parameter is r 12 = 2T 2 (q 12f 2 )ϕ(q,T ). Parameters q,r,μ(x) are found from the extremization of the free energy given in Appendix A. We observe that W (xy) is of the order of V on a distance xy equal to the typical size of the clump and sharply decreases at larger distances (Fig. 9). Therefore, the typical width of the barriers b is comparable to the size of the clump. A more quantitative comparison is obtained from the following quantities (computed for the parameters of Fig. 9): dx x W (x)/ dx W (x) = 0.057 and dx x ρ(x)/ dx ρ(x) = 0.082 in one dimension, and dx x W (x)/ dx W (x) = 0.088 and dx x ρ(x)/ dx ρ(x) = 0.097 in two dimensions. The overlap q 12 decreases on a similar typical distance; see Fig. 27 in Appendix B.

B. Numerical simulations

Activated diffusion

We ran Monte Carlo simulations of the model with multiple environments and measured the quantity D defined above (Sec. IV C). Results are plotted in Fig. 10. In agreement with the predictions above, we observe that the clump is trapped as soon as N exceeds a few hundreds or when T is too low or α too high. We nevertheless note that D is in general higher in two dimensions than in one: this effect will be discussed later (see Sec. VI). Interestingly, the crossover size N c [START_REF] Leutgeb | Pattern separation in the dentate gyrus and ca3 of the hippocampus[END_REF] is very robust to changes in parameters. Figure 11 shows that the constant-N c lines remain qualitatively unchanged with respect to the clump stability region as f and w vary, while the absolute location of the stability region in the (α,T ) plane varies; see Ref. [25].

Estimating the diffusion coefficient would require simulations long enough to allow the clump to move on distances larger than the environment size. The occurrence of transitions to other environments forbids such long simulation times for most parameter values (Fig. 10). As a consequence, the displacement of the clump during our simulations is generally smaller than the environment size. The values of D we measure are therefore indicative of the motion of the clump on a limited timescale and allow us to study the influence of parameters, e.g., the size N in Fig. 10, on this motion. Note that, in two dimensions, diffusion is easier, and the simulation times required to explore the environment are smaller.

It is interesting to notice that, due to disorder effects, the diffusion constant for a same set of stored environments varies with the environment the clump of activity is coherent with. In other words, in each attractor (stored map), the clump phase has a different diffusion dynamics. For some maps diffusion is relatively "easy," while the clump will remain trapped for very long times and hardly diffuse in other maps. This phenomenon is illustrated in Table I.

Transitions to other environments

Abrupt jumps between maps are often observed in Monte Carlo simulations with several environments. A detailed study of those transitions is postponed to a companion paper; hereafter we limit ourselves to briefly report the salient features of transitions, which are of interest to the dynamics of activity within one map studied in the present paper. An example of transitions is shown in Fig. 12. We observe that the activity configuration goes from being localized in the first environment (clump state) to being localized in the second environment, through an intermediary state which is weakly localized in both environments. This can be seen directly on the microscopic configuration σ , or, alternatively, by looking TABLE I. The diffusion constant D ( ) differs significantly from an environment to another within a same given system (set of couplings created from the L + 1 environments). The table shows the results obtained for one set of simulations with N = 1000 neurons, L + 1 = 4 randomly drawn environments, and T = 0.005. Each value is averaged over 100 simulations of 1000 rounds, initialized at different positions of space. The variations of D ( ) from environment to environment is larger than error bars. at the contributions of both environments to the log. probability P J (σ ) of the neural configuration, as shown in Fig. 13.

Transitions are less and less frequent as N increases. The decrease of rate of transitions with N is shown in Fig. 14. An important consequence is that the presence of transitions is in competition with diffusion. As N decreases the motion of the clump is facilitated, but so are transitions to other environments. We observe the existence of preferred "tunneling" locations, where map-to-map transitions are likely to take place. As transitions are made possible by the existence of intermediary activity configurations where the activity is partially localized in both maps, it is natural to expect that those preferred positions correspond to sites of local resemblance between the random permutations defining the maps. Such a similarity in the permutations at places where transitions happen most often is indeed observed [28]. A detailed study of those properties will be reported in a forthcoming publication.

C. Effects of partial activity

The study above can be repeated under the more realistic assumption that there exist many "silent" place cells, in the sense that only a fraction c (<1) of the neurons have place fields Evolution of E ≡ i<j J ij σ i σ j , for the same transition event as in Fig. 12. E is the contribution of environment to the logarithm of the probability of the neural configuration σ ; see (2). The crossing of E 1 and E 2 defines the transition between the two maps, as well as the intermediary state, where the activity is weakly localized in both maps. in a given environment. For instance, in one dimension, the variance V of the free energy, given by [START_REF] Park | Ensemble place codes in hippocampus: Ca1, ca3, and dentate gyrus place cells have multiple place fields in large environments[END_REF] in the case c = 1, becomes (see Ref. [25] for details about the c < 1 calculations) where

V c = -αrq + αT 2 c 2 (q -f 2 ) 2 ϕ c (q,T ) + T 2 c dxDz log 2 (1 + e βz
ϕ c (q,T ) = k 1 T πk sin(πkw) + c(q -f ) -2 (58)
and μ 2 is such that Dz[1 + e -βz √ αr-βμ 2 ] -1 = f . Having c < 1 quantitatively changes the stability region of the clump phase but does not have any qualitative effect on the static properties of the system [25]. Here we look at the effect of partial activity on the diffusion. Interestingly, it turns out that again the location of the contour lines for N c with respect to the stability domain of the clump phase remains essentially unchanged with c. This robustness phenomenon is illustrated in Fig. 15.

As a consequence, for given α,T , decreasing c, i.e., increasing the sparsity of the representation will have the effect of increasing the diffusion constant, mostly because the neural noise is relatively more important. The rate of transitions to other environments increases too. When c becomes too low, the clump is not stable anymore and disappears. Simulations are in good agreement with this prediction, as shown in Fig. 16. In dimension 2 the behavior with decreasing c is the same; see Fig. 17.

VI. MOTION UNDER AN EXTERNAL FORCE

We now investigate the motion of the clump under an external input.

A. Drift under an external force

We consider the behavior of the model when the environment is "tilted," i.e., when a force is applied to make the clump move in a given direction. In the absence of disorder in the interactions (single-environment case) the force is expected to move the clump with a positive and constant velocity. In the presence of disorder, the wrinkled energy landscape combined to the tilt will pin the activity. The motion will be strongly activated, with the clump trapped in minima most of the time, until the force exceeds some critical threshold, above which the clump will acquire a positive velocity.

This scenario is corroborated by simulations. We model the presence of a force through an increase of the probability of the two-neuron flip σ i = 1,σ j = 0 → σ i = 0,σ j = 1 with respect to σ i = 0,σ j = 1 → σ i = 1,σ j = 0, for i < j (1D case). This creates a bias in favor of motion to the right. More precisely, the Metropolis rate defined in Sec. IV A is modified as follows:

ω( E) = Ne -β( E-A x c ) if E A x c = N if E < A x c , (59)
where

x c = j -i + (i,j )N f N 2 (60)
is the displacement of the center of gravity of the clump when neuron i goes from being active to silent, and neuron j goes from being silent to active; (i,j ) ∈ {-1,0,+1} enforces periodic boundary condition. Parameter A denotes the intensity of the applied force.

Critical values of the force

Using the estimates F and b for, respectively, the height and the width of the free-energy barriers derived in Sec. V A, we evaluate the critical intensity A depin of the force above which the clump can overcome barriers. A rough estimate of this depinning force is obtained by imposing that the work of the force in moving the clump through the barrier, A depin × b , compensates the barrier height, F = √ V (α,T )N [START_REF] Leutgeb | Distinct ensemble codes in hippocampal areas ca3 and ca1[END_REF]. We obtain the typical value

A typ depin √ V (α,T )N b . (61)
Drift is mostly hindered by the highest barriers. The maximal height can be estimated by considering that barriers heights are Gaussian variables, drawn independently and at random for each one of the 1/ b segments of length b . Hence, according to extreme value theory, the maximal barrier heights is about √ 2 log(1/ b ) times the typical value computed above:

A max depin 2 log 1 b √ V (α,T )N b . ( 62
)
As the force is applied at the microscopic level on the neuron states, and not at the macroscopic scale on the clump itself, taking A too large will make the clump disintegrate. This will happen if the work of the force exceeds the cohesion energy of the clump. We estimate the critical intensity A break based on the following reasoning. Silencing a neuron within the clump and activating another neuron outside the clump costs on average (for the 1D case)

E μ inside -μ outside = 1 b |x-x c |< b /2 dx μ(x) - 1 1 -b |x-x c |> b /2 dx μ(x). ( 63
)
This energy cost is decreased by the work of the force, A x c , where x c is the change in the average position of the clump following a microscopic flip of two neuron states; see [START_REF] Wood | The global record of memory in hippocampal neuronal activity[END_REF]. The most favorable case, corresponding to the largest shift of the clump center, is x max = 1/(2f N). We conclude that the cost decreases linearly with A (and can even become negative at large A), leading to the breaking apart of the FIG. 18. Mobility of the clump in response to an external force, in the single-environment (α = 0), 1D case. The theoretical prediction for the effective mobility, μ 0,th , computed from Eq. ( 67), is plotted as a function of 1 N for T = 0.005 (dashed lines) and T = 0.006 (full lines) and compared to the results of Monte Carlo simulations μ 0sim . The agreement with the analytical prediction [which neglects terms smaller than O( 1N )] improves as N increases. Each point is averaged over 10 simulations, in which the clump, initially at location x = 0, had moved over four space bins (the environment is covered by 11 bins). localized collective activity. An estimate of the critical force at which this happens can be obtained from the comparison of the cost with the temperature of stability of the clump at zero force, T CL ; see Sec. II and Ref. [25]. We expect

E -A break x max T E T CL , (64)
or, equivalently,

A break 2 f N E 1 - T T CL . ( 65
)

Simulations

First, we tested the theoretical prediction (34) for the effective mobility of the quasiparticle in the 1D, singleenvironment case. We ran simulations for different values of N and A and measured the velocity of the center of the clump. Taking A(x) = -A x in Eq. (33) gives

V 0 = μ 0,th A, ( 66
)
where

μ 0,th ≡ β dx dy x D(x,y) u 0 (y) 2 dy dρ * (y) dy 2 (67)
is the predicted mobility of the clump. As expected, for a fixed number N of cells, the velocity increases linearly with A (up to A break ). The slope of this curve is our numerical estimate μ 0,sim for the mobility of the clump. This measure of the mobility is in very good agreement with theory, as shown in Fig. 18.

In one dimension, the pinning effect due to the environments other than the one in which the activity is localized is observed in simulations. An example is shown in Fig. 19 for one realization of the disorder. For the parameters values of the simulation of Fig. 19, we find, according to ( 61) and ( 62), depin as expected. In addition, note that the depinning force, A sim depin , is found to fluctuate from realization to realization, while our theoretical estimate is sample independent.

We also estimate the force at which the clump disintegrates, under the simulation conditions seen in Fig. 19. We find A break 1.82, in excellent agreement with the results of simulations, A sim break 1.8. In two dimensions, contrary to the 1D case, free-energy barriers can be bypassed. Drift can occur even with forces that are not strong enough to cross the barriers, and the value of A depin given above is not relevant. Simulations indeed show that the pinning of the clump is much weaker than in one dimension; see Fig. 19. This is an important point, which shows that the dynamics of the clump within one map strongly differ in the 1D and 2D cases. In two dimensions, contrary to one dimension, barriers can be bypassed by the clump trajectories. This phenomenon could explain the fact that the diffusion constants measured in 2D simulations are in general larger than their 1D counterparts (Fig. 10). We checked the existence of this bypassing mechanism by looking at trajectories of the clump in the (x,y) plane when an external force is applied along the x axis (Fig. 20, top). We observe displacements along the y axis, with preferred values for y, indicating that the overall rightward motion is the result of the clump motion around the barriers, instead of crossing them. We looked at the time spent in each position of the unit square (Fig. 20,bottom). Favored positions clearly appear, where the total time spent is several orders of magnitude greater than in other positions. The opposite of the logarithm of these residence times is an estimate of the free-energy landscape probed by the moving clump.

In experiments, place fields have been studied in both 1D and 2D environments ("one" referring to a linear track whose width is small compared to the length), but the 2D case is obviously of particular importance for natural environments.

B. Retrieval

In Hopfield's original model for attractor neural networks (ANN), a memory item corresponds to one activity configuration of the network. The retrieval phase consists in stabilizing the network activity in this configuration, starting from a different initial configuration. In contrast, in our ANN model for the hippocampus, a memory item corresponds to a map, i.e., a whole set of activity configurations corresponding to clumps centered around positions along the map. What does retrieval mean in this case? Two views are possible. First, it is of course possible to retrieve (in Hopfield's model sense) one particular activity pattern starting from a similar configuration, that is, a clump centered on one particular position in one particular environment. This retrieval mechanism, requiring a specific input, will be addressed in Sec. VI B 1. Second, one can focus on the broader issue of map retrieval. In this case one map would be retrieved, if the activity is coherent (localized) in the map, while the clump is free to wander in the environment; see Sec. VI B 2.

Retrieval of one position in a given environment

We investigate the dynamics of the model when one given position in a given environment is selected by a local field. The pattern to be retrieved is an activity configuration ξ corresponding to a clump centered on, say, position x 0 in environment 0. A local field h i is applied on the spins:

h i = h if i N -x 0 d 0 , 0 otherwise. (68)
Retrieval is detected by the measure of the overlap

m ≡ 1 f N i σ i ξ i . ( 69
)
An example of the retrieval process is given in Fig. 21: it occurs abruptly, as a global switching of the network activity to a configuration close to ξ . As expected, the time taken for retrieval is a decreasing function of h and d 0 (Fig. 22). It does not depend significantly on the initial conditions of the network.

Retrieval of one environment

In order to stabilize one particular map, e.g., of index , we ran simulations in which we increased the contribution J to the total synaptic matrix J . This artificial modification does not correspond to any physiological mechanism per se but could mimic the effect of a "context dependence" [29]. The synaptic matrix is modified as followed:

J ij → J ij + h J • J ij , ( 70
)
where h J > 0 and J ij = J 0 π (i)π (j ) (Sec. II). As expected, the time taken for retrieval is a decreasing function of h J (Fig. 23). Note that the retrieval is almost immediate as soon as the additional weight on the environment exceeds 10%. Interestingly, the retrieval is slightly slowed down if the initial state of the system is a clump in another environment, rather than a paramagnetic configuration. The global input is then in competition with the barriers opposing transitions between environments.

VII. EFFECTS OF OUT-OF-EQUILIBRIUM MECHANISMS ON CLUMP MOTION

A. Adaptation

An important biophysical process, which can be incorporated into the model, is spike frequency adaptation. The membrane voltage of frequently active neurons is hyperpolarized by potassium currents, and their firing rates decay to submaximal levels. Adaptation has been observed in hippocampal pyramidal cells [30]. This neural fatigue phenomenon has been proposed as a mechanism to make the clump, otherwise stationary, diffuse in the environment in the absence of external input (mental exploration) [26].

We introduce a mechanism for adaptation in the simulations, to see if it enhances the diffusion process as expected. At the cell level, adaptation can be modeled as an auto-inhibitory current that relaxes with a time constant τ adapt [26]. This autoinhibition was taken into account in the simulations by adding a local field on each spin whose value depends on the spin's past activity. The system is now out of equilibrium, but the time constant τ adapt is chosen to be large compared to thermalization times so that the fields vary slowly. More precisely, we add

h i (t) ≡ h adapt τ 0 e -τ/τ adapt σ i (t -τ ), (71)
where h adapt measures the intensity of the neural fatigue. We ran simulations with various time constants τ adapt and intensities h adapt . We used D defined in Sec. IV C as a measure of the clump square displacement per unit of time. Note that D does not correspond strictly speaking to a diffusion coefficient any more. As expected intuitively, we observe that increasing h adapt facilitates the motion of the clump (Fig. 24, top), but also tends to destabilize it. Transitions to other environments are more frequent (Fig. 24, bottom) as h adapt increases, and if h adapt is too large, the clump breaks apart.

These results support a recent work by Hopfield [26], according to which adaptation (and not the sole neural noise) could be the neural mechanism by which a bump of activity dynamically explores a continuous attractor manifold in the absence of visual or self-motion input. Such a spontaneous motion at the level of the neural activity, taking place without the animal's physically moving, appears useful in the realization of mental exploration tasks such as future trajectories planning or past trajectories remembering. These results also reveal the increasing occurrence of transitions between environments when out-of-equilibrium mechanisms are added to the model and stress the importance of this phenomenon in competition with clump motion within one map.

B. Fluctuations in the global inhibition

In our model the effect of inhibitory cells is modeled as a constant activity level f of pyramidal cells. However, in hippocampal recordings in rodents this level varies periodically across time, a phenomenon called theta rhythm [31]. These oscillations play a role in the position coding through the phase precession phenomenon [32,33] and have been proposed as a possible mechanism for resetting of the path integrator [9,34].

Here we address the issue of the effect of theta waves on the diffusing behavior of the clump. We know that changing f quantitatively changes the stability domain of the clump phase and correspondingly moves the N c contour lines. As a consequence, varying f at a given (α,T ) will have the effect of varying the diffusion constant, but in any case this constant remains quite low in the whole stability domain of the clump. The clump is not stable for stronger δf . For τ = 10 and τ = 100, each point is averaged over 100 simulations of length varying from a few tens of rounds to 1000 rounds depending on the frequency of transitions. For τ = 1000, longer simulations were necessary in order to cover several periods of f (t); each point is thus averaged over 10 simulations of duration up to 25 000 rounds.

So we do not expect the variations of f to improve dramatically the diffusion process.

We simulated the network at a given (α,T ) and activity level f (t) = f + δf sin(t/τ ) where δf is chosen small enough so that the clump phase remains stable at this (α,T ) and τ is large compared to the simulation unit time. As expected, there is no significant improvement of diffusion; see Fig. 25.

C. Asymmetric synapses

In the Hopfield model [15], couplings are given by Hebb's rule and are therefore symmetric. Our synaptic matrix (1) also follows a Hebbian prescription. Working with symmetric couplings ensures the existence of an equilibrium Gibbs measure over configurations [14], allowing us to use statistical mechanics tools in this framework. Nevertheless, in biological neural networks asymmetric synaptic plasticity exists [35]. In 1D environments for instance, where most place fields are directional [36], asymmetric learning may take place. In addition, in certain models of the hippocampus, asymmetric synapses have been proposed to play a critical role in some observed phenomena such as phase precession [37]. Attractor neural networks with asymmetric synapses and their storage capacity have been formally studied in Ref. [38].

To study the effect of asymmetric synapses on the dynamics of our model we randomly remove a fraction of the couplings J ij [14,38]. More precisely, if δ dil denotes the dilution fraction, for each i < j we choose

J ij → 0 J ji unchanged with probability δ dil 2 , J ij unchanged J ji → 0 with probability δ dil 2 , ( 72
)
J ij ,J ji unchanged with probability 1δ dil . We measured D defined in Sec. IV C, with the results shown in Fig. 26. We observe that the asymmetric dilution of synapses increases D. Nevertheless, because of the concomitant destabilization of the clump, the enhancement of D is here again in competition with more frequent transitions to other environments.

VIII. CONCLUSION A. Summary of results

In this work we have presented analytical and numerical results on the dynamics of a model for hippocampal place cells. Under certain conditions of noise and load, the activity is spatially localized in one of the stored environments (clump phase) [25]. Here we have focused on the motion of such a clump across space within one environment, under the influence of neural noise and of quenched disorder due to the other maps contributing to the couplings. In other words, we have studied the dynamics of an attractor neural network storing spatial maps within one of its attractors, with or without external input.

We have first addressed the issue of the macroscopic description of the clump. At equilibrium, the clump shape is described by the average density profile ρ(x). Here we have analytically shown, in the single-environment case, that a macroscopic description of its dynamical evolution within one map was also possible. More precisely the microscopic dynamics of the individual neurons produces an emergent, collective macroscopic motion of diffusion for the clump. The clump therefore acquires the status of a quasiparticle, with very weak fluctuations (for large sizes N) of shape, while moving in space. It is legitimate to say that the position of the center of the clump plays the role of a collective coordinate for the neural configurations. In their model of the hippocampus, Samsonovich and McNaughton [9] had already described the evolution of the clump by a collective coordinate that emerged from the microscopic dynamics in simulations, but the equivalence between both levels of description was not formally justified. Here, we have analytically demonstrated its soundness. We have, in addition, obtained an exact expression for the diffusion coefficient of the clump and its effective mobility as a function of the detailed dynamical rates of the single neurons used in the Monte Carlo simulations.

We have also considered the dynamical properties of the model in the presence of the quenched disorder caused by multiple-environment storage in the synapses. In this case, the free-energy landscape probed by the clump moving through space is rough. As soon as the number of units exceeds a few hundreds or even tens, the diffusion of the clump appears to be severely hindered by the free-energy barriers, especially in one dimension. This effect, predicted by the analytical study of the statistics of the free-energy landscape, is corroborated by Monte Carlo simulations. It is found to be very robust to changes in the parameters f , w, c. Therefore, noise alone is not enough for an efficient motion of the clump, and additional mechanisms must be taken into account. This point had already been underlined by Hopfield in a recent model for mental exploration in the hippocampus [26]. It is also related to the clustering effect predicted by Tsodyks and Sejnowski [8], who numerically observed that the presence of disorder in connections tends to make stable bumps collapse into positions corresponding to "places where the synaptic interaction between neurons is strongest," i.e., local minima of the energy. Interestingly, in the 2D case, the possibility of trajectories bypassing the freeenergy barriers leads to a larger coefficient constant than in one dimension. This effect is of particular relevance for biological cognitive maps, often thought to be 2D. Moreover, the crosstalk between environments also causes transitions from one map to the other, in competition with motion within one map.

We have then investigated the effect of a force on the network and have showed that a force could, indeed, help the clump overcome free-energy barriers and move across space. This setup allowed us to exhibit the bypassing of barriers in two dimensions.

The motion of the clump can also be enhanced by outof-equilibrium mechanisms. We have modified the model in order to incorporate spike-frequency adaptation, asymmetry in the synapses, and temporal fluctuations in the level of inhibition. For all mechanisms but the latter, motion is found to be facilitated.

B. Biological relevance

In order to perform exact, analytical calculations, and to reach a more controlled and accurate understanding of the phenomena at work than with simulations, we intentionally discard many biological features, of various degrees of importance, in our modeling.

We assume first that the learning process is complete (synapses are frozen) and perfect (the J ij perfectly reflect the topology of the environments, without distortion). In addition each new environment contributes additively to the synapses (Hebb's law). The separation of the learning and the retrieval processes is a common assumption. Quenched distortions in the synapses could be incorporated in the study, e.g., by making the matrix J 0 random rather than perfectly regular on a grid. We expect quenched distortions to have similar effects to the quenched interference noise coming from multiple map storage. Hebb's rule is also a common assumption; it has been shown, in the context of Hopfield's model, that the attractor dynamics is qualitatively robust against the choice of alternative, nonadditive rules [14]. We discussed the case of asymmetric synapses in Sec. VII C.

Another simplification of the present model is to assume that synaptic interactions code for the topology of the environments, i.e., spatial information only. We discard any additional "dimension," such as context dependence [29], as is the case in most models of place cells. Relaxing this assumption in a meaningful way is a tantalizing task in the absence of a clear experimental guidance. We have also assumed nondirectional place fields, in contrast with experimental observations (mostly in one dimension). Directionality could easily be incorporated in the model, and we do not expect it to have a significant effect on most of our results.

The effect of interneurons is modeled through a spatially homogenous inhibition, which maintains the global level of activity (fraction of active neurons) constant. We ignore spatial inhomogeneities in the inhibitory network, as well as fluctuations in the activity level, such as the theta rhythm. In Sec. VII B we have relaxed the latter hypothesis in simulations and have observed that a varying level of activity had no significant effect on the motility of the clump. Nevertheless fluctuations in the activity could have consequences on other phenomena, such as the transitions between maps [34].

Modeling neurons through binary units is also a big simplification. Realistic conductance-based models would be necessary to describe the dynamics of neurons in a accurate way from the biological point of view. However, such detailed models are intractable in the case of large networks. A majority of works on continuous attractor neural networks make use of rate models [8,10,12,39]. Here we choose to use binary units, as discussed in our previous study [25]. The use of binary units allows us to incorporate the noise in the neural response at the time scale of a spike, while rate variables usually represent the activity of neurons averaged over time, or over a population of neurons. In this respect, the binary description can be considered as more microscopic than rate-based models. Indeed, the rate-based macroscopic 032803-19 description naturally emerges in our calculation through the order parameters ρ(x) and μ(x); see also Sec. IIC in Ref. [25]. Our study therefore offers a microscopic basis for rate-based equations and for the properties of continuous attractors; see, for instance, the detailed description of the collective motion of the clump from the microscopic dynamical rules of individual neurons.

A drastic simplification in the present work is the absence of any input. Inputs, be they sensorial or the result of path integration, are indeed believed to be very important in biologically plausible situations. Yet, our work aims at studying the attractor dynamics. In this context, it is important to understand the spontaneous evolution of the network before taking any external input into consideration. Moreover, the precise form of the inputs to hippocampus, their timing, and their intensities are poorly known, which makes their effect on the hippocampal activity hard to model from a quantitative point of view.

Despite the restrictions listed above we expect that some of our results are quite general and would hold for more biologically oriented models. The effect of disorder on the motion of the collective clump within one map is a very robust feature of our model. In one dimension, the motion is drastically hindered, regardless of the parameter values. In two dimensions, this pinning effect is softened by the possibility of bypassing the barriers. We expect that, in higher dimensions, the motion of the clump would be even easier. This behavior, reminiscent of localization phenomena in condensed matter, is likely to remain true even for more realistic models from a biological point of view. A precise coding of position would therefore not be possible, in low dimensions, unless the clump is driven out of free-energy minima by strong enough inputs.

What clearly arises from this study is that, as a result of crosstalk, attractor manifolds coding for different maps are far from being flat, in contradistinction with the usual picture of continuous attractor neural network. Hence, distances in the space of hippocampal neural activities are distorted compared to the "true" distances in the real space. This finding is consistent with the assumption that the metric system of the brain is encoded in another region, while the hippocampus could serve as an associative system linking together places and other elements of memory.

C. Possible extensions

Our study could be extended along various directions, some of which are listed below.

An interesting feature of the model is the by-passing of barriers by the clump in 2D maps. To be more quantitative, we could imagine running drift simulations on a strip, that is, a 2D environment with periodic boundary conditions along the x axis and a finite size along the y axis. This would allow us to quantify the minimal "degree of two-dimensionality" for the motion of the clump, i.e., the minimal y width above which the clump can move around the barriers. We expect this width to be of the order of l b .

Our study of biologically motivated mechanisms possibly enhancing the motility of the clump is not exhaustive. For instance, synapse dynamics, that is, the short-term depression and/or facilitation of synapses, is another candidate. Its effect on the dynamics of a bump of activity in continuous attractor neural networks (in the absence of thermal and quenched disorder) has recently been studied by Fung et al. [39], who showed that short-term depression increases the motility of the clump.

In addition, it would be interesting to investigate further the issue of the response to inputs. How the hippocampus integrates the information conveyed by brain areas upstream CA3 is still not fully understood, in spite of a wealth of experimental results during the past ten years (notably the discovery of grid cells [40,41]). The hippocampus is not isolated but a part of a system of interacting regions [42]. The comprehension of the perforant pathway and mossy fibers inputs is a pivotal point. More generally, in the context of attractor network theory, reaching a deep understanding of the effect of these input sources of information on the attractor dynamics would be very important.

Last of all, a striking general result of our study is that diffusion is always in competition with transitions to other environments, whose main features were reported in Sec. V B 2. All the mechanisms we added to the model in order to make the clump move also increased the probability of these transitions. Two possible (and not mutually exclusive) explanations can be proposed. First, when the clump moves, it explores more positions in space and, thus, has a larger probability to find a "favorable" position for transitions, that is, a position where the energy barrier opposing a transition is not too large. Second, mechanisms enhancing the diffusion of the clump in one environment also tend to destabilize it, which makes transitions to another environment more likely. The study of these transitions is therefore a key issue, not only for the full understanding of the dynamics of our model, but also for the interpretation of experimental results, where manipulations of the visual cues resulted in abrupt swaps of the neural activity [22,24]. This question will be addressed in a forthcoming publication.

to the q ab , and

F n = αβ a<b r ab q ab + αT λ =0 Tr log[Id n -βλ(q -f 2 1 n )] - a λ a dxρ a (x) -f + a dxρ a (x)μ a (x) - 1 2 a dx dyρ a (x)J w (x -y)ρ a (y) -T dx log ⎡ ⎣ {σ a } e αβ 2 a<b σ a σ b r ab +β a μ a (x)σ a ⎤ ⎦ . (A2)
In Eq. (A2), α ≡ L N ; q, Id n and 1 n denote, respectively, the overlap matrix, the n-dimensional identity matrix and the n-dimensional matrix whose all entries are equal to one. The sum λ =0 runs over all the nonzero eigenvalues of the matrix J 0 .

Within the replica symmetric ansatz we assume

∀ a = b, ∀ x, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r ab = r q ab = q ρ a (x) = ρ(x) μ a (x) = μ(x) λ a = λ . (A3)
Finally, taking the n → 0 limit, we get β(qf 2 ) sin(kπw) kπβ(fq) sin(kπw) log 1 -β(fq) sin(kπw) kπ (A6) in one dimension, and

ψ 2D (q,β) ≡ 2 (k 1 ,k 2 )
=(0,0)

β(q -f 2 ) φ(k 1 ,k 2 ) -β(f -q) -log 1 - β(f -q) φ(k 1 ,k 2 ) (A7) with φ(k 1 ,k 2 ) ≡ k 1 k 2 π 2 sin(k 1 π √ w) sin(k 2 π √ w) (A8)
in two dimensions. The fixed-activity constraint is imposed through the parameter λ. When N → ∞ the integral is calculated through the saddle-point method. r, q, ρ(x), and μ(x) are found by writing the saddle-point equations where ϕ(q,T ) is defined by Eq. ( 46) in dimension 1 and Eq. ( 47) in dimension 2.

APPENDIX B: SPATIAL CORRELATIONS OF FREE-ENERGY FLUCTUATIONS

We consider n 2 copies of the system with a clump centered in x and n 2 other copies with a clump centered in y. In order to lighten notations, we take y = 0 (the problem is invariant by translation). Under this condition we have

∀ a < b n 2 ,∀x , ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
r ab = r ba = r 1 , q ab = q ba = q 1 , ρ a (x ) = ρ 1 (x ), μ a (x ) = μ 1 (x ),

λ a = λ 1 , ∀ n 2 < a < b, ∀x , ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
r ab = r ba = r 2 , q ab = q ba = q 2 , ρ a (x ) = ρ 2 (x ), μ a (x ) = μ 2 (x ), λ a = λ 2 , ∀ a n 2 < b, r ab = r ba = r 12 , q ab = q ba = q 12 .

(B1) (The dependence of q 12 and r 12 on |x| will be omitted to lighten notations.) By symmetry, r 1 = r 2 = r, q 1 = q 2 = q, λ 1 = λ 2 = λ, ρ 1 (xx) = ρ 2 (x ) = ρ(x ), and μ 1 (xx) = μ 2 (x ) = μ(x ). Replacing in Eq. (A2) and taking the 032803-21

small n limit, (A2) becomes

F n ∼ n→0 nF 0 + n 2 F 1 + O(n 3 ), ( B2
)
where F 0 = F given by (A5) and

F 1 = αβ 4
(rq + r 12 q 12 ) -α 2β q + q 12 2 f 2 2 + qq 12 2 2 ϕ(q,T ) where ϕ(q,T ) is given by [START_REF] Derdikman | A manifold of spatial maps in the brain[END_REF] in dimension 1 and (47) in dimension 2; κ(u,v) is given by [START_REF] Wilson | Reactivation of hippocampal ensemble memories during sleep[END_REF]. From Eq. ( 52) we have = 0, (B5) which give r 12 = 2T 2 (q 12f 2 )ϕ(q,T ), .

+
F 1 = - β 4 [V + W (x,
q 12 = dx DuDv κ(u,v) 1 + e -βu

(B6)

The overlap q 12 as a function of |x| is shown in Fig. 27. When the distance between the two clump centers increases, q 12 decreases from q (for x = y) to a saturation value lower than f 2 , on a typical distance roughly equal to the width of the clump. More precisely, in one dimension du uq(u) du q(u) = 0.113 and du uρ(u) du ρ(u) = 0.082; in two dimensions du uq(u) du q(u) = 0.125 and du uρ(u) du ρ(u) = 0.097. dans le cerveau permettent d'enregistrer in vivo l'activité simultanée de dizaines voire centaines de neurones individuels, c'est-à-dire quel neurone a émis un potentiel d'action à quel moment. C'est grâce à cette technique que O'Keefe & Dostrovsky ont fait en 1971 une découverte étonnante [17] : en enregistrant l'activité de neurones hippocampiques chez un rat se baladant dans un environnement familier, ils ont constaté que certains de ces neurones ne s'activaient que lorsque le rat était situé en des endroits précis. Chacun de ces neurones était ainsi associé à une position de l'espace et n'émettait des potentiels d'action que lorsque le rat était physiquement en cette position (voir par exemple la Figure 1.5). Ces neurones ont été baptisés cellules de lieu (place cells en anglais), et la région de l'espace associée à une cellule de lieu donnée est appelée son champ de lieu (place field ). Cette découverte a suscité une grande effervescence : on pensait avoir découvert le centre de la mémoire spatiale, de la navigation, voire le cadre spatial de toute mémoire épisodique (mémoire autobiographique). Un grand nombre de travaux expérimentaux et théoriques ont été menés depuis, explorant les propriétés des cellules de lieu sous toutes leurs coutures. Ici nous nous contenterons d'en donner un bref aperçu (pour plus de détails et de références, voir le paragraphe 1.4.2). D'abord, lorsqu'on dit que les cellules de lieu sont actives en des positions constantes, il faut préciser le référentiel dans lequel ces positions sont constantes. Or, il se trouve qu'en fonction des conditions expérimentales, les champs de lieu peuvent être "attachés" au référentiel du laboratoire, au référentiel de la plateforme sur laquelle évolue le rat, ou bien à des objets particuliers situés dans le champ de vision du rat (voir Figure 1.7). Il semble toutefois que le référentiel du laboratoire soit prédominant (par exemple, les champs de lieu sont maintenus dans l'obscurité [32]), et que ce n'est que lorsque le rat est désorienté ou quand les différents référentiels sont en conflit que les indices visuels peuvent être utilisés comme points de référence.

Ensuite, les champs de lieu se retrouvent lorsqu'on fait varier la dimension de l'espace physique : 1D pour un couloir linéaire, 2D pour un espace ouvert, 3D chez les espèces volant ou nageant.

Un autre fait notable est le caractère aléatoire de l'association champs de lieu -cellules de lieu. La position anatomique des cellules et la position géographique des champs récepteurs sont totalement décorrélées [41]. Ainsi, deux neurones voisins dans l'hippocampe peuvent avoir des champs de lieu très éloignés. De plus, une même cellule de lieu peut avoir plusieurs champs de lieu dans des environnements différents, et d'un environnement à l'autre les relations géométriques entre champs de lieu sont totalement modifiées, comme si dans chaque nouvel environnement on tirait au sort quel neurone serait actif où [42]. Ce phénomène est appelé remappage (remapping en anglais). Il est illustré dans la Figure 1.8. Précisons qu'ici, en faisant appel à la notion d'environnement, nous avons considéré l'espace physique comme morcelé en entités discrètes, un peu comme les pièces d'une maison. Ces entités sont appelées "environnements" (ou "cartes"). Il s'agit d'une image commode pour expliquer la notion de remappage, et aussi parce que dans la plupart des protocoles expérimentaux le rat est placé successivement dans plusieurs boîtes, qui sont bien des entités discrètes. Mais cette représentation d'un espace fragmenté n'est en toute rigueur pas nécessaire.

Terminons en mentionnant que des cellules de lieu ou leur équivalent ont été trouvées chez de nombreux mammifères, parmi lesquels le singe, la chauve-souris et l'homme. A ce propos, il est intéressant de noter que la recherche sur l'hippocampe humain, basée sur des données psychologiques et comportementales chez des patients plutôt que sur les enregistrements de neurones individuels, a suivi pendant des décennies une voie parallèle, quasi indépendante, à la recherche sur les rongeurs. C'est ainsi que chez l'homme, l'hippocampe a été désigné comme le centre de la mémoire épisodique, et sa composante spatiale a été pratiquement ignorée jusqu'à récemment. Chez le rat, au contraire, l'attention a été portée essentiellement sur les propriétés géométriques des cellules de lieu, et les composantes non-spatiales de leurs corrélats ont été plutôt laissés de côté. Mais les deux théories -mémoire épisodique et mémoire spatiale -ne sont pas incompatibles, comme le montrent des études plus récentes tendant à leur réconciliation. Néanmoins, dans la suite nous ferons référence principalement au rat, parce que c'est de là que viennent les résultats expérimentaux sur lesquels nous nous appuyons.

B.1.3 La théorie des attracteurs

Nous voudrions donc comprendre le fonctionnement des cellules de lieu. Cela implique d'une part de montrer comment les propriétés microscopiques du réseau cellulaire donnent naissance à cette activité localisée dans l'espace, stable et reproductible. D'autre part, en termes de fonction, puisque nous pensons que les cellules de lieu sont le support de la mémoire spatiale -ou de la composante spatiale d'une mémoire épisodique plus générale -nous devons proposer comment cette fonction de mémoire peut être remplie. Il se trouve, comme nous allons le voir, que ces deux objectifs reviennent à la même chose.

En effet, nous allons nous placer dans un cadre théorique dans lequel la fonction de mémoire est remplie justement par des configurations d'activité neuronale stables. Ce cadre théorique s'appelle la théorie des attracteurs. Il est basé sur le postulat suivant : Ce qui est mémorisé par un réseau de neurones est l'ensemble des états stables (ou attracteurs) de ce réseau. En d'autres termes, les éléments de mémoire sont les configurations d'activité qui sont stables sous la dynamique du réseau. La mémorisation d'une configuration est rendue possible par une modification adéquate des couplages synaptiques.

Ce postulat, qui peut dérouter à première vue, a été proposé par Donald Hebb en 1949 [7]. Il a depuis était étayé par de nombreuses observations expérimentales. L'idée est que, lorsque l'on a mémorisé quelque chose, par exemple la forme d'un objet, on est capable de maintenir cette image mentalement pendant un temps assez long. De plus, si on nous donne un indice partiel, comme une image tronquée de cet objet, notre cerveau va retrouver l'image entière, parce que le réseau aura convergé vers son attracteur le plus proche, qui est la configuration d'activité correspondant à cet objet mémorisé.

Depuis la proposition initiale de Hebb, de nombreux modèles de réseaux attracteurs ont été développés. D'après ce que nous venons de dire, un modèle de réseau attracteur est constitué des ingrédients de base suivants :

1. la définition des éléments à mémoriser : de quelles configurations d'activité veut-on faire des attracteurs ?

2. la définition de la dynamique du réseau, c'est-à-dire ses lois d'évolution temporelles, 3. la connectivité, c'est-à-dire quel neurone est connecté à quel neurone, et avec quelle intensité de couplage, 4. il faut enfin vérifier qu'avec cette connectivité et cette dynamique, les éléments que l'on voulait mémoriser sont bien des états stables.

Le roi des modèles de réseaux attracteurs, celui d'où sont partis la plupart des autres travaux, est le modèle de Hopfield, proposé par John Hopfield en 1982 [8]. Décrivons-le brièvement :

1. les configurations à mémoriser sont des configurations aléatoires de neurones binaires (chaque neurone est ou silencieux ou actif), indépendantes les unes des autres, 2. le réseau évolue de façon stochastique : un neurone est tiré au sort et son nouvel état est une fonction de l'état (silencieux ou actif) de ses voisins. Il peut y avoir de l'aléatoire (bruit) dans la réponse neuronale.

3. la connectivité est donnée par la règle de Hebb : l'intensité de la connexion entre deux neurones est égale à la somme sur toutes les configurations à mémoriser d'une contribution qui vaut 1 si les deux neurones ont le même état dans cette configuration, -1 sinon.

4. le fait que ces ingrédients permettent bien aux configurations d'être des attracteurs a été vérifié numériquement et analytiquement [8,[START_REF] Amit | Statistical mechanics of neural networks near saturation[END_REF]. Bien sûr, cela dépend de paramètres tels que le nombre de configurations à mémoriser et le niveau de bruit, qui doivent rester modérés. Si le niveau de bruit devient trop important, le rapport signal sur bruit est trop bas et l'activité devient totalement aléatoire. Si on "charge" trop les synapses en additionnant trop de configurations, la structure du réseau est trop floutée et le système ne parvient plus à retrouver les bonnes configurations.

Le modèle de Hopfield a déclenché une véritable avalanche de travaux dans la communauté des physiciens statisticiens [12]. La raison pour laquelle des physiciens se sont intéressés à ce sujet -en apparence très éloigné de leurs préoccupations habituelles -est qu'il existe en fait un lien très étroit, du point de vue formel, entre le modèle de Hopfield et certains systèmes magnétiques désordonnés appelés verres de spins, un domaine très actif depuis les années 1970 [START_REF] Mézard | Spin glass theory and beyond[END_REF]. En fait, la parenté entre neurosciences et physique statistique est beaucoup plus large que cela : la physique statistique s'attache à montrer comment des systèmes formés d'un grand nombre de constituants élémentaires microscopiques en interaction donnent lieu à des comportements collectifs à l'échelle macroscopique. C'est pourquoi la physique statistique, contrairement à ce que son nom semble indiquer, n'est pas du tout limitée à des problèmes de physique. Elle est une branche de la physique pour des raisons historiques, car elle est née de questions liées à la thermodynamique. Mais les comportements émergents sont un phénomène présent dans bien d'autres domaines : économie, sciences sociales, finance, biologie... Et bien sûr, les neurosciences : un grand nombre de neurones semblables, des interactions mutuelles et une évolution en partie aléatoire, tout cela constitue un terrain favorable au physicien statisticien. Après cette digression, revenons-en aux attracteurs. Il se trouve que plusieurs observations expérimentales suggèrent que les cellules de lieu de l'hippocampe forment un réseau attracteur, dont les configurations mémorisées ont une structure un peu particulière : au lieu d'être des états ponctuels du réseau, indépendants les uns des autres, elles forment un ensemble continu correspondant à l'espace physique sous-jacent. Les attracteurs sont ici dits "quasi-continus" : chaque attracteur correspond à un environnement (comme défini ci-dessus), qui est lui-même un ensemble continu de positions voisines. Les différents environnements sont décorrélés entre eux grâce au remappage. On a donc un ensemble discret d'attracteurs continus.

Nous étudions deux conséquences de cette hypothèse. Premièrement, on peut modéliser l'activité des cellules de lieu comme attracteurs d'un réseau neuronal (section B.2). Deuxièmement, on peut utiliser les propriétés des réseaux attracteurs pour décoder une activité mesurée expérimentalement. Il s'agit de retrouver, à partir de l'activité observée, dans quel environnement se trouve le rat. Cela sera détaillé en section B.3.

B.2 Modélisation

Nous avons donc proposé un modèle de réseau dont les états stables sont "localisés dans l'espace", c'est-à-dire qu'ils correspondent à des configurations dans lesquelles les neurones actifs ont des champs de lieu voisins dans l'espace physique. Nous avons considérés un espace physique à une ou deux dimensions, ce qui est le cas de la plupart des données expérimentales.

Notre modèle est une extension directe du modèle de Hopfield au cas d'attracteurs quasi-continus continus. Pour reprendre les éléments énumérés ci-dessus, les ingrédients de notre modèle sont les suivants :

1. dans chaque environnement à mémoriser, on attribue à chaque cellule de lieu une position -champ de lieu -tirée aléatoirement (remappage). Les configurations à mémoriser correspondent aux configurations dans lesquelles les neurones actifs sont ceux qui ont des champs de lieu voisins dans un des environnements.

2. la dynamique stochastique du réseau est la même que dans le modèle de Hopfield.

3. la connectivité est donnée par une version de la règle de Hebb adaptée au cas d'attracteurs quasi-continus, illustrée par la figure page 20. L'intensité de la connexion entre deux neurones est égale à la somme sur tous les environnements à mémoriser d'une contribution qui vaut 1 si les deux neurones ont des champs de lieu voisins dans cet environnement, 0 sinon. De plus, un terme d'inhibition globale est choisi de sorte que le nombre de neurones actifs à chaque instant soit égal à une fraction constante du nombre total de neurones. 4. vérifier que ces ingrédients permettent bien aux configurations d'être des attracteurs est l'un des objets de notre étude. La méthode employée est très similaire à celle des travaux précédents sur le modèle de Hopfield.

La première tâche, donc, est de vérifier que le réseau attracteur "marche", c'est-à-dire que ses états stables correspondent bien à des configurations localisées spatialement. Pour cela, il faut calculer les états stables.

Or, notre choix de la dynamique du réseau est tel que tout se passe comme si le système était décrit par une énergie E, qui est une fonction de la configuration adoptée par les neurones, et une température T , qui reflète le niveau de "bruit", c'est-à-dire le degré d'aléatoire dans la réponse des neurones. En d'autres termes, les états de notre système satisfont la loi de Boltzmann, qui dit qu'une configuration microscopique donnée -quels neurones sont actifs et quels neurones sont inactifs -apparaît avec une probabilité qui décroît avec son énergie (les configurations d'énergie basse sont les plus probables), et ce d'autant plus que la température est basse. Ceci décrit le système microscopique. Nous voulons le caractériser à une échelle macroscopique. La grandeur mesurant la probabilité des configurations macroscopique est l'énergie libre, qui contient un terme d'énergie et un terme d'entropie. Les états macroscopiques stables du système sont ceux dont l'énergie libre est minimale. En bref, nous devons calculer l'énergie libre de toute configuration du système et trouver pour quelles configurations elle est minimale.

Remarquons déjà que notre réseau n'est pas défini de façon exacte : nous avons dit que dans chaque environnement les champs de lieu étaient attribués aléatoirement. Il y a donc un grand nombre de possibilités, ou "réalisations possibles" du réseau (on parle de "désordre"), et dans chaque cas l'énergie libre que nous voulons calculer aura une valeur différente. Comme nous avons autre chose à faire que d'étudier chaque réalisation individuellement -ce qui n'aurait d'ailleurs aucun intérêt -nous calculons la moyenne de l'énergie libre sur toutes les réalisations possibles. En effet, nous savons que cette valeur moyenne est aussi une valeur typique, parce que la distribution des valeurs de l'énergie libre est très piquée (on dit que l'énergie libre est auto-moyennante). Donc il est tout à fait justifié de calculer la valeur moyenne, car elle donne une bonne idée de ce qu'il se passe en général, quelle que soit la réalisation que l'on considère. De plus, pour pouvoir mener le calcul à bien, on fait l'hypothèse que le nombre de neurones du réseau est tellement élevé qu'on peut le considérer comme infini. Il s'agit d'une approximation très classique en physique statistique. Dans le cas de l'hippocampe, qui contient des dizaines de milliers de neurones, elle est légitime.

Le calcul de l'énergie libre "moyennée sur le désordre" se fait par la méthode dite des répliques. Cette méthode de calcul, initialement développée pour l'étude des verres de spin [START_REF] Kirkpatrick | Infinite-ranged models of spin-glasses[END_REF], consiste à imaginer que l'on fait plusieurs copies -répliques -du système avec une même configuration du désordre. On peut alors accéder à des propriétés très fines du système en supposant que le nombre de répliques devient un nombre réel que l'on fait tendre vers zéro. Le calcul est présenté dans ses grandes lignes dans la section 2.3 page 21, et détaillé dans la référence [1]. Ici nous nous contenterons d'en donner les résultats principaux : en fonction du niveau de bruit T et du nombre d'environnements α que l'on veut mémoriser dans le réseau, trois types de configurations très différents sont les minima de l'énergie libre :

• pour T , α pas trop grands, les états stables sont des états où les neurones actifs ont des champs de lieu voisins dans un des environnements. C'est le régime où le système "marche" comme désiré. Ce régime est censé reproduire l'activité observée expérimentalement : lorsque le rat est situé en une position donnée dans un environnement donné, ses neurones actifs sont ceux dont les champs de lieu sont voisins de cette position dans cet environnement.

• pour T élevé, l'activité est dominée par le bruit : les neurones actifs changent tout le temps, sans rapport avec la position de leurs champs de lieu.

• pour α élevé, les synapses sont trop "chargées" et la capacité critique de mémoire est dépassée. Dans ce cas, les champs de lieu de neurones actifs forment des petits groupes localisés ça et là, mais sans cohérence globale.

Les états stables -ou "phases" -du système en fonction de T et α sont récapitulés dans le "diagramme de phase". Voir la figure 2.2 page 23. Ce diagramme de phase présente de nombreuses analogies avec celui du modèle de Hopfield. La parenté entre les deux modèles est discutée page 25.

On pourrait longuement disserter sur ce diagramme de phase -ce que nous avons fait en section 2.3 et référence [1]. Une vérification importante consiste à simuler numériquement le système, c'est-à-dire à faire tourner un algorithme dans lequel les neurones, leur connectivité et leur dynamique sont implémentés1 . On peut alors faire varier T et α dans les simulations et vérifier que les états stables correspondent bien à ceux prédits par le diagramme de phase. C'est effectivement le cas.

La forme des attracteurs est bien sûr une vérification cruciale -si les attracteurs ne sont pas localisés, le modèle ne peut plus guère prétendre à représenter de "vraies" cellules de lieu. Mais bien d'autres propriétés collectives intéressantes émergent des quelques ingrédients microscopiques simples décrits ci-avant. Ces propriétés a posteriori du modèle peuvent permettre de faire des prédictions sur ce qu'on devrait observer expérimentalement -à condition que le modèle soit assez fidèle. La vérification expérimentale de ces prédictions permet à son tour de voir ce qui va et ce qui ne va pas dans le modèle, et de l'améliorer. Une difficulté consiste alors à évaluer la portée du modèle, c'est à dire jusqu'à quel point il est censé reproduire la réalité. Mais pour l'instant, nous nous contentons d'explorer, autant que faire se peut, les propriétés du modèle.

Nous avons décrit les attracteurs comme des états stables. Cela veut dire que, une fois dans l'un de ces états, le système va y rester longtemps. Mais, à partir du moment où il y a du bruit -c'est à dire dès que T > 0, même très petit -le système ne peut pas y rester pour toujours. De temps en temps, il va "sauter" d'un attracteur à l'autre, passer d'une configuration où les neurones actifs ont des champs de lieu voisins dans l'environnement 1 à une configuration où les neurones actifs ont des champs de lieu voisins dans l'environnement 2 (par exemple). Nous pouvons analytiquement estimer la fréquence de ces transitions, ainsi que leur mécanisme. Il apparaît ainsi que les transitions se font le plus souvent en passant par un état intermédiaire dans lequel l'activité est à la fois localisée dans l'environnement de départ et dans l'environnement d'arrivée. Par conséquent, une transition a d'autant plus de chances d'avoir lieu que les positions de départ et d'arrivée se "ressemblent", dans le sens où les champs de lieu voisins de ces positions ont des neurones en commun. Cette propriété est très bien vérifiée dans les simulations : la fréquence des transitions entre deux positions données est directement corrélée à la similarité entre ces positions.

Lorsque le système ne saute pas entre deux environnements et qu'il reste localisé dans un même environnement, rien ne dit que la position en laquelle il est localisé doit rester constante. Et en effet, nous pouvons suivre dans les simulations le centre de la région où les champs de lieu ont leurs neurones actifs, et nous voyons que ce centre se promène dans l'environnement, comme si la région active était une seule particule (alors qu'elle est constituée d'un grand nombre de neurones qui ont leurs champs de lieu dans cette région, et que ces neurones changent lorsque la région bouge). Ce mouvement d'ensemble est typiquement un phénomène collectif, que l'on pourrait comparer, pour donner une image, à un essaim d'abeilles ou à une ola dans un stade. Grâce aux outils de la physique statistique, nous pouvons le caractériser avec un grand niveau de détail, et la comparaison avec les simulations nous permet de vérifier nos résultats.

Quel est le rapport de tout cela avec la réalité ? Il faut être conscient que notre modèle est une schématisation extrême du vrai hippocampe. Cela constitue d'ailleurs à la fois sa faiblesse et sa force, car c'est grâce à cette simplification que nous pouvons l'étudier aussi profondément. Mais beaucoup d'hypothèses drastiques ont été faites et beaucoup de détails ont été laissés de côté, par exemple dans la réponse des neurones ou dans la structure du réseau.

La simplification la plus radicale consiste à négliger les connexions avec les autres régions du cerveau, notamment les signaux afférents. Ces derniers sont pourtant supposés avoir le rôle très important d'indiquer au réseau attracteur vers quelle position il doit converger. En effet, dans notre modèle, toutes les positions de tous les environnements sont équivalentes et le système peut indifféremment se stabiliser en n'importe quelle position. Dans la réalité, il faut bien que quelque chose indique aux cellules de lieu où est le rat, sinon on n'observerait pas une telle corrélation entre position et activité neuronale. Mais, si ces signaux d'entrée sont importants pour la mise à jour de l'activité en fonction de la position du rat, ils sont plus secondaires du point de vue de la fonction de mémorisation. Pour que la théorie selon laquelle l'hippocampe est un réseau attracteur soit validée, il faut d'abord vérifier que les attracteurs ont bien les bonnes propriétés en l'absence de tout signal afférent. Nous avons d'ailleurs étudié quelques comportements du réseau lorsque l'on rajoutait des entrées externes (voir le chapitre 2), mais une compréhension profonde de l'effet de ces signaux demanderait une étude plus poussée.

De la même façon, le diagramme de phase ne veut pas dire, par exemple, que quelqu'un qui essaierait de mémoriser trop de lieux perdrait brusquement toute la mémoire lorsque son α dépasserait la valeur critique (entre autres parce que l'additivité des environnements dans la règle de Hebb n'est qu'une approximation)... Sa signification est que des attracteurs localisés peuvent apparaître, à partir d'hypothèses simples sur la structure du réseau, dans un certain régime de paramètres. Il est donc important, parmi toutes les propriétés du modèle, de faire la part des choses entre celles qui sont dues à une simplification trop importante (par exemple, la perte brutale de mémoire au α critique) et celles qui sont censées capter un phénomène "robuste à la simplification", c'est à dire que ces propriétés persisteraient si on complexifiait le modèle pour le rendre plus réaliste (c'est le cas par exemple des attracteurs localisés dans l'espace). Cette distinction n'est pas une tâche facile. En ce qui concerne les sauts entre environnements et la diffusion au sein d'un environnement, il semblerait qu'ils présentent une parenté avec des phénomènes observés expérimentalement.

B.3 Décodage

Nous nous sommes ensuite intéressés à un autre problème assez différent du premier, celui du décodage. L'idée est que l'activité des neurones est un code, qu'il devrait être possible de déchiffrer. Pour cela, il est nécessaire de partir d'hypothèses quant au contenu du message et à la façon dont il est encodé.

Sur le contenu, déjà, il nous faut essayer de deviner "à propos de quoi" les neurones "parlent". Si pour certaines régions du cerveau comme les aires sensorielles primaires, on peut raisonnablement restreindre les possibilités (par exemple, l'activité de la rétine reflète des évènements visuels), cela n'a rien d'évident dans le cas d'aires profondes telles que l'hippocampe. Nous avons un indice important qui est que l'activité des cellules de lieu est corrélée à la position spatiale. Mais cela ne veut pas dire que la première représente la seconde. Quant à la façon dont le message est encodé, le débat fait rage depuis de nombreuses années. Pour certains, l'unité sémantique est le potentiel d'action individuel. Pour d'autres, il faut considérer des taux de décharge moyens sur une population de neurones et / ou sur une fenêtre temporelle à préciser. Il y a ceux qui soutiennent que l'important est dans la donnée de quels neurones sont actifs et quels neurones ne le sont pas, et que le détail de leur activité est secondaire... En bref, il y a un grand nombre de possibilités et pas de consensus. En fait, il est possible qu'il n'y ait pas de réponse universelle et que cela dépende du système ou de l'aire cérébrale considérés.

Dans le cas de l'hippocampe, les travaux précédents ont presque tous fait l'hypothèse que l'activité des cellules de lieu représente la position, et que l'encodage se fait dans les taux de décharge moyennés sur quelques centaines de milisecondes. Le décodage d'un "vecteur de population" (la donnée du taux de décharge de tous les neurones enregistrés à un instant t) se fait alors simplement par comparaison de cette activité mesurée avec des activités de référence. Pour cela, il faut avoir auparavant enregistré l'activité des neu-rones lorsque le rat se trouve en toutes les positions possibles. Cela constitue la référence. Ensuite, lorsqu'on veut décoder, c'est-à-dire faire l'opération inverse et retrouver la position à partir de l'activité, il suffit de regarder en quelle position l'activité des neurones "ressemblait" le plus à celle que l'on cherche à décoder.

Nous avons défini le problème un peu différemment : ce que l'on cherche à décoder n'est plus la position du rat, mais dans quel environnement il se trouve. Quant à la façon dont l'information est encodée, nous avons fait l'hypothèse qu'elle l'était dans la co-activation simultanée de sous-populations de neurones. Il s'agit de l'hypothèse des assemblées cellulaires, qui n'est pas sans rappeler la théorie des attracteurs. Nous avons proposé plusieurs méthodes pour ainsi décoder, à partir de l'activité neuronale, dans quel environnement se trouve le rat.

Nous avons testé ces méthodes sur des activités neuronales enregistrées expérimentalement, grâce à K. Jezek et E. Moser de l'université de Trondheim en Norvège, qui ont eu l'obligeance de nous communiquer leurs données expérimentales. 34 cellules de lieu étaient enregistrées simultanément dans l'hippocampe d'un rat se baladant pendant 10 minutes dans un premier environnement familier A (une boîte carrée de 60 cm de côté avec un éclairage vert), puis pendant 10 autres minutes dans un environnement familier B (la même boîte carrée mais avec un éclairage blanc). Le rat ne sait pas qu'il s'agit de la même boîte, il croit que les deux environnements sont totalement distincts, comme en atteste le fait que dans chaque environnement ses neurones s'activent en des endroits très différents. Ensuite, le rat est placé dans l'environnement A et, après un petit moment, l'expérimentateur tourne un interrupteur qui change soudainement l'éclairage, de sorte que le rat se retrouve comme "téléporté" instantanément dans B. On renouvelle ce processus toutes les minutes environ, et pendant dix minutes, le rat est ainsi téléporté d'un environnement à l'autre. L'activité des neurones (c'est-à-dire les temps de tous les potentiels d'action des 34 cellules) et la trajectoire du rat sont enregistrées pendant tout ce temps. K. Jezek et ses collègues ont publié les résultats de cette expérience dans un article datant de 2011 [START_REF] Jezek | Theta-paced flickering between place-cell maps in the hippocampus[END_REF] : ils ont montré que, tout de suite après la téléportation, l'activité neuronale du rat "clignotait" entre deux états : l'un "ressemblant" à une activité typique dans A et l'autre "ressemblant" à une activité typique dans B. Ces allers-retours peuvent se prolonger pendant quelques secondes, après quoi l'activité neuronale se stabilise dans le bon environnement, signe que le rat a repris ses esprits après la téléportation.

Notre but n'était pas d'étudier précisément les téléportations elles-mêmes, mais de voir si nos méthodes basées sur l'hypothèse des assemblées cellulaires permettaient de décoder correctement dans quel environnement se trouve le rat à un instant donné, et de comparer ces résultats avec ceux de méthodes plus classiques. Il s'avère que les méthodes basées sur les assemblées cellulaires ont des performances similaires aux meilleures méthodes basées sur la comparaison de vecteurs de population. Cela constitue un élément encourageant en faveur de cette hypothèse. Toutefois, nous avons observé que les performances des différentes méthodes dépendaient fortement des caractéristiques de l'activité neuronale enregistrée. Nous devons donc tester ces méthodes sur d'autres données avec des caractéristiques différentes.

Notre approche nous permet en outre de dévoiler un "réseau effectif" entre les 34 cellules, c'est-à-dire la structure de couplages qui reproduit l'activité observée. Bien sûr il ne s'agit pas des vraies synapses, mais seulement de grandeurs effectives : nos 34 neurones sont connectés à des milliers d'autres neurones que nous n'observons pas mais qui contribuent à leur activité. Mais ce réseau effectif nous permet de voir que, au niveau des connexions effectives au moins, les deux composantes de la règle de Hebb -renforcement des connexions entre cellules co-actives et additivité des attracteurs -sont plutôt bien

1

 1 Methods in Computational NeuroscienceNeuroscience aims at understanding how the brain works. This introduction is devoted to clarifying what this means.

Figure 1 . 1 :

 11 Figure 1.1: Sketch of two neurons in synaptic contact.

Figure 1 . 2 :

 12 Figure 1.2: Classification of memory types.

Figure 1 . 3 :

 13 Figure 1.3: Location of the hippocampus in the human brain (left) and in the rodent brain (right), in blue.

Figure 1 . 4 :

 14 Figure 1.4: The hippocampal formation. Left: connectivity diagram. Blue boxes indicate the different regions (EC= entorhinal cortex, DG= dentate gyrus, Sub= subiculum). Orange arrows indicate the synaptic excitatory axons and their direction of propagation (PP= perforant pathway, MF= mossy fibers, RC= recurrent collaterals, SC= Schaffer collaterals). Right: drawing by Ramón y Cajal (1909) (adapted, with the same legend as the left figure).

Figure 1 . 5 :

 15 Figure 1.5: Spatially localized firing of a rat's place cell. The activity has been recorded in CA3 during a 14 minute exploratory session in a familiar 60 cm × 60 cm square box.Left: trajectory of the rat (black line) and positions where spikes were emitted by the cell (each red dot represents a spike). Right: corresponding smoothed rate map (i.e. average number of spikes per time unit as a function of position), displaying the characteristic bump of activity. The recording is taken from the dataset described in Section 3.2.

Figure 1 .

 1 Figure 1.6: a-c: Illustrative sketch of the firing fields of different cell types in a square environment, viewed from above. The portion of space where the cell is active is displayed in red. a: place cell. b: grid cell. c: border cell. d: firing rate of a head-direction cell (with preferential direction around 220˚), as a function of the orientation of the head.

Figure 1 . 8 :

 18 Figure 1.8: Illustrative diagram of the remapping of three place cells in three square environments. The place cells' place fields are displayed in red, blue and green respectively.Note the absence of systematic spatial relationship from one environment to the other. Also, place cells do not necessarily have a place field in all environments (there are "silent cells"), and place fields may overlap.

Figure 2 . 1 :

 21 Figure 2.1: Remapping and connectivity rule in the model, illustrated with three units and L + 1 two-dimensional environments. The place field centres of the units are displayed respectively in red, blue and green. Thick yellow lines indicate the excitatory couplings between cells with neighbouring place fields in each environment. These place fields overlap; here, for the sake of clarity, only the centres of the place fields are represented.

Figure 2 . 2 :

 22 Figure 2.2: Phase diagram in the (α, T ) plane in D = 1 with f = 0.1 and w = 0.05. Thick lines: transition between phases. Dashed-dotted line: T PM (α). Thin dashed line: CL phase's longitudinal stability regions. Dotted line: CL phase's line of replica-symmetry breaking. α CL : storage capacity at T = 0 of the replica-symmetric clump phase. α g : CL-SG transition load at T = 0. T CL : temperature of loss of stability of the clump at α = 0. T c : CL-PM transition temperature at α = 0. T PM = T PM (α = 0) (see text).

Figure 2 . 3 :

 23 Figure 2.3: Average energy for the unidimensional model with a single environment (α = 0), N = 3200 (triangles) and N = 6400 (circles). The average energy after a 10 N Monte Carlo steps depends on the initial condition (clump or uniform), which results in a clear hysteresis. Each point is averaged over 1000 simulations. Lines serve as a guide to the eye.

Figure 2 . 4 :

 24 Figure 2.4: Phase diagram of the Hopfield model, reproduced from Ref. [90]. T g : transition line between the paramagnetic and the glassy phases. T M : line of longitudinal stability of the ferromagnetic ("retrieval") phase. T C : transition line between the glassy and the retrieval phases. Inset: zoom on the low-T , high-α region. T R : line of replica-symmetry breaking of the retrieval phase.

Figure 2 . 5 :

 25 Figure 2.5: Time evolution of a 1D network with N = 1000 units and 2 stored environments observed in a Monte-Carlo simulation at T = 0.007, illustrating the coexistence of diffusion within one map and transitions between maps. Each black dot represents an active unit. Both panels represent the same data, they only differ by the ordering of the units along the x-axis. In the top panel, the units i are arranged according to their place field centres in environment 1. In the bottom panel, they are arranged according to their place field centres in environment 2. The y-axis represents time (in Monte-Carlo rounds).Between time 0 and time ≈ 1000 the activity is localized in map 2 and delocalized in map 1. Then it undergoes several transitions between 1 and 2. Between times ≈ 2000 and ≈ 3700, the network is in attractor 1 and the bump diffuses within this attractor. Finally, it ends up and diffuses in map 2. Note the abruptness of the transitions between maps.28

Figure 2 . 6 :

 26 Figure 2.6: Schematic representation of transitions between two 1D environments (red and blue) as a function of time. The insets represent the short-term average of the activity, where the spins have been ordered as their place field centres in map 1 (red line) and in map 2 (blue line), when the system is in map 1's attractor (left) and map 2's attractor (right).

Figure 2 . 7 :

 27 Figure 2.7: Rate of transitions between environments (number of transitions per Monte-Carlo round) as a function of α and T in the clump phase of the 1D model. The rates have been measured in Monte-Carlo simulations with N = 333 (left), N = 500 (centre) and N = 667 units (right). Each point is averaged over 100 simulations of 10000 rounds each. Note the decrease of the frequency transitions when N increases.

Figure 2 . 8 :

 28 Figure 2.8: Rate of transitions between environments (number of transitions per Monte-Carlo round) as a function of α and T in the clump phase of the 2D model. The rates have been measured in Monte-Carlo simulations with N = 32 × 32 units. Each point is averaged over 10 simulations of 10000 rounds each.

Figure 2 . 9 :

 29 Figure 2.9: Relationship between local similarity and transitions between two maps in a Monte-Carlo simulation with N = 333 spins, L + 1 = 2 environments, T = 0.006. Left: local resemblance Res 12 (x 1 , x 2 ) as a function of x 1 , x 2 . Right: logarithm of the number of observed transitions between maps 1 and 2 at the positions x 1 , x 2 . A relationship consistent with Eq. 2.10 appears.

Figure 2 . 10 :

 210 Figure 2.10: Trajectory of the centre of a freely diffusing bump in dimension 2, during 10 6 steps of a Monte Carlo simulation with N = 32 × 32 neurons, α = 0 and T = 0.005.

Figure 2 . 11 :

 211 Figure 2.11: Diffusion of a bump of neuronal activity in prefrontal cortex during the delay period in an oculomotor delayed response task. Left: spatio-temporal representation of the network activity. Neurons indices are ordered as their preferential firing direction (that would be the equivalent of our 1D place field centres). Right: same data on a plot arranged as our Figure 2.5. x-axis: neurons label. y-axis: time. Taken from Ref. [95].

Figure 2 . 13 :

 213 Figure 2.13: Percentage of cells having a place field as a function of the criterion (2.15). The graph bends around r 0 = 2.5 Hz.

Figure 2 . 14 :

 214 Figure 2.14: Width of the bump s as a function of w for c = c * , f = f * and for different values of T and α. We can see that the dependence in w virtually does not change with T and α: this allows us to directly estimate w * .

Figure 2 . 15 :

 215 Figure 2.15: Fluctuations of the shape of bump of activity δρ(x) in the one-dimensional case, for f = .01, w = 0.05, α = 0.01, c = 1. Left: T = 0.005. Right: T = 0.007. Here, we look at these fluctuations at the centre of the bump.

Figure 2 .

 2 16 shows the profile of the bump calculated with f * , c * , w * , T * , α * .

Figure 2 . 16 :

 216 Figure 2.16: Profile of the bump of activity with the estimated parameters f * , c * , w * , T * , α *

Figure 3 .

 3 1 shows examples of such auto-and cross-correlograms in our dataset.

Figure 3 . 1 :

 31 Figure 3.1: Examples of correlograms at resolution ∆ t = 10 ms. Left: auto-correlogram of neuron 17 in the reference session in B. Centre: cross-correlogram of two positively correlated neurons (neurons 8 and 13 in reference session in A). Right: cross-correlograms of two anti-correlated neurons (neurons 9 and 33 in reference session in A). In all cases, the global modulation by the theta rhythm (around 10 Hz) appears. Note the difference of scale on the y-axis.

Figure 3 . 2 :

 32 Figure 3.2: Fraction of bins with zero spike as a function of the bin length. This fraction is important for bins of less than 100ms.

Figure 3 . 3 :

 33 Figure 3.3: Rate map of neuron #3 in environment A, before (left) and after smoothing (right). The color scale for firing rates (in Hertz) is indicated by the bar on the right.

Figure 3 . 4 :

 34 Figure 3.4: Average error of the decoded position as a function of the time bin length ∆t.

Figure 3 . 5 :

 35 Figure 3.5: Couplings as a function of the normalized overlap O M (i, j) between the corresponding place fields r M i ( x) and r M j ( x). Left: raw inferred couplings J M ij . Right: rescaled couplings ĴM ij . Red: reliable couplings in map A (i.e. satisfying 3.28). Blue: reliable couplings in map B. The error bars on the inferred values are shown. Grey: unreliable couplings (not all shown).

Figure 3 . 6 :

 36 Figure 3.6: Effective couplings inferred in the test session as a function of the sum of the corresponding couplings in sessions A and B, weigthed by the relative time spent in each map during the test session. Left: raw inferred couplings J M ij . Right: rescaled couplings ĴM ij . Magenta: couplings reliable in all three sessions. Green: couplings reliable in at least one session (not all shown). Dashed line: y = x line.

Figure 3 .

 3 Figure 3.8 shows the performance of the different methods introduced in the case of the test session.As expected, the performance decreases at short timescales due to the large number of bins with zero spikes. This is particularly true for the dot-product methods because they are unable to decode those bins. The "local -dot product " performs better than the "dot product", meaning that the knowledge of the rat's position helps retrieving in which environment the rat is.A striking feature is that 4 methods ("rate -max. posterior", "rate -independent", "Ising -coupled" and "Ising -independent") have almost equal performances. There are two points in this result. First, the models based on rate maps and Poisson firing give results comparable to the Ising-based models. Though there is still ongoing work to fully understand this similarity, a tentative link between both classes of models is proposed hereafter (Section 3.8). Second, there is the fact that the independent models give similar results to their coupled counterparts -"rate -independent" vs "rate -max. posterior" on the one hand, "Ising -independent" vs "Ising -coupled" on the other hand. This means

Figure 3 . 7 : 8 ΠFigure 3 . 8 :

 37838 Figure 3.7: Ising-based "coupled" decoder in the test session: L A ( n(t)) -L B ( n(t)) as a function of time, for ∆t = 120 ms. The vertical lines indicate the teleportation times. The colors indicate the "true" environment, that is the configuration of the ligths: red for A and blue for B.

5 Figure 3 . 9 : 5 Figure 3 . 10 :

 5395310 Figure 3.9: Fraction Π of correctly decoded bins for the 7 decoded methods defined above as a function of the time bin length. Only the subpopulation of neurons with place fields in both A and B was used in the decoding (N c = 5).

Figure 3 . 11 :

 311 Figure 3.11: Transitions to the alternative environment (i.e. the one in which the rat is not physically) decoded by each method during the test session and considered as significant according to the criterion described in the text. The dots represent the bins of transitions for the methods indicated on the left. The x-axis represents the bins' indices (120 ms bins; only bins with more than 2 active cells included). The vertical lines indicate the times of switches.

Figure 3 . 12 :

 312 Figure 3.12: Zoom on a time segment of Fig. 3.11, with the same caption.

. 37 )

 37 J ab = log P Poisson ( S ab |M)P Poisson ( 0)|M P Poisson ( S a |M)P Poisson ( S b |M) = . . . (3.38) K abc = log P Poisson ( S abc |M)P Poisson ( S a |M)P Poisson ( S b |M)P Poisson ( S c |M) P Poisson ( S ab |M)P Poisson ( S bc |M)P Poisson ( S ac |M)P Poisson ( 0|M) = . . .

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: Left: fields in the "rate -max. posterior" model as a function of the corresponding fields in the "Ising -coupled" model. Right: two-point couplings. Red: in A (reliable couplings only); blue: in B (reliable couplings only). Gray: unreliable couplings (not all shown). Dashed line: y = x line.

Finally, an important

  part of the work -not necessarily the easiest one -just consists in defining what we do and why we do it. The distinction is sometimes blurred between what we assume, what we know, what belongs to the model and what belongs to reality. Thus it is important to keep in mind the motivations and the framework of our work, at the risk of endless analytical and numerical wanderings.

FIG. 1 .

 1 FIG. 1. (Color online) Example of remapping of the place field centers of N = 6 neurons (denoted by indices 1,...,6) in two different one-dimensional (1D) environments with periodic boundary conditions and w = 2 6 . Place fields in each environment are represented by colored dashed lines, place field centers are denoted by letters a,...,f.

FIG. 2 .

 2 FIG. 2. Average activity ρ(x) in dimension D = 1 in the paramagnetic phase (a) and in the clump phase ((b): temperature T = 0; (c): temperature T = 0.0073) for α = 0, computed with M = 2000 bins of discretization.

FIG. 4 .

 4 FIG. 4. Highest temperature at which the clump exists, T CL , as a function of the number M of discretization bins for three values of w. The average activity is f = 0.1 and the load vanishes, α = 0.

FIG. 6 .

 6 FIG. 6. Effect of the load α on the clump: average activity ρ(x) in dimension D = 1 in the clump phase at temperature T = 0.004 for α = 0 (left) and α = 0.02 (right).

FIG. 7 .

 7 FIG. 7. Domain of stability the clump phase, computed with M = 200 bins. Longitudinal and replicon instability lines correspond to, respectively, the full and dashed lines. Due to the computational effort required for the calculation of the replicon eigenvalues, only a few points (black dots) were computed.

FIG. 8 .

 8 FIG. 8. Phase diagram in the (α,T ) plane in D = 1. Thick lines: transitions between phases. Thin dashed lines reproduce stability regions described above. Critical lines were computed with M = 200.

FIG. 11 .

 11 FIG.11. Average energy for the unidimensional model with a single environment and for increasing sizes N . For each size, we plot the average energy obtained after thermalization for 10 N Monte Carlo steps starting from the uniform and from the clump configurations. Each point is averaged over 1000 simulations.

  FIG. 12. Correlation C(d) between spins at distance d (35) at low (left) and high (right) temperatures, and for various sizes N . (a) T = 0.004, (b) T = 0.01. Note the difference of logarithmic scale on the y axis between the two panels.

1 2

 1 FIG. 14. Contribution h i of environments1 to the local fields as a function of x = i/N -0.5 for the same model as in Fig.13. Inset: histogram of h i (rectangles) compared to the Gaussian distribution of mean f Lw and standard deviation √ αr (solid line). The value √ αr 6.98 × 10 -3 was obtained from the resolution of (28).

FIG. 16 .

 16 FIG. 16. Density of energy in the environment coherent with the clump for constant T = 0.004: results of Monte Carlo simulations for N = 2000 (circles) and N = 5000 (triangles) with error bars, compared to theoretical result computed with M = 1000 (line).

FIG. 17 .

 17 FIG. 17. Monte Carlo simulations around the clump-glass transition for T = 0.004: fraction of simulations found in the glassy phase after 100 rounds of 10N steps, as a function of α and for different N , with error bars. For each point the fraction was calculated from 50 simulations, half of which were started in a clump configuration and the other half in a uniform configuration.

FIG. 18 .

 18 FIG. 18. Influence of w on the clump phase: T CL (top) and α CL (bottom) as a function of w, for different fixed values of f . Note the maximum around w ∼ f in the latter graph. Computations were done with M = 1000. The numerical error is δα CL ∼ 0.005.

  FIG. 19. Influence of w on the first-order transitions: T c (top) and α g (bottom) as a function of w, for different fixed values of f . Computations were done with M = 1000.

FIG. 20 .

 20 FIG. 20. Influence of f on the clump phase: T CL (top) and α CL (bottom) as a function of f , for different fixed values of w. Note the maximum around f ∼ w in the latter graph. Computations were done with M = 1000. The numerical error is δα CL ∼ 0.005.

FIG. 22 .

 22 FIG. 22. Effect of partial activity: Influence of the fraction c of active cells on the clump domain: T c (top) and α g (bottom) as a function of c, for different fixed values of f and w. Computations were done with M = 200.

  the D = 2 case Eqs. (A4) can be simplified by exploiting the invariance by rotation: in polar coordinates μ(r) = 2

FIG. 23 .

 23 FIG. 23. Solid line: longitudinal stability region of the clump phase for D = 2. The D = 1 case is shown in thin dashed line for comparison.

FIG. 24 .

 24 FIG. 24. Two examples of pairings between 2k points: P A (left) and P B (right).

  neural network model of the hippocampus: Collective motion of the activity R. Monasson and S. Rosay Laboratoire de Physique Théorique de l'ENS, CNRS & UPMC, 24 rue Lhomond, 75005 Paris, France (Received 14 October 2013; revised manuscript received 18 December 2013; published 11 March 2014)

PHYSICAL REVIEW E 89 ,

 89 FIG. 1. (Color online) Sketch of the phase diagram in the plane of neural noise, T , and number of environments per neuron, α. Thick solid lines: transitions between phases. Thin dashed lines: stability region of each phase against fluctuations. Insets show the corresponding activity profiles in the 2D model (averaged over one round of Monte Carlo simulations after thermalization). In the clump phase we represent the same activity profile in the retrieved environment (top) and in another stored environment (bottom). See Ref. [25] (Fig. 8) for more quantitative details.

FIG. 2 .

 2 FIG. 2. Sketches of the clump of neural activity moving in space, shown at two subsequent times (only central parts are shown), in the 1D, single-environment case with T = 0.006. The dashed lines represent the equilibrium profile ρ * (x). Full lines correspond to average densities computed at the two times under consideration, which deviate from ρ * by terms of the order of N -1/2 . The horizontal dotted lines locate ρ = f and ρ = 1. Simulations parameters: N = 2000, activity averaged over short distance (10 spins) and time (5N Monte Carlo steps).

FIG. 3 .

 3 FIG.3. Sketch of the free-energy landscape probed by the clump of neural activity (dashed curve) moving through space. Fluctuations of the free energy are of the order of F and are correlated over a space scale equal to b .

FIG. 4 .

 4 FIG. 4. Position x c vs time t of a freely diffusing clump in dimension 1, for α = 0 and 50 000 rounds of Monte Carlo simulation with N = 333 neurons, and noise T = 0.006. Time unit = one round of 20N steps.

FIG. 5 .

 5 FIG. 5. Trajectory of a freely diffusing clump in dimension 2, for α = 0 and 50 000 rounds of Monte Carlo simulation with N = 32 × 32 neurons, and noise T = 0.005. Time unit = one round of 20N steps.

FIG. 7 .

 7 FIG. 7. Standard deviation β√ V of the free energy (in units of the temperature and divided by √ N ) as a function of the load α for fixed temperature T . Lines end at the clump instability limit.

FIG. 8 .

 8 FIG.8. Contour lines of constant N c in the phase diagrams of the 1D (top) and 2D (bottom) models. In one dimension, for a given N c , the area of the diagram above the contour line corresponds to free diffusion, while in the area below the diffusion is activated. In two dimensions, this distinction is less clear due to the possible by-passing of free-energy barriers (see text).

FIG. 9 .

 9 FIG. 9. Covariance W (xy) of the free energies of the clump centered on positions x and y, normalized by V . Results are shown for dimension 1, with T = 0.006, α = 0.01 (full line), and in dimension 2 with T = 0.004, α = 0.002 (dashed line).

FIG. 10 .

 10 FIG. 10. Logarithm of the diffusion constant D as a function of √ N with constant L + 1 = 2, measured in Monte Carlo simulations in both dimensions 1 and 2. For sufficiently large N , log 10 (D) seems to decrease linearly with √ N . The simulations length depends on the frequency of transitions: typically, of the order of 10-10 2 rounds for √ N = 26 and 1000 rounds for √ N > 35. Depending on the computational cost, each point is averaged over a number of simulations ranging from 5 (for large N ) to 100.

FIG. 11 .

 11 FIG. 11. Contour lines of constant N c in the 1D phase diagram for different values of w, f . Note the quantitative change in the T axis. The qualitative aspect is remarkably preserved.

7 ×FIG. 12 .

 712 FIG.12. Example of a transition observed in a Monte Carlo simulation with N = 1000 neurons, L + 1 = 2 environments, and T = 0.006. Neural configurations σ are shown at different times (black dots correspond to active neurons). Both panels show the same data, with the difference that neurons are ordered according to their place field centers π 1 (i) in environment 1 (top) and π 2 (i) in environment 2 (bottom). The transition takes place around time t 15 (time unit: one round of N steps).

  FIG.13. Evolution of E ≡ i<j J ij σ i σ j , for the same transition event as in Fig.12. E is the contribution of environment to the logarithm of the probability of the neural configuration σ ; see(2). The crossing of E 1 and E 2 defines the transition between the two maps, as well as the intermediary state, where the activity is weakly localized in both maps.

2 + T 2 ( 1 - 2 ) 2 ,FIG. 14 .FIG. 15 .

 221221415 FIG.14. Rate of transitions to other environments as a function of N for one realization of L + 1 = 2 1D environments and T = 0.006. Each point is averaged over 10 simulations of 1000 MC rounds. Time unit: 1 round of N steps. The decay of the rate is consistent with an exponentially decreasing function of N , hence with Arrhenius's law and the existence of free-energy barriers proportional to N .

FIG. 16 .

 16 FIG.16. Effect of partial activity on the theoretical free-energy barriers β √ V (top), on the diffusion constant D (bottom, left) and on the rate of transitions per round (bottom, right). Results correspond to the 1D case, T = 0.003, α = 0.003, N = 333. The dashed line indicates the limit of stability of the clump. The simulations length depends on the frequency of transitions: typically 1000 rounds for 1c = 0 and of the order of 10 2 rounds for 1c = 0.6. One round = 100N steps. Each point is averaged over 100 simulations.

FIG. 17 .

 17 FIG.17. Effect of partial activity on the diffusion constant D in the 2D case, with T = 0.002, α = 0.001, N = 45 × 45 units. The clump phase is not stable anymore when 1c exceeds 0.6. The simulation length depends on the frequency of transitions: typically, of the order of 10 3 rounds for 1c = 0 and of the order of 10 2 rounds for 1c = 0.6. 1 round = 100N steps. Each point is averaged over 30 simulations.

  th (T=0.006) μ 0,th (T=0.005) μ 0,sim (T=0.005) μ 0,sim (T=0.006)

FIG. 19 .

 19 FIG. 19. Velocity of the clump under a force A. Top: dimension one, T = 0.006, N = 1000 (the clump is not stable for larger A as indicated by the dashed line). Bottom: dimension two, T = 0.005, N = 32 × 32. Simulation time: 1000 rounds for α = 0; around 10 3 -10 4 rounds for α > 0. Each point is averaged over 10 simulations; one round = 20N steps.

FIG. 20 .

 20 FIG. 20. Bypassing of barriers in two dimensions. Top: Trajectory of the clump in the (x,y) plane under the effect of a force oriented rightward along the x axis, indicated by an arrow. Parameters are T = 0.004, α = 0.001, N = 32 × 32, A = 0.02, simulation time = 3.5 × 10 5 rounds of 20N steps. Bottom: Resulting contour plot of F ≡log τ tot (x,y), where τ tot (x,y) is the total time spent in position (x,y) after incorporation of periodic boundary conditions. F is an estimate of the free-energy landscape: deep local minima, hills and valleys appear; see gray-level scale on the right side.

FIG. 21 .

 21 FIG. 21. Evolution of the overlap with the retrieved pattern as a function of time, during two Monte Carlo simulations initialized in the clump phase in the same environment, at a position different from x 0 . N = 1000, T = 0.006, α = 0.01, d 0 = 0.05, time unit = 1 round of 20N steps.

FIG. 22 .

 22 FIG. 22. Average retrieval time in Monte Carlo simulations as a function of h (top) and d 0 (bottom). Each point is averaged over 10 simulations. N = 1000, T = 0.006, α = 0.01, time unit = 1 round of 20N steps. In the top panel, d 0 = 0.05.

FIG. 23 .

 23 FIG.23. Average retrieval time in Monte Carlo simulations as a function of h J , with two different initial conditions: a system in the paramagnetic phase (triangles) or in the clump phase in another environment (circles). Each point is averaged over 100 simulations. N = 1000, T = 0.006, α = 0.003, time unit = one round of 20N steps.

FIG. 24 .

 24 FIG.24. Adaptation: D (top) and frequency of transitions (bottom) as a function of the adaptation's intensity h adapt measured in a Monte Carlo in dimension 1 with N = 333 spins, α = 0.003, T = 0.004, and various values of τ adapt (time unit: 1 round of 20N steps). The clump is not stable for stronger h adapt . Depending on the frequency of transitions, simulation durations range from ∼10 to 1000 rounds; each point is averaged over 100 simulations. The estimated error on D varies between 5 × 10 -5 to a few 10 -3 when h adapt increases from 0 to 0.001.

FIG. 25 .

 25 FIG.25. Simulations of the case f (t) = f + δf sin(t/τ ): D as a function of δf measured in a Monte Carlo in dimension 1 with N = 1000 spins, α = 0.003, T = 0.005, w = 0.05, and various values of τ (time unit: 100N steps). The clump is not stable for stronger δf . For τ = 10 and τ = 100, each point is averaged over 100 simulations of length varying from a few tens of rounds to 1000 rounds depending on the frequency of transitions. For τ = 1000, longer simulations were necessary in order to cover several periods of f (t); each point is thus averaged over 10 simulations of duration up to 25 000 rounds.

FIG. 26 .

 26 FIG.26. Effect of asymmetric random dilution of synapses on D (top) and on the frequency of transitions to other environments (bottom): Monte Carlo simulations in dimension 1 with N = 333 spins, α = 0.003, T = 0.005. The clump is not stable for stronger dilution. Depending on the frequency of transitions, the simulations length varies between 1000 rounds and a few rounds. Each point is averaged over 1000 simulations. The estimated error on D varies between 5 × 10 -6 to 10 -4 when δ dil increases from 0 to 0.6.

  dr Dρ(x)Dμ(x) dλe -NβF , x)J w (xy)ρ(y)+ dxμ(x)ρ(x) -T dx Dz log(1 + exp{β[z √ αr + μ(x)]}). (A5) Dz ≡ exp(-z 2 /2)/ √ 2π is the Gaussian measure, ψ 1D (q,β) ≡ k 1

  dxρ(x) = f, which give r = 2T 2 (qf 2 )ϕ(q,T ), q = dx Du[1 + e -βu √ αr-βμ(x) ] -2 , ρ(x) = Du[1 + e -βu √ αr-βμ(x) ] -1 , (A10) μ(x) = dyJ w (xy)ρ(y) + λ, f = dxρ(x),

2 -

 2 Du log2 (1 + e β √ αru+βμ(x ) )+ Du log(1 + e β √ αru+βμ(x ) ) ( B 3 ) × Dv log(1 + e β √ αrv+βμ(x -x) ) -DuDv[κ(u,v) log(1 + e β √ α(r-r 12 )u+βμ(x ) )× log(1 + e β √ α(r+r 12 )v+βμ(x -x) )] ,

FIG. 27 .

 27 FIG.27. Overlap q 12 between two groups of replicas centered respectively on positions x and y, in dimension 1 with T = 0.006, α = 0.01 (full line) and in dimension 2 with T = 0.004, α = 0.002 (dashed line). Dotted lines indicate q 12 = f and q 12 = f 2 .

√ 1 × 1

 11 α(r-r 12 )-βμ(x ) -+ e -βv √ α(r+r 12 )-βμ(x -x) -1

Table 2 .

 2 1: Statistics of the bumps across cells having place fields (with the criterion 2.15), in environment A only, B only and both. The brackets . i indicate the average across cells. SD i (.) indicates the standard deviation across cells.

		only B only both
	nb. of cells with p.f.	22	18	34
	s i	0.047	0.044 0.045
	ρ max (Hz)	38.0	33.4	38.0
	h i (Hz)	17.2	14.5	16.0
	h i ρ max	0.45	0.43	0.42
	SD i (h) (Hz)	10.0	8.5	9.44
	SD i (h) ρ max	0.26	0.25	0.25

Table 3 . 1 :

 31 Summary of the environment-decoding methods introduced here.

	.1.
	rate-based

  .2), so we can rule out this last possibility.

	pair 1-17 3-11 3-31 3-33 9-13 9-15 9-26 11-33 20-29 25-29 27-30
	i, j
	J A ij	1.679 2.341 2.331 0.788 2.049 0.362 3.893 2.146 3.558 4.163 5.326
	J B ij	2.037 4.986 4.206 1.986 4.342 3.994 3.903 2.282 2.526 2.100 1.997
	J T ij	1.840 3.059 2.316 0.922 2.524 0.913 2.587 2.495 2.557 2.700 2.536
	Table

  Possible mechanisms enhancing mobility are proposed and tested. DOI: 10.1103/PhysRevE.89.032803 PACS number(s): 87.85.dq, 05.20.-y, 87.19.lv, 82.39.Rt

In the Hopfield model, the frustration comes from the Hebb rule (1.1), which leads both to excitatory and inhibitory couplings.

To be exact, there are also CA2 and CA4 regions, but they are very small.

Some of these experiments have been done on CA1, which is not a recurrent network (it has a feedforward connectivity). Nevertheless, it is fed by input from CA3 so its response in morphing experiments could reflect transitions occurring one synapse upstream in the presumed attractor network CA3[START_REF] Wills | Attractor dynamics in the hippocampal representation of the local environment[END_REF].

The same fact can be either in the model's ingredients or in its outcomes: for instance, the theta rhythm can be taken as a datum or it can be the consequence of a mechanism built to explain it.

We will hereafter use indifferently the terms "neuron", "place cell" and "spin", from the analogy with magnetic systems (see Section 1.2).

In this initial, basic version of the model, all place cells have place fields in all environments. The possibility of silent cells has been taken into account in an extension that will be described below.

Strictly speaking, one map is equivalent to N configurations (patterns in the Hopfield sense) with a high degree of correlation between them.

In the α > 0 case, we have observed it in simulations, but it is much more difficult to treat it analytically.

For this reason, the calculation by Battaglia & Treves with threshold linear units must be performed at zero temperature, even if the T > 0 case is technically possible to implement[START_REF] Treves | Graded-response neurons and information encodings in autoassociative memories[END_REF]: introducing stochasticity in the response would amount to count it twice, as this stochasticity is already present (on average) in the gain.

In fact, this opposition is not so fundamental, since what we call the neural code is but the result of microscopic laws. It is a code in the same extent as, for example, the shape of clouds codes for temperature, pressure, humidity and altitude. In principle, therefore, the correspondence between the environment and the neural activity could as well be entirely derived in a causal, mechanistical way. But the extreme complexity of the system and, above all, the fact that it is a living system shaped by evolution, make it convenient and reasonable to regard neural activity as an effective language.

More precisely, if we know how it has been encoded, then we can decode it by inverting the "coding function" ("model-based decoding"[START_REF] Paninski | Statistical models for neural encoding, decoding, and optimal stimulus design[END_REF]). Another approach is to build a dictionary between stimuli and responses, but this approach is limited first by the response variability aforementioned, and most of all by the technical impossibility to sample all possible stimuli.

Another way would be to record more cells simultaneously: this is made more and more possible by the improvement of recording techniques.

Note that the "rate -max. posterior" and "rate -independent" methods also describe the probabilities of neural configurations in terms of an underlying model. In this case, the underlying model consists in cells firing independently with a Poisson distribution with a parameter depending on the position in the former case, on the whole map in the latter. We will come back to the link between these models and the Ising model in Section 3.8.

A similar procedure has been followed by studies using information geometry, see e.g.[START_REF] Tatsuno | Investigation of possible neural architectures underlying information-geometric measures[END_REF].

Dans ce genre d'étude, on distingue les méthodes analytiques (par le calcul formel) des méthodes numériques (par la simulation). Dans le meilleur des cas, on est capable de faire l'étude analytique et on vérifie les résultats grâce aux simulations numériques. Parfois, le système est trop compliqué pour être traité analytiquement et on ne peut l'étudier que numériquement.

Remerciements

ACKNOWLEDGMENTS

We are deeply indebted to J. Hopfield for enlightening discussions, without which this work would not have been possible. We are grateful to S. Cocco, F. Stella, A. Treves, and J.-B. Zuber for very useful discussions, and to N. Brunel for a critical reading of the manuscript. R.M. acknowledges the hospitality and the financial support of the Simons Center for Systems Biology, at the Institute for Advanced Study, Princeton, where the initial part of this work was done. The work of S.R. was supported by a grant from Délégation Générale de l'Armement.

ACKNOWLEDGMENTS

We are indebted to J. Hopfield for very fruitful discussions, in particular on the emergence of the clump as a collective coordinate of the dynamics. We are grateful to F. Stella for useful discussions. The work of S.R. is supported by a grant from Délégation Générale de l'Armement.

Appendices

APPENDIX A: FORMULAS FOR TWO-DIMENSIONAL MAPS

The only difference in the replica computation lies in the eigenvalues of the coupling matrix. Thus, in dimension 2, the free energy functional writes

APPENDIX A: REMINDER ON THE FREE-ENERGY CALCULATION

In Ref. [25] we computed the average free energy of the system over random remappings. To do so, we used the replica method under the replica-symmetric assumption. In this appendix we remind the main results of this calculation.

The average partition function of the replicated system is

Généralités

Comprendre le fonctionnement du cerveau veut dire deux choses. D'une part, il s'agit d'expliquer ce que l'on observe (une activité électrique, métabolique...) de façon purement mécaniste, sans chercher à y assigner une fonction biologique. D'autre part, le cerveau est un système vivant et cela lui confère des propriétés particulières. Notamment, on peut lui assigner des fonctions, produits de l'évolution conférant à l'organisme un avantage sélectif. La seconde tâche du neuroscientifique est donc de déterminer quelles sont ces fonctions et comment elles sont implémentées.

Dans cette thèse, nous nous intéressons à la fonction de mémoire et à l'activité très particulière d'une région du cerveau appelée hippocampe. Nous allons d'abord préciser quelques notions et décrire cette activité.

Le cerveau est composé de neurones. Depuis un siècle, la thèse prévaut selon laquelle ces neurones sont l'unité de base de toutes les fonctions cérébrales. Ces cellules présentent une activité électrique qui peut se propager de neurone en neurone via des connexions appelées synapses (voir Figure 1.1). Cette activité a une forme stéréotypée quasi universelle -les potentiels d'actions. Un neurone reçoit des courants électriques provenant de ses voisins afférents. Si son potentiel électrique excède un certain seuil, le neurone envoie une décharge d'environ 100 mV sur 1 ms à ses voisins efférents (on dit qu'il émet un potentiel d'action). Ces voisins reçoivent ce signal, et ainsi de suite. Il y a environ 100 milliards de neurones chez l'homme (10 11 ), chacun est en moyenne connecté à 10 000 autres neurones (soit de l'ordre de 10 15 synapses au total). Certaines synapses sont excitatrices, d'autres sont inhibitrices. De plus, les synapses sont dites plastiques, c'est-à-dire variables au cours du temps (sur des échelles de temps très supérieures à celles des potentiels d'action) : en fonction de l'activité des neurones, certaines synapses vont être renforcées, d'autres affaiblies, et de nouvelles synapses vont être créées. Cette modification va engendrer à son tour des changements dans l'activité des neurones. La plasticité synaptique est donc à la base de l'apprentissage et de la mémoire : l'activité passée des neurones modifie la structure du réseau neuronal et influe ainsi sur l'activité future.

B.1.2 Les cellules de lieu

Venons-en à l'hippocampe. L'hippocampe est une structure subcorticale du cerveau située à la convergence de tous les circuits sensoriels. Les techniques de micro-électrodes insérées