Résumé

L'information visuelle est transmise de la rétine au cerveau par les cellules ganglionnaires. Il existe plusieurs types de cellules ganglionnaires, chaque type formant une mosaïque qui couvre l'intégralité de la scène visuelle. Comprendre la manière dont ces neurones encodent collectivement la scène visuelle est essentiel pour au moins deux raisons:

• La manière dont une population de neurones encodent collectivement une information sensorielle reste jusqu'à aujourd'hui mystérieuse. La rétine est un système idéal pour étudier cette question: elle a en effet une structure en couches 2D qui se prête idéalement à l'enregistrement d'une population complète de neurones à grande échelle, et elle opère une transformation complexe de la scène visuelle.

• Certaines maladies qui mènent à la cécité ne connaissent aujourd'hui pas de traitement. Plusieurs stratégies de restoration visuelle basées sur la stimulation directe de cellules ganglionnaires sont le sujet de recherches actives. Il pourrait être nécessaire de reproduire en imitant le code neural produit par la rétine pour optimiser les résultats de ces stratégies thérapeutiques.

Au cours de ma thèse, j'ai travaillé sur deux questions complémentaires, qui sont liées à ces deux sujets respectivement.

Dans une première partie, j'ai étudié la manière dont les cellules ganglionnaires encodent ensemble une scène visuelle complexe. Une hypothèse courante est que les cellules ganglionnaires d'un même type traitent la scène visuelle en se concentrant sur un seul aspect de la scène (par exemple les cellules sélectives à la direction), générant ensemble une version filtrée de la scène pour le cerveau. En procédant à des enregistrements de rétines de rat à l'aide d'une matrice d'électrode (MEA), nous montrons qu'une population homogène de cellules ganglionnaires de type "Fast OFF" combine dans sa réponse à une scène visuelle dynamique deux aspects radicalement différents de la scène. Les cellules dont le champs récepteur se trouve à proximité d'un objet qui bouge encodent linéairement sa position, tandis que les cellules distantes restent largement invariantes à la position de l'objet et encodent de manière non-linéaire les changements de vitesse de l'objet. Les cellules individuelles passent d'un calcul à l'autre en fonction du stimulus. Cette stratégie consistant à multiplexer les calculs pourrait jouer un rôle clé dans la réduction de dimensionnalité drastique effectuée par la rétine.

Dans une seconde partie, j'ai étudié la capacité d'une rétine dégénérée à transférer l'information visuelle après que les cellules ganglionnaires aient été rendues photo-sensibles par l'expression d'une protéine optogénétique. Nous avons mesuré les champs récepteurs de ces cellules ganglionnaires dans des explants de rétine de souris et de macaque. Nous avons montré qu'un modèle de réponse classique calculé à partir du champs récepteur (modèle LNP) est capable de prédire correctement la réponse de ces cellules à des stimuli complexes. En généralisant notre modèle pour une cellule unique à la population entière de cellules réactivées, nous avons pu simulé un test d'acuité in silico, en utilisant une approche de décodage Bayesian (utilisation de toute l'information présente par le décodeur). L'estimation de l'acuité visuelle obtenue avec cette méthode chez le macaque dépasse le seuil légal de cécité, un résultat prometteur en vue des futurs essais cliniques. Nous proposons ainsi une méthode indirecte pour estimer l'acuité maximale que pourrait retrouver un patient ayant bénéficié de cette thérapie. A l'avenir, notre méthode pourrait fournir une prédiction quantitative sur l'issue de nouvelles stratégies v thérapeutiques, avant même les essais cliniques sur des patients humains.

Pour conclure, notre travail montre une flexibilité inattendue du code neural dans la rétine. La compréhension de ce code pourraient être nécessaire pour reproduire une vision naturelle avec des techniques de restoration visuelle qui ciblent directement les cellules ganglionnaires.

Summary

Visual information is conveyed from the retina to the brain through ganglion cells. Ganglion cells are divided in different cell types, and each of them form a mosaic sampling the entire visual scene. Understanding how these neurons encode the visual scene is essential for at least two reasons:

• It is still unclear how a population of neurons collectively code sensory information. The retina is an ideal system to study this issue: while it performs complex processing on the visual stimulus, its 2-D structure makes it suitable for large-scale recordings of complete populations of neurons.

• Some diseases leading to blindness have currently no cure. Several visual restoration strategies based on the direct stimulation of ganglion cells are currently being investigated. Emulating the retinal code may be necessary to optimize the results of these therapeutic approaches.

In my thesis I have worked on two complementary questions, that are related to these two topics.

In a first part, I studied how retinal ganglion cells code together a complex visual scene. A common assumption is that ganglion cells of the same type extract a single feature of the scene (e.g. direction-selective cells), forming together a feature map. Using large-scale recordings in the rat retina, we show that a homogeneous population of fast OFF ganglion cells multiplexes two radically different features of a dynamical visual scene. Cells close to a moving object code linearly for its position, while distant cells remain largely invariant to the object's position and, instead, respond non-linearly to changes in the object's speed. Individual cells switch between these two computations depending on the stimulus. Therefore, a single ganglion cell type contains two feature maps rather than one. This strategy of multiplexing computations might play a key-role in the strong dimensionality reduction operated by the retina.

In a second part, I studied the ability of ganglion cells from a degenerated retina to transfer information when they are made light sensitive by optogenetic proteins. We measured the receptive fields of ganglion cells re-engineered to become light sensitive in explants of mouse and macaque retinae. We showed that a classical response model inferred from the receptive field profile (LNP model) predicted well the responses of these cells to complex stimuli. Using our model of the complete population of reactivated cells, we could estimate how well different letters of an acuity test could be discriminated by the brain, assuming an optimal use of information (ideal observer analysis). The acuity estimated with this method in macaque was above the level of legal blindness, a promising result for future clinical studies. We thus propose an indirect method to estimate the maximum acuity that could be obtained with an optogenetic visual restoration strategy. In the future, our method could provide a quantitative prediction for the outcome of novel therapeutic strategies before they are applied to human patients in clinical trials.

In conclusion, our work shows an unexpected flexibility in the way the retina encodes visual information. Understanding this flexible code might be necessary to emulate natural vision with restoration strategies targeting ganglion cells. The aim of this chapter is to provide the reader with an overview of the normal function (i.e. physiology) of retinal ganglion cells (RGCs), the output of the retina. The chapter starts with a brief description of the anatomical structure of the retina, followed by an extensive description of the responses of ganglion cells to various stimuli, a thorough description of the models and anatomical substrates underlying these responses, and a presentation of the different classifications of ganglion cells based on these responses.

Throughout the chapter, an emphasis will be given to studies showing how RGCs perform spatial integration of visual stimuli, and to examples of non-local processing by RGCs (i.e. processing of stimuli located outside the classically defined receptive field). This emphasis is meant to provide the reader with all the information required to assess the novelty of the contribution of the thesis presented in chapter III.

Starting with the most salient and universal features of ganglion cells physiology, we will then describe the more subtle aspects of the responses and the specificities of each cell type. As a consequence of this choice, we will often start by reviewing the early studies of retinal physiology and follow the chronological order of discoveries, as the most salient and universal features were often discovered first.

We will present results from different species in order to show the universality of the principles presented, but we will not emphasise the differences between species, although they certainly exist. Most of the data presented holds true for all vertebrates, and we tried to be explicit about characteristics that are found only in mammals or primates.

In part one, we will present the anatomy of the retina and the most salient features CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS of ganglion cells physiology. In part two, we will describe the spatial integration of visual scenes performed by retinal ganglion cells. In part three, we will focus on the specificities of each cell type and show that ganglion cells have a broad range of selectivity to complex features.

Basic facts about the retina

At the first stage of vision, light enters the eye through the pupil and is focused by the cornea and lens to form an image onto rod and cone photoreceptors.

Anatomical facts

General organization. Photoreceptors convert light into an electrical potential that modulates synaptic transmission to a second layer of neurons, the bipolar cells (fig. I.1). Bipolar cells typically pool inputs from multiple photoreceptors and convey a processed version of these inputs to ganglion cells. Ganglion cells axons (i.e. output terminals of neurons) form the optic nerve [START_REF] Demb | Functional Circuitry of the Retina[END_REF]. Visual information travels vertically from photoreceptors to bipolar cells and retinal ganglion cells, while horizontal and amacrine cells modulate this retinal processing by lateral interactions in the two plexiform layers.

Cell types. Mammalian retinal circuits are constructed from ª100 specific cell types [START_REF] Demb | Functional Circuitry of the Retina[END_REF][START_REF] Euler | Retinal bipolar cells: elementary building blocks of vision[END_REF][START_REF] Masland | The Neuronal Organization of the Retina[END_REF][START_REF] Sanes | The types of retinal ganglion cells: current status and implications for neuronal classification[END_REF]; Seung and Sümbül, 2014). These include 3 to 4 types of photoreceptor (1 rod for night vi-CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS overall subdivision of the IPL into On-layers, where bipolar cells excited by an onset of light stratify, and Off-layers, where bipolar cells excited by an offset of light stratify. Within the On-and Off-layers there are further subdivisions, such that at least 10 strata receive unique and substantively different excitatory and inhibitory inputs [START_REF] Roska | Vertical interactions across ten parallel, stacked representations in the mammalian retina[END_REF].

Mosaic-like tiling of the visual field. Ganglion cells of each type tile the retina, such that all regions are covered by at least one member of each type, with varying degrees of overlap (fig. I.3A-C, [START_REF] Borghuis | Design of a Neuronal Array[END_REF][START_REF] Masland | The Neuronal Organization of the Retina[END_REF][START_REF] Wassle | Cone Contacts, Mosaics, and Territories of Bipolar Cells in the Mouse Retina[END_REF][START_REF] Wässle | Parallel processing in the mammalian retina[END_REF]. The different types sample the world through apertures of a different size, partially determined by the size of their dendritic field (fig. I.3D). Within a cell type, cell size varies within the retina: neurons with smaller dendritic fields are found in areas devoted to high-acuity vision, such as the primate fovea [START_REF] Kolb | The midget pathways of the primate retina[END_REF].

Projections to the brain. The mouse retina projects to approximately 40 different brain regions [START_REF] Morin | Retinofugal projections in the mouse: Mouse Visual System[END_REF]. Individual ganglion cell types selectively connect to specific central targets dedicated to a range of functions: orienting attention, regulating circadian rhythms, controlling eye movements, and of course, generating visual perception [START_REF] Dhande | Contributions of Retinal Ganglion Cells to Subcortical Visual Processing and Behaviors[END_REF].

Functional responses

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS Aggregated responses. Adrian and Matthews (1927a,b) performed the first recordings from the optic nerve. At this time the recording technique only allowed to record the aggregated activity of a large number of optic nerve fibers. They found that, like any other sensory nerves, the optic nerve sends brief and stereotypical impulses of currents (ª1ms) to the brain, called spikes or action potentials. The size of spikes is not affected by light intensity, but the frequency of the spike train is. When the eye is illuminated, the discharge of impulses rises rapidly to a maximum frequency and then declines, at first rapidly and then more slowly. If the illumination has lasted a second or more, there is a renewed outburst of impulses when the light is turned off (fig. I.4A). When it became possible to record single optic nerve fibers, physiologists found that the different aspects of the responses described by Adrian and Matthews could be attributed to different cell types.

On, Off and On-Off cells. Using diffuse illumination in the bullfrog and other vertebrates, [START_REF] Hartline | The response of single optic nerve fibers of the vertebrate eye to illumination of the retina[END_REF] found that ganglion cells could be divided in three classes: cells that increased their firing rate in response to the onset of the light (On cells), cells that responded only to the offset of the light (Off cells), and those that responded to both onset and offset (On-Off cells) (fig. I.4B). Later, this classification was confirmed in mammals [START_REF] Gernandt | Single fibre analysis of inhibition and the polarity of the retinal elements[END_REF].

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

Discovery. The term receptive field (RF) was introduced by [START_REF] Sherrington | Observations on the scratch-reflex in the spinal dog[END_REF] to mark the area on a dog's skin from which a scratch-reflex could be elicited. Hartline applied the term to that area on the retina from which an excitatory response could be elicited in an optic nerve fiber of a frog [START_REF] Hartline | The receptive fields of optic nerve fibers[END_REF]. Hartline also noted that the sensitivity of a cell to light is not uniform over the receptive field.

Mapping techniques.

The receptive field of a cell can be mapped with several stimuli, and in particular:

• by probing different regions with a spot smaller than the whole RF [START_REF] Hartline | The receptive fields of optic nerve fibers[END_REF][START_REF] Kuffler | Discharge patterns and functional organization of mammalian retina[END_REF],

• by increasing the size of a spot centered on the RF until the response saturates [START_REF] Cook | Lateral inhibition in the inner retina is important for spatial tuning of ganglion cells[END_REF][START_REF] Chapter | The control of retinal ganglion cell discharge by receptive field surrounds[END_REF]Wiesel, 1960),

• by displaying a checkerboard stimulation where each check flips randomly between white and black at a fast rate (ª60Hz) and independently from the other checks [START_REF] Chapter | A simple white noise analysis of neuronal light responses[END_REF][START_REF] Hunter | The identification of nonlinear biological systems: Wiener and Hammerstein cascade models[END_REF][START_REF] Meister | The neural code of the retina[END_REF][START_REF] Sakai | White-noise analysis in neurophysiology[END_REF]. In this technique, the RF is obtained by averaging all the frames that triggered a spike (Spike-triggered average, see fig. I.5 for a schematic).

The extent to which these methods agree on the receptive field size of a cell is unclear, but to my knowledge no major discrepancies have been reported between different methods for mapping the RFs. • RGC types have not been well identified yet in the salamander. In mouse, more than 30 different cell types were recently identified [START_REF] Baden | The functional diversity of retinal ganglion cells in the mouse[END_REF], suggesting that the cell classes identified by [START_REF] Segev | Functional Organization of Ganglion Cells in the Salamander Retina[END_REF] might be composed of a combination of many more types.

Anatomical

• RGCs do not cluster in types in salamander, but rather constitute a continuum.

• A study [START_REF] Tkačik | Optimal population coding by noisy spiking neurons[END_REF] suggests that at high signal-to-noise ratios (SNR), redundancy between cells of a same type might become an advantage.

Temporal dynamics. Different cell types respond with different temporal dynamics to the stimulation of their receptive field. Some are fast ("brisk cells", peak response ª50ms after stimulation), some are rather slow ("sluggish cells", peak response ª100ms after stimulation) [START_REF] Cleland | Brisk and sluggish concentrically organized ganglion cells in the cat's retina[END_REF][START_REF] Segev | Functional Organization of Ganglion Cells in the Salamander Retina[END_REF]. Some cell types exhibit biphasic temporal profiles (fig. I.6E, [START_REF] Segev | Functional Organization of Ganglion Cells in the Salamander Retina[END_REF]). Under linear assumption, a biphasic cell responds mostly to the onset of its preferred stimulus, and as such can be classified as a transient cell in the taxonomy of [START_REF] Cleland | Sustained and transient neurones in the cat's retina and lateral geniculate nucleus[END_REF].

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

Classifications based on RF. Several attempts have been made to classify cells based on the properties of their receptive fields [START_REF] Chichilnisky | Functional asymmetries in ON and OFF ganglion cells of primate retina[END_REF][START_REF] Farrow | Physiological clustering of visual channels in the mouse retina[END_REF][START_REF] Segev | Functional Organization of Ganglion Cells in the Salamander Retina[END_REF]. Usually, these attempts have successfully separated up to 6 classes of cells. However, a recent functional classification showed that there are more than 30 different functional types of RGCs in mouse [START_REF] Baden | The functional diversity of retinal ganglion cells in the mouse[END_REF]. The volume of cells recorded (11,000 cells) and the use of a dense recording technique (calcium imaging) in the latter study can partly explain this discrepancy. Nevertheless, [START_REF] Baden | The functional diversity of retinal ganglion cells in the mouse[END_REF] have also used a number of other stimuli (moving objects, full-field varying in amplitude or in frequency, colors) to refine their classification of RGCs. Therefore this study suggests that cell types cannot be identified solely from their linear response properties to white noise or flashes.

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

2 Spatial integration in the retina 2.1 The center-surround structure

Phenomenology

Center-surround structure: concentric arrangement of regions of the receptive field characterized by an antagonism between center and surround.

Discovery.

By flashing small spots in different regions of the RF of ganglion cells in cat, [START_REF] Kuffler | Discharge patterns and functional organization of mammalian retina[END_REF] found that the spot could elicit an On response, an Off response, or an On-Off response depending on the region where it was flashed. He found a concentric arrangement of these different regions [START_REF] Kuffler | Discharge patterns and functional organization of mammalian retina[END_REF], fig. I.7A). Both center and surround could elicit an excitatory response of the cell.

Antagonistic interplay.

In the same year, [START_REF] Barlow | Summation and inhibition in the frog's retina[END_REF]

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

Universality. In cat (Wiesel, 1960) and macaque (De Monasterio, 1978a), most ganglion cells have center-surround organization. Each receptive field has a center more or less circular in shape, surrounded by a peripheral zone. RF centers varies from 0.125 mm (0.5 ± ) to 2 mm (8 ± ). Surround varies from 6 ± to 12 ± . This concentric arrangement of regions producing opposite and antagonistic effects was found in most ganglion cells of many species, including frog [START_REF] Barlow | Summation and inhibition in the frog's retina[END_REF] , mouse [START_REF] Stone | Response properties of ganglion cells in the isolated mouse retina[END_REF], rabbit [START_REF] Barlow | Retinal ganglion cells responding selectively to direction and speed of image motion in the rabbit[END_REF] and spider monkey [START_REF] Hubel | Receptive fields of optic nerve fibres in the spider monkey[END_REF]. The circular symmetry rule has some exceptions. In the rabbit, some RGCs have an anisotropic surround and respond mostly to elongated stimuli such as bars [START_REF] Levick | Receptive fields and trigger features of ganglion cells in the visual streak of the rabbits retina[END_REF]. These two early experiments are traditionally cited to define the classical surround of RGCs. Here is a list of interesting properties about the surround that we can already deduce form these experiments:

1. The surround can be excitatory. The experiment of Kuffler in cat (1953) shows that exclusive stimulation of the surround by small spots of lights can elicit spikes. This experiment also shows that the classical surround is not always subliminal, even for small objects.

2. The surround is not linear. In the experiment of [START_REF] Kuffler | Discharge patterns and functional organization of mammalian retina[END_REF], there is a region of the surround where both On and Off responses can be elicited.

3. The surround contribution is not strictly additive. If the surround was purely additive with the center, flashing the second spot alone in the experiment of [START_REF] Barlow | Summation and inhibition in the frog's retina[END_REF] would elicit a negative response of the ganglion cell. This is however not possible because the cell was already silent before stimulation (no spontaneous firing rate) and cannot produce a negative firing rate. We will see in the next section that CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS to account for the transient response of RGCs.

3. The result of the convolution is rectified so that the firing rate predicted is always positive.

After fitting the parameters of the DoG model to the response of a cat RGC to small spots of light, Rodieck and Stone (1965a,b) were able for the first time to predict quantitatively the firing rate of this cell in response to moving shapes of different sizes (fig. I.9B).

Issues with the DoG model

Spatio-temporal separability. The DoG model assumes that each elementary spatial component integrates the stimulus with the same temporal dynamics. Subsequent work showed that space-time separability is not quite satisfied in ganglion cell responses: for example, the response to light falling in the surround is delayed relative to the response in the center, owing to the time required for lateral signal flow through horizontal or amacrine cells, and transmission across an additional synapse [START_REF] Benardete | The receptive field of the primate P retinal ganglion cell, I: Linear dynamics[END_REF][START_REF] Enroth-Cugell | The receptive-field spatial structure of cat retinal Y cells[END_REF][START_REF] Sakai | Response dynamics and receptive-field organization of catfish ganglion cells[END_REF].

Global non-linearity. Another fundamental feature of Rodieck's model is the linearity of its response. Twice the intensity fluctuation will produce twice the firing rate fluctuation. Subsequently, it was found that a linear relation between stimulus and firing rate is true only when the modulations of the light intensity are small compared to the mean light intensity [START_REF] Benardete | The receptive field of the primate P retinal ganglion cell, I: Linear dynamics[END_REF][START_REF] Sakai | Response dynamics and receptive-field organization of catfish ganglion cells[END_REF]. RF shape is not Gaussian. RGC receptive fields in primate exhibit large-scale structure that deviates from either circular or elliptical Gaussian profiles (Gauthier et al., 2009a).

RFs of same type are coordinated to sample visual space uniformly (fig. I.10A). Moreover,

Anatomical substrate

How is the antagonistic surround generated? A first possibility that would involve only the feed-forward circuit (photoreceptors, bipolar cells, ganglion cells), would be that ganglion cells are directly fed by bipolar cells of opposite polarity in their center and in their surround. This hypothesis was ruled out because most ganglion cells stratify only in the On-bipolar strata or only in the Off-bipolar strata but still present the center-surround antagonism [START_REF] Wassle | Functional architecture of the mammalian retina[END_REF]. Two circuits, involving respectively horizontal cells and amacrine cells, were demonstrated to produce the antagonistic surround.

Horizontal cells. Horizontal cells transmit information laterally from photoreceptor cells

to other photoreceptors and to bipolar cells. They have been shown to contribute to the surround of ganglion cells in fish [START_REF] Naka | Dogfish ganglion cell discharge resulting from extrinsic polarization of the horizontal cells[END_REF] and rabbit [START_REF] Mangel | Analysis of the horizontal cell contribution to the receptive field surround of ganglion cells in the rabbit retina[END_REF].

However the extent of this contribution to the antagonistic surround is unclear, because the effect was studied in a rather unnatural setting in which single horizontal cells were electrically stimulated [START_REF] Cook | Lateral inhibition in the inner retina is important for spatial tuning of ganglion cells[END_REF] [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF]. Interestingly, amacrine cells have little or no contribution to the surround of cells with large RF centers in salamander [START_REF] Cook | Lateral inhibition in the inner retina is important for spatial tuning of ganglion cells[END_REF]. Consistently with this observation, parasol cells in primates, where the predominant influence CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS I.12). The divergence between these cell types appears when the spatial extent of the stimulus becomes much smaller than the RF-center size [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF]. For large drifting gratings (with the size of a stripe approximately equal to the size of the RF center), the response of a Y cell is dominated by the F1 Fourier component (i.e. the cell responds once during every cycle of the drifting grating). This is the signature of a linear behaviour. However, when the stripes become much smaller that the center of the RF, the Fourier F2 component of the response becomes dominant (the cell responds twice during every cycle of the drifting grating), which is indicating non-linear spatial integration. The transition between the linear and non-linear behaviour of Y cells with increasing spatial frequency can be obtained by stimulating the Y-cell in the center of its RF only, in the surround only and in both center and surround with a full-field grating [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF]. The presentation of a fine contrast-reversing grating in the center of a Y cell excites the cell, but the same stimulus presented in the surround inhibits the cell [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF].

For fine spatial frequencies, Y cells thus present an antagonistic centre-surround analogous to the X-cell, but instead of responding to a change in mean luminance, it responds to the absolute amount of change in local luminance by an excitation in its center and an inhibition in its surround.

Response to texture motion. Y-type cells respond to texture motion, largely independently of the exact texture and direction of motion [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF] Universality. Hochstein and [START_REF] Chapter | Quantitative analysis of retinal ganglion cell classifications[END_REF] confirmed that cells could be well separated in X and Y classes in the cat. Y-cells were found in many other species such as guinea pig [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF][START_REF] Zaghloul | Functional circuitry for peripheral suppression in mammalian Y-type retinal ganglion cells[END_REF], rabbit [START_REF] Caldwell | New properties of rabbit retinal ganglion cells[END_REF][START_REF] Famiglietti | Class I and class II ganglion cells of rabbit retina: a structural basis for X and Y (brisk) cells[END_REF][START_REF] Hamasaki | Properties of X-and Y-cells in the rabbit retina[END_REF], and monkey [START_REF] Crook | Y-cell receptive field and collicular projection of parasol ganglion cells in macaque monkey retina[END_REF]De Monasterio, 1978b;[START_REF] Petrusca | Identification and characterization of a Y-like primate retinal ganglion cell type[END_REF]. Note that the separation between these two classes does not always appear clear-cut and may in some systems rather represent the extremes of a continuum with different degrees of nonlinear integration, as reported, for example, in the mouse retina [START_REF] Carcieri | Classification of Retinal Ganglion Cells: A Statistical Approach[END_REF] and guinea pig retina [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF]. Anatomical substrate. Bipolar cells receptive fields match subunit size of Y-cells [START_REF] Dacey | Center surround receptive field structure of cone bipolar cells in primate retina[END_REF][START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF], they can be transient, and they transmit a rectified response to light steps at their synaptic terminal [START_REF] Werblin | Six different roles for crossover inhibition in the retina: Correcting the nonlinearities of synaptic transmission[END_REF]. For all these reasons bipo- CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS lar cells are thought to be the substrate of non-linear subunits in Y-cells [START_REF] Demb | Bipolar cells contribute to nonlinear spatial summation in the brisk-transient (Y) ganglion cell in mammalian retina[END_REF][START_REF] Gollisch | Eye smarter than scientists believed: neural computations in circuits of the retina[END_REF].

Subunit Model and Anatomical substrate

If bipolar cells systematically operate a rectification of their input at their synaptic output, how could X-cells perform a linear spatial summation of their inputs in response to spatial patterns such as a reversing grating? Werblin (2010) reveals a circuit in which a rectified On bipolar and a rectified Off bipolar cell combine to produce a linear integration of positive and negative steps of light at the level of the ganglion cell (fig. I.15E,F). This circuit is called 'crossover inhibition' because it takes an inhibitory amacrine cell to propagate a polarity-reversed version of the response of one of the bipolar cells to the target ganglion cell.

Parameter Fitting. The parameters of the subunit model consist of the coefficients of the linear filters at each stage (filters of bipolars and of the RGC), and the nonlinearity parameters at each stage. Techniques for estimating the subunit model parameters include:

• Choosing biophysically relevant parameters by hand for the bipolar stage and fit only the RGC parameters (all bipolar of a same type have identical parameters and tile the visual field) [START_REF] Chen | Alert Response to Motion Onset in the Retina[END_REF]. This goes back to fitting the parameters of an LN model to a pre-filtered version of the stimulus.

• Alternating between optimizing the bipolar filters and bipolar non-linearities, in each case holding the remaining set of parameters constant (block gradient ascent,

McFarland et al. (2013))

. The ganglion cell parameters can be optimized simultaneously during either (or both) optimization stages. For a fixed set of bipolar nonlinearities, the bipolar filters can be optimized, although the resulting likelihood surface will not in general be convex because the filters operate inside the bipolar nonlinearities. Nevertheless, [START_REF] Mcfarland | Inferring Nonlinear Neuronal Computation Based on Physiologically Plausible Inputs[END_REF] found that in practice their op-CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS timization is well-behaved and that local minima can be avoided with appropriate optimization procedures. [START_REF] Vintch | A Convolutional Subunit Model for Neuronal Responses in Macaque V1[END_REF][START_REF] Vintch | Efficient and direct estimation of a neural subunit model for sensory coding[END_REF] has proposed a similar iterative coordinate descent scheme.

• Other techniques include finding classes of equivalence with other optimization problems, such as non-negative matrix factorization (Gollisch et al, unpublished) or spike-triggered-covariance [START_REF] Wu | Convolutional spike-triggered covariance analysis for neural subunit models[END_REF]. These frameworks are equivalent to the subunit model under certain constraints (e.g. quadratic shape of the bipolar non-linearity, positivity of all filter weights). largely orthogonal to On/Off/On-Off classification. However, we found no plausible subunit model for an On-Off X cell. To our knowledge, there was no report of On-Off X cells in the literature. Therefore we conclude that Y cells are a priori not all On-Off cells, but that X cells can probably not be On-Off.

X/Y cells and previous classifications

Transient/sustained classification. [START_REF] Cleland | Brisk and sluggish concentrically organized ganglion cells in the cat's retina[END_REF] suggested that Y cells are always transient and X cells are always sustained in cat. However, subsequent studies showed that ganglion cells in cat and rabbit cannot be classified as X or Y simply on the basis of the transient or sustained nature of their response [START_REF] Caldwell | New properties of rabbit retinal ganglion cells[END_REF].

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

I.17A). The shift-effect was also present is some X cells but the responses were weaker and sluggish (time to peak is ª0.5s) [START_REF] Barlow | The effects of remote retinal stimulation on the responses of cat retinal ganglion cells[END_REF][START_REF] Derrington | The mechanism of peripherally evoked responses in retinal ganglion cells[END_REF][START_REF] Fischer | Quantitative aspects of the shift-effect in cat retinal ganglion cells[END_REF].

Are these responses truly different from the classical antagonistic surround? Yes. First, the polarity of this far surround is opposite to the polarity of the classical surround [START_REF] Ikeda | The outer disinhibitory surround of the retinal ganglion cell receptive field[END_REF]. Stimulation with a light spot in the far surround gives an increase in firing in response to an On flash in On-center cells and to an Off flash in Off-center cells (fig.

I.17B,C). The far surround is also not antagonistic to the center. A spot presented in the far surround causes an enhancement of the central response when flashing in phase with the center spot, while it causes an inhibition of the central response when presented 180 ± out of phase [START_REF] Ikeda | The outer disinhibitory surround of the retinal ganglion cell receptive field[END_REF]. The sensitivity of the far surround is much weaker than the sensitivity of the center [START_REF] Ikeda | The outer disinhibitory surround of the retinal ganglion cell receptive field[END_REF], and these responses are suppressed by anaesthetics [START_REF] Barlow | The effects of remote retinal stimulation on the responses of cat retinal ganglion cells[END_REF][START_REF] Mcilwain | Receptive fields of optic tract axons and lateral geniculate cells: peripheral extent and barbiturate sensitivity[END_REF]. This might explain why the far surround was not discovered in early experiments [START_REF] Barlow | Summation and inhibition in the frog's retina[END_REF][START_REF] Kuffler | Discharge patterns and functional organization of mammalian retina[END_REF]. Second, shift responses are unlike responses from the main receptive field in that the magnitude of the response to gratings is not proportional to their contrast, as is the case with stimuli applied to the classical receptive field [START_REF] Barlow | The effects of remote retinal stimulation on the responses of cat retinal ganglion cells[END_REF][START_REF] Fischer | Quantitative aspects of the shift-effect in cat retinal ganglion cells[END_REF]. Increasing the contrast of the reversing grating has almost no influence above the threshold where it triggers a response (all-or-none behavior, fig. I.17,D).

Mechanism. The far surround is suppressed by barbiturates [START_REF] Mcilwain | Receptive fields of optic tract axons and lateral geniculate cells: peripheral extent and barbiturate sensitivity[END_REF], suggesting that GABA is involved in the transmission of these distant activations to the RGC (Eggers and Lukasiewicz, 2010). The non-linear aspect of these activations (Y-like) suggests that transmission happens at the level of the IPL (Werblin, 1972b). The exact circuits involved are still a matter of discussion, although it appears clearly that a disinhibitory CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS circuit must mediate these distant activations [START_REF] Eggers | Interneuron circuits tune inhibition in retinal bipolar cells[END_REF][START_REF] Ikeda | The outer disinhibitory surround of the retinal ganglion cell receptive field[END_REF][START_REF] O'brien | Inhibitory network properties shaping the light evoked responses of cat alpha retinal ganglion cells[END_REF].

Universality. The shift-effect is found in most Y cells of the cat, and in some X cells.

Watanabe (1980) also reported the shift effect in a subset (17%) of rabbit RGCs. He also found that some RGCs in rabbit present an inhibitory shift effect. It is however unclear whether this inhibitory shift effect is different from the classical inhibitory surround of a Y-cell.

Non-linear interactions between center and surround

In 1972, Werblin (1972a) found that stimulating the surround of a mudpuppy RGC with a rotating windmill stimulus reduced the response of the cell to a spot flashing in its center (fig. I.18A). The experiment was soon reproduced successfully in turtle [START_REF] Schwartz | Organization of on-off cells in the retina of the turtle[END_REF], rabbit [START_REF] Caldwell | New properties of rabbit retinal ganglion cells[END_REF], cat (Enroth-Cugell and Jakiela, 1980) and macaque [START_REF] Solomon | Suppressive Surrounds and Contrast Gain in Magnocellular-Pathway Retinal Ganglion Cells of Macaque[END_REF] with rotating windmills or reversing gratings in the periphery. Interestingly, the effect was also present in X cells. The rotating windmill was also found to reduce responses to flashes in the surround (Enroth-Cugell and Jakiela, 1980). [START_REF] Shapley | Nonlinear spatial summation and the contrast gain control of cat retinal ganglion cells[END_REF] characterized carefully the interaction between center and surround. They found that activation of the surround:

• diminished the responses to low frequency stimuli in the center, while maintaining the responses to high frequency stimuli intact.

• diminished the response latency to high frequency stimuli. [START_REF] Shapley | Nonlinear spatial summation and the contrast gain control of cat retinal ganglion cells[END_REF] also found that stimulation of the center had no non-linear impact on the response of the cell to the surround, although the central stimulus modulated

Other characteristics of the surround

Many other non-linear effects in the surround of RGCs were reported, that are not accounted by the DoG model, the LN model or the subunit model. Here is a non-exhaustive list aiming at showing the diversity of these effects:

• In cat X cells, the diameter of the RF center increases as retinal illumination is reduced, while the surround region becomes relatively ineffective (Enroth-Cugell and Robson, 1966).

• In cat X cells, the extent of the surround depends on the temporal frequency of the stimulus [START_REF] Dawis | The receptive field organization of X-cells in the cat: Spatiotemporal coupling and asymmetry[END_REF].

• In cat X and Y cells, the mean firing rate is increased by a coarse moving grating in the periphery and decreased by a fine moving grating in the periphery [START_REF] Passaglia | Effects of remote stimulation on the mean firing rate of cat retinal ganglion cells[END_REF]. This study suggests that there are non-linear subunits of different size accounting for each of these effects.

Diverse computations are performed by retinal ganglion cells

Until now we have focused on the physiological properties shared by most retinal ganglion cells: the receptive field, the center-surround structure, the linear and non-linear properties of spatial summation (X and Y-cells). In this section we will give examples of diverse computations performed by different types of ganglion cells. For each computation, we will present the current knowledge about the retinal circuit implementing it and list the species in which it was reported.

CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

In the last section we will see how these computations can be related to specific behaviors.

Local edge detectors

Properties. In an article called "What the frog's eye tells the frog's brain", Lettvin et al.

(1968) found an intriguing type of RGCs. Cells of this type did not respond to change in general illumination, but responded to a small object (3 ± or less) passing through their receptive field. Furthermore, the cells would not respond to a wider object moving through their receptive field. Moving a checked or dot texture (with dots no further apart than half of the width of the RF) did not elicit any response either. However, if any dot within the RF moved differentially with respect to the background texture, the cell would respond to that dot as if it were alone. Cells presenting very similar properties were also described in it directly inhibits the RGC and it inhibits the pre-synaptic bipolar cells. Finally they found that inhibitory responses from the center and excitatory responses from the surround arrive with the same lag at the RGC, whereas the surround signal flows through at least one CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS more neuron, the amacrine cell. They suggest that the central bipolars must be of a slow type to allow the surround signals to catch up with the center. This suggestion is consistent with the slow response dynamics of these cells (ª30ms time to peak).

In rabbit, it was shown that inhibition that produces the antagonistic surround of LEDs is mediated largely presynaptically [START_REF] Van Wyk | Local edge detectors: a substrate for fine spatial vision at low temporal frequencies in rabbit retina[END_REF], by GABAergic amacrine cells inhibiting bipolar cell terminals [START_REF] Russell | Retinal synaptic pathways underlying the response of the rabbit local edge detector[END_REF]. Furthermore, a recent study suggests that the surround is only partly mediated by spiking amacrine cells (Venkataramani et al., 2014).

Universality. Cell types presenting these properties, known as local edge detectors, were found in frog [START_REF] Maturana | Anatomy and Physiology of Vision in the Frog (Rana pipiens)[END_REF], pigeon [START_REF] Maturana | Directional Movement and Horizontal Edge Detectors in the Pigeon Retina[END_REF]

Orientation selective cells

Properties. [START_REF] Levick | Receptive fields and trigger features of ganglion cells in the visual streak of the rabbits retina[END_REF] Mechanism. In rabbit, the circuit of orientation selectivity RGCs was first studied by [START_REF] Caldwell | New properties of rabbit retinal ganglion cells[END_REF], who showed that the GABAA and GABAC receptor antagonist picrotoxin abolished orientation selectivity. This result led them to propose a model in which a circular excitatory receptive field center is flanked by an oriented inhibitory surround. However, subsequent work combining intracellular recordings with dye fills of orientation selective ganglion cells suggested that oriented dendrites of ganglion cells could also contribute to orientation selectivity, enhancing the effect of inhibitory flanks [START_REF] Amthor | Morphologies of rabbit retinal ganglion cells with complex receptive fields[END_REF][START_REF] Bloomfield | Orientation-sensitive amacrine and ganglion cells in the rabbit retina[END_REF]. Another study using voltage-clamp [START_REF] Chapter | Orientation selectivity in rabbit retinal ganglion cells is mediated by presynaptic inhibition[END_REF] showed that orientation selectivity in horizontal and vertical Off-OSGCs in rabbit relies on presynaptic GABAergic inhibition, but that the two cell types differ in that horizontal Off-OSGCs receive direct OS inhibition (tuned to the null orientation), whereas vertical Off-OSGCs are instead influenced by a disinhibitory circuit that reduces tonic inhibition in the preferred orientation. In mouse, a recent study [START_REF] Nath | Cardinal Orientation Selectivity Is Represented by Two Distinct Ganglion Cell Types in Mouse Retina[END_REF] using voltage-clamp suggests that both excitatory and inhibitory conductances in On-OSGCs are orientation selective. Furthermore, orientation selective inhibition is resistant to either GABA or glycine receptor antagonists applied alone but not to the combination of both antagonists, whereas orientation selective excitation is resistant to all inhibitory receptor blockers tested. These results suggest that multiple orientation selective pathways converge onto On-OSGCs.

Universality. Orientation selective cells were found by many studies in rabbit [START_REF] Amthor | Morphologies of rabbit retinal ganglion cells with complex receptive fields[END_REF][START_REF] Bloomfield | Orientation-sensitive amacrine and ganglion cells in the rabbit retina[END_REF][START_REF] Caldwell | New properties of rabbit retinal ganglion cells[END_REF][START_REF] Levick | Receptive fields and trigger features of ganglion cells in the visual streak of the rabbits retina[END_REF][START_REF] Chapter | Orientation selectivity in rabbit retinal ganglion cells is mediated by presynaptic inhibition[END_REF]. Some degree of orientation selectivity has been reported in macaque midget CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS jects that extend beyond the center of the receptive field: they rather signal local motion arising from objects moving within the visual field [START_REF] Vaney | Direction selectivity in the retina: symmetry and asymmetry in structure and function[END_REF]. Interestingly, DS-GCs are not exclusively selective to motion: they also respond to the onset and offset of stationary spots of light in their receptive field [START_REF] Barlow | Selective sensitivity to direction of movement in ganglion cells of the rabbit retina[END_REF][START_REF] Barlow | Retinal ganglion cells responding selectively to direction and speed of image motion in the rabbit[END_REF][START_REF] Im | Directionally selective retinal ganglion cells suppress luminance responses during natural viewing[END_REF].

Mechanism. On-Off DSGCs exhibit small, bistratified dendritic trees, accounting for their response to both light onset and offset [START_REF] Vaney | Direction selectivity in the retina: symmetry and asymmetry in structure and function[END_REF]. GABAergic wide-field amacrine cells connecting onto the presynaptic bipolar terminals of DSGCs mediate the inhibitory surround of the cells but not the direction selectivity [START_REF] Hoggarth | Specific wiring of distinct amacrine cells in the directionally selective retinal circuit permits independent coding of direction and size[END_REF].

Starburst amacrine cells are the main actors of direction selectivity [START_REF] Vaney | Direction selectivity in the retina: symmetry and asymmetry in structure and function[END_REF].

Direction selectivity in DSGCs is abolished when starburst cells are ablated or temporarily inactivated [START_REF] Vaney | Direction selectivity in the retina: symmetry and asymmetry in structure and function[END_REF][START_REF] Vlasits | Visual stimulation switches the polarity of excitatory input to starburst amacrine cells[END_REF]. A recent histological study of starburst cells suggested a mechanism for direction selectivity [START_REF] Kim | Space-time wiring specificity supports direction selectivity in the retina[END_REF]. This study showed that starburst cell receives inputs from sustained bipolar cell near the soma and inputs from transient bipolar near the dendritic tips. When motion moves outward from the center toward the tip, the transient and sustained bipolar inputs would maximally depolarize the release sites at the tips (fig. I.21C,D). DSGC simply inherits their direction selectivity from starburst amacrine cells by selectively wiring with the starburst cell dendrites located in the adequate direction relative to the amacrine cell soma [START_REF] Briggman | Wiring specificity in the direction-selectivity circuit of the retina[END_REF]. For example, a DS cell that prefers upward motion receives inhibitory inputs that prefer downward motion, which are the downward-pointing starburst amacrine cell dendrites.

Intrinsic properties of DSGCs, most notably the generation of dendritic action potentials in response to local excitation, amplify direction selectivity [START_REF] Oesch | Direction-selective dendritic action potentials in rabbit retina[END_REF]Sivyer and

Other DSGCs

On DSGCs. A second population of DSGCs responds only at light on. These On-DSGCs have much larger receptive fields than the On-Off DSGCs, they have only a weak inhibitory surround, and they only respond to slow image motion [START_REF] Vaney | Direction selectivity in the retina: symmetry and asymmetry in structure and function[END_REF]. On-DSGCs respond preferentially to object motion in one of three directions aligned with the vestibular axes: anterior, superior with a posterior component and inferior with a posterior component. It has been shown in rabbit retina that there are in fact two distinct types of On DSGCs, one is sustained and the other transient. On-DSGCs provide the major retinal projection to the medial terminal nucleus (MTN), a nucleus of the accessory optic system. The accessory optic system plays a critical role in moving the eyes smoothly to compensate slow, global visual motion [START_REF] Simpson | The accessory optic system[END_REF]. A recent study [START_REF] Gauvain | Projection-Specific Characteristics of Retinal Input to the Brain[END_REF] shows that On-DSGC also innervate the Superior Colliculus (SC).

Off DSGCs. Off-DSGCs (or J-RGCs) were recently discovered in mouse retina isolated genetically [START_REF] Kim | Molecular identification of a retinal cell type that responds to upward motion[END_REF]. They have a single preferred direction, which is upward motion. Their dendritic arbor is oriented according to their preferred direction (fig. I.22). J-RGCs receive little input from starbust amacrine cells and thus must rely on other mechanisms for direction selectivity [START_REF] Kim | Molecular identification of a retinal cell type that responds to upward motion[END_REF].

Suppressed by-Contrast cells

Suppressed by-Contrast (SbC) cells. RGCs of this type have a high level of spontaneous activity, and they respond by decreasing their firing rate to any change in their receptive field (lights flashed on or off, movement of dark or light shapes). Since their discovery in the rabbit retina [START_REF] Levick | Receptive fields and trigger features of ganglion cells in the visual streak of the rabbits retina[END_REF], SbC RGCs have been recorded in cat [START_REF] Mastronarde | Two types of cat retinal ganglion cells that are suppressed by contrast[END_REF] CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS [START_REF] Troy | Responses to sinusoidal gratings of two types of very nonlinear retinal ganglion cells of cat[END_REF], rabbit [START_REF] Sivyer | Uniformity detector retinal ganglion cells fire complex spikes and receive only light-evoked inhibition[END_REF][START_REF] Sivyer | A novel type of complex ganglion cell in rabbit retina[END_REF], mouse [START_REF] Tien | Genetically identified suppressed-by-contrast retinal ganglion cells reliably signal selfgenerated visual stimuli[END_REF] and macaque retina (De Monasterio, 1978b).

Behaviorally relevant computations

The goal of the visual system is to extract useful information from the visual scene [START_REF] Marr | Vision: A computational approach[END_REF]. Many ganglion cell types seem to already extract behaviorally relevant features of the visual scene (Gollisch and Meister (2010) for a review). In this section we present examples of cell types which computation appears to be relevant to a specific behavior.

We also present a case where a cell type was causally associated to a behavior (DS cells and visual pursuit).

Predator/Prey detection W3 ganglion cells in mouse act as local-edge detectors [START_REF] Zhang | The most numerous ganglion cell type of the mouse retina is a selective feature detector[END_REF]. As such, they respond specifically to small moving targets in their receptive field on a featureless or stationary background. Their preferred stimulus makes them ideally suited to detect a hawk in the sky, a stimulus of particular relevance for a mouse and to which it should react promptly. Furthermore, the distribution of W3 cells across mouse retina shows a peak in ventral retina, the region corresponding to the upper visual field of the mouse, where the sky is.

Local-edge detectors are also found in rabbit [START_REF] Levick | Receptive fields and trigger features of ganglion cells in the visual streak of the rabbits retina[END_REF] and frogs [START_REF] Lettvin | What the Frog's Eye Tells the Frog's Brain[END_REF].

In frogs, it has been suggested [START_REF] Maturana | Anatomy and Physiology of Vision in the Frog (Rana pipiens)[END_REF]) that local-edge detectors act as bug detectors and trigger feeding behavior. Another cell type, the dimming detector, would be well suited to detect threats and trigger escape behavior [START_REF] Maturana | Anatomy and Physiology of Vision in the Frog (Rana pipiens)[END_REF].

Eyes movements.

A visual scene drifting on the retina triggers the eye to follow it, thus keeping the image stable on the retina [START_REF] Yonehara | Congenital Nystagmus Gene FRMD7 Is Necessary for Establishing a Neuronal Circuit Asymmetry for Direction Selectivity[END_REF]. This reflex is called the nystagmus, or optokinetic reflex. A recent study has shown that the mutation of gene FRMD7, a gene that is defective in human congenital nystagmus, also provoke the loss of direction selectivity in mouse direction-selective ganglion cells (DSGCs). Interestingly, the gene mutation only affects horizontal nystagmus and horizontal DSGCs. A subset of DSGCs project to the accessory optic system (Gauvain and Murphy, 2015), a brain region involved in the generation of smooth eye movements that keep the image stable on the retina in case of slow, global visual motion [START_REF] Simpson | The accessory optic system[END_REF]. Taken together, these elements strongly suggest a direct causal relation between the activity of a type of RGCs and a behavior, the nystagmus.

In primate, a careful characterization of responses of broad thorny ganglion cells has lead to the suggestion that they might be ideally suited to guide eye movements involved in visual pursuit of moving objects [START_REF] Puller | Broad Thorny Ganglion Cells: A Candidate for Visual Pursuit Error Signaling in the Primate Retina[END_REF]. However, causality between the activity of this cell type and behavior has not been established. But optogenetic reactivation also has the problem that the retinotopy is lost in this region, and the spatial translation of each pixel towards the right bipolar of ganglion cells is unknown and might vary from one individual to the next.

Chapter II

Methods of visual restoration and acuity

Finally, photoreceptors adapt to 8 or 9 log units of light intensity between day and night conditions [START_REF] Roska | La rétinopathie pigmentaire : restauration visuelle par thérapie optogénétique[END_REF]. This is not the case of reactivated cells that do not benefit from these adaptation phenomena and can only cover ranges of 2 or 3 logarithmic units [START_REF] Roska | La rétinopathie pigmentaire : restauration visuelle par thérapie optogénétique[END_REF]. Consequently, even for optogenetic strategies a visual stimulator will be necessary to send the visual information in the sensitivity range of the protein used.

Addressing problem

In order to obtain a good resolution, visual restoration techniques need to target neurons precisely. Each technology comes with different targeting issues and potential solutions to these issues. ON/OFF pathways. In the normal retina, bipolar cells can be separated in two classes:

Confinement

ON bipolars that respond primarily to an onset of light, and OFF bipolars that respond primarily to an offset of light. Ganglion cells largely inherit this division between ON and OFF pathways.

Single electrodes of retinal implants (subretina or epiretinal) stimulate hundreds or thousands of cells, and thus send the same signal to the ON and OFF pathway. This unnatural encoding of visual scene may or may not result in an intuitively interpretable percept for human patients [START_REF] Fine | Pulse trains to percepts: the challenge of creating a perceptually intelligible world with sight recovery technologies[END_REF]. A solution to this problem might be provided by a recent study by [START_REF] Twyford | Differential responses to high-frequency electrical stimulation in ON and OFF retinal ganglion cells[END_REF]. This study shows that it is possible to stimulate ON and OFF RGCs differentially, by modulating the amplitude of a high-frequency electrical stimulation.

Optogenetic strategies face the same issue because there are currently no specific promoters to distinguish ON from OFF RGCs (

Acuity estimation from receptor spacing

Several studies find a good adequacy between the visual acuity obtained with a prosthesis And yet, many examples show that the spacing between receptors of a sensory system is not a limit for the acuity of the system. For example, the normal retina of humans is able to distinguish changes in the separation of two lines with a resolution of 1 second of arc in a Vernier task, whereas photoreceptors in the fovea are arranged in a lattice corresponding to an angular spacing of 30 seconds of arc [START_REF] Klein | Hyperacuity thresholds of 1 sec: theoretical predictions and empirical validation[END_REF]. This phenomenon is called hyperacuity and has been extensively documented in the retina (for a review, [START_REF] Westheimer | Visual hyperacuity[END_REF]). Another example comes from a visual restoration study [START_REF] Stingl | Artificial vision with wirelessly powered subretinal electronic implant alpha-IMS[END_REF] that measured the acuity of blind patients implanted with the Alpha-IMS implant.

One patient was able to distinguish the orientation of gratings with stripe width of 0.15 ±

(3.3 cpd), whereas the spacing between two electrodes of the implant corresponded to a visual angle of 0.23 ± (70 microns).

CHAPTER II. METHODS OF VISUAL RESTORATION AND ACUITY

In fact, there is no fundamental physical principle justifying that acuity should be limited by the receptor density [START_REF] Bialek | Spikes: exploring the neural code[END_REF], chapter Hyperacuity, p223). Many other parameters are necessary to estimate the resolution limit of a lattice of receptors [START_REF] Burak | Bayesian model of dynamic image stabilization in the visual system[END_REF], such as signal-to-noise ratio of the receptors, spatial profile of their receptive field, time of exposure of the visual scene, ability of the visual system to displace the image on the receptor lattice during the time of exposure (with eye micro movements for example).

Chapter III

Multiplexed computations in retinal ganglion cells of a single type Contribution

Using multi-electrode recordings in the rat retina, we find a cell type that is performing a linear computation in its center (like an X cell) and a non-linear computation in its surround (like a Y cell). By studying the interactions between center and surround, we find that a global gain control is suppressing the response of the surround in case of a central stimulation. We argue that this cell type is effectively multiplexing two radically different features, which is contradictory with a popular view of the retinal code according to which each cell type is extracting one feature of the visual scene (cf chapter I for an introduction on the physiology of RGCs and chapter V for a discussion on the multiplexing hypothesis).

Summary

In the early visual system, cells of the same type perform the same computation in different places of the visual field. How these cells code together a complex visual scene is unclear. A common assumption is that cells of the same type will extract a single stimulus feature to form a feature map, but this has rarely been observed directly. Using large-scale recordings in the rat retina, we show that a homogeneous population of fast OFF ganglion cells simultaneously encodes two radically different features of a visual scene. Cells close to a moving object code linearly for its position, while distant cells remain largely invariant to the object's position and, instead, respond non-linearly to changes in the object's speed. Cells switch between these two computations depending on the stimulus. We developed a quantitative model that accounts for this effect and identified a likely disinhibitory circuit that mediates it. Ganglion cells of a single type thus do not code for one, but two features simultaneously. This richer, flexible neural map might also be present in other sensory systems.

Introduction

A major challenge of the visual system is to extract meaningful representations from com- receptive field center overlapped with an object performed a linear computation that was highly sensitive to the position of the object. In contrast, cells of the same cell type that were far from any moving object responded nonlinearly to fast motion, and were largely invariant to the exact position of distant objects. Individual cells switched from one computation to the other when their receptive field center was stimulated. We constructed a model that quantitatively accounted for these findings, and determined that the observed scheme of distal activation is implemented by a disinhibition circuit of amacrine cells.

plex

Results

Cells of a single cell type respond to very distant moving objects

We recorded large ensembles of ganglion cells from the rat retina using a micro-electrode

Linear computation inside and non-linear computation outside of the receptive field center

We asked if the observed ganglion cell responses to motion outside their receptive field

Switching between two modes of computation

Since the cells perform distinct computations in their center and in their distant surround, we studied how these computations interact when both center and surround are stimulated at the same time. What happens to distant cells if another bar is simultaneously shown inside their receptive field center? One possibility is that central and distant

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

Global gain control explains the gradual suppression of distant responses

Our previous results indicated that the influence of distant inputs is suppressed when the receptive field center is stimulated. To elucidate further how central inputs suppress distant ones, we asked if the suppression increases gradually as central inputs become progressively stronger, or if the suppression is only activated once the strength of central inputs exceeds a threshold.

To test this we displayed a series of stimuli where two bars were oscillating over the visual field at incommensurable frequencies (see methods). By averaging over the oscillation period of each bar, we could isolate the responses due to each bar. Our analysis focused on neurons for which one of the bars was within the receptive field center, while the other bar was outside. The central bar was displayed at several luminances, ranging from zero contrast (i.e., at background gray level) to maximally dark bar. We observed that responses to the distant bar decreased gradually as the luminance of the central bar went from gray to full dark, implying that the suppression of distant inputs was gradual Suppression of distant responses by strychnine showed that the weights assigned to distant subunits in our model are mediated by glycinergic amacrine cells. How could these weights be positive, while glycinergic amacrine cells have an inhibitory effect on their post-synaptic targets? One explanation is a disinhibitory loop, where one amacrine cell inhibits its post-synaptic amacrine cell target, which in turn disinhibits the ganglion cell (directly or through bipolar cell) (see also [START_REF] Manu | Disinhibitory gating of retinal output by transmission from an amacrine cell[END_REF]). Such a disinhibitory circuit could involve serial connections between GABAergic and glycinergic amacrine cells [START_REF] Eggers | Interneuron circuits tune inhibition in retinal bipolar cells[END_REF], or, alternatively, serial connections between different types of glycinergic cells. Glycinergic amacrine cells can ultimately inhibit OFF bipolar cells [START_REF] Eggers | Multiple pathways of inhibition shape bipolar cell responses in the retina[END_REF] or ganglion cells (O' Brien et al., 2003).

The net effect of such a disinhibitory circuit is a distant excitation of ganglion cells (fig.

III.6D).

Discussion

We have shown that two representations of a stimulus coexist, at the same time, within "neural map" where there is a one-to-one correspondence between one cell type and one visual feature: here we show that a "neural map" can contain more than one "feature map" at the same time. Multiplexing two computations in a single neural type could enable optimal use of coding resources: if ganglion cells don't have an object inside their receptive field center, rather than staying silent, they are put to use to code for a different feature of

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

far surround ("shift-effect": [START_REF] Barlow | The effects of remote retinal stimulation on the responses of cat retinal ganglion cells[END_REF][START_REF] Cleland | Sustained and transient neurones in the cat's retina and lateral geniculate nucleus[END_REF][START_REF] Fischer | Quantitative aspects of the shift-effect in cat retinal ganglion cells[END_REF][START_REF] Ikeda | The outer disinhibitory surround of the retinal ganglion cell receptive field[END_REF][START_REF] Mcilwain | Receptive fields of optic tract axons and lateral geniculate cells: peripheral extent and barbiturate sensitivity[END_REF]). Here we constructed a model that can accurately predict how fast OFF ganglion cells would respond to distant, complex stimuli, and how these distant stimuli would be integrated with other stimuli simultaneously displayed inside the receptive field center. Previous models mostly focused on how the surround modulates responses to central stimuli. However, how responses to distant stimuli can modulate ganglion cells themselves, and how they could be affected by center stimulation, has received less attention [START_REF] Shapley | Nonlinear spatial summation and the contrast gain control of cat retinal ganglion cells[END_REF]. [START_REF] Demb | Functional Circuitry of the Retinal Ganglion Cell's Nonlinear Receptive Field[END_REF] found that inputs from center and surround stimulation were summed linearly, while we found a non-linear suppression of distant inputs. This discrepancy could be due to a difference of species, cell type, or recording technique. [START_REF] Passaglia | Effects of remote stimulation on the mean firing rate of cat retinal ganglion cells[END_REF] showed that distant stimulation could be suppressed by center stimulation, but the timescale of the modulation was much longer than in our work. Interestingly, Jadzinsky and Baccus (2015) suggested a model to predict how stimulation of the surround can affect the selectivity to the center stimulation that bears some similarity with our model. In most studies, the stimulus employed to modulate activity from the surround was very large. In our study, we showed that the same stimulus triggered two different types of responses, a central one and a distant one, within the same type of ganglion cell, demonstrating the coexistence of the two representations.

Our results suggest that the retinal network implements the activation of ganglion was reduced. This reduction of surround inhibition was mediated by a disinhibitory circuit similar to the one we uncovered.

We have shown that a single cell type mosaic can simultaneously multiplex several fundamentally distinct computations. Our findings considerably enrich the classical view of ganglion cell types as being tightly linked to their corresponding feature maps, and uncover the flexibility of the retinal code when stimulated with complex, dynamical stimuli.

The notion of a feature map is central to most sensory structures. Flexible computations, where several features are represented by a cell type simultaneously in response to complex stimuli, might also be implemented in other sensory areas. It remains to be understood whether this flexibility can be seen as arising from some efficient coding principle [START_REF] Tkačik | Information Processing in Living Systems[END_REF], and how such flexible coding schemes can be interpreted by the downstream areas (Botella-Soler et al., 2016).

Material and methods

Unless stated otherwise, all error bars in figures and text are standard error of the mean (SEM). SD stands for standard deviation.

Retinal recordings

Recordings were performed on the Long-Evans adult rat. Animals were euthanized according to institutional animal care standards. The retina was isolated from the eye under dim illumination and transferred as quickly as possible into oxygenated AMES medium.

The retina was then lowered with the ganglion cell side against a multi-electrode array whose electrodes were spaced by 60 µm, as previously described [START_REF] Marre | Mapping a complete neural population in the retina[END_REF][START_REF] Yger | Fast and accurate spike sorting in vitro and in vivo for up to thousands of electrodes[END_REF]. Raw voltage traces were digitized and stored for off-line analysis using a 252-channel preamplifier (MultiChannel Systems, Germany). The recordings were sorted using custom spike sorting software developed specifically for these arrays [START_REF] Marre | Mapping a complete neural population in the retina[END_REF][START_REF] Yger | Fast and accurate spike sorting in vitro and in vivo for up to thousands of electrodes[END_REF]. We extracted the activity of a total of 810 neurons over 5 experiments with satisfying standard tests of stability and limited number of refractory period violations.

Visual stimulation

Our stimulus was composed of one or two black bars moving randomly on a gray background. Each bar was animated by a Brownian motion, with additional feedback force to stay above the array, and repulsive forces so that they do not overlap. The bars stayed within an area that covers the whole recording array. The amplitude of the bar trajectories allowed them to sweep the whole recording zone. The trajectories of the bars x 1 and x 2

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

are described by the following equations [START_REF] Mora | Dynamical criticality in the collective activity of a population of retinal neurons[END_REF]:

dv 1 dt = °v1 τ + sign(x 1 °x2 ) µ R |x 1 °x2 | ∂ 6 °ω2 0 (x 1 °µ1 ) + σ W 1 (t ) (III.1) dv 2 dt = °v2 τ + sign(x 2 °x1 ) µ R |x 2 °x1 | ∂ 6 °ω2 0 (x 2 °µ2 ) + σ W 2 (t ) (III.2)
where W 1 (t ) and W 2 (t ) are two Gaussian white noises of unit amplitude, µ 2 °µ1 = 600µm

is the shift between the means, ω 0 = 1.04 Hz, τ = 16.7 ms, R = 655µm and σ = 21.2µm•s °3/2 .

The width of one bar is 100µm. The stimulus was displayed using a Digital Mirror Device and focused on the photoreceptor plane using standard optics. For receptive field mapping, a random binary checkerboard was displayed for 1 hour at 50 Hz (check size: 60 µm).

All the other stimuli used (for classification of cells, fitting the gain control model and pharmacological study) are described in the corresponding method section. For all stimuli, the level of light of the gray background was between 10 12 and 10 13 photons.cm °2.s °1.

Typing

We performed cell classification based on the response of the cells to a set of stimuli and on their temporal receptive field.

Full field flicker: this stimulus consisted of a 15-seconds sequence of a full-field stimulus, repeated 100 times. The stimulus was generated by selecting a random row of pixels from a natural image and displaying subsequently at 40Hz the intensity of these pixels uniformly on the entire screen.

Shifting barcode: this stimulus consisted of an alternation of white and black stripes of

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

width 70 µm chosen randomly, moving at a constant speed of 1000 µm/s in the 4 cardinal directions. For each direction, the 17-seconds sequence was repeated 30 times.

For each cell, we created a vector by concatenating the PSTH in response to the full field flicker stimulus, the 4 PSTHs in response to the shifting barcode stimulus corresponding to the 4 cardinal directions, the temporal receptive field and the auto-correlogram of the cell in response to the checkerboard stimulus. The PSTHs of the shifting barcode were temporally realigned beforehand according to the receptive field location of each cell. PSTH for each stimulus was normalized such that they all had a mean of 0 and a variance of 1.

We then performed PCA on this collection of vectors. We kept the projections on the first eigenvectors in order to explain 95% of the total variance. We then performed clustering on these vectors using the peak density algorithm (Rodriguez and Laio, 2014). The threshold parameters of the algorithm were manually adjusted in order to select the outliers as centroids of the clusters. This method allowed us to identify reliably an OFF type of ganglion cells across all experiments. The receptive fields (RF) were regularly tiling the visual field, with little overlap between them. This mosaic property, often observed in the retina, was used here as a validation of our typing procedure, as we did not use the position of the RFs in the clustering procedure.

Synchrony between cells

To quantify the synchrony between cells, we displayed a 10-second bar movie to the retina, repeated 54 times. A maximum of 25 cells of the same type recorded simultaneously were subdivided in two groups, the distant cells, that were more than 200 µm away from the

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

central bar position, and the central cells, that were less than 200 µm away from the central bar position. For all cells we computed the PSTH with a time bin of 20 ms. We computed the Pearson coefficient between all pairs of PSTHs of distant cells, and all pairs of PSTHs of central cells respectively. We grouped the pairs based on the distance between their receptive field centres along the bar motion axis.

Linear model and subunit model

Subunit model

The subunit model is a two-layer model that predicts the response of a ganglion cell to the moving bar. Each layer performs a linear combination of its inputs followed by a nonlinear transformation. The first layer is a collection of identical and translated Linear-Non-Linear (LN) units. The second layer is a unique LN unit taking the output of the first layer as an input.

In the first layer, we tiled the space with 200 bipolar-like ON and OFF subunits on a onedimensional lattice, with subunits equally spaced at 20µm interval. Each unit had a receptive field with a Gaussian spatial profile of the right polarity and a biphasic temporal profile, modelled by a sinusoid. All units of a same polarity are identical up to a translation. The non-linearity was a rectified square function, h. The output of the first layer was therefore:

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE F(x, t )=h( Z t t °Tsubunit Z 1 °1 sin(2πω(t °t 0 ))e °(x °x0 ) 2 2σ 2 s(x 0 , t 0 )dx 0 dt 0 ) (III.3)
where h(x)=x 2 if x ∏ 0, and 0 otherwise. T subunit = 0.3 s, ω = 1/T subunit , σ = 30 µm

The stimulus movie s(x, t ) was one-dimensional in space because the stimulus was a long bar, whose length can be considered infinite. We used a temporal binning of 17ms, corresponding to the refresh rate (60Hz) of the screen used to project the movie on the retina.

The second layer consisted of a single Linear-Non-Linear Poisson unit. The unit pooled linearly its inputs from all the subunits of the first layer according to a kernel K, with an extension in time of 0.5 seconds. To obtain the firing rate r (t ) of the cell, the weighted sum was passed through a non-linearity of the form f (x) = log(1 + exp(x)). The spikes were then generated according to a Poisson process.

r (t )=α log(1 + exp(βG(t ) + θ)) (III.4)
where

G(t )= Z T filter 0 Z x K(x, t 0 )F(x, t °t 0 )dxdt 0 (III.5)

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

with T filter = 0.5 s, and α, β, θ are parameters of the non-linearity that are fitted to the data.

The linear model (LN) was built using the same architecture as the subunit model, except that the rectified square non-linearities in the subunits were replaced by the identity.

Fitting

For both models we used the same fitting procedure. The parameters of the kernel K and the parameters of the spiking non-linearity α, β, θ were the only parameters fitted to the data. The kernel parameters and the spiking non-linearity parameters were fitted alter-

C= X t °LL(s obs (t )|s pred (t )) + λ XT kL Kron Kk 2 + λ 1 kKk 1 (III.6)
where LL is the loglikelihood of the observed spike train s obs (under Poisson assumption), K is the kernel defined above, λ XT = 300 is the penalty term enforcing smoothness of the kernel, L Kron is the Kronecker sum of discrete Laplacians, λ 1 = 400 is the L1 penalty term enforcing sparseness of the filter coefficients.

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

explainable variability was defined as the average Pearson coefficient between pairs of PSTHs generated by instantiations of a Poisson process with mean firing rate equal to the real firing rate of the cell estimated from the PSTH. We divided the performance of our model (defined as the Pearson coefficient between real and predicted PSTH) by this explainable variability to obtain the ratio of explainable variability predicted by our model.

Calculation of the average linear filters in the subunit model.

To For all cells and in both bar conditions, only a portion of the extended receptive field center was visited by the bars, therefore inducing a bias in the filters fitted on these movies.

To compute the weights of the average filter without bias, we first realigned the filter of each cell relative to the center of its receptive field. Then for each coordinate (x, t ) of the average filter we averaged the corresponding subunit weights for the subset of cells for which the coordinate was visited more 200 times/hour by the bar.

Suppression index

In figure III.3B, we quantified the suppression of the response to the distant bar when there was another bar moving inside its receptive field center. For this we defined the We then fitted a single model on all contrast conditions. The model was of the form:

R gain (t )= R(t ) 1 + H R t 0 t °τ R(t 0 )dt 0 (III.10)
where τ = 1 s is the time constant of integration of the gain control and H is the gain. R(t )

is the total response before application of the gain control, given by the equation: Note that the information about the future of the stimulus was not always zero. This is because the successive positions of the bar are correlated in time, so that part of the information conveyed by the cell response about the past position of the bar is also informative about the future position of the bar. We then defined the stimulus as the speed of the bar S with different lags δt relative to the cell response. The speed was defined as:

R(t )=α 0 • c • r c (t ) + α 1 (c • r c (t )) 2 + β 0 • r d (t ) + β 1 • r d (t ) 2 (III.
S(t )= |P(t ) °P(t °τ)| τ (III.14)
where τ = 100 ms. We discretized linearly the space of S in 10 bins and we computed mutual information between R(t ) and S(t + δt ). To estimate the information rate in the insets of figure III.5D, we divided the mutual information by the bin size (10 ms). For each cell, we finally computed the ratio between the maximum of I(R(t ), S(t +δt )) and the maximum of I(R(t ), P(t + δt )) over all time lags tested.

Pharmacology

To block glycinergic transmission, we added 1 microMol strychnine (Sigma-Aldrich ref. For the population analysis, we flashed a bar 100 µm wide in random locations relative to the receptive fields of the cells, 20 times at each location. For each cell recorded of the type under study (17 cells), we selected the flashes that were less than 80 µm away from the receptive field center to study the effect of central stimulation. To study the effect of distant stimulation, we selected the flashes that were between 200 µm and 500 µm away from the cell receptive field center. For each stimulus and each cell, significant responses were determined based on a z-score analysis. We estimated the mean and standard deviation (SD) of the activity prior to stimulus and considered that a response was detected if the activity exceeded the mean by more than five times the SD in the second following the onset of the stimulus (for a bin size of 40 ms). To estimate the percentage of responding cells in fig. III.6, we estimated means and standard errors of mean by pooling together all stimulus conditions across all the cells. We performed a one-tailed two-sample t-test to assess the reduction of responses to the distant flash after drug was added to the bath.

The p-value was less than 10 °3. Here we developed an approach mixing experiments, data analysis and modeling to estimate precisely the spatial resolution of retinas reactivated with optogenetics. This spatial resolution gives an upper bound on the expected acuity of a blind patient treated with this strategy. In both mouse and macaque, optogenetic proteins were expressed in retinal ganglion cells following an intravitreal adeno-associated virus (AAV) injection in vivo. We then recorded the activity of populations of ganglion cells with micro-electrode recordings. We measured the receptive field of each cell, and characterized precisely their light sensitivity.

Based on these data, we designed a model of the processing performed by the population of reactivated ganglion cells on the stimulus. This model precisely predicts the cells' responses to complex stimulation patterns. We used this model to predict the response of the ganglion cells to letters of different sizes, and estimated how well the different letters could be discriminated by decoding the ganglion cell activity, which quantifies the spatial resolution of the reactivated retina.

CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY

The spatial resolution that we calculate gives a precise prediction of the best acuity that could be reached by treated patients. This acuity is above the limit of legal blindness.

This result suggests that therapy based on optogenetic reactivation is a promising avenue to restore high-resolution vision in blind patients. Our model also makes interesting predictions about different possible strategies to increase acuity of treated patients.

Results

Receptive field estimation and LN model

For the mouse experiments, we targeted retinal ganglion cells (RGCs) of blind rd1 mice

Acuity estimation of the reactivated retina

Our goal was to estimate how much information from a visual scene is conveyed through the reactivated retinas. Since the LN model predicted accurately the response of ganglion cells activated by an optogenetic protein, we could then build a model of the entire ganglion cell population expressing the optogenetic protein based on these data.

First we extended the model for one neuron to a model of the full retina. For this we assumed that ganglion cells were placed on a squared grid, with a density equal to the density of transfected cells in the experiment. In a previous study, we found that around 40% of ganglion cells were transfected in the macaque foveal ring (measured from confocal imaging in Chaffiol et al. ( 2016)), and the density of ganglion cells in the macaque fovea has been estimated to 51,108 cells/mm 2 [START_REF] Ahmed | Cell density ratios in a foveal patch in macaque retina[END_REF]. Each neuron was simulated with an identical LN model, with the parameters (STA, non-linearity) chosen to be equal to the average parameters found in the experimental data. Each neuron in our model was thus identical up to a translation of its receptive field. We assumed there was no noise correlation between ganglion cells, an assumption which is justified by the fact that we did not find any shared variability (i.e. noise correlation) between cells in the data (data not shown -note that this is consistent with previous studies suggesting that most of the noise correlation come from photoreceptors (Ala-Laurila et al., 2011)).

We then used our model to simulate the spiking response of the reactivated retina to an

Influence of model parameters on acuity

An advantage of this mode is that it allows playing with the different parameters to predict how they will influence the acuity. There are mostly three relevant parameters in this model: the density of cells, the size of the receptive field, and the non-linearity. These three parameters will change depending on the therapeutic strategy adopted. First, the density of cells will vary if the ratio of transfected cells is varied. This could happen if, for example, the AAV dose is changed. Second, the size of the receptive field could change depending on how protein expression is engineered.

Discussion

We built a model of the information transmitted by reactivated ganglion cells to the brain.

This model was realistic and properly fitted to the data, and allowed quantitative predictions of the best acuity we can expect in a treated patient. Our model predicted that a patient should be able to discriminate letters of size corresponding to a visual acuity of 20 minutes of visual angle.

One frequently heard criticism of vision restoration with optogenetic reactivation of ganglion cells is that the whole axon of the neuron could become light sensitive, thus creating an unnaturally elongated receptive field. In practice, the limited size of the measured receptive fields demonstrates that the reactivated ganglion cells are mostly sensitive to the stimulation of their soma and proximal dendritic field, and not to the stimulation of their axon: otherwise we would have measured large and elongated receptive fields encompassing the axon image. Our study also emphasizes that the spatial resolution of the whole reactivated retina does not depend solely on the receptive field size: other factors such as the density of reactivated cells and the steepness of the non-linearity also play a key role in the resulting maximum acuity.

Optimality assumption

We measured the spatial resolution of the reactivated retina, and assumed that this will correspond to the acuity of the patient. The main assumption in this approach is that the brain can make the best use of the information transmitted by the retina. The region reactivated optogenetically forms a ring around the photoreceptors of the fovea, and this novel geometry also needs to be learned by the brain. Many studies have shown that a reorganization of the adult visual cortex is possible following a lesion in the retina [START_REF] Keck | Synaptic scaling and homeostatic plasticity in the mouse visual cortex in vivo[END_REF][START_REF] Keck | Massive restructuring of neuronal circuits during functional reorganization of adult visual cortex[END_REF]. Additionally, a promising strategy is to pre-process the visual input before sending a stimulation pattern to the ganglion cells. This pre-processing can be optimized to help the brain make the best use of the information transmitted by the retina. Both pre-processing of the visual input and brain plasticity could help to achieve a perceptual performance close to the optimal spatial resolution estimated here, but only direct tests on patients will determine how close to optimal they can be.

Predicting efficacy of future treatments

The interest of this modeling approach is to predict how different factors in the therapeutic strategy can affect acuity. Thanks to the high density of ganglion cells in the fovea, the coverage factor (i.e. average number of receptive fields overlapping with one point of the visual space) in this region is very high. As a result, the ratio of transfected cells can be decreased without strongly affecting the expected acuity.

CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY

The merit of this precise modeling approach is to show that several factors can influence the expected acuity. Some approaches have assumed that the size of the receptive field of individual cells are the main determinant of acuity [START_REF] Lorach | Neural stimulation for visual rehabilitation: Advances and challenges[END_REF]). Here we show that this is not necessarily the case, and that other factors, such as the density of active cells, might also play a key role there. This kind of modelling approach could be used in the future to optimise the therapy delivered to the patient.

Material and methods

Unless stated otherwise, all error bars in figures and text are standard error of the mean (SEM). SD stands for standard deviation.

AAV production and injection

Details of the gene delivery and optogenetic protein expression in mice has been detailed elsewhere (Chaffiol et

Multielectrode array recordings

Recordings were made using a multielectrode array (MEA) comprised of 256 extracellular electrodes spaced at 100 µm on a square grid (MEA256 100/30 iR-ITO; Multi-Channel Systems, Reutlingen, Germany). The retinal area covered by the MEA was 1.6 mm 2 . Once a piece of retina had been isolated, it was placed ganglion cell side down onto the array.

A perforated dialysis membrane was used to hold the retina in place on the array. The array was superfused with Ames solution (3 ml/minute, gassed with 95% O2-5% CO2) and maintained at 34 ± C. Raw RGC activity was amplified and sampled at 20kHz. Resulting data was stored and filtered with a 200 Hz high pass filter for offline analysis. The recordings were sorted using custom spike sorting software developed specifically for these arrays [START_REF] Marre | High accuracy decoding of dynamical motion from a large retinal population[END_REF][START_REF] Yger | Fast and accurate spike sorting in vitro and in vivo for up to thousands of electrodes[END_REF]. We extracted the activity of a total of 810 neurons over 5 experiments with satisfying standard tests of stability and limited number of refractory period violations.

Visual stimulation and receptive field estimation

The stimulus was displayed using a Digital Mirror Device and focused on the photoreceptor plane using standard optics.

The receptive field (RF) of a retinal ganglion cell (RGC) is the particular region of the visual field in which a stimulus will trigger the firing of that cell, and the temporal delay with which the response will occur. Here we characterized the spatial and temporal components of the RFs using spike-triggered average (STA) on a white noise stimulus.

CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY

We fitted a non-linear function to these responses and used it for the model. We chose this way of computing the non-linearity to take advantage of the full range of responses of the ganglion cells in our simulation. This allowed us to model the response of the cells to very bright intensities.

Acuity test in silico

To simulate the test, we picked one out of the 4 possible orientations and flashed the letter E at a new random position every 60 ms (corresponding to the decay time constant of the temporal RF, fitted with an exponential decay). The purpose of this random renewal of the position is to mimic the fixational eye movements of the patient, which would displace the letter over the retinal surface (see also [START_REF] Burak | Bayesian model of dynamic image stabilization in the visual system[END_REF]). The letter was white on a black background (100% contrast).

We then simulated the entire retinal output as a collection of ganglion cells with receptive fields regularly spanning the visual field. Our experiments have determined the radius of each receptive field to be 56 microns on average for the macaque. This average value was thus used as the 1-std of the Gaussian RF of the neurons. Each time a stimulus was presented, it was convolved with the receptive field of each cell and the result was characterized through the non-linear function experimentally predicting the firing rate for each cell.

We then assumed that ganglion cells emitted spikes according to a Poisson process, as in a classical Linear-Non-linear Poisson model, previously used in the retina [START_REF] Pillow | Spatio-temporal correlations and visual signalling in a complete neuronal population[END_REF]. PATIENTS TREATED WITH OPTOGENETIC THERAPY

We then decoded which letter was presented from these spike trains. For this we adopted a maximum likelihood strategy: we assumed perfect knowledge of the model and tested for which stimulus the observed spiking response was the most likely. First we computed the firing rates of the whole population of cells in response to every possible position and every letter f(cell, position, letter). Then, assuming a Poisson distribution of firing rates, we calculated the log-likelihood of the firing rates observed given any letter [START_REF] Doya | Bayesian brain: Probabilistic approaches to neural coding[END_REF]:

log p( f obs |letter)= X position,cell f obs (cell) log f (cell, position, letter) °∆t X position,cell f (cell, position, letter) + c (IV.2)
where f obs are the firing rates simulated for all cells in response to the letter presented, ∆t is the time of presentation of the letter in a given position (60ms), and c is a constant.

We then chose the letter with the highest log-likelihood as the prediction of our model.

The decoding was performed at each time step. Over the time course of the presentation, the decoding could benefit from evidence accumulated at previous time steps. So the decoding performance got better and better over time. We repeated this test 100 times with random letters and averaged the performance over time of our decoder. The performance was defined as the percentage of letters correctly guessed. We defined the acuity as the smallest letter for which the performance was 95% after a time exposure of 1 second.

CHAPTER feature from the visual scene. However, there is an increasing body of evidence that a spike emitted by a ganglion cell does not always carry the same meaning about the visual scene. This leads us to the hypothesis that one ganglion cell type is multiplexing several features.

After presenting the arguments in favor of the multiplexing hypothesis, we will discuss arguments against it and suggest future developments that might lead to a definitive answer to the question.

1 Arguments in favor of multiplexing

Experimental evidence

On-Off DSGCs respond to flashes. On-Off direction selective ganglion cells (DSGCs)

have often been reported to spike robustly in response to changes in luminance without motion [START_REF] Barlow | Selective sensitivity to direction of movement in ganglion cells of the rabbit retina[END_REF][START_REF] Barlow | The mechanism of directionally selective units in rabbit's retina[END_REF][START_REF] Fried | Mechanisms and circuitry underlying directional selectivity in the retina[END_REF][START_REF] Trenholm | Parallel mechanisms encode direction in the retina[END_REF]. Therefore, a burst of spikes emitted by a DSGC mean one of two things: either there is a object moving in one direction, or there is a flash. [START_REF] Barlow | Selective sensitivity to direction of movement in ganglion cells of the rabbit retina[END_REF] suggested that the brain infers the direction of motion by reading out responses from a Our result. We find a type of RGCs in rat that performs two different computations depending on the location of a moving object. When the object moves in the center of the cell's receptive field, the cell performs a linear (X-like) computation that extracts the position of the object. When the object moves in the surround of the RF, the cell performs a non-linear (Y-like) computation that extracts the speed of the object. This particular setting allowed us to explore the interplay between these two multiplexed features. We find that one computation is prioritized on the other -which is suppressed-with a non-linear gain control mechanism.

Multiplexing features as a way of reducing dimensionality

The optic nerve can be seen as a physical information channel [START_REF] Shannon | A mathematical theory of communication[END_REF], and as such has a finite capacity to transmit information. The retina is performing a huge dimensionality reduction of its input in the optic nerve. In human, there are 130 millions photoreceptors for only 1.3 million fibers in the optic nerve (factor of 100). In mouse, there are 6.6 millions photoreceptors for only 45,000 thousand fibers in the optic nerve (factor of 100) (Webvision Utah). Here, we suggest that multiplexing is a coding strategy that contributes to this dimensionality reduction [START_REF] Meister | Multineuronal codes in retinal signaling[END_REF]. Here are the main theories of efficient coding in the retina.

Adaptation to luminosity and contrast. Vision operates under a huge range of lighting conditions, from a night without moon in the countryside to a sunny day in a ski resort.

The light intensities of visual scenes can span 9 orders of magnitude in different contexts, and 4 orders of magnitude within a single visual scene [START_REF] Dunn | Light adaptation in cone vision involves switching between receptor and post-receptor sites[END_REF]. And yet, a fiber of the optic nerve has a much narrower dynamic range, with firing rates spanning 2 or 3 orders of magnitude, depending on species. Redundancy reduction in space and time. Nearby pixels in natural images have highly correlated light intensities [START_REF] Srinivasan | Predictive coding: a fresh view of inhibition in the retina[END_REF]. As a consequence, encoding each pixel intensity independently in a separate channel would result in a highly redundant code. By convolving the visual scene with an antagonistic center-surround filter, retinal ganglion cells avoid this redundancy and encode only local changes in intensity (Atick andRedlich, 1991, 1992;[START_REF] Srinivasan | Predictive coding: a fresh view of inhibition in the retina[END_REF]. Similarly, biphasic temporal filters, which are found in many RGC types, reduce the temporal redundancy inherent to natural visual scenes [START_REF] Van Hateren | Spatiotemporal contrast sensitivity of early vision[END_REF]. Compared to pixel encoding, these strategies reduce considerably the number of spikes that needs to be sent through single optic nerve fibers, thus optimizing the first constraint evoked by Barlow.

Arguments against multiplexing

Precise definition of a feature lacking

In order to prove that ganglion cells are multiplexing different features, it is necessary to define precisely what we mean by feature extraction. An interesting lead is to define a feature as an aspect of visual scenes that is relevant to the animal's behavior [START_REF] Barlow | Possible principles underlying the transformations of sensory messages[END_REF][START_REF] Gollisch | Eye smarter than scientists believed: neural computations in circuits of the retina[END_REF][START_REF] Marr | Vision: A computational approach[END_REF]. According to this definition, a feature detector would make explicit the information that is relevant to behavior, and should be invariant

Unnatural stimuli

In the examples of multiplexing cited above, we mentioned the case of On-Off DSGCs, that respond preferentially to movement in a preferred direction, but also to flashes of light.

Interestingly, a recent study in rabbit [START_REF] Im | Directionally selective retinal ganglion cells suppress luminance responses during natural viewing[END_REF] showed that natural movies radically suppress the response to those luminance changes that occur without associated motion. Further testing with controlled stimuli revealed that this suppression is triggered by a global motion within the receptive-field surround, regardless of the direction of motion. DS cells would thus act as true feature detectors in a natural context. A limit of this study is that it used only natural movies with permanent global motion and rich spatial contrast. One could argue that a still background with wide regions of poor contrast (e.g. sky) is also a natural setting.

Remapping of features in different contexts

The However, this definition of an explicit representation neglects another type of decoding that immediate downstream neurons are perfectly suited for: coincident spike detection [START_REF] Barlow | Models of the visual cortex[END_REF][START_REF] Chapter | Conditions for versatile learning, Helmholtz's unconscious inference, and the task of perception[END_REF][START_REF] Meister | Multineuronal codes in retinal signaling[END_REF]. This non-linear operation, performed on neu-

  Retinal Ganglion Cell DSGC: Direction-Selective Ganglion Cell LED: Local-Edge Detector RF: Receptive Field STA: Spike-Triggered Average LN: Linear-Non-linear model LNP: Linear-Non-linear-Poisson model AAV: Adeno-Associated Virus ChR2: Channelrhodopsin-2 NpHR: Halorhodopsin ix x CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS

  Transient and sustained cells. Orthogonally to this early classification, Cleland et al. (1971) found that ganglion cells in cat could be divided as transient and sustained (fig. I.4C). Transient cells encode changes in light intensity around a mean, whereas sustained cells encode the mean light level. The sustained/transient classification applied to On, Off and On-Off cells.

  substrate. The receptive field of a ganglion cell is widely believed to match the location and extent of its dendritic arbor (fig. I.6A-C, Baden et al. (2016); Demb et al. CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS Regular tiling. Consistently with anatomy, it is often found that the receptive fields of a given type of RGC tile the visual field in a regular lattice (fig. I.6D, Chichilnisky (2001); Devries and Baylor (1997)). However, this mosaic organisation of receptive fields of a single type might not be the rule for all species. In salamander, Segev et al. (2006) isolated 6 distinct types of ganglion cells based on their temporal receptive field profile. Interestingly he found a vast amount of overlap between cell RFs of a same type and no evidence pointing to a mosaic organisation. This observation could be explained by three different hypotheses:

  discovered the inhibitory influence of the surround on the center. He first flashed a spot in the RF center of a frog ganglion cell, and obtained spiking responses to both onset and offset of the stimulus (On-Off cell). In a second trial, he flashed simultaneously to the first spot a second light spot about 0.5 mm away from the central spot. The second spot suppressed both On and Off responses to the first spot (fig. I.7B). Note that flashing the second spot alone did not elicit any response (cell stayed silent).

  . More recently, Davenport et al. (2008) showed that H1 horizontal cells are responsible for the surround antagonism in parasol ganglion cells in the macaque retina (fig. I.12A). Interestingly, the pathway involved was not the classical GABAergic one but a novel pathway that directly modulates the calcium current in cones. Amacrine cells. Amacrine cells are responsible for the surround antagonism of some ganglion cell types in salamander (fig. I.12B, Cook and McReynolds (1998)) and guinea pig

  In a beautiful experiment (presented in fig. I.14), Demb et al. (1999) demonstrates that the linear aspect of the response of Y-cells (at the origin of its linear center-surround structure) is generated by the same pathway that is responsible for the non-linear responses at fine spatial frequencies (fig. I.14 for an explanation).

Subunit model.

  Hochstein and Shapley (1976) found a circuit that explains qualitatively how the Y-cell responds to moving textures regardless of the direction or the spatial pattern (fig. I.15A,B). Small features of the texture activate different subfields as they move around. The subfields have strongly transient responses, and are thus sensitive to local changes, but not to static patterns. A nonlinear rectification associated to each subfield then allows accumulation of signals from many activated subfields while preventing cancellation from other subfields that experience non-preferred stimulus changes. The subunit model was later found to give a precise quantitative description of Y-cell responses to various stimuli, including reversing gratings (Enroth-Cugell and Freeman, 1987; Olveczky et al., 2003; Victor and Shapley, 1979).

Figure I. 15 :

 15 Figure I.15: Microcircuits of X and Y cells. (A) Y-cells show activation when a fine grating shifts in either direction over the receptive field (circle), even though the average illumination remains constant. (B) The underlying microcircuit of Y-cells. Each shift of the grating excites some bipolar cells and inhibits others. The bipolar cells have biphasic dynamics (see impulse response in inset) and thus respond transiently. Only the depolarized bipolar cells communicate to the ganglion cell, because of rectification in synaptic transmission. Thus, the ganglion cell fires transiently on every shift (reprinted from Gollisch 2010). Notations: triangle = neuron; rectangle = temporal filter function; oval = instantaneous rectifier; closed/open circle = sign-preserving/inverting synapse. (C) Circuit diagram to the center-surround structure of a Y cell. When a grating reverses contrast in the periphery, it evokes asynchronous responses in adjacent cones and thus in their postsynaptic OFF bipolar cells. The latter release transmitter asynchronously onto an OFF wide-field spiking amacrine cell. Assuming that the nonlinearity arises at the bipolar-amacrine synapse, it is then transmitted via the spiking amacrine cell to the ganglion cell and/or its presynaptic bipolar cell. The spiking amacrine cell releases an inhibitory transmitter, such as GABA, and hyperpolarizes the ganglion cell at each contrast reversal, creating the characteristic nonlinear response. (D) Simplified diagram of the Y-cell including center-surround structure. Amacrine cells are showed to contact the ganglion cell directly but they could also contact bipolar cells. (E) Compensation for nonlinearities in X-cells mediated by crossover inhibition. Scheme for signal flow of crossover inhibition at a generalized synapse in the retina. (A,B) Voltage responses in ON and OFF presynaptic cells to a bright step of light. (C) Excitatory currents generated in the postsynaptic ON cells showing rectification where presynaptic depolarization elicits a large inward current, while presynaptic hyperpolarization elicits a smaller outward current. (D) Excitatory currents generated in a postsynaptic OFF cell. (E) Crossover current to an ON postsynaptic cell derived from the OFF pathway carried by an OFF amacrine cell (blue arrow). (F) Crossover current to an OFF postsynaptic cell. (G) Voltage in an ON postsynaptic cell generated by the addition of ON excitation and OFF crossover inhibition. (H) Postsynaptic voltage in an OFF postsynaptic cell generated by OFF excitation and ON crossover inhibition. (reprinted from Werblin 2010). (F) Circuit diagram of an X cell implementing cross-over inhibition.

  On/Off/On-Off classification. Y cells respond to both polarity reversals of a fine grating (fig. I.13B). Does this imply that Y cells are always On-Off cells? No. We present in figure I.16 examples of biophysical microcircuits showing that X/Y cell classification is in theory

  mouse retina (W3 cells, Zhang et al. (2012)) (fig. I.19A-C). Mechanism. Zhang et al. (2012) investigated the circuit underlying Local Edge Detectors (LEDs) (fig. I.19D). The On-Off center of these cells and the stratification of their dendritic arbor in the IPL suggest that they receive synapses from both On and Off bipolar cells. Adding an action potential blocker (TTX) to the bath deactivated the inhibition produced by global motion entirely, suggesting that the inhibitory surround is mediated by spiking amacrine cells. Using voltage-clamp, they could measure the excitatory and inhibitory currents received by the RGC during global motion and differential motion. Interestingly they found that the inhibitory surround acts by two complementary pathways:

  and rabbit (Levick, 1967). Local edge detectors comprise approximately 15% of ganglion cells in rabbit (van Wyk et al., 2006). Interestingly, it was shown recently that LEDs are the most numerous ganglion cell type in mouse RGCs (W3 cells, Zhang et al. (2012)).

  first reported the existence of orientation selective (OS) RGCs in rabbit retina (fig. I.20A). These RGCs are selective for either horizontal or vertical orientations of stationary or moving bars. Many other cell types present an oval-shaped RF. The specificity of these cells is the complete absence of response for a bar presented orthogonally to the preferred orientation (Levick, 1967) . OSGCs can be On-center or Offcenter. Weak responses can be elicited from stimulating the flanking region with a small spot of the opposite polarity of the center. Wide stimulation of the inhibitory flanks can produce large responses. There is mutual inhibitory interactions between the center and the flanking regions, as shown by the experiment presented in figure I.20B: simultaneous CHAPTER I. PHYSIOLOGY OF RETINAL GANGLION CELLS illumination of both surround and center with a full-field flash results in the suppression of both On and Off responses.

  Other behaviorally relevant computations. Cell types computation have been linked to many other behaviorally relevant features. Y cells without antagonistic surround could detect global motion[START_REF] Gollisch | Eye smarter than scientists believed: neural computations in circuits of the retina[END_REF]. Y cells with an antagonistic surround could detect objects moving differentially from the background ("Object Motion Sensitive" cells, Olveczky et al. (2003)). "Sensitizing cells", found in rabbit, mouse and salamander retinas, could keep track of the location of camouflage objects after they stopped moving(Kastner and Baccus, 2011, 2014). Cells responding to approaching motion were described in mouse retina[START_REF] Chapter | Approach sensitivity in the retina processed by a multifunctional neural circuit[END_REF].

  of electrical stimulation. Retinal prostheses use an array of electrodes to electrically stimulate the retina. The resolution of the electrode-tissue interface is a critical issue[START_REF] Lorach | Neural stimulation for visual rehabilitation: Advances and challenges[END_REF][START_REF] Weiland | Electrical Stimulation of the Retina to Produce Artificial Vision[END_REF]. It is controlled by several factors, such as density of electrodes, electrical current spread in extra-cellular medium, and how neurons respond to electrical stimulation. Optimizing these variables in retinal stimulators is the subject of much of the ongoing research in the field[START_REF] Weiland | Electrical Stimulation of the Retina to Produce Artificial Vision[END_REF]. Solutions to improve the confinement of the electrical fields include:• using local return electrodes[START_REF] Lorach | Photovoltaic restoration of sight with high visual acuity[END_REF],• tailoring adapted spatial patterns of stimulation[START_REF] Jepson | Spatially patterned electrical stimulation to enhance resolution of retinal prostheses[END_REF],• optimizing stimulus pulse duration[START_REF] Weitz | Improving the spatial resolution of epiretinal implants by increasing stimulus pulse duration[END_REF],• creating microscopic cavities in the implant in which the current stays confined (Djilas et al. (2011), 3D implant). Axonal stimulation. Epiretinal arrays stimulate axon fibers of cells that are located far from the area of stimuation (fig. II.7, Behrend et al. (2011); Rizzo et al. (2003); Wilms and Eckhorn (2005)). Argus II (epiretinal implant) patients typically report that phosphenes often appear in the form of highly elongated ellipses. While the increased distance of the stimulating electrodes from the axonal fibres means that axonal stimulation may be less of a concern for subretinal devices, there are indications that in a subset of patients a CHAPTER II. METHODS OF VISUAL RESTORATION AND ACUITY certain amount of axonal stimulation occurs (Tsai et al., 2009; Wilke et al., 2011). Solutions to reduce the perceptual effect of axonal stimulation include: • carrying out an image processing that signal only edges of objects to minimize the number of stimulated electrodes (Fine and Boynton, 2015), • incorporating the perceptual effects of axonal stimulation in the encoding model (Fine and Boynton, 2015), • developing encoding models that include anodic (suppressive) stimulation on electrodes located on the axonal pathway (Fine and Boynton, 2015). It is unclear whether optogenetic activation of RGCs will also create an elongated RF due to the expression of light sensitive proteins on the axon of ganglion cells. If this was the case, a solution would consist in targeting the expression of the opsin to somas by taking advantage of intrinsic localization mechanisms of target cells (Baker et al., 2016; Greenberg et al., 2011). Fading of electrically evoked percepts. Retinal prosthesis users have reported that percepts tend to fade within seconds of stimulus onset (Pérez Fornos et al., 2012; Stingl et al., 2015; Zrenner et al., 2011). The prevailing theory for percept fading in normal vision is that retinal neurons adapt to the unchanging stimulus[START_REF] Weiland | Electrical Stimulation of the Retina to Produce Artificial Vision[END_REF]. In visual prosthetics with head-mounted cameras, such as the Argus II, patients tend to move their head to change the pattern of electrical stimulation on the retina. In prosthetics with embedded photodiodes, such as Alpha IMS, eye movements move the captured images over the stimulated retina, much like they do in natural vision. However, there are still reports of image fading with this prosthesis[START_REF] Hafed | Oculomotor behavior of blind patients seeing with a subretinal visual implant[END_REF][START_REF] Stingl | Subretinal Visual Implant Alpha IMS-Clinical trial interim report[END_REF]. A potential explanation for the persistent fading despite utilizing microsaccades is that the electrical CHAPTER II. METHODS OF VISUAL RESTORATION AND ACUITY lation to the light sensitivity of the optogenetic protein, by controlling light income with an external device[START_REF] Fine | Pulse trains to percepts: the challenge of creating a perceptually intelligible world with sight recovery technologies[END_REF]. Alternatively, photodiodes of the Alpha IMS implant adjust their sensitivity across a wide range of luminance (5 log units), allowing patients to use the implant in dimly illuminated rooms, as well as outside on a bright sunny day[START_REF] Stingl | Artificial vision with wirelessly powered subretinal electronic implant alpha-IMS[END_REF].

  and the spacing between stimulating electrodes (da Cruz et al., 2013; Lorach et al., 2015; Stingl et al., 2013). The Nyquist-Shannon sampling theorem is evoked to justify this adequacy (Palanker et al., 2005; Stingl et al., 2013), and also to predict the maximum acuity reachable with untested prosthesis designs (Palanker et al., 2005; Tochitsky and Kramer, 2015; Weiland et al., 2016).

  array of 252 electrodes[START_REF] Marre | Mapping a complete neural population in the retina[END_REF][START_REF] Yger | Fast and accurate spike sorting in vitro and in vivo for up to thousands of electrodes[END_REF]. We measured the receptive field center of each cell with binary checkerboard noise. To separate ganglion cells into different types, we displayed several stimuli (full field flicker, drifting textures) and grouped together cells with similar responses (see Methods). In the following, we focus on a single group composed of well-isolated fast OFF cells. Their responses to spatially uniform stimuli were nearly identical (fig. III.1A), and their receptive fields clearly tiled the visual space (fig. III.1B). We then displayed a bar moving randomly over the visual field. This dark bar over a gray background was animated by a Brownian motion with a feedback force to keep the CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE bar positioned over the array. As expected, ganglion cells whose receptive field center overlapped with the bar position responded reliably to a repeated trajectory, as shown by their PSTHs in fig. III.1C. More surprisingly, reliable responses were also elicited in cells whose receptive field centers were far away from the bar. The receptive field center diameter was on average 287 ± 23 µm (mean ± SD, n=25), and cells as far as 670µm from the closest bar position responded to the moving bar. These distant cells fired synchronously to the moving bar, largely independently of the location of their receptive field, while central cells did not. Central cells were only synchronous when they were very close to each other. The mean cross-correlation between the responses of pairs of central cells was 0.02 ± 0.04 (mean ± SEM, n = 20 pairs) for cells separated by more than 200 µm along the axis perpendicular to the bar. In comparison, distant cells remained synchronous over large distances (fig. III.1D). The mean cross correlation was 0.53 ± 0.03 (mean ± SEM, n = 35 pairs, Pearson correlation r ) for distant cells separated by more than 200 µm. This distant activation had a profound effect on the structure of the retinal activity: while the bar covered a region 0.4 mm wide, ganglion cells were activated over an area wider than 1.4 mm (fig. III.1E).

Figure

  Figure III.1: A single cell type responds synchronously to distant moving objects. A: Raster of 25 cells of the same type responding to a full field uniform flicker. Each line corresponds to a repeat of the stimulus, and each cell is indicated by a different color (alternating pink and blue). The black curve indicates the light intensity of the flicker over time. B: Receptive fields of a population of ganglion cells of the same type. Each ellipse represents the position and shape of the spatial receptive field associated with one cell (1-SD contour of the 2D Gaussian fit to the spatial profile of the RF). Inset: temporal profiles of the receptive fields of the same cells. C: PSTHs of multiple ganglion cells responding to repeated presentations of a randomly moving bar. Gray shade: position of the bar as a function of time (shade width corresponds to the bar width). Blue traces: PSTHs of individual ganglion cells, with baselines positioned to scale relative to the bar. Blue and red vertical rectangles indicate central and distant cells, respectively. Black ellipse shows an example synchronous firing event of the distant cells. D: Average ± SE cross-correlation between PSTHs of pairs of cells, as a function of their pairwise distance measured along the bar motion axis. Curves shown separately for cells whose receptive field center either was (blue) or was not (red) stimulated by the bar. E: Schematic diagram shows central cells (blue) and distant cells (red) that respond synchronously.

Figure III. 2 :

 2 Figure III.2: OFF ganglion cells perform a linear computation in their receptive field center, and a non-linear computation in the surround. A: Schematic of the LN model, composed of a linear filter and static non-linearity. B: Response (PSTH, gray) of a ganglion cell whose receptive field center is stimulated by the bar, is predicted by the LN model (blue). r = 0.89. C: Response (PSTH, gray) of the same ganglion cell when the bar is far from the receptive field center, is not predicted well by the LN model (blue). r = 0.02. D: Schematic of the subunit model, composed of a first stage (each subunit linearly filters the stimulus and applies a static nonlinearity), followed by weighted linear pooling and a second non-linearity. E: Response (PSTH, gray) of the same ganglion cell (as in B and C) to distant stimulation is predicted well by the subunit model (red). r = 0.83. F: Performance of the LN (blue) and subunit (red) models in predicting ganglion cell responses, as a function of the distance of the cell to the bar. Blue shade: position distribution of the bar. G: Schematic showing that cells whose receptive field center is on top of the moving bar perform a linear computation while distant cells perform a non-linear computation.

(Figure III. 5 :

 5 Figure III.5: Central computation codes for position, while distant computation is invariant to position and codes for stimulus change. A: Distribution of the bar positions for the complete stimulus trajectory ("prior distribution", gray) and 100 ms before the spike of a central cell (blue). Zero corresponds to the location of the cell's receptive field (RF) center. B: Same as A for a distant cell with its receptive field center far from the bar. C: Distribution of the absolute speed of the bar for the complete stimulus trajectory (gray) and 100 ms before the spike of a distant cell (red). D: Ratio between the information individual cells carry about bar speed vs about bar position, as a function of the average distance to the bar. Distribution of bar positions is shown as a blue shade. For selected cells, the insets indicate the mutual information between the spiking response and the position (blue) or the speed (red) at different time delays. E: Schematic showing that central cells code for bar position while distant cells are nearly invariant to it.

a

  neural population formed by ganglion cells of a single type. We constructed a mathematical model that recapitulated the multiplexing of the two relevant computations. To that end, the model required nonlinear summation within the receptive field as well as a gain control mechanism. The model predicted precisely the responses of the fast OFF ganglion cells to a bank of dynamical stimuli which included complex, spatio-temporal stimulation in the far surround. Finally, our experiments suggested that a disinhibitory retinal circuit composed of two amacrine cells could mediate the distant computation.When an object is moving randomly, neurons whose receptive field centers overlap with the object code for its position, while distant neurons code for general, large-scale changes in the stimulus. Each neuron can switch from one computation to the other depending on the visual context. Recent works have shown that the feature extracted by a cell can change when the average luminance changes[START_REF] Smirnakis | Adaptation of retinal processing to image contrast and spatial scale[END_REF][START_REF] Tikidji-Hamburyan | Retinal output changes qualitatively with every change in ambient illuminance[END_REF], or during saccadic exploration of the visual scene[START_REF] Geffen | Retinal ganglion cells can rapidly change polarity from Off to On[END_REF]. Here we show that, in a single visual scene, the same cell type can be used to extract two features simultaneously. Feature extraction does not change only with the average luminance of the visual scene. Rather, two features can be extracted at the same time by a single cell type in a single visual scene. These findings expand the traditional view of a

  cells by distant stimuli through a disinhibitory circuit in which intermediary amacrine cells are activated by bipolar cells and subsequently inhibit glycinergic amacrine cells. This release of glycinergic inhibition can affect both OFF bipolar cells (Eggers and Lukasiewicz, 2010) and OFF ganglion cells (O'Brien et al., 2003), and results in OFF ganglion cell activation. It is unclear if this disinhibitory relay is composed of GABAergic and glycinergic CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE cells, or only of glycinergic cells. Attempts to disentangle the two hypotheses by blocking GABAergic transmission triggered large oscillations in the retina, making the results difficult to interpret (Demb et al., 1999). A similar disinhibitory circuit might also be involved in other kinds of complex processing taking place in the ganglion cell surround. When large visual features stimulate distant regions of the surround, the inhibitory input to bipolar cells (Eggers and Lukasiewicz, 2010) and ganglion cells (O'Brien et al., 2003)

  natively using block gradient descent (McFarland et al., 2013) across 6 iterations. The repeated parts of the stimulus were held back during fitting and were used to cross-validate the model. The parameters of the kernel were optimized to maximize the log-likelihood function of the spike train under Poisson assumption (McFarland et al., 2013). For this optimization we performed Limited-memory BFGS gradient descent on the parameters of the kernel (McFarland et al., 2013). In order to avoid overfitting, we imposed two regularisation constraints: spatio-temporal smoothness and sparseness of the kernel. The cost function C was of the form:

  compute the average filter in the one-bar condition (fig III.3D), we selected only the cells stimulated outside of their receptive field (RF) center. Our criterion was that the bar central position should be more than 200 µm away from the RF center. To compute the average filter in the two-bar condition (fig III.3C), we selected only the cells that were stimulated inside their receptive field centers by at least one of the bars. Our criterion was that the bar central position should be less than 200 µm away from the RF center.

  CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPErandomly and ranging linearly from 0 to 1. For each contrast, the two bars were oscillating during an uninterrupted sequence of 50 seconds, so that the central bar had traveled exactly 100 periods and the distant bar exactly 99 periods during a sequence. At the end of a sequence, all possible phase shifts between the two bars had been visited exactly once.This trick allowed us to average out the influence of one bar when computing the PSTH on the period of the other bar.To show the gradual suppression of the distant response in fig. III.4B, we normalized the amplitude of the response to the distant bar by the amplitude of the response to the distant bar alone (i.e. zero contrast for the central bar).

  11) where r c is the response to the central bar alone at full contrast, r d the response to the distant bar alone, c is the contrast. r c and r d were estimated from the PSTHs in response to the central bar and to the distant bar played alone respectively. We needed to introduce quadratic terms because the PSTH for the central bar condition depended quadratically on the contrast of the central bar. This is consistent with our subunit model, where the first layer contained a rectified quadratic function h.CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPEinstantaneous position of the bar with a lag δt ranging from -1 second (information about the past stimulus) to 1 second (information about the future stimulus):I(R(t ), P(t + δt )) = H(P(t )) °[p(R(t ) = 1)H(P(t + δt )|R(t ) = 1)+ p(R(t ) = 0)H(P(t + δt )|R(t ) = 0)] (III.13)

S8753) to the

  bath (Curtis et al., 1971; Lee et al., 2016b; Menger and Wassle, 2000; Schaeffer and Anderson, 1981). To generate the rasters and PSTHs in response to the central bar, we flashed a dark bar of width 100 µm in the center of the receptive field of the cell for 0.5 CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE s 40 times, separated by 0.5 s of gray screen. For the distant responses, we used 230 µm wide bars flashed for 1 s, in a region 0.5 to 1 mm away of the cell's receptive field center.

( 4 /

 4 5 weeks old) with an AAV2 encoding ReaChR-mCitrine (a variant of Channel Rhodospin with red-shifted sensitivity) under a pan-neuronal hSyn promoter via intravitreal injections. Details of the gene delivery and optogenetic protein expression has been detailed elsewhere (Chaffiol et al., 2016; Sengupta et al., 2016). Retinas were harvested after 4 weeks for MEA recordings. To estimate the size of the region whose stimulation can activate a ganglion cell, we displayed a random checkerboard stimulus for about an hour (details in methods). We estimated the Spike Triggered Average (STA) by averaging over the frames that evoked a spike (see methods). Many cells showed a well-defined receptive field (n=22, fig. IV.1A). The average radius of receptive fields was 58±4 SEM µm (n=22), which is slightly smaller than what is usually measured in normal retinas. The temporal time course of the receptive field (fig. IV.1B) was consistent with a direct light-activation of ganglion cells: there latency of the peak response was only 33ms. CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY

  mixture of DSGCs encoding different preferred directions. DSGCs of different types responding together would unequivocally signal a change in luminance without motion. In this coding scheme, a single DSGC type is multiplexing different features (flash, motion in one direction). Polarity switch in response to a change in luminance. In the mouse retina, two recent studies (Pearson and Kerschensteiner, 2015; Tikidji-Hamburyan et al., 2015) found that many RGCs switch polarity with changes in ambient luminance. Across each luminance transition, most ganglion cells exhibited polarity switching. The response changes occurred for a variety of stimuli, including full-field contrast steps, localized steps, and naturalistic movies. Interestingly, the polarity of the linear filter estimated from a white noise stimulus was found to be stable across all conditions (Tikidji-Hamburyan et al., 2015). Various different circuits were identified that mediate the change in selectivity in different types. These studies illustrate a case where RGCs are extracting different features depend-CHAPTER V. DISCUSSION ing on the context of the visual scene (e.g. day or night). Polarity switch in response to a peripheral shift. In the salamander retina, Geffen et al. (2007) described a subset of On-Off RGCs that transiently switch their polarity from OFF to ON after a sudden shift of a grating presented in their periphery (fig. V.1A), a stimulus reminiscent of a saccade. The switch is fast and transient (ª100ms). During this brief interval, an amacrine cell circuit suppresses the OFF bipolar pathway and strengthens the ON bipolar pathway (fig. V.1B). This example of multiplexing is particularly interesting because the switch in polarity is the result of a complex circuit involving active suppression, and not just a passive property of the circuit such as vesicle depletion. It thus suggests a deliberate coding strategy rather than a failure of the neural circuit to signal a feature robustly in all contexts.

  How does this hypothesis relate to previous work on efficient coding in the retina ? In an article of 1961, Barlow mentions two constraints on the capacity of the optic nerve:1. the capacity of a single fiber (function of the average firing rate on the fiber) 2. the number of fibers composing the nerve Subsequent studies suggesting efficient coding strategies in the retina have focused on the first constraint(Atick and Redlich, 1991, 1992;[START_REF] Buchsbaum | Trichromacy, Opponent Colours Coding and Optimum Colour Information Transmission in the Retina[END_REF][START_REF] Dunn | Light adaptation in cone vision involves switching between receptor and post-receptor sites[END_REF][START_REF] Dunn | The impact of photoreceptor noise on retinal gain controls[END_REF][START_REF] Smirnakis | Adaptation of retinal processing to image contrast and spatial scale[END_REF][START_REF] Srinivasan | Predictive coding: a fresh view of inhibition in the retina[END_REF][START_REF] Van Hateren | Spatiotemporal contrast sensitivity of early vision[END_REF], but few studies have considered the second constraint[START_REF] Gjorgjieva | Benefits of pathway splitting in sensory coding[END_REF][START_REF] Ratliff | Retina is structured to process an excess of darkness in natural scenes[END_REF].

  Photoreceptors and RGCs adapt to the level of luminance by only encoding the fluctuations around the mean (Dunn et al., 2007; Dunn and Rieke, 2006). The retina also adapts its dynamic range to the range of these fluctuations (contrast adaptation: Baccus and Meister (2002); Kastner and Baccus (2014); Olveczky et al. (2003); Smirnakis et al. (1997)). By filtering out luminance and contrast information, this adaptive neural code considerably reduces the amount of information to be transmitted onto single optic nerve fibers.

  to irrelevant causes of variation. Unfortunately, in most examples of multiplexing presented above (except for the case of DS cells, see chapter I for behavioral relevance), there is no clear relation between the cell computation and a specific behavior. Without this link to behavior or perception, it is impossible to rule out completely that the cell -apparently multiplexing different features -might in fact be robustly extracting a single feature in very different ways in different contexts. Some direction-selective cells have been recently linked to a behavior, the optokinetic reflex[START_REF] Yonehara | Congenital Nystagmus Gene FRMD7 Is Necessary for Establishing a Neuronal Circuit Asymmetry for Direction Selectivity[END_REF], cf chapter I). It would be interesting to study if these cells can change their selectivity depending on the visual scene that needs to be encoded. A recent study[START_REF] Vlasits | Visual stimulation switches the polarity of excitatory input to starburst amacrine cells[END_REF] carried on explanted retinas showed that DS cells can change their preferred direction following a prolonged global motion stimulation. However, this effect was not reversible, suggesting that it was caused by a deterioration of the retina that would not happen in physiological conditions. Further experiments on DS cells in vivo would be interesting to confirm this change in selectivity.

  studies of Tikidji-Hamburyan et al. (2015) and Geffen et al. (2007) do not preclude the possibility that RGCs act largely as uniform feature detectors, and only remap their feature selectivity in different contexts (saccade versus fixation, day versus night). The idea of uniform feature detectors would thus remain largely true in any given visual context. However, our study shows that in the very same context, cells of a same type extract different features of the visual scene. These results thus show another level of flexibility of the neural code in the retina. CHAPTER V. DISCUSSION2.4 The decoding problemA major concern arising from the multiplexing hypothesis is the decoding question: how can the brain attribute a given spike to the right feature? In the example provided by Geffen et al.(2007), it is reasonable to assume that the brain can detect the saccadic context and infer the change of polarity of specific types of RGCs. It is less clear how the brain can distinguish multiplexed features in the case presented in our study. We find that cells performing the non-linear computation in their surround respond synchronously across large regions of the retina. This synchrony might act as a disambiguating signal for the brain.The decoding problem is tightly associated to the question of what an explicit representation of information in a neural population is.[START_REF] Gollisch | Eye smarter than scientists believed: neural computations in circuits of the retina[END_REF] propose the following definition: "In a distributed representation, the answer of interest should be obtainable through linear decoding, namely by a simple weighted summation over the single-neuron activities. Such a decoding could be achieved in a single step by a downstream neuron that samples from the population." Using this definition, a population of RGCs that multiplexes features does not provide the brain with an explicit representation of any of the features, which defeats the purpose of extracting features.

  rons of the same type or of different types, could well disambiguate multiplexed signals in a single step. Furthermore, there is a huge amount of synchrony in the retina (Schnei-CHAPTER V. DISCUSSION dman et al., 2006), which is an ideal setting for a combinatorial code based on coincident spiking. Finally, an unpublished study by Roska et al ("Different modes of visual integration in the lateral geniculate nucleus revealed by targeted, single-cell-initiated transsynaptic tracing") shows that single cells in the lateral geniculate nucleus (LGN) of mouse typically receive inputs from two or more different RGC types. The precise description of branching patterns of neurons of the LGN is an exciting perspective, that could provide new insights on the way to disentangle features multiplexed in the activity of a population of retinal ganglion cells, and more generally on the nature of the neural code in the retina.

3 Optogenetic therapy VS Retinal prostheses

  Some diseases leading to blindness have currently no cure. This is the case of retinitis pigmentosa (RP), a disease affecting 1 million persons worldwide. In RP, photoreceptors degenerate, resulting in a progressive reduction of the visual field that often leads to blindness (fig. II.1, Lorach et al. (2013)).The photoreceptors are often the first victims of diseases leading to blindness. This observation has lead to the development of two methods for vision restoration that pro-CHAPTER II. METHODS OF VISUAL RESTORATION AND ACUITY layer (bipolar cells, amacrine cells) as well as in the ganglion cell layer, even in late stages of the disease[START_REF] Humayun | Morphometric analysis of the extramacular retina from postmortem eyes with retinitis pigmentosa[END_REF][START_REF] Santos | Preservation of the inner retina in retinitis pigmentosa: a morphometric analysis[END_REF]. Subretinal implants are placed between the choroid and the retina, so they primarily target the remaining neurons in the inner nuclear layer of the retina (bipolar cells, amacrine cells). Another issue is the absence of promoters specific to ON versus OFF cells for RGCs. Consequently, for strategies targeting RGCs, there is currently no other option than using ubiquitous promoters and to commute all cells in either ON cells with ChR2, or in OFF cells with NpHR, or in a mix between the two[START_REF] Roska | La rétinopathie pigmentaire : restauration visuelle par thérapie optogénétique[END_REF]. However, this field moves rapidly and new viral vectors with specific promoters and improved transduction efficiencies are under development[START_REF] Roska | La rétinopathie pigmentaire : restauration visuelle par thérapie optogénétique[END_REF].

	Electronic prostheses have already produced partial restoration of vision in blind hu-
	mans. With these implants, some blind patients were able to read letters again, and even
	sometimes words (Roska et al., 2013). However, in all treated patients the restored visual
	Alpha IMS implant. The only commercialized subretinal implant is the Alpha IMS im-acuity was low. The healthy human retina contains 1.2 million RGCs, but current retinal
	plant, created by the German start-up Retinal Implant AG (Weiland et al., 2016). Each chips contain only hundreds or thousands of electrodes spaced 50-500 microns apart, up
	"pixel" of the implant consists of a canonical circuit consisting of a photodetector, ampli-
	1.2 Optogenetic therapy
	Description. Optogenetic therapy aims at restoring light sensitivity in degenerated reti-
	nas with the help of light sensitive proteins. These proteins directly transform light into
	an electrical signal in the target cell (fig. II.5). One of the most widely used optogenetic
	protein is channelrhodopsin-2 (ChR-2), originally found in the unicellular green algae
	Chlamydomonas reinhardtii (Mutter et al., 2014). Another optogenetic tool with antago-

Contents 1 Visual restoration strategies .......................... 1.1 Retinal Prostheses ............................. 1.2 Optogenetic therapy ............................ 1.3 Optogenetic therapy VS Retinal prostheses ............... 2 Current challenges in restoring a high acuity vision ............. 2.1 Addressing problem ............................ 2.2 Encoding problem ............................. 3 Acuity estimation ................................. 3.1 Acuity estimation from receptor spacing ................ CHAPTER II. METHODS OF VISUAL RESTORATION AND ACUITY vide visual information directly to neurons of the inner retina, bypassing degenerated photoreceptors (Strazzeri et al., 2014). The first method consists in a prosthesis conveying an electrical representation of visual stimuli to retinal ganglion cells via an array of stimulating electrodes placed above or below the ganglion cell layer. A second, optogenetic method, consists in inserting light-gated channels such as channelrhodopsin into surviving cones somas, bipolar or retinal ganglion cells in order to render these cells lightsensitive and thus able to transduce visual to neural signals in place of the degenerated photoreceptors. fier, and electrical stimulator. Light activates the photodiodes, and the connected electrodes stimulate the retina (fig. II.4). The implant has 1,500 electrodes of size 50x50µm, with 70µm center-to-center spacing. It spans a region of 3x3mm on the retina, corresponding to a visual angle of 10 ± .

Acuity. In clinical trials, the best visual acuity restored with this implant was 20/546

[START_REF] Stingl | Artificial vision with wirelessly powered subretinal electronic implant alpha-IMS[END_REF]

, which is also the best performance obtained so far with retinal prostheses

[START_REF] Weiland | Electrical Stimulation of the Retina to Produce Artificial Vision[END_REF]

. This acuity is still bellow the level of legal blindness.

nistic action is halorhodopsin (eNpHR), a light-activated chloride pump that hyperpolarizes the target cell. Both ChR-2 and eNpHR have fast kinetic properties with deactivation 1.

to 10-fold wider that the packing density of RGCs

[START_REF] Tochitsky | Optopharmacological tools for restoring visual function in degenerative retinal diseases[END_REF]

. The stimulated area of the retina is also limited by the physical size of the chip, which typically only covers the region corresponding to 20 degrees of visual angle. Larger chips with higher electrode densities can be manufactured, but may result in problems with power delivery, crosstalk between neighbouring electrodes and current spread in the extra-cellular medium

[START_REF] Wilke | Spatial resolution and perception of patterns mediated by a subretinal 16-electrode array in patients blinded by hereditary retinal dystrophies[END_REF]

.

Optogenetic therapies have not been tested on patients yet but provide a promising alternative to implants. First these therapies could be advantageous in terms of cost (the current implant costs 150,000$, not including surgery), and contrarily to the implant they do not require an invasive surgery. The critical difference between these techniques is the way by which the current generated by light stimulates the cells. With implants, the current is distributed in the extra-cellular medium and activates the cells at proximity. In optogenetics the cells themselves become light sensitive. In principle, such treatments can confer light-sensitivity to all neurons of a particular cell type, allowing for high visual acuity. However, in practice, the efficiency of viral transduction tends to be low, resulting in the expression in a minority of targeted cells, for example around 5% of mouse bipolar cells

[START_REF] Lagali | Light-activated channels targeted to ON bipolar cells restore visual function in retinal degeneration[END_REF]

or 5-10% of marmoset RGCs

[START_REF] Ivanova | Evaluation of AAV-mediated expression of Chop2-GFP in the marmoset retina[END_REF]

.

Some challenges are common to the two approaches. One of them is the reactivation of the central retina, the region of high acuity in human vision. In this region, RGCs, bipolar cells and even photoreceptor somas are deported from the foveal pit. Consequently, RGCs are not spread in a monolayer but rather pile on several layers, making selective electrical stimulation of individual cells difficult if not impossible with retinal implants.

  [START_REF] Roska | La rétinopathie pigmentaire : restauration visuelle par thérapie optogénétique[END_REF][START_REF] Weiland | Electrical Stimulation of the Retina to Produce Artificial Vision[END_REF]. However, some studies using specific promoters for ON bipolar cells have successfully restored As seen in chapter one, RGCs can be classified as transient or sustained. It is unclear how visual restoration techniques targeting ganglion cells can reproduce these properties[START_REF] Weiland | Electrical Stimulation of the Retina to Produce Artificial Vision[END_REF]. Another hallmark of retinal processing is center-surround antagonism in RGCs. Strategies targeting upstream layers can restore this antagonism (Busskamp et al., 2010; Lagali et al., 2008). Alternatively, Greenberg et al. (2011) has been able to recreate antagonistic center-surround interactions in an explanted rabbit retina, by targeting specifically somatic and dendritic compartements of RGCs with NpHR and ChR2 and convolving the stimulus with an appropriate filter (fig. II.8). there is currently no model of blind primate, on which visual restoration therapies could be tested (but see Strazzeri et al. (2014) for a recent model of photoreceptor degeneration based on focal laser damage). These obstacles have led researchers to estimate acuity indirectly with theoretical approaches.

	Center-Surround.

ON responses

[START_REF] Lagali | Light-activated channels targeted to ON bipolar cells restore visual function in retinal degeneration[END_REF]

and OFF responses (mediated by amacrine cells, Macé et al. (2015)) in at least some ganglion cells. CHAPTER II. METHODS OF VISUAL RESTORATION AND ACUITY Transient/Sustained RGCs. DS cells. The nystagmus is a reflex eye movement that stabilizes the visual scene projection on the retina. A recent study has shown that DS cells are required for this reflex in mouse (Yonehara et al., 2016). Apart from the strategy of Busskamp et al. (2010), which consists in reactivating remaining photoreceptors genetically, no other visual restoration strategy has been able to recreate direction-selectivity in RGCs. and

  visual scenes. Feature maps, where the same computation is applied repeatedly across different sub-regions of the entire visual scene, are essential building blocks for this task, for both sensory networks[START_REF] Fitzpatrick | Editorial overview: neural maps[END_REF][START_REF] Ohki | Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex[END_REF] and artificial vision systems[START_REF] Lecun | Deep learning[END_REF]. Ganglion cells, which form the retinal output, can be divided into different types[START_REF] Baden | The functional diversity of retinal ganglion cells in the mouse[END_REF][START_REF] Field | Functional connectivity in the retina at the resolution of photoreceptors[END_REF][START_REF] Wassle | Functional architecture of the mammalian retina[END_REF]. In the classical view of retinal function, cells of the same type extract a single feature from the visual scene and generate a feature map that is then sent to the Hamburyan et al., 2015), so that feature extraction will be influenced by the global parameters of the visual scene, e.g., by its luminance and contrast. Furthermore, ganglion cell activity can be modulated by stimulation outside of the cells' classically-defined receptive fields[START_REF] Marre | High accuracy decoding of dynamical motion from a large retinal population[END_REF][START_REF] Mcilwain | Receptive fields of optic tract axons and lateral geniculate cells: peripheral extent and barbiturate sensitivity[END_REF][START_REF] Passaglia | Effects of remote stimulation on the mean firing rate of cat retinal ganglion cells[END_REF][START_REF] Passaglia | Effects of remote stimulation on the modulated activity of cat retinal ganglion cells[END_REF][START_REF] Roska | Rapid global shifts in natural scenes block spiking in specific ganglion cell types[END_REF], implying that feature extraction may not be entirely local, especially when presented with complex, dynamical stimuli. As a result, it is not clear how irregular trajectories of moving objects, which are ubiquitous in natural scenes(Branson et al., 2009; CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE ferent features from a visual scene composed of irregularly moving bars. Within a homogeneous population of fast OFF ganglion cells recorded simultaneously, cells whose

brain (da Silveira and

[START_REF] Da Silveira | Cell types, circuits, computation[END_REF]

. This "one type = one feature" view is well illustrated in the retina when objects move across the visual field at constant speed. In this case, previous work has shown that a single type indeed represents a single feature of the scene (Berry et al., 1999; Leonardo and Meister, 2013; Trenholm et al., 2013; Vaney et al., 2012). However, processing by ganglion cells also depends on the visual context (Farrow et al., 2013; Geffen et al., 2007; Shapley and Enroth-Cugell, 1984; Smirnakis et al., 1997; Tikidji-Ezenman et al., 1985), are represented by ganglion cells of the same type.

Here we show that a single ganglion cell type extracts simultaneously two very dif-

Chapter IV Predicting the visual acuity reachable for blind patients treated with optogenetic therapy Contribution

  We argue that this technique could provide a fast, cheap and accurate estimate of the acuity that can be expected from upcoming strategies of visual restoration.CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPYextra-cellular medium (cf chapter II). Vision restoration based on optogenetics aims atexpressing light sensitive proteins in cells of the retina, and stimulate these newly lightsensitive cells with patterned light to evoke visual perception. This is a promising alternative, but it is unclear what acuity can be expected with this strategy. Measuring the acuity with behavioral experiments on non-human primates is made difficult by the fact that the light stimulation necessary to activate the transfected cells will also activate the photoreceptors, and the effect of photoreceptor versus optogenetic activation cannot be separated. There is no model of blind primates that could circumvent this issue.

	The spacing between receptors of a visual system (e.g. photoreceptors, reactivated cells,
	photodiodes, electrodes, etc.) is only one of many parameters necessary to estimate the
	maximum acuity reachable by the system. In this chapter, we propose a principled way to
	measure the best acuity reachable by blind patients from extra-cellular recordings of reti-
	nal ganglion cells.

  acuity test (fig. IV.3A). We chose a classical acuity test used in ophthalmology, the random E test, where the letter 'E' is presented in 4 possible directions. The test consisted in presenting randomly a letter to the retina in silico for 1 second, animated by a random jitter mimicking eye movements (see methods). We then predicted which letter was presented CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY from the spike trains. The success rate of the prediction depended on two main factors (see methods): the size of the letter and the time of exposure of the letter. To be realistic and consistent with real in-situation acuity tests, we defined the acuity score as the smallest letter for which the success rate was above 95% with a time exposure of 1 second. With this definition of acuity, we predicted that the acuity reached by the macaque treated with Catch in our experiment was 20 minutes of visual angle, which is exactly the threshold of legal blindness (fig. IV.3B,C).

  [START_REF] Greenberg | Differential targeting of optical neuromodulators to ganglion cell soma and dendrites allows dynamic control of centersurround antagonism[END_REF] have shown that it was possible to express combinations of opsins selectively in the somas, dendrites and axons to effectively reduce the size of the receptive field. Finally, the amplitude of the non-linearity (i.e. the maximal reachable firing rate) is related to the maximal current possibly generated by the opsin. Depending on the choice of opsin, this amplitude could

	CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND
	PATIENTS TREATED WITH OPTOGENETIC THERAPY
	vary. For example, red-shifted opsins can be used at high light intensities without harm-
	ing the biological tissue, and thus can be used to trigger larger currents in RGCs (Sengupta
	et al., 2016).

  Even for late CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY blind patients who have already experienced visual stimulation, the stimulation received from the reactivated retina will have a novel structure that needs to be learned. For example, former OFF ganglion cells now respond to light onset, and need to be processed like ON ganglion cells. Learning to use this novel retinal code will require a reorganization of visual cortices, where ON and OFF subregions have a distinct topographic arrangement (Kremkow et al., 2016; Lee et al., 2016a).

  IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPYIn the human eye, if we assume that 5 microns is approximately equal to 1 arcmin of visual angle, then discriminating a letter of 250 microns which edges are separated by 50 microns is equivalent to a Snellen acuity of 20/200. This is exactly the limit for legal blindness, and it is above the best acuity ever reached with a retinal prosthesis(20/546,

	Lorach et al. (2013), review), and it is also above the acuity predicted for recent prosthesis Chapter V
	designs (20/250, Lorach et al. (2015)).
	Discussion
	New facts make no sense without
	fitting them together with old
	facts, and then what matters is
	whether they change people's
	ideas.
	Horace Barlow

A popular view of the retina

[START_REF] Barlow | Possible principles underlying the transformations of sensory messages[END_REF]

; Gollisch and Meister, 2010; Lettvin et al., 1968; Münch et al., 2009; Zhang et al., 2012) is that each ganglion cell type extracts one

Acknowledgments

First I would like to thank heartfully Olivier Marre for the huge investment of time and energy he spent in my formation. I learned as much from his deep knowledge in neuroscience than from the exemple of hard work and rigor that he gave me every day.

CELLS OF A SINGLE TYPE

The penalty terms were chosen to minimize overfitting. To fit the linear model (LN), we divided by 10 these two penalty terms as it slightly improved the performance of the model for distant cells. The parameters of the non-linearity were fitted by minimizing the cost function with the active-set method. The following constraints were enforced: α > 0, β > 0, θ has an upper bound. β and θ were redundant with the kernel parameters but adding them accelerated the convergence of the optimization [START_REF] Mcfarland | Inferring Nonlinear Neuronal Computation Based on Physiologically Plausible Inputs[END_REF].

Quantification of the performance of the LN model and of the subunit model

We fitted the model on the unrepeated part of the stimulus and we tested the performance of the model on the repeated part of the stimulus (54 repetitions of a 10 second sequence).

For each cell we then computed the Pearson coefficient r between the real PSTH and the predicted PSTH (time bin: 17 ms). Population averages are indicated in the text as mean ± standard error of the mean. In figure III.2F, we set to zeros all negative Pearson coefficients for readability.

In order to show that the LN model was performing significantly better for central stimulation than for distant stimulation, we selected only the cells that were less than 300 µm away from the bar in one condition and more than 400 µm away from the bar in the other condition. We then performed a paired t-test comparing the performance of the LN model in both conditions for each cell. A possible explanation for why the linear model performed poorly for distant cells could be that distant stimulation evoked less reliable responses. In order to exclude this possibility, we computed the ratio of explainable variability predicted by the model. The We then computed the suppression index, defined as: where PSTH 1 distant bar and PSTH 2 distant bar were computed on two different sets of trials. We performed this quantification on the 25 cells recorded and plotted the mean and SEM of the suppression index for the real data and for the equivalent linear model. A suppression index higher than I linear supp indicates a true suppression that cannot be explained by noise.

Gain control model

We displayed two bars of width 300 µm and separated by 800 µm, oscillating with a sine wave trajectory at slightly different frequencies: the central bar was oscillating at 2 Hz and the distant bar at 1.98 Hz. The central bar was played at 8 different contrasts interleaved

CHAPTER III. MULTIPLEXED COMPUTATIONS IN RETINAL GANGLION CELLS OF A SINGLE TYPE

We fitted the parameters α 0 , α 1 , β 0 and β 1 and H so as to maximise the log-likelihood of the spike train under Poisson assumption (bin size: 17 ms). To adjust the parameters we used the active set method. However, we fixed the parameter τ to 1 second because the periodicity of the stimulus did not allow us to explore thoroughly the time constant of integration of the gain. To test our model, we measured for each cell (n=21) and each contrast the amplitude of the response to the distant bar (defined as max(PSTH)-min(PSTH), bin: 100 ms) and compared it to the amplitude predicted by our model. We then estimated the percentage of variance explained by our model across all cells and conditions using bootstrapping.

Information estimation

The information conveyed by the cell response R about the stimulus X (i.e. mutual information between R and X) is equal to the reduction in entropy of the distribution of X provided by the knowledge of R.

In our case we first defined the stimulus as the position P(t + δt ) of the moving bar for different lags δt relative to the cell response R(t ) (in fig. III.5D, δt is the x-axis of the insets). The lags were introduced to account for the delay in the neural response. We discretized linearly the space of P in 10 bins in order to have a well-sampled distribution with our finite dataset. We discretized the spike train in 10 ms bins and we binarized it by setting to 1 all the bins where there was at least one spike and to 0 the other bins. Changing the discretization steps used to bin P and the spike train did not change qualitatively our results. Then we computed the mutual information between the cell response and the CHAPTER IV. PREDICTING THE VISUAL ACUITY REACHABLE FOR BLIND PATIENTS TREATED WITH OPTOGENETIC THERAPY

Summary

When trying to restore vision in blind patients, a major issue with current approaches based on retinal prosthesis is that the visual acuity of implanted patients remains below the level of legal blindness (cf chapter II). Expressing an optogenetic protein in ganglion cells is a promising alternative, but the acuity that can be expected has not been quantified. This acuity cannot be measured with behavioral experiments due to the lack of blind primate models. Here we combine experiments on mouse and macaque retina, with modeling and ideal observer analysis to estimate the spatial resolution of retinas reactivated with optogenetics, and therefore the theoretical acuity that could be reached by a patient treated with optogenetic therapy. We measured the receptive fields of these lightsensitive ganglion cells from the explanted retinae and showed that a classical Linear-Non-linear-Poisson (LNP) model predicted well the responses of these cells to complex stimuli. We obtained a complete model of how the population of reactivated ganglion cells responds to complex stimuli. We then performed an ideal observer analysis of this model to estimate how well letters in different orientations could be discriminated by the brain, assuming an optimal use of information. The obtained acuity is above the limit of legal blindness, a promising result for future clinical studies.

Introduction

Retinal prostheses can restore a basic vision, but the visual acuity of the implanted patients remains below the level of legal blindness [START_REF] Humayun | Interim results from the international trial of Second Sight's visual prosthesis[END_REF][START_REF] Lorach | Neural stimulation for visual rehabilitation: Advances and challenges[END_REF][START_REF] Stingl | Artificial vision with wirelessly powered subretinal electronic implant alpha-IMS[END_REF]. This is mostly due to the fact that the size of the region activated by the electrical stimulation is too large, due to the electrode size and the current spread in the The stimulus is a flickering black-and-white checkerboard where at every stimulus frame the intensity of each checker is drawn from a binary distribution. Computing the STA consists in selecting and averaging the frames in a 200ms time window preceding each spike, to form a 3 dimensional description of the receptive field (2 dimensions are space, 1 dimension is time). The spatial RF is defined as the temporal slice of the STA that contains the maximal value of the whole STA. The temporal RF is defined as the temporal evolution of the check of the STA with the maximal average value.

The check size of the checkerboard was chosen to be a fraction (ª1/3) of the whole RF size (for macaque : 67 µm, for mouse: 50 µm). The stimulus was displayed using a

Digital Mirror Device and focused on the ganglion cell plane using standard optics (refresh rate for macaque : 30 Hz; mouse : 40 Hz). The light source was an epi-fluoresence lamp with a white spectrum ranging from 380 to 780nm and a total light intensity of 10 16 photons.cm °2.s °1.

Linear Non-linear model

We fitted the responses of the ganglion cells with a Linear-Non-linear model (fig IV .4). In this model the stimulus is first convolved with the receptive field of the cell. Then the result goes through a non-linearity to predict the firing rate over time. The non-linearity relates the amount of light in the receptive field to the firing rate of the cell.

We estimated the shape of the receptive field by fitting a two-dimensional Gaussian to the measured spatial receptive field.

To measure the non-linearity, we first calculate the result of the convolution of the checker-