Leonardo L Gollo
email: leonardo@ifisc.uib.es

Claudio Mirasso
email: claudio@ifisc.uib-csic.es

Alessandro E P Villa
email: avilla@neuroheuristic.org

Dynamic control for synchronization of separated cortical areas through thalamic relay

Keywords: dynamic relaying, thalamocortical circuit, zero-lag synchronization, correlation, firing pattern, thalamus, reticular thalamic nucleus

. Our results show that the model circuit is able to generate fast oscillations in frequency ranges like beta and gamma bands triggered by an external input to the thalamus formed by independent Poisson trains. We propose a control mechanism to turn "On" and "Off" the synchronization between cortical areas as a function of the relative rate of the external input fed into dorsal and ventral thalamic neuronal populations. The current results emphasize the hypothesis that the thalamus could control the dynamics of the thalamocortical functional networks enabling two separated cortical areas to be either synchronized (at zero-lag) or unsynchronized. This control may happen at a fast time scale, in agreement with experimental data, and without any need of plasticity or adaptation mechanisms which typically require longer time scales.

Introduction

In the central nervous system (CNS) it is assumed that the information is mainly represented by the activity of neurons transmitted to other neurons through synaptic links. The extent of the neural network activated by a specific "piece of information" is a never ending matter of investigation but it is accepted that both average levels of discharges, firing rate [START_REF] Gollo | Active dendrites enhance neu-ronal dynamic range[END_REF], and precise spike timing contribute to neural coding. Spatiotemporal firing patterns (Villa et al., 1999b;[START_REF] Hayon | A model for representing the dynamics of a system of synfire chains[END_REF] and coherent oscillatory neural activity [START_REF] Fries | The gamma cycle[END_REF] associated to sensory and behavioral events support the hypothesis that temporal information plays a key role in brain processing. Empirical phenomena and extensive experimental data validated across different species [START_REF] Gray | Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties[END_REF][START_REF] Engel | Synchronization of oscillatory neuronal responses between striate and extrastriate visual cortical areas of the cat[END_REF][START_REF] Castelo-Branco | Neuronal synchrona correlates with surface segregation rules[END_REF][START_REF] Tiesinga | Regulation of spike timing in visual cortical circuits[END_REF] emphasize the importance of emerging cortico-cortical synchrony as a major phenomenon for binding features distributed neural activity [START_REF] Von Der Marlsburg | Self-organization of orientation selective cells in the striate cortex[END_REF][START_REF] Fries | A mechanism for cognitive dynamics: neuronal communication through neuronal coherence[END_REF][START_REF] Desbordes | Timing precision in population coding of natural scenes in the early visual system[END_REF]. Despite the success of physical models to reproduce oscillatory patterns of neural activity it is not clear whether the synchronization is the result of network processing exclusively limited to cortico-cortical interactions or subcortical structures might also intervene [START_REF] Contreras | Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback[END_REF][START_REF] Traub | A mechanism for generation of long-range synchronous fast oscillations in the cortex[END_REF][START_REF] Vicente | Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays[END_REF][START_REF] Chawla | Zero-lag synchronous dynamics in triplets of interconnected cortical areas[END_REF], for a recent review please refer to [START_REF] Uhlhaas | Neural synchrony in cortical networks: history, concept and current status[END_REF].

The thalamus is a structure of CNS that could play an important role to let the emergence or to control cortico-cortical synchronization because the exchange of information between the thalamus and cerebral cortex is a general feature of all ascending sensory pathways but olfaction [START_REF] Jones | The Thalamus[END_REF][START_REF] Sherman | Thalamic relays and cortical functioning[END_REF]. The connectivity pattern between thalamus and cortex is usually viewed as been characterized by thalamocortical integration and corticothalamic feedback [START_REF] Steriade | The functional states of the thalamus and the associated neuronal interplay[END_REF]Villa et al., 1999a;[START_REF] Villa | Cortical modulation of auditory processing in the thalamus[END_REF].

Multiple thalamocortical modules characterized by the same basic connectivity may be assumed to work in parallel and include three main components (see Fig. 1): (i) dorsal thalamic neurons (e.g. from the medial geniculate body for the auditory pathway or from the lateral geniculate body for the visual pathway) recipient of the sensory input from the periphery; (ii) cells of the thalamic reticular nucleus (R), a major component of the ventral thalamus;

(iii) the cortical area receiving the corresponding thalamic input. The thalamic reticular nucleus receives collateral inputs from both thalamocortical and corticothalamic fibres and sends its inhibitory projections to the dorsal thalamus, thus regulating the firing mode of the thalamocortical neurons. The thalamic reticular nucleus receives inputs also from several forebrain and midbrain areas known to exert modulatory functions [START_REF] Mccormick | Sensory gating mechanisms of the thalamus[END_REF], in particular from nerve growth factor responsive basal forebrain cholinergic cells [START_REF] Villa | Nerve growth factor (ngf) modulates information processing in the auditory thalamus[END_REF] that are involved in many cognitive functions and whose dysfunction is associated to Alzheimer's Disease. In the auditory system evidence exist that corticofugal activity regulates the response properties of thalamic cell assemblies by changing their bandwidth responsiveness to pure tones [START_REF] Villa | Corticofugal modulation of information processing in the auditory thalamus of the cat[END_REF] thus allowing to selectively extract information from the incoming sensory signals according to the cortical activity (Villa et al., 1999a). This model suggests that the thalamocortical circuit carries embedded features that enable the build-up of combined supervised and unsupervised information processing akin to produce an adaptive filter [START_REF] Tetko | Efficient partition of learning datasets for neural network training[END_REF] aimed to select behaviorally relevant information processing [START_REF] Von Kriegstein | Task-dependent modulation of medial geniculate body is behaviorally relevant for speech recognition[END_REF].

The current study is not aimed at simulating any detailed thalamocortical circuit, but rather to assess the role of simple variables that could play a major role in controlling the emergence and maintenance of synchronized activity in distributed cortical areas that project to the same thalamic nuclei. Our model predicts that small changes in the cortical neurons firing rate, due to non-correlated background synaptic activity in the thalamic region, is capable of generating single or multi-frequency oscillations along with zero-lag synchronization between distant cortical regions. We quantify this synchronized state by measuring the signal-to-noise ratio which does not monotonically increase with the firing rate. According to our model, thalamic activity plays a key role in controlling the appearance of lag free synchronization between cortical areas. In addition, despite its simplification, the model provides hints about the conditions necessary to achieve that synchronization. We report an efficient control set as the ratio of dorsal over ventral thalamus external input activity to switch on thalamocortical synchronous dynamics. That switch occurs at a fast time scale, without any need of synaptic plasticity which would require longer time scales [START_REF] Fries | A mechanism for cognitive dynamics: neuronal communication through neuronal coherence[END_REF]. The type of control that we suggest is not limited to an "On"-"Off" switch, but it allows to control the appearance of synchronous activity over an extended range of frequencies despite the delays involved in the long-range cortico-cortical interactions [START_REF] Ringo | Time is the essence: A conjecture that hemispheric specialization arises from interhemispheric conduction delay[END_REF][START_REF] Vicente | Coherent Behavior in Neuronal Networks[END_REF] C RR corresponding to the inhibitory connections within the reticular and perigeniculate nucleus of the thalamus (R). Note the excitatory input from the ascending sensory pathway to the dorsal thalamus, the excitatory projection from the thalamus to the cortex with a collateral to R, and the excitatory projection of the cortex to the thalamus with a collateral to R. The only output of R is an inhibitory backprojection to the thalamus. (b) Explicit connections within one thalamocortical module.

Methods

To study the synchronization of cortical activity facilitated by the thalamic relay we conducted extensive numerical simulations of a reduced thalamocortical model of spiking integrate-and-fire neurons subject to background noise and an external driving. The model includes both local synapses and longrange interactions with different delays according to functional connectivity in a four populations motif [START_REF] Milo | Simple building blocks of complex networks[END_REF] (Fig. 2). The simulations were performed using NEST, the neuronal simulation tool [START_REF] Brette | Simulation of networks of spiking neurons: a review of tools and strategies[END_REF] with the PyNEST interface [START_REF] Eppler | Pynest: a convenient interface to the nest simulator[END_REF].

Neuronal model. The integrate-and-fire neuron model [START_REF] Brunel | Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons[END_REF] for each neuron i satisfies the following dynamical equation for the membrane potential V i (t):

τ mem (m) dV i (t) dt = -V i (t) + RI i (t) , (1)
where τ mem (m) is the membrane time constant of neuron i belonging to the population m (as in Fig. 2); I i (t) is the total current arriving to the soma.

The last term in the above equation is given by the sum of all postsynaptic potentials (PSP) of neurons belonging to the network plus the total postsynaptic potentials of all external neurons, the latter being modeled as a Poisson process. Thus,

RI i (t) = τ mem (m) j J(j) k δ(t -t k j -τ (z, m)) + V ext . (2)
The first sum is taken over all presynaptic neurons j, each neuron receives C e (m, z) excitatory synapses and C i (m, z) inhibitory synapses and they depend on the inter-population (long-range) connections z if both neurons belong to different populations or otherwise on the population m to whom they belong. t k j is the time of the k -th spike received by neuron i from its neighbor j. The axonal conduction delay is given by τ (z, m), which corresponds to a spike of a presynaptic neuron j that reaches neuron i. J(j) stands for the PSP and depends on whether its presynaptic neighbor neuron j is excitatory (J(j) = J e ) or inhibitory (J(j) = J i ). V ext is the postsynaptic potential generated by neurons from outside the thalamocortical network. It is given by an independent and homogeneous Poisson process of N ext external neurons, each one firing with a fixed average rate ν(m). The external spike contributes with a change of the membrane potential by J ext whenever it impinges upon neuron i. The dynamics of the neurons can be described as following: the neurons start at a rest potential V r (m) which can be changed by the synaptic current.

If the potential V i (t) of the i-th neuron reaches the threshold θ(m) a spike is generated and its membrane potential is reset to V r (m) after an absolute refractory period (τ rp = 2 ms).

After a brief parameter search and according to the range of values described in the literature we have set characteristic parameters for each population m presented in Table 1. The rational of our choice was to preserve the simplicity of an oversimplified model of the thalamocortical circuit, though retaining the main dynamical features. The values of the threshold, the resting membrane potential, and the membrane time constants were selected such that the neurons in R were the most excitable and those in T were the least excitable because T neurons are meant to receive the external input arising from the ascending sensory pathways. For the sake of simplicity, the refractory period and the excitatory/inhibitory postsynaptic efficacies were chosen to be the same for all neurons. Thalamocortical model. The topology of the model is characterized by two thalamic and two cortical neural populations [START_REF] Shepherd | The Synaptic Organization of the Brain[END_REF][START_REF] Huguenard | Thalamic synchrony and dynamic regulation of global forebrain oscillations[END_REF]. The overall layout of our model is depicted in Fig. 2. Table 2 Parameters for inter-population (long-range) connections z between any two regions.

Each neuron of the target population receives input from a randomly selected neuron belonging to the efferent population. The connectivity parameter values described in Table 2 were set arbitrarily in order to maintain the relative proportion of cell types usually described in the literature [START_REF] Jones | The Thalamus[END_REF][START_REF] Sherman | Thalamic relays and cortical functioning[END_REF]. The number of connections were set to keep 160 afferences to each neuron of C, 75 afferences to each neuron of T and 150 afferences to each neuron of R. This pattern of convergence-divergence is meant to preserve the known anatomical thalamocortical and corticothalamic pattern of connectivity [START_REF] Jones | The Thalamus[END_REF][START_REF] Sherman | Thalamic relays and cortical functioning[END_REF]. The specific proportion of afferences generated by each population is indicated in the boxes at the bottom of Fig. 2. The delays were set to account for typical axonal delays described in the thalamus and cortex of mammals [START_REF] Swadlow | Information flow along neocortical axons[END_REF][START_REF] Knoblauch | Spike-timing-dependent synaptic plasticity can form "zero lag links" for cortical oscillations[END_REF]. Despite the fact that we have not systematically investigated all ranges of axonal delays, we observed that the results are robust against these delays. The most critical parameter is the delay between the thalamus and the cortical areas (τ T C ) which must be kept identical for all ascending projections. If this delay is not the same for all TC connections the maximum number of coincident spikes in the cross-correlograms does not occur at zero-lag but at a lag that depends on the difference between the TC time delays. It is worth mentioning that a constant latency between thalamus and cortex irrespective of the distances has been reported due to regional myelination differences that compensate for the conduction velocities [START_REF] Salami | Change of conduction velocity by regional myelination yields to constant latency irrespective of distance between thalamus to cortex[END_REF]. (+). Notice that the inhibitory projections are represented with a rounded shape tip. The boxes at the bottom of the figure show the pattern of the afferences of a cortical area (C 1 , bottom left), of principal thalamic neurons (T, bottom centre), and of thalamic reticular neurons (R, bottom right). The thalamus is formed by two neuronal populations, the excitatory thalamocortical projecting neurons (T) and the inhibitory reticular and perigeniculate neurons (R) which are reciprocally interconnected (C T R ,C RT ). In addition, there are local excitatory connections (C T T ) between thalamic principal cells and local inhibitory connections (C RR ) between reticular thalamic cells. Two cortical "areas" (C 1 and C 2 ) are connected to the same thalamic region. Each cortical area includes both excitatory (80%) and inhibitory (20%) neurons. The cortical excitatory neurons send feedback projections to the thalamus (C CT ,C CR ), and establish long range corticocortical projections (C CC ) and local connections (Ce). The cortical inhibitory neurons establish only local connections (Ci). The inter-population connectivity is described by the parameters of Table 2. The background activity at rate ν 0 and the external input at rate ν T consist of independent Poisson trains with parameters of Table 3. Neurons in T are the only ones receiving an external input meanwhile all other neurons receive background activity. The external input is uncorrelated and defines the key parameter: ν T ν 0 .

Table 3 Parameters of the background and external afferences.

Results

We have simulated the activity of large populations of interacting neurons with delayed connections. We used a simple integrate and fire (I&F) neuronal model in order to keep the problem easily computationally tractable.

The model retains threshold dynamics and if the membrane potential reaches the threshold a spike is fired. The membrane potential is reset after the firing to its resting potential with an absolute refractory period (2 ms). The spike is transmitted to all target neurons which receive an excitatory or in- Thalamocortical circuit dynamics. In the most symmetrical case, the T region is set in order to receive external driving with the same rate as the other populations (ν T = ν 0 ). The firing rate in R is higher than in the cortex which is also higher than in T. For a typical number of cortico-cortical interaction, say C CC < 40, due to the network connectivity and the difference in the neuronal parameters, there is no correlation among the different areas, and the activity is random and irregular. For ν T > ν 0 other scenario takes place. The raster plots of 150 neurons randomly chosen among all neuronal populations illustrate the network dynamics. Such a typical raster plot is depicted in Fig. 3a. It shows the case in which the cortico-cortical connections are set as C CC = 40 and the thalamus is receiving an external input of mean rate ν T = 7/3ν 0 . The neurons within the populations T and R are synchronized at a high frequency. The two cortical areas exhibit a large number of coincidences at zero-lag, meaning that they are synchronized and in-phase.

The cross-correlograms (see Methods section for details) between the cortical areas and between the thalamus and one cortical area are shown in Fig. 3b,c.

The graphic clearly indicates in-phase correlation among cortical areas while the thalamus and the cortical area are out of phase (with the cortical area delayed by 6 ms).

The synchronization of the cortical regions depends on the external input to T. Fig. 4 shows the raster plot of a single trial characterized at t = 50 ms by a sudden increase of the T activity from the mean rate ν 0 to 7/3ν 0 . The synchrony does not occur in the system for low values of input ν T , for instance ν T = ν 0 , from 0 ms to 50 ms or after the input is switched off, say for time of the cortical oscillation frequency as a function of ν T /ν 0 is shown in Fig. 5b for directly interconnected (C CC = 40) and disconnected (C CC = 0) cortical areas. The frequencies are determined from the power spectrum analysis of the cross correlograms. Only those components whose power is larger than 20% of the maximum power are considered here. In the disconnected case, the cortical areas oscillate at a single frequency close to the thalamic firing rate (see rate in single frequency component dominates the dynamics for ν T = 5/3ν 0 (Fig. 6d).

The observation of the raster plots and of the cross-correlograms illustrates further the dynamics emerging from the interaction between the cortical areas.

In Fig. 7a, b it can be observed that for C CC = 60 and ν T = 5/3ν 0 the slow frequency component related to the cortical firing frequency is predominant.

The peak is not sharp, at ±4 ms from the zero-lag, and a "master-slave" dynamics can be observed in the region of high instantaneous firing rate (say from 50-80 ms after the external input onset). With parameters of C CC = 100 and ν T = 7/3ν 0 multiple frequencies are observed in the raster plot and in the cross-correlogram (Fig. 7c,d). In this case, both the zero-lag cortical synchronization and the leader-ladder dynamics present a strong competition.

At very large values C CC = 110 the cortico-cortical connection dominates and gives rise to an out-of-phase cortical synchronized dynamics between the two areas (Fig. 7e,f) The signature of this dynamics appears both in a double peak at ±6ms (corresponding to the cortico-cortical coupling time in the cross correlation function) and in the raster plot where zero-phase synchronization does not occur between the cortical areas.

Discussion

We have presented the dynamics of a simplified thalamocortical circuit. Our results suggest that the thalamus could be a central subcortical area that is able to trigger the emergence of zero-lag synchrony between distant cortical areas due to a dynamical relaying [START_REF] Fischer | Zero-lag long-range synchronization via dynamical relaying[END_REF][START_REF] Vicente | Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays[END_REF]. According to this phenomenon a central element can enable two populations to synchronize at zero-lag. Other subcortical areas such as the brainstem [START_REF] Scheller | General anesthesia increases temporal precision and decreases power of the brainstem auditory-evoked response-related segments of the electroencephalogram[END_REF] and the hippocampus are likely to play a similar role in dynamical relaying. However, the peculiar recurrent connections of the thalamic reticular nucleus [START_REF] Jones | The Thalamus[END_REF][START_REF] Sherman | Thalamic relays and cortical functioning[END_REF] might provide the thalamocortical circuit with specific features that do not account just for the synchronized pattern, but also for switching "on" or "off" the asynchronous state. Furthermore, considering that large scale integration may occur as a consequence of neuronal coherence, the critical question about how the dynamical selection of integrated areas is achieved remains open [START_REF] Salinas | Correlated neuronal activity and the flow of neuronal information[END_REF][START_REF] Fries | A mechanism for cognitive dynamics: neuronal communication through neuronal coherence[END_REF][START_REF] Vicente | Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays[END_REF][START_REF] Uhlhaas | Neural synchrony in cortical networks: history, concept and current status[END_REF]. We suggest that an increase in the external activity fed into the T population with respect to that of R yields the cortical areas synchronize at zero-phase lag as depicted in Fig 3 . That means the thalamus would be able to control the cortical synchronous state and regulate large scale integration. This control can occur at a fast time scale in agreement with experimental data and without any need of plasticity or adaptation mechanisms which typically require longer time scales. The main input sources to T are the ascending sensory input and the descending cortico-fugal pathway, thus suggesting that both inputs may play an important role in controlling cortical synchrony. This hypothesis for the cortico-petal projections is complementary to the hypothesis of "adaptive filtering" suggested elsewhere for the cortico-fugal projections [START_REF] Villa | Corticofugal modulation of information processing in the auditory thalamus of the cat[END_REF](Villa et al., , 1999a;;[START_REF] Tetko | Efficient partition of learning datasets for neural network training[END_REF].

According to our model, see Figs. 5b, 6b, the thalamocortical circuit is able to generate fast oscillations in frequency ranges like beta and gamma bands triggered by an external input to the thalamus formed by independent Poisson trains. The question of how to generate such fast oscillations has been largely discussed in the literature [START_REF] Traub | A mechanism for generation of long-range synchronous fast oscillations in the cortex[END_REF][START_REF] Doiron | Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli[END_REF][START_REF] Doiron | Oscillatory activity in electrosensory neurons increases with the spatial correlation of the stochastic input stimulus[END_REF][START_REF] Börgers | Background gamma rhythmicity and attention in cortical local circuits: A computational study[END_REF][START_REF] Marinazzo | Input-driven oscillations in networks with excitatory and inhibitory neurons with dynamic synapses[END_REF][START_REF] Börgers | Gamma oscillations mediate stimulus competition and attentional selection in a cortical network model[END_REF] but, as recently pointed out [START_REF] Nikolić | Model this! seven empirical phenomena missing in the models of cortical oscillatory dynamics[END_REF], empirical phenomena like the cycle skipping were not satisfactorily described. The cycle skipping is observed experimentally in the current thalamocortical model when each cortical neuron spikes according to a gamma frequency modulation but with a smaller firing rate. In the raster plots of Fig. 3a it is possible to observe that few neurons spike at a given gamma cycle. Then, the oscillations are in fact shared by a whole population while single neurons skip cycles. As shown in Figs. 5a,b the cortical oscillations, for instance at a signal-to-noise ratio local maximun ν T ≃ 7 3 ν 0 , occur at frequencies near 80 Hz for disconnected areas and in multiple frequencies for C CC = 40, while the average firing rate is approximately 1 4 of it, 20 spikes/s. In general, the firing rate of the cortical populations (see Figs. 5a,6a) were found to be related to the lowest frequency component in case of multiple frequency oscillations. Otherwise the cortical firing rate tends to be much lower than the single frequency of oscillations (e.g., Fig. 5b) or close to it for low external driving (ν T = 5 3 ν 0 ), e.g., Fig. 6b.

The current results emphasize the hypothesis that the thalamus could control the dynamics of the thalamocortical functional networks enabling two separated cortical areas to be either synchronized (at zero-lag) or unsynchronized.

Correlations in the output firing rate of two neurons have been shown to increase with the firing rate (de la [START_REF] De La Rocha | Correlation between neural spike trains increases with firing rate[END_REF]. Indeed we observed that for increasing input rates (ν T ) the firing rate of all populations increase monotonically, accordingly to an expected sigmoidal function (Fig. 5a). König and collaborators [START_REF] König | Relation between oscillatory activity and long-range synchronization in cat visual cortex[END_REF] reported physiological evidence of long-range synchrony with oscillations, whereas short-range synchrony may occur with or without oscillations. Our results, especially for low number of cortico-cortical inter-population synapses (say smaller than the internal connectivity), are in agreement with this finding. However, synchrony without oscillations in local circuit may appear due to extensive sharing of common excitatory inputs which typically generate the zero-lag coincidence observed when neurons are fire at high rates (de la [START_REF] De La Rocha | Correlation between neural spike trains increases with firing rate[END_REF]. Conversely, neurons correlated by long-range connections are likely to share very few synaptic driving, such that synchrony without oscillations should be very rare.

In order to suggest an insight of the model with the anatomical pattern of the circuit one should consider that the thalamocortical and corticothalamic projections are reciprocal to a great extent but corticothalamic projections are characterized by a dual pattern of synapses on the thalamic neurons. Small endings formed the major corticothalamic terminal field, whereas giant terminals were less numerous and formed additional terminal fields together with small terminals. [START_REF] Rouiller | A comparative analysis of the morphology of corticothalamic projections in mammals[END_REF][START_REF] Takayanagi | Microtopography of the dual corticothalamic projections originating from domains along the frequency axis of the cat primary auditory cortex[END_REF].

The modal switch of corticothalamic giant synapses controlled by background activity was recently reported [START_REF] Groh | Driver or coincidence detector: Modal switch of a corticothalamic giant synapse controlled by spontaneous activity and short-term depression[END_REF]. We speculate that this finding and our results may suggest that each pattern of corticothalamic synapse might correspond to a different function. One synaptic type might be involved in assessing the circuitry necessary for the build-up of cortico-cortical synchronization. The other synaptic type would be involved in transmitting stimulusrelated information. Which is which is a question that the current study is unable to answer. We must also consider the fact that our model of individual dynamics of the integrate-and-fire neurons does not produce burst discharges [START_REF] Sherman | Tonic and burst firing: dual modes of thalamocortical relay[END_REF][START_REF] Krahe | Burst firing in sensory systems[END_REF]. This is a clear limitation and the inclusion of a more physiologically realistic model as well as greater neuronal diversity [START_REF] Buia | Role of Interneuron Diversity in the Cortical Microcircuit for Attention[END_REF] are scheduled for our future work. Despite the simplification of our circuitry and the neuronal network modeling in general the robustness of our model is an interesting outcome of this study. The zero-lag synchrony between the cortical areas depends only on the identical axonal delays τ (T C). If these delays are not the same for all TC connections the maximum number of coincident spikes in the cross-correlograms does not occur at zero-lag but at a lag that depends on the difference between the TC time delays. However, it is worth mentioning that regional myelination that compensates for changes in the conduction velocity has been reported as a mechanism that could keep constant latency between thalamus and cortex irrespective of the distances. Moreover, our results are in agreement with the suggestion reported by [START_REF] Chawla | Zero-lag synchronous dynamics in triplets of interconnected cortical areas[END_REF] about the key role of the thalamus favoring the zero-lag synchronization.

We have arbitrarily kept the external input ν 0 over R and the cortex populations fixed but we might have kept fixed T and the cortex populations with a variable external input into R (ν R ). In fact it is the dependency on the variable ν T ν R which represents the control key of the dynamic activity of the system as both rates of external inputs (ν T , ν R ) are varying over time [START_REF] Mcalonan | Guarding the gateway to cortex with attention in visual thalamus[END_REF][START_REF] Yu | Change detection by thalamic reticular neurons[END_REF]. The importance of uncorrelated inputs can be viewed as emphasizing the role of so-called "background activity", which was already reported to play an important role in controlling the thalamocortical circuit dynamic state [START_REF] Wolfart | Synaptic background activity controls spike transfer from thalamus to cortex[END_REF]. We are convinced that further simulations with more accurate details of the neuronal models and with embedded models of the dual cortico-fugal connectivity may provide critical clues for better understanding the mechanisms of the dynamical control subserving the synchronization of cortico-cortical distributed activity.

Fig. 1 .

 1 Fig. 1. (a)A functional scheme of the modular organisation of the typical thalamocortical sensory pathway (somatosensory, visual, sensory). The signs indicate the nature of the connections, (+) excitatory and (-) inhibitory. Notice the big arrows labeled C CC corresponding to long-range excitatory cortico-cortical connections and C RR corresponding to the inhibitory connections within the reticular and perigeniculate nucleus of the thalamus (R). Note the excitatory input from the ascending sensory pathway to the dorsal thalamus, the excitatory projection from the thalamus to the cortex with a collateral to R, and the excitatory projection of the cortex to the thalamus with a collateral to R. The only output of R is an inhibitory backprojection to the thalamus. (b) Explicit connections within one thalamocortical module.

  The thalamus is composed by two separate populations, one of excitatory thalamocortical principal cells (T) and another of inhibitory neurons corresponding to the thalamic reticular and perigeniculate nuclei (R). The two thalamic populations are also characterized by recurrent intrathalamic connections. The cortical populations are formed by an excitatory cell type with local, long range cortical, and feedback corticothalamic projections and by an inhibitory type characterized by only local efferent projections. In addition, the two cortical populations are distributed in two "areas" (C 1 and C 2 ) which may or may not be interconnected (following the value of parameter C CC ).It is a hierarchical network, with both an intra-population random structure and a simple inter-population pattern of connectivity with longer delays. The populations have both internal and external connectivity. Then, the topology satisfies the following constrains: both R (C CR ) and T (C CT ) populations receive cortical feedback, the cortical populations are innervated by T (C T C ) but do not receive inhibitory feedback from R. There are also direct connections from R to T (C RT ) and from T to R (C T R ). Long range cortico-cortical connections are determined by C CC . Assuming that the thalamus is composed by both R and T populations, the thalamocortical model may also be reduced to a three populations network formed by a central thalamic region (T and R) and two balanced cortical areas. Each neuron of a given population receives the same amount of postsynaptic connections. The presynaptic neurons are set randomly, therefore, the postsynaptic distribution is binomial for each type of neuron (excitatory or inhibitory) within a given population.

Fig. 2 .

 2 Fig.2. Circuit layout. The sign at the arrow tip indicate the effect of the connection either excitatory (+) or inhibitory(+). Notice that the inhibitory projections are represented with a rounded shape tip. The boxes at the bottom of the figure show the pattern of the afferences of a cortical area (C 1 , bottom left), of principal thalamic neurons (T, bottom centre), and of thalamic reticular neurons (R, bottom right). The thalamus is formed by two neuronal populations, the excitatory thalamocortical projecting neurons (T) and the inhibitory reticular and perigeniculate neurons (R) which are reciprocally interconnected (C T R ,C RT ). In addition, there are local excitatory connections (C T T ) between thalamic principal cells and local inhibitory connections (C RR ) between reticular thalamic cells. Two cortical "areas" (C 1 and C 2 ) are connected to the same thalamic region. Each cortical area includes both excitatory (80%) and inhibitory (20%) neurons. The cortical excitatory neurons send feedback projections to the thalamus (C CT ,C CR ), and establish long range corticocortical projections (C CC ) and local connections (Ce). The cortical inhibitory neurons establish only local connections (Ci). The inter-population connectivity is described by the parameters of Table2. The background activity at rate ν 0 and the external input at rate ν T consist of independent Poisson trains with parameters of Table3. Neurons in T are the only ones receiving an external input meanwhile all other neurons receive background activity. The external input is uncorrelated and defines the key parameter: ν T ν 0 .

  0.1 mV J ext external synaptic efficacy 10.0 Hz ν 0 external driving Poisson mean rate to C and R 8.0-45.0 Hz ν T overall external driving Poisson process to T 450 N ext number of external afferences Background activity and external input. To model the background activity we assume that each neuron in the network is connected with N ext excitatory external neurons subject to an independent random Poisson processes with average rate ν 0 for neurons of all regions. The thalamic region (T) receives the background activity combined with an external input also modeled by independent Poisson process, such that both the overall external input to T is a process characterized by rate ν T . The parameters used for the Poisson background and the external driving are presented in Table 3. Cross-correlation analysis. We run extensive simulations and analyze the spike trains over several trials. In order to quantify the results from the numerical simulations, we define two values from the cross-correlogram: a) its mean value representing the "noise" level quantifying the expected number of coincidences by chance; b) the peak of the cortico-cortical cross-correlogram (typically at zero-lag) that stands for the "signal". Those quantities are used to compute the signal-to-noise ratio for different values of ν T and different strengths of cortical interconnectivity (C CC ). The results are averaged over 100 trials during 2, 000 ms in a stationary regime after 500 ms of transient dynamics. The averaged result is condensed in a single cross-correlogram, which measures the mean number of coincidences (in a 2 ms bin) of 3, 000 randomly selected neuron pairs belonging to different populations and also averaged over the trials. This procedure allows us to assess the mean behavior of the dynamics and eliminate single trial fluctuations. The "noise" is determined by the mean over the time lag in the averaged crosscorrelogram. It can also be calculated analytically considering the activity of the two populations just as been independent: Let F (p) be the mean firing rate of a population p and b the bin size of the computed cross-correlogram, therefore the mean cross-correlogram (noise) of two arbitrary populations i and j is given by XCOR i-j = F (i)F (j)b . For a typical thalamocortical circuit the two cortical areas have either maximum synchrony at zero-lag or no synchrony (unless C CC is greater than the number of internal excitatory cortical connections C eC ). Thus the "signal" of the cortico-cortical dynamics is defined as the number of coincidences in the cross-correlogram at zero time lag.

  hibitory postsynaptic potential according to the type of synapse. The spike is transmitted with a delay depending on the connection type. Large delays are associated with inter-population connections and short delays with local connections within each population. The results analyze the firing rate, crosscorrelation indicators, oscillation and synchronization information calculated from the spike trains of individual neurons and neuron populations. It is worth mentioning that the neuronal spike times were reliably reproduced despite the simplicity of the I&F model.

Fig. 3 .Fig. 5 .

 35 Fig. 3. Thalamocortical dynamics. (a) Raster plots of 150 neurons randomly chosen (50 from each cortical population and 25 neurons from R and T). The firing times of the local cortical inhibitory neurons are represented by grey dots. R, C 1 , C 2 receive a background Poissonian noise at rate ν 0 Hz. T receives a Poissonian noise at rate ν T = 7 3 ν 0 . (b) Averaged cross-correlogram of 3,000 randomly selected neuronal pairs of different C 1 and C 2 populations averaged over 100 trials. Bin size 2 ms. The horizontal line correspondings to the mean value stands for the noise. The peak at zero-lag stands for the signal. These values are used to compute the signal-to-noise ratio (see text for details). (c) Averaged cross-correlogram of 3,000 randomly selected neuronal pairs of different T and C 1 populations averaged over 100 trials. Same labels as panel (b). Notice that the maximun of C 1 -C 2 crosscorrelation occurs exactly at zero-time lag while the maximum of T-C 1 occurs at a lag of 6 ms.

Fig. 5a )Fig. 6 .

 5a6 Fig.5a). In the interconnected case (C CC = 40) a single frequency dominates the oscillatory dynamics only if ν T < 2ν 0 . Beyond this threshold at least two frequencies of oscillation appear. For ν T = 7 3 ν 0 three different frequencies are observed (as in Fig.3b). The lowest frequency is related to the firing rate of the neurons within the cortical areas. The intermediate frequency is related to the thalamic firing rate like in the disconnected case. An increase of the oscillatory frequency in the cortical areas is due to greater interaction between

Fig. 7 .

 7 Fig. 7. Dynamics of the cortical area as a function of the cortico-cortical interaction strength. (a,b) The uper panel shows the cross-correlogram for ν T = 5/3ν 0 and c cc = 80. The lower panel shows the corresponding raster plot for all cortical neurons. C 1 neurons are indexed from 1 to 1000 and C 2 neurons are indexed from 1001 to 2000. (c,d) Same as previous for ν T = 7/3ν 0 and c cc = 100. (e) Same as previous for ν T = 7/3ν 0 and c cc = 110.

Acknowledgments

We thank Raul Vicente and Javier Iglesias for useful comments and suggestions. The authors acknowledge financial support from the European Commission Project GABA (FP6-NEST Contract 043309), LLG and CM also acknowledge the MEC (Spain) and Feder under project FIS2007-60327 (FISICOS).


Marco Caponigro
email: marco.caponigro@inria.fr

Ugo Boscain
email: ugo.boscain@polytechnique.edu

Thomas Chambrion
email: thomas.chambrion@iecn.u-nancy.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

Mario Sigalotti Control

Control of the bilinear Schrödinger equation for fully coupling potentials

Keywords: Quantum Control, Control of the Schrödinger equation, Bilinear control systems on Lie groups

teaching and research institutions in France or abroad, or from public or private research centers.

= (H 0 + u(t)H 1 )ψ.

(1)

Here ψ belongs to the Hilbert sphere of a complex Hilbert space H and H 0 , H 1 are self-adjoint operators on H. The control u is scalar-valued and represents the action of an external field. The most studied case is the one in which H 0 = -∆ + V (x), H 1 = W (x), where x belongs to a domain D ⊂ R n with suitable boundary conditions and V, W are functions (identified with the corresponding multiplicative operators) characterizing respectively the autonomous dynamics and the coupling of the system with the control u. However, equation (1) can be used to describe more general controlled dynamics. For instance, a quantum particle on a Riemannian manifold subject to an external field (in this case ∆ is the Laplace-Beltrami operator) or a two-level ion trapped in a harmonic potential (the so-called Eberly and Law model [START_REF] Law | Arbitrary control of a quantum electro-magnetic field[END_REF], [START_REF] Bloch | Finite controllability of infinite-dimensional quantum systems[END_REF]). In the last case, as in many others relevant physical situations, the operator H 0 cannot be written as the sum of a Laplacian plus a potential.

The controllability problem consists in establishing whether, for every pair of states ψ 0 and ψ 1 , there exist a control u(•) and a time T such that the solution of (1) with initial condition ψ(0) = ψ 0 satisfies ψ(T ) = ψ 1 . Unfortunately the answer to this problem is negative when H is infinite dimensional. Indeed, Ball, Marsden, and Slemrod proved in Ball et al. [1982] a result which implies (see [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF]) that equation ( 1) is not controllable in (the Hilbert sphere of) H. Moreover, they proved that in the case in which H 0 is the sum of the Laplacian and a potential in a domain D of R n , equation ( 1) is neither controllable in the Hilbert sphere S of L 2 (D, C) nor in the natural functional space where the problem is formulated, namely the intersection of S with the Sobolev spaces H 2 (D, C) and H 1 0 (D, C). Hence one has to look for weaker controllability properties as, for instance, approximate controllability or controllability between the eigenstates of H 0 (which are the most relevant physical states).

However, in certain cases one can describe quite precisely the set of states that can be connected by admissible paths. Indeed in [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF], [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] the authors prove that, in the case in which H 0 is the Laplacian on the interval [-1, 1], with Dirichlet boundary conditions, and H 1 is the operator of multiplication by x, the system is exactly controllable near the eigenstates in H 7 (D, C) ∩ S (with suitable boundary conditions). This result was then refined in [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF], where the authors proved that the exact controllability holds in H 3 (D, C)∩S, for a large class of control potentials (see also [START_REF] Nersesyan | Global exact controllability in infinite time of Schrödinger equation[END_REF]).

In dimension larger than one (for H 0 equal to the sum of the Laplacian and a potential) or for more general situations, the exact description of the reachable set appears to be more difficult and at the moment only approximate controllability results are available.

In [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] an approximate controllability result for (1) was proved via finite dimensional geometric control techniques applied to the Galerkin approximations. The main hypothesis is that the spectrum of H 0 is discrete and without rational resonances, which means that the gaps between the eigenvalues of H 0 are Q-linearly independent. Another crucial hypothesis appearing naturally is that the operator H 1 couples all eigenvectors of H 0 .

The main advantages of that result with respect to those previously known are that: i) it does not need H 0 to be of the form -∆ + V ; ii) it can be applied to the case in which H 1 is an unbounded operator; iii) the control is a bounded function with arbitrarily small bound; iv) it allows to prove controllability for density matrices and it can be generalized to prove approximate controllability results for a system of Schrödinger equations controlled by the same control (see [START_REF] Chambrion | Simultaneous approximate tracking of density matrices for a system of Schroedinger equations[END_REF]).

Results similar to those presented in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] have been obtained, with different techniques, in [START_REF] Vahagn Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF] (see also [START_REF] Ito | Optimal bilinear control of an abstract Schrödinger equation[END_REF], [START_REF] Beauchard | Semi-global weak stabilization of bilinear Schrödinger equations[END_REF], [START_REF] Vahagn Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF], [START_REF] Nersesyan | Global exact controllability in infinite time of Schrödinger equation[END_REF]). They require less restrictive hypotheses on the spectrum of H 0 (which is still assumed to be discrete) but they do not admit H 1 unbounded and do not work for the density matrices. However, it should be noticed that [START_REF] Vahagn Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF] proves approximate controllability with respect to some Sobolev norm H s , while the results given in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] permit to get approximate controllability in the weaker norm L 2 . As it happens for the results in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF], the sufficient conditions for controllability obtained in [START_REF] Vahagn Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF] are generic.

In this paper we prove the approximate controllability of (1) under less restrictive hypotheses on the spectrum of H 0 than those in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF]. More precisely, assume that H 0 has discrete spectrum (λ k ) k∈N , that the operator H 1 couples all eigenvectors of H 0 , and that |λ j -

λ k | = |λ l -λ m | for every (j, k), (l, m) ∈ N 2 \ {(p, p) :
p ∈ N} such that {j, k} = {l, m}, then the system is approximately controllable.

The idea of the proof is the following. We recover approximate controllability for the system defined on an infinite-dimensional Hilbert space through fine controllability properties of the finite-dimensional Galerkin approximations, which allow us to pass to the limit as N → ∞. More precisely, we prove that, for n, N ∈ N with N ≫ n ≫ 1, for given initial and final conditions ψ 0 , ψ 1 in the Hilbert sphere of H which are linear combinations of the first n eigenvectors of H 0 , it is possible to steer ψ 0 to ψ 1 in the Galerkin approximation of order N in such a way that the projection on the last N -n components has arbitrarily small norm along the trajectory. This kind of controllability for the Galerkin approximation of order N is proved in two steps: firstly, thanks to a time-dependent change of variables we transform the system in a driftless one, nonlinear in the control, and we prove the result up to phases. The change of variables was already introduced in Agrachev and [START_REF] Agrachev | An estimation of the controllability time for single-input systems on compact Lie groups[END_REF], [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF]; the technical novelty of this paper is the convexification analysis for the transformed system, which allows to conclude the controllability with less restrictive non-resonance hypotheses. Secondly, the control of phases is obtained via a classical method, using as pivot an eigenstate of H 0 and exploiting the time-reversibility properties of the Schrödinger equation.

FRAMEWORK AND MAIN RESULTS

Settings and notations

As in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF], we use an abstract framework instead of a presentation in terms of partial dif-ferential equations. The advantage is that this presentation is very versatile and applies without modification for Schrödinger equation on a (possibly not bounded) domain of R n or on a manifold as S 1 . Hereafter N denotes the set of strictly positive integers. Definition 1. Let H be an Hilbert space with scalar product •, • , (A, B) be a pair of (possibly unbounded) linear operators of H, with domains D(A) and D(B), and δ > 0. Let us introduce the formal controlled equation dψ dt

(t) = (A + u(t)B)ψ(t), u(t) ∈ [0, δ]. (2)
We say that (A, B, Φ, δ) satisfies (A) if the following assumptions are verified:

(A1) Φ = (φ k ) k∈N is an Hilbert basis of H made of eigen- vectors of A associated with the family of eigenvalues (iλ k ) k∈N ; (A2) φ k ∈ D(B) for every k ∈ N; (A3) A + uB : span{φ k | k ∈ N} → H is essentially skew- adjoint for every u ∈ [0, δ]; (A4) φ j , Bφ k = 0 for every j = k, j, k ∈ N; (A5) |λ j -λ k | = |λ l -λ m | for every (j, k), (l, m) ∈ N 2 \ {(p, p) : p ∈ N} such that {j, k} = {l, m}.
A crucial consequence of assumption (A3) is that, for every constant u in [0, δ], A + uB generates a group of unitary transformations e t(A+uB) : H → H. The unit sphere of H is invariant for all these transformations. Definition 2. Let (A, B, Φ, δ) satisfy (A). For any piecewise constant control function u :

[0, T ] → [0, δ], the solution of (2) with initial condition ψ 0 ∈ H is ψ(t) =e (t-j-1 l=1 t l )(A+uj B) • e tj-1(A+uj-1B) • • • • • • e t1(A+u1B) (ψ 0 ), (3)
where

j-1 l=1 t l ≤ t < j l=1 t l and u(τ ) = u j if j-1 l=1 t l ≤ τ < j l=1 t l .
Notice that a solution ψ(•) of (2) as in (3) satisfies, for every n ∈ N and almost every t ∈ [0, T ] the differential equation

d dt φ n , ψ(t) = -(A + u(t)B)φ n , ψ(t) . (4)
As already recalled, exact controllability is hopeless in general. Definition 3. Let (A, B, Φ, δ) satisfy (A). We say that (2) is approximatly controllable if for every ψ 0 , ψ 1 in the unit sphere of H and every ε > 0 there exist k ∈ N, t 1 , . . . , t k > 0, and

u 1 , . . . , u k ∈ U such that ψ 1 -e t k (A+u k B) • • • • • e t1(A+u1B) (ψ 0 ) < ε.

Main results

Theorem 2.1. Let (A, B, Φ, δ) satisfy (A). Then ( 2) is approximately controllable. Theorem 2.2. Let (A, B, Φ, δ) satisfy (A). For every ε > 0 and for every j, k ∈ N, j = k, there exist a piecewise constant control u : [0, T u ] → [0, δ] and θ ∈ S 1 such that the trajectory ψ(t) of ( 2) with initial condition φ j and corresponding to the control u satisfies ψ(T u )-e iθ φ k < ε and

u L 1 ≤ 5π 4| φ j , Bφ k | .

CONVEXIFICATION PROCEDURE

Time-reparametrization

We denote by P C the set of piecewise constant functions u : [0, ∞) → [0, ∞) such that there exist u 1 , . . . , u p > 0 and 0

= t 1 < • • • < t p+1 = T u for which u : t → p j=1 u j χ [tj ,tj+1) (t).
Let us identify u = p j=1 u j χ [tj ,tj+1) with the finite sequence (u j , τ j ) 1≤j≤p where τ j = t j+1 -t j for every 1 ≤ j ≤ p.

We define the map

P : P C → P C (u j , τ j ) 1≤j≤p → 1 u j , u j τ j ,
which satisfies the following easily verifiable properties. Proposition 4. For every u ∈ P C, P • P(u) = u and P(u) L 1 = p i=1 τ j . Assume that (A, B, Φ) satisfies (A). In analogy with Definition 2, we define, for every u

= p j=1 u j χ [tj ,tj+1) ∈ P C such that u(t) ∈ [0, δ] for every t ≥ 0, the solution of dψ dt (t) = (u(t)A + B)ψ(t), (5)
with initial condition ψ 0 ∈ H as

ψ(t) = e (t-t l )(u l A+B) • • • • • e t1(u1A+B) (ψ 0 ) , where t l ≤ t ≤ t l+1 .
System (5) is the time reparametrization of system (2) induced by the transformation P, as stated in the following proposition. Proposition 5. Let u = (u j , τ j ) 1≤j≤p belong to P C and ψ 0 be a point of H. Let ψ be the solution of (2) with control u and initial condition ψ 0 , and ψ be the solution of (5) with control P(u) and initial condition ψ 0 . Then

ψ (T u ) = ψ ( u L 1 ).
Proof. It is enough to remark that, if u = 0, for every t ∈ [0, ∞), e t(A+uB) = e tu( 1 u A+B) . Remark 6. As a consequence of Proposition 5 it is equivalent to prove controllability for (2) with control u ∈ [0, δ] and to prove controllability for system (5) with control u ∈ [1/δ, ∞).

Convexification

For every positive integer N let the matrices A (N ) = diag(iλ 1 , . . . , iλ N ) and

B (N ) = ( Bφ j , φ k ) N j,k=1 =: (b jk ) N j,
B(N) = B (N ) -d(B). Then y : t → e -v(t)A (N ) -td(B) ψ(t), is a solution of ẏ(t) = e -v(t)A (N ) -td(B) B(N) e v(t)A (N ) +td(B) y(t). (Θ N )
Let us set ϑ N (t, v) = e -vA (N ) -td(B) B(N) e vA (N ) +td(B) . Lemma 7. Let K be a positive integer and γ 1 , . . . , γ K ∈ R \ {0} be such that |γ 1 | = |γ j | for j = 2, . . . , K. Let ϕ(t) = (e itγ1 , . . . , e itγ K ). Then, for every t 0 ∈ R, we have conv ϕ([t 0 , ∞)) ⊇ νS 1 × {(0, . . . , 0)} , where ν = ∞ k=2 cos π 2k > 0. Moreover, for every R > 0 and ξ ∈ S 1 there exists a sequence (t k ) k∈N such that t k+1 -t k > R and

lim h→∞ 1 h h k=1 ϕ(t k ) = (νξ, 0, . . . , 0) .
Proof. Since ϕ(t -t 0 ) = (e -it0γ1 e itγ1 , . . . , e -it0γ K e itγ K ), (6) it is enough to prove the lemma for t 0 = 0. We can suppose that |γ 1 | = 1 and, up to a reordering of the indexes, that there exist n and ñ such that 1

≤ n ≤ ñ ≤ K, |γ i | = |γ j | for every i, j ∈ {1, . . . , n}, γ 2 , . . . , γ ñ ∈ Z, γ ñ+1 , . . . , γ K ∈ R \ Z, and {|γ n+1 |, . . . , |γ ñ|} ⊂ {|γ 2 |, . . . , |γ n |}.
Consider the 2 n-1 real numbers defined as follows: let t1 = 0, and for k ∈ {1, . . . , n -1} and j ∈ {1, . . . ,

2 k-1 }, t2 k-1 +j = tj + π |γ k+1 | .
Up to a reordering of the tj , we can suppose that 0 = t1 < t2 < • • • < t2 n-1 . Take an integer r larger than R/2π, then set t j = tj + 2πr(j -1), in such a way that t k -t k-1 > R for every k = 2, . . . , 2 n-1 . Now consider the arithmetic mean of the l-th (complex) coordinates of ϕ(t 1 ), . . . , ϕ(t 2 n-1 ). We show that this quantity is zero for l = 2, . . . , ñ. Indeed, from the definition of t j , we have

2 n-1 j=1 e itj γ l = n-1 k=1 1 + e iπγ l /|γ k+1 | ,
which is zero since so is the k-th factor when |γ l | = |γ k+1 |.

On the other hand, the arithmetic mean of the first coordinate is uniformly bounded away from zero. Indeed

1 2 n-1 2 n-1 j=1 e iγ1tj = n-1 k=1 1 + e iπγ1/|γ k+1 | 2 = n k=2 cos π 2|γ k | = exp n k=2 log cos π 2|γ k | ≥ exp ∞ k=2 log cos π 2k = ν . (7)
Since log cos π 2k ∼ -π 2 8k 2 as k tends to infinity, then the sum k≥2 log cos π 2k converges to a (negative) finite value l.

Therefore we have found a sequence of numbers t j such that the arithmetic mean of the first coordinate of ϕ(t 1 ), . . . , ϕ(t 2 n-1 ) is uniformly bounded away from zero and the arithmetic means of following ñ -1 coordinates are zero. According to (6), the role of t 1 , . . . , t 2 n-1 can equivalently be played, for every k ∈ N, by the 2 n-1 -uple t k j = t j + 2πmk , j = 1, . . . , 2 n-1 , where the integer m is larger than r + t 2 n-1 /2π. Now, let l ∈ {ñ + 1, . . . , K}, so that γ l / ∈ Z. For every h ∈ N, the arithmetic mean of the l-th coordinate of the points ϕ(t k j ) (k = 0, . . . , h, j = 1, . . . , 2 n-1 ) is 1 2 n-1 (h + 1)

2 n-1 j=1 h k=0 e it k j γ l = = 1 2 n-1 (h + 1) 2 n-1 j=1 e itj γ l h k=0 e i2πmkγ l =   1 2 n-1 2 n-1 j=1 e itj γ l   1 (h + 1) 1 -e i2πm(h+1)γ l 1 -e i2πmγ l h→∞ -→ 0 .
Therefore, we found a sequence of points in the convex hull of ϕ([0, ∞)) converging to (2 1-n 2 n-1 j=1 e iγ1tj , 0, . . . , 0). The lemma follows from (7) and by rotation invariance (see ( 6)). Remark 8. The constant ν can be easily computed numerically. One finds ν ≈ 0.430 > 2 5 . as the n × n matrix whose entries are all zero but the one of index (j, k) which is equal to 1.

An auxiliary system

The control system (Σ n ) is linear in x. For every θ in S 1 and every 1 ≤ j, k ≤ n, j = k, the matrix e iθ e (n)

jk -e -iθ e (n) kj is skew-adjoint with zero trace. Hence the control system (Σ n ) leaves the unit sphere S n of C n invariant. In order to take advantage of the rich Lie group structure of group of matrices, it is also possible to lift this system in the group SU (n), considering x as a matrix.

PROOF OF THEOREM 2.1

4.1 Controllability of (Σ n ) Proposition 9. Let (A, B, Φ, δ) satisfy (A). Then, for every n in N, the finite dimensional control system (Σ n ) is completely controllable both in S n ⊂ C n and in SU (n).

Proof. Note that e

(n)

jk -e

(n) kj , i(e (n) jk + e (n) kj ) = 2i(e (n) jj -e (n)
kk ) . Hence the Lie algebra generated by the linear vector fields x → ν|b jk |(e iθ e jk -e -iθ e kj )x contains the whole tangent space su(n)x of SU (n) at x. The controllability in S n follows by projection.

Tracking

(Σ n ) via (Θ N )
Next proposition states that, for every N ≥ n system (Θ N ) can track, in projection on the first n components, without time reparametrization, every trajectory of system (Σ n ). Call Π (N ) n the projection mapping a N ×N complex matrix to the n × n matrix obtained by dropping the last N -n columns and the last N -n rows. Proposition 10. For every n, N ∈ N, N ≥ n, ε, δ > 0, and for every solution X : [0, T ] → SU (n) of system (Σ n ) with initial condition X(0) = I n there exist a piecewise constant control u :

[0, T ] → [1/δ, +∞) such that the solution Y : [0, T ] → SU (N ) of system (Θ N ) with initial condition Y (0) = I N corresponding to v(t) = t 0 u(s)ds satisfies Π (N ) n Y (t) -X(t) < ε, for every t ∈ [0, T ].
Proof. Given a trajectory X(t) of system (Σ n ) with initial condition X(0) = I n , denote by (j, k) = (j(t), k(t)) ∈ N 2 , j = k and θ = θ(t) ∈ S 1 its corresponding control functions. Being these function piecewise constant, it is possible to write [0, T ] as union of q intervals [0, T ] = q p=0 [t p , t p+1 ] in such a way that j, k, and θ are constant on [t p , t p+1 ) for every p = 0, . . . , q.

We are going to construct the control u by applying recursively Lemma 7. Let δ > 1/δ. Fix p ∈ {0, . . . , q} and j, k, θ such that (j(t), k(t)) = (j, k) and θ(t) = θ on [t p , t p+1 ). Apply Lemma 7 with

γ 1 = λ j -λ k , {γ 2 , . . . , γ K } = {λ l -λ m | l, m ∈ {1, . . . , N }, {l, m} = {j, k}, and l = m}, R = max 1≤l<m≤n |b ll -b mm | |λ l -λ m | T + δT,
and t 0 = t 0 (p) to be fixed later depending on p. Then, for every η > 0, there exist h = h(p) > 1/η and a sequence (w p α ) h α=1 such that w p 1 ≥ t 0 , w p α -w p α-1 > R, and such that 1

h h α=1 e i(λ k -λj )w p α -ν bjk |b jk | e iθ < η, and 1 h h α=1 e i(λ l -λm)w p α < η,
for every l, m ∈ {1, . . . , N } such that {l, m} = {j, k}, and l = m.

Set τ p α = t p + (t p+1 -t p )α/h, α = 0, . . . , h, and define the piecewise constant function

v η (t) = q p=0 h α=1 w p α + iτ p α b jj -b kk λ j -λ k χ [τ p α-1 ,τ p α ) (t) . (8)
Note that by choosing t 0 (p) = w p-1 h(p-1) + R for p = 1, . . . , q and t 0 (0

) = R we have that v η (t) is non-decreasing. Set M (t) = ν|b j(t)k(t) | e iθ(t) e (N ) j(t)k(t) -e -iθ(t) e (N ) k(t)j(t) . The definition of v η is such that t 0 ϑ N (s, v η (s))ds η→0 -→ t 0 M (s)ds, (9)
uniformly with respect to t ∈ [0, T ]. This convergence guarantees (see for example [Agrachev and Sachkov, 2004, Lemma 8.2]) that the projection Π (N ) n of the solution Y η (t) of system (Θ N ) with control v η and initial condition I N converges uniformly in [0, T ] to X(t) as η tends to 0. In particular it is possible to choose η sufficiently small in such a way that Π

(N ) n Y η (t) -X(t) < ε/2 for every t ∈ [0, T ]. If v η (t) were of the type t 0 u(s)ds for u : [0, T ] → [1/δ, ∞)
piecewise constant then we would be done. Although v η is piecewise constant, it is possible to construct a continuous piecewise linear approximation v ′ η of v η with slope not smaller than δ and such that

t 0 ϑ N (τ, v η (τ )) -ϑ N (τ, v ′ η (τ )) dτ η→0 -→ 0,
uniformly with respect to t ∈ [0, T ]. This is possible thanks to the choice of R and since the size of every interval [τ p k , τ p k+1 ] is smaller that 1/h < η. In particular there exists η sufficiently small in such a way that, if Y ′ η is the solution of system (Θ N ) with control v ′ η and initial condition I N , then Y η (t) -Y ′ η (t) < ε/2 for every t ∈ [0, T ]. Finally, construct the control u as as the derivative of v ′ η .

Tracking (Σ n ) in modulus via the original system

Next proposition extend the result of tracking for the system (Θ N ) to the infinite dimensional system (5). Proposition 11. For every ε > 0, for every integer n, and for every trajectory x : [0, T ] → S n of system (Σ n ) there exists a piecewise constant control function u : [0, T ] → [1/δ, +∞) and a corresponding trajectory ψ(t) of ( 5) satisfying | φ j , ψ(t) | -|x j (t)| < ε for every t in [0, T ], j = 1, . . . , n and φ j , ψ(0) -x j (0) < ε for every j = 1, . . . , n.

Proof. Consider µ > 0. For every j ∈ N the hypothesis that φ j belongs to D(B) implies that the sequence (b jk ) k∈N is in ℓ 2 . It is therefore possible to choose N ≥ n such that k>N |b jk | 2 < µ for every j = 1, . . . , n.

Let X : [0, T ] → SU (n) be the solution of system (Σ n ) corresponding to the control associated with the trajectory x and with initial condition X(0) = I n . By Proposition 10, for every η > 0 there exist a piecewise constant control

u η : [0, T ] → [1/δ, +∞) such that the solution Y : [0, T ] → SU (N ) of system (Θ N ) with initial condition I N corresponding to v η (t) = t 0 u η (τ )dτ satisfies Π (N ) n Y (t) -X(t) < η , for every t ∈ [0, T ]. Notice that Y (t) = R η (t, 0) is the resolvent of (Θ N ) associated with the control v η .
Let ψ(t) be the solution of (5) corresponding to u η with initial condition ψ(0) = x 1 (0)φ 1 + • • • + x n (0)φ n . Call q η j (t) = e -iλj v η (t)-tbjj φ j , ψ(t) , j ∈ N . For almost every t ∈ [0, T ] and for every j ∈ N,

qη j (t) = ∞ k=1 b jk e i(λ k -λj )v η (t)+t(b kk -bjj ) q η k (t). ( 10
)
Therefore Q η N (t) = (q η 1 (t), . . . , q η N (t)) T satisfies the timedependent linear equation

Qη N (t) = ϑ N (t, v η (t))Q η N (t) + P η N (t), with Π (N ) n P η N (t) ≤ √ µn. Hence Q η N (t) = R η (t, 0) x(0) 0 + t 0 R η (s, t)P η N (s)ds.
Consider the projection on the first n coordinates. By (9), R η (s, t) converges uniformly, as η tends to 0, to a time dependent operator from C N into itself which preserves the norm of the first n components, then there exists η sufficiently small such that

Π (N ) n t 0 R η (t, s)P η N (s)ds < 2T √ µn.
Hence Q η n (t) -x(t) < ε, for η and µ small enough. In particular, for j = 1, . . . , n

| φ j , ψ(t) | -|x j (t)| = |q η j (t)| -|x j (t)| < ε.

Approximate controllability

For n ∈ N, let π n : H → C n be the map associating with ψ ∈ H the n-uple ( φ 1 , ψ , . . . , φ n , ψ ).

Approximate controllability follows from Proposition 11 and Remark 6 applied both to (2) and to its time-reversed version.

Proof of Theorem 2.1. Fix ψ 0 , ψ 1 ∈ S, ε > 0, and n sufficiently large such that ψ 0 is ε-close to n j=1 φ j , ψ 0 φ j . Without loss of generality we can assume that λ 1 = 0. Thanks to Proposition 9, there exists an admissible trajectory x of (Σ n ) steering π n (ψ 0 ) ε-close to π n (φ 1 ) = (1, 0, . . . , 0). Applying Proposition 11 and Remark 6, there exists a piecewise constant control function u : [0, T ] → [0, δ] and a corresponding trajectory ψ(t) of (2) satisfying | φ j , ψ(t) | -|x j (t)| < ε for every t in [0, T ], j = 1, . . . , n and φ j , ψ(0) -x j (0) < ε for every j = 1, . . . , n.

In particular, taking t = 0, T , we have that both ψ(0)ψ 0 < 2ε and ψ(T ) -e iθ φ 1 < Cε, for some θ ∈ S 1 , with C independent of n, ε and ψ 0 .

Hence, thanks to the unitarity of the evolution of (2), u steers ψ 0 to a (C + 2)ε-neighborhood of e iθ φ 1 .

Note that if (A, B, Φ, δ) satisfies (A), then the same is true for (-A, -B, Φ, δ). Notice, moreover, that, by unitarity of the evolution of (2), if u :

[0, T ] → [0, δ] steers ψ 0 to a ε-neighborhood of ψ 1 for the time-reversed control system ψ(t) = -(A + u(t)B)ψ(t), then u(T -•) : [0, T ] → [0, δ]
steers ψ 1 ε-close to ψ 0 for the original system (2).

Hence there exists a control function ũ : [0, T ] → [0, δ] steering (2) from e i θ φ 1 to a (C + 2)ε-neighborhood of ψ 1 , for some θ ∈ S 1 .

Let τ > 0 be such that e τ A (e iθ φ 1 ) = e i θ φ 1 .

Finally, the concatenation of u, of the control constantly equal to zero for a time τ , and of ũ( T -•) steers (2) from ψ 0 to a 2(C + 2)ε-neighborhood of ψ 1 .

ESTIMATES OF THE L 1 NORM

We derive now estimates of the minimal L 1 norm of the control u steering φ j to a ε-neighborhood of φ k .

The strategy to get L 1 estimates is the following. Recall that, instead of considering the control system ẋ = (A + uB)x driven by a piecewise constant function u : [0, T u ] → [0, δ], we have defined the function P(u) : [0, u L 1 ] → [1/δ, ∞) and considered the control system ẋ = (P(u)A + B)x. To estimate the L 1 norm of u, it is enough to estimates the time needed to transfer the system ẋ = (uA + B)x from a given source to a given target. We observe that, in view of Proposition 11, the time needed for the transfer of ẋ = (uA + B)x from φ j to an εneighborhood of φ k is smaller than or equal to the time needed to transfer system (Σ n ) from π n (φ j ) to an εneighborhood of π n (φ k ) (n ≥ j, k).

Such time estimates have been given for general trajectories (i.e., trajectories that do not necessarily steer an eigenstate of A to another) in [START_REF] Agrachev | An estimation of the controllability time for single-input systems on compact Lie groups[END_REF].

We proceed now to the proof of the Theorem 2.2.

Proof of Theorem 2.2. The solution x : R → S n of the differential equation

ẋ = ν|b jk | e (n)
jk -e

(n) kj

x with initial condition x(0) = π n (φ j ) is a trajectory of (Σ n ).

For every t in R, x(t) = exp tν|b jk | e

(n)

jk -e

(n) kj π n (φ j ).

In particular, one checks that x π 2ν|b jk | = -π n (φ k ).

In other words, the control system (Σ n ) can exchange (up to a phase factor) the eigenstates j and k of A, in time less than π 2ν|b jk | < 5π 4|b jk | . Remark 12. Notice that the bound given in Theorem 2.2 does not depend on ε. However, it is possible that the time T u needed to achieve the transfer of system (2) grows to infinity as ε tends to zero. When only some |b jk | are small, the bound given by Theorem 2.2 may be too conservative. It is possible to find other estimates by comparing the time needed to steer an eigenstate to another with ad-hoc metrics on SU (n). The resulting expressions are somewhat intricate and can be found in [Agrachev and Chambrion, 2006, Section 5].

  Control of the bilinear Schrödinger equation for fully coupling potentials M. Caponigro * U. Boscain * * T. Chambrion * M. Sigalotti * 1. INTRODUCTION In this paper we are concerned with the controllability problem for the Schrödinger equation i dψ dt

  k=1 , be the Galerkyn approximations at order N of A and B, respectively. Let t → ψ(t) be a solution of ψ = (uA (N ) + B (N ) )ψ , corresponding to a control function u and consider v(t) = t 0 u(τ )dτ . Denote by d(B) the diagonal of B (N ) and let

  Let (A, B, Φ) satisfy (A). With every n ∈ N we associate the control system on C n ẋ = ν|b jk | e iθ e θ(t) ∈ S 1 and (j, k) = (j(t), k(t)) ∈ N 2 \ {(p, p) : p ∈ N} are piecewise constant controls. Set e (n) jk

ACKNOWLEDGEMENTS

This work was supported by the ANR project GCM, program "Blanche", project number NT09 504490 and by the European Research Council, ERC StG 2009 "GeCoMethods", contract number 239748. The article has been partially written while attending the trimester "Control of Partial Differential Equations and Applications" held at IHP in Paris in 2010.