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SUMMARYNeuronal cultures are a prominent experimental tool to understand complex functional organization in neuronal assemblies. However, neurons grown on flat surfaces exhibit a strongly coherent bursting behavior with limited functionality. To approach the functional richness of naturally formed neuronal circuits, here we studied neuronal networks grown on polydimethylsiloxane (PDMS) topographical patterns shaped as either parallel tracks or square valleys. We followed the evolution of spontaneous activity in these cultures along 20 daysin vitrousing fluorescence calcium imaging. The networks were characterized by rich spatiotemporal activity patterns that comprised from small regions of the culture to its whole extent. Effective connectivity analysis revealed the emergence of spatially compact functional modules that were associated to both the underpinned topographical features and predominant spatiotemporal activity fronts. Our results show the capacity of spatial constraints to mold activity and functional organization, bringing new opportunities to comprehend the structure-function relationship in living neuronal circuits.

Cultures of dissociated neurons are an invaluable experimental tool in studying neuronal networks at an intermediate scale in an in vitro controlled physico-chemical environment. Moreover, current micro-fabrication techniques allow the design of a custom connectivity between subpopulations, which could make it possible to carry out computations with devices involving living cells. The quorum percolation (QP) model has been designed in the context of neurobiology to describe bursts of activity occurring in neuronal cultures from the point of view of collective phenomena rather than from a dynamical synchronization approach. Such a model is well suited to describe triggered activity in neuronal devices, and its generic character points at the necessity of heavily structured devices to go beyond collective bursting. We derive a continuous extension of the QP model, seen as information propagation on a non-metric directed graph, and discuss how its critical behavior might give insight on the connectivity of neuronal networks. The link with metric graphs, embedded in a two-dimensional space, is tackled by the introduction of a geometrical model based upon a random walk, where axon growth is constrained by obstacles such as walls and channels. This provides a starting point for the construction of neuronal devices in vitro capable of more complex behaviors. Lastly, we show how simulations of bursts with a dynamical adaptive integrate-and-fire model can be interpreted in terms of QP, confirming the robustness of this synchronized behavior.

Experimental and numerical studies have revealed that isolated populations of oscillatory neurons can spontaneously synchronize and generate periodic bursts involving the whole network. Such a behavior has notably been observed for cultured neurons in rodent's cortex or hippocampus. We show here that a sufficient condition for this network bursting is the presence of an excitatory population of oscillatory neurons which displays spike-driven adaptation. We provide an analytic model to analyze network bursts generated by coupled adaptive exponential integrate-and-fire neurons. We show that, for strong synaptic coupling, intrinsically tonic spiking neurons evolve to reach a synchronized intermittent bursting state. The presence of inhibitory neurons or plastic synapses can then modulate this dynamics in many ways but is not necessary for its appearance. Thanks to a simple self-consistent equation, our model gives an intuitive and semi-quantitative tool to understand the bursting behavior. Furthermore, it suggests that after-hyperpolarization currents are sufficient to explain bursting termination. Through a thorough mapping between the theoretical parameters and ion-channel properties, we discuss the biological mechanisms that could be involved and the relevance of the explored parameter-space. Such an insight enables us to propose experimentally-testable predictions regarding how blocking fast, medium or slow after-hyperpolarization channels would affect the firing rate and burst duration, as well as the interburst interval.

The Quorum Percolation model has been designed in the context of neurobiology to describe bursts of activity occurring in neuronal cultures from the point of view of statistical physics rather than from a dynamical synchronization approach. It is based upon information propagation on a directed graph with a threshold activation rule; this leads to a phase diagram which exhibits a giant percolation cluster below some critical value mC of the excitability. We describe the main characteristics of the original model and derive extensions according to additional relevant biological features. Firstly, we investigate the effects of an excitability variability on the phase diagram and show that the percolation transition can be destroyed by a sufficient amount of such a disorder; we stress the weakly averaging character of the order parameter and show that connectivity and excitability can be seen as two overlapping aspects of the same reality. Secondly, we elaborate a discrete time stochastic model taking into account the decay originating from ionic leakage through the membrane of neurons and synaptic depression; we give evidence that the decay softens and shifts the transition, and conjecture than decay destroys the transition in the thermodynamical limit. We were able to develop mean-field theories associated with each of the two effects; we discuss the framework of their agreement with Monte Carlo simulations. It turns out that the the critical point mC from which information on the connectivity of the network can be inferred is affected by each of these additional effects. Lastly, we show how dynamical simulations of bursts with an adaptive exponential integrateand- fire model can be interpreted in terms of Quorum Percolation. Moreover, the usefulness of the percolation model including the set of sophistication we investigated can be extended to many scientific fields involving information propagation, such as the spread of rumors in sociology, ethology, ecology.

In this thesis, I provide models and numerical tools to better understand and predict the behavior and development of neuronal cultures and devices.Neuronal cultures have proven invaluable in improving our understanding of how the brain processes information, by enabling researchers to investigate neuronal and network response functions to various perturbations and stimuli.Furthermore, recent progress in microfluidics have opened the gate towards more elaborated neuronal devices, bringing us one step closer to complex signal processing with living in vitro neurons.In a first part, I propose a mechanism to explain the epileptiform bursts of activity present in cultures, mechanism which I formulate as a concise theoretical model. I subsequently test the predictions of this model on cultures and show that they are indeed compatible with the behavior observed in vitro.I further develop this description in the second part of the thesis, where I analyze its spatiotemporal dynamics and the fact that burst nucleate in specific areas in the network.Since predictions and analysis of these nucleation centers strongly depends on the network structure, I develop a simulation platform to enable efficient modeling of the network development. This software takes into account the interactions between the neurons and their environment and is the first platform to provide versatile and complete models to simulate the entire growth process of neurons. I demonstrate that this simulator is able to generate valid neuronal morphologies, then use it to propose new network topologies to describe neuronal cultures, as well as to reproduce existing neuronal devices. I then show that the activities sustained by these structures are compatible with the experimental recordings.Eventually, I discuss several future directions for which the use of neuronal devices would enable to circumvent current limitations of neuronal cultures, thus providing new information on the processes which underlie brain development and plasticity.; Dans cette thèse, je propose plusieurs modèles et outils numériques afin de mieux comprendre et prédire le comportement et le développement de cultures et dispositifs neuronaux.Les cultures de neurones ont en effet été un outil précieux durant les 20 dernières années : elles ont permis de mieux comprendre la manière dont le cerveau traite les différentes informations qui lui parviennent en donnant aux scientifiques la possibilité de tester les effets de médicaments sur les neurones, ainsi que d'obtenir leurs réponses détaillées à diverses perturbations et stimuli.De plus, de récentes avancées en microfluidiques ont ouvert la voie à la conception de dispositifs neuronaux plus élaborés, rapprochant encore un peu plus la perspective du traitement de signaux complexes via des neurones in vitro.Dans une première partie, je propose un mécanisme pour expliquer les bouffées d'activité épileptiformes présentes dans les cultures, mécanisme que je formule via un modèle théorique concis. J'effectue ensuite une vérification expérimentale des prédictions du modèle sur des cultures et montre que celles-ci sont effectivement compatibles avec le comportement observé in vitro.Dans une seconde partie, je décris plus en détail la description de la dynamique spatio-temporelle du phénomène, notamment le fait que les bursts nucléent en des zones bien précises du réseau neuronal.Comme les prédictions et analyses effectuées dépendent fortement de la structure de ce réseau, je présente ensuite la réalisation d'une plateforme de simulation afin de permettre de modéliser efficacement le développement des réseaux neuronaux. Ce logiciel prend en compte les interactions entre les neurones et leur environnement et constitue la première plateforme à fournir des modèles polyvalents et complets pour décrire l'intégralité du processus de croissance neuronal. Je montre ensuite que ce simulateur est capable de générer des morphologies valides et l'utilise pour proposer des nouvelles topologies de réseaux afin de décrire les cultures de neurones. Je reproduis également des dispositifs neuronaux existants et montre que les activités entretenues par ces structures sont compatibles avec les observations expérimentales. Enfin, je discute plusieurs directions de recherche possibles, pour lesquelles l'utilisation de dispositifs neuronaux spécifiques permettrait de contourner les limitations des cultures neuronales et fournirait ainsi de nouvelles informations sur les processus sous-tendant le développement et la plasticité cérébrale