Cette thèse traite de modèles mésoscopiques de cortex appelés champs neuronaux. Les équations des champs neuronaux décrivent l'activité corticale de populations de neurones, ayant des propriétés anatomiques/fonctionnelles communes. Elles ont été introduites dans les années 1950 et portent le nom d'équations de Wilson et Cowan. Mathématiquement, elles consistent en des équations intégro-différentielles avec retards, les retards modélisant les délais de propagation des signaux ainsi que le passage des signaux à travers les synapses et l'arbre dendritique. Dans la première partie, nous rappelons la biologie nécessaire à la compréhension de cette thèse et dérivons les équations principales. Puis, nous étudions ces équations du point de vue des systèmes dynamiques en caractérisant leurs points d'équilibres et la dynamique dans la seconde partie. Dans la troisième partie, nous étudions de façon générale ces équations à retards en donnant des formules pour les diagrammes de bifurcation, en prouvant un théorème de la variété centrale et en calculant les principales formes normales. Nous appliquons tout d'abord ces résultats à des champs neuronaux simples mono-dimensionnels qui permettent une étude détaillée de la dynamique. Enfin, dans la dernière partie, nous appliquons ces différents résultats à trois modèles de cortex visuel. Les deux premiers modèles sont issus de la littérature et décrivent respectivement une hypercolonne, /i.e./ l'élément de base de la première aire visuelle (V1) et un réseau de telles hypercolonnes. Le dernier modèle est un nouveau modèle de V1 qui généralise les deux modèles précédents tout en permettant une étude poussée des effets spécifiques des retards / This thesis deals with mesoscopic models of cortex called neural fields. The neural field equations describe the activity of neuronal populations, with common anatomical / functional properties. They were introduced in the 1950s and are called the equations of Wilson and Cowan. Mathematically, they consist of integro-differential equations with delays, the delays modeling the signal propagation and the passage of signals across synapses and the dendritic tree. In the first part, we recall the biology necessary to understand this thesis and derive the main equations. Then, we study these equations with the theory of dynamical systems by characterizing their equilibrium points and dynamics in the second part. In the third part, we study these delayed equations in general by giving formulas for the bifurcation diagrams, by proving a center manifold theorem, and by calculating the principal normal forms. We apply these results to one-dimensional neural fields which allows a detailed study of the dynamics. Finally, in the last part, we study three models of visual cortex. The first two models are from the literature and describe respectively a hypercolumn, i.e. the basic element of the first visual area (V1) and a network of such hypercolumns. The latest model is a new model of V1 which generalizes the two previous models while allowing a detailed study of specific effects of delays

This thesis deals with mesoscopic models of cortex called neural fields. The neural field equations describe the activity of neuronal populations, with common anatomical / functional properties. They were introduced in the 1950s and are called the equations of Wilson and Cowan. Mathematically, they consist of integro-differential equations with delays, the delays modeling the signal propagation and the passage of signals across synapses and the dendritic tree. In the first part, we recall the biology necessary to understand this thesis and derive the main equations. Then, we study these equations with the theory of dynamical systems by characterizing their equilibrium points and dynamics in the second part. In the third part, we study these delayed equations in general by giving formulas for the bifurcation diagrams, by proving a center manifold theorem, and by calculating the principal normal forms. We apply these results to one-dimensional neural fields which allows a detailed study of the dynamics. Finally, in the last part, we study three models of visual cortex. The first two models are from the literature and describe respectively a hypercolumn, i.e. the basic element of the first visual area (V1) and a network of such hypercolumns. The latest model is a new model of V1 which generalizes the two previous models while allowing a detailed study of specific effects of delays

Normalization is important for a large range of phenomena in biological neural systems such as light adaptation in the retina, context dependent decision making and probabilistic inference. In a normalizing circuit the activity of one neuron/-group of neurons is divisively rescaled in relation to the activity of other neurons/­­groups. This creates neural responses invariant to certain stimulus dimensions and dynamically adapts the range over which a neural system can respond discriminatively on stimuli. This thesis examines whether a biologically realistic normalizing circuit can be implemented by a spiking neural network model based on the columnar structure found in cortex. This was done by constructing and evaluating a highly structured spiking neural network model, modelling layer 2/3 of a cortical hypercolumn using a group of neurons as the basic computational unit. The results show that the structure of this hypercolumn module does not per se create a normalizing network. For most model versions the modulatory effect is better described as subtractive inhibition. However three mechanisms that shift the modulatory effect towards normalization were found: An increase in membrane variance for increased modulatory inputs; variability in neuron excitability and connections; and short-term depression on the driving synapses. Moreover it is shown that by combining those mechanisms it is possible to create a spiking neural network that implements approximate normalization over at least ten times increase in input magnitude. These results point towards possible normalizing mechanisms in a cortical hypercolumn; however more studies are needed to assess whether any of those could in fact be a viable explanation for normalization in the biological nervous system. / Normalisering är viktigt för en lång rad fenomen i biologiska nervsystem såsom näthinnans ljusanpassning, kontextberoende beslutsfattande och probabilistisk inferens. I en normaliserande krets skalas aktiviteten hos en nervcell/grupp av nervceller om i relation till aktiviteten hos andra nervceller/grupper. Detta ger neurala svar som är invarianta i förhållande till vissa dimensioner hos stimuli, och anpassar dynamiskt för vilka inputmagnituder ett system kan särskilja mellan stimuli. Den här uppsatsen undersöker huruvida en biologiskt realistisk normal­iserande krets kan implementeras av ett spikande neuronnätverk konstruerat med utgångspunkt från kolumnstrukturen i kortex. Detta gjordes genom att konstruera och utvärdera ett hårt strukturerat rekurrent spikande neuronnätverk, som modellerar lager 2/3 av en kortikal hyperkolumn med en grupp av neuroner som grundläggande beräkningsenhet. Resultaten visar att strukturen i hyperkolumn­modulen inte i sig skapar ett normaliserande nätverk. För de flesta nätverks­versioner implementerar nätverket en modulerande effekt som bättre beskrivs som subtraktiv inhibition. Dock hittades tre mekanismer som skapar ett mer normaliserande nätverk: Ökad membranvarians för större modulerande inputs; variabilitet i excitabilitet och inkommande kopplingar; och korttidsdepression på drivande synapser. Det visas också att genom att kombinera dessa mekanismer är det möjligt att skapa ett spikande neuronnät som approximerar normalisering över ett en åtminstone tio gångers ökning av storleken på input. Detta pekar på möjliga normaliserande mekanismer i en kortikal hyperkolumn, men ytterligare studier är nödvändiga för att avgöra om en eller flera av dessa kan vara en förklaring till hur normalisering är implementerat i biologiska nervsystem.