FAUGERAS , Olivier

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I discuss how the notion of neural fields, a phenomenological averaged description of spatially distributed populations of neurons, can be used to build models of how visual information is represented and processed in the visual areas of primates. I describe in a pedestrian way one of the basic principles of operation of these neural fields...

We review in a pedestrian way some recent mathematical results concerning the solutions to general delayed neural field equations and some of their bifurcations. We indicate how these results can be brought to bear on models of visual perception.

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Many tissue level models of neural networks are written in the language of nonlinear integro-differential equations. Analytical solutions have only been obtained for the special case that the nonlinearity is a Heaviside function. Thus the pursuit of even approximate solutions to such models is of interest to the broad mathematical neuroscience...

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The CNS, like all complex systems, features a large variety of spatial and temporal scales. A given scale is usually accessible through a class of measurement modalities, e.g. electro-encephalography gives us access to the "mean" activity of very large populations of neurons whereas a micro-electrode can record from a single neuron. It is...

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I will discuss some of the relations between mathematics and neuroscience, in effect the neuroscience of visual perception of edges and textures. Starting from the question of knowing how the cortical representation of retinal images is organized, I will show that it leads in a very natural way to geometric concepts grounded in hyperbolic...

We derive the mean-field equations of completely connected networks of excitatory/inhibitory Hodgkin-Huxley and Fitzhugh-Nagumo neurons and prove that there is propagation to chaos, i.e. that in the limit the neurons become a) independent (this is the propagation to chaos) and b) a copy (with the same law) of a new individual, the mean field...

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I will discuss some of the relations between mathematics and neuroscience, in effect the neuroscience of visual perception of edges and textures. Starting from the question of knowing how the cortical representation of retinal images is organized, I will show that it leads in a very natural way to geometric concepts grounded in hyperbolic...

I discuss how ideas coming from various branches of mathematics such as Group Theory and Riemannian Geometry have influenced Mike Brady's work in the area of visual perception and its applications to robotics, resulting in significant progress in these fields and in the identification of fascinating open questions.

We develop a framework for the study of delayed neural fields equations and prove a center manifold theorem for these equations. Specific properties of delayed neural fields equations make it difficult to apply existing methods from the literature concerning center manifold results for functional differential equations. Our approach for the...

We present a new derivation of the classical action underlying a large deviation principle (LDP) for a stochastic hybrid system, which couples a piecewise deterministic dynamical system in R d with a time-homogeneous Markov chain on some discrete space Γ. We assume that the Markov chain on Γ is ergodic, and that the discrete dynamics is much...

In this paper, we consider neural field equations with space-dependent delays. Neural fields are continuous assemblies of mesoscopic models arising when modeling macroscopic parts of the brain. They are modeled by nonlinear integro-differential equations. We rigorously prove, for the first time to our knowledge, sufficient conditions for the...

We study the asymptotic law of a network of interacting neurons when the number ofneurons becomes infinite. The dynamics of the neurons is described by a set of stochasticdifferential equations in discrete time. The neurons interact through the synaptic weightsthat are Gaussian correlated random variables. We describe the asymptotic law of...