Section: New Results

Macroscopic limits of stochastic neural networks and neural fields

Pinwheel-Dipole configuration in cat visual cortex

Participants : Jérôme Ribot [CIRB] , Alberto Romagnoni [CIRB] , Chantal Milleret [CIRB] , Daniel Bennequin [CIRB] , Jonathan Touboul.

One fascinating aspect of the brain is its ability to process information in a fast and reliable manner. The functional architecture is thought to play a central role in this task, by encoding efficiently complex stimuli and facilitating higher level processing. In the early visual cortex of higher mammals, information is processed within functional maps whose layout is thought to underlie visual perception. The possible principles underlying the topology of the different maps, as well as the role of a specific functional architecture on information processing, is however poorly understood.

• In [25] , we show that spatial frequency representation in cat areas 17 and 18 exhibits singularities around which the map organizes like an electric dipole potential. These singularities are precisely co-located with singularities of the orientation map: the pinwheel centers. We first show, using high resolution optical imaging, that a large majority (around 80%) of pinwheel centers exhibit in their neighborhood semi-global extrema in the spatial frequency map. These extrema created a sharp gradient that was confirmed with electrophysiological recordings. Based on an analogy with electromagnetism, a mathematical model of a dipolar structure is proposed, that was accurately fitted to optical imaging data for two third of pinwheel centers with semi-global extrema.

• Mathematically, this pinwheel-dipole architecture is fascinating. We demonstrated mathematically in [26] that two natural principles, local exhaustivity of representation and parsimony, would indeed constrain the orientation and spatial frequency maps to display co-located singularities around which the orientation is organized as a pinwheel and spatial frequency as a dipole. Moreover, using a computational model, we showed that this architecture allows a trade-off in the local perception of orientation and spatial frequency, but this would occur for sharper selectivity than the tuning width reported in the literature. We therefore re-examined physiological data and show that indeed the spatial frequency selectivity substantially sharpens near maps singularities, bringing to the prediction that the system tends to optimize balanced detection between different attributes.

These results shed new light on the principles at play in the emergence of functional architecture of cortical maps, as well as their potential role in processing information.

Absorption properties of stochastic equations with Hölder diffusion coefficients

Participants : Jonathan Touboul, Gilles Wainrib [ENS] .

In [29] , we address the absorption properties of a class of stochastic differential equations around singular points where both the drift and diffusion functions vanish. According to the Hölder coefficient alpha of the diffusion function around the singular point, we identify different regimes. Stability of the absorbing state, large deviations for the absorption time, existence of stationary or quasi-stationary distributions are discussed. In particular, we show that quasi-stationary distributions only exist for alpha < 3/4, and for alpha in the interval (3/4, 1), no quasi-stationary distribution is found and numerical simulations tend to show that the process conditioned on not being absorbed initiates an almost sure exponential convergence towards the absorbing state (as is demonstrated to be true for alpha = 1). Applications of these results to stochastic bifurcations are discussed.

On a kinetic FitzHugh-Nagumo model of neuronal network

Participants : Stéphane Mischler [CEREMADE] , Cristóbal Quiñinao [CIRB] , Jonathan Touboul.

We investigate in [33] the existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behavior of interacting Fitzhugh-Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We proved existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrated uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we showed exponential nonlinear stability in the small connectivity regime.