Mathematical Neuroscience and Applications |

The stochastic Hodgkin-Huxley neurons considered in this paper replacetime-constant deterministic input $a dt$ of the classical deterministic modelby increments $\vartheta dt + dX_t$ of a stochastic process: $X$ isOrnstein-Uhlenbeck with volatility $\sigma>0$ and backdriving force $\tau>0$,and we call $\vartheta>0$ the signal. We have ergodicity and strong laws oflarge numbers for various functionals of the process, and characterize 'quietbehaviour' and 'regular spiking' as events whose probability depends on theparameters $(\tau,\sigma)$ and on the signal $\vartheta$. The notions of quietbehaviour and regular spiking allow for a construction of circuits ofinteracting stochastic Hodgkin-Huxley neurons, combining excitation withinhibition according to a bloc structure along the circuit, on whichself-organized rhythmic oscillations can be observed.

We analyze the interplay between contact geometry and Gabor filters signalanalysis in geometric models of the primary visual cortex. We show inparticular that a specific framed lattice and an associated Gabor system isdetermined by the Legendrian circle bundle structure of the $3$-manifold ofcontact elements on a surface (which models the V1-cortex), together with thepresence of an almost-complex structure on the tangent bundle of the surface(which models the retinal surface). We identify a scaling of the lattice, alsodictated by the manifold geometry, that ensures the frame condition issatisfied. We then consider a $5$-dimensional model where receptor profilesalso involve a dependence on frequency and scale variables, in addition to thedependence on position and orientation. In this case we show that a proposedprofile window function does not give rise to frames (even in a distributionalsense), while a natural modification of the same generates Gabor frames withrespect to the appropriate lattice determined by the contact geometry.

In this article, we are interested in the behavior of a fully connectednetwork of $N$ neurons, where $N$ tends to infinity. We assume that the neuronsfollow the stochastic FitzHugh-Nagumo model, whose specificity is thenon-linearity with a cubic term. We prove a result of uniform in timepropagation of chaos of this model in a mean-field framework. We also exhibitexplicit bounds. We use a coupling method initially suggested by A. Eberle(arXiv:1305.1233), and recently extended in (1805.11387), known as thereflection coupling. We simultaneously construct a solution of the $N$-particlesystem and $N$ independent copies of the non-linear McKean-Vlasov limit in sucha way that, considering an appropriate semi-metric that takes into account thevarious possible behaviors of the processes, the two solutions tend to getcloser together as $N$ increases, uniformly in time. The reflection couplingallows us to deal with the non-convexity of the underlying potential in thedynamics of the quantities defining our network, and show independence at thelimit for the system in mean field interaction with sufficiently smallLipschitz continuous interactions.

We take the testing perspective to understand what the minimal discriminationtime between two stimuli is for different types of rate coding neurons. Ourmain goal is to describe the testing abilities of two different encodingsystems: place cells and grid cells. In particular, we show, through the notionof adaptation, that a fixed place cell system can have a minimum discriminationtime that decreases when the stimuli are further away. This could be aconsiderable advantage for the place cell system that could complement the gridcell system, which is able to discriminate stimuli that are much closer thanplace cells.