The stochastic Hodgkin-Huxley neurons considered in this paper replace
time-constant deterministic input $a dt$ of the classical deterministic model
by increments $\vartheta dt + dX_t$ of a stochastic process: $X$ is
Ornstein-Uhlenbeck with volatility $\sigma>0$ and backdriving force $\tau>0$,
and we call $\vartheta>0$ the signal. We have ergodicity and strong laws of
large numbers for various functionals of the process, and characterize 'quiet
behaviour' and 'regular spiking' as events whose probability depends on the
parameters $(\tau,\sigma)$ and on the signal $\vartheta$. The notions of quiet
behaviour and regular spiking allow for a construction of circuits of
interacting stochastic Hodgkin-Huxley neurons, combining excitation with
inhibition according to a bloc structure along the circuit, on which
self-organized rhythmic oscillations can be observed.
We analyze the interplay between contact geometry and Gabor filters signal
analysis in geometric models of the primary visual cortex. We show in
particular that a specific framed lattice and an associated Gabor system is
determined by the Legendrian circle bundle structure of the $3$-manifold of
contact elements on a surface (which models the V1-cortex), together with the
presence of an almost-complex structure on the tangent bundle of the surface
(which models the retinal surface). We identify a scaling of the lattice, also
dictated by the manifold geometry, that ensures the frame condition is
satisfied. We then consider a $5$-dimensional model where receptor profiles
also involve a dependence on frequency and scale variables, in addition to the
dependence on position and orientation. In this case we show that a proposed
profile window function does not give rise to frames (even in a distributional
sense), while a natural modification of the same generates Gabor frames with
respect to the appropriate lattice determined by the contact geometry.
In this article, we are interested in the behavior of a fully connected
network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons
follow the stochastic FitzHugh-Nagumo model, whose specificity is the
non-linearity with a cubic term. We prove a result of uniform in time
propagation of chaos of this model in a mean-field framework. We also exhibit
explicit bounds. We use a coupling method initially suggested by A. Eberle
(arXiv:1305.1233), and recently extended in (1805.11387), known as the
reflection coupling. We simultaneously construct a solution of the $N$-particle
system and $N$ independent copies of the non-linear McKean-Vlasov limit in such
a way that, considering an appropriate semi-metric that takes into account the
various possible behaviors of the processes, the two solutions tend to get
closer together as $N$ increases, uniformly in time. The reflection coupling
allows us to deal with the non-convexity of the underlying potential in the
dynamics of the quantities defining our network, and show independence at the
limit for the system in mean field interaction with sufficiently small
Lipschitz continuous interactions.