On the long time behaviour of single stochastic Hodgkin-Huxley neurons with constant signal, and a construction of circuits of interacting neurons showing self-organized rhythmic oscillations

The stochastic Hodgkin-Huxley neurons considered in this paper replace time-constant deterministic input $a dt$ of the classical model by increments $\vartheta dt + dX_t$ a process: $X$ is Ornstein-Uhlenbeck with volatility $\sigma>0$ and backdriving force $\tau>0$, we call $\vartheta>0$ signal. We have ergodicity strong laws large numbers for various functionals process, characterize 'quiet behaviour' 'regular spiking' as events whose probability depends on parameters $(\tau,\sigma)$ signal $\vartheta$. notions quiet behaviour regular spiking allow construction circuits interacting neurons, combining excitation inhibition according to bloc structure along circuit, which self-organized rhythmic oscillations can be observed.