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

TL;DR

This paper studies the long-term behavior of stochastic Hodgkin-Huxley neurons driven by Ornstein-Uhlenbeck noise, characterizing their activity states and constructing circuits that exhibit self-organized rhythmic oscillations.

Contribution

It introduces a stochastic neuron model with Ornstein-Uhlenbeck input and constructs interacting neuron circuits showing emergent rhythmic activity.

Findings

Ergodicity and strong laws of large numbers established for the model

Characterization of quiet and spiking behaviors based on parameters

Construction of neuron circuits with self-organized oscillations

Abstract

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.