Reinhard Höpfner - 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

mna:9279 - Mathematical Neuroscience and Applications, January 30, 2023, Volume 3 - https://doi.org/10.46298/mna.9279
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 oscillationsArticle

Authors: Reinhard Höpfner 1

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.


Volume: Volume 3
Published on: January 30, 2023
Accepted on: December 7, 2022
Submitted on: March 31, 2022
Keywords: Mathematics - Probability,60J25 (primary), 60F15, 60K35, 62M99 (secondary)

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