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Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input. (English. French summary) Zbl 1335.60091
Summary: We consider a model describing a neuron and the input it receives from its dendritic tree when this input is a random perturbation of a periodic deterministic signal, driven by an Ornstein-Uhlenbeck process. The neuron itself is modeled by a variant of the classical Hodgkin-Huxley model. Using the existence of an accessible point where the weak Hörmander condition holds and the fact that the coefficients of the system are analytic, we show that the system is non-degenerate. The existence of a Lyapunov function allows to deduce the existence of (at most a finite number of) extremal invariant measures for the process. As a consequence, the complexity of the system is drastically reduced in comparison with the deterministic system.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J25 Continuous-time Markov processes on general state spaces
60H07 Stochastic calculus of variations and the Malliavin calculus
60H30 Applications of stochastic analysis (to PDEs, etc.)
92C20 Neural biology
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