On periodic signals in stochastic Hodgkin-Huxley models

27.04.2018, 10:15-11:45  –  Haus 28, Raum 0.108
SFB-Kolloquium

Reinhard Höpfner, Universität Mainz

We consider a stochastic Hodgkin-Huxley model where dendritic input –modelled as an autonomous SDE which depends on a deterministic T-periodic signal t→S(t) encoded in its drift– is the only source of noise. This amounts to a 5d random system driven by 1d Brownian motion. We do have criteria to prove positive Harris recurrence (ergodicity) for systems of this type ([2], [3], [1]). As a consequence, we dispose of strong laws of large numbers for the system. In particular, we can describe the spiking activity of the neuron in the long run using strong laws of large numbers.
Let τ_n denote the beginning of the n-th spike. Whereas successive interspike times ξ_n:=τ_n+1−τ_n have no reason to be independent, there is a Glivenko-Cantelli theorem ([2]) for the sequence (ξ_n)_n≥1: empirical distribution functions F_n converge as n→∞ to some honest limit distribution function F. This limit F characterizes the spiking behaviour of the neuron in the long run. It depends on the modelization of the dendritic input, in particular on the signal t→S(t) encoded in its drift.
We are interested in statistical inference on the unobserved deterministic signal t→S(t), assuming that the Hodgkin Huxley neuron can be observed over a long time interval.
Large parts of the talk are joint work with Eva Löcherbach (Cergy-Pontoise) and Michele Thieullen (Paris VI).

References:
[1] R. Höpfner, E. Löcherbach, M. Thieullen: Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input. AIHP 52 (2016), 483–501
[2] R. Höpfner, E. Löcherbach, M. Thieullen: Ergodicity and limit theorems for degenerate diffusions with time periodic drift ... stochastic Hodgkin-Huxley model. ESAIM P+S 20 (2016), 527–554
[3] R. Höpfner, E. Löcherbach, M. Thieullen: Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Bernoulli 23 (2017), 2587–2616