Polynomials under Ornstein-Uhlenbeck noise and an application to inference in stochastic Hodgkin-Huxley systems

Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels

In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the n…

关于具有恒定信号的单随机霍奇金-赫胥黎神经元的长时间行为，以及自组织节律性振荡的相互作用神经元回路构建

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

该论文研究带恒定信号的随机霍奇金-赫胥黎神经元的长时间行为，证明遍历性与大数定律，并利用安静与规则尖峰概念构建兴奋-抑制块结构回路，实现自组织节律振荡。

Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model

We formulate simple criteria for positive Harris recurrence of strongly degenerate stochastic differential equations with smooth coefficients when the drift depends on time and space and is periodic in the time argument. There is no time dependence in the diffusion coefficient. Our criteria rely on control systems and the support theorem, existence of an attainable inner point of full weak Hoermander dimension and of some Lyapunov function. Positive Harris recurrence enables us to prove limit theorems for such diffusions. As an application, we consider a stochastic Hodgkin-Huxley model for a s…

Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population t…

用于随机Hodgkin-Huxley型系统的分裂方法及一个局部基本均方收敛定理

Splitting methods for stochastic Hodgkin-Huxley type systems and a localized fundamental mean-square convergence theorem

本文提出了一种适用于局部Lipschitz系数的随机微分方程局部均方收敛定理，并基于此构造了分裂方法，通过局部一致性证明和矩有界性，展示了其在Hodgkin-Huxley型系统中的优越性。

Transition densities for strongly degenerate time inhomogeneous random models

In this paper we study the existence of densities for strongly degenerate stochastic differential equations whose coefficients depend on time and are not globally Lipschitz. In these models neither local ellipticity nor the strong Hörmander condition is satisfied. In this general setting we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hörmander condition is satisfied. We also exhibit regions where these densities remain positive. We then apply these results to stochastic Hodgkin-Huxley models as a first step towards the study of ergodicit…

Analysis and Approximation of Stochastic Nerve Axon Equations

We consider spatially extended conductance based neuronal models with noise described by a stochastic reaction diffusion equation with additive noise coupled to a control variable with multiplicative noise but no diffusion. We only assume a local Lipschitz condition on the nonlinearities together with a certain physiologically reasonable monotonicity to derive crucial $L^\infty$-bounds for the solution. These play an essential role in both the proof of existence and uniqueness of solutions as well as the error analysis of the finite difference approximation in space. We derive explicit error e…