具有FitzHugh-Nagumo输运的随机双域方程的全局强适定性

无界域上具有分布依赖性的非自治随机 FitzHugh-Nagumo 系统的拉回测度吸引子

Pullback measure attractors for non-autonomous stochastic FitzHugh-Nagumo system with distribution dependence on unbounded domains

本文研究无界域上依赖分布的非自治随机 FitzHugh-Nagumo 系统，基于 Banach 不动点定理证明解的适定性，并结合分裂技巧、尾端估计与 Vitali 定理证明拉回测度吸引子的存在唯一性。

Splitting schemes for FitzHugh--Nagumo stochastic partial differential equations

We design and study splitting integrators for the temporal discretization of the stochastic FitzHugh--Nagumo system. This system is a model for signal propagation in nerve cells where the voltage variable is solution of a one-dimensional parabolic PDE with a cubic nonlinearity driven by additive space-time white noise. We first show that the numerical solutions have finite moments. We then prove that the splitting schemes have, at least, the strong rate of convergence $1/4$. Finally, numerical experiments illustrating the performance of the splitting schemes are provided.

随机FitzHugh-Nagumo模型中的混合模振荡和尖峰间隔统计

Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

我们研究随机FitzHugh-Nagumo方程，模拟神经元动作电位的动力学，在参数制度的特点是混合模式振荡。峰间时间间隔与分隔连续峰的小振幅振荡的随机数有关。证明了该数具有渐近几何分布，其参数与次随机马尔可夫链的主特征值有关。我们提供了严格的界限上的本征值在小噪声制度，并推导出一个近似的依赖于系统的参数为一个大范围的噪声强度。这产生了一个精确的描述所观察到的混合模式模式和interspike间隔的概率分布。

An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh-Nagumo model

In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component only. We prove moment stability of the discretised processes and a strong convergence rate of $1/2$. Several numerical examples, centred around a mean-field model for FitzHugh-Nagumo neurons, illustrate that the standard uniform scheme fails and that t…

Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions

We prove local existence of solutions for a class of suitably renormalised coupled SPDE-ODE systems driven by space-time white noise, where the space dimension is equal to 2 or 3. This class includes in particular the FitzHugh-Nagumo system describing the evolution of action potentials of a large population of neurons, as well as models with multidimensional gating variables. The proof relies on the theory of regularity structures recently developed by M. Hairer, which is extended to include situations with semigroups that are not regularising in space. We also provide explicit expressions for…

Corrigendum to "Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions"

Lemma 4.8 in the article [Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions, Electronic J. Probability 21 (18):1-48 (2016), arXiv:1504.02953] contains a mistake, which implies a weaker regularity estimate than the one stated in Proposition 4.11. This does not affect the proof of Theorem 2.1, but Theorems 2.2 and 2.3 only follow from the given proof if either the space dimension $d$ is equal to $2$, or the nonlinearity $F(U,V)$ is linear in $V$. To fix this problem and provide a proof of Theorems 2.2 and 2.3 valid in full generality, we consider an alt…

FitzHugh-Nagumo神经元平均场网络中的混沌传播

Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

本文研究全连接FitzHugh-Nagumo神经元网络在$N$趋于无穷时的行为，利用反射耦合方法在平均场框架下证明了时间上一致的混沌传播，并给出显式界限，解决了底层势能非凸性问题。

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequenci…

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüg…

Hodgkin-Huxley和Fitzhugh-Nagumo神经元递归多种群网络的平均场描述和混沌传播

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

我们推导出的平均场方程所产生的相互作用尖峰神经元的网络的限制，神经元的数量趋于无穷大。神经元属于固定数量的种群，可以用Hodgkin-Huxley模型或其简化版本Fitzhugh-Nagumo模型来表示。神经元之间的突触要么是电的，要么是化学的。假设网络完全连接。最大电导随机变化。在所有的神经元的初始条件是独立的，从同一个法律，只取决于他们所属的人口的条件下，我们证明了混沌现象的传播发生，即在平均场极限，任何有限数量的神经元成为独立的，在每个人口，有相同的概率分布。这个概率分布是一组隐式方程的解，或者是类似于McKean-Vlasov方程的非线性随机微分方程，或者是类似于McKean-Vlasov-Fokker- Planck方程的非局部偏微分方程。我们证明了这些方程的适定性，即解的存在性和唯一性。我们还显示了一些初步的数值实验的结果表明，平均场方程是一个很好的表示的平均活动的有限大小的网络，即使是适度的大小。这些实验还表明，McKean-Vlasov-Fokker- Planck方程可能是理解平均场动力学的好方法，例如，分叉分析