Splitting methods for stochastic Hodgkin-Huxley type systems and a localized fundamental mean-square convergence theorem
Pierre Étoré, Anna Melnykova, Irene Tubikanec
TL;DR
The paper develops a localized mean-square convergence framework for SDEs with locally Lipschitz coefficients and applies it to stochastic Hodgkin–Huxley type systems featuring a conditional linear drift. It introduces Lie-Trotter and Strang splitting schemes that exploit this structure, proves local mean-square convergence with rate $q-1$ for locally MS-consistent schemes and global convergence under bounded $2p$th moments, and demonstrates key properties such as state-space preservation and geometric ergodicity. Through rigorous moment-boundedness, local consistency analyses, and a broad convergence theory, the authors show that the splitting methods provide reliable, structure-preserving approximations that outperform Euler–Maruyama variants in preserving dynamics like spiking and invariant measures. Numerical experiments corroborate the theory, highlighting larger allowable time steps and substantial computational savings while maintaining accurate qualitative behavior. The framework and splittings have potential applicability beyond HH-type models to a wide class of locally Lipschitz SDEs and other conditionally linear systems, offering practically meaningful tools for accurate and efficient simulations.
Abstract
Existing fundamental theorems for mean-square convergence of numerical methods for stochastic differential equations (SDEs) require globally or one-sided Lipschitz continuous coefficients, while strong convergence results under merely local Lipschitz conditions are largely restricted to Euler-Maruyama type methods. To address these limitations, we introduce a novel localized version of the fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients, which naturally arise in a wide range of applications. Specifically, we show that if a numerical scheme is locally consistent in the mean-square sense of order $q>1$, then it is locally mean-square convergent with rate $q-1$. Building on this result, we further prove that global mean-square convergence follows, provided that both the exact solution and its numerical approximation admit bounded $2p$th moments for some $p>1$. These new convergence results are illustrated on a class of locally Lipschitz SDEs of Hodgkin-Huxley type, characterized by a conditionally linear drift structure. For these systems, we construct different Lie-Trotter and Strang splitting methods exploiting their conditional linearity. The proposed convergence framework is then applied to these schemes, requiring innovative proofs of local consistency and boundedness of moments. In addition, we establish key structure-preserving properties of the splitting methods, in particular state-space preservation and geometric ergodicity. Numerical experiments support the theoretical results and demonstrate that the proposed splitting schemes significantly outperform Euler-Maruyama type methods in preserving the qualitative features of the model.