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

par Pierre Etoré (LJK (Université Grenoble Alpes))

mardi 31 mars 2026, 09:50 → 10:50 Europe/Paris

Amphi Schwartz

Motivated by the study of splitting schemes for the Hodgkin-Huxley model, we introduce a novel localized version of the fundamental mean-square convergence theorem for numerical schemes for SDEs with locally Lipschitz coefficients. Indeed existing fundamental theorems for mean-square convergence for schemes for stochastic differential equations (SDEs) usually require globally or one-sided Lipschitz continuous coefficients. Those assumptions are not satisfied by the Hodgkin-Huxley model. Specifically, we show that if the coefficients of the SDE are locally Lipschitz and 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 2pth moments for some p>1.

 

Having at hand these new convergence results we perform fully the study of convergence of splitting schemes for the Hodgkin-Huxley model. This model describes the neural activity of a giant squid and is characterized by a conditionally linear drift structure. We perform  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. 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.