Splitting integrators for stochastic FitzHugh-Nagumo models

We focus on the numerical analysis of eminent stochastic partial differential equations (SPDEs) from neurosciencecalled stochastic FitzHugh-Nagumo systems.

These models consist in a system of two SPDEsand describe the electrical activity in excitable cells, such as neurons.Due to the challenges of solving SPDEs analytically, robust numerical methodsare essential to obtain approximated solutions to SPDEs on computers.These numerical methods should be both computationally efficient and mathematically well-founded.The numerical methods developed and studied in this project are (explicit) splitting integrators.They will provide efficient ways to approximate solutions to several stochastic FitzHugh-Nagumo systems such assystems with spatio-temporal noise in both components of the system,systems with multiplicative noise having (almost-surely) positive solutions,and systems of SPDEs in dimension 1,2, and 3.

The accuracy of the splitting integrators will be provedin terms of strong convergence, where one looks at the mean of the error,and also in terms of weak convergence, where one looks at the error of the means.This has never been done before.The results of this project will enable researchers to, e.g., simulate and analyse the dynamics in brains.Ultimately, the results of the project will lead to a better understandingof how neurons interact and how the brain functions.The proposal asks for the funding of one doctoral student, research time for the applicant and travel money.