Open-source tools for dynamical analysis of Liley's mean-field cortex model

TL;DR

This paper presents open-source computational tools for analyzing the complex spatio-temporal dynamics of Liley's mean-field cortex model, enabling detailed numerical investigation of cortical activity patterns.

Contribution

It introduces an implementation of implicit time-stepping and linearization techniques for Liley's model using PETSc, facilitating direct analysis of its full spatio-temporal behavior.

Findings

Successfully implemented parallelized numerical methods for the model.

Able to compute equilibria and periodic solutions of cortical dynamics.

Demonstrated capability to simulate cortical tissue behavior at high resolution.

Abstract

Mean-field models of the mammalian cortex treat this part of the brain as a two-dimensional excitable medium. The electrical potentials, generated by the excitatory and inhibitory neuron populations, are described by nonlinear, coupled, partial differential equations, that are known to generate complicated spatio-temporal behaviour. We focus on the model by Liley {\sl et al.} (Network: Comput. Neural Syst. (2002) 13, 67-113). Several reductions of this model have been studied in detail, but a direct analysis of its spatio-temporal dynamics has, to the best of our knowledge, never been attempted before. Here, we describe the implementation of implicit time-stepping of the model and the tangent linear model, and solving for equilibria and time-periodic solutions, using the open-source library PETSc. By using domain decomposition for parallelization, and iterative solving of linear problems, the code is capable of parsing some dynamics of a macroscopic slice of cortical tissue with a sub-millimetre resolution.