Codimension Two Bifurcations and Rythms in Neural Mass Models

TL;DR

This paper investigates the bifurcation structures in neural mass models to understand the emergence of epileptic seizures and brain rhythms, linking mathematical bifurcations to physiological oscillations.

Contribution

It provides a detailed bifurcation analysis of classical cortical models, highlighting how parameter changes lead to different neural rhythms and epileptiform activity.

Findings

Identification of bifurcation points leading to seizure-like activity

Correlation of model oscillations with sleep wave frequencies

Insights into parameter effects on epileptiform dynamics

Abstract

Temporal lobe epilepsy is one of the most common chronic neurological disorder characterized by the occurrence of spontaneous recurrent seizures which can be observed at the level of populations through electroencephalogram (EEG) recordings. This paper summarizes some preliminary works aimed to understand from a theoretical viewpoint the occurrence of this type of seizures and the origin of the oscillatory activity in some classical cortical column models. We relate these rhythmic activities to the structure of the set of periodic orbits in the models, and therefore to their bifurcations. We will be mainly interested Jansen and Rit model, and study the codimension one, two and a codimension three bifurcations of equilibria and cycles of this model. We can therefore understand the effect of the different biological parameters of the system of the apparition of epileptiform activity and observe the emergence of alpha, delta and theta sleep waves in a certain range of parameter. We then present a very quick study of Wendling and Chauvel's model which takes into account GABA A inhibitory postsynaptic currents.