Dynamics of a neuron–glia system: the occurrence of seizures and the influence of electroconvulsive stimuli

Abstract

In this paper, we investigate the dynamics of a neuron-glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro-scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single-cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant paramenters. Keywords Nonlinear dynamics • Bifurcation theory • Neuron-glia cell system • Monodomain and bidomain model • Seizure • Electroconvulsive therapy (ECT) JS was supported by RCN no.273077.

Simulating Epileptic Seizures using the Bidomain Model

“…It continues to alternate between bursts and periods of quiescence. This is similar to what was observed in the study by Erhardt et al 4 .…”

“…In this paper we simulated the electrical potential in a human brain during a seizure using the Bidomain model with a Cressman 6,7 excitation model in 10 geometries (7 idealized 2D models and 3 MR based 3D models). The Bidomain-Cressman model resulted in electrical waves propagating through the brain from unstable to stable regions as also observed in Erhardt et al 4 . The PSD of these waves follow a power-law distribution similar to what is found in EEGs.…”

“…As such, Model B is in a seizure-like state, and alternates between periods of spike trains (bursts) and quiescence, seen in Figure 2. We compared models by counting the number of spikes, as also done in Erhardt et al 4 , as well as computing the power-spectra.…”

“…A further experiment with a Morris-Lecar model 37 in the WM in a 3D simulation had negligible effect in the simulated potentials when compared with no WM cell model. A previous study of the coupling of the Cressman and Bidomain models showed that there is a balance between the conductivity and the size of the unstable region, i.e., the right hippocampus 4 . If the conductivity is too high, or the unstable region too small, the stable regions of the model dominates and spiking stops.…”

“…An epileptic seizure is due to excessive or synchronous neural activity in the brain 2,3 . Extensive modeling has been performed on the neuronal level, but only few works 4,5 , have to the authorst' knowledge, scaled up the neuronal modelling to brain-wide simulations. The so-called Bidomain model, widely used for cardiac modeling, allows for this upscaling.…”