Alpha-Delta Transitions in Cortical Rhythms as grazing bifurcations

TL;DR

This paper investigates the transition between alpha and delta brain oscillations in the Jansen-Rit cortical model, identifying it as a grazing bifurcation caused by excitatory feedback collapse.

Contribution

It introduces a novel bifurcation perspective on alpha-delta transitions, modeling them as grazing bifurcations in a simplified neuronal activation framework.

Findings

Alpha-delta transition is a grazing bifurcation.

Discontinuity in feedback causes oscillation collapse.

All-or-nothing activation models are appropriate.

Abstract

The Jansen-Rit model of a cortical column in the cerebral cortex is widely used to simulate spontaneous brain activity (EEG) and event-related potentials. It couples a pyramidal cell population with two interneuron populations, of which one is fast and excitatory and the other slow and inhibitory. Our paper studies the transition between alpha and delta oscillations produced by the model. Delta oscillations are slower than alpha oscillations and have a more complex relaxation-type time profile. In the context of neuronal population activation dynamics, a small threshold means that neurons begin to activate with small input or stimulus, indicating high sensitivity to incoming signals. A steep slope signifies that activation increases sharply as input crosses the threshold. Accordingly in the model the excitatory activation thresholds are small and the slopes are steep. Hence, a singular limit replacing the excitatory activation function with all-or-nothing switches, eg. a Heaviside function, is appropriate. In this limit we identify the transition between alpha and delta oscillations as a discontinuity-induced grazing bifurcation. At the grazing the minimum of the pyramidal-cell output equals the threshold for switching off the excitatory interneuron population, leading to a collapse in excitatory feedback.