ALPHA–DELTA TRANSITIONS IN CORTICAL RHYTHMS AS GRAZING BIFURCATIONS

Abstract The Jansen–Rit model of a cortical column in the cerebral cortex is widely used to simulate spontaneous brain activity (electroencephalogram, 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, we replace the excitatory activation function with its singular limit, which is an all-or-nothing switch (a Heaviside function). 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.