A neuronal transition probability model for the evolution of power in the sigma and delta frequency bands of sleep...

Although the time-courses of power in the delta and sigma frequency bands over the NREM episode in the human sleep EEG have been studied for several years, and their detailed forms have been well measured, no mathematical model has yet been formulated to account for the relation between them. The model presented here attempts to explain the form and relative timing of these curves by a consideration of the behavior of the thalamocortical neuronal populations that are believed to play a part in their generation. The model applies the mathematics of the cascade radioactive decay process, adapted to a finite population of thalamocortical neurons oscillating initially in the beta mode. At the beginning of the NREM episode, each neuron of this population is assumed to acquire a constant probability of transitionning to the sigma oscillation mode and, at the same time, each neuron of the newly created sigma population is assumed to acquire a constant probability of transitionning to the delta oscillation mode. This simple model is sufficient to explain the main characteristics of the first half of the time-courses of the sigma and delta powers: the initial positive correlation as they increase together, followed by the sigma peak and the subsequent negative correlation. At the end of this first phase, the model initiates an identical, but reverse, process that reproduces the observed delta maximum and sigma plateau, followed by the concomitant fall of both sigma and delta power. The time-course of the beta power and the overall negative correlation between beta and delta are also reproduced as integral consequences of the model.