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Simulating brain rhythms using an ODE with stochastically varying coefficients

The brain produces rhythms in a variety of frequency bands. Some are likely by-products of neuronal processes; others are thought to be top-down. Produced entirely naturally, these rhythms have clearly recognizable beats, but they are very far from periodic in the sense of mathematics. They produce signals that are broad-band, episodic, wandering in magnitude, in frequency and in phase; the rhythm comes and goes, degrading and regenerating. Rhythms with these characteristics do not match standard dynamical systems paradigms of periodicity, quasi-periodicity, or periodic motion in the presence of a Brownian noise. Thus far they have been satisfactorily reproduced only using networks of hundreds of integrate-and-fire neurons. In this paper, we tackle the mathematical question of whether signals with these properties can be generated from simpler dynamical systems. Using an ODE with two variables inspired by the FitzHugh-Nagumo model, and varying randomly three parameters that control the magnitude, frequency and degree of degradation, we were able to replicate the qualitative characteristics of these natural brain rhythms. Viewing the two variables as Excitatory and Inhibitory conductances of a typical neuron in a local population, our model produces results that closely resemble gamma-band activity in real cortex, including the moment-to-moment balancing of E and I-currents seen in experiments.