Weakly Connected Quasiperiodic Oscillators, FM Interactions, and Multiplexing in the Brain

SIAM Journal on Applied Mathematics (1999), 59:2193-2223

Eugene M. Izhikevich

Systems Science Center, Box 7606,
Arizona State University,
Tempe, AZ 85287-7606.

Abstract. We prove that weakly connected networks of quasiperiodic (multi-frequency) oscillators can be transformed into a phase model by a continuous change of variables. The phase model has the same form as the one for periodic oscillators with the exception that each phase variable is a vector. When the oscillators have mutually nonresonant frequency (rotation) vectors, the phase model has an uncoupled form. This implies that such oscillators do not interact even though there are connections between them. When the frequency vectors are mutually resonant, the oscillators interact via phase deviations. This mechanism resembles that of FM radio, with an additional feature -- multiplexing of signals. Possible applications to neuroscience are discussed.

Keywords: Weakly connected neural networks, invariant manifolds, quasiperiodic oscillators, chaos, phase model, resonances, FM interactions, multiplexing, thalamo-cortical system

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