Weakly Connected Quasi-periodic Oscillators, FM Interactions, and Multiplexing in the Brain

Abstract

We prove that weakly connected networks of quasi-periodic (multifrequency) oscil- lators 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 uncouples. This implies that such oscillators do not interact even though there might be physical connections between them. When the frequency vectors have mutual low-order resonances, the oscillators interact via phase deviations. This mechanism resembles that of the FM radio, with a shared feature|multiplexing of signals. Possible applications to neuroscience are discussed. Key words. weakly connected neural networks, invariant manifolds, quasi-periodic oscillators,