*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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