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posted on 25.01.2006by S. Coombes
We use geometric dynamical systems methods to derive phase equations for networks
of weakly connected McKean relaxation oscillators. We derive an explicit
formula for the connection function when the oscillators are coupled with chemical
synapses modeled as the convolution of some input spike train with an appropriate
synaptic kernel. The theory allows the systematic investigation of the way in
which a slow recovery variable can interact with synaptic time scales to produce
phase-locked solutions in networks of pulse coupled neural relaxation oscillators.
The theory is exact in the singular limit that the fast and slow time scales of the
neural oscillator become effectively independent. By focusing on a pair of mutually
coupled McKean oscillators with alpha function synaptic kernels, we clarify the role
that fast and slow synapses of excitatory and inhibitory type can play in producing
stable phase-locked rhythms. In particular we show that for fast excitatory synapses
there is coexistence of a stable synchronous, a stable anti-synchronous, and one stable
asynchronous solution. For slower synapses the anti-synchronous solution can
lose stability, whilst for even slower synapses it can regain stability. The case of
inhibitory synapses is similar up to a reversal of the stability of solution branches.
Using a return-map analysis the case of strong pulsatile coupling is also considered.
In this case it is shown that the synchronous solution can co-exist with a continuum
of asynchronous states.