Synaptic organizations and dynamical properties of weakly connected neural oscillators.
I. Analysis of a canonical model

Biological Cybernetics (1996) 75:117-127

Frank C. Hoppensteadt and Eugene M. Izhikevich

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

Abstract. We study weakly connected networks of neural oscillators near multiple Andronov-Hopf bifurcation point. We analyze relationships between synaptic organizations (anatomy) of the networks and their dynamical properties (function).

Our principal assumptions are:
1) Each neural oscillator comprises two populations of neurons: excitatory and inhibitory ones.
2) Activity of each population of neurons is described by a scalar (one-dimensional) variable.
3) Each neural oscillator is near a non-degenerate supercritical Andronov-Hopf bifurcation point.
4) The synaptic connections between the neural oscillators are weak.

All neural networks satisfying these hypotheses are governed by the same dynamical system, which we call the canonical model. Studying the canonical model shows that

Using the canonical model we can illustrate self-ignition and autonomous quiescence (oscillator death) phenomena. That is, a network of passive elements can exhibit active properties and vice-versa.

We also study how Dale's principle affects dynamics of the networks, in particular, the phase differences that the network can reproduce. We present a complete classification of all possible synaptic organizations from this point of view.

The theory developed here casts some light on relations between synaptic organization and functional properties of oscillatory networks. The major advantage of our approach is that we obtain results about all networks of neural oscillators, including the real brain. The major drawback is that our findings are valid only when the brain operates near a critical regime, viz. multiple Andronov-Hopf bifurcation.

Keywords: Weakly connected neural networks -- Neural oscillators -- Multiple Andronov-Hopf bifurcation -- Canonical model -- Natural Phase differences -- Self-ignition -- Oscillator death -- Synaptic organizations -- Dale's principle

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