**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

- A neural oscillator can communicate only with those oscillators which have roughly the same natural frequency. That is, synaptic connections between a pair of oscillators having different natural frequencies are functionally insignificant.
- Two neural oscillators having the same natural frequencies might not communicate if the connections between them are from among a class of pathological synaptic configurations. In both cases the anatomical presence of synaptic connections between neural oscillators does not necessarily guarantee that the connections are functionally significant.
- There can be substantial phase differences (time delays) between the neural oscillators, which result from the synaptic organization of the network, not from the transmission delays.

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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