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Excitatory-Inhibitory Neural Network 1 From: Theoretical Neuroscience, by Peter Dayan and Larry Abbott, MIT Press, 2005 pp. 266-269 The system studied here is one the simplest types of neural networks to exhibit oscillatory activity. It can be regarded as a simplified model of a fully-connected network comprised of a large number of excitatory and inhibitory neurons. Alternatively, it can be regarded as a model comprised of a single excitatory neuron and a single inhibitory neuron. All synaptic weights of a given type are given identical values: the excitatory-toexcitatory, inhibitory-to-excitatory, excitatory-to-inhibitory, and inhibitory-to-inhibitory weights are, respectively 1.25, –1, 1, and 0. (These are dimensionless parameters.) Similarly, all firing thresholds of a given type are identical: the excitatory and inhibitory thresholds are, respectively, –10 Hz and 10 Hz. Finally, all time constants of a given type are identical: the excitatory and inhibitory time constants are, respectively, 10 ms and I. The system has a single parameter: I. The excitatory and inhibitory firing rates are denoted, respectively, vE and vI. The system then reads: 10vE' = – vE + [1.25vE – vI + 10]+ IvI' = – vI + [ vE – 10]+ The function [x]+ denotes the positive part of x: [x]+ = x if x > 0 [x]+ = 0 if x 0 Note that the use of the positive-part function makes the system non-linear, and indeed it is the only nonlinearity in the system. With a single parameter, the system exhibits a limited range of behaviors. Of interest is the bifurcation from a stable-equilibrium behavior (attracting spiral point) to a limit-cycle (periodic attractor) behavior, which occurs when I becomes larger than 40 ms. T = 0.25/10 – 1/I D = 0.75/10I With parameter I = 20: With parameter I = 60: