Fixed Points in Two--Neuron Discrete Time Recurrent Networks: Stability and Bifurcation Considerations

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The position, number and stability types of fixed points of a two--neuron recurrent network with nonzero weights are investigated. Using simple geometrical arguments in the space of derivatives of the sigmoid transfer function with respect to the weighted sum of neuron inputs, we partition the network state space into several regions corresponding to stability types of the fixed points. If the neurons have the same mutual interaction pattern, i.e. they either mutually inhibit or mutually excite themselves, a lower bound on the rate of convergence of the attractive fixed points towards the saturation values, as the absolute values of weights on the self--loops grow, is given. The role of weights in location of fixed points is explored through an intuitively appealing characterization of neurons according to their inhibition/excitation performance in the network. In particular, each neuron can be of one of the four types: greedy, enthusiastic, altruistic or depressed. Both with and without the external inhibition/excitation sources, we investigate the position and number of fixed points according to character of the neurons. When both neurons self-excite (or self-inhibit) themselves and have the same mutual interaction pattern, the mechanism of creation of a new attractive fixed point is shown to be that of saddle node bifurcation. (Also cross-referenced as UMIACS-TR-95-51)