Fixed Points in Two--Neuron Discrete Time Recurrent Networks:
Stability and Bifurcation Considerations
Fixed Points in Two--Neuron Discrete Time Recurrent Networks:
Stability and Bifurcation Considerations
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Date
1998-10-15
Authors
Tino, Peter
Horne, Bill G.
Giles, C. Lee
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Abstract
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)