Existence and Stability of Standing Pulses in Neural Networks : I Existence

Details Existence and Stability of Standing Pulses in Neural Networks : I Existence Existence and Stability of Standing Pulses in Neural Networks : I Existence

2004-07-08

arxiv.org/abs/q-bio/0407013v1

Biology

Quantitative Biology

Neurons and Cognition

31 pages, 29 figures, submitted to SIAM Journal on Applied Dynamical Systems

Scientific paper

10.1137/040609483

We consider the existence of standing pulse solutions of a neural network integro-differential equation. These pulses are bistable with the zero state and may be an analogue for short term memory in the brain. The network consists of a single-layer of neurons synaptically connected by lateral inhibition. Our work extends the classic Amari result by considering a non-saturating gain function. We consider a specific connectivity function where the existence conditions for single-pulses can be reduced to the solution of an algebraic system. In addition to the two localized pulse solutions found by Amari, we find that three or more pulses can coexist. We also show the existence of nonconvex ``dimpled'' pulses and double pulses. We map out the pulse shapes and maximum firing rates for different connection weights and gain functions.

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