Abstract
In this paper we consider the dynamics of a recurrent analogue neural network consisting of identical compartmental model neurons (Rall,1964). We show how the associated set of ordinary differential equations can be reduced to a much smaller set of Volterra integro-differential equations in which the state of each neuron is represented by a single scalar variable. The kernel of the Volterra equations is of the convolution type, and is determined by the single neuron response function. Each neuron in the network effectively performs a temporal summation over all previous inputs to that neuron as determined by the convolution integral. Thus one can consider the network as having infinite or continuously distributed delays. The reduction presented here provides a compact and analytically tractable way of incorporating dendritic structure into neural network models. It should be noted that Poggio and Torre (1977) have also considered the representation of dendritic structure in terms of functional equations, however their analysis is purely at the single neuron level.
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References
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© 1994 Springer-Verlag London Limited
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Bressloff, P.C. (1994). Integral equations in compartmental model neurodynamics. In: Marinaro, M., Morasso, P.G. (eds) ICANN ’94. ICANN 1994. Springer, London. https://doi.org/10.1007/978-1-4471-2097-1_37
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DOI: https://doi.org/10.1007/978-1-4471-2097-1_37
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Print ISBN: 978-3-540-19887-1
Online ISBN: 978-1-4471-2097-1
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