Dirac distributions and threshold firing in neural networks
type: Item
instance of: scholarly article
title: Dirac distributions and threshold firing in neural networks
zbMATH Open document ID: 0729.92006
DOI: 10.1016/0893-9659(89)90103-1
author: Thomas L. Saaty
author: Luis Vargas
published in: Applied Mathematics Letters
publication date: 1989-01-01T00:00:00Z
review text: We extend our earlier work on positive reciprocal kernels of Fredholm integral operators [J. Math. Psychol. 31, 83-92 (1987; Zbl 0609.92038)] to study firings and their synthesis in neural networks. First we show that, in general, neural response in both spontaneous and nonspontaneous firing give rise to generalized functions of the Dirac type. For spontaneous unstimulated firing we solve a homogeneous eigenvalue equation and obtain a family of gamma type functions. Finite linear combinations of these functions are dense in Sobolev spaces. The solution of the inhomogeneous equation representing nonspontaneous firing belongs to these spaces. Next we show that according to known facts about neural networks, the forcing function of the inhomogeneous equation is a linear combination of the above functions and can be used to represent the synthesis of stimuli within a neuron causing it to fire. We also show that the solution of the inhomogeneous equation can be expressed as a linear combination of the basic functions describing the neural response to those stimuli. The need for a firing threshold, characteristic of the Dirac distribution, emerges as a necessary condition for the existence of a solution. Second, we study the synthesis of the response of several neurons in both hierarchic and feedback network arrangements. The analysis is then briefly generalized to examine response to several stimuli and represent it as a direct sum of topological spaces. One observation is that generalized functions are appropriate representations of neural firings. Another is that understanding the structure of this representation is facilitated by the inevitable use of a fundamental set of dense functions to deal with the operations of a very complex system.
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 46N60
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 4207816
zbMATH Keywords: membrane potential
zbMATH Keywords: positive reciprocal kernels of Fredholm integral operators
zbMATH Keywords: neural networks
zbMATH Keywords: homogeneous eigenvalue equation
zbMATH Keywords: gamma type functions
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: firing threshold
zbMATH Keywords: Dirac distribution
zbMATH Keywords: neural firings
MaRDI profile type: Publication
full work available at URL: 0893-9659(89)90103-1
OpenAlex ID: W1966981451
cites work: Q3250680
cites work: Q3857069
cites work: Q4693168
cites work: Q3044280
cites work: Stimulus-response with reciprocal kernels: The rise and fall of sensation
cites work: Q4045174
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description: scientific article; zbMATH DE number 4207816