Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The neuron as a Lie group germ and a Lie product

Author: William C. Hoffman
Journal: Quart. Appl. Math. 25 (1968), 423-440
DOI: https://doi.org/10.1090/qam/99871
MathSciNet review: QAM99871
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Abstract | References | Additional Information

Abstract: A basic Lie group derived earlier that describes the ``constancies'' of visual perception is extended to one more dimension to take into account the nonlinear flow of an afferent volley of nerve impulses through the layers of the visual cortex. One is then led, via the usual determination of the solutions of a Lagrange partial differential equation in terms of an associated Pfaffian system of ordinary differential equations, to a correspondence between neuron cell body, Lie group germ, and critical point of the system of ordinary differential equations governing the orbits. The local phase portraits of the latter bear a marked resemblance to one or the other of the neuron types defined by Sholl. Since ``brains are as different as faces", the concept of structural stability plays an important role in analyzing the connectivity of the neural network. Finally, Lukasiewicz's theory of parentheses is used to obtain a graph-theoretic representation of the Jacobi identity, which then serves to explain the branching of neuronal processes (dendrites).

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Additional Information

DOI: https://doi.org/10.1090/qam/99871
Article copyright: © Copyright 1968 American Mathematical Society

American Mathematical Society