Derivation of Hebb's rule

TL;DR

This paper derives a biologically plausible learning rule consistent with Hebb's postulate, demonstrating convergence to pattern-storing weights and linking biological and mathematical solutions for neural network learning.

Contribution

It introduces a new local learning rule that aligns with Hebb's principle and converges to mathematically optimal weights for pattern storage.

Findings

Weights converge to finite values with repeated pattern presentation

Final weights match those obtained via a modified pseudo-inverse method

Biological learning rule achieves mathematically optimal pattern storage

Abstract

On the basis of the general form for the energy needed to adapt the connection strengths of a network in which learning takes place, a local learning rule is found for the changes of the weights. This biologically realizable learning rule turns out to comply with Hebb's neuro-physiological postulate, but is not of the form of any of the learning rules proposed in the literature. It is shown that, if a finite set of the same patterns is presented over and over again to the network, the weights of the synapses converge to finite values. Furthermore, it is proved that the final values found in this biologically realizable limit are the same as those found via a mathematical approach to the problem of finding the weights of a partially connected neural network that can store a collection of patterns. The mathematical solution is obtained via a modified version of the so-called method of the pseudo-inverse, and has the inverse of a reduced correlation matrix, rather than the usual correlation matrix, as its basic ingredient. Thus, a biological network might realize the final results of the mathematician by the energetically economic rule for the adaption of the synapses found in this article.