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J. Phys. A: Math. Gen. 22 (1989) 2031-2038. Printed in the UK Universality in the space of interactions for network models
 

Summary: J. Phys. A: Math. Gen. 22 (1989) 2031-2038. Printed in the UK
Universality in the space of interactions for network models
L F Abbott and Thomas B Kepler
Physics Department, Brandeis University, Waltham, MA 02254, USA
Received 26 January 1989
Abstract. By modifying the measure used to sum over coupling matrices, we generalise
Gardner's calculation of the fractional interaction-space volume and storage capacity of
neural network models. We also compute the local field distribution for the network. The
generalised measure allows us to consider networks with a wide variety of properties away
from saturation, but we find that the original results for saturated networks are universal
for all well behaved measures. Other universality classes including those containing Hebb
matrices and pseudo-inverse matrices are obtained by considering singular measures.
1. Introduction
One of the most impressive and imaginative of Elizabeth Gardner's many contributions
to neural network research was her analysis [11of the space of interactions for network
models. This pioneering work allows us to compute several important properties of
neural network memories in a model-independent way. Consider an N-node network
designed to store and recall ON uncorrelated patterns .$' ( i = 1,. .. ,N, p = 1,. ..,a N )
using the couplings J,,. Define
Gardner computed the fractional volume in the space of all coupling matrices occupied

  

Source: Abbott, Laurence - Center for Neurobiology and Behavior & Department of Physiology and Cellular Biophysics, Columbia University

 

Collections: Biology and Medicine