What determines the capacity of autoassociative memories in the brain?

Abstract

Threshold-linear (graded response) units approximate the real firing behaviour of pyramidal neurons in a simplified form, suited to the analytical study of large autoassociative networks. Here we extend previous results on threshold-linear networks to a much larger class of models, by considering different connectivities (including full feedback, highly diluted and multilayer feedforward architectures), different forms of Hebbian learning rules, and different distributions of firing rates (including realistic, continuous distributions of rates). This allows an evaluation of the main factors which may affect, in real cortical networks, the capacity for storage and retrieval of discrete firing patterns.

In each case a single equation is derived, which determines both α_c, the maximum number of retrievable patterns per synapse, and I_m, the maximum amount of retrievable information per synapse. It is shown that:

• 1. Non-speeific effects, such as those usually ascribed to inhibition, or to neuramodulatary afferents, alter the overall response, but do not affect the capacity for retrieval.

• 2. The crucial parameter which affects α_c is a, the sparseness of the neural code. Results previously obtained with binary distributions of rates are shown to hold in general, namely that as a → 0 (sparse coding) α_c grows proportionally to (aln(l/a))⁻¹, while I_m depends only weakly on a.

• 3. When the coding is sparse, α_c and I_m become independent of the connectivity, and in particular the relative disadvantages of fully connected feedback networks, which are prominent with non-sparse codes, disappear.

• 4. The precise form of the distribution of rates, and that of the learning rule used, turn out to have rather limited effects on α_c and I_m. It is to be noted, however, that when non-binary patterns are stored using nonlinear learning rules, such as those possibly modelling the action of NMDA receptors, information distortion occurs in retrieval, and results in a marked decrease of I_m.

These results may be applied to help understand the organization of a specific network in the hippocampus thought to operate as an autoassociative memory.