Retrieval-time properties of the Little-Hopfield model and their physiological relevance

RISAU GUSMAN, SEBASTIAN; IDIART, MARCO A. P.

PHYSICAL REVIEW E - STATISTICAL PHYSICS, PLASMAS, FLUIDS AND RELATED INTERDISCIPLINARY TOPICS

American Physical Society

Año: 2005 vol. 72 p. 419131 - 419137

1063-651X

We perform an extensive numerical investigation on the retrieval dynamics of the synchronous Hopfield model, also known as Little-Hopfield model, up to sizes of  $2^{18}$  neurons. Our results correct and extend much of the early simulations on the model. We find that the average convergence time has a power law behavior for a wide range of system sizes, whose exponent depends both on the network loading and the initial overlap with the memory to be retrieved. Surprisingly, we also find that the variance of the convergence time grows as fast as its average, making it a non-self-averaging quantity. Based on the simulation data we differentiate between two definitions for memory retrieval time, one that is mathematically strict, $\tau_c$, the number of updates needed to reach the attractor whose properties we just described, and a second definition correspondent to the time, $\tau_{\eta}$, when the network stabilizes within a tolerance threshold $\eta$ such that the difference of two consecutive overlaps with a stored memory is smaller than $\eta$. We show that the scaling relationships between $\tau_c$ and $\tau_{\eta}$ and the typical network parameters as the memory load $\alpha$ or the size of the network N vary greatly, being $\tau_{\eta}$  relatively insensitive to system sizes and loading. We propose $\tau{\eta}$ as the physiological realistic measure for the typical attractor network response.