Amplitude equations of associative memory patterns in spatially distributed systems

TL;DR

The resulting coupled amplitude equations describe the spatio-temporal dynamics of memory recall and show that short-range connections can sustain spatio-temporal memory pattern dynamics in the form of propagating patterning fronts.

Abstract

Evolution equations are derived for the amplitudes of associative memories: heterogeneous states stored in the connectivity of distributed systems with non-local interactions. The resulting coupled amplitude equations describe the spatio-temporal dynamics of memory recall. They capture pattern completion and selection, and show that short-range connections can sustain spatio-temporal memory pattern dynamics in the form of propagating patterning fronts. The derived amplitude equations are of the same form as those describing classical pattern-forming instabilities, indicating a universality of the dynamics of memory recall and pattern formation in non-equilibrium systems.

Figures (2)

• Figure 1: Bulk dynamics with $M=3$ stored patterns, comparing the pattern amplitudes $\hat{A}_i = \langle u(t,x), \mu_i\rangle$ obtained from simulating the full system [Eq. (\ref{['eq:du']}), grey markers], and the amplitudes obtained by numerically integrating the system of amplitude equations [Eq. (\ref{['eq:ampleq']}), black lines].
• Figure 2: Spatio-temporal profile of a pattern forming wave-front invading a spatially homogeneous domain. Colour intensity indicates the local pattern amplitude $\hat{a}_k(r,t) = 2L \left[u(r, t) \odot \mu_k(r)\right]$. The theoretical front positions $r_\pm^{(k)}(t) = r_\pm^{(k)}(0) \pm \varepsilon \sqrt{2 \alpha_{kk} D_{kk}}t$ are shown as white dashed lines.