Temporal Self-Organization: A Reaction–Diffusion Framework for Spatiotemporal Memories

Self-organizing maps (SOMs) find numerous applications in learning, clustering, and recalling spatial input patterns. The traditional approach in learning spatiotemporal patterns is to incorporate time on the output space of a SOM along with heuristic update rules that work well in practice. Inspired by the pioneering work of Alan Turing, who used reaction–diffusion equations to explain spatial pattern formation, we develop an analogous theoretical model for a spatiotemporal memory to learn and recall temporal patterns. The contribution of the paper is threefold: 1) using coupled reaction–diffusion equations, we develop a theory from first principles for constructing a spatiotemporal SOM and derive an update rule for learning based on the gradient of a potential function; 2) we analyze the dynamics of our algorithm and derive conditions for optimally setting the model parameters; and 3) we mathematically quantify the
temporal plasticity
effect observed during recall in response to the input dynamics. The simulation results show that the proposed algorithm outperforms the SOM with temporal activity diffusion, neural gas with temporal activity diffusion and spatiotemporal map formation based on a potential function in the presence of correlated noise for the same data set and similar training conditions.