Entropy-based Stability of Fractional Self-Organizing Maps with Different Time Scales

Abstract

The behavior of self-organizing neural maps, which develop through a combination of long and short-term memory, involves different time scales. Such a neural network's activity is characterized by a neural activity equation representing the fast phenomenon and a synaptic information efficiency equation representing the slow part of the neural system. The work reported here proposes a new method to analyze the dynamics of self-organizing maps based on the flow-invariance principle, considering the performance of the system's different time scales. In this approach, the equilibrium point is determined based on the estimate for the entropy at each iteration of the learning rule, which is generally sufficient to analyze existence and uniqueness. In this sense, the viewpoint reported here proves the existence and uniqueness of the equilibrium point on a fractional approach by using a Lyapunov method extension for Caputo derivatives when 0 < < 1. Furthermore, the global exponential stability of the equilibrium point is proven with a strict Lyapunov function for the flow of the system with different time scales and some numerical simulations.