Analytical conditions for synchronization for a simple memristive neural network

The memristor manifests qualities and behaviors naturally observed in biological synapses. It exists at the nanoscale, consumes very little power, is ideally suited for parallel processing, may process and store information, and exhibits a fluxdependent conductivity. Further, the memristor may reproduce rules governing the process of neural learning, including the Hebbian rule, Spike-Time-Dependent-Plasticity and Spike-Rate-Dependent- Plasticity. Hence memristive nano-structures are suitable candidates for the design of core elements of novel neuromorphic systems. It is then of interest to investigate models of memristive neural networks in order to gain some insight into the mechanisms underlying the complex dynamics emerging in biological systems. This manuscript presents an analytical treatment, based on the contraction mapping theorem, which allows the derivation of analytical expressions for the conditions of synchronization in a simple memristive neural network consisting of two Hindmarsh-Rose neurons interacting through a unidirectional memristive coupling arrangement. The results of the study confirm the conclusions from recent numerical investigations which revealed the strong impact of the dynamics occurring on the memristive coupling path on the emergence of synchrononization among the neuron oscillators.