Collective dynamics of adaptive memristor synapse-cascaded neural networks based on energy flow

The collective dynamics regulated by the field energy has garnered significant attention at the level of biological neuron networks, but this has not yet been involved in the brain-like functional regions represented by the Hopfield neural network (HNN). To realize the interaction between electric and magnetic fields, the magnetic-flux controlled memristor is used as a bionic synapse to connect the three-neuron-based HNN. Three topological brain-like functional region networks, including the memristor-cascaded dual HNN, chain HNN and lattice HNN, are systematically proposed. Firstly, the generalized Lyapunov energy function of the single HNN is constructed based on the electric field energy to establish an energy evaluation framework. Under the premise of energy boundedness, the intricate firing activities and energy evolution within the single HNN are revealed through two-dimensional (2-D) dynamical distributions and 1-D bifurcation diagrams. Significantly, to simulate the plasticity and self-controllability of the synapse, an adaptive regulation scheme for the memristor synapse controlled by the energy difference between adjacent HNNs is proposed. When the energy flow drives the memristor synapse to adaptively activate, grow and saturate, the collective behaviors of three HNNs are visualized through spatiotemporal patterns and local curves. The study reveals that the adaptive regulation mechanism of the memristor synapse enables the homogeneous HNNs to adaptively achieve energy balance and complete synchronization. When considering the difference and neuronal deformation of local functional regions, heterogeneous HNNs exhibit rapid accumulation or release of energy within the heterogeneous region, while maintaining energy balance outside this region, thereby forming diverse gradient distributions regarding energy, the coupling strength of memristor synapses, and membrane potential. This study demonstrates the reliability of energy flow-based adaptive memristor synapses, thereby enhancing the controllability of neural networks and deepening the understanding of the operating mechanism of brain-like functional region networks.