Discrete second-order locally active memristor-based neuron map with grid-like attractor coexistence

Higher-order memristors offer a compact approach to representing multiple internal memory states in neuronal models. In this work, a discrete second-order locally active memristor (SO-LAM), featuring two internal states and a bounded periodic memductance, is integrated into the Aihara neuron map, yielding a memristive neuron with two history-dependent internal states. The membrane potential drives the evolution of both memristive states, while the two states jointly determine the memductance and feedback current. The discrete SO-LAM exhibits memristive characteristics, local activity, and edge-of-chaos regions. Through the combined effects of neuronal nonlinearity and memristive coupling, the SO-LAM-based neuron map generates regular and complex firing dynamics. The periodicity of the two memristive states gives rise to two-dimensional translational equivariance, indicating that the coexisting attractors in the grid-like array are spatially shifted copies of the same fundamental attractor rather than dynamically independent states. Representative hardware trajectories corresponding to the numerically identified hyperchaotic regime and symmetry-related attractor copies are reproduced using a field-programmable gate array (FPGA), demonstrating the feasibility of digital implementation. These results provide a compact framework for studying multistate history-dependent modulation and symmetry-induced coexistence in discrete memristive neurons.