Establishing a realistic and multiplier-free implemented biological neuron model is significant for recognizing and understanding natural firing behaviors, as well as advancing the integration of neuromorphic circuits. Importantly, memristors play a crucial role in constructing memristive neuron and network models by simulating synapses or electromagnetic induction. However, existing models lack the consideration of initial-boosted extreme multistability and its associated energy analysis. To this end, we propose a multiplier-free implementation of the Rulkov neuron model and utilize a periodic memristor to represent the electromagnetic induction effect, thereby achieving the biomimetic modeling of the non-autonomous memristive Rulkov (mRulkov) neuron. First, theoretical analysis demonstrates that the stability distribution of the time-varying line equilibrium point is determined by both the parameters and the memristor's initial condition. Furthermore, numerical simulations show that the mRulkov neuron can exhibit parameter-dependent local spiking, local hidden spiking, and periodic bursting firing behaviors. In addition, based on the periodic characteristics of the memductance function, the topological invariance of the mRulkov neuron is comprehensively proved. Therefore, local basins of attraction, bifurcation diagrams, and attractors related to extreme multistability can be boosted by switching the memristor's initial condition. Significantly, the novel boosted extreme multistability is discovered in the Rulkov neuron for the first time. More importantly, the energy transition associated with the boosting dynamics is revealed through computing the Hamilton energy distribution. Finally, we develop a simulation circuit for the non-autonomous mRulkov neuron and confirm the effectiveness of the multiplier-free implementation and the accuracy of the numerical results through PSpice simulations.
