This research article introduces a novel switchable memristor that can be configured as a nonvolatile discrete memristor, a nonvolatile continuum memristor, or a volatile memristor by adjusting its internal parameter. This memristor is used to simulate the autapse of the Hindmarsh-Rose (HR) neuron, a model known for its complex bifurcation phenomena and diverse firing patterns. Additionally, a flux-controlled memristor is introduced to model the effect of external electromagnetic radiation on the HR neuron, leading to the development of an improved 4D HR neuron model without equilibrium points. The study reveals hidden firing activities related to the strength of autapse and electromagnetic radiation intensity through various nonlinear analysis methods, including phase diagrams, time series, bifurcation diagrams, Lyapunov exponent spectra, and two-parameter dynamical maps. The results show that the memory attributes of the memristive autapse play a crucial role in neuronal firing activities, inducing transitions between periodic spiking with different frequencies and chaotic firing. Furthermore, the transition between periodic and chaotic firing is observed when the switchable memristor is configured as three different types of memristors. A hardware circuit using current-mode devices is designed to improve accuracy and reduce power consumption. Multisim simulation results validate the correctness of the neuron model and the effectiveness of the numerical analysis. The study concludes that the firing activity of neurons can be more accurately simulated by considering the memory switching of memristive synapses, highlighting the potential for further research on the impact of different memory attributes on collective firing behavior.This research article introduces a novel switchable memristor that can be configured as a nonvolatile discrete memristor, a nonvolatile continuum memristor, or a volatile memristor by adjusting its internal parameter. This memristor is used to simulate the autapse of the Hindmarsh-Rose (HR) neuron, a model known for its complex bifurcation phenomena and diverse firing patterns. Additionally, a flux-controlled memristor is introduced to model the effect of external electromagnetic radiation on the HR neuron, leading to the development of an improved 4D HR neuron model without equilibrium points. The study reveals hidden firing activities related to the strength of autapse and electromagnetic radiation intensity through various nonlinear analysis methods, including phase diagrams, time series, bifurcation diagrams, Lyapunov exponent spectra, and two-parameter dynamical maps. The results show that the memory attributes of the memristive autapse play a crucial role in neuronal firing activities, inducing transitions between periodic spiking with different frequencies and chaotic firing. Furthermore, the transition between periodic and chaotic firing is observed when the switchable memristor is configured as three different types of memristors. A hardware circuit using current-mode devices is designed to improve accuracy and reduce power consumption. Multisim simulation results validate the correctness of the neuron model and the effectiveness of the numerical analysis. The study concludes that the firing activity of neurons can be more accurately simulated by considering the memory switching of memristive synapses, highlighting the potential for further research on the impact of different memory attributes on collective firing behavior.