Effect of external excitation on the isolated and collective dynamics of a generic FitzHugh-Rinzel neuron

This contribution considers the isolated and collective dynamics of a FitzHugh-Rinzel (FHR) neuron obtained by adding a third variable to the generic FitzHugh-Nagumo neural circuit. From the Kirchhoff electrical circuit laws, the state equations of the model are derived; afterward, the stability around a zero time constant is investigated, and the hidden dynamics of the model is revealed. Analytical and theoretical investigation of the energy is done to support the various firing activities, such as bursting and spiking, captured in the model. Using the well-known modulation instability theory, the collective behavior of a network made of 50 neurons is analyzed in a chain configuration. As a result, regular patterns consisting of alternate bright and dark bands that are almost periodic and localized in space and time are found. Also, the brighter regions correspond to the regions where the neurons fire, while in the dark regions, the neurons are quiescent. Finally, the brighter regions could be pictured as individual spikes within a burst.

Automated summary

This article studies the isolated and collective dynamics of a FitzHugh-Rinzel neuron by adding a third variable to the generic FitzHugh-Nagumo neural circuit. It investigates the stability around a zero time constant and reveals the hidden dynamics of the model. The energy is analytically and theoretically investigated to support the various firing activities captured in the model, and the collective behavior of a network made of 50 neurons is analyzed using modulation instability theory.