From Spiking to Bursting: A Third-Order Piecewise-Linear Neuron with Adaptive Dynamics

This work presents a novel third-order piece-wise linear neuron model (P3FAN) that can replicate a broad spectrum of dynamical behaviours, such as bursting, tonic spiking, low-frequency spiking, and plateau potentials. The proposed model enables precise characterization of the temporal features associated with each oscillatory regime, while remaining both analytically and computationally tractable. Analytical results are first rigorously drawn and then validated through numerical simulations. The approach provides a versatile and efficient framework for the analysis and design of neural-inspired control systems.