Dissecting the Phase Response of a Model Bursting Neuron

We investigate the phase response properties of the Hindmarsh-Rose model of neuronal bursting using burst phase response curves (BPRCs) computed with an infinitesimal perturbation approximation and by direct simulation of synaptic input. The resulting BPRCs have a significantly more complicated structure than the usual Type I and Type II PRCs of spiking neuronal models, and they exhibit highly timing-sensitive changes in the number of spikes per burst that lead to large magnitude phase responses. We use fast-slow dissection and isochron calculations to analyze the phase response dynamics in both weak and strong perturbation regimes.

💡 Research Summary

This paper provides a thorough investigation of the phase‑response properties of the Hindmarsh‑Rose (HR) bursting neuron model. The authors compute burst phase response curves (BPRCs) using two complementary approaches: an infinitesimal PRC (iPRC) derived from the adjoint of the linearized system, and direct numerical simulations of synaptic inputs that produce finite, strong perturbations. The iPRC analysis reveals that the HR model’s phase sensitivity is highly non‑uniform, with steep gradients near burst onset and termination and pronounced spikes in the curve when a single spike is added to or removed from a burst. Direct simulations show that strong excitatory or inhibitory inputs can dramatically alter the number of spikes within a burst, leading to phase advances or delays that are an order of magnitude larger than those predicted by the iPRC.

To interpret these findings, the authors employ a fast‑slow decomposition, separating the rapid voltage‑activation variables (x, y) from the slow recovery variable (z). The slow variable governs the overall burst period, and its crossing of a critical threshold triggers the start or end of a burst. By numerically constructing isochrons—surfaces of equal phase—in the three‑dimensional state space, the study identifies regions where small variations in z produce negligible phase changes and narrow “boundary” zones where minute changes cause a switch in spike count, manifesting as the sharp, nonlinear features observed in the BPRC.

The paper also explores the effect of timing‑specific synaptic perturbations. Inhibitory inputs delivered mid‑burst can abort the ongoing spike train and initiate a new burst, effectively resetting the phase not merely by a temporal delay but by restructuring the entire bursting pattern. This behavior underscores the importance of considering both weak (linear) and strong (non‑linear) perturbation regimes when characterizing bursting neurons.

Overall, the work extends traditional PRC theory—normally limited to Type I (monotonic) and Type II (non‑monotonic) curves for tonic spiking—to accommodate the richer dynamics of bursting. By integrating iPRC calculations, direct strong‑perturbation simulations, fast‑slow analysis, and isochron mapping, the authors present a unified framework that captures the continuous transition from linear phase shifts to large, spike‑count‑driven phase jumps. The results have implications for understanding synchronization and desynchronization mechanisms in networks of bursting neurons, for modeling pathological bursting in neurological disorders, and for designing neuromorphic circuits that exploit burst‑based coding.