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. Our findings include the following: (1) The phase response during the active segment of the burst displays considerable sensitivity (even for very weak perturbations) closely associated with spike times in the perturbed burster. (2) The isochron geometry of the fast subsystem explains the shape and sensitivity of the recorded phase response during the active bursting segment. (3) The onset of the homoclinic bifurcation in the fast subsystem during the burst cycle strongly affects the isochron geometry. (4) Strong perturbations may trigger changes in spike number per burst, sometimes accompanied by early burst initiation or termination, leading to large magnitude phase responses that cannot be accounted for using the theory of weakly coupled oscillators.