Describing Functions and Phase Response Curves of Excitable Systems

TL;DR

Classical describing function (DF) and phase response curve (PRC) analyses assume oscillatory, harmonic dynamics that poorly describe excitable neurons. The paper introduces an event-based extension with the event describing function (eDF) and event phase response curve (ePRC), defined on discrete spike trains implemented via Dirac impulses, and demonstrates their use on Hodgkin–Huxley neurons. It shows how 1:1 event-locked behavior enables predicting ring-network oscillations (e.g., half-center oscillators) and elucidates entrainment properties under periodic perturbations; parameter shifts such as $\\tau_{decay}$ and $g_{syn}$ modulate timing without eliminating qualitative predictions. This framework provides practical tools for designing and controlling central pattern generators and neuromorphic circuits built from excitable components.

Abstract

The describing function (DF) and phase response curve (PRC) are classical tools for the analysis of feedback oscillations and rhythmic behaviors, widely used across control engineering, biology, and neuroscience. These tools are known to have limitations in networks of relaxation oscillators and excitable systems. For this reason, the paper proposes a novel approach tailored to excitable systems. Our analysis focuses on the discrete-event operator mapping input trains of events to output trains of events. The methodology is illustrated on the excitability model of Hodgkin-Huxley. The proposed framework provides a basis for designing and analyzing central pattern generators in networks of excitable neurons, with direct relevance to neuromorphic control and neurophysiology.

Figures (6)

• Figure 1: Excitable neuronal circuit and discrete-event model of fundamental excitable node.
• Figure 2: eDF simulation samples ($\tau_{decay}=1$). The onset is significantly longer for inhibitory than for excitatory nodes.
• Figure 3: Absolute onset curves of an excitatory (E) and inhibitory (I) excitable node, and synaptic parameter variations of the inhibitory node.
• Figure 4: Event Describing Functions (eDF): relative event onset in function of input period enables graphical prediction of network oscillation period and existence.
• Figure 5: ePRC simulation for a baseline inhibitory rebound spike.