Kyle C. A. Wedgwood,

Neurons have long been known to exhibit periodic oscillations in their membrane potential when exposed to constant stimuli. Such oscillations are key to the function of many neural systems, for example in the supra chiasmatic nucleus, which produces circadian rhythms, and the pre-Bötzinger complex, responsible for regulating breathing. If these rhythms are disrupted, say be changing time zone, the oscillatory response will often be preserved but may be shifted in phase. This phase shift is dependent on the phase at which the perturbation occurs.

Under the assumption of weak perturbations and strongly attracting limit-cycles, the use of the phase response curve, which relates the phase shift to the phase of the perturbation allows for dimension reduction in neural networks. It has, for example, been used to study animal gaits and swimming in tadpoles, where further simplication can be made through the use of equivariant bifurcation theory.

In general, we do not expect perturbations in a biological context to be weak, nor do we expect instantaneous relaxtion back to the underlying oscillation. In many situations the reduced system belies the true nature of the original one. Through case studies of a common neural model, we investigate scenarios in which this is true and try to develop a framework that can overcome this using phase-amplitude coordinates.