The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

A synthesis is presented of two recent studies on modelling the nonlinear neuro-mechanical hearing processes in mosquitoes and in mammals. In each case, a hierarchy of models is considered in attempts to understand data that shows nonlinear amplification and compression of incoming sound signals. The insect’s hearing is tuned to the vicinity of a single input frequency. Nonlinear response occurs via an arrangement of many dual capacity neuro-mechanical units called scolopidia within the Johnston’s organ. It is shown how the observed data can be captured by a simple nonlinear oscillator model that is derived from homogenization of a more complex model involving a radial array of scolopidia. The physiology of the mammalian cochlea is much more complex, with hearing occurring via a travelling wave along a tapered, compartmentalized tube. Waves travel a frequency-dependent distance along the tube, at which point they are amplified and “heard”. Local models are reviewed for the pickup mechanism, within the outer hair cells of the organ of Corti. The current debate in the literature is elucidated, on the relative importance of two possible nonlinear mechanisms: active hair bundles and somatic motility. It is argued that the best experimental agreement can be found when the nonlinear terms include longitudinal coupling, the physiological basis of which is described. A discussion section summarizes the lessons learnt from both studies and attempts to shed light on the more general question of what constitutes a good mathematical model of a complex physiological process.

A growing body of literature testifies to the importance of quantitative reasoning skillsin the 21st-century biology curriculum, and to the learning benefits associated withactive pedagogies. The process of modeling a biological system provides an approach thatintegrates mathematical skills and higher-order thinking with existing course contentknowledge. We describe a general strategy for teaching model-building in an introductorybiology course, using the example of a model of an infectious disease outbreak.Preliminary assessment data suggest that working through the formal process of modelconstruction may help students develop their scientific reasoning and communicationskills.