2011/11/28 by Kevin Lin, Kevin K. Lin, Lin, Kevin K. +6
Biochemistry, Genetics and Molecular Biology · Computer Science · Neuroscience · Physics and Astronomy · #37N25 #92B25 #FOS: Biological sciences #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #Nonlinear Dynamics and Pattern Formation #msc:37N25 #msc:92B25 #q-bio.NC #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1111.6353
openalex publication_date 2011/11/28 · arxiv created 2012/01/18 · arxiv updated 2012/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are "sufficiently weak", an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of "sticky" phase-space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.