2011/07/14 by John Burke, Burke, John, Mathieu Desroches +8
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Biological sciences #FOS: Mathematics #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #Nonlinear Dynamics and Pattern Formation #math.DS #q-bio.NC #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1107.2834
arxiv created 2011/07/14 · openalex publication_date 2011/07/14 · arxiv updated 2011/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Rapid action potential generation --- spiking --- and alternating intervals of spiking and quiescence --- bursting --- are two dynamic patterns observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which was originally observed in a simple biophysical model of a Purkinje cell. In this article, we expand on that original result by showing that torus canards arise in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.