Abstract
Rapid action potential generation - spiking - and alternating intervals of spiking and quiescence - bursting - are two dynamic patterns commonly 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 we show arises 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.
Highlights
The primary unit of brain electrical activity - the neuron - generates a characteristic dynamic behavior: when excited sufficiently, a rapid increase decrease in the neuronal voltage occurs, see for example [1]
As k increases above the torus bifurcation, the system exhibits amplitude modulated (AM) spiking as the trajectory winds around the torus near the saddle-node of periodic orbits of the fast system
Torus canards were originally identified in a fifth order model of a Purkinje cell [21], where it was shown that the torus canard explosion occurs within the transition region between tonic spiking and bursting
Summary
The primary unit of brain electrical activity - the neuron - generates a characteristic dynamic behavior: when excited sufficiently, a rapid (on the order of milliseconds) increase decrease in the neuronal voltage occurs, see for example [1]. Recent research has led to a number of classification schemes of bursting, including a scheme by Izhikevich [7] based on the bifurcations that support the onset and termination of the burst’s active phase. We show that they arise in well-known neuronal models exhibiting three different classes of bursting: subHopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting These models are all third order dynamical systems with two fast and one slow variable. We propose that torus canard explosions are a commonly-occurring transition mechanism from spiking to bursting in neuronal models The organization of this manuscript is as follows. Bursting trajectories and torus canards are found using direct numerical simulations with a stiff-solver suited to multiple time scale systems, starting from arbitrary initial conditions, and we disregard transients in the figures
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