On transition to bursting via deterministic chaos

We study statistical properties of the irregular bursting arising in a class of neuronal models close to the transition from spiking to bursting. Prior to the transition to bursting, the systems in this class develop chaotic attractors, which generate irregular spiking. The chaotic spiking gives rise to irregular bursting. The duration of bursts near the transition can be very long. We describe the statistics of the number of spikes and the interspike interval distributions within one burst as functions of the distance from criticality. Bursting oscillations are ubiquitous in the experimental and modeling studies of excitable cell membranes. Many models generating bursting have been subject to intensive research due to their physiological significance and dynamical complexity (see [1, 5, 7, 9, 10, 13, 15] and references therein). Under the variation of parameters even minimal 3D models of bursting neurons exhibit a rich variety of periodic and aperiodic dynamical patterns corresponding to different spiking and bursting regimes. The transitions between these patterns may contain complex dynamical structures such as period-doubling cascades and deterministic chaos. In particular, it was shown that the transition from tonic spiking to bursting in a class of bursting neuron models, so-called square-wave bursters, contains windows of chaotic dynamics [1, 2, 10, 15]. In view of the complex bifurcation structure of this class of problems, it is important to identify the universal features pertinent to different dynamical patterns and transitions between them. In the present Letter, we describe statistical features of the irregular bursting arising in a class of neuronal models close to the transition from spiking to bursting. Prior to transition to bursting, the systems in this class develop chaotic attractors, which generate irregular spiking. The chaotic spiking gives rise to irregular bursting. The duration of bursts near the transition can be extremely long (see Figure 1). We analyze the statistics of the number of spikes and the interspike intervals within one burst as functions of the distance from criticality. To describe our results, we use a three variable model of a bursting neuron introduced in [5]. The model is based on three nonlinear conductances: persistent sodium, INaP , the delayed rectifier, IK , a slow potassium M -current, IKM , and a passive current, IL. In spite of a number of ∗Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, [email protected]