Voltage interval mappings for activity transitions in neuron models for elliptic bursters

We performed a thorough bifurcation analysis of a mathematical elliptic bursting model, using a computer-assisted reduction to equationless, one-dimensional Poincaré mappings for a voltage interval. Using the intervalmappings, wewere able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally mixed-mode oscillations and quiescence in the FitzHugh–Nagumo–Rinzel model. We illustrate the wealth of information, qualitative and quantitative, that was derived from the Poincaré mappings, for the neuronal models and for similar (electro)chemical systems. © 2011 Elsevier B.V. All rights reserved. 1. Pointwise Poincaré mappings and elliptic bursting models Activity types of isolated neurons and their models may be generically classified as hyperand depolarizing quiescence, subthreshold and mixed mode oscillations, endogenous tonic spiking and bursting. Bursting is an example of composite, recurrent dynamics comprised of alternating periods of tonic spiking oscillations and quiescence. The type of bursting in which tonic spiking oscillations alternate with sub-threshold oscillations is often referred to as Mixed Mode Oscillations (MMO). Various endogenous bursting patterns are the natural behavior rhythms generated by central pattern generators (CPG) [1]. A CPG is a neural network, or a mini circuit, controlling various vital repetitive locomotive functions of animals and humans [2]. We contend that understanding all plausible transitions of the activity patterns of individual neuron models would allow for better understanding of networked models. In this study we elaborate on the transition mechanisms by revealing the underlying bifurcations between neuronal activities on the elliptic bursting models of (inter) neurons which are used as the building blocks in the CPG circuitry. Bursting represents direct evidence of multiple time scale dynamics of a neuron. Deterministic modeling of bursting neurons ∗ Corresponding author. E-mail address: [email protected] (A. Shilnikov). 0167-2789/$ – see front matter© 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2011.04.003 was originally proposed and done within a framework of threedimensional, slow–fast dynamical systems. Geometric configurations of models of bursting neurons were pioneered by Rinzel [3,4] and enhanced in [5–8]. The proposed configurations are all based on the geometrically comprehensive dissection approach, or the time scale separation which have become the primary tools in mathematical neuroscience. The topology of the slow motion manifolds is essential to the geometric understanding of neurodynamics. Through the use of geometric methods of the slow–fast dissection, where the slowest variable of the model is treated as a control parameter, it is possible to detect and follow the manifolds made of branches of equilibria and limit cycles in the fast subsystem. Dynamics of a slow–fast system are determined by, and centered around, the attracting sections of the slowmotion manifolds [9–16]. The slow–fast dissection approachworks exceptionally well for a multiple time scale model, provided the model is far from a bifurcation in the singular limit. On the other hand, a bifurcation describing a transition between neuron activities may occur from reciprocal interactions involving the slow and fast dynamics of the model. Such slow–fast interactions may lead to the emergence of distinct dynamical phenomena and bifurcations that can occur only in the full model, not in either subsystem of the model. As such, the slow–fast dissection fails at the transition where the solution is no longer constrained to stay near the slow motion manifold, or when the time scale of the dynamics of the fast subsystem slows to that of the slow system, near the homoclinic and saddle node bifurcations, for example. J. Wojcik, A. Shilnikov / Physica D 240 (2011) 1164–118