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

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Abstract

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 interval mappings, we were 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.

Highlights

► Elliptic burster models are studied via reduction to equationless Poincaré mappings. ► The approach is applicable to elliptic bursters and other electrochemical systems. ► Transitions between tonic spiking, mixed mode oscillations and bursting are examined. ► Emergence and bifurcations of torus are reported in such elliptic slow–fast models.

Section snippets

Pointwise Poincaré mappings and elliptic bursting models

Activity types of isolated neurons and their models may be generically classified as hyper- and depolarizing quiescence, sub-threshold 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

FitzHugh–Nagumo–Rinzel model

The mathematical FitzHugh–Nagumo–Rinzel model of the elliptic burster is given by the following system of equations with a single cubic nonlinear term: v=vv3/3w+y+I,w=δ(0.7+v0.8w),y=μ(cyv); here we fix δ=0.08,I=0.3125 an applied external current, and μ=0.002 is a small parameter determining the pace of the slow y variable. The slow variable, y, becomes frozen in the singular limit, μ=0. We employ c as the primary bifurcation parameter of the model, variations of which elevate/lower the

Voltage interval mappings

Methods of the global bifurcation theory are organically suited for examinations of recurrent dynamics such as tonic spiking, bursting and subthreshold oscillations [41], [48], [49], as well as their transformations. The core of the method is a reduction to and derivation of a low dimensional Poincaré return mapping with an accompanying analysis of the limit solutions; fixed, periodic and homoclinic orbits each representing various oscillations in the original model. Mappings have been actively

Qualitative analysis of mappings

The family of mappings, given in Fig. 8, allows for global evolutionary tendencies of the model (1) to be qualitatively analyzed. One can first see the flat mappings in grey have a single fixed point corresponding to the tonic spiking state. We can further deduce the saddle–node bifurcation that gives birth to the two unstable fixed points, at the mapping and bisectrix crossing. The fixed points diverge from each other and one fixed point moves towards the stable fixed point in the upper

Quantitative features of mappings: kneadings

In this section we discuss quantitative properties of the interval mappings for the dynamics of the model (1). In particular, we carry out the examination of complex dynamics with the use of calculus-based and calculus-free tools such as Lyapunov exponents and kneading invariants for the symbolic description of MMOs.

Chaos may be quantitatively measured by a Lyapunov exponent. The Lyapunov exponent is evaluated for one-dimensional mappings as follows: λ=limN+1Ni=1Nlog|T(vi)|, where T(vi) is

Discussion

We present a case study for an in-depth examination of the bifurcations that take place at activity transitions between tonic spiking, bursting and mixed mode oscillations in the FitzHugh–Nagumo–Rinzel model. The analysis is accomplished through the reduction to a single-parameter family of equationless Poincaré return mappings for an interval of the “voltage” variable. We stress that these mappings are models themselves for evaluating the complex dynamics of the full three-dimensional model.

Acknowledgments

We would like to thank Paul Channell, Alex Neiman, Bryce Chung and Summer (Xia) Hu for useful comments. This work was supported in part by the GSU Brains and Behavior program, RFFI grant No. 050100558, “Grant opportunities for Russian scientists living abroad” project #14.740.11.0919, and NSF grant DSM 1009591.

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