Applications of the Poincaré mapping technique to analysis of neuronal dynamics

doi:10.1016/j.neucom.2006.10.091

Neurocomputing

Volume 70, Issues 10–12, June 2007, Pages 2107-2111

https://doi.org/10.1016/j.neucom.2006.10.091 Get rights and content

Abstract

A single neuron can demonstrate different spiking and bursting patterns which can be elicited naturally depending on modulation status or artificially due to disturbances caused by distinct recording techniques. For example, when pharmacologically isolated with bicuculline a leech oscillatory heart interneuron can show an endogenous bursting activity while recorded extracellularly, or the periodic tonic spiking activity while recorded intracellularly. Transitions between these oscillatory patterns are in general non-local and could not be understood using only the local analysis of the neuron's rest states, but the global theory tools such as the Poincaré return mapping analysis. The mappings constructed then predict the temporal characteristics of the spiking and bursting patterns and allow one to study transitions between them. The technique is directly applicable to neuronal models of various types, as well as is aimed to be employed in neurophysiological experiments.

Introduction

Exploration of generic mechanisms of transitions between distinct types of neuronal activity is a fundamental task for determining the basic principles of a neuron's functioning. Commonly, neuronal networks controlling rhythmic movements (central pattern generators) produce bursting patterns of activity [4]. In systems like the leech, the central pattern generator controlling heartbeat, invariability of the pattern is important for survival. The ability of an endogenous bursting of single oscillatory interneurons brings robustness to the system versus variations of the strength of the intra-network coupling [3]. Thus, keeping the system away from the transitions could be vital for this system. Previously, we have developed a powerful averaging technique for the analysis of oscillatory spiking and bursting modes in neuronal models without their slow–fast decomposing. It allowed us to discover two new scenarios of generic transitions between bursting and tonic spiking activities [7], [6].

Here, we present a computationally more efficient technique, which is based on the measurements of the minimum values of voltage in spiking cycles and does not involve the averaging, which gives a number of advantages especially for experimental implementations.

Section snippets

Model

In this paper we employ a three-dimensional model of a pharmacologically reduced oscillatory heart interneuron [1], [7], [6]. It is given by $\dot{V} = - 2 [30 m_{K 2}^{2} (V + 0.07) + 8 (V + 0.046) + 200 f_{\infty}^{3} (- 150, 0.0305, V) h_{Na} (V - 0.045)], {\dot{m}}_{K 2} = 4 [f_{\infty} (- 83, V_{1 / 2} V_{K 2}^{shift}, V) - m_{K 2}], {\dot{h}}_{Na} = 24.69 [f_{\infty} (500, 0.0333, V) - h_{Na}],$ where $V$ , $m_{K 2}$ , and $h_{Na}$ are the membrane potential, activation of the persistent potassium current, $I_{K 2}$ , and inactivation of the fast sodium current, $I_{Na}$ , respectively; $f_{\infty} (k, V_{1 / 2}, V) = 1 / (1 + e^{k (V_{1 / 2} + V)})$ is a Boltzmann function

Central manifolds of slow motion

One may notice from (1) that the time constant of the potassium current in the model is several times slower than those of the other variables in the system. Hence, due to this disparity of time scales, Eqs. (1) could be considered within a framework of fast–slow systems. The feature of such a system is that its dynamics is centered around the manifolds of slow motions. In other words, no matter how system is perturbed for a short time, it will be found again near the stable manifolds of slow

Poincaré mapping toolkit

A straightforward approach for constructing a Poincaré mapping is the following: one needs a relatively long recoding of the membrane potential where pairs $(V_{i}, V_{i + 1})$ of successive minima (or maxima) in $V$ are singled out; these points comprise the graph of a one-dimensional Poincaré mapping: $T : V_{i} \to V_{i + 1}$ . A drawback of this approach is obvious: if the number of distinct pairs is relatively small, the graph is sparse, and gives limited information about the prevailing type of neuronal activity

Conclusions

New tools are developed allowing for a reduction to a one-dimensional Poincaré mapping revealing the hidden organizing centers of dynamics of the membrane potential in Hodgkin–Huxley-type models. The method is applicable for a broad class of neuronal models with fast–slow dynamics including square-wave type [5]. This assertion is supported in Fig. 4 illustrating two-dimensional projections of the spiking and quiescent central manifolds onto a 3D projection of the phase space of the

Acknowledgement

This work was supported by RFFI Grant # 050100558, NIH Grant NS-043098 and GSU Brains and Behavior program.

Paul Channell Jr. is a MS graduate student in the Department of Mathematics and Statistics at Georgia State University. His GRA fellowship is supported by GSU Brains and Behaviors program.

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Paul Channell Jr. is a MS graduate student in the Department of Mathematics and Statistics at Georgia State University. His GRA fellowship is supported by GSU Brains and Behaviors program.

Gennady S. Cymbalyuk is an assistant professor in the Physics and Astronomy Department at Georgia State University. He received a Ph.D. from the Physics Department at Moscow State University and the Institute of Mathematical Problems in Biology in Pushchino, Russia in 1996. His research interests include neuromics, dynamics of neuronal systems, and neuronal control of rhythmic behaviors.

Andrey Shilnikov is an associate professor in the Department of Mathematics and Statistics at Georgia State University. He received a Ph.D. in Mathematics and Mathematical Physics from Gorky State University in 1990. He does research in dynamical systems and bifurcation theory in application to mathematical neuroscience.

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