Applications of the Poincaré mapping technique to analysis of neuronal dynamics
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 bywhere , , and are the membrane potential, activation of the persistent potassium current, , and inactivation of the fast sodium current, , respectively; 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 of successive minima (or maxima) in are singled out; these points comprise the graph of a one-dimensional Poincaré mapping: . 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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One dimensional map-based neuron model: A phase space interpretation
2020, Chaos, Solitons and FractalsCitation Excerpt :In one-dimensional map-based models, the only variable is the membrane voltage [42]. Some models can be analyzed based on the general status of the phase space [43–46]. For example, it has been proved that the phase space of a recursive map-based model should be near the identity line in the rest state and contain a saddle-node, homoclinic or flip bifurcation [47].
The intrinsic phase response properties of an interneuron model
2012, NeurocomputingCitation Excerpt :As mentioned above, the shape of BPRC for the interneuron has closely relationship with the minimal membrane potentials on the periodic orbits, and the onto Poincaré return mappings presented in [17,21,22] were constructed based on the minimal membrane potentials on the periodic orbits, so we naturally think of constructing the onto Poincaré return mappings. The numerically construct method can be found in [17,21,22], and the results are shown in Fig. 3. According to the return mapping, we find the amplitudes of the minimal membrane potentials are increased gradually from peak 1 to 5 (x-axis).
Voltage interval mappings for activity transitions in neuron models for elliptic bursters
2011, Physica D: Nonlinear PhenomenaCitation Excerpt :The properties of MMOs, or broadly the current description of transitions between bursting, tonic spiking and subthreshold oscillations in elliptic bursters is incomplete and presents a challenging problem for mathematical neuroscience and the dynamical systems theory in general. In this paper we refine and expound on the technique of creating a family of one-dimensional mappings, proposed in [18,46,47] for the leech heart interneuron, into the class of elliptic models of endogenously bursting neurons. We will show a plethora of information, both qualitative and quantitative, that can be derived from the mappings to thoroughly describe the bifurcations as such a model undergoes transformations.
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.