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Research Paper | DOI: https://doi.org/10.31579/2692-9422/005
1 Department of General Physics, Physics Faculty, Lomonosov Moscow State University, Moscow, Russia
2 Laboratory of dynamic systems, Physics Faculty, Lomonosov Moscow State University, Moscow, Russia.
*Corresponding Author: Sergey Belyakin, Department of General Physics, Physics Faculty, Lomonosov Moscow State University, Moscow, Russia
Citation: Sergey B, Sergey S (2020) Neural Networks In A Generalized Model Of The N-Pacemaker Phase Response Curve, 3(1): Doi: 10.31579/2692-9422/005
Copyright: © 2020 Sergey B , This is an open-access article distributed under the terms of The Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Received: 16 February 2020 | Accepted: 05 March 2020 | Published: 16 March 2020
Keywords: topological degree; oscillating system; purkinje fibers
In this publication, we generalize the proposed model of two interacting oscillators in the case of a strong difference in their periods (when the pacemaker pulses do not alternate) and propose a General model describing a network of oscillators coupled globally. Our goal is to make the model as simple as possible and enter the minimum number of parameters. Therefore, we will fully characterize the pacemaker of their internal lengths of the cycle and re-present them as pulse oscillators. Interaction of pacemakers is described by PRC.
The circle map curve the phase response and the Arnold tongues of
Consider some physical quantity ξ, which reflects the internal state of the biological oscillator. Let the eigenfrequency of the oscillator be equal T0. Let's call a marker any event that can be clearly seen in the experiment, which is reached by the value ξ only once per period. Such a marker may be, for example, the beginning of the action potential in the cardiac preparation. Let's define the oscillator phase as follows. The phase of an arbitrarily selected marking event (for example, the maximum value of ξ) is assumed to be zero. At any next time t, 0 < t xss=removed>
Suppose that an external periodic perturbation acts on a nonlinear oscillator. Then each external influence shifts the state of the system to a new state (1):
φn+1 = φn + f (φn) (mod1). (1)
The function f (φn) is called the phase response curve (PRC) [1] and determines the phase change after the stimulus. It is convenient to represent the points f (φn) of the system state lying on the circle of the unit radius. Then, by iterating the mapping (1), one point of the circle is converted to another point of the same circle. If the circle map is continuous, then it can be characterized by a number called the topological degree and equal to the number of passes through φn+1 the unit circle during f (φn) the time it passes once. In periodic perturbations of self-oscillations with a stable limit cycle, the dynamics is often described by maps of a circle with a topological degree 0 (when the over-threshold response gives rise to a new cycle) or 1 (which expresses a sub-threshold response to stimulation). The different types of circle maps are shown.
Along with the topological degree, an important characteristic of the circle display is the number of rotations. We define it as the time average ratio of the external perturbation period to the period of the perturbed oscillator. If the rotation number is rational, ρ = M/N (here M is the number of cycles of the stimulator, and N is the number of cycles of the nonlinear oscillator), then the dynamics of the system will be periodic with the capture of the multiplicity phase N/M. If the rotation number is irrational, the system demonstrates quasi-periodic or chaotic behavior.
In many cases, the disturbance by a single pulse of a spontaneously oscillating system leads to a phase shift of the current rhythm (see, for example, [2] and references there). The magnitude of the shift depends on both the magnitude of the stimulus and its phase in the cycle. The graph of the dependence of the new phase on the previous phase (i.e., PRC) is either a continuous circle map with a topological degree of 1 or 0, or a discontinuous function. Phase shift experiments were performed for a large number of different systems. We are interested, first of all, in the phase response curve, experimentally obtained in the study of cardiac drug. In [3] the duration of the cycle of spontaneous oscillations of Purkinje fibers after stimulation by short pulses of electric current was measured. The obtained phase response curve of the Biphase form is shown in Fig.2. Based on the study of this experimental material, the following generalizations can be made [3].
After the disturbance, the rhythm is usually restored (after the transition process) with the same frequency and amplitude as before the disturbance, and its phase is shifted. Depending on the phase, a single stimulus can result in either an elongation (early stimulus) or a shortening (late stimulus) of the duration of the perturbed cycle. At some amplitudes of the stimulus, obvious discontinuities are observed. To further study the dynamics of any constructed model, it is necessary, having experimentally obtained PRC, to find a good analytical approximation of this curve. This will allow to investigate the main features of the behavior of the system. The main characteristic of the desired function is the need to directly depend on only two physical parameters: the amplitude of the stimulus and the phase of the applied perturbation. All other (so-called "internal") parameters describing the course of the curve should (ideally) be reduced to these two.
One of the simplest (and coarsest) approximations of a given PRC is the sinusoidal function, which ultimately results in a map of the form (2):
φn+1 = f (a,b,φn) = φn + a + b sin 2πφn (mod1). (2)
Where a and b are constants. However, despite its simplicity, this approximation correctly reflects the qualitative structure of the phase portrait of the system under study.
The analysis of bifurcations of reversible circle maps was undertaken in the last century by A. Poincare and still attracts much -by V. I. Arnold [4] (see also [5] and the references given there). For fig.3 the bifurcation diagram of the circle diffeomorphism on the parameter plane (b, a) is shown. This diagram is divided into areas called language (or horns) of Arnold, which correspond to the sustainable capture phase ratio N/M (i.e., N cycles of the stimulator has M cycles of a nonlinear oscillator). Arnold languages exist for all rational relations N/M, where N and M are mutually Prime numbers. This means that there are an infinite number of Arnold languages that correspond to all possible ratios of frequencies of the stimulator and the perturbed oscillator. Between any two languages corresponding to N/M and N*/M* phase captures, there is another capture region corresponding to the capture of multiplicity phases (N+N*)/(M+M*). The structure shown in Fig.3, is the usual behavior for low stimulus amplitudes in simple theoretical models discussed below. However, as the amplitude of the periodic effect increases, this structure collapses.
Relaxation model of Poincare oscillator
A widely used idealization of some periodically stimulated oscillators is the Gelfand and cetlin model [6], or the relaxation model [2,7-9]. In this model, the value referred to as activity increases to the upper threshold, leading to some event. Then the activity returns to the lower threshold. If the rates of rise and fall of activity to the thresholds are fixed, and the thresholds are also fixed, then a periodic sequence of events is generated, the frequency of which is easy to determine. Periodic perturbation in relaxation models can be included in the form of threshold modulation, usually sinusoidal.
In some works, instead of sinusoidal modulation of the threshold, other functions were considered, for example, Delta function peaks, rectangular and triangular pulses, etc. (see references in [10]). Arnold in [4] briefly discussed the possibility of using obtained model in the relaxation mapping to study the rhythms Wenkebach. Subsequent researchers found that piecewise linear monotone, discontinuous maps (Fig.1c), similar to those found in the relaxation model, appear in theoretical models of atrial-ventricular communication in AB blockade [11,12]. Such mappings can be experimentally measured and used to predict complex rhythms observed in humans [13].
Despite the wide application, the relaxation model too simplistically describes the interaction of the oscillator and the external perturbation; much more vital is the use of models that take into account the individual response of the system to the external perturbation. Cardiologists usually assume that the nonlinear ODE that contains oscillation with a stable limit cycle, represent a suitable model for the generation of periodic activity of the heart [14]. In the case where the limit cycle is quickly achieved after a single stimulus, and the action of a single stimulus is known, it is possible to calculate the effect of periodic stimulation. The prototype of the model with a periodically perturbed limit cycle is the van der Pol equation with a sinusoidal perturbation.
Consider the effect of a periodic sequence of short pulses on the oscillations described by a system with a limit cycle (see, for example, [10]). The simplest model is Poincare oscillator. In this model, a stable limit cycle is a circular trajectory. The perturbation is a horizontal displacement of magnitude b, and after stimulation, the system rapidly approaches the limit cycle along its radius.
If it is the phase φn immediately preceding the nth stimulus, then the phase preceding (n +1) the nth stimulus is simply
φn+1 = τ + g(φn, b) (mod1)
Where τ is the time interval between periodic stimuli normalized for the eigenfrequency of the autogenerator, and PRC g(φ) is easily calculated [14-16].
This theoretical model for periodically perturbed limit cycles was independently proposed by several researchers [15-18]. Since for a simple model with a limit cycle RPC is calculated quite easily, it is possible to use analytical and numerical methods to determine the detailed structure of the phase capture zones as a function of the amplitude b and frequency a of the stimulus. In this example, for low stimulus amplitudes (b ≤ 1), the capture zone topology has a classical Arnold structure (Fig.3), and the circle display is a reversible degree 1 display. However, for b > 1, the dynamics is described by displaying a circle of zero topological degree. The extensions of Arnold's languages have a more complex form. There are bifurcations in the system, leading to doubling of the period and chaos.
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