Dynamics of spiral waves driven by a rotating electric field

doi:10.1016/j.cnsns.2013.03.010

Communications in Nonlinear Science and Numerical Simulation

Volume 19, Issue 1, January 2014, Pages 60-66

https://doi.org/10.1016/j.cnsns.2013.03.010 Get rights and content

Highlights

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We found rotation-synchronization between a spiral and a rotating electric field.
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The influence of chirality, and polarized modes of the rotating electric field on synchronization is studied.
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The phase diagram describing the controllable region in excitability-period plane is presented.

Abstract

We study the behavior of spiral wave under the driving of a rotating electric field. The rotating electric field can drive a spiral wave to be synchronous, depending on four factors: its frequency and amplitude, chirality, and polarized modes. Rotation-synchronization characterized by the rotating direction is focused on. We discuss the behavior of synchronization, such as the dependence of angle-differences between the spiral tip and the electric field on ratio of frequency, the influences of different polarized modes of the electric field, the radius of synchronous spiral wave, and so on. A circularly polarized electric fields (CPEF) can suppress meandering spiral to rigid one and prevent breakup of spiral in medium with low excitability. The phase diagram describing the controllable region in excitability-period plane is presented. The influences of polarized modes of electric field on minimum excitability of medium are also studied.

Introduction

Spiral waves are typical examples of spatiotemporal patterns far from thermodynamic equilibrium [1], [2]. They have been observed in various biological and chemical systems: platinum with oxidation of CO [3]; slime mould dictyostelium discoideum [4]; the prototypical Belousov–Zhabotinsky (BZ) reaction [5], [6], [7]; the heart muscle where spiral waves associate with cardiac arrhythmia [8]; in neuronal networks and coupled oscillators [9], [10], [11]. Dynamics of spiral waves under the control of various external fields has attached much attention [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. Examples include illuminations [12], [13], [14], magnetic [15] and electric field [16], [17], [18], [19], [20], noise [21], pacing [22], [23], feedback [24], [25], etc. Consequent interesting behavior of spiral wave, such as drifting [17], [26], breakup [18], [26], elimination [27], and so on, has been studied.

A slaved spiral wave may respond to a external force by changing its frequency. It is observed that feedback stimuli [29] and periodic modulation of excitability can induced petal-synchronization of meandering spiral [14], [30]. Periodic illumination [31] and feedback mediated method [32] can also suppress a meandering spiral into a rigidly rotating one. Compared with those widely studied scalar quantity fields, vector quantity fields are much more interesting when studying dynamics of spiral, since the direction of the external field importantly affects the spiral behavior. For example, the angle between the spiral tip and direction of the electric field determines the direction of drift [33], [34], [35]. Recently, synchronization of a spiral by a circularly polarized electric field (CPEF) is studied in the light that both the spiral waves and CPEF possess the rotation symmetry [35]. Consider that the direction of CEPF rotating all the time while its amplitude is kept, it is worthwhile to further study the nontrivial synchronization by focusing on the information between the rotation of CPEF and spiral tip motion. Investigation on the direction-rotation synchronization is a interesting study, which is not only quite different from the case of scalar fields but also the cases of DC and AC vector electric field. Questions include information about angle difference between the CPEF and spiral tip, chirality-dependent synchronization, radius of tip trajectory determined by the amplitude and rotating frequency of the CPEF, and so on. It is also shown that the CPEF can prevent the spiral breakup when the excitability of medium is weak [35]. The second frequency which results from Hopf bifurcation and leads to spiral meander can be eliminated by the CPEF with suitable amplitude. However, the influence of frequency of the CPEF on preventing spiral breakup is still unexplored. Further more, the modes of polarization of the rotating electric field will play a important role on preventing spiral breakup. Motivated by the above mentioned points, we study the dynamics of spiral waves driven by a rotating electric field.

Section snippets

Model

Here, we use a two-dimensional (2D) Bär model (a modified FitzHugh–Nagumo model) in the presence of a electric field to describe the interaction of an activator $u (x, y, t)$ with an inhibitor $v (x, y, t)$ [17], [36]: $\begin{matrix} \frac{\partial u}{\partial t} = f (u, v) + \nabla^{2} u + E_{x} \frac{\partial u}{\partial x} + E_{y} \frac{\partial u}{\partial y}, \\ \frac{\partial v}{\partial t} = g (u) - v . \end{matrix}$ The local reaction kinetics take the following form $f (u, v) = \frac{1}{ε (x, y)} u (1 - u) (u - \frac{v + b}{a}),$ $g (u) = \{\begin{matrix} 0 & 0 ⩽ u < 1 / 3, \\ 1 - 6.75 u {(u - 1)}^{2} & 1 / 3 ⩽ u < 1, \\ 1 & 1 ⩽ u . \end{matrix}$

A rotating electric field $E (t) = (E_{x} i, E_{y} j)$ is employed with $E_{x} = γ \cos (ω_{E} t + ϕ_{x})$ and $E_{y} = γ \cos (ω_{E} t + ϕ_{y})$ . Through changing the phase shift $ϕ = ϕ_{y}$

Synchronization of rigidly rotating spiral

When a circularly polarized electric field (CPEF) (with $ϕ = 3 π / 2$ , anticlockwise rotation) is applied on a spiral, the rotation frequency of the entrained spiral is found to be synchronized with that of the electric field in a 1:1 ratio [35]. This behavior seems just like the well-known synchronization by a external scalar field. Here, we focus on rotation-synchronization in a class by itself characterized by the rotating direction. In Fig. 1(a), we draw a sketch showing the CPEF and the core

Conclusions

In conclusion, we have studied the dynamics of spiral waves driven by a rotating electric field. A spiral wave can be synchronized by using a circularly polarized electric field (CPEF) with suitable amplitude and frequency. We focus on rotating synchronization characterized by the direction of rotation. The angle-differences $α$ between the spiral tip and the electric field shows a tongue-like curves in $α - ω_{E} / ω_{0}$ plane. When the frequency of CPEF is close to that of the spiral, the $α$ is minimal.

Acknowledgements

This work is partially supported by the National Nature Science Foundation of China (Nos.11005026, 11105074, and 11274271) and Qianjiang researcher plan in Zhejiang Province (Nos. 2012R10057).

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View full text

Dynamics of spiral waves driven by a rotating electric field

Highlights

Abstract

Introduction

Section snippets

Model

Synchronization of rigidly rotating spiral

Conclusions

Acknowledgements

J Theor Biol

Commun Nonlinear Sci Numer Simul

Nonlinear Sci Numer Simul

Nonlinear Sci Numer Simul

Prog Biophys Mol Bio

Chaos Solitons Fractals

Commun Nonlinear Sci Numer Simul

Chem Phys Lett

Rev Mod Phys

Science

J Chem Phys

Chemical waves and patterns

Nature

Phys Rev Lett

Phys Rev Lett

Phys Rev E

Phys Rev Lett

Nature