# Drift of spiral waves

Post-publication activity

Spiral waves will drift in response to symmetry breaking perturbations.

## Overview

Drift of a spiral wave is directed change of its location with time in response to perturbations, as opposed to their meander which is spontaneous variation of the spiral wave rotation due to its internal instabilities. This article considers a few types of drift, such as

• resonant drift, occurring in response to time-dependent forcing with a period close to the period of the spiral wave,
• inhomogeneity induced drift which occurs when properties of the system vary in space,
• anisotropy induced drift which occurs when properties of the system differ in different directions,
• boundary induced drift, observed in bounded media when core of the spiral wave is close to the boundary,
• drift due to interaction of spirals with each other when their own periods are approximately equal and cores are near to each other.

Some people consider in the same class the "induced drift" (we shall call it "high frequency induced drift" here) of a spiral wave in excitable media which is observed when the spiral wave is overwhelmed by another, more frequent source of excitation waves.

## Symmetry of unperturbed spiral waves

A reaction-diffusion system on a plane is invariant with respect to the Euclidean group. A spiral wave is therefore a symmetry-breaking solution. A rigidly rotating spiral only retains one symmetry: a shift in time is equivalent to a rotation in space. Hence one spiral wave solution implies a three-dimensional family of such solutions, varying by the location of the rotation centre, and the initial phase of rotation. These solutions are at most neutrally stable with respect to each other (i.e. orbitally stable, but not asymptotically stable). That is, a small perturbation applied to one such solution will typically result to the the slight shift of the spiral wave in space and in phase. This neutral stability is due to the Euclidean symmetry of the reaction-diffusion system of equations. When this symmetry is broken, the location and phase of a spiral wave is liable to changes, i.e. drift.

## Resonant drift

Figure 1: Resonant drift. Here and in other movies: fragments of simulations of FitzHugh-Nagumo model, red is $$u$$ and blue is $$v\ .$$ Here, green is external forcing.

Periodic variations of system properties in time, say by varying illumination in a light-sensitive Belousov-Zhabotinsky reaction (BZ) medium or another external forcing, leads to directed long-distance displacement of the spiral (Agladze et al. 1987; Davydov et al. 1988). The trajectory of the instant rotation centre is a circle with the radius $$R_{r.d.} = \frac{v}{|\tilde{\omega}-\Omega|}$$ where drift velocity $$v$$ depends on the nature (e.g. amplitude) of the forcing, $$\Omega$$ is the angular frequency of the forcing and $$\tilde{\omega}$$ is the angular frequency of the drifting spiral (which may differ from that of the free spiral, $$\omega\ ,$$ due to the forcing). In case of exact equality $$\Omega=\tilde{\omega}$$ the spiral drifts along a straight line, and the direction is determined by the relationship of initial phases of the spiral rotation and the forcing.

An easy way to understand resonant drift is to consider a periodic series of short "shocks", say flashes of lights for the light-sensitive BZ medium. Due to stability and symmetry of a spiral wave, one such shock generically results in a displacement of the rotation centre of a spiral. If a series of shocks is timed so that each leads to a displacement in the same direction, this produces a drift.

Perturbation theory described below gives a more detailed description of the resonant drift, for the case when the medium and/or external forcing are spatially inhomogeneous and/or time dependent, in terms of a third-order ODE system.

The resonant drift of spirals has been discussed as a possible mechanism of low-voltage defibrillation, however, for the moment this article is written, no direct tests of the idea has been performed in heart experiments.

## Inhomogeneity induced drift

Figure 2: Inhomogeneity induced drift. Green: excitability (an extra factor at the kinetic term of the activator equation)

If the excitable or oscillatory medium supporting spiral waves is spatially inhomogeneous, i.e. the right-hand sides of the reaction diffusion system depend explicitly on space variables, then the rotation of the spirals is typically not stationary, and a directed drift is observed.

Perturbation theory for small inhomogeneity described below gives the following qualitative predictions:

• drift velocity is a function of distribution of the inhomogeneity with respect to the current position of the spiral. As a consequence, in a fixed distribution of inhomogeneity, spiral waves obey a second-order ODE system.
• In the first order, the drift velocity is proportional to the inhomogeneity.
• Hence, superposition principle for different types of inhomogeneities.
• A consequence of the mirror-symmetry of the reaction-diffusion system of equations: the sensitivity of the drift component along the parameter gradient is a scalar (does not change if clockwise rotating spiral is replaced with counter-clockwise one), whereas the component perpendicular to the gradient is a pseudo-scalar (reverses its sign if direction of rotation of spiral is changed).

Most important are variations of the medium properties near the core of the spiral (see about "wave-particle duality" below), hence it is convenient to describe the drift in terms of the movement of the tip.

Two easy-to-follow mechanisms of the drift:

Figure 3: Mechanisms of inhomogeneity-induced drift. Left: due to refractoriness gradient (Krinsky, 1968). Right: due to excitability gradient (Pertsov and Ermakova, 1988)
• Drift of "tight" spirals in a gradient of the refractory period: see Figure 3 left. The medium inside the circle has a longer refractory period than outside. The tip of the spiral moves along the outer side of the circle until the inside recovers, for a long interval (clockwise in the picture). Then it moves along the inner side, waiting for the outside to recover, for the shorter interval (counter-clockwise in the picture). Overall result is a clockwise drift.
• Drift of "sparse" spirals in a gradient of excitability: see Figure 3 right. The medium is more excitable in the lower part of the square than in the upper part of the square. The more excitable spiral tip turns round quicker and its path has higher curvature. Overall result is a rightward drift.

These pictures only explain the component of the drift across the gradient. A more formal explanation can be done in terms of kinematic theory of excitable media (see Mikhailov et al. 1994).

## Anisotropy induced drift

Figure 4: Drift induced by constant electric field (directed downwards) in BZ reaction. Montage of photos from Agladze and De Kepper, 1992

If the medium supporting spiral waves is anisotropic, e.g. the right-hand sides of the model depend explicitly on spatial derivatives and are not invariant under rotations, this also could cause drift of spirals. Two examples:

• Drift due to an advection field, such as induced by electric field in a chemical reaction involving charged components (Agladze and De Kepper, 1992; Steinbock et al, 1992, see Figure 4). If all species have the same electrophoretic velocity the spiral will obviously drift with the same velocity. Perturbation theory for small anisotropy described below predicts that if electrophoretic velocity of species is proportional to their diffusivities, then the components of drift velocity along and across the electric field are related to the rate of collapse and speed of axial drift of a scroll ring. Note however that experiment shown on Figure 4 is beyond the linear perturbation theory as the direction of the drift depends on the electric field strength, contrary to that theory.
• Drift due to anisotropic diffusion. Simple symmetry considerations show that a standard anisotropy, e.g. non-invariance of system properties with respect to swap $$x \leftrightarrow y$$ can not produce drift. However a stronger type of anisotropy, non-invariance with respect to swap $$x \leftrightarrow -x$$ is akin to advection mentioned above and can produce drift. Such drift has been observed in a metamaterial constructed of "chemical diodes" (Dupont et al, 1998).

## Boundary induced drift

Figure 5: Boundary induced drift. Shown is a fragment of the medium; green: an impermeable boundary

Non-flux, or perhaps other types other passive types of boundaries, can affect spiral waves in a similar way to inhomogeneities (although of course by no means small ones). There are examples of analytical treatment of this type of drift in some models by asymptotic matching techniques, although general theory is still to be done.

Phenomenologically, properties of this drift are similar to inhomogeneity induced drift, i.e.

Figure 6: Mechanisms of boundary-induced drift: tip path has higher curvature nearer to the border (Ermakova and Pertsov, 1986)
• Drift velocity is a function of the position of the boundary with respect to the current position of the spiral. As a consequence, in a fixed distribution of inhomogeneity, spiral waves obey a second-order ODE system.
• The velocity of the boundary-induced drift decreases quickly with the distance from the spiral to the boundary.
• A superposition principle for different types of boundaries.
• Component of the velocity towards the boundary remains unchanged, whereas the component along the boundary reverses, if direction of rotation of the spiral is reversed.

Two main types of behaviour of spirals near a boundary have been observed: it

• either drifts along the boundary at a certain stable distance from it (if the component towards the boundary changes sign, from attraction when far to the repulsion when near),
• or drifts away from the boundary until it stops feeling it (if the repulsion takes place at all distances at which the drift is observed).

## Interaction of spirals

Figure 7: Drifting bound pair of counter-rotating spirals drift. Dashed green line is the shock structure and a symmetry axis
Figure 8: Repulsion of counter-rotating spirals (Ermakova et al. 1989)
Figure 9: Asymmetric bound pair of co-rotating spirals might be perceived as a meandering two-armed spiral (Zemlin et al. 2006).

If there are more than one spiral wave in the medium and they are not too close to each other, they divide the medium to domains of influence. The boundaries of the domains are "shock structures". The shock structures act similar to no-flux boundaries. If such an internal quasi-boundary is close enough to a spiral wave, it can affect its drift in a similar way as the real boundary does. Correspondingly, there are two most common types of interaction of two spirals. If a single spiral is repelled from a no-flux boundary and freezes at some distance $$d_f$$ from it, then two spirals will repel each other to $$2d_f$$ (see Figure 8). If a single spiral drifts along the boundary at a stable distance $$d_d\ ,$$ then two spirals would tend to form a pair at a distance of $$2d_d$$ from each other (see Figure 7). Sometimes the mutual symmetry of the interacting pair of spirals is unstable, and one spiral may "enslave" the other. It is not clear if such asymmetric regimes can be interpreted in terms of drift of spirals, or are more adequately described as a peculiar meandering regime of a double-armed spiral (see Figure 9)..

The above applies to interaction of spiral waves in a medium that is homogeneous in its properies. Spatial inhomogeneities can bring in new phenomena, see below under "High frequency induced drift".

## "Wave-particle duality" of spiral waves

In all the above cases, it is convenient to consider spiral waves as "particles", interacting with each other or reacting to external perturbations as localized, point-wise objects. This is in a seeming contradiction with the wave-nature of these waves. The spiral waves look like non-localized objects, filling up all available space, but behave like localized objects. The mathematical nature of this paradox have been brought to the forefront by Biktasheva and Biktashev (2003) in terms of perturbative dynamics of spiral waves, i.e. resonant and inhomogeneity-induced drifts . Consider a reaction diffusion system

$\tag{1} \partial_t\mathbf{u}=\mathbf{f}(\mathbf{u})+\mathbf{D}\nabla^2\mathbf{u}+\epsilon\mathbf{h}, \qquad \mathbf{u}=\mathbf{u} (\mathbf{r},t),\mathbf{f},\mathbf{h}\in\mathbb{R}^{\ell}, \; \ell\geq2, \qquad \mathbf{r}\in\mathbb{R}^2$

with perturbation $$\epsilon\mathbf{h}=\epsilon\mathbf{h}(\mathbf{r},t,\mathbf{u},\nabla\mathbf{u},\dots)\ ,$$ and assume existence, at $$\epsilon=0\ ,$$ of stationary rotating spiral solutions

$\tag{2} \mathbf{u}(\mathbf{r},t) =\mathbf{U}(\rho(\mathbf{r}-\mathbf{R}),\theta(\mathbf{r}-\mathbf{R})+\omega t-\Phi)$

where $$\rho(),\theta()$$ are polar coordinates and $$\mathbf{R}=(X,Y),\Phi$$ are arbitrary constants, location of the core of the spiral and its phase.

Then the first order perturbation theory in $$\epsilon$$ gives solutions close to (2) with $$\mathbf{R},\Phi$$ slowly varying according to motion equations

$\tag{3} \dot{R}=\dot{X}+i\dot{Y}=\epsilon H_1(\mathbf{R},\Phi,t), \qquad \dot\Phi=\epsilon H_0 (\mathbf{R},\Phi,t) .$

In particular, for the drift under resonant forcing in a spatially inhomogeneous medium, this leads to (Biktashev and Holden 1994)

$\tag{4} \dot{R}=C(X,Y)+v(X,Y) e^{i\Theta}, \qquad \dot{\Theta}=\Omega(t)-\tilde\omega(X,Y)$

where $$\Theta$$ is the phase difference between the spiral and the resonant forcing, $$\Omega$$ is the forcing frequency, and $$\tilde\omega(X,Y)$$ is the own (perturbed) spiral angular frequency possibly depending on the current spiral location.

In motion equations (3), the velocities of spatial drift, $$H_1\ ,$$ and temporal/phase drift, $$H_0\ ,$$ are linear functionals of the perturbations,

$\tag{5} H_n=\oint_{t-\pi/\omega}^{t+\pi/\omega} \frac{\omega d\tau}{2\pi} \iint\limits_{\mathbb{R}^2} d^2\mathbf{r} \; e^{in(\Phi-\omega\tau)} \left\langle \mathbf{W}_n( \rho(\mathbf{r}-\mathbf{R}), \theta(\mathbf{r}-\mathbf{R}+\omega\tau-\Phi) ),\mathbf{h} \right\rangle.$

where $$\mathbf{h}$$ is evaluated at the unperturbed solution (2).

Figure 10: Response functions (colour) of a spiral wave (elevation) in the FitzHugh-Nagumo model.

The kernels $$\mathbf{W}_{0,1}$$ of these functionals, sometimes called response functions (RFs), are critical eigenfunctions of the adjoint linearized problem. These eigenfunctions are dual to the eigenfunctions of the linearized problem produced by the generators of the symmetry group, sometimes called Goldstone Modes (GMs). The "wave-particle" duality then reduces to the difference between these eigenfunctions. The GMs, constructed from spatial derivatives of the spiral wave solution, are non-localized as that solution. The RFs, however, are essentially localized, i.e. exponentially decay far from the core of the spiral. This is of course is only possible because the linearized problem is not self-adjoint.

Apparently, the defining property is the direction of the group velocity: the spiral wave will have a localized RF and behave as a localized object if and only if it is a source of waves, so far enough from the core, the group velocity is directed outwards.

The mathematical aspects of particle-like behaviour in non-perturbative cases, such as boundary-induced drift or interaction of spirals with each other, are less clear. In the few examples where analytical answers are known, this seems to be associated with the exponential growth of solutions of inhomogeneous linearized problem with the free term given by the spatial gradient of the spiral wave solution. This is also dependent on the outward direction of the group velocity.

Both localization properties seem therefore be equivalent. That is, if a spiral wave does not feel a non-flux boundary when far from it, it will not feel a weak inhomogeneity at the same distance, and vice versa. Although quite plausible physically, mathematically this is still an open question.

## High frequency induced drift

Figure 11: The faster spiral enslaves the slower one, turns it into a dislocation and causes its drift. Green: excitability

Figure 12: Mechanisms of high-frequency drift of a dislocation (Krinsky, Agladze, 1983)

Interaction of spiral waves in a spatially nonuniform excitable medium gets new features including a drift-like phenomenon. If two sources of excitation waves in the same medium have different frequencies, the shock structure demarcating their domains of influence will move away from the faster source towards the slower one. If the slow source is a spiral wave and the shock structure reaches its core, the spiral wave turns into a wavebreak, translocating from one incoming wave to another. The location of the wavebreak changes with time, i.e. drifts. If the high-frequency source stops, the wavebreak develops into spiral wave at the new place.

## References

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• Mikhailov, A.S., Davydov, V.A. and Zykov, V.S. "Complex dynamics of spiral waves and motion of curves". Physica D, 70:1-39, 1994
• Biktashev, V.N. and Holden, A.V. "Design Principles of a Low-Voltage Cardiac Defibrillator Based on the Effect of Feed-Back Resonant Drift", Journal of Theoretical Biology, 169(2): 101-113, 1994
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