Scroll Wave Turbulence
Self-sustained regime in excitable media mediated by three-dimensional instability of scroll waves.
Overview
Compared to drift of spiral waves in two spatial dimensions, the scroll waves in three spatial dimensions have more degrees of freedom. Scroll waves rotate around filaments, which can not only move in space, but also change shape. The phase of rotation may vary along the filament, the feature known as twist of the scroll wave. Twist of a scroll wave and curvature of its filament are factors of its dynamics that are specifically three-dimensional. Sometimes interaction of all the factors can make the steady rotation of scroll waves unstable and lead to apparently chaotic regime in which the scroll filaments are bent, increase in length, break up and multiply. Such regime is known as scroll wave turbulence (SWT). This is akin to spiral wave break-up in two spatial dimensions. However, scroll wave turbulence is an essentially 3D phenomenon in that it may be observed in a system which does not show any turbulent behaviour whilst in 2D. This may happen for cases where 2D spiral waves are stationarily rotating or meandering. At the time of writing this article, two mechanisms of SWT in excitable media have been well described. One is mediated by negative filament tension which is an intrinsic property of a scroll wave that makes a curved filament to curve even more. The other is observed in non-uniformly anisotropic media and is related to development of twist instabilities.
The concept of filament tension
The concept of filament tension can be understood in terms of a perturbative dynamics of a scroll wave. Consider a reaction-diffusion system
\[\tag{1} \partial_t\mathbf{u}=\mathbf{f}(\mathbf{u})+\mathbf{D}\nabla^2\mathbf{u}, \qquad \mathbf{u}=\mathbf{u} ({\vec r},t), \, \mathbf{f}=\mathbf{f}(\mathbf{u})\in\mathbb{R}^{\ell}, \; \ell\geq2, \qquad {\vec r}\in\mathbb{R}^2 \text{ or } \mathbb{R}^3 \]
and assume existence of stationarily
rotating spiral solutions in \(\mathbb{R}^2\ ,\)
\[\tag{2} \mathbf{u}({\vec r},t) =\mathbf{U}(\rho,\theta+\omega t), \]
where \(\rho,\theta\) are polar coordinates in \(\mathbb{R}^2\ ,\)
\(\omega\) is a fixed constant.
A simple extension of spiral wave solution to the third spatial dimension
is called a straight scroll wave. More generically,
a scroll wave solution in \(\mathbb{R}^3\) may be viewed
as a solution of the form
\[ \mathbf{u}({\vec R}+{\vec N}\rho\cos\theta + {\vec B}\rho\sin\theta,t) = \mathbf{U}(\rho,\theta+\omega t-\Phi) + \mathcal{O}(\varepsilon), \]
where \(\varepsilon\) is a formal small parameter measuring deformation of a scroll wave compared to the straight scroll, \({\vec R}={\vec R}(\sigma,t)\) is the parametric equation of filament position at time \(t\ ,\) \(\Phi=\Phi(\sigma,t)\) is the rotational phase distribution, \({\vec N}={\vec N}(\sigma,t)\) and \({\vec B}={\vec B}(\sigma,t)\) are the unit principal normal and binormal vectors to the filament at point \({\vec R}(\sigma,t)\ .\) Vectors \({\vec N}\) and \({\vec B}\) together with tangent vector \({\vec T}\) make a Frenet-Serret triple which can be defined using arclength differentiation operator \( \mathcal{D}_s{u}(\sigma)=|\partial_{\sigma}{\vec R}|^{-1}\partial_{\sigma}u(\sigma). \) Then the tangent vector is \({\vec T}=\mathcal{D}_s{ {\vec R} }\ ,\) the curvature \(\kappa\) and the normal unit vector \({\vec N}\) are defined by \( \kappa{\vec N}=\mathcal{D}_s{ {\vec T} } \ ,\) and the binormal vector \( {\vec B}={\vec T}\times{\vec N} \) completes the triad. The Frenet-Serret description is easy to understand but it has a significant technical disadvantage: it becomes degenerate at the points of zero filament curvature, \(\kappa=0\ .\) An alternative description, free from this disadvantage, is in terms of Fermi coordinates. Either way, the equation of motion of the filament, in the assumption of small filament curvature, \(\kappa=\mathcal{O}(\varepsilon)\ ,\) and small twist, \(\mathcal{D}_s\Phi=\mathcal{O}(\varepsilon)\ ,\) can be written as
\[\tag{3} \partial_t{\vec R} = \alpha \mathcal{D}_s^2{\vec R} + \beta \left[ \mathcal{D}_s{\vec R} \times \mathcal{D}_s^2{\vec R} \right] + \mathcal{O}(\varepsilon^2) \]
#filmotion is written in the assumption that parameterization of
the filament is chosen in such a way that a point with a fixed
\(\sigma\) moves orthogonally to the filament.
The coefficients in the equation of motion are calculated
using the response functions \(\mathbf{W}_1\) (see Drift of spiral waves) as
\[ \alpha + i \beta = -\frac12 \int\limits_0^{\infty}\oint \left[\mathbf{W}_1(\rho,\theta)\right]^+ \mathbf{D} e^{-i\theta} \left( \partial_{\rho}-\frac{i}{\rho}\partial_{\theta} \right) \mathbf{U}(\rho,\theta) \;\mathrm{d}\theta \;\rho\,\mathrm{d}\rho . \]
Now let us consider the total length of the filament, defined at each \(t\ :\)
\[\tag{4} S(t)=\int \mathrm{d}s = \int \left| \partial_{\sigma}{\vec R} \right| \,\mathrm{d}\sigma, \]
where the integral is taken over the whole filament.
Differentiation of #totlength, with account of #filmotion and using
integration by parts, reveals that neglecting boundary effects (which are absent
for closed filaments and vanish for smooth impermeable boundaries),
the rate of change of the total length is described by
\[\tag{5} \frac{\mathrm{d}S}{\mathrm{d}t} = - \alpha \int \kappa^2 \,\mathrm{d}s + \mathcal{O}(\varepsilon^2). \]
This implies that unless the filament is straight, and within the
applicability of the perturbation theory, the total length of the
filament will decrease if
\(\alpha>0\)
and increase if
\(\alpha<0\ .\)
Hence this coefficient is often called filament tension.
SWT mediated by negative filament tension
With negative filament tension, the straight scroll (\(\kappa\equiv0\ ,\) \(\Phi=\mathrm{const}\)) is unstable. Any small deviation from the straight shape grows. The constant increase in filament shape due to curvature-induced drift is stopped when other factors, not accounted for by the perturbation theory quoted above, become significant. These include high curvatures, high twists, interaction of different scroll waves or different parts of the same scroll wave with each other and with medium boundaries. One possible outcome is that mutual interaction of scrolls may stabilise them so that double or multiple scrolls are observed. Alternatively, if stabilization does not occur, then the SWT can onset. A third possibility is extinction of all scroll waves. Empirical studies of SWT suggest that in smaller media, extinction or mutual stabilization prevail, whereas for sufficiently large media, SWT becomes more prevalent. However, it is now known if stabilization and extinction may still be possible, just less probable, in larger media. The chaotic character of SWT for now is currently only a conjecture. Note that for \(\beta=0\) and planar filament, #filmotion is similar to Kuramoto-Sivashinsky equation without its regularization term, and chaotic nature of solutions of Kuramoto-Sivashinsky equation is known.
The above quoted perturbation theory is based on the assumption that 2D spirals are stationarily rotating, formalized by #Spiral. At the time of writing this article, an extension to the meandering case has not been done yet; however, filament tension still can be defined via the rate of change of average radius of axisymmetric scroll rings. For cases where meandering scroll rings expanded, the full 3D simulations also show SWT.
SWT due to twisted anisotropy
A 3D regimes phenomenologically very similar to negative tension SWT have been observed in simulations of scroll waves in media with spatially non-uniform anisotropy where the diffusion matrix components are not scalars but tensors, and vary in space:
\[\tag{6} \partial_t u_j = f_j (u_k) + \sum\limits_{k=1}^{\ell} \sum\limits_{\mu,\nu=1}^{3} \frac{\partial}{\partial x_{\mu}} \left( D^{j,k}_{\mu,\nu}({\vec r}) \frac{\partial}{\partial x_{\mu}} u_k \right), \qquad j,k\in\{1,\dots,\ell\}, \qquad \mu,\nu\in\{1,2,3\}. \]
In cardiac excitation models, \( D^{j,k}_{\mu,\nu}({\vec
r})=P^{i,j} Q_{\mu,\nu} \) where matrix component
\(P^{1,1}=1\)
(assuming 1 is the index of the transmembrane voltage field)
and all other \(P^{i,j}=0\ ,\) and
tensor \(Q_{\mu,\nu}\)is related to local direction of fibres
(see Models of heart). A number of numerical studies have been
done with "rotational anisotropy": in a rectangular volume, the
filament direction is within \((x,y)\) plane, and this
direction linearly changes depending on the \(z\)-coordinate.
This roughly corresponds to the structure of ventricular walls. In
such numerical studies, fiber rotation could cause scroll wave
breakup. A more detailed analysis revealed a typical motif: fiber
rotation creates regions of highly localized filament twist, called
"twistons". These twistons migrate along filaments, and their
collision with boundaries causes a scroll wave filament to break,
producing daughter scroll waves. At the time of writing this article,
there is no published theory of this mechanism of SWT, although the
perturbation theory described above has been recently extended to
cover non-uniform anisotropic models (#RDSaniso).
A brief history of the question
- Yakushevich (1984) derived #filmotion for scroll waves in complex Ginzburg-Landau equation with scalar diffusion. In that case, \(\alpha\) coincides with the diffusion coefficient and \(\beta=0\ .\) This implied, in particular, that circular filaments must contract. She also introduced the concept of filament tension (as "filament elasticity").
- Panfilov and Rudenko (1987) have found in simulations assuming axial symmetry, that scroll rings in a piece-wise linear modification of FitzHugh-Naumo model can expand as well as contract.
- Brazhnik et al (1987) proposed a semi-phenomenological "kinematic" theory which indicated that expansion, rather than contraction, of circular filaments is associated with lower excitability. They also noted that if the dynamics of filaments of arbitrary shape is such that every small piece moves as if it was a piece of circular filament (thus in fact postulating #filmotion), then expansion of circular filaments would signify growth of filament length and instability of straight shape of filaments, which may lead to complicated and possibly random behaviour.
- Keener (1988) derived equations of motion of curved and twisted scrolls, using singular perturbation theory for a generic reaction-diffusion system (#RDS), in Frenet-Serret coordinates. These equations were different from #filmotion in that the evolution of scroll filament shape and its twist were linked.
- Biktashev et al. (1994) showed that terms in Keener's equations, linking filament dynamics to twist, do in fact vanish, and obtained #filmotion as written. They also demonstrated relationship between coefficient \(\alpha\) and rate of change of filament length (#lengthrate), confirmed that prediction by numerical simulations in FitzHugh-Nagumo model, and mentioned possible links to cardiac fibrillation.
- Winfree (1994a, 1994b) used word "turbulent" when describing dynamics of persistent tangled filaments in his 3D of FitzHugh-Nagumo model and speculated on nature of SWT based on those, and its possible link to cardiac fibrillation.
- Panfilov and Keener (1995) showed that fiber rotation (#RDSaniso) could cause scroll wave breakup in a piecewise linear variant of FitzHugh-Nagumo model.
- Biktashev (1998) observed SWT mediated by negative filament tension, in simulations of 3D FitzHugh-Nagumo model.
- Fenton and Karma (1998a and 1998b) simulated the effect of fiber rotation on the stability of scroll waves, using FitzHugh-Nagumo model and Fenton-Karma models and described the twiston mechanism of scroll wave break-up.
- Hakim and Karma (1999) have shown analytically that negative filament tension is observed in the large core (i.e. suppressed excitability), stiff limit of generic FitzHugh-Nagumo type models.
- Henry and Hakim (2000) numerically predicted negative tension in the Barkley model.
- Alonso et al. (2004) demonstrated SWT in 3D numerical simulations of Barkley model.
- Alonso et al. (2006) numerically predicted negative filament tension in the Oregonator model of the Belousov-Zhabotinsky reaction and observed the SWT in 3D simulations, for cases where 2D spirals are both stationary and meandering spirals.
- Alonso and Panfilov (2007) numerically predicted negative filament tension and observed SWT in 3D simulations for the Luo-Rudy phase 1 model of cardiac tissue at low excitability parameters.
- Verschelde et al. (2007) derived #filmotion using Fermi coordinates.
- Alonso and Panfilov (2008) have found numerically negative filament tension in the Luo-Rudy phase 1 model of cardiac tissue for parameters far from large core, low excitability limit, and for which 2D spirals are meandering.
- Dierckx et al. (2009) extended the perturbative dynamics of scroll waves based on Fermi coordinates, to models with non-uniform anisotropy (#RDSaniso).
References
- Alonso, S. and Panfilov, A.V. (2008). Negative filament tension at high excitability in a model of cardiac tissue. Physical Review Letters 100(21): 213101.
- Alonso, S. and Panfilov, A. V. (2007). Negative filament tension in the Luo-Rudy model of cardiac tissue. Chaos 17(1): 015102.
- Alonso, S.; Kahler, R.; Mikhailov, A. S. and Sagues, F. (2004). Expanding scroll rings and negative tension turbulence in a model of excitable media. Physical Review E 70(5): 056201.
- Alonso, S.; Sagues, F. and Mikhailov, A. S. (2006). Negative-tension instability of scroll waves and Winfree turbulence in the Oregonator model. Journal of Physical Chemistry A 110(43): 12063-12071.
- Biktashev, V. N. (1998). A three-dimensional autowave turbulence. International Journal of Bifurcation and Chaos 8(4): 677-684.
- Biktashev, V. N.; Holden, A. V. and Zhang, H. (1994). Tension of organizing filaments of scroll waves Phil. Trans. Roy. Soc. London, ser. A 347: 611--630.
- Brazhnik, P. K.; Davydov, V. A.; Zykov, V. S. and Mikhailov, A. S. (1987). Vortex rings in excitable media. Zh. Eksp. Teor. Fiz. 93: 1725-1736.
- Fenton, F. and Karma, A. (1998a). Fiber-rotation-induced vortex turbulence in thick myocardium. Physical Review Letters 81(2): 481-484.
- Fenton, F. and Karma, A. (1998b). Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos 8(1): 20-47.
- Dierckx, H.; Bernus, O. and Verschelde, H. (2009). A geometric theory for scroll wave filaments in anisotropic excitable media. Physica D 238(11-12): 941-950.
- Henry, H. and Hakim, V. (2000). Linear stability of scroll waves Physical Review Letters 85(25): 5328-5331.
- Keener, J. P. (1988). The dynamics of 3-dimensional scroll waves in excitable media. Physica D 31(2): 269-276.
- Panfilov, A. V and Keener, J. P. (1995). Re-entry in three-dimensional FitzHugh-Nagumo medium with rotational anisotropy. Physica D 84: 545-552.
- Panfilov, A. V. and Rudenko, A. N. (1987). Two regimes of the scroll ring drift in the three-dimensional active media. Physica 28D: 215-218.
- Hakim, V. and Karma, A. (1999). Theory of spiral wave dynamics in weakly excitable media: Asymptotic reduction to a kinematic model and applications. Physical Review E 60(5): 5073-5105.
- Verschelde, H.; Dierckx, H. and Bernus, O. (2007). Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue Physical Review Letters 99(16): 168104.
- Winfree, A. T. (1994b). Electrical turbulence in three-dimensional heart muscle. Science 266: 1003-1006.
- Winfree, A. T. (1994a). Persistent tangled vortex rings in generic excitable media. Nature 371: 233-236.
- Yakushevich, L. V. (1984). Vortex filament elasticity in an active medium Studia Biophysica 100: 195-200.
