View source for Scroll wave turbulence

<strong>Scroll wave turbulence</strong> is a self-sustained regime in
a three-dimensional 
[[autowaves|autowave]] (e.g. 
[[excitable media|excitable]] or
oscillatory) medium
mediated by an
instability of [[scroll waves]] and including persistent
multiplication and annihilation of scroll filaments.

== Overview ==

[[Image:ScrollTurb.gif|thumb|500px|right|ScrollTurbMovie|
Scroll wave turbulence developed from a scroll with a curved
filament.  Simulation of [[FitzHugh-Nagumo model]] in 3D, parameters
as in [[#Biktashev-1998|(Biktashev 1998)]].  Red surfaces are fronts 
<math>(u=0, \; v<0)</math> 
blue surfaces are backs 
<math>(u=0, \; v>0)</math> 
of excitation waves, and yellow lines are singularities
<math>(u=0, \; v=0)</math>
rotating around slowly moving scroll filaments, where
<math>u,v</math> 
are respectively the activator and inhibitor fields.
]]

Compared to [[drift of spiral waves]] in two spatial dimensions, the
[[scroll waves]] in three spatial dimensions have more degrees of
freedom.
While spiral waves rotate around ''cores'', 
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 breakup|spiral wave break-up]] in two spatial dimensions.
However, scroll wave turbulence is a 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 [[Meander of spiral waves|meandering]].  At
the time of writing this article, three specifically three-dimensional
mechanisms of SWT in excitable and oscillatory 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. Another is the finite-wavelength instability of scroll
filaments, which causes them to spontaneously deform into ''wrinkled''
or ''zig-zag'' shape.  The third one is observed in non-uniformly
anisotropic media and is related to development of twist
instabilities. Apart from that, any mechanism that could produce
turbulence in two dimensions would, of course, reveal itself in three
dimensions as well (and these mechanisms are not considered in this
article).  SWT is thought to be relevant for understanding of
[[Cardiac arrhythmia|fibrillation of the heart]]. 
Cardiac tissue is excitable and three-dimensional, and reentrant waves
may take the form of scroll waves. Scroll waves can produce
tachycardia and unstable 3D waves are associated with fibrillation.

== The concept of filament tension ==

The concept of filament tension can be understood in terms of 
perturbative dynamics of a scroll wave
([[#Yakushevich-1984|Yakushevich, 1984]];[[#Keener-1988|Keener, 1988]];[[#Biktashev-etal-1994|Biktashev et al., 1994]]). Consider a reaction-diffusion system

:<math>\label{RDS}
\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
</math>

and assume existence of stationarily
rotating spiral solutions in <math>\mathbb{R}^2\ ,</math> 

:<math>\label{Spiral}
\mathbf{u}({\vec r},t)
  =\mathbf{U}(\rho,\theta+\omega t),
</math>


where <math>\rho,\theta</math> are polar coordinates in <math>\mathbb{R}^2</math>,
and the rotation frequency 
<math>\omega</math> 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 <math>\mathbb{R}^3</math> may be viewed
as a solution of the form

:<math>
  \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), 
</math>

where 
<math>\varepsilon</math>
is a formal small parameter measuring
deformation of a scroll wave compared to the straight scroll, 
<math>{\vec R}={\vec R}(\sigma,t)</math>
is the parametric equation of filament
position at time 
<math>t\ ,</math> 
<math>\Phi=\Phi(\sigma,t)</math> 
is the distribution of the rotation phase along the filament,
<math>{\vec N}={\vec N}(\sigma,t)</math> 
and 
<math>{\vec B}={\vec B}(\sigma,t)</math> 
are the unit principal normal and binormal vectors to the filament at point
<math>{\vec R}(\sigma,t)\ .</math>
Vectors
<math>{\vec N}</math> and
<math>{\vec B}</math>
together with tangent vector
<math>{\vec T}</math>
make a [http://en.wikipedia.org/wiki/Frenet–Serret_formulas Frenet-Serret triple] which can be defined using arc length
differentiation operator 
<math>
  \mathcal{D}_s{u}(\sigma)=|\partial_{\sigma}{\vec
    R(\sigma)}|^{-1}\partial_{\sigma}u(\sigma).
</math>
Then the tangent vector is 
<math>{\vec T}=\mathcal{D}_s{ {\vec R} }\ ,</math>
the curvature
<math>\kappa</math>
and the normal unit vector
<math>{\vec N}</math> 
are defined by 
<math>  \kappa{\vec N}=\mathcal{D}_s{ {\vec T} } \ ,</math>
and the binormal vector 
<math>
{\vec B}={\vec T}\times{\vec N}
</math>
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, <math>\kappa=0\ .</math> 
An alternative description, suggested by [[#Verschelde-etal-2007|Verschelde et al. (2007)]],
(see also [[#Dierckx-Verschelde-2013|Dierckx and Verschelde, 2013]])
free from this disadvantage, is in terms
Fermi-Walker coordinates, corresponding to a [[Levi-Civita connection|Levi-Civita (torsion-free metric)
connection]] along the filament (hereafter called torsion-free coordinates for
brevity). Either way, the
equation of motion of the filament, in the assumption of small filament
curvature, <math>\kappa=\mathcal{O}(\varepsilon)\ ,</math> and small twist,
<math>\mathcal{D}_s\Phi=\mathcal{O}(\varepsilon)\ ,</math> can be written as

:<math>\label{filmotion}
\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)
</math>

Equation \eqref{filmotion} is written in the assumption that parameterization of
the filament is chosen in such a way that a point with a fixed
<math>\sigma</math> moves orthogonally to the filament. 
The coefficients in the equation of motion are calculated
using the response functions <math>\mathbf{W}_1</math> (see [[Drift of spiral waves]]) as 

:<math>
 \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
   .
</math>

Following [[#Biktashev-etal-1994|Biktashev et al. (1994)]], let us now
consider the total length of the filament, defined at each
<math>t</math>:

:<math>\label{totlength}
S(t)=\int \mathrm{d}s
  = \int \left| \partial_{\sigma}{\vec R} \right| \,\mathrm{d}\sigma, 
</math>

where the integral is taken over the whole filament. 
Differentiation of \eqref{totlength}, with account of \eqref{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

:<math>\label{lengthrate}
\frac{\mathrm{d}S}{\mathrm{d}t} \approx - \alpha \int \kappa^2
  \,\mathrm{d}s .
</math>

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
<math>\alpha>0</math>
and increase if 
<math>\alpha<0\ .</math>
Hence the coefficient <math>\alpha</math> is often called ''filament tension''.

Filament tension can be defined via the asymptotic rate of collapse or
expansion of 
scroll rings of large radius and
exact axial symmetry, in which
case the problem becomes mathematically equivalent to that of a
[[Drift_of_spiral_waves#Anisotropy_induced_drift|drift of a spiral 
wave in an electric field]]
([[#Hakim-Karma-1999|Hakim, Karma 1999]];[[#Henry-Hakim-2002|Henry, Hakim 2002]]).
This definition can be formally
extended to the case when unperturbed spiral waves are 
[[Meander of spiral waves|meandering]],
even though at the moment of writing this
article, an asymptotic theory of evolution of meandering scroll does
not yet exist. 

Filament tension depends on parameters of the medium, typically
becoming negative in media with lower excitability, where spiral waves
have larger radii ([[#Panfilov-Rudenko-1987|Panfilov and Rudenko,
1987]]; [[#Brazhnik-etal-1987|Brazhnik et al., 1987]];
[[#Krinsky-etal-1996|Krinsky et al., 1996]];
[[#Hakim-Karma-1999|Hakim and Karma (1999)]]), although there are
exceptions [[#Alonso-Panfilov-2008|(Alonso and Panfilov,
2008)]] (this is important as cardiac tissue is typically highly
excitable). 
[[#Henry-2004|Henry (2004)]] has noted a correspondence
between the sign of filament tension and the [[Meander of spiral
waves|spiral wave meandering]]: namely, that the manifold
<math>\alpha=0</math> in the parametric space of \eqref{RDS} in 3D
correlates with the locus of spontaneously drifting spirals in
\eqref{RDS} in 2D, separating "outward petal" and "inward petal"
meandering patterns, which is related to the resonance between the
Hopf and the Euclidean frequencies.

== SWT mediated by negative filament tension ==

As was noted already by [[#Brazhnik-etal-1987|Brazhnik et al (1987)]],
negative filament tension should make a straight scroll 
unstable, as any sufficiently smooth small deviation from the straight
shape will grow. The constant increase in filament length due to
curvature-induced drift as predicted by \eqref{totlength} 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 develop ([[#Biktashev-1998|Biktashev (1998)]],
[[#Alonso-etal-2004|Alonso et al. (2004)]],
[[#Alonso-etal-2006b-JPCA|Alonso et al. (2006b)]],
[[#Alonso-Panfilov-2007|Alonso and Panfilov (2007)]]
see also 
[[#Alonso-etal-2013|Alonso et al. (2013)]]
for a review). 
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 not known if stabilization and extinction
may still be possible, just less probable, in larger media. The
chaotic character of SWT is currently only a conjecture. Note
that for <math>\beta=0</math> and planar filament, equation
\eqref{filmotion} is similar to [[Kuramoto-Sivashinsky equation]]
without its regularization term, and chaotic nature of solutions of
Kuramoto-Sivashinsky equation is known.  Comparative statistical
analysis of numerical solutions of SWT due to negative tension and
due to the two-dimensional spiral break-up instabilities, see
e.g. [[#Zaritsky-etal-2004|Zaritsky et
al. 2005]],[[#Clayton-2008|Clayton
2008]],[[#Davidsen-etal-2008|Davidsen et al. 2008]], may be considered
as indirect confirmation of their chaotic nature. At the moment of
writing this article, such studies have been mostly empirical,
although attempts have been made to summarise the findings in terms of
Markov models [[#Davidsen-etal-2008|(Davidsen et al. 2008)]].

The above quoted perturbation theory is based on the assumption that
2D spirals are stationarily rotating, formalized by \eqref{Spiral}. At the
time of writing this article, an extension to the 
[[Meander of spiral waves|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.

The search for possible low-voltage [[defibrillation]] methods
motivates control of SWT with the view of elimination of all scroll
waves. Alonso et al. ([[#Alonso-etal-2003|2003]],
[[#Alonso-etal-2006a-C|2006a]]) explored periodic
time-dependent variation of parameters of the mathematical model,
representing repetitive low-voltage electrical stimulation. They
focused on ''super-resonant'' case with stimulation frequency higher
than that of the scrolls. Such stimulation can sometimes effectively
reverse the sign of filament tension from negative to positive, thus
stopping the breeding of scroll waves and "taming" the SWT. This,
however, does not guarantee elimination of already existing scroll
waves if they are not scroll rings topologically. 
Such elimination can be achieved by ''resonant'' stimulation, i.e.
with the frequency equal to that of the scroll waves, exploiting
[[Drift of spiral waves#Resonant drift|resonant drift]] phenomenon
[[#Morgan-etal-2008|(Morgan et al. 2008)]].
Super-resonant ''localized'' stimulation can cause
[[drift of spiral waves#High_frequency_induced_drift|induced drift]] of scroll waves, which can also be used for
suppressing the SWT ([[#Zhang-etal-2005|Zhang et al. 2005]]). 


== SWT due to twisted anisotropy ==

Three-dimensional 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: 

:<math>\label{RDSaniso}
\frac{\partial u_j}{\partial t} = 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\}.
</math>


In cardiac excitation models, <math> D^{j,k}_{\mu,\nu}({\vec
r})=P^{i,j} Q_{\mu,\nu} </math> where matrix component
<math>P^{1,1}=1</math> (assuming 1 is the index of the transmembrane
voltage field) and all other <math>P^{i,j}=0\ ,</math> and tensor
<math>Q_{\mu,\nu}</math>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 <math>(x,y)</math> plane, and this direction
linearly changes depending on the <math>z</math>-coordinate
([[#Panfilov-Keener-1995|Panfilov and Keener, 1995]]; Fenton and Karma
[[#Fenton-Karma-1998a|1998a]] and [[#Fenton-Karma-1998b|1998b]]).
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 can be extended to
cover non-uniformly anisotropic models \eqref{RDSaniso} 
([[#Verschelde-etal-2007|Verschelde et al. 2007]], 
[[#Dierckx-etal-2009|Dierckx et al. 2009]]). 

== SWT due to finite-wavenumber instability ==

A peculiar mechanism of three-dimensional instability of scroll waves
that can also lead to SWT has been described in the [[complex
Ginzburg-Landau equation]] (see e.g.  [[#Aranson-Bishop-1997|Aranson
and Bishop (1997)]], [[#Aranson-Kramer-2002|Aranson and Kramer
(2002)]], [[#Rousseau-etal-2008|Rousseau et al. (2008)]],
[[#Reid-etal-2011|Reid et al. (2011)]] and references therein).

:<math>\label{CGLE}
  \partial_t A = A + (1+ib)\nabla^2 A - (1+ic) |A|^2 A ,
  \qquad
  A \in \mathbb{C} .
</math>

Although the filament tension in this oscillatory autowave model is
always positive, <math>\alpha=1+b^2</math>, so a straight scroll is
stable against perturbations that are sufficiently smooth in space, at
some combinations of parameters <math>b,c</math> it still can develop
a three-dimensional instability with respect to finite wavenumber
(related to two-dimensional "accelerating" instability, which is in
turn related to near-Galilean invariance of \eqref{CGLE} in the limit
<math>b \gg 1</math>). This instability can lead to spontaneously
twisted vortices with helical-shaped filaments, or to SWT (called
"spatiotemporal intermittency").

Notably, a finite-wavenumber three-dimensional instability of scroll
waves, leading to helical filaments, was also described for 
[[Barkley model]] by [[#Aranson-Mitkov-1998|Aranson and Mitkov (1998)]] and
[[#Henry-Hakim-2002|Henry and Hakim (2002)]]; however they did not
report that this instability could lead to SWT in that model.

== "Winfree turbulence" ==

In literature, SWT is sometimes referred to as "Winfree turbulence",
due to high-impact publications by A.T. Winfree
([[#Winfree-1994a|1994a]] and [[#Winfree-1994b|1994b]]). In the
[[#Winfree-199b|Science paper]], Winfree discussed plausible
mechanisms of cardiac fibrillation, including its possible links with
three-dimensional instabilities of scroll waves. In that context,
Winfee mentioned the topological opportunities allowed by the 3D space
for filaments to "lie on their sides" and to "snake about, fragment
and close in rings". As to possible causes of "turbulent" behaviour,
Winfree mentioned inhomogeneity and anisotropy, but also referred to
the cases of "turbulent" behaviour in a perfectly homogeneous and
isotropic medium, presented in his earlier [[#Winfree-199a|Nature
paper]]. These were in the form of numerical solutions in the
[[FitzHugh-Nagumo]] model with knotted or entangled filaments, which
were topologically persistent but did not remain periodic in time
hence called "turbulent". The parameters of those "turbulent" examples
corresponded to meandering rather than rigidly rotating spirals, which
was the apparent cause of the non-periodicity.

== References ==

* {{Bibitem article 2|Negative filament tension at high excitability in a model of cardiac tissue.|Physical Review Letters|100(21)|2008|213101|Alonso|S.|Panfilov|A.V.|label=Alonso-Panfilov-2008}}
* {{Bibitem article 2|Negative filament tension in the Luo-Rudy model of cardiac tissue.|Chaos|17(1)|2007|015102|Alonso|S.|Panfilov|A. V.|label=Alonso-Panfilov-2007}}
* {{Bibitem article 3|Taming Winfree Turbulence of Scroll Waves in Excitable Media.|Science|299|2003|1722-1725|Alonso|S.|Sagues|F.|Mikhailov|A.|label=Alonso-etal-2003}}
* {{Bibitem article 4|Expanding scroll rings and negative tension turbulence in a model of excitable media.|Physical Review E|70(5)|2004|056201|Alonso|S.|Kahler|R.|Mikhailov|A. S.|Sagues|F.|label=Alonso-etal-2004}}
* {{Bibitem article 3|Periodic forcing of scroll rings and control of Winfree turbulence in excitable media.|Chaos|16|2006|023124|Alonso|S.|Sagues|F.|Mikhailov|A. S.|letter=a|label=Alonso-etal-2006a-C}}
* {{Bibitem article 3|Negative-tension instability of scroll waves and Winfree turbulence in the Oregonator model.|Journal of Physical Chemistry A|110(43)|2006|12063-12071|Alonso|S.|Sagues|F.|Mikhailov|A. S.|letter=b|label=Alonso-etal-2006b-JPCA}}
* {{Bibitem article 3|Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics.|Bulletin of Mathematical Biology|75|2013|1351-1376|Alonso|S.|B&auml;r|M.|Panfilov|A. V.|label=Alonso-etal-2013}}
* {{Bibitem article 2|Instability and Stretching of Vortex Lines in the Three-Dimensional Complex Ginzburg-Landau Equation|Physical Review Letters|79(21)|1997|4174-4177|Aranson|I. S.|Bishop|A. R.|label=Aranson-Bishop-1997}}
* {{Bibitem article 2|The world of the complex Ginzburg-Landau equation.|Reviews of Modern Physics|74|2002|99-143|Aranson|I. S.|Kramer|L.|label=Aranson-Kramer-2002}}
* {{Bibitem article 2|Helicoidal instability of a scroll vortex in three-dimensional reaction-diffusion systems|Physical Review E|58|1998|4556|Aranson|I.|Mitkov|I.|label=Aranson-Mitkov-1998}}
* {{Bibitem article 1|A three-dimensional autowave turbulence.|International Journal of Bifurcation and Chaos|8(4)|1998|677-684|Biktashev|V. N.|label=Biktashev-1998}}
* {{Bibitem article 3|Tension of organizing filaments of scroll waves|Phil. Trans. Roy. Soc. London, ser. A|347|1994|611--630|Biktashev|V. N.|Holden|A. V.|Zhang|H.|label=Biktashev-etal-1994}}
* {{Bibitem article 4|Vortex rings in excitable media.|Zh. Eksp. Teor. Fiz.|93|1987|1725-1736|Brazhnik|P. K.|Davydov|V. A.|Zykov|V. S.|Mikhailov|A. S.|label=Brazhnik-etal-1987}}
* {{Bibitem article 1|Vortex filament dynamics in computational models of ventricular fibrillation in the heart.|Chaos|18|2008|043127|Clayton|R. H.|label=Clayton-2008}}
* {{Bibitem article 3|Filament-induced surface spiral turbulence in three-dimensional excitable media|Physical Review Letters|101|2008|208302|Davidsen|J.|Zhang|M.|Kapral|R.|label=Davidsen-etal-2008}}
* {{Bibitem article 3|A geometric theory for scroll wave filaments in anisotropic excitable media.|Physica D|238(11-12)|2009|941-950|Dierckx|H.|Bernus|O.|Verschelde|H.|label=Dierckx-etal-2009}}
* {{Bibitem article 2|Effective dynamics of twisted and curved scroll waves using virtual filaments.|Physical Review E|88|2013|062907|Dierckx|H.|Verschelde|H.|label=Dierckx-Verschelde-2013}}
* {{Bibitem article 2|Fiber-rotation-induced vortex turbulence in thick myocardium.|Physical Review Letters|81(2)|1998|481-484|Fenton|F.|Karma|A.|letter=a|label=Fenton-Karma-1998a}}
* {{Bibitem article 2|Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation.|Chaos|8(1)|1998|20-47|Fenton|F.|Karma|A.|letter=b|label=Fenton-Karma-1998b}}
* {{Bibitem article 2|Theory of spiral wave dynamics in weakly excitable media: Asymptotic reduction to a kinematic model and applications.|Physical Review E|60(5)|1999|5073-5105|Hakim|V.|Karma|A.|label=Hakim-Karma-1999}}
* {{Bibitem article 2|Linear stability of scroll waves|Physical Review Letters|85(25)|2000|5328-5331|Henry|H.|Hakim|V.|label=Henry-Hakim-2000}}
* {{Bibitem article 1|Spiral wave drift in an electric field and scroll wave instabilities|Physical Review E|70|2004|026204|Henry|H.|label=Henry-2004}}
* {{Bibitem article 1|The dynamics of 3-dimensional scroll waves in excitable media.|Physica D|31(2)|1988|269-276|Keener|J. P.|label=Keener-1988}}
* {{Bibitem article 3|Dense and Sparse Vortices in Excitable Media Drift in Opposite Directions in Electric Field|Physical Review Letters|76|1996|3854-3857|Krinsky| V.|Hamm|E.|Voignier|V.|label=Krinsky-etal-1996}}
* {{Bibitem article 3|Control of scroll wave turbulence using resonant perturbations|Physical Review E|78|2008|046207|Morgan|S. W.|Biktasheva|I. V.|Biktashev|V. N.|label=Morgan-etal-2008}}
* {{Bibitem article 2|Re-entry in three-dimensional FitzHugh-Nagumo medium with rotational anisotropy.|Physica D|84|1995|545-552|Panfilov|A. V|Keener|J. P.|label=Panfilov-Keener-1995}}
* {{Bibitem article 2|Two regimes of the scroll ring drift in the three-dimensional active media.|Physica|28D|1987|215-218|Panfilov|A. V.|Rudenko|A. N.|label=Panfilov-Rudenko-1987}}
* {{Bibitem article 3|Filament turbulence in oscillatory media|Europhysics Letters|94|2011|68003|Reid|J. C.|Chat&eacute;|H.|Davidsen|J.|label=Reid-etal-2011}}
* {{Bibitem article 3|Twisted vortex filaments in the three-dimensional complex Ginzburg–Landau equation|Chaos|18|2008|026103|Rousseau|G.|Chat&eacute;|H.|Kapral|R.|label=Rousseau-etal-2008}}
* {{Bibitem article 3|Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue|Physical Review Letters|99(16)|2007|168104|Verschelde|H.|Dierckx|H.|Bernus|O.|label=Verschelde-etal-2007}}
* {{Bibitem article 1|Persistent tangled vortex rings in generic excitable media.|Nature|371|1994|233-236|Winfree|A. T.|letter=a|label=Winfree-1994a}}
* {{Bibitem article 1|Electrical turbulence in three-dimensional heart muscle.|Science|266|1994|1003-1006|Winfree|A. T.|letter=b|label=Winfree-1994b}}
* {{Bibitem article 1|Vortex filament elasticity in an active medium|Studia Biophysica|100|1984|195-200|Yakushevich|L. V.|label=Yakushevich-1984}}
* {{Bibitem article 3|Intermittent self-organization of scroll wave turbulence in three-dimensional excitable media|Physical Review Letters|92|2004|168302|Zaritsky|R. M.|Mironov|S. F.|Pertsov|A. M.|label=Zaritsky-etal-2004}}
* {{Bibitem article 5|Suppress Winfree turbulence by local forcing excitable systems|Physical Review Letters|94|2005|188301|Zhang|Hong|Cao|Zhoujian|Wu|Ning-Jie|Ying|He-Ping|Hu|Gang|label=Zhang-etal-2005}}

== Recommended reading ==

* {{Bibitem article 2|Control of waves, patterns and turbulence in chemical systems|Physics Reports|425|2006|79–194|Mikhailov|A. S.|Showalter|K.}}

== See also ==

[[Autowaves]], [[Excitable media]], [[Scroll waves]], [[Spiral waves]], [[Drift of spiral waves]], [[Cardiac arrhythmias]], [[Belousov-Zhabotinsky reaction]]


[[Category: Cardiac Dynamics]]
[[Category:Pattern Formation]]