# Stability Analysis

Two
important questions in the investigation of cardiac arrhythmias are
how these activation patterns develop and how their complexity can be
characterized. What properties of the tissue determine its
susceptibility to arrhythmias? Where and when are certain activation
patterns most sensitive to perturbations? What makes irregular
activity easy or difficult to terminate? We expect that the answers
will yield valuable information for the prevention of arrhythmias and
about strategies to terminate them. One of our approaches to tackle
these questions is to view cardiac tissue as a high-dimensional
non-linear dynamical system. Using a number of different numerical
models of cardiac tissue, we carry out linear stability analysis of
different activation patterns to assess their (ir)regularity. As a
very general method that can be applied to arbitrary attractors of
the system, this procedure yields growth rates (so-called Lyapunov
exponents) of perturbations localized at specific sensitive spots
(given by Lyapunov vectors) and allows us to estimate the attractor
dimension – a quantitative measure for the complexity of the
dynamics. Abrupt changes of excitation patterns (Fig. 1) and
symmetries in the system can be detected as well as the onset of
chaotic activity.

### Figure 1

Detection
of bifurcations via Lyapunov exponents. The bifurcation parameter ε is a model parameter, which turns a rigidly rotating spiral wave
(left) into a meandering spiral (right). Transitions between
qualitatively different activation patterns (bifurcations) are
indicated by one or more Lyapunov exponents becoming zero, which is
the case at ε≈0.06
here. Lyapunov exponents (i.e. growth rates of perturbation modes) are ordered by size, with λ_{1}
being the largest. All Lyapunov exponents are less or equal to zero,
since both the rigidly rotating spiral wave and the meandering spiral
wave are stable activation patterns.

In
extended media instabilities often occur first locally and then grow
and spread out. Such scenarios can be investigated and characterized
by means of Lyapunov vectors. For spatio-temporal systems Lyapunov
vectors are given as patterns evolving in time and may thus be
interpreted as generalizations of the concept of (active) modes [1].
In state space (more precisely, in the tangent space of the state
space) Lyapunov vectors point in the directions of characteristic
growth of perturbations (on average quantified by Lyapunov
exponents). Usually two types of “Lyapunov vectors” have to be
distinguished: (i) the orthogonal set of vectors occurring with the
standard algorithm for computing Lyapunov exponents (based on
QR-decomposition or Gram-Schmidt reorthogonalisation) and (ii)
so-called covariant Lyapunov vectors, that are not orthogonal but
possess several desirable (invariant) features. Covariant Lyapunov
vectors became practically available only recently due to novel
numerical algorithms proposed by Ginelli et al. [2] and Wolfe and
Samuelson [3]. With a view to applications to excitable media we
prepared a detailed study of the theory and computation of covariant
Lyapunov vectors including new alternatives for their efficient
computation [4].

In
order to transfer the knowledge we gain about the nature of certain
instabilities to experiments, the next step will be to connect these
rather abstract measures to quantities that can be measured in real
cardiac tissue, e.g., the number of phase singularities or the
propagation velocity. Preliminary results show that there is a close
connection between the complexity of spatio-temporal chaos and the
number of rotating waves. Another focus is on the relationship
between the stability of wave patterns and the degree of cellular
heterogeneity (e.g.,
fibrotic tissue).

### References

- P.V.
Kuptsov and U. Parlitz,
*Phys.
Rev. E* **81**,
036214 (2010).
- F. Ginelli et al.,
*Phys.
Rev. Lett.* **99**,
130601 (2007).
- C.L.
Wolfe and R.M. Samuelson,
*Tellus* **59A**,
355-366 (2007).
- P.V.
Kuptsov and U. Parlitz, (submitted for publication).