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\shtitle{Complex Dynamics of the Cardiac Rhythms } % short title line
\title{Complex Dynamics of the Cardiac Rhythms }
\author{Filippi S., Cherubini C.~\affil{Faculty of Engineering\\
University Campus Bio-Medico of Rome\\
via E.
Longoni 47, 00155 Rome, Italy\\s.filippi@unicampus.it}}
\abstract{Many biological systems which appear complex both in
space and time and result still not understood, require new
theoretical approaches for the nonlinear dynamics. In particular
we focus on the theoretical analysis of the underlying mechanisms
of heart dynamics. This could clarify the (apparently) chaotic
behaviour of the normal heart-beat and especially the control of
the bifurcations of dynamics arising in situations of disease. The
principal target is to find a possible clear distinction between
normal and pathological regimes. In the literature, the complex
Ginzburg Landau equation (CGLE), used as the amplitude equation
valid near a stationary bifurcation, is considered as the
prototype for heart dynamics and the related spatial and temporal
chaos. This is a very general model exhibiting spiral wave
solutions. In this article we clarify the links between the
phenomenology of the heart and the Landau-Ginsburg models.}
\begin{document}
\maketitle
\section{Introduction}
Complex systems, as observed in nature and modeled by mechanical
or electrical approximations, have the property that their
dynamics depends on many competing effects. Nevertheless a complex system
is composed, so that can be recognized more or less clearly,
depending on the system itself, a hierarchy of structures over a
wide range of time and/or length scales and the presence of some
form of coherent structures.
The description of such interrelated structures with reciprocal
influence and peculiar individual behaviors requires mathematical
models based on nonlinear differential equations. Spiral waves are
a form of self-organization observed in various excitable
biological systems. In particular in heart pathologies such as
arrhythmias and fibrillation, the wave of excitation are moving
spirals. The simplest class of mathematical models generating
spiral waves is the reaction diffusion systems on the plane. The
study in two dimensions the generation of stable and self-sustained
spiral waves (SpW) is believed to be a preliminary step towards
the comprehension of the high-complexity three dimensional patterns. In cardiac
tissue models, the role of reagents concentration transmembrane
voltage, and the role of the diffusion is played by intercellular
conductivity. In Sec.\ \ref{features} considerations concerning
features and behaviour of the cardiac rhythms are presented. In Sec.\
\ref{waves}, we focus on the phenomenological way to approach the
study of traveling waves in an excitable media. In Sec.\
\ref{landau} we explain the reasons that suggest the use of the
Landau-Ginzburg equation for the heart dynamics. Finally in Sec.\
\ref{whylandau} some ideas are exposed furnishing a theoretical
explanation of the phenomenological use of the LG equation, which has
been so successful in modeling and fitting the experimental data.
\section{Features of the Cardiac Rhythms} \label{features}
The heart is primarily composed of muscle tissue
over which a network of nerve fibers coordinates the contraction
and relaxation of the whole muscle tissue to obtain an efficient,
wave--like pumping action. The sinuatrial node (SA) is the natural
pacemaker for the heart. In the upper area of the right atrium,
it sends the electrical impulse that triggers each heartbeat. The
impulse spreads through the atria, prompting the cardiac muscle
tissue to contract
in a coordinated wave--like manner.\\
The impulse that originates from the SA node strikes the
atrioventricular (AV) node which is situated in the lower portion
of the right atrium. The AV node in turn sends an impulse through
the nerve network to the ventricles, initiating then the same
wave--like
contraction of the ventricles.\\
The electrical network serving the ventricles leaves the
atrioventricular node through the Right and Left Bundle Branches.
These nerve fibers send impulses that cause the cardiac muscle
tissue to contract. Even the simplest theoretical model shows the
enormous complexity that can arise from periodic stimulation of
non-linear oscillations. In fact many different effects can arise
from the specific kinds of stimulations and responses. In
principle can be evident a periodic synchronized rhythm, in which
appears evidence of {\it regularity }, but the most frequent ones in
biological systems are aperiodic rhythms (whose oscillation
frequency is not measurable univocally), quasi--periodic rhythms
(when two rhythms with different frequencies march through each
other with little interaction) and the most general case of chaotic
ones\cite{glass01}.
It is known that some features of normal heart rate (NHR)
variability are related to chaotic dynamics. In this view, the
long--range correlations in NHR serve as organizing principle for
highly complex, nonlinear processes that generate fluctuations on
a wide range of time scales involved in the cardiac rhythm. In
fact, the lack of a characteristic scale prevents excessive
mode--locking that restrict the functional plasticity of the
living organism.
For instance in the case of fibrillation the beating becomes
highly regular, manifesting thus the breakdown of long--range
correlations which were characterizing the multi-system
interrelation. This fact has to be interpreted as the emergence of
a dominant frequency mode leading to a highly periodic behavior,
so that rather than being a chaotic process, cardiac fibrillation
involves an unexpected amount of spatial and temporal patterning
and order \cite{holden98,gray98}.
\section{Traveling Waves in Excitable Media.}\label{waves}
A phenomenological way to approach the evolution of the cardiac
muscle lies in the direct study of traveling waves in an excitable
media (as the heart tissue has to be considered) and in finding
models in order to mimic such excitable media as systems in which a wave
propagates. The complexity of the signals which are involved in any
heart--beat has lead to inquire about the effect of a very general
kind of wave propagating over the heart. It has been shown that
spiral waves provide one of the most striking examples of pattern
formation in nonlinear active media and have attracted attention
since their description in the Belousov-Zhabotinsky (BZ) chemical
reaction medium \cite{belm}. Successively they have been described
in many other systems and models, coming soon under study for what
regards the cardiac activity, being of potentially vital interest,
as they underly some lethal pathologies.
The construction of models for complex systems spatially extended
requires also to take care of the strong irregularity which is
found both in space and time for a complex dynamical systems such
as heart tissue. Depending on the scales considered, in the
fibers' detail or in the whole heart as for few milliseconds
(chemical chain reactions) up to minutes and days (typical
duration of global rhythmic analysis) are always found new
behaviors. To this purpose, the identification of the dynamical
degrees of freedom and the instability mechanisms leading to
disorder are of great importance. The very general spiral waves
solutions of CGLE in oscillatory media under the effect of
inhomogeneity are well studied in this perspective\cite{ott}.
\section{Landau-Ginzburg equations}\label{landau}
Analogies between bifurcations in dynamical regimes
of complex systems and the phenomenology of phase transitions
suggest to investigate more deeply the reasons of interest for the
Landau-Ginzburg equation in the context of the heart dynamics.
Usually in the textbooks, the LG formulation is introduced in the
context of the ferromagnetic systems~\cite{Peskin,Zinn}. In the
microscopic scale, one typically starts the discussion presenting
an Ising-like ferromagnetic system described by a certain
d-dimensional lattice with an attractive, translation invariant,
short range two body interaction. The spin variable on the i-th
site is defined as $S_i$, with the additional symmetry $S \to -S$.
Defining ${\cal H}$ as the energy of a spin configuration,
$d\rho(S)$ the one spin
configuration which weights the spin configuration at each site, we
can define a partition function whose energy will be given by
\begin{equation}
{\cal H} (S)=-\sum_{ij}V_{ij}S_i S_j-\sum_i H_i S_i \,.
\end{equation}
The first term couples the various spins, and the second one
instead represents the coupling with an external magnetic field
$H$ whose application will favor or not one of the possible
states of magnetization.
In particular, if we require a short range ferromagnetic interaction with two
possible states of magnetization only (up and down), and we choose
$d=3$, we can approximate the theory close to the phase transition
(at temperature $T=T_C$) with a spin density $s(\vec x)$ whose
integral furnishes the magnetization M, i.e. $M=\int d^3x s(\vec
x)$. The dynamics is entirely codified in the Gibbs free energy
\begin{equation}
\int d^3x \left[ \frac12(\nabla s)^2+V(s)-Hs\right]\,,\qquad
V(s)=b(T-T_c)s^2+cs^4\
\end{equation}
where $b>0, c>0$ and $H$ is again the external magnetic field.
The variation of $G$ furnishes the well know (real) LG
equation
\begin{equation}
-\nabla^2s+2b(T-T_C)s+4cs^3-H(\vec x)=0\,.
\end{equation}
We point out that, depending by the fact that $T>T_C$ or $TT_C$, the non linear partial derivative equation is
approximated by the linear one because the macroscopic
magnetization must vanish, i.e.
\begin{equation}
-\nabla^2s+2b(T-T_C)s=H(\vec x)\,.
\end{equation}
We assume to turn the magnetic field in a point, i.e. $H(\vec
x)=H_{0}\delta^{(3)}(\vec x)$ and we compute the Green function
(zero field spin-spin correlation function)
\begin{equation}
D(\vec x)=~~=\frac{H_{0}}{4\pi r}e^{-r/\xi}
\,,\qquad \xi=[2b(T-T_C)]^{-1/2}\,.
\end{equation}
The quantity $\xi$ is know as the correlation length (the range of
correlated spin fluctuations) which becomes infinite on the
transition temperature. If one wants to improve this result, it is necessary to
adopt the methods of euclidean quantum field theory (and consequently the renormalization group).
Clearly there will be additional counter-terms which will furnish a more complicate effective
equation than the LG one. In general, adopting such theoretical framework, one will obtain that
every computed physical quantity will be related to the phase
transition by certain power laws (critical exponents) which result more precise than the one computed using the standard LG theory. For the
magnetization for example one will have $M\propto (T_C-T)^\beta$.
The interesting result will be that these indices will take a
fixed value for all systems in a given universality class. For
both single-axis magnets and for fluids $\beta=0.313$ for example.
This is the universality in the critical exponents, which links
totally (apparently) different systems using a unique theory. We stress that the
reason why such a miraculous thing happens is because the basic
idea of the LG theory is that the Gibbs energy of the different
physical system can be always expanded in powers of an order
parameter (in our ferromagnetic case this is the temperature) \cite{Peskin,Zinn}.
One can generalize the previous LG scheme introducing a temporal dependence on the equations and assuming
that the various quantities result defined in the complex domain.
In this case one obtains the complex Landau-Ginzburg equation
\begin{equation}
\partial_t A=\mu A-(1+i\alpha)\vert A\vert^2 A+(1+i\beta)\nabla^2A\,,
\end{equation}
which describes, again, a large quantity of phenomena ranging from
nonlinear waves to second-order phase transitions, from
superconductivity, superfluidity, and Bose-Einstein condensation
to liquid crystals. One can easily recognize the presence of a
diffusion linear equation plus a ``mass term" and non-linearities.
The study of ``wave-like solutions" shows the existence of a
characteristic frequency $\omega_0$ which manifests an Hopf
bifurcation between two different regimes. In this sense the
quantity $\omega$ can be considered as the {\it order parameter}.
A detailed discussion of the properties of such an equation can be
found in ref.~\cite{Vulpiani}. It can be shown in particular that
this non linear PDE can generate rotating spiral-like
solutions~\footnote{As a remark, based on a preliminary and yet
unpublished analysis, we would like to say that this peculiar
morphology of the solutions is not necessarily due to the
non-linearities of the LG equation.}.
\section{Landau-Ginsburg and Heart dynamics}\label{whylandau}
The previous section concerning the Landau-Ginsburg theory suggests some considerations for the heart dynamics. Considering the real LG equation first, we can present the following comments.
\begin{itemize}
\item
As the simplest possible modelization, we assume plausible that
the heart dynamics, as any thermodynamical system, possesses a
Gibbs free energy which will be expandable in series of a certain
order parameter, now different from temperature. We think natural
to establish some analogies between this cardiac system and the
magnetic LG one. First of all we point out that the contraction of
the cardiac muscle cells is triggered by electrical excitations,
consequently we think that the counterpart of the spin-density now
will be a density of polarization. The external magnetic field
will be replaced by an electric external excitation (pacemaker
cells).
\item Again, in this simplest real case, we expect that such a macroscopic behavior must come from a
microscopic Ising-like structure in which instead on the
magnetizations $\uparrow\downarrow$, there will be two different
states for the heart cells, i.e. polarized/depolarized.
\item We expect that in analogy with modern statistical mechanics, once
one has understood the role of the order parameter and the microscopic
structure, the use of the renormalization group will be necessary
to improve the LG modelizations.
\item We expect that in analogy with the statistical mechanics one
could perform the exact evolution of the model using the Lattice
Field Theory. This will require the use of Montecarlo simulations.
\item In relation with the previous points, it is well known that a macroscopic non-linear electric circuit
can be a model of the heart dynamics. The idea of
considering a network of many small non-linear similar electric
circuits in interaction could be an approximation of the realistic
lattice of polarizing-depolarizing cells of the exact statistical
models. Modern engineering has many tools to simulate this type
dynamics: we have in mind for example the numerical simulations on huge supercomputers of
microprocessors (having millions of interacting elements) which test the project before its physical implementation.
\end{itemize}
These comments furnish a theoretical justification for the
phenomenological use in the literature of the LG equation which is so successful in modeling and fitting the experimental
data. It would be a fantastic result to show that the universality
classes of the phase transitions could be extended to biological
models too, manifesting in an elegant way the simplicity of
Nature. However the considerations presented above are clearly an approximation. In fact if there is an electric wave
propagating, we expect by classical electrodynamics to find
magnetic waves too. But from electromagnetism, we have two degrees of freedom now and consequently
we expect that the complete description will be possible by using a complex partial derivative equation, which contains in its real and imaginary parts all the necessary informations.
In this sense the complex Landau-Ginsburg equation results the best candidate, although the microscopic considerations previously presented become in this case much more complicate.
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c) We conclude with the last remark.
~~