# Synchronization and Information Transmission in Spatio-Temporal Networks of
Deformable Units

F. M. Moukam Kakmeni1,2 and M. S. Baptista1,3

###### Abstract

We study the relationship between synchronization and the rate with which
information is exchanged between nodes in a spatio-temporal network that
describes the dynamics of classical particles under a substrate Remoissenet-
Peyrard potential. We also show how phase and complete synchronization can be
detected in this network. The difficulty in detecting phase synchronization in
such a network appears due to the highly non-coherent character of the
particle dynamics which unables a proper definition of the phase dynamics. The
difficulty in detecting complete synchronization appears due to the spatio
character of the potential which results in an asymptotic state highly
dependent on the initial state.

###### pacs:

05.45.-a; 05.45.Gg; 05.45.Pq; 05.45.Xt

##  1\. Introduction

The sine-Gordon potential and similar others have been used to model the
dynamics of many systems in physics, biology and engineering
[?,?,?,?,?,?,?,?]. However, in real physical systems, the shape of the
substrate potential can deviate from the standard one with a direct incidence
on the stability properties of the system. In physical situations, such as
charge-density waves, Josephson junctions, or crystals with dislocations, the
application of the standard sine-Gordon model becomes too restrictive. In
recent years, a number of potentials whose shapes can be turned at wish have
appeared in the literature of nonlinear dynamical systems [?,?,?,?,?,?,?].
These more realistic potentials certainly provide richer insights onto the
physics of reals systems than what is predicted using the conventional, rigid
models such as the sine-Gordon, double-sine-Gordon and $\phi^{4}$ potentials.
In particular, we can expect a more rich and complex synchronization phenomena
in models of nonlinear oscillators involving them.

The purpose of the present paper is to study networks formed by oscillators
under realistic shape deformable potentials. To model the network, we use the
Remoisnet-Payrar potential, which has been extensively used in the literature
to describe the disturbance of the sinusoidal shape of the substrate periodic
potential of the Sine Gordon equation [?,?,?].

We are mainly interested in the complex relationship between synchronization
and transmission of information. By synchronization, we mean complete
synchronization (CS) [?,?,?,?] and chaotic phase synchronization (PS) [?]. The
information point-of-view will be provided by the procedure described by
Baptista et al. in Refs. [?]. As we shall show synchronization and information
are directly related in such an active network. The larger the synchronization
is the larger the rate with which information is exchanged between nodes in
the network, the so called mutual information rate (MIR).

Such relationship can be experimentally explored when one needs to observe how
nodes are attached to each other in a real network. For situations where the
nodes of the network are neither completely synchronous nor phase synchronous,
the MIR provides one the level of connectivity. In addition, the MIR limits
the amount of information that can be retrieved in some point of the network
about an arbitrary external stimulus.

Due to the spatio character of the studied network, both approaches, the ones
in Refs. [?,?,?,?] and the ones in Refs. [?] might face difficulties to be
implemented and this work resolves many of them. In particular, we study
networks which have node trajectories departing from randomly initial
conditions. That creates a situation similar to the one observed in networks
constructed with nodes presenting different parameters, when the methods in
Ref. [?] should be used with precaution.

Note that a quite number of physical objects allowing a model description with
the aid of the sine-Gordon equation are known: arrays of forced damped
pendula, vortices in long Josephson junctions, charge-density waves in quasi-
one-dimensional conductors etc… [?,?]. For real physical systems, the account
of various disturbances and of a more complex character of atomic interactions
breaks the exact integrability of the initial sine-Gordon equation, leaving
the possibility for describing the system dynamics in terms of the same quasi-
particles which now interact with each other.

The rest of the paper is organized as follows: in Sec. 2. we explore the
dynamics of the network in consideration and analyze the effect of the
deformability parameter in the substrate potential on the stability
synchronization of the network. In Sec. 3. we analyze phase synchronization in
such networks, and Sec. 4. is devoted to the study of information transmission
within the nodes of the network. Finally, we present the conclusions in Sec.
5..

##  2\. Synchronization dynamics of the networks

###  2.1 Description of the networks

We first investigate the dynamical properties of a single particle in a
deformable substrate potential. If we define the variable $x$ as the
displacement of the particle in the potential well, then the equation of
motion describing its dynamics reads

$\ddot{x}+\lambda\dot{x}+\omega^{2}\displaystyle\frac{\partial
V(x,r)}{\partial x}=\eta_{0}\cos\Psi t.$ (1)

In this work we consider the following fixed set of parameters
$\lambda=0.01,\omega=1,\eta_{0}=0.19$. The parameters $\Psi$ and $r$ will be
varied.

Recall that $x$ is the coordinate variable which characterizes the behavior of
the particle in the potential well $V(x,r)$. The new issues of our model under
consideration are the following: we apply an AC force
$\eta(t)=\eta_{0}\cos\Psi t$ to the particle and assume also the external
viscous damping with a coefficient $\lambda$. In this work, $V(x,r)$ is a
nonlinear potential with a deformable shape introduced by Remoissenet-Payrard
to study the coherent structure in a network formed by a similar system. There
are many versions of this potential, but we concentrate our analyses on the
most general case defined as [?,?,?,?,?,?,?]

$V(x,r)=(1-r)^{2}\displaystyle\frac{1-\cos x}{1+r^{2}+2r\cos x}$ (2)

Figure  1.: Form of the potential as a function of $r$ and the corresponding
periodic orbits for a free particle ($\eta_{0}$=0 and $\lambda=0$). The
pictures in the left (right) column shows $x$ vs. $V(x,r)$ ($x$ vs.
$\dot{x}$).

where the deformability parameter $r$ fulfills the condition $|r|<1$.

The advantageous feature of this potential can be summarized in the fact that
it reproduces the sine Gordon ($r=0$) while avoiding most of it shortcomings.
A shape of broad wells separated by narrow barriers can be obtained for $r>0$
and for $r<0$, a shape of deep narrow wells separated by broad gently sloping
barriers can be obtained.

Figure 1 shows the form of the potential and the corresponding phase plane as
a function of the parameter $r$, for $r>0$. One can observe that the larger
the parameter $r$ is, the flatter the bottom of the potential.

In real physical systems, such potential can be produced by the interaction of
an adatom with substrate atoms, where the parameter $r$ could account for the
temperature or pressure dependence, or for the geometry of the surface of the
metallic surface. It can be calculated from the first principles as described
in Refs. [?,?,?] (and reference therein). However it is more reliable to
determine the parameter $r$ from experimental data. Estimates for e.g., a H/W
adsystem (hydrogen atoms absorber on a tungsten surface), yield $r\approx-0.3$
[?,?,?]

Figure  2.: Parameter space plot of the frequency $\psi$ and the deformability
parameter $r$. Points represent chaotic behavior (positive KS-entropy and
continuous Fourier spectrum).

Typically, if periodic oscillators are subjected to a periodic force,
different phase-locking phenomena as well as chaos may be observed. And
chaotic oscillators when subject to a periodic force give rise to a series of
bifurcation phenomena.

Figure 2 shows the parameter space diagram of the oscillator in Eq. (1).
Points (blank space) indicate values of the frequency $\Psi$ and the
deformability parameter $r$ for which the oscillator in Eq. (1) is chaotic
(periodic).

The $(r,\Psi)$ space is characterized by the predominance of periodic
solutions. The chaotic solutions appear only for the value of the
deformability parameter approaching the limit $1$. However for high frequency,
the chaotic motion appears earlier, that is at $r\simeq 0.4$. For larger $r$
and $\Psi$ the parameter space presents a complex pattern whose chaotic
regions appear side-by-side with periodic regions. For the specific narrow
band of the frequency $\Psi$ around $0.70$ and $0.75$ a deep band of chaotic
motion can be found for $r$ between $0.1$ and $0.4$. This confirms the chaotic
behavior of deformable models systems as first suggested in references
[?,?,?].

We now consider a network of $N$ dynamical units of oscillators described by
equations (1) and (2). The governing equation for the network is given by:

$\displaystyle\dot{p_{i}}$ $\displaystyle=$ $\displaystyle n_{i}$ (3)
$\displaystyle\dot{n_{i}}$ $\displaystyle=$ $\displaystyle-\lambda
n_{i}-\omega^{2}\displaystyle\frac{\partial V(p_{i},r)}{\partial
p_{i}}+\eta_{0}\cos\Psi t$ (5)
$\displaystyle+g_{l}(p_{i+1}-2p_{i}+p_{i-1})\>\>\>\>\>\>\>\textrm{with}\>\>\>\>\>\>i=1,2...,N$

where $V(p_{i},r)$ is given by Eq. (2). The constant parameter $g_{l}$
determines the strength of the coupling and $N$ the number of oscillators
coupled. This equation is known as the Frenkel-Kontorova(FK) model with
harmonic interaction and non-sinusoidal substrate potential. It has been
extensively studied in the research of static characteristics of kinks
(topological solitons) such as the effective mass, shape, and amplitude of the
Peierls potential, the interaction energy of kinks, and the creation energy of
kink-antikink pairs. The applicability of the extended Frenkel-Kontorova model
for describing diffusion characteristics of a quasi-one-dimensional layer
adsorbed on a crystal surface has also been discussed in Ref. [?]. For real
physical systems, the account of various disturbances and of a more complex
character of atomic interactions break the exact integrability of the initial
Sine Gordon equation, leaving the possibility for describing the system
dynamics in terms of the same quasi-particles which interact with each other.
This interaction, which is due to the departure from complete integrability,
results in the following effects. The kolmogorov-Sinai entropy becomes nonzero
and the Fourier spectrum of excited states of the system becomes continuous.
Both characteristics of chaos.

###  2.2 Stability of the synchronization

Our analysis will be limited to networks of identical units. Since the $N$
systems are identical, it exists an exactly synchronized solution of Eq. (3),
and the synchronization manifold is defined by
$\mathcal{M}$=$\\{p_{1}=p_{2}=....=p_{N}=p_{s};n_{1}=n_{2}=....=n_{N}=n_{s}\\}$.

In the study of synchronization, a very relevant problem is to assess the
conditions for the stability of the synchronous behavior for the networks and
for the coupling configuration. The master stability function approach was
originally introduced for arrays of coupled oscillators [?], and it has been
latter extended to the case of complex networks of dynamical systems [?,?]. To
use this, let us consider $N$ coupled dynamical units, each of them giving
rise to the evolution of 2-dimensional vector fields $x_{i}$ ruled by a local
set of ordinary differential equations $\dot{x}_{i}=\textbf{F}(x_{i})$. The
equations of motion using the new variable can be written as

$\displaystyle\dot{x}_{i}=\textbf{F}(x_{i})+g_{l}\sum_{j=1}^{N}G_{ij}\textbf{H}(x_{j}),\qquad
i=1,2,...,N,$ (6)

where $\dot{x}_{i}=\textbf{F}(x_{i})$ governs the local dynamics of the $i$th
node. $x_{i}=[p_{i},n_{i}]^{T},$ and $\textbf{F}(x_{i})=\left[n_{i},-\lambda
n_{i}-\omega^{2}\frac{\partial V(p_{i},r)}{\partial p_{i}}+\eta_{0}\cos\Psi
t\right]^{T}$ with $V(p_{i},r)$ as in Eq.(2), the output function
$\textbf{H}(x_{i})$ is a vectorial function defined trough the matrix
$\mathbf{E}$=$\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right)$ by
$\textbf{H}(x_{i})=$ $\mathbf{E}$ $x_{i}$ , and $G(t)$ is a symmetric
Laplacian matrix ($\sum_{j}G_{ij}=0$) describing the networks connection and
given by

$\mathbf{G}=\left(\begin{array}[]{ccccc}-2&1&0&\ldots&1\\\ 1&-2&1&\ldots&0\\\
0&1&-2&\ldots&0\\\ \vdots&\vdots&\vdots&\ddots&1\\\
1&0&\ldots&1&-2\end{array}\right)$

The stability of the synchronization state can be determined from the
variational equations obtained by considering an infinitesimal perturbation
$\delta x_{i}$ from the synchronous states, $p_{i}=\delta p_{i}+p_{s}$,
$n_{i}=\delta n_{i}+n_{s}$. The equations of motion for the perturbation
$\delta x_{i}$ can be straightforwardly obtained by expanding the Eq. (6) in
Taylor series of first order around the synchronized state which gives

$\displaystyle\delta{\dot{x}}_{i}$ $\displaystyle=$ $\displaystyle
D\textbf{F}(x_{s})\delta
x_{i}+g_{l}\sum_{j=1}^{N}G_{ij}D\textbf{H}(x_{s})\delta
x_{i},\>\>\>\>\>\>\>\>i=1,2,...,N,$ (7) $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\left[D\textbf{F}(x_{s})\delta_{ij}+g_{l}G_{ij}D\textbf{H}(x_{s})\right]\cdot\delta
x_{i},\>\>\>\>\>\>i=1,2,...,N,$ (8)

where $D\textbf{F}$ and $D\textbf{H}$ are the Jacobians of the vector field
and the output function respectively.

Equation (5) is referred to as the variational equation and is often the
starting point for stability determination. This equation is rather
complicated since given arbitrary coupling $G$ it can be quite high
dimensional. However, we can simplify the problem by noticing that the
arbitrary state $\delta x_{i}$ can be written as $\delta
x_{i}=\sum_{i=1}^{N}\textbf{v}_{i}\bigotimes\xi_{i}(t)$ with
$\xi_{i}(t)=(\xi_{1,i},\xi_{2,i})$ where $\gamma_{i}$ and $\textbf{v}_{i}$ are
the set of real eigenvalues and the associated orthogonal eigenvector of the
matrix $G$ respectively, such that $G\textbf{v}_{i}=\gamma_{i}\textbf{v}_{i}$
and $\textbf{v}_{i}^{T}\textbf{v}_{i}=\delta_{ij}$. By applying
$\textbf{v}_{i}^{T}(t)$ (and $v_{i}$) to the left (right) side of each term in
Eq. (7) one finally obtains a set of N blocks for the coefficients
$\xi_{i}(t)$. The first term with the Kronecker delta remains the same. This
results in a variational equation in the eigenmode form

$\displaystyle\dot{\xi}_{k}=\left[D\textbf{F}(x_{s})+g_{l}\gamma_{k}D\textbf{H}(x_{s})\right]\xi_{k},k=0,1,2,...,N-1,$
(9)

We recall that $\gamma_{k}$ are the eigenvalues of $G$, and are given by
$\gamma_{k}=-4\sin^{2}(\pi k/N)$ for the diffusive coupling [?]. Note that
each equation in Eq. (9) corresponds to a set of 2 conditional Lyapunov
exponents $\lambda_{k}^{j}$ (j=1,2) along the eigenmode corresponding to the
specific eigenvalue $\gamma_{k}$. For $k=0$, we have the variational equation
for the synchronization manifold $(\gamma_{0}=0)$ and its maximum conditional
Lyapunov exponent $\lambda^{1}_{0}$ corresponds to the one of the isolated
dynamical unit. The remaining variations $\xi_{k}$, k=1,2,…,N-1 are transverse
to $\mathcal{M}$, and describe the system’s response to small deviations from
the synchronization manifold. Any deviation from the synchronization manifold
will be reflected in the growth of one or more of these variations. The
stability of the synchronized state is ensured if arbitrary small transverse
variations decay to zero. So, CS exists if $\lambda^{1}_{k}<0$, for $k\geq 1$.

We also calculate the condition for the synchronization in the network by
using the Lyapunov spectra, calculated directly from Eq. (7). Complete
synchronization in the generalized sense as defined in Refs. [?,?] exists if
the second largest Lyapunov exponent is negative.

Due to the periodic potential in Eq. (2), the active network in Eq. (6) is
highly sensitive to initial conditions. As a consequence, networks whose
elements have random initial conditions that differ only slightly completely
synchronize for a coupling strength smaller than the coupling strength needed
to completely synchronize networks that have elements whose initial conditions
differ moderately. Often, the network never complete synchronizes, and one can
only have that $|x_{k}-x_{l}|<\vartheta$, and so, the trajectory is never
perfectly along the synchronization manifold. Even thought $\vartheta$ might
be small, it is sufficiently large in order to mislead the statement that
complete synchronization appears by only checking the conditional exponents.
This discrepancy is due to the fact that, in this system, when the initial
conditions are not too close, the systems goes to different attractors and the
approximation made to obtain the conditional lyapunov exponents [Eq. (9)] is
no longer completely valid, thought it provides still approximate results. The
effect of having nodes with different initial conditions in the studied
network is similar to having networks with different parameters.

In Fig. 3, we show the parameter spaces (coupling $g_{l}$ vs. deformability
parameter $r$) of the complete synchronization regime. Points show the values
of $g_{l}$ and $r$ for which all the transversal ($k\geq 1$) conditional
exponents are negative [Figs. 3(A),(C)] or when the second largest Lyapunov
exponent becomes negative [Figs. 3(B),(D)].

When the initial conditions differ by no more than 0.01 [Figs. 3(A-B)] the two
conditions to predict complete synchronization provide the same surface in the
parameter space. However, when these initial conditions differ by no more than
0.5 [Figs. 3(C-D)], the conditional exponents predict the appearance of
complete synchronization for a coupling strength smaller than the strength for
which it really appears, as predicted by the value of the second largest
Lyapunov exponents [Figs. 3(D)].

One can also observe from these figures that as the deformability parameter
increases, the system becomes more and more unstable. When $r>9.5$, it is
almost not possible to find complete synchronization in the network for low
values of coupling strength $g_{l}$. So, when the potential $V(P_{i},r)$ has a
flat bottom, the particles are almost non-synchronizable in the network.

Figure  3.:  Appearance of complete synchronization in a network of $N$=5
diffusively coupled oscillators. Points represent $g_{l}$ and $r$ values for
which the conditional exponent $\lambda^{1}_{1}$ is negative (A,C) and for
which the second largest Lyapunov exponent is negative (B,D). In (A,B), the
initial conditions differ by at most 0.01 and in (C,D) the initial conditions
differ by at most 0.5.

##  3\. Phase synchronization

Phase synchronization [?,?,?] is a phenomenon defined by

$|\phi_{k}-m\phi_{l}|\leq\epsilon,$ (10)

where $\phi_{k}$ and $\phi_{l}$ are the phases of the nodes $x_{k}$ and
$x_{l}$ in the network [Eqs.(3)] and $m=\omega_{l}/\omega_{k}$, where
$\omega_{k}$ and $\omega_{l}$ are the average frequencies of oscillation of
these nodes, and $\epsilon$ is a finite number. In this work, we have used in
Eq. (10) $m=1$, which means that we search for $\omega_{k}:\omega_{l}$=1:1
(rational) phase synchronization. If another type of
$\omega_{k}:\omega_{l}$-PS is present, the methods in Refs. [?] can detect.

The phase $\phi$ is a function constructed on a good 2D subspace, whose
trajectory projection has proper rotation, i.e, it rotates around a well
defined center of rotation. Often, a good 2D subspace is formed by the
velocity space. In the oscillator considered in this work, one can use the
results of [?], and define the phase of the oscillator $x_{i}$ in Eqs. (3) as

$\phi(t)=\int_{0}^{t}\frac{\ddot{n}_{i}\dot{p}_{i}-\ddot{p}_{i}\dot{n}_{i}}{{(\dot{p}_{i}^{2}+\dot{n}_{i}^{2})}}dt.$
(11)

However, the oscillators in Eqs. (3) for the considered parameters have not a
well defined phase, and even in a state where complete synchronization is
achieved, one cannot use Eq. (11) to verify whether PS exists.

In short, if PS exists, in a subspace, then the points obtained from
observations of the position of one node’s trajectory at the time another node
makes any physical event do not visit the neighborhood of a special curve
$\Gamma$, in this subspace. A curve $\Gamma$ is defined in the following way.
Given a point $x_{0}$ in the attractor projected onto the subspace of one
oscillator where the phase is defined, $\Gamma$ is the union of all points for
which the phase, calculated from this initial point $x_{0}$ reaches $n\langle
r\rangle$, with $n=1,2,3,\ldots,\infty$ and $\langle r\rangle$ a constant,
usually 2$\pi$. Clearly an infinite number of curves $\Gamma$ can be defined.

Figure  4.: The appearance of phase synchronization in two bidirectionally
coupled oscillators. (A) There is no phase synchronization and the conditional
observations are not localized with respect to the curve $\Gamma$ pictorially
represented in the figure. (B) There is evidence of phase synchronization and
the conditional observations are localized. Simulations are done considering
initial conditions no more than 0.01 apart.

Formally, for non-coherent dynamical systems for which phase is still not well
defined, PS implies localization of the conditional sets [?], but the contrary
is not always true. Therefore, finding localized sets should be considered a
strong evidence that PS exists.

As an example, consider Eqs. (3) with two coupled oscillators, $r$=0.9, and
$\phi=0.08$. For a small coupling $g_{l}$=0.01, in Fig. 4(A), we show a
situation that PS is not present for $g_{l}$=0.01 and in Fig. 4(B), an
evidence that PS exists, for $g_{l}$=0.05. The curve $\Gamma$, a continuous
curve transversal to the trajectory, is pictorially represented by the
straight line $\Gamma$. In 4(A), the conditional observations are not
localized and thus there is no PS in this subspace. The light gray line (green
online) represents the attractor projection on the subspace $(p_{i},n_{i})$ of
the oscillator $x_{1}$, and filled gray circles (red online) represent the
points obtained from the conditional observations of the oscillator $x_{1}$
whenever the oscillator $x_{2}$ makes an event. An event is considered to be
the crossing of the trajectory to the line $n_{2}=0$, for $p_{2}>0$.

To have a general picture of when PS might appear in the two coupled
oscillators, we show in Fig. 5(A) the quantity $\kappa$ with respect to
$g_{l}$, defined as

$\kappa=\frac{\max{(p_{1}^{i})-\min{(p_{1}^{i})}}}{\max{(p_{1}(t))-\min{(p_{1}(t))}}}$
(12)

where $p_{1}^{i}$ represents the value of $p_{1}$ at the instant the
trajectory of oscillator $x_{2}$ makes an event. Therefore, $\kappa$ is
related to how broad the conditional observations visit the attractor. In Fig.
5(B) we show a few values of $p_{1}^{i}$ with respect to $g_{l}$. For
$g_{l}\geq 0.06$, CS takes place.

Figure  5.: The appearance of phase synchronization in two bidirectionally
coupled oscillators. (A) Occupation of the conditional observations with
respect to the attractor, $\kappa$, and in (B) the position variable
$p_{1}^{i}$ when the oscillator $x_{2}$ makes the $i$th crossing with the
section $n_{2}$=0, for $p_{2}>0$.

For large networks composed of $N$ nodes, this analysis is straightforward and
PS between two nodes can be stated if the conditional observations realized in
one node, whenever the other node makes an event, produces a localized set.

##  4\. Information Transmission in the Network

In order to study the way information is transmitted in active networks, we
introduce quantities and terminologies that assist us to better present our
ideas and approaches.

The mutual information rate (MIR) is the rate with which information is being
exchanged between two oscillation modes or elements in the active network.

The channel capacity, $\mathcal{C}_{C}$, is defined as the maximal possible
amount of information that two oscillation modes or nodes within the network
with a given topology can exchange, a local measure that quantifies the point-
to-point rate with which information is being transmitted.

The Kolmogorov-Sinai entropy offers an appropriate way of obtaining the
entropy production of a dynamical system. In chaotic systems, the entropy
equals the summation of all the positive Lyapunov exponents ( [?]). Here, it
provides a global measure of how much information can be simultaneously
transmitted among all pairs of oscillation modes or nodes. Therefore, the KS-
entropy, $H_{KS}$, of an active network, calculated for a given coupling
strength, bounds the MIR between two oscillation modes, $I$, calculated for
the same coupling strength. Thus,

$I\leq H_{KS}$ (13)

An active network is said to be self-excitable (non-self-excitable) when
$\mathcal{C}_{C}$ $>H_{KS}^{(0)}$ (when $\mathcal{C}_{C}$ $\leq
H_{KS}^{(0)}$), with $H_{KS}^{(0)}$ representing the KS entropy of one of the
$N$ elements forming the active network, before they are coupled.

According to [?], the upper bound for the MIR between two oscillation modes in
an non-self-excitable active network, denoted as $I$, can be calculated by

$I^{k}\leq\lambda_{0}^{1}-\lambda_{k}^{1}$ (14)

where $\lambda_{0}^{1}$ and $\lambda_{k}^{1}$ ($k=1,\ldots,N-1$) are the
positive largest conditional exponent [?], numerically obtained from Eq. (9),
with the oscillators possessing equal initial conditions. $\lambda_{0}$
measures the exponential divergence of trajectories along the synchronization
manifold and $\lambda_{k}$ along the transversal modes. The units used for the
MIR is [bits/unit time], which can be obtained by dividing Eq. (14) by
$\log_{e}(2)$.

The networks as in Eq. (6) are predominantly of the non-self-excitable type.
Only for a very small coupling strength, and a larger number of nodes, the
network has a negligible increase of the KS-entropy, which we will disregard.

As can be seen from the $H_{KS}$ curve in Fig. 6, the two coupled oscillators
are of the non-self-excitable type, since $H_{KS}^{(0)}=H_{KS}(g_{l}=0)/2$
which is approximately equal to $\mathcal{C}_{C}$. In this figure, we also
show the MIR exchanged between the two coupled oscillators. As typically
happens for non-excitable networks, the channel capacity is reached when the
network complete synchronizes. Since the network is composed of two
bidirectionally coupled systems, the MIR between the only two existing modes
is actually the MIR between the two oscillators.

Comparing Figs. 5(A) and 6, one can see that there is a direct relationship
between synchronization and information. The larger the amount of
synchronization the larger the MIR, again another typical character of non-
excitable networks.

Figure  6.: [Color online] (green) Circles show the KS-entropy $H_{KS}$ and
(blue) squares the MIR, $I^{1}$, for two bidirectional coupled oscillators.
(red) Diamonds show $I^{1}$ for a network of $N$=5 diffusively coupled
oscillators.

For larger networks with arbitrary topologies, the MIR between oscillation
modes is just a rescaled version of the MIR between two coupled oscillators.
Given that $g^{(2)}_{l}$ is the coupling strength for which complete
synchronization takes place in two coupled oscillators, and therefore this
coupled system operates with its channel capacity, the coupling strength for
which complete synchronization takes place in a whole network composed of $N$
nodes with a certain topology is given by $g^{(N)}_{l}$

$g^{(N)}_{l}=2\displaystyle\frac{g^{(2)}_{l}}{\gamma^{1}_{1}(N)}\>\>\>\>\>$
(15)

At the parameter $g^{(N)}_{l}$, every pair of oscillators operate with the
channel capacity. Equation (15) means that having the curve for the MIR for
two coupled oscillators, the curve of the MIR for larger networks is rescaled
by the second largest conditional Lyapunov exponent of the Laplacian matrix
$\gamma^{1}_{1}(N)$.

As an illustration of Eq. (15), we show in Fig. 6, the MIR for a network
composed of 5 oscillators coupled diffusively. In this figure, we show the
quantity $\langle I\rangle$ defined as $\langle
I\rangle=1/(N-1)\sum_{1}^{N-1}I^{k}$. Note that even though $\langle I\rangle$
might change its values according to the network topology and $N$, its maximal
value is bounded by the channel capacity, which do not depends on the $N$ and
the topology, another typical characteristic of non-self-excitable networks.

##  5\. Conclusion

We study the relationship between synchronization and the rate with which
information is exchanged between nodes in a spatio-temporal network which
describes the dynamics of classical particles under a substrate Remoissenet-
Peyrard potential. In particular, we study networks formed by Frenkel-
Kontorova(FK) oscillators suffering the action of harmonic interaction and
non-sinusoidal substrate potential.

We show that such networks are predominantly of the non-self-excitable type,
i.e. as the coupling strength among the nodes increases the KS-entropy
decreases. Other additional characteristics of non-self-excitable networks
are: the mutual information rate (MIR) and the synchronization level increase
simultaneously as the KS-entropy decreases; the channel capacity, the maximal
of the MIR, is achieved for the same coupling strength for which complete
synchronization appears.

We have overcome two difficulties concerning the detection of phase and
complete synchronization in this complex spatio-temporal network. Even though
the phase dynamics of each oscillator is not well defined, we have implement a
technique which allows to evidence the presence of phase synchronization, by
detecting the presence of localized sets obtained by the conditional
observations. The more localized the sets are (which implies larger amount of
phase synchrony) the larger the MIR. Concerning complete synchronization, we
show that the master stability equation which provides the stability of the
normal transversal modes (providing conditions to state complete
synchronization) should be used with caution in such a network. The reason is
that the final state is highly dependent on the initial conditions, a
consequence of the spatio character provided by the potential. For that
reason, in case the nodes have sufficiently different initial conditions, one
should only state complete synchronization using the master stability equation
in an approximate sense. A more rigorous condition to state complete
synchronization is provided by the verification that the second largest
Lyapunov exponent is negative.

Finally, we have shown how one can calculate the MIR between oscillation modes
in larger networks with different topologies using as the only input
information the curve of the MIR with respect to the coupling strength for two
bidirectionally coupled oscillators. Having the curve for the MIR for two
coupled oscillators, the curve of the MIR for larger networks is rescaled by
the second largest conditional Lyapunov exponent of the Laplacian matrix of
the larger network, the matrix that describes the way the nodes are connected
in the network. That enables one to construct larger networks based on the
dynamical characteristics of only two coupled oscillators.

## Acknowledgments

Both authors acknowledge the wonderful time spend in the Max Planck Institute
for the Physics of Complex Systems (MPIPKS) and thank the financial support
provided by this Institute. We also thank Sara P. Garcia for a critical
reading of the manuscript.

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