# Correlation and collective behaviour in Adler-type locally coupled
oscillators at the edge of chaos

Ernesto Estevez-Rams<EMAIL_ADDRESS>Facultad de Física-Instituto de
Ciencias y Tecnologías de Materiales, Universidad de la Habana, San Lázaro y
L, CP 10400 Habana, Cuba  K. Garcia-Medina  Facultad de Física, Universidad de
la Habana, San Lázaro y L, CP 10400 Habana, Cuba  B. Aragón-Fernández
Universidad de las Ciencias Informáticas. Carretera a San Antonio, Boyeros. La
Habana. Cuba

###### Abstract

Dynamical systems can be analyzed as computational devices capable of
performing information processing. In coupled oscillators, enlarged
capabilities are expected when the set of units is formed by subsets with
collective behaviour within them but weak correlation between the subsets. A
system of non-linear oscillators weakly coupled in the phase approximation is
studied. The informational distance maps show different regimes of collective
behaviour, ranging from independent, local, and global, as the control
parameters of the system are changed. This rich set of behaviours happens
despite the simple nature of the model used. Complex hierarchies between
oscillators can be identified near the edge of chaos region in the
informational distance dendrograms. We identify the emergence of local
collective regimes, further corroborated by the spatiotemporal maps.
Correlation diagrams also exhibit non-trivial dependence for the system at the
edge of chaos.

oscillators $|$ entropy $|$ complexity $|$

Introduction. Non-linear coupled oscillators have been used to model several
natural phenomena guckenheimer83 ; mosekilde02 . They exhibit a wide range of
behaviours from trivial periodic dynamics to chaotic, including instances with
all the fingerprints of a complex system alonso17 ; alonso18 ; estevez18 ;
burylko22 ; song23 .

Defining complexity remains an elusive task despite the considerable
literature on the subject. One of many possible approaches is to consider the
emergence of global dynamics in systems composed of many interacting elements,
each simple when isolated estevez18 . However, everyone seems to agree that
lasting correlations along the components of the system are a necessary
feature grassberger86 . Such correlations can be recognized by the emergence
of patterns persistent in time, yet it still allows for unpredictability or
randomness. The interaction between randomness and structured dynamics makes
optimal prediction of the system time evolution a formidable task zambella88
but always better than mere guessing.

Related, but not quite a measure of complexity, is Kolmogorov or algorithmic
complexity kolmogorov65  $K(s)$. The idea is to measure randomness by the
length of the smallest algorithm $s^{*}$ that, in a Universal Turing Machine
(UTM), can reproduce the system’s output $s$ ($K(s)=|s^{*}|$). Appealing to a
UTM makes this measure absolute up to a constant or, at most, a slowing
increasing function of the system size vitanyi93 . Finding the smallest
algorithm is undecidable, a consequence of the halting problem vitanyi93 , but
it is unnecessary in many cases. For most evaluations, it suffices to learn
how the length of the shortest algorithm scales with system size, a rate
equivalent to the Kolmogorov-Sinai entropy or entropy density $h$ of a system
zurek89 . $h$ measures the unpredictability of the system’s output or the
ability to generate new information cover06 .

Kolmogorov complexity is, therefore, a measure of disorder, not complexity.
However, it is the starting point for the formal derivation of other
magnitudes that measure correlations and are better suited for quantifying
complexity in some way. One of these magnitudes is the effective complexity
($E$) introduced by Grassberger grassberger86 , also known as excess entropy
crutchfield03 . $E$ measures the amount of correlation in a system at all
scales. Formally, it is the mutual information between the two halves of a bi-
infinite one-dimensional output crutchfield03 .

From a computational point of view, several related but different approaches
can be taken to recognize complexity. The degree of complexity of a dynamical
system is related to the class of computational tasks that it can perform
wolfram02 . In this approach, the system is seen as a computational device
capable of performing effective computations. Computation, as used in this
context, must not be considered an anthropocentric concept. Effective
computation in nature can include but is not limited to human understandable
algorithmic perspective, such as that involved in computer programming as
understood in computer science. However, a broader scope is to consider that
any system capable of generating, modifying, transmitting, and storing
information in some way is a computational device.

The increasing evolution of neural networks has accommodated the use of
computational devices in the sense explained above, where they are not viewed
as algorithmic programable devices but as entities with information processing
and storage ability appeltant10 . This greatly increases the interest in
finding and characterizing such systems.

The degree of universality in the ability to compute goes through answering
the question of how adaptable the system is to a change in the initial input
conditions. Suppose a system is indifferent to the initial conditions, not
changing its behaviour as the input dramatically changes. In that case, it has
a limited computational capacity, i.e., it always performs the same task
regardless of the initial stimulus. On the other extreme, if it has a widely
varying behaviour with small changes in the initial conditions, its
reliability as a computational device is questionable, as its robustness
against noise is rather weak. Between those two extreme cases, the most
universal type of computing occurs.

Another important consideration is the ability of the system to correlate its
components in such a way as to enhance its collective strength beyond some
trivial manner. It is well known that it is in the collective dynamics of
neurons that thought emerges gerstner14 . Also, in terms of memory, the system
must be able to retain, even in a modified way, some of the initial or the
generated information during its evolution. Again, memory is known to be a
result of the collective behaviour of neurons.

It is clear by now that measuring non-trivial correlations between the units
of a system is an important step toward understanding it as a computational
device.

The fact that non-linear coupled oscillators have been used to model to some
degree the brain activity gerstner14 ; ashwin16  is a statement of the kind of
complex behaviour such systems can simulate. Nevertheless, it is unclear a
priori which specific instance of coupled oscillators is capable of what
degree of computation.

A rich set of behaviours has been found in a simple model of locally coupled
non-linear oscillators alonso17 ; estevez18 ; estevez23 . By changing the
control parameters of the system, the set of coupled oscillators can exhibit
trivial, chaotic, and complex behaviour. More interesting is the claim that
enhanced computation capability can be seen right in the border between the
chaotic and the complex region, the so-called edge of chaos (EOC) region
estevez19 .

The edge of chaos hypothesis refers to the emergence, in non-linear systems of
different nature chua05 ; estevez18 ; estevez19 ; garcia23b , of a surge in
self-organization along the transition between order and chaos, a region well-
defined and fundamentally unstable, with a stark competition between order and
disorder langton90 . Some authors even link this hypothesis to the so-called
”criticality” under which the brain seems to operate alexander96 , or
evolution and life itself bilder14 ; lewin99 .

In this contribution, we use informational distance li04  to study the
correlation between oscillators. The informational distance is based on how
innovative two sequences $s$ and $p$ are, one with respect to the other in a
symmetrical way, in terms of pattern production, when the other is taken into
account

$d=\frac{\max\\{K(s|p),K(p|s)\\}}{\max\\{K(s),K(p)\\}},$ (1)

where $K(x|y)$ is the Kolmogorov complexity of $x$ conditioned in knowing $y$
(the length of the shortest algorithm reproducing $x$, if the shortes
algorithm reproducing $y$, is given). The normalized distance $d$ is between
$0$ and $1$.

This distance is not a Hamming-type distance because it does not measure the
number of different bits between the sequences but how far they are in
algorithmic space.

As defined in (1), the distance is uncomputable for the same reason as the
algorithmic complexity. Therefore, an estimation procedure is needed. There
have been several approaches to such estimation, but we will be following
estevez15  (See the methods section for details).

As already stated, entropy density

$h=\lim_{|s|\rightarrow\infty}\frac{H[s]}{|s|},$

where $H[s]$ is the Shannon block entropy of the string $s$, and effective
complexity measure

$E=\sum_{L=1}^{\infty}(h(L)-h),$

with $h(L)=H[s(1,L)]-H[s(1,L-1)]$, will be used as a measure of randomness and
the levels of correlation at different scales in the system, respectively.

$h(L)$ can be taken as a finite-size estimate of $h$, and therefore, $h(L)-h$
quantifies how much we are overestimating the actual randomness of the
sequence when only correlations only at scales smaller than $L$ are accounted
for. There are several interpretations for the effective measure complexity
grassberger86 ; crutchfield03 , and the reader is encouraged to read the
relevant literature. It will suffice to say that it measures the amount of
patterns in a signal when noise has been accounted for. In this sense, knowing
that is the mutual information between past and present, equivalently, between
two halves of a bi-infinite sequence, it measures the persistence of
structural features in a sequence at all time (length) scale. As with any
other single magnitude, it can not capture the full scope of what we
understand by complexity, but together with the other entropic magnitudes; it
moves in the direction of characterizing it quantitatively.

First, we show the emergence of local collective behaviour in the EOC region.
The local nature of the collective behaviour indicates a complex scenario
where different groups of oscillators interact. Second, we show that the local
collective behaviour emerging at the EOC changes as the system parameters
change towards the interior of the needle region, gaining in scope until
nearly global informational coherence is achieved.

It is also interesting to notice that, despite being such a simple model,
dynamics expected in more involved models already emerge in this simple setup.

We should expect very limited computational capacity when each oscillator is
independent of the others, and no collective behaviour is seen. Conversely,
the computational capacity should also be reduced if all oscillators are
synchronized in a phase-lock regime. For rich processing capability, one
should expect that some local correlation between the oscillator must be
present while global synchronization is not attained. The interaction between
those local subsets and collective behaviour can lead to higher information
processing capacity.

The model. Let us consider a system of locally coupled Adler-type oscillators.
Each oscillator in isolation is governed by a law of the type adler73

$\frac{d\theta}{dt}=\dot{\theta}=\omega+\gamma\cos\theta,$ (2)

where $\omega$ is the intrinsic frequency of the oscillator, and $\gamma$
determines the strength of the self-feedback. Equation (2) has been the
subject of research in the past as an oscillator with an injection of a
sinusoidal signal li10 , as given by the $\theta$-dependent feedback term in
the right-hand side, that separates it from the harmonic oscillators. The
feedback is a control mechanism to phase-lock the oscillator synchronized to a
frequency source. Phase locking refers to the output phase of the oscillator
being related to the input phase, in this case, the injection signal. Adler
systems, as described by equation (2), is just but one of a family of
oscillators with injection feedback that can be realized electronically li10 .

Analytical treatment of equation (2) results in critical points
$\dot{\theta}=0$, given by $\cos(\theta^{*})=-\omega/\gamma$, which can only
happen if $|\omega/\gamma|\leq 1$. Two points of different stability exist if
the strict inequality happens, and one marginally stable critical point when
equality happens ($|\omega/\gamma|=1$). For $\omega=\gamma$, oscillations
starting with a lower $\theta$ phase will be attracted to the critical point,
while those starting above the critical value $\theta^{*}$ will be repelled.
For $|\omega/\gamma|<1$, a saddle-node bifurcation leads to two critical
points, one stable and one unstable. In the asymptotic $t\rightarrow\infty$
time, the oscillator settles into
$\sin\theta=\pm\sqrt{1-(\omega/\gamma)^{2}}$. When $|\omega/\gamma|>1$, the
phase speed $\dot{\theta}$ has a $\theta$ interval where a slowdown occurs,
whose magnitude is related to the value of $|\omega/\gamma|-1$. The solution
is periodic for all time values with frequency
$\omega_{a}=1/2\sqrt{\omega^{2}-\gamma^{2}}$.

For the coupled oscillator system, the following local coupling introduced by
Alonso alonso17  will be used

$\frac{d\theta_{i}}{dt}=\dot{\theta_{i}}=\omega+\gamma\cos\theta_{i}+(-1)^{i}\left(\cos\theta_{i-1}+\cos\theta_{i+1}\right),$
(3)

which has been thoroughly studied alonso17 ; estevez18 ; estevez23  . The
coupling is taken to depend on the absolute value of neighbouring phases, and
its alternating sign answers to the local balancing of positive and negative
injections. This choice of coupling differs from the usual Kuramoto coupling
based on phase difference and not absolute values. There is no priory
justification for such coupling other than its possible realization as an
electronic circuitry. The fact that it has shown complex structured behaviour
in some regions of the parameter space resembling other systems with
computational abilities points to its potential usefulness estevez23 .
Considering an even total number of oscillators guarantees global balancing of
interactions. The overall effect of this balance can be thought of as the
absence of a net injection to the system, thus avoiding the presence of a
drift velocity for the entire system in the phase circle.

Equation (3) splits the oscillators into two classes, given by the sign of the
neighbourhood feedback. The parameter space of this system shows regions of
periodic, chaotic, and complex behaviour. It is especially noteworthy that it
has been reported, in one of the borders between the chaotic and the needle
complex region (figure 1a), a surge in magnitudes such as $E$, for some
initial conditions estevez19  (See figure 1c). In this edge of chaos (EOC)
border, there is a drop in entropy density $h$ (figure 1b) as the system
enters the complex region, an expected behaviour as the system moves out of
the chaotic regime. The reader can find further details in alonso17 ;
estevez18 ; estevez23 .

Results and discussion. In the region where a drop of the $h$ values and a
surge in $E$ values is observed (Figure 1c-d), the correlation among
oscillators should exhibit a more involved picture. This is what is shown in
figure 1b, where the informational distance matrix or map between oscillators
is shown at the EOC (point 2: $\omega=2.225$, $\gamma=1.200$) compared to
neighbouring points in the parameter space ($\omega=2.225$,
$\gamma=1.150,1.205,1.210,1.350$, points 1, 3, 4 and 5).

Figure 1b-above shows the distance map among the even oscillators, while
figure 1b-below corresponds to the odd oscillators. The split between odd and
even oscillators was made to take into account that each class has a different
sign feedback, given by the $(-1)^{i}$ term, as shown in equation (3). In the
chaotic regime (point 1, $\gamma=1.150$), the oscillators are essentially free
in their behaviour, and no collective dynamics are found. This is shown in the
distance map and the spatiotemporal plot of figure 2, where the evolution of
the oscillators’ phase at discrete time steps is shown as a colour map. This
is in clear contrast with the distance matrix of the EOC region (point 2,
$\gamma=1.200$), where non-trivial correlations between oscillators emerge and
can be seen in the patterns formed in the distance map. The spatiotemporal
diagram of figure 2 also shows the formation of patterns persistent in time
and travelling along the oscillators: collective behaviour has emerged. Yet,
the presence of incoherent boundaries between the coherent oscillators points
to the local character of this coherency. Another interesting feature is that
collective regions do not necessarily come from the oscillators’ initial
(random) phase values, as coherent regions can emerge or die during time
evolution: the emergence and persistence of patterns is a finite time event.

For points 3 and 4, near the EOC but more into the needle region, the distance
map shows less patterning (figure 1b), while well into the needle region
(point 5), decoherence seems to be lost for any oscillator and coherence has
become global. The spatiotemporal maps support this picture; as the system
moves from the EOC region into the needle region, coherence becomes
increasingly global and persistence in time becomes longer, which can be seen
in figure 2 for $\gamma=1.205,1.210$, and $\gamma=1.350$ it already shows a
complete pattern driven spatiotemporal behaviour.

The dendrogram from the distance matrix (Figure 1e) further emphasizes the
above analysis. Whereas for the chaotic region (point 1), the corresponding
tree is shallow, with all pairs of oscillators showing a maximum distance
between them, well within the needle region (point 5), the dendrogram shows
that within a given oscillator class (odd or even, according to the sign of
$(-1)^{i}$ in equation (3), the distance between oscillators has decreased
significantly. This decrease in distance points to global correlations between
the oscillators. In contrast, at or near the EOC (points $2$, $3$, $4$), the
dendrogram shows complex hierarchies among the oscillators, with several
levels of deepness in the tree within a class of oscillators. Local
communities of oscillators can be identified as sub-trees in the dendrogram
for these three points. Enhancement is not a property just of the EOC at point
$2$, although it seems to be an optimal value as shown by the surge in $E$.
Suboptimal improvement, with respect to the neighbouring regions, is still a
feature for points $3$ and $4$, as can be seen by the fact that the dendrogram
(figure 1e) at points $2$, $3$ and $4$, shows no qualitative difference and
just slight quantitative ones, (e.g., the mean distance between odd
oscillators shown in figure 1 are, respectively, $0.797$, $0.826$ and $0.843$.
In contrast, for point $5$ is $0.391$). Fine-tuning the control parameters to
a very tight interval of control parameter values is unnecessary. Of course,
this could be related to the finite number of oscillators far from any
thermodynamic limit, yet infinity is never achieved in real systems.

Simulations were performed with several initial phase values, random and non-
random. The distance matrix shows an evident dependence on the initial
condition, a feature of an adaptable computational device (See supplementary
material).

Figure 3 shows the plots of the average value of $\Delta
h_{ij}=|h_{i}-h_{i+2j}|$ and $\Delta E_{ij}=|E_{i}-E_{i+2j}|$, averaged over
$i$ as a function of $j$, for odd and even oscillators. For independent
oscillators, both magnitudes should be constant with $j$, the observed
behaviour for the chaotic region $\gamma=1.150$. A constant behaviour should
be observed for globally coupled oscillators, such as all oscillators in
phase. This is almost the functional dependence seen for $\gamma=1.210,1.350$.
The fact that there is a weak dependence with $j$ between the oscillators
indicates that complete phase lock has not been attained. For locally
collective behaviour, with decoupling between these regions of collective
oscillators, a more involved functional dependence of $\Delta h_{ij}$ and
$\Delta E_{ij}$ should be expected, and this is the case for $\gamma=1.205$,
even more so for the EOC point, $\gamma=1.200$. For the EOC plot, it must be
noted that the non-trivial function of both magnitudes includes monotonicity
changes and plateaus, indicating some correlation between oscillators that are
not necessarily close to each other. $\Delta h_{ij}$ and $\Delta E_{ij}$ vs
$j$ is, for $\gamma=1.200$, a result of the balance between pattern or
collective behaviour and unpredictability or lack of correlation.

At EOC, the value of $E$ for the oscillators is between $4$ and $10$, while
the entropy density $h$ is between $0.1$ and $0.5$. Both intervals are
intermediate compared to the points outside this region. $h$ interval at EOC
is lower than in the chaotic region and larger than inside the needle region,
while the $E$ interval at EOC is above the interval for the chaotic regime and
below the needle region (See supplementary material).

Conclusions. The picture that emerges from the analysis is that, indeed, at
the EOC, the surge of $E$ with the drop of $h$ is witnessing the emergence of
complex dynamics. The observed balance between structuring and
unpredictability is precisely the fingerprint of complexity. The non-trivial
local collective behaviour discovered by the patterns in the distance map, the
dendrogram, and the spatiotemporal map indicates this complex dynamic. The
reported result argues for enhanced computational power in the EOC region.
However, this optimal improvement can still be suboptimal for neighbouring
points on the side of the needle region.

Current research is being done to see if similar findings hold for other
systems of non-linear coupled oscillators. In other coupled oscillator
systems, different coupling forms and topologies are being considered
garcia23a ; garcia23b , and results will be reported elsewhere. It should be
emphasized that the procedure presented is of general application and is not
limited to the type of systems reported. As long as a time series output can
be achieved from a given model, entropic measures can be estimated, and
distance metrics can follow.

Methods. In all cases, integration of the system defined by equation (3) was
performed using a fourth-order Runge-Kutta method as implemented on GNU
Scientific Library (GSL). In random instances, the results were compared to
the solution given by the numerical solver of Mathematica Mathematica ; in all
cases, both solutions were identical within the numerical error. Also,
different integration steps were used until further refinement of the time
step did not change the reported result.

Since all entropic markers introduced are defined for discrete-state systems,
a discretization of the original continuous activity is needed. For the sake
of simplicity, a binary alphabet was chosen. Furthermore, given the bounded
nature of the $sin(\theta)$ function, the average activity at any given time
becomes a convenient choice for the threshold value. Binarization was thus
carried, assigning a $1$ state to all oscillators with activity higher than
the mean value and $0$ otherwise. This has been done before in similar systems
estevez18 ; estevez23 . The question of generating partition is one with no
universal answer; one expects that the chosen discretization keeps sufficient
information on the relevant features in the analyzed system. If the
discretization proves to be successful in doing so, it can only be evaluated a
posteriori. In any case, the thresholding procedure does not introduce
artefacts in the data processing but merely reduces the amount of information
carried by the data.

Entropy density $h$ was estimated through a Lempel-Ziv factorization as
explained in estevez19 . Briefly, Lempel-Ziv factorization lz76  creates an
exhaustive history from the input by defining a factor every time an unseen
substring happens while scanning the input string from left to right.

The LZ76 complexity $C(s)$ of the sequence $s$ is defined as the number of
factors in its exhaustive history.

Let the entropy rate be given by

$h(s)=\lim_{N\rightarrow\infty}\frac{H[s(1,N)]}{N},$ (4)

where $H[s(1,N)]$ is the Shannon block entropy cover06  of the string
$s(1,N)\in s$. Then, defining

$c(s)=\frac{C(s)}{N/\log{N}}.$ (5)

Ziv ziv78  proved that, if $s$ is the output from an ergodic source, then

$\limsup_{N\rightarrow\infty}c(s)=h(s).$ (6)

This is the base of using $c(s)$ as an estimate of $h$ for $N\gg 1$.

Equation (6) is valid in the infinite limit; in practical cases, equation (5)
is used as an estimate for the entropy density. Assuming an i.i.d. Gaussian
distribution of the word length, the error estimate for the entropy density
estimated by Lempel Ziv can be calculated from amigo06

$\sigma=c^{3/2}\frac{\varsigma}{\sqrt{N\log N}},$ (7)

$\varsigma$ is the standard deviation of the word length. For a $10^{4}$
length sequence, which is the one used in this study, equation (7) gives an
order of magnitude for the error bound around $10^{-2}$. Lesne et al. lesne09
have further discussed the estimation of error in the use of entropy
estimators in short sequences. Using numerical simulations, they showed that
Lempel-Ziv estimators of entropy density show good agreement even for
sequences as short as $2\times 10^{3}$. As stated, sequence length will be at
least $10^{4}$, and errors in estimating entropy density should be
insignificant.

From the entropy density, a magnitude related to the effective complexity can
readily estimated by a random shuffle algorithm estevez19 .

As Kolmogorov randomness is uncomputable, a practical alternative to equation
(1) is needed estevez15 . $K(s|p)$ is roughly equal to $K(sp)-K(p)$ (there is
a logarithmic correction term which is not significant for long enough
sequences), and $sp$ denotes the concatenation of both strings. The
informational distance can then be estimated from

$d(s,p)=\frac{K(sp)-min\\{K(s),K(p)\\}}{max\\{K(s),K(p)\\}}.$ (8)

If $s$ and $p$ have the same length, then we can cast equation (8) in terms of
the entropy density

$d(s,p)=\frac{h(sp)-min\\{h(s),h(p)\\}}{max\\{h(s),h(p)\\}}.$ (9)

The normalized LZ76 complexity is now used to estimate the entropy density,

$d(s,p)=\frac{c_{LZ}(sp)-min\\{c_{LZ}(s),c_{LZ}(p)\\}}{max\\{c_{LZ}(s),c_{LZ}(p)\\}}.$
(10)

For the distance matrix, simulations were performed for $200$ oscillators with
circular boundary conditions, random initial phase values were taken, and
$6000$ time steps, the first $1000$ time steps were dropped before the
similarity distance was calculated. Dendrograms were built using the average
intercluster dissimilarity as criteria from the distance matrix. The
Dendrogram function of Wolfram Mathematica was used Mathematica .

$5000$ oscillators were also used with periodic boundary conditions for the
spatiotemporal maps. Again, the first $1000$ time steps were dropped, and the
next $1000$ time steps are shown. The $\langle\Delta h_{j}\rangle$ and
$\langle\Delta E_{j}\rangle$ plots were calculated for $500$ oscillators, as
before the first $1000$ time steps were dropped.

Acknowledgment. EER and KGM would like to thank Alexander von Humboldt
Stiftung for financial support. EER would like to thank the MPI-PKS for a
visiting grant. EER thanks H. Kantz at MPI-PKS for the in-depth discussion and
the excellent working environment.

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Figure 1: (a) Entropy density map in parameter space for the locally coupled
oscillators of equation (3). The needle region is the low $h$ (black) region.
Five points have been chosen, all with $\omega=2.225$, and (1) $\gamma=1.150$,
(2) $\gamma=1.200$, (3) $\gamma=1.205$, (4) $\gamma=1.210$ and (5)
$\gamma=1.350$. The high $h$ (red) region is in the chaotic regime, where
point (1) is taken. Point (2) is at the edge of chaos boundary. Points (3) and
(4) are in the needle region close to the EOC, while point (5) is well within
the needle region. (b) The distance matrix for a 200 oscillator configuration
in all five points described. The distance map is shown separately for even
and odd oscillators. (c) The entropy density $h$ as a function of $\gamma$ for
$\omega=2.225$, as $\gamma$ increase the systems leaves the chaotic region and
gets into the needle region, the boundary is seen as a sudden drop in the $h$
value. (d) The effective complexity measure $E$ as a function of $\gamma$, the
same conditions as in (c) applies. A surge in the $E$ value can be seen at the
EOC boundary. (e) The dendrogram built from the corresponding distance matrix.
In the chaotic region (point 1), oscillators are independent between them, and
the dendrogram is a trivial tree with a shallow deepness; within the needle
region (point 5), the information distance between oscillators has dropped
significantly within oscillators of one type (same sign), a more global
collective behaviour has emerged. In the intermediate points (2, 3 and 4), at
or near the EOC, a complex dendrogram shows the emergence of local collective
behaviours with non-trivial hierarchies. At the same time, the global coupling
is not achieved.  Figure 2: Spatio-temporal maps for the five points depicted
in figure 1. The activity $\sin\theta$ values are shown as colours ranging
from black for $0$, green for $1/2$, and red for $1$. Time runs horizontally
while the oscillator index runs vertically. On the left is the whole
spatiotemporal map; on the right, the spatiotemporal map is decomposed into
odd and even oscillators. For $\gamma=1.1500$, a salt and pepper contrast
shows that each oscillator behaves independently of the others in the chaotic
region. No pattern can be discerned. Collective behaviour as time-persistent
patterns emerges from EOC at $\gamma=1.200$ and becomes more dominant as
$\gamma$ increases to $1.35$ where total global coherence can be seen. At EOC,
there is a balance between pattern and incoherent regions in the oscillator
dynamics.  Figure 3: Plot for the five described points, of the mean value of
$\Delta h_{ij}=|h_{i}-h_{i+2j}|$ (left) and $\Delta E_{ij}=|E_{i}-E_{i+2j}|$
(right) as a function of $j$ (averaged over $i$). For independent oscillators,
both magnitudes should not depend on $j$, the behaviour observed for
$\gamma=1.150$. Global coherence with near-phase locking should also show a
slow dependence with $j$ as seen for $\gamma=1.210,1.350$. Local coherent
regions and incoherent oscillators should show a non-trivial function of $j$
as seen for $\gamma=1.200,1.205$.