Papers in Physics, vol. 7, art. 070002 (2015)

Received: 20 November 2014, Accepted: 10 March 2015
Edited by: C. A. Condat, G. J. Sibona
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070002

www.papersinphysics.org

ISSN 1852-4249

Two distinct desynchronization processes caused by lesions in globally
coupled neurons

Fabiano A. S. Ferrari,1∗ Ricardo L. Viana1

To accomplish a task, the brain works like a synchronized neuronal network where all
the involved neurons work together. When a lesion spreads in the brain, depending on
its evolution, it can reach a significant portion of relevant area. As a consequence, a
phase transition might occur: the neurons desynchronize and cannot perform a certain
task anymore. Lesions are responsible for either disrupting the neuronal connections or, in
some cases, for killing the neuron. In this work, we will use a simplified model of neuronal
network to show that these two types of lesions cause different types of desynchronization.

I. Introduction

The neuronal dynamics can be represented as a dy-
namical system and a population of neurons as a
neuronal network. The mean electrical field ampli-
tude of a population of neurons has neglected values
when they are uncoupled or weakly coupled. This
amplitude is enhanced when the coupling between
them is high enough to make them synchronized
among themselves [1]. At the synchronized state,
it is possible to measure the mean electrical activ-
ity of a large number of closed neurons using EEG
[2, 3]. Abnormalities or absence of synchronization
have been reported as a consequence of neurode-
generative diseases [4, 5]. This dynamical effect is
a consequence of topological changes caused by le-
sions spreading in the brain. However, every dis-
ease has its own features and, here, we propose to
study the different dynamical effects caused by dif-
ferent types of lesions.

Measures of neuronal functional activity using

∗Email: fabianosferrari@gmail.com

1 Physics Department, Universidade Federal do Paraná,
Curitiba, Brazil.

EEG [6] and fMRI [7] have shown spatiotemporal
patterns formation. An explanation for this be-
havior is the emergence of a critical state in the
neuronal dynamics providing conditions for a for-
mation of distinct clusters at the functional level
[8]. When the neuronal population is considered as
a complex network structure with a hierarchical-
modular architecture, this high heterogeneity is re-
lated to a stretching of criticality and consequently
increased functionality [9]. Recent papers have
shown functional differences between healthy and
unhealthy patients with different neuropathologies
[4, 10, 11]. Schizophrenia, for example, has been
related to neuronal decoupling [12].

Unfortunately, many papers are constrained to
the study of functional connections and the com-
parison between healthy and unhealthy patients.
Efforts have been made to explain the dynami-
cal changes caused by lesions and different mod-
els have been proposed to connect what happens in
the neuronal level to what happens in the macro-
scopic level [8, 13]. Nevertheless, a complete under-
standing about the dynamical effect of lesions in
the brain is still missing.

From the point of view of electrical activity, when
the brain needs to execute a specific task there is

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Papers in Physics, vol. 7, art. 070002 (2015) / F. A. S. Ferrari et al.

a group of neurons that synchronize and work to-
gether to perform it. When a lesion spreads in the
brain, depending on its size, it can disrupt impor-
tant connections and certain tasks cannot be done
anymore. Based on this hypothesis, we present
here a simplified neuronal network model of glob-
ally coupled neurons to study the effects of the
desynchronization induced by a lesion spreading
randomly in the brain. We focus on two main cases:
one in which the connections among neurons are
disrupted and a second one in which the lesion kills
the neurons. Despite of the simplicity of the model,
the observed phase transitions from synchronized
to the desynchronized state have shown different
properties for these two types of lesions.

II. Model

In this work, we consider a network of Rulkov neu-
rons globally coupled (mean field). However, other
neuronal models could be used and provide similar
results, for example: Kuramoto [14], Hindmarsh-
Rose [15] and Morris-Lecar [16]. Rulkov neurons
are described by a fast variable x and a slow vari-
able y. The dynamic associated with each neuron
in the network can be described as

x
(j)
n+1 =

α(j)

(1 + (x
(j)
n )2)

+ y(j)n +
ε

N

N∑
i=1

x(i)n , (1)

y
(j)
n+1 = y

(j)
n −σx

(j)
n −β, (2)

where σ = β = 0.001, α(j) is a bifurcation pa-
rameter randomly chosen in the interval [4.1, 4.3],
exhibiting bursts, N is the network size and ε is the
coupling strength [17].

The first step in our analysis is to choose an ap-
propriate coupling strength such that the network
becomes synchronized. To characterize phase syn-
chronization, we will define a geometric phase for
each neuron. Considering one period of oscillation
and the distance between two successive bursts, the
phase of each neuron j is given as [1],

ϕ(j)n = 2πk + 2π
n−n(j)k

n
(j)
k+1 −n

(j)
k

, (3)

where k is the k − th burst and nk is the time in
which the k − th burst started. The phase syn-
chronization can be found through the Kuramoto’s

(a) (b) (c)

Figure 1: Network representation. Panel (a) shows
a fully connected network, panels (b) and (c) show
the effect of lesions type 1 and type 2, respectively.
The dashed lines show where the damage caused
by the lesion type is.

order parameter,

rn =
1

N

∣∣∣∣∣∣
N∑
j=1

expiϕ
(j)
n

∣∣∣∣∣∣ , (4)
when this value is one, it means the network is fully
synchronized in phase and when this value is zero,
it means the network is fully desynchronized [14]. It
is known that neuronal networks described by Eqs.
(1) and (2) exhibit phase synchronization when the
coupling strength is increased up to a certain value
[1], as we will show in the next section.

The second step in our analysis is to study how
lesions spreading in the network cause desynchro-
nization. To study this fact, we will assume two
different types of lesions:

Type 1. Lesions that disrupt the connection be-
tween the neurons.

Type 2. Lesions that kill neurons.

In Fig. 1 (a), we show a representation for a
network of globally coupled neurons; the effect of
lesions type 1 are represented in soutthe Figure Fig.
1 (b), while the effect for lesions type 2 are shown
in Fig. 1 (c).

We also consider that for each type of lesion, the
coupling strength can be affected by three different
situations:

Reinforced coupling. For every new damaged
neuron, the coupling strength is increased by
ε(t) = ε0/(N −Nd).

Invariant coupling. The coupling strength does
not change with the lesion size, so ε(t) = ε0/N.

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Papers in Physics, vol. 7, art. 070002 (2015) / F. A. S. Ferrari et al.

0 0.01 0.02 0.03 0.04

ε

0

0.2

0.4

0.6

0.8

1

<
R
>

N=1000

N=2500

N=5000

N=10000

Figure 2: The mean order parameter as a func-
tion of the coupling strength. The different colors
represent different network sizes. Here, 〈R〉 is the
order parameter averaged over the whole network
for a time series of 10000 discrete steps after 80000
transient times.

Reduced coupling. For every new damaged neu-
ron, the coupling strength is decreased by
ε(t) = ε0/(N + Nd).

Here, Nd is the number of disconnected neurons
and ε0 is the initial coupling strength.

III. Results and Discussion

The first step is to find the values for the coupling
strength in which the network shows phase synchro-
nization. In Fig. 2, we show that when the cou-
pling strength is below εc = 0.02 then the system
is completely desynchronized (disregarding fluctu-
ations ∼ 1/

√
N). Above this critical value, the

order parameter increases, and close to ε = 0.04,
the network can be considered fully synchronized.
Based on that for our results, we will use as initial
coupling strength ε0 = 0.04. From Fig. 2, we can
also see that the transition toward synchronization
is invariant with respect to the network size.

For lesions type 1, when ε(t) is reinforced af-
ter each new damaged neuron, the synchroniza-
tion (characterized by the mean order parameter)
decays linearly with the number of disconnected

neurons (Nd) but the network just becomes com-
pletely desynchronized when all the neurons are le-
sioned, see Fig. 3 (a). For the cases where ε(t)
is invariant or reduced, we observe a roughly first
order phase transition where the order parameter
decreases linearly up to a critical size of lesioned
neurons Nd,critical and the whole network desyn-
chronizes. This fact is absent for the reinforced case
because increasing the coupling strength increases
Nd,critical such that the first order phase transition
is never observed. The three color lines in Fig. 3
(a) indicate that the smaller the coupling strength
becomes, the faster the complete phase desynchro-
nization happens.

An interesting effect occurs for lesions type 2,
shown in Fig. 3 (b). When the coupling strength
is reinforced after each lesion, we do not observe
desynchronization. This phenomenon is caused by
the fact that when neurons die they do not con-
tribute to the global effect in the network and the
remaining neurons being more strongly connected
remain synchronized. This fact is not observed for
the cases in which the coupling strength remains
the same (green line) or is reduced (blue line).
For lesions type 2, when the coupling strength de-
creases toward Nd,critical, the observed decay fol-
lows a roughly second order phase transition, see
Fig. 3 (b).

IV. Conclusions

Here, we have investigated two types of lesions and
their effects. The presence of phase transition from
synchronized to desynchronized state was observed
for all cases except for lesions type 2 when the net-
work is reinforced. We have observed that lesions
type 1 obey a roughly first order phase transition
while lesions type 2 obey a roughly second order
phase transition. For both types of lesions, when
the coupling strength is continuously reduced af-
ter each new damage, then the network desynchro-
nization is faster. Based on that, increasing the
coupling strength can be a strategy to compensate
the desynchronization effect induced by lesions, but
this strategy is more effective for lesions type 2.

The two distinct phase transitions allow us to
define a characterization scheme: if the synchro-
nization decays linearly, we could say that we are
dealing with a lesion type 1 while if the synchro-

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Papers in Physics, vol. 7, art. 070002 (2015) / F. A. S. Ferrari et al.

0 2000 4000 6000 8000 10000

N
d

0

0.2

0.4

0.6

0.8

1
<

R
>

ε(t)=ε
0
/(N-N

d
)

ε(t)=ε
0
/N

ε(t)=ε
0
/(N+N

d
)

linear fit

0 2000 4000 6000 8000

N
d

0

0.2

0.4

0.6

0.8

1

<
R

>

ε(t)=ε
0
/(N-N

d
)

ε(t)=ε
0
/N

ε(t)=ε
0
/(N+N

d
)

(b)(a)

Figure 3: The desynchronization process induced by lesions. Panel (a): lesions type 1, panel (b): lesions
type 2. The different colors indicate the three different coupling effects: reinforced, invariant and reduced
(black, green and blue, respectively). Here, Nd is the number of affected neurons (disrupted for (a) and
killed for (b)) and N = 10000.

nization decays non-linearly, then a lesion type 2
could be the case. However, a mixture of events
could also be observed and then a characterization
would become difficult to achieve. Despite this fact,
our results show that even simplified models are
useful to understand and classify types of lesions
and advances in this segment could be helpful to
understand the progress of neurodegenerative dis-
eases.

Acknowledgements - This work has financial sup-
port from the Brazilian research agencies CNPq
and CAPES. We acknowledge Carlos A. S. Batista
for relevant discussions.

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