doi: 10.1016/j.procs.2015.11.004
Multi-chimera states in the Leaky Integrate-and-Fire model
N.D. Tsigkri-DeSmedt1, J. Hizanidis1,2,P.H¨ovel3,4,andA.Provata
1∗
1Institute of Nanoscience and Nanotechnology, National Center for Scientific Research “Demokritos”
Athens, Greece
2Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of
Crete Heraklion, Crete, Greece.
3Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Berlin, Germany
4Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universit¨at zu Berlin, Berlin,
Germany
Abstract
We study the dynamics of identical leaky integrate-and-fire neurons with symmetric non-local
coupling. Upon varying control parameters (coupling strength, coupling range, refractory pe-
riod) we investigate the system’s behaviour and highlight the formation of chimera states. We
show that the introduction of a refractory period enlarges the parameter region where chimera
states appear and affects the chimera multiplicity.
Keywords: Chimera state, synchronisation, leaky integrate-and-fire, neuron models, refractory period
1 Introduction
The study of the dynamics and in particular collective behaviour of coupled oscillators has
received great interest from scientists in different fields varying from chemical and mechanical
systems to neuroscience and beyond [19]. A very interesting and unexpected synchronisation
phenomenon that was first observed in identical coupled oscillators is the so-called chimera
state. This is a dynamical scenario in which part of the oscillators are synchronised, while
simultaneously others are not synchronised. These states were first observed in 2002 by Ku-
ramoto and Battogtokh [9], while the term “chimera” was coined later, in 2004, by Abrams and
Strogatz [3]. Potential applications of chimera states include the unihemispheric sleep that ap-
pears in dolphins and some birds, which sleep with one eye open meaning that half of the brain
is synchronised and half is not synchronised, power grids and social systems [18]. On one hand,
this surprising phenomenon has been observed numerically in various neuron models such as
leaky integrate-and-fire, Kuramoto phase oscillators, Hindmarsh-Rose, FitzHugh-Nagumo, and
∗Corresponding Author
Procedia Computer Science
Volume 66, 2015, Pages 13–22
YSC 2015. 4th International Young Scientists Conference on
Computational Science
Selection and peer-review under responsibility of the Scientific Programme Committee of YSC 2015
c
The Authors. Published by Elsevier B.V.
13
SNIPER/SNIC model [2, 1, 4, 7, 9, 15, 16, 23]. On the other hand, experimental verifications
[6, 10, 13, 21, 24] do not include examples from neuroscience so far. This gives rise to an even
greater interest to study chimera states as it may lead to a better understanding of information
processing in neuron networks.
In this study we examine the effect of different control parameters on the appearance of
chimera states for Leaky Integrate-and-Fire (LIF) neuronal oscillators that are arranged in a
1-dimensional regular ring topology. We compare the behaviour of coupled LIF units with and
without refractory period and we find that in both cases chimera states appear. We show that
when the refractory period is introduced the chimera states are enhanced and their multiplicity
increase.
In the next section we introduce the single and coupled LIF models. Subsections 2.2 and 2.3
describe the coupled LIF model with and without a refractory period, respectively. In Sec. 3 we
show the development of chimera states in a network of coupled LIF neurons. We demonstrate
the differences in the form of chimera states between a network of coupled LIF neurons with
and without a refractory period. Finally, the main conclusions are recapitulated in Sec. 4.
2 The leaky integrate-and-fire model
2.1 The single neuron model
The LIF model is a simple model for spiking neurons [5] which was introduced in 1907 by Louis
Lapicque. It describes the dynamical evolution of the membrane potential of a single neuron.
Figure 1 depicts the spiking behaviour of the membrane potential of a single LIF neuron in
time.
The membrane potential u(t) evolves according to the following equation
˙u(t)=−u(t)+μ(1)
with a reset condition
∀u(t)=uth ⇒lim
ε→0u(t+)=urest (2)
where uth is the threshold of the potential and μ>u
th denotes a constant. In LIF, whenever
the membrane potential reaches the threshold u(t)=uth,aspikeisfiredandthemembrane
potential is instantaneously reset to the rest state urest. In this study the potential in the rest
state is set equal to zero, urest =0.
2.2 Non-locally coupled LIF neurons
Neurons “fire” electrical signals as a result of receiving inputs from other neurons. This obser-
vation sets the need of studying a network of coupled neurons. We study a network of NLIF
neurons that are arranged in a regular ring topology with non-local connections, that is, each
element is coupled to Rnearest neighbours on either side, as schematically depicted in Fig. 2.
The dynamic evolution in time of this system is determined by
˙ui(t)=−ui(t)+μ+σ
2R
i+R
j=i−R
[ui(t)−uj(t)] (3)
with the same reset mechanism for each element as described in Sec. 2.1. Here, σis the coupling
strength and Rdenotes the coupling range. The index ihas to be taken modulo N. The network
nodes are considered identical, that is, they have the same system and coupling parameters.
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
14
02040
time units
0
0.2
0.4
0.6
0.8
1
u
Figure 1: (Colour online) Dynamic evolution of a single neuron in time according to Eq. (1).
Parameters: μ=1,uth =0.99, urest =0.
Figure 2: Topology the considered one-dimensional ring network.
The study of a system of coupled oscillators [22, 11, 17] involves the identification of pa-
rameter regions where synchronisation occurs. In the next sections, we investigate the effect of
the coupling strength σand the coupling range Ron synchronisation phenomena with special
focus on chimera states.
2.3 Coupled neurons with a refractory period
In many neuron models, the neuron stays in its rest state for a certain period of time after
firing. In order to take this into account, we consider a refractory period pr[8]. The refractory
period is a time interval, during which a neuron remains at rest after firing and is not able to
trigger an additional spike.
The dynamics of the refractory LIF model is described by the equations of the coupled LIF
neurons system Eq. (3), except that after firing each neuron remains at the rest state for time
pr. Figure 3 depicts the spiking behaviour of the membrane potential of a single LIF neuron
with a refractory period pr=1.
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
15
02040
time units
0
0.2
0.4
0.6
0.8
1
ui
Figure 3: (Colour online) Dynamic evolution of a LIF neuron with a refractory period of pr=1.
Other parameters as in Fig. 1.
3 Chimera states in coupled leaky integrate-and-fire neu-
rons
We investigate the appearance of chimera states of a network of LIF neurons with and without
refractory period. In references [18, 20, 12, 25, 14] chimera states appearintheLIFsystem,for
different realisations of the model and of the coupling geometry. In reference [14] the authors
have shown the existence of chimera states in coupled LIF systems with delay dynamics. In this
study, we show that the presence of a refractory period favours their appearance, while at the
same time has an effect on their multiplicity. The refractory period is different from delayed
self-feedback in the sense that the former introduces a dead (resting) time after firing while in
the latter each neuron receives input not only by its neighbours but also by its past states.
3.1 Without refractory period
In the following, starting from random initial conditions ui,{i=1, ..., N}distributed over the
interval [0,1], we investigate the appearance of chimera states for a finite network of N= 1000
neurons by varying the coupling strength σand the coupling range R. See Fig. 4. We observe
that for R= 100 chimera states do not appear for very small values of the coupling strength
σ≤0.5norforσ≥0.6. They are found for intermediate values of the coupling strength, such
as σ=0.565, as shown in Fig. 4(e). Notice that the chimera states observed in this case are
transient and disappear for longer times. See red curves in Fig. 4.
The appearance of chimera states at a certain value of the coupling strength also depends on
the value of the coupling range R. More specifically, we show that as we increase the coupling
range R, chimera states appear for a larger value of σ, as shown in Fig. 5. Thus the range of the
values of the coupling strength that favour the appearance of chimera states, shifts following
the change of the coupling range.
Chimera states are highly dependent on initial conditions. Figure 6 shows the temporal
evolution of chimera states starting from two different random initial conditions in columns (a)
and (b), respectively. All other parameters are the same. We find that when the system starts
from an initial state (a) it reaches complete synchronisation, while when it starts from initial
state (b) a chimera state is formed as shown in the plots.
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
16
0.993
0.996
0.999
ui
0
0.5
1
0
0.5
1
ui
0
0.5
1
ui
0 200 400 600 800 1000
ith oscillator
0
0.5
1
ui
0 200 400 600 800 1000
ith oscillator
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4: (Colour online) Snapshot of the membrane potential uifor different values of the
coupling strength: (a) σ=0.4, (b) σ=0.52, (c) σ=0.54, (d) σ=0.56, (e) σ=0.565, (f)
σ=0.57, (e) σ= 58, (g) σ=0.6. The blue line corresponds to t= 1000 time units and the red
linetot= 9000 time units. Other parameters: N= 1000, uth =0.98, R= 100 and μ=0.99.
0
0.5
1
ui
(a) (b) (c)
0500 1000
ith oscillator
0
0.5
1
ui
0500 1000
ith oscillator
0500 1000
ith oscillator
Figure 5: (Colour online) Snapshots of the membrane potential uifor different values of the
coupling parameters Rand σ. The upper panel corresponds to σ=0.565, while the lower one
to σ=0.7. The coupling range is (a) R= 200, (b) R= 300 and (c) R= 400. Other parameters
as in Fig 4.
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
17
0
0.2
0.4
0.6
0.8
1
ui
(a) (b)
0
0.5
1
ui
0500 1000
ith oscillator
0
0.5
1
ui
0500 1000
ith oscillator
t=1000
t=12000
t=9000
t=12000
t=60
t=270
Figure 6: (Colour online) Snapshots of the membrane potential uifor different time units and
for different initial conditions in panels (a) and (b). Parameters: N= 1000, uth =0.98,
σ=0.565, R= 350 and μ=1.
3.2 With refractory period
The study of the network of coupled LIF neurons shows that this network displays the phe-
nomenon of chimera states which are mostly transients. We now examine the effect of the
refractory period in their spatial form and temporal evolution. Using values of the coupling
strength σand the coupling range Rfor which we observed chimera states in the original
coupled LIF system, we now consider the influence of a non-zero refractory period pr.
In Fig. 7 we demonstrate that the refractoryperiod enhances the appearance of chimera
states. When varying the refractory period it is natural to use a time scale for comparison
that is intrinsic to the system. As this reference, we use the period Tof the oscillations of the
coupled LIF unit.
In Fig. 8 we compare the system of the N= 1000 oscillators behaviour with and without a
refractory period. For pr= 0, as show in Fig. 8(a), the chimera state has one coherent and one
incoherent region. On the contrary, in Fig. 8(b) the chimera state that is formed in the system
with pr=0.5Thas four incoherent and four coherent regions.
In the following, we elaborate on the effect of the refractory period. As shown in Fig. 9
the chimera multiplicity changes as prvaries from 0.1Tto T. Notice that the chimera states
appear only for intermediate values of the refractory period and that the number of coherent
and incoherent regions for fixed values of the coupling strength and coupling range remains
constant. A potential interpretation of this behaviour is that the neurons by sections slightly
differ in phase. Intuitively, the small but substantial values of the refractory period facilitate the
grouping because the condition ui(t) = 0 forces neighbouring elements to synchronise locally in
the rest state. The grouping of neurons in sections, influences the neurons on the boundaries
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
18
0
0.5
1
ui
(a) (b) (c)
0500 1000
ith oscillator
0
0.5
1
ui
0500 1000
ith oscillator
0500 1000
ith oscillator
Figure 7: (Colour online) Snapshots of the membrane potential uifor different values of the
coupling parameters Rand pr. The upper panel corresponds to pr= 500 time units and the
lower panel corresponds to pr= 1000. The coupling range is (a) R= 200, (b) R= 300 and
(c) R= 400. Other parameters are N= 1000, uth =0.98, σ=0.565, μ=0.99, t= 9000 time
units.
0500 1000
ith oscillator
0
0.5
1
ui
(a)
0500 1000
ith oscillator
0
0.5
1
ui
(b)
Figure 8: (Colour online) Snapshots of the membrane potential ui: panel (a) depicts the
formation of chimera states in a network without a refractory period and panel (b) shows
the formation of chimera states in a network with pr=0.5T. Other parameters are: N= 1000,
σ=0.565, uth =0.98 , R= 300 and μ=0.99.
between sections, which destabilise and become asynchronous.
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
19
0
0.5
1
ui
0
0.5
1
ui
0
0.5
1
ui
0
0.5
1
ui
0500 1000
ith oscillator
0
0.5
1
ui
0500 1000
ith oscillator
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure 9: (Colour online) Snapshots of the membrane potential uiin space for different values
of the refractory period pr,(a)pr=0.1T,(b)pr=0.2T,(c)pr=0.3T,(d)pr=0.4T,(e)
pr=0.5T,(f)pr=0.6T,(g)pr=0.7T.(h)pr=0.8T,(i)pr=0.9T,(j)pr=T.Other
parameters are: N= 1000, uth =0.98, R= 300 and μ=0.99.
4 Conclusions
Chimera states on a non-locally coupled network of LIF neurons highly depend on the combi-
nation of coupling strength, coupling range and refractory period pr. The analysis of a network
of N= 1000 neurons has shown that chimera states appear for intermediate values of the
coupling strength. Furthermore, the emergence of chimera states also depends on the value of
the coupling range. More specifically, we have observed that as the coupling range increases,
the range of coupling strengths that favour the appearance of chimera states shifts to higher
values. Additionally, we have noticed that a crucial control parameter for the occurrence of
chimera states is the refractory period, a resting period between two consecutive excitations of
a neuron. We have shown that the refractory period helps the chimera states survive for longer
periods, while at the same time is responsible for the formation of multiple coherent and inco-
herent regions. The number of coherent and incoherent regions for fixed values of the coupling
strength and the coupling range, does not depend on the value of the refractory period.
Our results represent only a first approach to the study of the effect of control parameters
in a LIF network and to the phenomenon of partial synchronisation (more specifically chimera
states). Future work should address quantitative investigation of the parameter regions which
favour chimera states and could include additional parameters related to the experimentally
measured time-scales of biological neurons.
Multi-chimera states in the LIF model Tsigkri-DeSmedt, Hizanidis, H¨ovel and Provata
20
5Acknowledgements
This work was supported by the German Academic Exchange Service (DAAD) and the Greek
State Scholarship Foundation IKY within the PPP-IKYDA framework. This research has
been cofinanced by the European Union (European Social FundESF) and Greek national funds
through the Operational Program Education and Lifelong Learning of the National Strategic
Reference Framework (NSRF) – Research Funding Program: THALES. Investing in knowledge
society through the European Social Fund. Funding was also provided by NINDS R01-40596.
The research work was partially supported by the European Union’s Seventh Framework Pro-
gram (FP7-REGPOT-2012-2013-1) under grant agreement n316165. PH acknowledgez support
by DFG in the framework of the Collaborative Research Center 910.
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