Mass synchronization: Occurrence and its control with possible applications to brain dynamics

7.9 MB·English

by  V. K. Chandrasekar

Mass synchronization: Occurrence and its control with possible applications to brain
dynamics
V. K. Chandrasekar,1,∗ Jane H. Sheeba,1,† and M. Lakshmanan1,‡
1Centre for Nonlinear Dynamics, School of Physics,
Bharathidasan University, Tiruchirappalli - 620 024, Tamilnadu, India
(Dated: January 10, 2011)
Occurrence of strong or mass synchronization of a large number of neuronal populations in the
braincharacterizesitspathologicalstates. Inordertoestablishanunderstandingofthemechanism
underlying such pathological synchronization we present a model of coupled populations of phase
oscillatorsrepresentingtheinteractingneuronalpopulations. Throughnumericalanalysis,wediscuss
theoccurrenceofmasssynchronizationinthemodel,whereasourcepopulationwhichgetsstrongly
1
synchronizeddrivesthetargetpopulationsontomasssynchronization. Wehypothesizeandidentify
1
apossiblecausefortheoccurrenceofsuchasynchronization,whichissofarunknown: Pathological
0
synchronization is caused not just because of the increase in the strength of coupling between the
2
populations but also because of the strength of the strong synchronization of thedrive population.

n Weproposeademand-controlledmethodtocontrolthispathologicalsynchronizationbyprovidinga
a delayed feedback where the strength and frequency of the synchronization determines the strength
J and the time delay of the feedback. We provide an analytical explanation for the occurrence of

7 pathological synchronization and its control in thethermodynamic limit.

] PACSnumbers:
C
N
Strongsynchronizationofseveralgroupsofneu- ronal mass synchronization.
.
o ronal oscillators occurs during pathological states
i like Parkinson’s tremor, epileptic seizures, etc.
b
- This synchronization first occurs locally in a par- I. INTRODUCTION
q ticular region of the brain due to synchroniza-
[
tion of a group of neurons. This group then
Synchronization is an ubiquitous phenomenon and a
1 acts as a pacemaker triggering synchronization to
topic of active research in recent years [1–3]. Given that
v various other neuronal populations resulting in a
it is not always a desirable phenomenon, synchrony con-
8 mass synchronization. Deep brain stimulation is
5 a standard therapy for patients with this disor- trolmechanismare alwaysrequired. For example, in the
3 brain,when synchronizationsupports cognitionvia tem-
der. However since the actual cause and mech-
1 poralcoding of information[4, 5], it is desirable while at
anism of the occurrence of mass synchronization
1. in the brain is not yet fully known, there is al- the same time it is undesirable when synchronization of
0 a mass of neuronal oscillators occurs at a particular fre-
ways a need to understand the same in order to
1 quencybandresultinginpathologiesliketrauma,Parkin-
be able to control it effectively. In this article
1 son’s tremor and so on. Other examples include lasers
we attempt to gain this understanding by study-
:
v ing a system of coupled populations of phase os- andJosephsonjunctionarrays[6],emissionofmicrowave
i frequencies by coupled spin torque nano oscillators [7]
X cillators representing interacting neuronal popu-
where synchronization is desirable, while in the cases of
lations in the brain. We find the occurrence of
r epileptic seizures [8], Parkinson’s tremor [9], event re-
a mass synchronization in such a system where the
lated desynchronization [10, 11], or pedestrians on the
occurrence ofsynchronization inthesourcepopu-
Millennium Bridge [12], it is undesirable.
lation triggers synchronization in the target pop-
Pathological strong (or mass) synchronization occurs
ulations also. We hypothesize a possible cause
when several groups of neurons enter into synchroniza-
for the occurrence of mass synchronization in the
tion and the strength of the synchronization is large
brain, which is so far not known. We propose a
because of the large number of neuronal networks that
demand controlled method to control this mass
are engaged in synchrony. In particular, tremors in the
synchronization by providing a delayed feedback.
case of Parkinson’s disease (PD), essential tremor and
We provide numerical evidenceand analytical ex-
several other neurological diseases like epileptic seizures
planation for the occurrence and control of neu-
are caused by mass synchronization of oscillating neu-
ronal networks. Synchronization in such cases may also
be called abnormal synchronization, since under normal
conditions mass synchronization does not occur. Clus-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] ters of neurons fire in a synchronized manner with a fre-
‡Electronicaddress: [email protected] quency similar to that of the tremor and act like a pace-
maker driving the pre-motor and motor cortical areas of
2
the brain causing mass synchronization [13, 14]. In fact is assumed to be the same in all the populations, that
it has been shown that the local field potentials in the is equal to N. σηη′ is the strength of the coupling be-
thalamic and the sub-thalamic nucleus (STN) drive the tween the neurons in η′ and those in η. Note that
tremorinthelimb[15–17]. Inthecaseofhealthysubjects η =η′(1,2,...N′)correspondstointerconnectionamong
these neurons in the STN oscillate in a desynchronized theneuronsofthesamepopulation. Hereω(η) isthenat-
k
manner [18]. Deep brain stimulation is a standard ther- uralfrequencyofthe kthneuroninthe populationη and
apy for such patients with such tremors where a train 0 ≤ |αηη′| < π/2 is the phase lag. In reality every oscil-
of strong high frequency electric pulse is administered lator has different α, but we have taken them to be the
to the areas like STN where there is mass synchroniza- same for all of them, for simplicity.
tion. Inspiteofthegreatamountofinterestandresearch For a better understanding, a schematic diagram of
focussed on pathological tremors, the actual mechanism the modelis giveninFig. 1forN′ =2,wherethesource
that cause these tremors are still unknown [19–23]. Fur- populationisrepresentedbyadottedcircle(θ(1))andthe
ther,controllingofpathologicaltremorsisalsoaproblem i
target is represented by a solid circle (θ(2)). In normal
which has recently gained much attention [24–27]. i
states it is the low frequency signal that drives the high
Giventhattheinteractionmechanismandthetopolog-
frequency signal [28, 29], but in the case of pathological
ical structure of the neuronal networks are highly com-
synchronization, high frequency signal from the source
plex and are not completely known, and that there is a
drives the target populations that are oscillating at (rel-
lackoftechnologytomeasuresynchronizationineveryre-
atively)lowfrequenciesontosynchronization. Therefore
gionofneuronalnetwork,itishighlycomplicatedtogain
we have chosen σ = µσ , where 0 ≤ µ < 1, since the
a complete understanding of the underlying mechanism 12 21
coupling from the source to the target (σ ) is stronger
in these pathological states. In fact, the actual mecha- 21
than the coupling from the target to the source (σ ) in
nism for synchronous oscillations in Parkinson’s disease 12
pathologicalstates. Oscillatorsinside the sourceand the
remains unknown. Nevertheless, one can always repre-
target populations are coupled among themselves with
sentthesystemofinteractingneuronsasamathematical
strengths σ and σ , respectively. In order to quantify
modelandsimulatethechallengingcaseofapathological 11 22
synchronization within a population, we can define the
synchronizationto understandtheunderlyingdynamical
order parameter as
aspects. With this motive, in this paper, we hypothesize
the possible cause for the occurrence of mass synchro-
N
nization in PD tremors and also provide a method to zη =rηeiψ(η) = 1 eiθj(η). (2)
control it. N
j=1
The plan of the paper is as follows: In the following X
Sec. II we introduce a model of coupled populations When r = 1, there is complete synchronization within
η
of phase oscillators which represents interacting popu- the η population since in this state the phases of all the
lations of neurons. We present our numerical finding of oscillators in that population are the same. When r
η
the occurrenceofmass synchronizationwherethe source takes values in between 0 and 1 there is partial desyn-
population drives the target in Sec. III. In Sec. IV we chronization and when r = 0 there is complete desyn-
η
proposeamethodtocontroltheoccurrenceofpathologi- chronizationinthe η population. The time averageofr
η
calsynchronizationby providingadelayedfeedback. We is given by
analytically explain the occurrence of mass synchroniza-
tion and the method to control it in Sec. V. Finally in 1 T
R =<r >= r dt, (3)
Sec. VI we present our conclusions. η η η
T
Z0
which is used to characterize the occurrence of strong
II. THE MODEL synchronization in a population. Numerically, strong
synchronizationis characterizedby R >0.8 and desyn-
η
Weconsideramodelofinteractingpopulationsofneu- chronization is characterized by R ≤ 0.3 for T = 105
η
ronswhichisrepresentedbypopulationsofcoupledphase time units. When strong synchronization occurs in all
oscillators. One of the populations is considered to be a the populations due to the strong coupling strength in
sourcewhichactsasapacemakerandpossiblydrivesthe the source it is called mass synchronization. From equa-
otherpopulations(namedastargets)ontomasssynchro- tions (1) and (2) the model equations can be rewritten
nization. The model equations read as as
θ˙i(η) =ωi(η)−ηN′=′1σNηηη′′ Nηj=′=1Nsin(θi(η)−θj(η′)+αηη′), θ˙i(η) =ωi(η)−ηN′=′1σηη′rη′sin(θi(η)−ψ(η′)+αηη′). (4)
X X X
i=1,2,...,Nη′ =N,(1)
In order to find out the cause for the occurrence of mass
where η = 1,2,...N′, N′ is the number of populations synchronizationandalsotofindwaystocontrolit,wenu-
andNη′ is the numberofneuronsinpopulationη′ which merically simulate system (4) using Runge-Kutta fourth
3
8 4
θi(1) σ θi(2) (a) R1=0.2 (c) R1=0.99
21
µσ21 (1)θi4 (1)θi3
σ σ
11 22 0 2
0 50 100 0 50 100
t t
FIG. 1: Schematic representation of system (1) for N′ = 2. 8 4
(b) R=0.3 (d) R=0.98
θ(1) is the source population and θ(2) is the target. The cou- 2 2
i i
pling strengths within the populations are quantified by the
parameters σ and σ . The coupling strengths from the (2)θi4 (2)θi3
11 22
source to the target and the target to the source are quanti-
fied by theparameters σ =µσ and σ , respectively.
12 21 21 0 2
0 50 100 0 50 100
t t
FIG.2: Thetimeevolutionofthephasesinthesource(panels
order routine with a time step of 0.01. We have set
(a) and (c)) and the target (panels (b) and (d)) populations
N = 1000 and have assumed a Lorentzian distribution are plotted for N = 1000. Panels (a) and (b) represent the
for the oscillator frequencies given by initial desynchronized state characterized by R = 0.2 and
1
R = 0.3 and panels (c) and (d) represent the occurrence of
2
g(ω(η)) = γη (ω(η)−ω )2+γ −1, (5) strongsynchronizationinthetargetθi(2)(R2=0.98)whenthe
π η η source population θ(1) gets synchronized (R = 0.99). Here
(cid:20) (cid:21) i 1
σ22 =0.1, σ21 =1.5, µ=0.3, γ1,2 =0.05, ω1=1.5, ω2 =0.5,
whereγ is the half widthathalfmaximumandωη is the αij =0,i,j =1,2,andσ11 =0.1in(a)and(b)andσ11 =2.5
central frequency. We have discarded transients of the in (c) and (d).
order of 105 in our simulations. Simulations lasted for
2×105stepsandtheresultsareshownforasmallwindow
of time (which we label to start with 0 in the figures
for convenience) towards the end of the total simulation populations are desynchronized(σ =σ =0.1)with a
11 22
time. The initial phases are considered to be randomly very low value of the averageorder parameters R =0.2
1
distributed between 0 and 2π. In the following sections and R = 0.3, even with a strong coupling between the
2
we will discuss how the occurrence of synchronization source and the target (σ = 1.5 and µ = 0.3). Upon
21
in the source population causes mass synchronization in increasingthe coupling strengthwithin the sourcepopu-
the targetpopulations and will also presenta method to lation,thatisσ =2.5,wefindthatstrongsynchroniza-
11
control this mass synchronization. tion occurs in the source (characterized by R = 0.99)
1
Themodelweconsiderhereisofmassmodeltype and which in turn induces strong synchronization in the tar-
focusses on the macroscopic properties such as the syn- get (characterized by R = 0.98). Note that this strong
2
chronization of large population of neurons. The model synchronization occurs in the target due to its coupling
thereforedoesnotincludeadetailedmicroscopicdescrip- with the source, that is σ , and not because of the cou-
21
tion to incorporate the effects of hierarchal connections, plingwithinitself,sinceσ =0.1only. Thiscanbeseen
22
spiking activities of individual neurons and populations, from Fig. 2 where we have plotted the time evolution
theroleofneurotransmittersandsoon[30–34]. Wemake of the phases in the source (panels (a) and (c)) and the
useofthemacroscopicmodelingapproachtostrikeabal- target (panels (b) and (d)) populations. Panels (a) and
ancebetweenabstractanddetailedmodelingwhilekeep- (b) show the initial desynchronization state and (c) and
ing the number of variables and parameters to a reason- (d)depicttheoccurrenceofmasssynchronizationdueto
able minimum. Thus we attempt to make a qualitative the increasein the coupling (σ ) within the sourcepop-
11
concurrence between the model results and the occur- ulation. Fig. 3showsthesnapshotsofthedistributionof
rence of pathologicalmass synchronization. the phases in the (x ,y )=(cosθ ,sinθ ) plane where the
i i i i
panels (a)-(d) correspond to those in Fig. 2. In panels
(a) and (b) the oscillators in the source and the target
III. OCCURRENCE OF MASS populations are desynchronized initially and hence the
SYNCHRONIZATION phases are distributed throughout the unit circle. Pan-
els (c) and (d) depict the distribution of phases in the
In order to illustrate the occurrence of mass synchro- source and the target populations, respectively, in the
nization, we have taken N′ = 2, so that we have a sys- mass synchronization state; the phases occupy a rather
tem of one target and one source populations, each hav- narrow region on the unit circle.
ing 1000 oscillators. We have verified our results for the Thusitbecomesevidentthatfortheonsetofpatholog-
N′ = 3 case also (see Fig. 4 below). To start with, icalsynchronizationtheneuronsinthesourcepopulation
we consider a state when both the source and the target arestrongly coupledamong themselves. Due to this, the
4
1(a) 1(b) 1
SS
(1)yi 0 (1)yi 0 0.8
R (N’=2)
−1 −1 1
−1 0 1 −1 0 1 R (N’=2)
(1) (1) Rη 2
x x
i i R (N’=3)
1
0.3
1(c) 1(d) R2(N’=3)
DS
R (N’=3)
2) 0 2) 0 3
(yi (yi
0
0.5 1.5 2.5
σ
−1 −1
11
−1 0 1 −1 0 1
(2) (2)
xi xi FIG. 4: The time average of rη for increasing σ11 is plotted
for two cases, N′ =2 (dotted lines) and N′ =3 (solid lines).
Here σηη = 0.05, ση1 = 1.5, σ1η = µση1, for η = 2,3, µ =
FIG. 3: (a)-(d) are the snapshots of the phase portraits on 0.2, αij = π/2−0.1, i,j = 1,2,3, γ1,2,3 = 0.05, ω1 = 1.5,
the(xi,yi) plane corresponding to Fig. 2 (a)-(d),where xi = ω2,3 = 0.5, and σ23 = σ32 = 0.01. The regions DS and
cosθi and yi=sinθi. SS denote the desynchronization and strong synchronization
states, respectively.
IV. METHOD TO CONTROL MASS
occurrenceofastrongsynchronizationinthesourcepop-
SYNCHRONIZATION
ulationispropagatedontothetargetpopulationsbecause
of an increase in the order parameter r of the source,
1 In order to control the occurrence of mass neuronal
seeequation(4). Asaconsequenceσ r increasesfaster
21 1 synchronizationweapplyadelayedfeedbackforcetothe
than µσ r and hence the source drives the target and
21 2 populations in which case the model equation (1) be-
makes it to get synchronized with itself. Under normal
comes
conditions, however σ does not attain higher values to
11
cause strong synchronizationin the source. Thus for the N′
occurrenceofmasssynchronizationnotonlythecoupling θ˙i(η) = ωi(η)− σηη′rη′sin(θi(η)−ψ(η′)+αηη′)
strength between the populations is crucial but also the
η′=1
strength of the synchronizationof the source (character- X
(η)
ized by r1 or R1) plays a vital role. This is evident from ±FRτsin(θi −φτ), i=1,2,...,N, (6)
Fig. 4wherewehaveplottedthetimeaverageofr (R )
η η
where R = R(t − τ) and φ = φ(t − τ) and Z(t) =
for varying coupling strength σ of the source for the τ τ
two cases, N′ =2 and N′ =3. 11 Nη′′=1zη′ = Reiφ = N1′ Nη′′=1rη′eiψη′ is the global or-
der parameter. The feedback can be either positive or
For the case N′ = 3 we have considered three pop- nPegative as denoted by Pthe ± sign in the above equa-
ulations of coupled phase oscillators each having 1000 tion since the dynamics of the system does not change
oscillators with σ = 0.05, σ = 1.5, σ = µσ , for qualitatively upon using either + or − signs [35]. Also,
ηη η1 1η η1
η = 2,3, µ = 0.2, γ = 0.05, ω = 1.5, ω = 0.5, one can note that the ± sign can be interchanged upon
1,2,3 1 2,3
and σ = σ = 0.01. One of the populations is the replacing φ with φ−π.
23 32
source (with average order parameter R ) and there are TheglobalorderparameterR(t)measuresthestrength
1
two target populations (with average order parameters of synchronization of all the populations (combined) in
R and R ) in the system. For both the cases N′ = 2 the system while the order parameter r measures the
2 3 η
and N′ = 3, as σ increases the strength of the syn- strength of synchronization of the ηth population. The
11
chronizationofthesourceincreasesandasaconsequence schematic representation of this mass synchronization
synchronization in the target populations also increases. control set up is shown in Fig. 5. The strength of the
Interestingly, as one can see from the figure, for a given global synchronization R(t) and the corresponding fre-
σ , the target population achieves strong synchroniza- quency Ω (= dφ/dt) are measured from the populations
11
tion with the source even before the source population of neurons denoted by filled circles. A delayed feedback
achieves complete synchronization. Our model results of strength F and a time delay of τ is fed back to the
provetobe validforasystemoflargenumberofcoupled neuronal populations in order to control mass synchro-
populations also and we have presented here the case of nization. This method is effective since one can have
three populations for illustration. control over certain parameters like time delay and the
5
R & Ω F & τ 8 (a) R1=0.23
1) 4
(θi
0
0 100 200
t
8
(b) R=0.28
2
2) 4
(θi
0
FIG. 5: (Color online) Schematic representation of the con- 0 100 200
t
trol set up. Filled circles represent theneuronalpopulations.
Thestrengthoftheglobalsynchronization(R)andthecorre-
spondingfrequency(Ω) aremeasured. A delayedfeedbackof FIG.6: (a)and(b)showtheoccurrenceofdesynchronization
strength F and a time delay of τ is fed back to the neuronal for τ = 1.0 and F = 5.5 in the source (θ(1)) and the tar-
i
populations in order to cause desynchronization. get (θ(2)) populations, respectively due to delayed feedback,
i
following panels (c) and (d) of Fig. 2. HereN=256.
strength of the feedback. There is also no need to know V. ANALYTICAL EXPLANATION
about the interactions amongthe neurons or the charac-
teristicsofindividualneuronaloscillators,butallweneed Inorderto analyticallyexplainthe occurrenceofmass
is the macroscopic information. Further, the feedback is synchronization and its control, we analyze system (1)
demand controlled because F and τ are fixed depending in the continuum limit N → ∞. In this limit, a prob-
upon the values of R and Ω. In addition, there is no ability density for the oscillator phases can be defined
need to seek for a new controlling signal that stimulates as ρ(η)(ω,θ,t), which describes the number of oscillators
the neuronalpopulations,butthe measuredpathological withphaseswithin[θ,θ+dθ]andnaturalfrequenciesbe-
signal is used to controlthe same. We choose to provide tween[ω,ω+dω]at time t. This distribution ρ(η)(ω,θ,t)
delayedfeedback because it canaffect the activity of the obeys the evolution equation
individual neurons by shifting their phases.
∂ρ(η) ∂
The numerical simulation of equation (6) showing the =− (ρ(η)v(η)),η =1,2,...,N′ (7)
∂t ∂θ
control of mass synchronization is given in Fig. 6 which
followsthestateofmasssynchronizationshowninFig. 2 where from equation (4) (in the continuum limit) the
(panels (c) and (d)) after applying the delayed feedback velocity v(η) is given by
(τ =1.0andF =5.5). Duetothedelayedfeedbackboth
the source(panel(a)) andthe target(panel(b)) popula- N′ ∞ 2π
tionsaredesynchronized. Thecorrespondingdistribution v(η) = ω(η)− σηη′ sin(θ(η)−θ′(η′)+αηη′)
of phases on the (xi,yi) plane is shown in Fig. 7 where ηX′=1 Z−∞Z0
the phases are distributed throughoutthe unit circle de- ×ρ(η′)(ω(η′),θ′(η′),t)dθ′(η′)dω(η′),
noting the desynchronization states of the source (panel
η =1,2,...,N′. (8)
(a)) and the target (panel (b)) populations. For numer-
ical simulations including delayed feedback we have set
Also, in the continuum limit the order parameter (2)
N =256. We have also checked that the results are size
for the population η becomes
independent by calculating them for N =128 and 512.
∞ 2π
In the following section we analytically show the oc- z = eiθ(η)ρ(η)(ω(η),θ(η),t)dθ(η)dω(η). (9)
currence of mass synchronization and its control with η
Z−∞Z0
delayedfeedback in supportofthe abovenumericalfind-
ings as follows: (i) We will show that when the source is Nowwerewritethevelocityintermsoftheorderparam-
desynchronizedthe target population will also be desyn- eters to get
chronized. (ii) Then whenstrongsynchronizationoccurs
iInntthheessoturrocnegthsyenscahmroeniiszaintidouncesdtaitne,thwehteanrgdeetlaaylseod.f(eieiid) v(η) = ω(η)− N′ σ2ηiη′ ei(θ(η)+αηη′)zη∗′
backisintroducedthesynchronizationisbrokenforsuit- ηX′=1 (cid:18)
aobfltehevatliumeseodfeltahye.strengthof the feedback and the value −e−i(θ(η)+αηη′)zη′ . (10)
(cid:19)
6
1(a) 1(b) Now we express the amplitude equation (14) in polar
coordinates zη =rηeψη as (see equation (2))
(1)yi 0 (2)yi 0 1−rη2 N′
r˙η = −γηrη+( ) σηη′rη′cos(ψη−ψη′ +αηη′),
2
η′=1
X
−1 −1 1+r2 N′
−1 x0i(1) 1 −1 x0i(2) 1 ψ˙η = ωη−( 2rηη)η′=1σηη′rη′sin(ψη−ψη′ +αηη′),
X
η =1,2,...,N′. (15)
FIG. 7: (a) and (b) are the snapshots of the phase portraits
on the (x,y) plane corresponding to Fig. 6 (a) and (b) for Using these equations we will explain the occurrence of
N=256.
pathologicalsynchronizationwhenthesourceissynchro-
nized, and its control by applying delayed feedback, in
the following subsections.
Upon introducing the ansatz of Ott and Antonsen [36]
weconsideraspecialclassofthe density functionsinthe
A. Occurrence of mass synchronization
form of Poissonkernel as
ρ(η)(ω(η),θ(η),t) = g(ω(η)) 1+ ∞ (a (ω(η),t)eiθ(η))k AsnotedearlierinSec. III,wehavenumericallyfound
2π η theoccurrenceofmasssynchronizationforthecaseN′ =
(cid:20) (cid:18)kX=1 2 (as well as N′ =3) in Eq. (1). In order to explain this
+c.c ,η =1,2,...,N′, (11) analytically we rewrite the amplitude equation (15) for
the case N′ =2 as
(cid:19)(cid:21)
where g(ω(η) is the distribution of the oscillatorfrequen- 1−r2
r˙ = −γ r +( 1)(σ r cos(α )
cies. Here c.c denotes the complex conjugate of the pre- 1 1 1 2 11 1 11
ceding term. The main advantage of using this ansatz is +µσ r cos(ψ −ψ +α )),
21 2 1 2 12
that it has the same function aη in all the Fourier har- 1+r2
monics, except that a is raised to the nth power in the ψ˙ = ω −( 1)(σ r sin(α )
η 1 1 11 1 11
2r
nth harmonic. Thus one can reduce the infinite set of 1
+µσ r sin(ψ −ψ +α )), (16)
dynamical equations exactly into a set of finite number 21 2 1 2 12
of equations. Using the above expression (11) for ρ(η) 1−r2
r˙ = −γ r +( 2)(σ r cos(α )
into the evolution equation (7), we get the equation for 2 2 2 2 22 2 22
aη as +σ21r1cos(ψ2−ψ1+α21)),
1+r2
a˙η + iω(η)aη ψ˙2 = ω2−( 2)(σ22r2sin(α22)
2r
N′ 2
= ση2η′ eiαηη′zη∗′ −e−iαηη′zη′a2η , +σ21r1sin(ψ2−ψ1+α21)). (17)
ηX′=1 (cid:18) (cid:19) To start with we will consider the case µ = 0 in equa-
η =1,2,...,N′. (12) tions (16) and (17) for analytical simplicity to identify
pathological synchronization in neuronal populations so
Now, substituting the Poisson kernel (11) into the order that the source strongly (completely) drives the target
parameter equation (9) we get populations. However,ingeneralonecanobservetheoc-
currence of pathological synchronization for µ < 1 also
∞
z = a (ω(η),t)g(ω(η))dω(η). (13) asinthe caseofnumericalsimulationswhereµ=0.2and
η η
Z−∞ 0.3. Forthispurpose,wewillalsoconsiderbelowthecase
µ6=0 inequations (16) and (17) so as to providea theo-
Assuming a Lorentzian distribution (5) for g(ω(η)) and
reticalbasisforthenumericalfindings. Consequently,we
evaluating the integral in equation (13) by contour inte-
can conclude that when the strength of the synchroniza-
gration we get z∗ = a (ω −iγ ,t). Thus equation (12)
η η η η tion increasesin the source it induces synchronizationin
becomes
the target.
From equation (16), for µ = 0, the synchronization of
z˙ + (γ −iω )z
η η η η
the source is characterized by the stability of the fixed
= N′ ση2η′ e−iαηη′zη′ −eiαηη′zη∗′zη2 , potohinetr hr1san=d, th1e−de2sγy1n/cσ¯h1r1o,nwizhaetrioenσ¯st=ateσcisosc(hαa)r.acOtenrizthede
ηX′=1 (cid:18) (cid:19) by the stabiplity of the fixed point r1d = 0. When σ¯11 <
η =1,2,...,N′. (14) 2γ , the fixed point rd becomes stable and there is no
1 1
7
synchronizationin the source. In this state the equation (ω +ω )/2. The stability of the fixed point (r ,r ) =
1 2 1 2
for the target population is given as (0,0) is determined by the real part, Re(λ ) = −γ +
±
1−r2 κ4 ±21(ξ)12 for ∆ω =0, ofequation(20) (thatis the fixed
r˙ = −γ r +( 2)(σ r cos(α ) points are stable when Re(λ ) < 0). In this connection
2 2 2 2 22 2 22 ±
we wish to note that when σ = σ and µ = 1 the
1+r2 11 22
ψ˙ = ω −( 2)σ r sin(α ). (18) stabilityequationreducestotheonestudiedbyMontbrio
2 2 22 2 22
2r2 et al. [38] for a system of two interacting populations.
Again one can check that for σ¯ < 2γ the fixed point In this case the stability diagram for equation (20) will
22 2
r2 = 0 is stable and the target population is desynchro- be similar to Fig. 1 of [38]. When σ11 increases, the
nized. Thus when the source is desynchronized, the tar- condition γ < κ ± 1(ξ)12 is satisfied and hence the fixed
4 2
get population is also desynchronized. This result holds points(r ,r )=(0,0)becomeunstable. Thismeansthat
1 2
goodforµ<1alsoasisevidentfromournumericalfind- the increase in σ to a sufficiently high value induces
11
ings as depicted in panels (a) and (b) of Figs. 2 and 3 synchronization in the target also apart from inducing
for µ=0.3. synchronizationin the source.
On the other hand, when the coupling strength in the
source σ¯ increases so that σ¯ >2γ the fixed point rd
11 11 1 1
becomes unstable and rs becomes stable thus establish- B. Controlling mass synchronization
1
ing synchronization in the source. The synchronization
strength of the source increases as 1−2γ1/σ¯11. Af- In Sec. 3 we have shown from numerical analysis that
tersynchronizationinthe sourceisestablished,equation pathological synchronization can be controlled by pro-
p
(17) reduces to viding a delay feedback. Therefore including the delay
feedback term in Eq. (6) provides the following ampli-
1−r2
r˙ = −γ r +( 2)(σ r cos(α ) tude equation,
2 2 2 22 2 22
2
+σ21 1−2γ1/σ¯11cos(ψ+α21)), N′ F
ψ˙ = ω¯ −(p12+rr22)(σ22r2sin(α22) z˙η + (γη−iωη)zη+ηX′=1 2(cid:18)zη′τ −zη∗′τzη2(cid:19)
2 N′
+σ21 1−2γ1/σ¯11sin(ψ+α21)), (19) = ση2η′ e−iαηη′zη′ −eiαηη′zη∗′zη2 , (21)
whereψ =ψ2−ψ1 apndω¯ =ω2−ω1+(σ11−γ1)tan(α11). ηX′=1 (cid:18) (cid:19)
This equation does not admit the fixed point r = 0.
This means that when synchronization emerges2in the where zη′τ =zη′(t−τ). In the pathologicalsynchroniza-
tionstateallthepopulationsareinsynchronizationwith
source, the targetpopulation also begins to get synchro-
thesourceandhencethedynamicsofallthe populations
nized. The strength of the synchronization in the target
are similar to that of the source and are completely syn-
increasesaccordingtoσ 1−2γ /σ¯ ,eventuallylead-
21 1 11 chronizedwithit. Thusallthepopulationshavethesame
ingtosynchronizationwiththesource(wenoteherethat
in [36, 37] an equation simpilar to (19) has been analyzed synchronization strength and phase, that is rη ≃ R and
ψ ≃ φ, η = 1,2,...,N′. With this reasoning equation
in connection with the stability of the synchronized and η
(21) becomes
desynchronized states). This results in the pathological
strong synchronizationwhich corresponds to our numer- 1−R2
ical findings as depicted in panels (c) and (d) of Figs. 2 R˙ = −γR+( )(ν¯R−FRτcos(φ−φτ)),
2
and 3.
1+R2
Even for µ 6= 0, from equations (16) and (17), we can φ˙ = ω−( )(νˆR−FR sin(φ−φ )), (22)
τ τ
2R
establish that when the source is completely desynchro-
nisizwedhetnhethtearfigxeetdispaolsinotcrom=ple0teelxyisdtes,syrnc=hro0naizlseod.exTihstast. for γ = γη, ω = ωη, ν¯ = Nη′=′ 1σηη′cos(αηη′) and
When σ11 increases, syn1chronization i2s induced in the νˆ = Nη′′=1σηη′sin(αηη′), η P= 1,2,...,N′. Here R
source. Hence, the fixed point r = 0 does not exist as and φ are the global order parameters defined by Z =
a result of which r2 = 0 also bec1omes nonexistent. This Reiφ =PN1′ Nη′′=1rη′eiψη′. Now with a chage of variable
meansthatwhensynchronizationarisesinthesource,the φ=Ωt+φˆequation (22) can be rewritten as
target also spontaneously becomes synchronized. This P
can be described as follows: The eigenvalue of the fixed 1−R2
R˙ = −γR+( )(ν¯R−FR cos(φˆ−φˆ +Ωτ)),
point (r ,r )=(0,0) is τ τ
1 2 2
λ± =−γ+ κ4 ± 21(ξ−iAˆ∆ω−∆ω2)12 −iω¯, (20) φˆ˙ = ωˆ −(1+2RR2)(νˆR−FRτsin(φˆ−φˆτ +Ωτ)), (23)
where γ1,2 = γ, αi,j = 0, i,j = 1,2, κ = σ11 + σ22, where ωˆ = ω −Ω. The fixed point R = 0 of the above
Aˆ=σ −σ , ξ =(1Aˆ2+µσ2 ),∆ω =ω −ω andω¯ = equation, which characterizes the fully desynchronized
11 22 4 21 1 2
8
state, is not of physical interest here. So we are now tion: When strong synchronization occurs in the source
interestedinthepathologicalsynchronizationstateofthe this in turn increases the coupling strength between the
system which is represented by the fixed point solution 1
R 6= 0 and φˆ = constant. Consequently from equation
(23) we have
2γ
R = 1− ,
s ν¯−Fcos(Ωτ)
R0.5
τ=0.05,ν=2.0
νˆ−F sin(Ωτ)
Ω = ω+(γ−ν¯+F cos(Ωτ)) . (24) τ=0.10,ν=2.5
ν¯−F cos(Ωτ)
τ=0.05,ν=2.0
Forsuitable values ofF andτ the strengthofthe patho- τ=0.10,ν=2.5
logical synchronization R can be reduced for a given set
of system parameters. This is evident from Fig. 8 where 0
0 1.7 3.4
we have plotted the order parameter R as a function of
F
the strength of the feedback F for two different values
of the time delay parameter τ. From Fig. 8, we observe FIG. 8: The occurrence of desynchronization denoted by a
that in the absence of feedback, that is F =0, there is a decrease in R due to delayed feedback. The solid and the
strongsynchronizationinthe systemandhence R≃0.9. dottedcurvesaretheanalyticallyobtainedsolutionsgivenby
Upon introducing the feedback, the synchronization in equation(24). ∗and⋆curvesarethecorrespondingnumerical
the system is destroyed for a suitable values of F for a simulations of equation (1) for N′ = 2, γ = 0.02, ω = 1.5
given τ. The solid and the dotted curves are the analyt- σi1 =ν and αij =0, i,j =1,2.
ically obtained solutions given by equation (24) and the
curves denoted by ∗ and ⋆ are the correspondingnumer-
ical simulations of equation (1) for N′ = 2. Thus our
analytical findings are in reasonable agreement with the source and the target. In this state the source strongly
numerical results. drives the targetand as a result the targetgets synchro-
Thus we find that the above analytical consideration nized with the source. Thus not only the coupling be-
correlates our earlier numerical study and provides a tween the source and the target causes the pathological
strong theoretical basis to identify the hypothesis that synchronizationbutalsothestrengthofthesynchroniza-
when strong synchronization occurs in the source it in- tion in the source plays a vital role.
ducessynchronizationinthetargetalsobyincreasingthe We have proposed a possible method to control the
couplingstrengthbetweenthesourceandthetarget. We occurrence of this pathologicalsynchronization. We find
furtherconcludethatstrongsynchronizationcanbecon- that a delayed feedback given to the system of neuronal
trolled by a suitable delayed feedback control. populations destroys the strong synchronization and es-
tablishes desynchronization. The method is demand-
controlled since the strength and the time delay of the
VI. CONCLUSION feedback is determined from the strength and the fre-
quency of the synchronization. We have analytically an-
Occurrence of strong pathological synchronization in alyzed the model in the thermodynamic limit and have
neuronalpopulationsseverelyimpairsbrainfunctionand deduced the amplitude equation using the ansatz of Ott
causes pathological tremors. In general, during patho- and Antonsen [36]. From this equation we have shown
logical synchronization, a particular group of neurons in theexistenceofsynchronizedsolutionsinthetargetwhen
certain regionof the brain gets synchronizedand acts as strongsynchronizationoccursinthesource. Uponapply-
a pacemaker which drives other cortical neuronal popu- ing delayed feedback we have found that the strength of
lations onto mass synchronization. The synchronization the synchronizationcanbe reduced. We believe thatour
frequency in such casesis close to orequal to the tremor hypothesisforthecauseoftheoccurrenceofpathological
frequency [13, 14]. synchronization will be useful for a better understand-
In this paper we have considered a model of coupled ing and further research in this direction. The proposed
populations of phase oscillators to study the occurrence method will help in the effective stimulation of neuronal
of strong synchronization. We consider one of the popu- populations in deep brain stimulation therapy.
lations to be the source and the other populations that The work is supported by the Department of Science
are driven to be the targets. We find the occurrence of and Technology (DST)–Ramanna program (ML), DST–
pathological synchronizationin the model, and from the IRHPA research project (ML and VKC) and an INSA
dynamicalpropertiesweareabletohypothesizethepos- Senior Scientist fellowship (ML). JHS is supported by a
siblecausefortheoccurrenceofsuchstrongsynchroniza- DST–FAST TRACK Young Scientist research project.
9
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