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Metastable states and transient activity in ensembles of excitatory and inhibitory elements

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2010 EPL 91 20006

(http://iopscience.iop.org/0295-5075/91/2/20006)

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July 2010

EPL, 91 (2010) 20006 www.epljournal.org

doi: 10.1209/0295-5075/91/20006

Metastable states and transient activity in ensembles of excitatoryand inhibitory elements

M. A. Komarov1(a), G. V. Osipov

1 and J. A. K. Suykens2

1Department of Control Theory, Nizhny Novgorod University - Gagarin Avenue, 23, 603950 Nizhny Novgorod, Russia2K.U. Leuven, ESAT-SCD/SISTA - Kasteelpark Arenberg 10, B-3001 Leuven (Heverlee), Belgium, EU

received 20 May 2010; accepted in final form 12 July 2010published online 5 August 2010

PACS 05.45.Xt – Synchronization; coupled oscillators

Abstract – Complex activity in biological neuronal networks can be represented as a sequentialtransition between complicated metastable states. From a dynamical systems theory point of viewsequential activity in neuronal networks is associated with the existence of stable heterocliniccontours in the phase space of the corresponding neuronal model. Previously, the conditions ofexistence and stability of these contours have been studied in networks consisting of only inhibitorysynaptically coupled cells. In this paper the effect of excitatory neurons is studied. In contrastto early studied cases, in the considered model the heteroclinic trajectories are located on two-dimensional manifolds. This leads to the existence of an infinitely large number of transitionsbetween metastable states. The conditions for the existence of metastable states and arisingsequential dynamics are presented.

Copyright c© EPLA, 2010

Introduction. – Recent theoretical and experimentalstudies indicate that the sequential transient activityplays a crucial role in neuronal circuits functioning [1–3].Such type of activity takes place in important func-tions of the neural system of animals and concernsproducing spatio-temporal patterns in motor systemsof mollusks [4,5], dynamics in the high vocal center ofbirds [6], and fast recognition of stimuli in sensory corticalensembles [7]. Experiments show that such a sequence ofactive states in neural ensembles is strongly determinedby external stimuli. Simultaneously, this sequential activ-ity is a non-periodic transient process reproducing itselffrom trial to trial. In [8] it was shown that reproducibleand highly depending on external stimuli sequentialdynamics is based on a winnerless competition principle.The models which are able to demonstrate winnerlesscompetition dynamics were suggested and investigatedin [5,9–12]. The key property of such models is theexistence of a heteroclinic contour in the phase space ofthe model. Such a contour corresponds to the sequenceof heteroclinic orbits connecting saddle points or saddlecycles. The transition of a phase point in the vicinity ofa certain singular saddle trajectory corresponds to theactivation of a certain neuron or cluster of neurons. At thesame time such a sequence has a large basin of attraction.

(a)E-mail: makomar@mail.ru

It is strongly determined by external stimuli and bythe topology of inhibitory connections between neurons.The heteroclinic contour and sequential activation in theconsidered models arise when there is a sufficient levelof asymmetry in the inhibitory interactions between theelements in the network. On the other hand, real neuronalnetworks have a large number of excitatory synapsesand the influence of the stimulating interaction betweenelements on the sequential activity and heterocliniccontour formation is still poorly understood. In thisarticle we show that the arising of a heteroclinic contourand sequential activity can be caused due to excitatoryconnections even in the case of symmetric inhibitionbetween elements. In the first section we investigatethe basic model of sequential activity proposed in [9],but we include both excitatory and inhibitory typesof connections. In the second part, based on analyti-cal results described in the first section, we show thegeneration of sequential activity produced by excitatoryconnections in the network of biologically relevant spikingmodels.

Generalized Lotka-Volterra model. – Let usconsider the basic model of sequential activity introducedin [9]. The modified model includes both excitatory aswell as inhibitory elements. This model is described by

20006-p1

M. A. Komarov et al.

Fig. 1: Scheme of coupling between entities.

the following ODE system:

Ei =Ei(σEi +∑j

ρEEij Ej −∑j

ρEIij Ij −Ei),

Ii = Ii(σIi +∑j

ρIEij Ej −∑j

ρIIij Ij − Ii),

i= 1, . . . , N.

(1)

Such a system can be considered as N interacting entitiesof neurons. The variables Ei and Ii denote the averagedspiking rate of excitatory and inhibitory popula-tions which form the i-th entity [9]. The nonnegativecoupling coefficients ρEEij � 0, ρIEij � 0 correspond to theexcitatory connection formed from the excitatory popu-lation of the j-th entity to the excitatory and inhibitorypopulations of the i-th entity, correspondingly. The terms∑Nj=1 ρ

EEij Ej in the first equation and

∑Nj=1 ρ

IEij Ej in the

second one of the system (1) characterize the stimulatingforce (influence) of the excitatory neurons of the network.The terms

∑j ρEIij Ij ,

∑j ρIIij Ij and nonnegative coupling

coefficients ρEIij � 0, ρIIij � 0 describe the inhibitory inter-action between the neurons of the network. The constantsσE,Ii > 0 describe the external stimuli applied to theneurons [5] (fig. 1).The case of only inhibitory interactions between the

neurons (Ei = 0) was studied in the papers [5,9] and itwas shown that under certain conditions on the couplingmatrix {ρIIij } the existence of a stable heteroclinic channeland sequential dynamics is possible. In this paper bothexcitatory and inhibitory types of neurons are presented(Ei �= 0, Ii �= 0).First, we consider the case when two subsystems of each

entity are identical. It means that i) external inhibitoryand excitatory force to the different neurons of the entityare identical:

σEi = σIi = σi,

ρEEij = ρIEij = ρ

Eij ,

ρEIij = ρIIij = ρ

Iij ,

and ii) elements inside each entity are noninteracting:

ρIEii = ρEIii = 0,

ρEEii = ρIIii = 0.

In the case of zero excitation (ρIE,EEij = 0) and strong

symmetric inhibitory couplings (ρEI,IIij = ρEI,IIji > 1) theexcitatory neurons do not influence the inhibitory neuronsand the dynamics of the system is governed by theinhibitory network Ii with inhibitory connections only.Such type of dynamics is well studied in [9,13,14]. Thesystem has a number of 2N stable nodes Pi : {Ei = Ii =σi;Ej = Ij = 0; i, j = 1, . . . , N ; j �= i} and all phase trajec-tories of the system approach the stable equilibrium pointsPi. The most interesting dynamics occur when introduc-ing asymmetric inhibitory [5,9] or excitatory connections.Each steady state Pi has N − 1 pairs of equal eigenvaluesλ1,2i = σj +(ρ

Eji− ρIji)σi and two eigenvalues λ1,2i =−σi.

All Pi are stable equilibrium points in the case of stronginhibitory connections ρIji and sufficiently weak excitatory

interaction ρEji:

ρIji >σj

σi+ ρEji. (2)

Suppose that there exists a certain set of steady statesPi such that for every point Pi there exists such a numberni ∈ {1, . . . , N} that condition (2) is not satisfied. It meansthat the two eigenvalues λ1,2ni = σni +(ρ

Enii− ρInii)σi > 0

and that the equilibrium Pi has a two-dimensional unsta-ble manifold. This can be realized if a more strict conditionis satisfied: ρEnii− ρInii > 0. Therefore, we can conclude:

ρIji− ρEji >σj

σi, j �= ni,

ρInii− ρEnii <σniσi< 0.

(3)

It is necessary to note that we consider the case ofasymmetric interactions between neurons, i.e. if for thei-th saddle ρInii− ρEnii <

σniσithen for the ni-th equilibrium

one hasρIini − ρEini > σi

σni. (4)

Now we will show that conditions (3) and (4) provide theexistence of a heteroclinic channel in the system (1). Toprove this we consider the saddle point P1 (E1=I1=σ1,Ek = Ik = 0) and the system (1) on an invariant mani-fold M1 : {Ej = Ij = 0, j �= {1, n1}} where the dynamics isdescribed by the following four-dimensional system:

E1 =E1(σ1+ ρE1kEk − ρI1kIk −E1),

I1 = I1(σ1+ ρE1kEk − ρI1kIk − I1),

Ek =Ek(σk + ρEk1E1− ρIk1I1−Ek),

Ik = Ik(σk + ρEk1E1− ρIk1I1− Ik).

(5)

Here we denote k= n1. Due to the conditions (3), (4) thefixed point Pk : {E1=I1=0 Ek=Ik=σk} in the system (5)has negative eigenvalues λ1,2k =σ1+(ρ

E1k − ρI1k)σk<0,

λ3,4k =−σk < 0 and P1 : {E1 = I1 = σ1, Ek = Ik = 0} hastwo positive λ1,21 = σk+(ρ

Ek1− ρIk1)σ1 > 0 and two nega-

tive λ3,41 =−σ1 < 0 eigenvalues. Thus Pk onM1 is a stablenode and P1 is a saddle with a two-dimensional unstable

20006-p2

Metastable states and transient activity in ensembles of excitatory and inhibitory elements

manifold. After the anzatz E∗k = σk −Ek, I∗k = σk − Ik thesystem can be rewritten as follows:

E1 =E1(−β− ρE1kE∗k + ρI1kI∗k −E1),I1 = I1(−β− ρE1kE∗k + ρI1kI∗k − I1),E∗k = (E

∗k −σk)(ρEk1E1− ρIk1I1+E∗k),

I∗k = (I∗k −σk)(ρEk1E1− ρIk1I1+ I∗k),

(6)

where β =−(σ1+σk(ρE1k − ρI1k))> 0. In the system (6) thestable node Pk is located at the origin and the coordinatesof the saddle P1 are E1 = I1 = σ1, E

∗k = I

∗k = σk. Let us

show now that all trajectories starting from the vicinityof the saddle point P1 approach point Pk. To show thiswe consider region G which includes points P1 and Pkand prove that all trajectories from this area approachthe attractor P1. Let us consider region G:{

E1, I1 � 0, E∗k , I∗k � σk,γ ≡ ρEk1E1− ρIk1I1 > 0.

(7)

The boundaries of region G have the following features:

i) the boundaries E1 = 0, I1 = 0, Ek = σk, Ik = σk areinvariant planes,

ii) while I1 =ρEk1ρIk1E1 (γ = 0), the inequality I1 � ρ

Ek1

ρIk1E1

holds.

Features i), ii) of the G boundaries provide for the factthat all trajectories starting in region G remain inside it.The third equation of system (6) determines the dynamicsof E∗k . One can see that in G the variable E

∗k tends

to −γ = ρIk1I1− ρEk1E1 � 0 with increasing time. In ananalogous way we can claim that in G the value of I∗kalso tends to −γ � 0. Therefore E∗k and I∗k decrease in thearea G at least to zero. Let us consider the right-hand sideof the first equation of system (6):

E1 =E1(−β− ρE1kE∗k + ρI1kI∗k −E1).One can conclude that in area G:

−E∗k < ρEk1E1− ρIk1I1 < ρEk1E1.Thus,

E1(−β− ρE1kE∗k + ρI1kI∗k −E1)<E1(−β+ ρI1kI∗k −E1(1− ρE1kρEk1)).

Let us suppose that

1− ρE1kρEk1 > 0. (8)

Condition (8) is the condition of asymmetric excitatoryconnections between entities. In our case it emphasizesthat ρE1k has to be sufficiently small. Taking into accountthat −β < 0,−E1(1− ρE1kρEk1)� 0 and I∗k decreases at leastto zero in area G, we can claim that all trajectories

02

4

024−0.5

0

0.5

Ek

P1

Pk

E1−I

1

Ik

0 2 4 6 8 10

0

1

2

3

4

5

Ek*+I

k*

Pk

P1

E1

(a)

(b)

Fig. 2: (Colour on-line) (a) Projection of phase trajectoriesstarting from the saddle point P1 and approaching the stablenode Pk in (5); (b) Projection on the plane (E1, E

∗k + I

∗k)

of phase trajectories starting from the saddle point P1 andapproaching the stable node Pk in (6).

from the area G go inside the region G′ ∈G, where−β+ ρI1kI∗k < 0 and

E1 =E1(−β− ρE1kE∗k + ρI1kI∗k −E1)< 0in all points of G′ except for P1 where the derivative isequal to zero. Thus, in the area G′ the following inequali-ties take place: E1 � 0 and I1 � ρ

Ek1

ρIk1E1 which guarantee the

convergence of E1, I1 to zero with increasing time. Takinginto account that E∗k and I

∗k tend to −γ = ρIk1I1− ρEk1E1

we can state that all trajectories from the area G approachthe stable node P1.Hence, the analysis of the system (1) on the invari-

ant manifold M1 shows that conditions (3), (4) withconditions of asymmetric excitatory connections (8)provide the existence of heteroclinic trajectories betweenequilibriums P1 and Pk (fig. 2) located on the unstabletwo-dimensional manifold of the saddle point P1. Inthe 4-dimensional subsystem Pk is a stable node but inthe whole 2N -dimensional phase space Pk might haveunstable directions. In the latter case, we can then provethe existence of heteroclinic trajectories between the

20006-p3

M. A. Komarov et al.

saddle points Pk and Pnk , by considering the invariantmanifold Mk.The heteroclinic contour is locally stable when all

saddles that are forming a contour are dissipative [9]. Thisholds when the maximal stretching along the unstablemanifold is less than the minimal contraction along the

stable manifold: min{−λs}

max{λu} > 1 [15]. Let us consider for defi-niteness min{−λs}=−λs0 = σi. In this case the conditionfor the local stability of the heteroclinic contour can bewritten as follows:

σni +σi+(ρEnii− ρInii)σi > 0. (9)

Hence, under certain conditions on the coupling coeffi-cients, a locally stable heteroclinic contour exists in thesystem (1). One can see that conditions (3) always providethe local stability of the heteroclinic contour. In contrastwith [5,9], saddles are connected via trajectories lying ontwo-dimensional unstable manifolds. Time series and theprojection of the phase trajectory in the vicinity of theheteroclinic contour are shown in fig. 3.According to conditions (3), (4), (8) the existence of a

stable heteroclinic contour firstly can be associated withcoupling coefficients ρE,Iij and parameters σj correspond-ing to the stimuli applied to the elements of the network.In articles [5,8–12] it was shown that sequential dynam-ics and heteroclinic contours arise when in the systemthere is a sufficient level of asymmetry in the inhibitory

couplings between the elements, i.e. whenρIiniρInii> ρ∗ where

ρ∗ is a certain constant value depending on the model andparameters of the network. The above-described analyticalstudy shows that the sequential dynamics and metastablestates can arise also in the network with symmetric inhibi-tion. In fact conditions (3), (4), (8) can be satisfied whileρIini = ρ

Inii= ρ0 >

σniσi, but now asymmetry in the interac-

tions can be carried due to asymmetric excitatory connec-tions:

ρEnii > ρ0−σniσi> ρEini . (10)

While condition (10) holds for the system (1), a stableheteroclinic contour exists at symmetric inhibitoryconnections. Figure 3 shows the dynamics in the vicinityof the heteroclinic contour which arises exactly due toasymmetric excitatory connections.

Spiking model. – A similar organization of connec-tions between populations of excitatory and inhibitoryneurons leads to sequential dynamics also in the caseof biologically relevant spiking models. Using the resultsobtained above, we now present the effect of generating asequential activity due to asymmetric excitatory connec-tions in a network of spiking neuron models [16]. The

dynamics of an isolated neuron (SE,Ii,j = 0) is described by

01

23

45

0 1 2 3 4 5

012345

E1

Ek

P1

Pk

Pm

Em

0

5

10

0

5

10

0 100 200 300 400 500 600 7000

5

10

Time

E1+I

1

Ek+I

k

Em

+Im

(a)

(b)

Fig. 3: (Colour on-line) (a) Time series in the vicinity ofthe heteroclinic contour (noise added). (b) Projection ofthe phase trajectory (blue line) in the vicinity of hetero-clinic trajectories (red lines) lying on a two-dimensionalmanifold (k= n1, m= nk). Transition of the phase point inthe vicinity of a certain saddle point corresponding to theactivation of a certain entity in the network. Parameters:σ1,k,m = 5, ρEk1 = ρ

Emk = 14.5, ρEk1 = ρ

Ekm = ρ

E1m = ρ

Em1 = 0,

ρIk1 = ρI1k = ρ

Im1 = ρ

I1m = ρ

Ikm = ρ

Imk = 10.

the following system of equations:

Ei,j = 0.04E2i,j +5Ei,j +140−ui,j +SEi,j +σEi,j ,

ui,j = a(bEi,j −ui,j),Ii,j = 0.04I

2i,j +5Ii,j +140− vi,j +SIi,j +σIi,j ,

vi,j = a(bIi,j − vi,j).

(11)

Here Ei,j and Ii,j denote membrane potentials of i-thexcitatory and inhibitory neurons in a population withnumber j. ui,j and vi,j are the recovery variables of theneurons. According to [16], when the membrane potentialEi,j(Ii,j) reaches the value 30mV the recovery variableui,j(vi,j) shifts by the value d and Ei,j(Ii,j) resets tothe value c. In our simulation we use the following

20006-p4

Metastable states and transient activity in ensembles of excitatory and inhibitory elements

parameters: a= 0.1, b= 0.2, c=−65, d= 2. ParametersσE,Ii,j correspond to the external stimuli applied to the

cells. Terms SE,Ii,j represent the external synaptic currentcaused by the action potentials of the other neurons in thenetwork:

SEi,j =∑m,n

(ρE,Ei,j,m,nREm,n− ρE,Ii,j,m,nRIm,n)

SIi,j =∑m,n

(ρI,Ei,j,m,nREm,n− ρI,Ii,j,m,nRIm,n).

(12)

As before ρI,Ei,j,m,n are the strengths of excitatory andinhibitory connections directed from the neurons of theentity with number n to the neurons of the j-th entity.RE,Im,n are the variables governed by the nonlinear first-order differential equations:

REm,n = αEF (Em,n)(1−REm,n)−βEREm,n,

RIm,n = αIF (Im,n)(1−RIm,n)−βIRIm,n,

(13)

where F (x) = 1/(1+ exp(−10x)) is a sigmoid activationfunction and αI,E = 1, βI,E = 0.01 the synaptic timeconstants. If the membrane potential Em,n (Im,n) reacheszero value than the variable REm,n (R

Im,n) increases and

changes the total synaptic currents SE,Ii,j (according

to (12)) of the postsynaptic neuron. Therefore eqs.(11)–(13) determine the model of the network. Weconsider three (j, n= 1, . . . , N , N = 3) entities. As beforein the system (1) each entity contains two populations byten (i,m= 1, . . . ,M , M = 10) neurons of excitatory andinhibitory type. Within the entity the elements are slightlynonidentical (parameters σE,Ii,j are randomly distributed

in the range [5.0, 5.1]) and they are not interacting

with each other (ρE,Ii,j,i,n = 0). Inhibitory interactions are

symmetric (ρI(E),Ii,j,m,n = ρ

I(E),Im,n,i,j = for all m �= j). In the

case of excitation absence (ρE,Ei,j,m,n = 0) and sufficientlystrong inhibition in the system, the multistability of threestationary states is possible: one entity due to stronginhibition suppresses spiking oscillations in other entities(fig. 4(a)). By introducing asymmetric excitatory connec-tions one satisfies conditions (3),(4),(9) and a sequentialswitching between the activity of the entities becomespossible. Sufficiently strong excitatory connections causemetastable states and transient activation of the entitiesin the network (fig. 4(b)).

Conclusion. – The generation of sequential activity isa major property of neural networks at all levels of theorganization of the animals nervous system. One of thedynamical principles underlying the sequential transientactivity is the winnerless competition principle [8]. Themathematical picture of sequential activity is given bythe stable sequence of heteroclinic orbits between saddletrajectories in the phase space of the dynamical modeldescribing the activity of the network. The transition ofthe phase point in the vicinity of heteroclinic contour leads

(a)

(b)

Fig. 4: (Colour on-line) (a) The case of strong and symmet-ric inhibitory connection with excitation absence between theelements (ρE,Ei,i,j,m,n = ρ

I,Ei,i,j,m,n = 0, ρ

E,Ii,i,j,m,n = ρ

I,Ii,i,j,m,n = 4.3

for all i,m= 1, . . . , N and j, n= 1, . . . ,M). (b) Introducingasymmetric excitatory connections leads to the sequentialswitching between the entities of neurons ((ρE,Ei,j,i+1,n) =

ρI,Ei,j,i+1,n = 3.9).

to the consecutive visiting in time of certain saddle pointsor saddle cycles which corresponds to the consecutive acti-vation of the neurons in the network. In articles [5,8–12]it was shown that a heteroclinic contour and transientactivity arises only in the case of asymmetric inhibitoryinteractions between elements in neural network. However,excitatory synapses are a prevailing type of interactionin neural ensembles and in this paper the influence ofexcitatory interactions between neurons on the forma-tion of stable heteroclinic contours and sequential dynam-ics was studied. It is shown that excitatory connectionsbetween neurons can lead to the formation of metastablestates and sequential switching of activity also in the caseof symmetric inhibitory connections. We will note that,in contrast to [5,9], in the considered model the hetero-clinic trajectories are located on two-dimensional unstable

20006-p5

M. A. Komarov et al.

manifolds. The conditions for the sequential transientactivity formation were investigated in a phenomenolog-ical neuronal network model. It was demonstrated thatasymmetric excitatory synaptic connections can also leadto the formation of sequential dynamics in the case of bio-logically relevant neuronal and synaptic coupling models.

∗ ∗ ∗We thank M. I. Rabinovich for fruitful discussions.

JAKS acknowledges the support from K. U. Leuven,the Flemish government, FWO and the Belgian federalscience policy office (FWO G.0226.06, CoE EF/05/006,GOA AMBioRICS, IUAP DYSCO, BIL/05/43). MAKand GVO acknowledge the support from the FederalProgram “Scientific and Scientific-educational brainpowerof innovative Russia” for 2009–2013 (contracts Π2018,Π15, Π2308, 02.740.11.5138, Π942, 02.740.11.5188), andfrom the RFBR (grants 08-02-92004, 08-02-970049, 10-02-00940).

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