Proceedings of lnternational Joint Conference on Neural Networks, Montreal, Canada, July 31 - August 4, 2005

Stability Conditions of the full MI Model of

Excitatory and Inhibitory Neural PopulationsRoman Ilin

Department of Mathematical SciencesThe University of Memphis

Memphis, TN 38152E-mail: rilin@memphis.edu

Abstract-We consider the model of interacting neural pop-ulations to be the main building block of K-sets, as suggestedby WJ. Freeman. The full KII set's dynamics is understoodthrough building the system up from the reduced KII. Theoreticalcondition for stability of the intermediate K1I model is derivedand the regions of structural stability of the full KII model areidentified based on numeric data.

I. INTRODUCTION

It is only natural to consider the properties of the biologicalbrain when designing new types of neural networks. After all,the field of neuro-computing was inspired by the discoveriesin neuroscience.

Studying the EEG produced by the animal brain, Walter J.Freeman proposed a mathematical model aiming to reproducecomplex spatio-temporal oscillations observed in cortical EEGsignals ([1], [2]). He further suggested that such a modelcan be used as a novel type of neural network (NN) withnonconvergent encoding principles capable of performing thesame tasks as conventional NN and much more ([3], [4], [5],[6]). Freeman called the model's components K-sets, and thusthe new type of NN was born. Originally, K-sets modeledthe olfactory system of rabbits, but they have later beengeneralized to other regions of the brain.

The olfactory system consists of several layers of inter-connected neural populations projecting activations from theolfactory receptors to the cortical part of the brain; see Fig.1. An important part of the olfactory system is the olfactorybulb (OB). The OB consists of interacting neural populationsthat are mutually excitatory, mutually inhibitory, or coupledinto a negative feedback loop; see Fig. 2. The three typesof interactions are necessary in order to create the complexdynamics of the system. These dynamics are characterized bya chaotic basal state which jumps into different attractors uponreceiving stimuli from the outside world [5].

Each circle in Fig. 1 represents a population of thousandsof neurons. It is called a KO set. Its dynamics are described bythe following 2nd order ordinary differential equation (ODE):

1( d2t) +ad+ + 3y(t)) = P(t)/3 dt2 d (1)

Robert KozmaDepartment of Mathematical Sciences

The University of MemphisMemphis, TN 38152

E-mail: rkozma@memphis.edu

where a and : are constants derived from biological mea-surements.The right hand side represents input received from other

populations. Before one population's output becomes anotherpopulation's input, it is transformed through the followingnonlinear transfer function [2]:

(1Q eli)Q(y) =qm,,(1 -e q. (2)

This is an asymmetric nonlinear sigmoid where qm is aconstant that determines its maximum value for large y's. Thederivative Q'(y) is positive for all y's and has its maximumat Ymax > 0-

Thus the interacting populations are described by coupledODE's with a nonlinear r.h.s. The coupling is achieved throughconnection weights which model the gain at synaptic con-nections of neurons. These weights cannot be determinedexperimentally and it is our task to understand how the choiceof coupling weights affects the dynamics of K-sets and toselect the most biologically plausible weight parameters.The ODE's under consideration cannot be solved analyti-

cally and understanding their behavior requires using varioustechniques from the field of dynamic systems.

The basic unit of the OB or cortical model consists offour interacting populations, as shown in Fig. 2. It has beensuggested that the simplified version of the model with thebasic unit consisting of only two interacting populations asshown on Fig. 3 has all the essential properties of the originalmodel, since all three types of interactions are present whensuch units are linked in a layer using excitatory to excitatoryand inhibitory to inhibitory connections. This is called thereduced KII model (RKII). Several studies of this system havebeen conducted ([7], [8]).

In this paper we consider the original model, with the basicunit consisting of four populations. It is called the full KII orsimply KII. It consists of two excitatory and two inhibitorypopulations coupled together. We build on the results ofprevious studies of reduced KII sets to obtain further insightinto full KII dynamics.

II. REVIEW OF RESULTS FOR REDUCED KIIA proven engineering approach for analyzing complex

systems is to initially simplify the system, understand its

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Olfactory System I

2 g g ~~~El-Receptors|

: > ~~~~~~~Bulb(OB)|

Ah< * |TractCortex(inner links

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Fig. 1. Schematic representation of the olfactory system. Receptors aretopographically mapped to the olfactory bulb, which consists of interactingexcitatory and inhibitory populations. The bulb in turn projects to the olfactorycortex. White circles correspond to the excitatory populations and black circlescorrespond to the inhibitory populations.

P (t)

Fi F21 nN

Fig. 2. Olfactory bulb can be modelled as a layer of interacting excitatory andinhibitory populations. Each unit is the full KII set, responsible for complexdynamical properties of the system. The lateral connections between the topand bottom populations are responsible for memory. Pi (t)-PN (t) are inputsfrom the olfactory receptors.

properties, and extend it by adding more complex componentsand interactions. This is the approach taken by Xu and Principe[8], who considered the reduced KII as shown in Fig. 3. Thisis a 4th order system described by two second order coupledODE's and it depends on two weight parameters:

j1 + aY1j + !3Y = 3(-WieQ(Y3) + P(t))l 3 + a&3 + !Y3 = 3(WeiQ(MY))

Here and everywhere else in this paper we assume that theweights are positive numbers.The main results concerned the existence of unique equi-

librium, and the existence of two parameter regions where theequilibrium is stable and unstable. The condition of stabilityhas been obtained using a generalization of the Poincare-Bendixon theorem. The stability condition, in the absence ofexternal input, is given by:

lWeiWieI < (4)

The condition of oscillations in the presence of externalinput is given as follows:

IWeiWieI > Q 1(*)Q'(y*) , (5)

where y* and y* are the equilibrium coordinates. It has beenhypothesized that the unstable region corresponds to limitcycle oscillations. The analytical results have been validatedwith numerical simulation of the solution.

P(v

we iVIWie

Fig. 3. Reduced KIl set is the simplest model of mixed excitatory-inhibitorypopulation. It has only two parameter weights.

III. ANALYTICAL STABILITY CONDITIONS FORINTERMEDIATE KII

Our goal is to generalize the previous results obtained forthe reduced KII. The approach is to build the more andmore complicated models, until the complexity of the full KIIis achieved. We begin by adding two lateral excitatory andinhibitory weights wii and wle. The resulting system is shownin Fig. 4. We call this an "intermediate" KII.

This is an 8th order system as it is described by 4 ODE'sgiven below:

l+01 + !Y1 = 1(-WieQ(Y3) - WeQ(4)| ~~~~+P(t))

Y2 + aY2 + 1Y2 = 0( W!eQ(Y3))Y3 +013 +1Y3 = /(WeiQ(yl) + WeiQ(y2))t 4 + ay4 +±y4=/3(W4iQ(y

(6)

First we find equilibria of this system, by setting the lst and2nd order derivatives to zero and solving the resulting systemof algebraic equations. Expressing Y2 and y4 in tenns of theother two variables, we obtain the following two simultaneousequations.

{ Y =-ie * Q(Y3) - WI * Q(wei * Q(Yl)) + P(t)lY3 = Wei * Q(Y1) + wei * Q(_Wfe * Q(y3)) (7)

Next we prove that the above system has a unique equi-librium. Note that the r.h.s. of Eq. (7) is defined for all realvalues of Y1, Y3, Wei, Wj, Wie, Wfe. In order to show that thereis at most one equilibrium, let us differentiate both sides ofEq. (7) w.r.t. Yi. The first and second conditions in Eq. (7)will give the following derivatives

dy3 _

dyl(8)1 +w eQ'WieQ(y3l))weiQ (Y1)

WieQ (Y3)

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dy3 WeiQ'(Yl) (9)dy, 1 + WeiQ'(-W!eQ(Y3))WLeQ (Y3)'

Using the fact that the derivative of the Q(y) functionis always positive, we can see that the r.h.s. of Eq. (8)is always negative, meaning that the first equation (7) is amonotonously decreasing function. Similarly, the r.h.s. of Eq.(9) is always positive, meaning that the second equation (7)is a monotonously decreasing function. Therefore the twofunctions can intersect at most once. Moreover, in the absenceof external input the equilibrium equals to zero. Q.E.D.As an illustration, consider Fig. 5; the curves correspond

to implicit plots of our two equations where Wei = 0.5,Wie = 17Wli =1e W = 1, P(t) = 0.5. The lower of thetwo decreasing curves corresponds to the first equation whereP(t) = 0.

In the following considerations we outline the derivation ofthe analytical condition for the stability of an intermediate KIIset. For simplicity we consider the case of P(t) = 0. Previ-ously we have shown that there is unique zero equilibrium inthis case. We need to find conditions when this equilibriumis stable. This can be done by considering the eigenvalues ofthe Jacobian of our system. We follow the approach used by[8] and find the condition when the maximum real part of theeight eigenvalues equals to zero. This gives us the boundarybetween the regions of stable and unstable equilibrium.The Jacobian Df ofthe intermediate KII is displayed in Fig.

7. Note that in the Jacobian matrix Q'i stands for the dQ/dyevaluated at Yi. The equilibrium value for the intermediateKII will be zero, therefore in the next considerations we useyi = 0. The eigenvalues of the Jacobian can be found bysolving the equation det(Df-A*I) = 0, where I is the 8 by 8identity matrix. This is an 8th degree polynomial, however, inthis special case the exact analytical solution can be obtained.The expression for the 8 eigenvalues is given in Fig. 8.We need to find when the largest real part ofthe eigenvalues

is equal to zero. In general there are four weight parameters.We will consider the case where w = Wei and W1e = Wie,when the eigenvalues are given by the equation in Fig. 9.

Solving max(Re(Aj-8)) = 0, we obtain the following finalresult.

Wei * Wie = -1/2at2(v'5 - 3) /,B (10)

It is remarkable that this result is the same as the case ofthe reduced KII (RKII) given in Eq. (4), except for a constantfactor of (3- -)/2.We implemented a procedure to evaluate the eigenvalues of

the Jacobian in the case where wli £w#j and w,e #4 wi,. Onesuch case is shown in Fig. 6. In this figure you can see the twocases for which we have theoretical solutions. The shape ofthe border for the case where wi = 0.5wei and wle = 05Wiedoes not seem to change, which means that the condition isstill a hyperbola. It appears that the increase in the strength ofwI and wl continuously shifts the border down making theregion of stable zero equilibrium smaller.

.eK. Ie

I

Fig. 4. Intermediate KII is obtained by setting Wee two weights of the fullKII to zero. Setting two lateral weights wt. and wUe to zero will transformit into RKII.

Fig. 5. Graphical solution confirms the existence of single equilibrium.The intersection of two curves corresponding to y3 and y, is the uniqueequilibrium. The two yl curves correspond to P(t)=0.5 and P(t)=O.

w.ie

Fig. 6. Stability boundaries of intermediate KII with various level of w1eand wii parameters. Dotted lines are analytical results, while solid lines marknumerical evaluation of eigenvalues.

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Df =

0-'000

13We,iQ',i0

OPeiQyl

1-a

000000

000-00ey200

001-a0000

013WieQy3

0-ow

ieQfY30

-100

00001-a00

0-3WieQY4

00000-13

0000001-a

Fig. 7 Jacobian matrix of the intermediate KII.

Al = -1/2a 1/2 a2-4,3 + 2,3 /2 ?WW2i + 4WieWeiWliWle + (2WieWei+ 4WeIite)

Fig. 8 Eigenvalues of the Jacobian Matrix at zero equilibrium.

Al-8 =-1/2a + 1/2 2-4 *13 ± 2,3-6 * WieWei ± 2V5WieWei.Fig. 9 Eigenvalues of the Jacobian Matrix at zero equilibrium where wei = Wei and wJe = Wze

IV. STRUCTURAL CHANGES IN THE EQULIBRIA OF FULLKII

A. Numeric ToolsNewton's method can be used to find equilibria of a dy-

namical system, given in a general form as:

x= f(x). (I1

Finding the equilibria is equivalent to solving the equationf(x) = 0. It is solved using the iterative formula

Xn+ = Xn - Df-1(xn) * f(xn), (12)

starting with the initial guess xO. Df is the Jacobian matrix off(x) evaluated at xn. Usually several guesses are tested, sincedepending on xo Newton's method may converge to differentequilibrium points or may not converge at all.When studying higher order systems, numerical solutions

and simulations are an essential part of the research. We haveimplemented a library of Matlab functions which (i) simulatethe KII set with and without external input; (ii) calculate theJacobian matrix of the system around a given point of the statespace; (iii) numerically solve the equilibrium equations usingNewton's method. We are interested so far in autonomousbehavior, so in simulations we always set P(t) to zero exceptfor the first time step.

B. ExperimentsThe next step in building up our system is to introduce

one of the remaining two weights, Wee. The system is nowrepresented by Fig. 10. From here on we assume that weiWei and We = Wie-

We scan the 3 dimensional parameter space changing eachweight in the interval [0..2] with step size 0.1. For each tripletof weights we find equilibriums using Newton's method andevaluate the eigenvalues of the Jacobian matrix. To visualize,we fix wee and vary the other two parameters.

It appears that the properties of our system change sig-nificantly. We have multiple equilibria now; some of themstable and some unstable. For example, consider Fig. 11 wherewee = 1.5 . There is still area in the upper right corner withzero stable and one unstable equilibrium, which, probably,corresponds to limit cycle oscillations. There is also an arearight below with one stable zero equilibrium. However, thereare areas with three equilibria and five equilibria.

If we increase wee to 2, the area with 5 equilibria spreadsout. This result can be seen in Fig. 12.

P(t

Fig. 10. Introducing the excitatory weight into the intennediate KII

V. CONTINUATION OF EQUILIBRIA AND BIFURCATIONSStudies of the equilibria show that the excitatory weight

wee introduces richer dynamical properties into our system.We use the continuation tool called "Content" [9] to obtainsome insight into the local properties of KIT. Content allowsto "continue" equilibrium points of a dynamical system, con-stantly monitoring the eigenvalues [10] of the system. Based

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.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

W.1e

find three equilibria, two stable and one unstable, with thezero equilibrium being stable. At BP we again are down totwo stable equilibria. As we keep increasing wee, there are

again two stable and one unstable equilibria, but it is the zero

equilibrium that is unstable now.

2.1 2.3

Fig. I 1. Number of equilibria, shown as total(stable), as a function of wejand Wie. The excitatory weight wee is kept constant at 1.5.

2.3

2.1

1.9

1.7

1.5 3(1)

1.3 1(0)

1. 5(2)

0.9

0.7

056

0.3 3(2)0.1

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3

W.ie

Fig. 12. Number of equilibria, shown as total(stable), as a function of wejand wie. The excitatory weight wee is kept constant at 2.

on their values, we detect different types of bifurcation points,which are points of structural change in the behavior of thesystem.We would like to understand how multiple equilibria are

generated. Continuation shows that the system has

a limit point (LP) bifurcation, where two equilibria -

stable and unstable - collide and disappear. The conditionfor LP is the appearance of a zero real eigenvalue [10].We also study branching point (BP) bifurcation. We de-tect zero real eigenvalue when there are two equilibriumcurves corresponding to stable and unstable equilibriapassing through this point. As the two curves meet, thestable curve becomes unstable and the unstable becomesstable.

One example of continuation is given in Fig. 13. Here,weights wei and wie are kept constant at 0.4 and 0.3, re-

spectively. The inhibitory weight wii is turned off. We startwith an arbitrary value of wee and continue the equilibriumcorresponding to this value in both directions. When a specialpoint is encountered, we take different actions based on thetpe of the point. Analysis of this diagram gives a betterinsight into the results of numerical studies. For example, wehave obtained numerically two stable equilibria at the pointmarked by LP on the diagram, which are zero and nonzero,respectively. If we move a little bit to larger wee values, we

543210

-1

-2-30 .5 1 1.5 2 2.5 3

Fig. 13. Continuation of Equilibria using Content 1.5. Wei=0.4, wieO.3,wii=O. As the excitatory weight wtee increases, the system undergoes struc-tural changes. For small Wee, there is only one stable zero equilibrium. Atwee=1.00041 a new stable equilibrium appears. As Wee increases more,another, unstable equilibrium appears. At wee=1.13339 a Branch Point isdetected. Here two equilibria meet and the unstable equilibrium becomesstable whereas the stable zero equilibrium becomes unstable.

Finally, we introduce parameter wii and scan the parameterspace looking for all possible equilibria and their stabilitycharacteristics. Since the parameter space is now larger, werestricted the weights to the interval [0.4..2]. The picturebecomes much more mixed as can be seen in Fig. 14. Weobserve up to 9 equilibria with up to 4 of them stable. Thereis still the familiar pattern that the upper right corner has fewerequilibria. However notice that the region with one unstableequilibrium has shrunk significantly. Moreover it is boundedfrom above as well.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

W.ie

Fig. 14. Number of equilibria of the full KII, shown as total(stable), as afunction of wej and Wie. The excitatory and inhibitory weights wee and wiiare kept constant at 2

VI. DISCUSSION OF THE RESULTS

In order to illustrate the obtained stability conditions, we

ran software simulations of the system using Runge-Kutta 4thorder method with step size fixed at 0.1Ims.The small region labelled "1(0)" in Fig. 14 contains no

stable equilibria. Since the system's state space is bounded by

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2.3

2.1

1.9

1.7

1.5

1.3

0.9

0.7

0.5

0.3

0.1

1(0)5(3)

1(1)

3(2)

I_LP HH

H..-Hw

.... ....0.1

Io1 .

IID.

the sigmoid function Q, we can reasonably suggest that thebehavior is a limit cycle. This has not been proven, howevernumerous simulations confirm this hypothesis.We suppose that the unstable region plays an important role

in successful implementations of K-systems. It is expected thatthe region where we are guaranteed to have non-convergentbehavior, as shown in Fig. 14, can be beneficial for under-standing the dynamics of living systems. This is a relativelysmall but non-zero measure region ofthe parameter space; lessthan 15% of the parameter combinations tested belong to thisregion.

Since the unstable region has oscillations, it is reasonableto try to characterize them further. Simulations in the regionconfirmed the existence of limit cycles. To estimate the periodof oscillations, the average distance between successive peaks(m(T)) has been computed, along with its standard deviationor(T). The inverse of the oscillation period gives the oscillationfrequency. We can distinguish between two types of oscilla-tions: one with low a(T) and another one with high a(T).Regions corresponding to the two types can be seen in Fig.15, for wee = 2 and wii = 2.High a(T) corresponds to a more irregular form of the

cycle, as shown in Fig. 16 A. In this case the system'sactivations tend to increase and are stopped when they reachthe saturation boundary imposed by the sigmoid. In the caseof low a(T), oscillations are regular with smaller amplitude,which means that the saturation boundary is probably notreached. This is shown in Fig. 16 B.

1.7

1.6

15

1.4

1.3

14 1.5 1.6 1.7 10 19

Wie

Fig. 15. Region with zero stable equilibrium (see Fig. 14). The numbers ineach square are the characteristic oscillation frequency in Hz. Dark squarescorrespond to high oa and light squares correspond to low a.

The other regions with both stable and unstable equilibriaare subject to further analysis. We do not know where wecan expect the appearance of limit cycles. For instance, theparameter set of Fig. 16 C contains five equilibria, two ofthem stable. We could not detect oscillations simulating thesystem with different initial conditions. However, as it wasshown in our previous work [111, there are parameter setswith both limit cycle and equilibria observed under differentinitial conditions.

'0*

I1 A

0 200 400 600 80 1000 1200 1400 1000 1000 2000

B0 200 400 0o0 eo0 1o 1200 1400 1000 1D00 2000

20 ,

IC-201

0 200 400 600 1000 1200 1400 6W 100 2000time, ms.

Fig. 16. Simulation of KIt. The four node activations are plotted. Theparameter sets are A) [2, 1.4, 1.6,2], B) [2, 1.6, 1.6, 2], C) [2, 1.6, 0.8,21

VII. CONCLUSIONSIn this contribution we applied methods of dynamic systems

analysis to gain insight into the behavior of KIl sets. Wehave extended the stability condition derived for reduced KIT([8]) to the case of intermediate KII. The number and stabilityof equilibria in different parameter regions demonstrates richdynamical properties ofthe full KII. The regions where we canexpect sustained limit cycle oscillations have been identified,and one of them has been further divided into two regionswith high- and low-amplitude oscillations. The latter regionproduces limit cycle with activations staying within the highnonlinearity region of the sigmoid, which may be a goodcandidate for higher order K sets parameter selection.

VIII. ACKNOWLEDGEMENTSThis work is supported in part by NASA Research Grant

No. NCC-2-1244, and by NSF Grant No. EIA-0130352.

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[6] W.J. Freeman, How Brains Make up Their Minds, Columbia UniversityPress, 2000.

[71 W.J. Freeman and S. Jakubith, Bifurcation analysis of continuous timedynamics oscillatory neural networks, Brain Theory, A. Aertsen (Ed.),Elsevier Science PubI. B.V., 1993.

[8] D. Xu and J.C. Principe, "Dynamical analysis of neural oscillators in anolfactory cortex model," IEEE Transactions on Neural Networks, vol.15,no.5, pp.1053-1062, 2004.

[91 Y.A. Kuznetsov, "Continuation of stationary solutions to evolutionproblems in content," CWI-Report AM-R961 1, Centrumn voor Wiskundeen Infonnatica, Amsterdam, 1996.

[10] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, second ed.,Applied Mathematical Sciences, vol.112, Springer-Verlag New York,Inc, 1998.

[1 1] R. Ilin, R. Kozma, and W.J. Freeman, "Studies on the conditions of limitcycle oscillations in the kii models of neural populations," pp.151 1-1516, IJCNN 04, IEEE Press.

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