Biol. Cybern. 66, 319-325 (1992) Biological Cybemetics �9 Springer-Verlag 1992

Bifurcation analysis of a neural network model

Roman M. Borisyuk t and Alexandr B. Kirillov 2

1 Research Computing Center of the USSR Academy of Sciences, Pushchino, Moscow Region, 142292, USSR 2 Department of Cell Biology and Neuroscience, UT Southwestern Medical Center, 5323 Harry Hines Boulevard, Dallas, TX 75235-9039, USA

Received October 10, 1991/Accepted in revised form September 9, 1991

Abstract. This paper describes the analysis of the well known neural network model by Wilson and Cowan. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inhibitory popula- tions of neurons. We analyze the dependence of the model's behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied. In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.

I Introduction

The neural network models are now widely used to simulate brain functions and to solve numerous prob- lems of information processing. The evolution of neural networks is often described by systems of differential equations. These systems can have different types of attractors: steady states, limit cycles and strange attrac- tors. Steady states are the easiest to work with; this may be the reason why they are so often used to model neural networks. For example, Hopfield's model of associative memory (Hopfield 1984) is a dynamical system with stable stationary states corresponding to stored patterns. Another example is the adaptive reso- nance network (Carpenter and Grossberg 1987) which consists of two layers of neurons and can solve classifi- cation problems. Like the Hopfield model, this one makes use of multiple stable steady states, typical in such competitive systems.

The other two types of attractors are also widely used in neurophysiological modeling, since periodic and aperiodic oscillations are of great importance for living neurophysiological systems (Freeman and Scarda 1985; Gray and Singer 1989). Baird (1989) suggested a neural network, which is based on harmonic oscillators of a certain frequency with regulated amplitudes and phases, to memorize sequences of events. In a model of the "semantic computer" Shimizu et al. (1988) constructed a neural network consisting of Van der Pol oscillators. Nicolis (1985) used strange attractors to memorize and process information. In a number of papers, variations of the model developed by Wilson and Cowan (1972) were used to study the synchronization of oscillations in the visual cortex (Schuster and Wagner 1990; Buhmann and vonder Malsburg 1991) and to model hippocampus (Sbitnev et al. 1982; Bragin and Sbitnev 1980).

This paper describes a detailed analysis of a model of a neural oscillator by Wilson and Cowan (1972). The model consists of a system of two nonlinear differential equations that describe the interactions between excita- tory and inhibitory populations of neurons. Wilson and Cowan found hysteresis phenomena and limit cycle activity in the model. Hysteresis was used to account for short-term memory, and oscillations were related to studies of periodically changing neural activity of the thalamic somatosensory neurons and to studies of EEG rhythms. We analyze here the model's behavior when two of the model parameters are varied (Borisyuk and Kirillov 1982). One of these parameters is the external input to the excitatory population, the other parameter is the connectivity coefficient that indicates the extent to which the excitatory population affects the inhibitory one. The parameter plane is paritioned into regions of equivalent behavior bounded by bifurcation curves where qualitative changes take place. The representative phase diagram is constructed for each region. This allows us to describe qualitatively the model's behavior in each region and to predict changes in the model dynamics as parame- ters are varied. In particular, we show that for some parameter values the system can exhibit long period

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oscillations when the limit cycle approaches the saddle separatrix loop. A new type of dynamical behavior is also found when the system activity is either stationary or oscillatory, depending on the initial point at fixed values of parameters.

The paper is organized as follows. In Sect. 2 we describe the model and interpretation of the model parameters. In Sect. 3 we present the partition of the plane of two parameters (the structural portrait of the system). And in Sect. 4 we discuss the types of behavior the system can exhibit and show how they switch under parametric variation. In Sect. 5 we present the discus- sion of results and their possible applications.

The computations of bifurcation curves have been carried out using the F O R T R A N program package developed under supervision of Prof. E.E. Shnol at the Pushchino Research Computing Center of the USSR Academy of Sciences (Khibnik and Shnol 1982; Bala- baev and Lunevskaya 1978; Zarhin and Kovalenko 1978; Khibnik 1979; Borisyuk 1981; Kuznetsov 1983). The IBM PC interactive versions of some of these programs are now available: the TRAX program to compute and plot the trajectories of dynamical systems and the LOCBIF program that analyzes bifurcations of equilibria in dynamical systems (Levitin 1989; Khibnik 1990a, b). Some additional computations and diagrams have been done using a SUN Sparc 2 workstation.

2 The model

We consider a localized neural population composed of an excitatory subpopulation and an inhibitory subpop- ulation. The evolution of the activity of the neural population is described by the system of two nonlinear differential equations (Wilson and Cowan 1972):

dE/d t = - E + (k~ - reE)Se(ClE - - CzI + P) (1)

dI /d t = - I + (ki - riI)S~(c3E - c4I + Q)

Here E(t) and I(t) are the averaged activities of the excitatory and the inhibitory subpopulations, re and ri are the refractory constants, P and Q are the external inputs to the excitatory and the inhibitory subpopulations, ke and k i a r e constants, and cl, c2, c3, c4 are the strengths of connections between subpopulations (Fig. 1).

Cl

C2

C4

Fig. l. A scheme of connections of the neural network composed of an excitatory subpopulation and an inhibitory subpopulation

Following Wilson and Cowan (1972), we assume that

S(x) = 1/(1 + exp( - b ( x - 0 ) ) ) - 1/(1 + exp(b0))

where b and 0 are parameters, so that the function S(x) has the following properties:

1) S(x)--* - 1[(1 + exp(b0)) as x ~ - ~ ; 2) S(x) --. exp(b0)/( 1 + exp(b0)) as x ~ + ~ ; 3) s (0 ) = 0.

The state E = 0, ! = 0 is chosen to be the state of low level background activity. Therefore, E = 0, I = 0 must be a steady state solution of(1) in the absence of external input (P(t) = 0 and Q(t) = 0). Small negative values of E(t) and I(t) represent depression of resting activity.

The choice of S(x) determines the parameters ke and k i : k e --- Se( -~ 030) and k i = S i ( -~ oo). We assume that all parameters are fixed, except for P and c3. The values of the fixed parameters are taken from the paper of Wilson and Cowan (1972): 0 e = 4 , b e = l . 3 , 0 i=3 .7 , b, .=2, r e = 1 , r i = 1, Q(t) = 0, c I = 16, c 2 = 12 and Ca = 3. The parameter P is assumed to be independent of time, that is P(t) = const, and c3 > 0.

3 Creating the structural portrait

Here we recall what a two-parameter structural portrait is, and we create such a portrait for system (1) for a pair of parameters P and c3. We wish to keep track of the evolution of the phase diagram of (1) when we vary the constant input P to the excitatory subpopulation and the connectivity coefficient e3, which represents the average number of excitatory synapses per cell in the inhibitory subpopulation.

By the structural portrait of system (1) we mean the plane (P, c3) partitioned into a number of regions by the four following types of curves:

I) fold point bifurcation curve, 2) Andronov-Hopf bifurcation curve, 3) saddle separatrix loop curve, 4) double limit cycle curve.

The phase diagrams are qualitatively identical for the values of P and c 3 within one region, and change dramatically when the parameter values pass through the boundary between regions: the system bifurcates. For instance, fixed points and limit cycles may appear or disappear or change stability type. For each region of the structural portrait we construct a representative phase diagram showing typical behavior of the system when parameter values belong to this region.

The structural portrait gives an idea of all types of behavior that the model could exhibit under variation of two parameters, and shows all possible transitions between them. It explains, for instance, how limit cycles appear and disappear, which is not quite clear from the presentation of Wilson and Cowan.

Before we turn to the actual structural portrait of the Wilson and Cowan model, we define each of the four bifurcation curves we mentioned above.

3.1 FoM point bifurcation curve

When parameters pass through this curve two steady states appear or disappear. So this curve separates a region on the plane (P, c3) where the phase diagram contains two steady states, from a region where these steady states disappear. Mathematical definition of the fold point bifurcation curve is given in Appendix A1.

3.2 Andronov-Hopf bifurcation curve

A fixed point undergoes Andronov-Hopf bifurcation when the Jacobian matrix of the system has the sum of eigenvalues equal to zero. If the steady state is a focus (complex eigenvalues), the Andronov-Hopf bifurcation curve separates regions on the (P, c3) plane where the phase diagram contains a stable or an unstable focus. When the parameters pass through that curve a limit cycle appears or disappears. If the steady state is a saddle (real eigenvalues), the curve is not strictly bifur- cational since qualitative behavior does not change (the curve separates areas where the saddle has a positive eigenvalue sum, from areas where the eigenvalue sum is negative). Nevertheless, it is desirable that the structural portrait contains this curve, since, for example, its intersection with the curve of the saddle separatrix loop is the end point of the double limit cycle. In other words, the separatrix loop of the neutral saddle may give birth to a double limit cycle. For the mathematical definition of the Andronov-Hopf bifurcation curve, see Appendix A2.

3.3 Saddle separatrix loop curve

This curve separates a region on the (P, c3) plane, where the phase diagram contains a limit cycle close to the saddle separatrix loop, from a region, where there is no limit cycle in the phase diagram. As the parameters (P, c3) approach the curve, the limit cycle gets closer to the separatrix loop, and the period of oscillations un- boundedly increases. Then the cycle becomes "glued" to the loop and disappears. For the mathematical defin- ition of the saddle separatrix loop bifurcation curve, see Appendix A3.

3.4 Double limit cycle curve

When parameters pass through this curve two limit cycles appear or disappear. So this curve separates a region on the plane (P, c3) where the phase diagram contains two limit cycles, from a region where these limit cycles disappear.

3.5 Structural portrait

Now we are ready to describe the structural portrait of the system (see Fig. 2). The fold point bifurcation curve is depicted as a thick line in Fig. 2. At parameter values on one side of that curve (regions 1 and 7), the system has one fixed point, whereas, at parameter values on the other side of the curve, the system has three fixed points. The Andronov-Hopf bifurcation curve is de- picted as a thin line in Fig. 2 (solid thin line corre-

321

1

10 �84

2/ I f . . . . . . .

-2

20 t e:~ 2,,

2

. . . . i . . . . i -1 0 2

P

11.5

10.5-

Detail

% i 4 I 4

I . /

6 . . j ~

f -

C

0.45 0,55 0.65 0.75 p

Fig. 2. The structural portrait of system (1). The fold point bifurca- tion curve depicted as a thick curve. The Andronov-Hopf bifurcation curve is shown as a thin cont inuous curve for complex eigenvalues (i.e., for neutral focus) and dashed thin curve for neutral saddle (note that dashed thin curve is not actually a bifurcation curve). The dot-dash line is the saddle separatrix loop curve. The double limit cycle bifurcation curve is depicted as a dotted curve

sponding to a neutral focus and dashed thin line corre- sponding to a neural saddle).

Points A and B are tangencies of the fold point bifurcation curve and Andronov-Hopf bifurcation curve. If the parameters of system (1) coincide with one of these points, the system has a fold point and an Andronov-Hopf bifurcation at the same time, i.e., both eigenvalues are equal to zero. The neighborhood of such a bifurcation point in the generic case was studied by Bogdanov (1976) (see also Arnold 1978; Bautin and Leontovitch 1976). The studies showed that every such point is the end of one more curve - the saddle separatrix loop curve. Thus, points A and B are the ends of the separatrix loop curve depicted by the dot- dash line in Fig. 2. When the parameters move along the fold point curve, the nonzero eigenvalue reverses its sign at points A and B. Numerical simulations show that as parameter values pass the fold point curve

322

between points A and B, an unstable fixed point and a saddle fixed point appear. In other cases a stable and a saddle fixed points appear.

At point C, lying on the Andronov-Hopf bifurca- tion curve, the first Lyapunov value L is equal to zero. L < 0 to the right of the point, and on the left of C the value of L is negative. If L < 0, the focus, becoming unstable on the Andronov-Hopf bifurcation curve, gives birth to a stable limit cycle. If L < 0, an unstable limit cycle disappears.

The point C is the end of the double limit cycle curve, shown as dotted line in Fig. 2. The other end of this curve is at the point D, where the saddle separatrix loop curve and thc Andronov-Hopf bifurcation curve meet. The separatrix loop is neutral at D, and the sum of the eigenvalues is positive to the right of D, and is negative to the left of D. This means that, as parameter values pass through the saddle seperatrix loop curve to the right of D, only a stable limit cycle can appear or disappear. To the left of D, this can only be an unstable limit cycle.

All the curves partition the plane of parameters (P, c3) into regions of equivalent behavior. There are seven different types of behavior. Figure 3 gives their representative phase diagrams.

4 Types of behavior and how they change under parametric variation

4. I. The phase diagram typical of region 1 has a single stable fixed point to which all the trajectories con- verge. At P = 0 this fixed point corresponds to the low level background activity of the network (see Fig. 3.1).

4.2. In region 7 the phase diagram also has a single fixed point, but it is unstable. The fixed point is sur- rounded by a stable limit cycle to which all the trajecto- ries converge as t---, or. This phase diagram is shown in Fig. 3.7.

The other phase diagrams, numbered 2 to 6, have three fixed points.

4.3. In region 2 the phase diagram has one saddle and two stable fixed points (see Fig. 3.2). The separa- trix of the saddle is the border between the domains of attraction of two stable fixed points. One of these fixed points has a low level (LL) of total activity E(t) + I(t), and the other has a high level (HL) of the population activity.

4.4. In region 5 the phase diagram also has two stable fixed points and a saddle. In addition, there is an unstable cycle around the HL stable fixed point (Fig. 3.5). Therefore, the domain of attraction of the HL fixed point is bounded by the unstable cycle and trajec- tories outside the limit cycle converge to the LL fixed point as t ~ .

In regions 3, 4 and 6 the phase diagram has a stable and an unstable fixed points and a saddle fixed point.

4.5. The phase diagram of region 4 has a simple structure: all the trajectories converge to the LL stable fixed point as t ~ (Fig. 3.4).

0.4

0.2 ~

O.C

0.0 0.2 0.4

3

E 0.6

2

4

5 6

7

Fig. 3. Phase diagrams corresponding to the regions on the structural portrait (Fig. 2). Axes labels are the same for all diagrams and are shown for diagram 1 only. The saddle separatrices and stable limit cycles are depicted as thick curves. Dashed thin curves correspond to unstable limit cycles

4.6. The unstable fixed point is surrounded by the stable limit cycle in the phase diagram of region 3, so that some trajectories converge to the LL stable fixed point, and others converge to the limit cycle. So, the system may settle down either to a fixed point, or to a limit cycle, depending on the initial point on phase plane (Fig. 3.3). This behavior is an interesting addition to those described in (Wilson and Cowan 1972). We think that it may be of interest to neurophysiologists, since with identical network parameters the model can exhibit both stationary activity and oscillations, de- pending on the initial values.

4. 7. The phase diagram of region 6 is similar to that of region 3, with the difference that the domain of attraction of the stable cycle is bounded by an un- stable limit cycle. The trajectories inside the unstable cycle converge to the stable limit cycle, as t ~ ~ , while

323

c3

3 16 1 .~\

IS M K A D~

/ 14 4

I / / , il

0.7

7

C A

112 ' 117 ' P

Fig. 4. Switches of the network dynamics upon decreasing the exter- nal input P (c 3 = 15). From A to M we cross regions 1, 7, 3 and 4 consecutively and return to region 1

other trajectories converge to the stable LL fixed point (Fig. 3.6).

Now we describe some transitions between the types of dynamical behavior which occur when parameters are varied. Assume that the interconnection strength c3 is 15, and the external input P gradually decreases from 3 to 0. In this case we pass through the regions 1, 7, 3, 4 consecutively and return to 1 (Fig. 4).

For Pc[B, A] the system settles down to the high activity fixed point. Its perturbations on the phase plane result in the current state first being repelled away from the fixed point and then evolving back towards it. The time required to reach the fixed point grows as the external input P approaches the point B (t ~ 1/la I where a < 0 is the real part of the eigenvalues, and [al--*0 as P"~,B). In this way for some external inputs the system would take a long time to approach the steady state after the system state is perturbed. At P = B the fixed point loses stability, so that a = 0, and gives birth to a stable limit cycle, since the first Lya- punov value L is negative. For Pc[C, B] the system oscillates. For P close to B (but P < B) the oscillations have small amplitude (soft Andronov-Hopf bifurca- tion), and as P is decreased further, the amplitude of oscillations increase.

At P = C a pair of fixed points appear far from the limit cycle. One of the fixed points corresponds to LL background activity, another one is a saddle fixed point.

Therefore, for Pc[D, C] the system may both oscil- late or be in the LL fixed point. For fixed parameters the system may change f rom one of these modes to the other if it is perturbed. The perturbation should be strong enough to force the system across the boundary between the domains of attraction of the fixed point and the limit cycle. This boundary is the saddle separa- trix (see Fig. 3.2).

At P = D a saddle separatrix loop appears. The stable limit cycle which exists for P > D merges with the loop and disappears for P < D. So, for Pc[K, D] the system is in the LL fixed point (the H L fixed point is unstable).

A

0.2-

0.1-

0 10 20 Time

0.3

0 10 20 Time

B

0.3

0 " 2 1

0.1

0 10 20 Time

0 10 20 Time

C D 30 0.2.

20 .g

--~.0.1 �9 t

. . . . . . . . . J . . . . . . . . . i , , , 1.05 1.45 1.85

P

Fig. 5A-D. Unboundedly increasing period of oscillations as P".~D (D is a point in Fig. 4). A and B: the plots of E(t) and l(t) for P = 1.4 and P = 1.075 near the point D. C and D: the period of oscillations and the cycle amplitude versus P

. . . . . . . . . t . . . . . . . . . i , , , 1.05 1.45 1.85

P

Note that for P",~D the oscillation period T in- creases unboundedly (T ,-~ l n { 1 / ( P - D)}), and the am- plitude does not change much. Figure 5 shows the plots E(t) and I(t) for various P near D, and plots of the oscillation period T and amplitude V versus P.

At P = K the unstable HL fixed point and the saddle merge and for P c [M, K] the system is again in the LL fixed point. The plot of total network activity (Eo + Io) at the fixed point, versus P is shown in Fig. 6. Other switches between types of behavior may be de- scribed in a similar manner.

0.6-

0.4

...o *o LII

0.2 I

M K DC B p

Fig. 6. Total activity of the network in the steady state (Eo + Io) versus P. Thick curve corresponds to stable steady states and thin curve corresponds to unstable ones

324

30

20

10

/ 2

-2 -1

/ i 4

,

[ , ' , I , i . . . . i . . . . i . . . . J . . . . r . . . . i ,

2 3 4 5 6 P

Fig. 7. Structural portrait of the system contains other branches of bifurcation curves for large values of P and c 3

To conclude, we should note that for large values of P and c3 the structural portrait may show other branches of the bifurcation curves (Fig. 7). I f we change the parameters that we have assumed to be fixed, these branches may get closer and start ot inter- act, giving rise to new interesting phenomena. A three- parameter structural portrait of the system will be considered elsewhere.

5 Discussion

The present qualitative study not only permits the enumeration of all possible dynamical regimes of a two parameter system, but also shows their switches upon parameter variation. This method may be used in the design of model neural networks, since varying the parameters may result in different types of behavior, i.e. parameter control may be used to select behavioral modes. An example is the changing of interconnection strengths during learning, which in turn may change external input.

Borisyuk and Urshumtseva (1990) studied neural oscillators in a circle with excitatory subpopulations being locally interconnected. In the absence of intercon- nections or for weakly connected oscillators, the system behaves as a pool of independent neural oscillators. Increased interconnection strengths lead to various complex types of behavior like stochastic oscillations, travelling waves, etc. In this case the results of a qualitative study of an isolated oscillator help predict the system's behavior. The predictions could be checked further by numerical experiments.

Borisyuk (1990) considered a model of selective attention. It is essential for the model that the neural oscillator can exhibit behavior with two stable attrac- tors simultaneously existing in the phase space. These may be two stable fixed points, or a stable steady state and a limit cycle. The model of selective attention, composed of a chain of locally connected oscillators, can differentiate between two steady identical stimuli presented at different points along the chain.

There is one additional application of the results of our qualitative study: the development of memory mod- els to store time-varying sequences of events. Our anal- ysis has not only given information about the types of dynamical behavior themselves but has also allowed their control, as well as the manipulation of jumps from one mode to another. These techniques promise to be useful in the study of neural networks processing spatiotemporal patterns.

Appendix A1

Rewrite the system (1) in the form

dE/dt = F(E, I, P) (2)

dI/dt = G(E, L c3)

Let (E 0, I0) be fixed point of the system (2), that is

F(Eo, I0, P) = 0 G(Eo, Io, c3) = 0 (3)

Assume that for P = P ' and c3 = c' at least one of the eigenvalues of the linearization matrix (the Jaco- bian matrix) calculated at E = E 0 and I = I 0 is zero. This is valid if and only if the matrix determinant is zero, so that

r ~?G

A(Eo, Io, P ' , c') = = 0 OG c~G

Consider a system of three equations

F(E, L P) = 0

G(E, L c 3) = 0

A(E, L P, c3) = 0

This system defines a curve in the four-dimensional space (E, L P, c3). Projection of this curve onto the plane (P, c3) we call the fold point bifurcation curve.

Appendix A2

Let (Eo, I0) be a fixed point of the system (2), that is (3) is satisfied. Assume that the sum of eigenvalues 2] and 22 of the linearization matrix is zero. This means that either 21.2 = _+ i~, or 2] and 22 are real and 2, = - 22. The sum 2] + 22 is zero if and only if the trace of the matrix of linear part of system (2), Sp(E, L P, c3) is zero. In an N-dimensional system of differential equa- tions, (N > 0), the necessary and sufficient condition for a sum of two eigenvalues to be zero is that (N - 1)th Gurvitz determinant is equal to zero (Gantmakher 1967).

Consider a system of three equations

F(E, L P) = 0

G(E, L c3) = 0

Sp( E, L P, C3) = 0

325

These equations define a curve in the four-dimen- sional space (E, / , P, C3). We call the projection of the curve onto the plane (P, c3) the Andronov-Hopf bifur- cation curve.

Appendix A3

Assume that the fixed point (E o, I0) is a saddle. Also assume that at P = P ' and c3 = c' one of its separatrices forms a loop. Note that this bifurcation is not local. Let J(P, c3) be the separatrix split function defined in a neighborhood of the point (Eo, I0, P' , c'). Then the separatrix loop exists if

F(E, L P) = 0

G(E, I, c3) = 0

J(E, I, P, c3) = 0

T h i s s y s t e m def ines a c u r v e in t h e s p a c e (E, L P , c3). T h e p r o j e c t i o n o f t h i s c u r v e o n t o t h e p l a n e (P , c3) we ca l l t h e c u r v e o f t h e s a d d l e s e p a r a t r i x l oop .

Appendix A4

The double limit cycle curve is defined analogously to the fold point bifurcation curve. Here instead of the fixed point of the system (2) we consider a fixed point of the Poincare map. The condition that the point (E , / , P, c3) belongs to the double limit cycle curve is written as a system of three equations. Two of them define that the point is the fixed point of Poincare map, and the third one defines that the point is a multiple fixed point of this map. We call the projection of this curve onto the plane (P, c3) the double limit cycle curve.

Acknowledgements. The authors thank A. I. Khibnik for many valu- able discussions, and Yu A. Kuznetsov for critically reading the manuscript. ABK acknowledges support from the Texas Advanced Research Program and Biological Humanics Foundation and Grants MH44337, DA02338 and AFOSR 90-146.

References

Arnold VI (1978) Supplementary chapters of the theory of ordinary differential equations. Nauka, Moscow (in Russian)

Baird B (1989) A bifurcation theory approach to the programming of periodic attractors in networks models of olfactory cortex. In: Touretzky S (ed). Advs Neural Inf Process Syst 1:459-467

Balabaev NK, Lunevskaya LV (1978) Motion along a curve in an n-dimensional space. FORTRAN algorithms and programs. Is- sue 1. Pushchino, Acad Sci USSR (in Russian)

Bautin NN, Leontovitch EA (1976) Methods and techniques of the qualitative analysis of dynamical systems on the plane. Nauka, Moscow (in Russian)

Bogdanov RI (1976) Versal deformation of a singular point of the vector field on the plane for a nonzero eigenvalue. In: Works of IG Petrovsky Seminar, Issue 2. Moscow State University Press, Moscow: 37-65 (in Russian)

Borisyuk RM (1981) Stationary solutions of a system of differential equations depending on parameters. FORTRAN algorithms and programs. Issue 6. Acad Sci USSR, Pushchino (in Russian)

Borisyuk RM (199(~) Interacting neural oscillators can imitate the selective attention. In: Holden AV, Kryukov VI (eds) Neuro- computers and attention. Manchester University Press, Manchester (in press)

Borisyuk RM, Kirillov AB (1982) A qualitative study of a model of neural network consisting of two homogeneous populations. Preprint. Acad Sci USSR, Pushchino (in Russian)

Borisyuk RM, Urzhumtseva LM (1990) Dynamical regimes in a system of interacting neural oscillators. In: Holden AV, Kryukov VI (eds) Neural Networks: theory and architecture. Manchester University Press, Manchester (in press)

Bragin AG, Sbitnev VI (1980) The septal input to the CA 3 hippocampal field. 1. Identifying the states. Biofizika 25:190 (in Russian)

Buhmann J, v o n d e r Malsburg C (1991) Sensory segmentation by neural oscillators. In: International Joint Conference on Neural Networks - 1991, 2:603-608

Carpenter GA, Grossberg S (1987) A massively parallel architecture for a self-organizing neural pattern recognition machine. Corn- put Vision Graph Image Process 37:54-115

Freeman WJ, Scarda CA (1985) Spatial EEG patterns, nonlinear dynamics and perception: the Neo-Sherringtonian view. Brain Res Rev 10:147-175

Gantmakher PhR (1967) The theory of matrices. Nauka, Moscow (in Russian)

Gray CM, Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc Natl Acad Sci USA 86:1698-1702

Hopfield J (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA, 81:3088-3092

Khibnik AI (1979) Periodic solutions of a system of differential equations. FORTRAN algorithms and programs. Issue 5. Acad Sci USSR, Pushchino (in Russian)

Khibnik AI (1990a) Using TRAX: A tutorial to Accompany TRAX, a program for simulation and analysis of dynamical systems. Exeter Software, Setauket, NY

Khibnik AI (1990b) LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three. In: Roose D (ed) Continuation and bifurcations: numerical tech- niques and applications. Kluwer, London, New York

Khibnik AI and Shnol EE (1982) Software for qualitative analysis of differential equations. Acad Sci USSR, Pushchino (in Russian)

Kuznetsov UA (1983) One-dimensional separatrix of the system of ordinary differential equations with parameters. FORTRAN al- gorithms and programs. Issue 8. Acad Sci USSR, Pushchino (in Russian)

Levitin VV (1989) TRAX: Simulation and analysis of dynamical systems. Version 1.1. Exeter Software, Setauket, NY

Nicolis JS (1985) Chaotic dynamics of information processing with relevance to cognitive brain functions. Kybernetes 14:167-172

Sbitnev VI, Gulayev AA and Bragin AG (1982) Dynamical models of the functional organization of hippocampus. In: Memory and leafing mechanisms. Nauka, Moscow (in Russian)

Schuster HG and Wagner P (1990) A model for neuronal oscilla- tions in the visual cortex. Biol Cybern 64:77-85

Shimizu H, Yamaguchi Y and Satoh K (1988) Holovision: a seman- tic information processor for visual perception. In: Kelso JAS, Mandel AJ, Shlesinger MF (eds) Dynamic patterns in complex systems. World Scientific, Signapore

Wilson HR and Cowan JD (1972) Excitatory and inhibitory interac- tions in localized populations of model neurons. Biophys J 12:2-24

Zarhin YuG and Kovalenko VN (1978) Finding solutions of a system of two algebraic equations. FORTRAN algorithms and programs. Issue 2. Acad Sci USSR, Pushchino (in Russian)

Dr. Alexandr B. Kirillov Dept. of Cell Biology and Neuroscience UT Southwestern Medical Center 5323 Harry Hines Boulevard Dallas, TX 75235-9039 USA