strange nonchaotic attractor in high-dimensional...

International Journal of Bifurcation and Chaos, Vol. 13, No. 1 (2003) 251–260c© World Scientific Publishing Company

STRANGE NONCHAOTIC ATTRACTORIN HIGH-DIMENSIONAL

NEURAL SYSTEM

J. W. SHUAI∗ and D. M. DURAND†

Department of Biomedical Engineering, Case Western Reserve University,Cleveland, OH 44106, USA∗[email protected]

†[email protected]

Received July 11, 2001; Revised November 5, 2001

A general method for the creation of strange nonchaotic attractor is proposed. As an exam-ple, the strange nonchaotic attractor in a high-dimensional globally coupled neural system isstudied numerically. For such an attractor, the time interval of continuously positive finite-time Lyapunov exponent must be smaller than the period of the driving stimulus, althoughit has a long-time positive tail. The intermittency between laminar and burst behavior is acharacteristic dynamic of the strange nonchaotic attractors. Simulation results show that thechaotic phase occurs only within a small region around the origin in the parameter space. Morethan half of the large nonchaotic region is the strange nonchaotic phase. The chaotic phaseis typically surrounded by strange nonchaotic attractors. This result also suggests that somebiological signals that have a strange structure may be nonchaotic rather than chaotic.

Keywords: Chaos; fractal; spatiotemporal system; neural networks; EEG.

1. Introduction

The dynamical properties of spatially extendednonlinear systems such as neural networks, mul-timode lasers and charge-density waves have at-tracted great interest in the past decade. Such sys-tems can be described by a set of coupled differentialequations or iterated maps. The neural networksmodels and the coupled map lattices characterizedby discrete space and time variables are two simple,computationally tractable dynamical systems thatdisplay behaviors qualitatively similar to those ofmore complicated models [Hopfield, 1982; Kaneko,1990]. Rich spatiotemporal complex behaviors in-cluding chaos, patterns, traveling waves, memoryand synchronization are revealed in such systems.

Often, highly complex behavior appears in chaoticspatiotemporal systems. However, with the studyof the unidirectional coupling map lattices, it hasbeen shown that complex phenomena can also takeplace in some nonchaotic spatiotemporal systems[Vergni et al., 1997].

A type of complex behavior occurring in non-chaotic systems is strange nonchaotic attractor(SNA) that is geometrically complicated but whichtrajectory does not exhibit sensitive dependenceon initial conditions asymptotically [Grebogi et al.,1984; Ding et al., 1989; Ditto et al., 1990]. Dif-ferent mechanisms for the creation of SNA havebeen proposed in low-dimensional quasiperiodicallyforced systems [Heagy & Hammel, 1994; Nishikava

∗Author for correspondence. Current address: Department of Physics and Astronomy, Ohio University, Athens, OH 45701-2979.

251

252 J. W. Shuai & D. M. Durand

& Kaneko, 1996; Yalcinkaya & Lai, 1996; Prasadet al., 1997; Brown & Chua, 1998; Shuai & Wong,1998, 1999]. An interesting question is whether themechanism discussed for the emergence of SNA inlow-dimensional systems can be generalized to high-dimensional systems. Most of the SNA dynamicsinvolve a collision between a stable and an unstableinvariant region [Heagy & Hammel, 1994; Nishikava& Kaneko, 1996; Yalcinkaya & Lai, 1996; Prasadet al., 1997]. In fact, the system often becomeschaotic when the control parameter has a smallchange after such a collision. So the resultant SNAtypically occurs in the narrow vicinity along thetransition boundary between chaotic and periodicphases in the parameter space. There is little dis-cussion whether such a narrow vicinity still remainsor can be detected when such a collision happens fora quasiperiodically forced high-dimensional system.

The existence of SNA in the quasiperiodicallyforced circle map lattices with unidirectional cou-pling has been investigated [Sosnovtseva et al.,1998], however the unidirectional coupling map lat-tices are too simple and special for high-dimensionalsystems. The goal of this paper is to present ageneral method for the creation of SNA in high-dimensional spatiotemporal systems. Shuai andWong [1998] pointed out that the low-frequencyquasiperiodically driven logistic map can occur asSNA if the dynamics become expanding for longtime intervals, repeatedly. Based on this, a gen-eral SNA method that is applicable to any chaoticsystem is presented in Sec. 2, which is applied tohigh-dimensional systems in the present paper. Itshows that the complex behavior in spatiotempo-ral systems can be due to the strange nonchaoticdynamics, rather than chaotic dynamics. An ex-ample of a globally coupled neural network drivenby a quasiperiodical signal is presented in Sec. 3.It is the first example of SNA in high-dimensionalneural systems. In Sec. 4, a simulation example isstudied in detail. Its phase diagram of the exampleis discussed in Sec. 5. We show that in the param-eter space the chaotic phase occurs only within asmall region around the origin. A notable result isthat more than half of the large nonchaotic regionis SNA phase. Another interesting observation isthat the chaotic attractors are typically surroundedby SNAs. Conclusions are drawn out in Sec. 6. Therelevancy of our results to real neural systems is alsodiscussed in Sec. 6. We suggest that some biolog-ical signals that have a strange structure may notbe chaotic.

2. Creation of SNA in ChaoticSystems

For an autonomous discrete map x(t + 1) =G(x(t), ξ) with the control parameter ξ, the depen-dent function of the maximum Lyapunov exponenton ξ, i.e. Λξ, can be obtained. Varying ξ, the systemcan be chaotic or periodic. Suppose when ξ is in re-gion C, the system is chaotic; when ξ is in regionP, the system is periodic. Now let the parameterξ be replaced by a quasiperiodic function given asfollows:

ξ(t) = A0 +A1 sin(2πωt) (1)

where ω is irrational and small, and A1 > 0. Notethe driving period T = 1/ω.

The driving force ξ(t) slowly changes with timein region R = (A0 − A1, A0 + A1) quasiperiodi-cally. During a short time interval around time t,the force ξ(t) can be approximated by the constantforce Ft = A0 + A1 sin(θt) with θt = 2πωt [Shuai& Durand, 2000]. Thus, whether the dynamics ofthe system x(t + 1) = G(x(t), ξ(t)) are expandingor contracting within this short time interval canbe estimated by the dynamics of the autonomoussystem x(t + 1) = G(x(t), Ft), which is driven bythe constant force Ft. Suppose part of region Ris in C and the other in P. According to the ap-proximation of constant force, the dynamics of thesystem are expanding and the trajectory runs intoa bursting state when the driving force is in regionC; while the dynamics are converging and the tra-jectory runs into a laminar state when the force isin P. By the name of laminar state, the trajectorystays in a small invariant region. The bursting statemeans that the trajectory chaotically oscillates inthe phase space with a random amplitude.

A low enough frequency ensures that the timeinterval of expanding dynamics, i.e. the time inter-val that the force ξ(t) stays in chaotic region C, canbe sufficiently long within a driving period. Theirrational frequency means that different values ofξ(t) are always provided at different time t. So,with such a time series of ξ(t), different expand-ing orbits can be obtained under repeatedly ap-pearing long-time expanding dynamics. A strangegeometric structure thus occurs. If the convergingdynamics are stronger than the expanding dynam-ics, a negative maximum Lyapunov exponent canbe obtained asymptotically. The resultant attractoris nonchaotic but strange. Accordingly, the power

Strange Nonchaotic Attractor in High-Dimensional Neural System 253

spectrum of the trajectory is a broadband spectrumcombined with sharp peaks.

The dynamics of the system can be discussedwith the finite-time Lyapunov exponent. In par-ticular, the maximum nontrivial Lyapunov expo-nent λτ (t) quantifies the expanding or contract-ing exponent that the trajectory experiences fromtime t to t + τ . One can discuss the distributionof λτ (t) in two ways. The first way is to con-sider λτ (t) versus the observing time window τ withfixed starting time t0. Such distribution is dis-cussed for some SNA systems [Prasad et al., 1997].The finite-time Lyapunov exponent in the SNA sys-tems is characterized with a long-time positive tail.Here we show that the distribution of λτ (t0) hasan additional interesting property in SNA systemdriven by a low-frequency force: typically λτ (t0)become negative periodically with period T in thelong-time positive tail. The dynamics of systemx(t + 1) = G(x(t), ξ(t)) in a driving period canbe approximated by a number of autonomous sys-tems x(t+ 1) = G(x(t), Ft). With different drivingperiods, the same autonomous systems can be ap-plied. Thus, the dynamics of the system with dif-ferent driving periods are similar. Nonchaotic sys-tem guarantees that contracting dynamics shouldbe stronger than expanding dynamics in each driv-ing period. The maximum time interval of contin-uously positive λτ (t0) must then be smaller thanthe driving period T , although the maximum timeinterval having positive λτ (t0) can be large enough.

The second way is to discuss λτ (t) versus t witha fixed small τ [Shuai & Wong, 1998]. In this casethe observing time window τ is fixed at a smallvalue τ0. The distribution of λτ0 (t) versus t forfixed τ0 can give detailed information about whichtime intervals the dynamics are expanding, i.e. astrange structure occurs. Suppose that the trajec-tory runs into a pure expanding region in the longtime interval (t0, t0 + TA). Then all λτ0 (t) witht1 < t < t1 + TA are positive. In fact, any smallτ � TA can be applied to discuss the dynamicsof the system. Although the exact value of λτ (t)varies with the choice of τ , once the trajectory runsinto the pure expanding interval (t0, t0 +TA), λτ (t)and λ2τ (t) typically have the same signs because wehave

λ2τ (t0) =1

2(λτ (t0) + λτ (t0 + τ)) (2)

Based on the approximation of constant force [Shuai& Durand, 2000], the sign of the finite-time Lya-punov exponent λτ (t) of the system x(t + 1) =

G(x(t), ξ(t)) within a small window τ can be es-timated by the maximum Lyapunov exponent ΛFof the autonomous system x(t+ 1) = G(x(t), Ft).

In short, if a spatiotemporal system is drivenby a low-frequency quasiperiodic force to experi-ence expanding dynamics repeatedly with long timeintervals, while its asymptotic dynamics are non-chaotic, an SNA occurs. This method can be ap-plied to any chaotic system because one can alwaysfind a parameter in such a manner that the systemis chaotic or periodic corresponding to different val-ues of the parameter. In fact, the dimensionalityof the mapping G is neither mentioned nor neededhere for the creation of SNA. In the present paperwe focus on the high-dimensional neural system.

3. SNA in Neural Model

As an application, a globally coupled neural net-work is discussed in the paper. The model consistsof N analog neurons {Si(t)}, i = 1, . . . , N , with−1 ≤ Si ≤ 1, where every neuron Si is connectedto all other neurons Sj by random couplings {Jij}.The following parallel dynamics are used for updat-ing of the neurons:

Si(t+ 1) = g(hi(t)) , i = 1, . . . , N . (3)

The sigmoidal function which is a typical transferfunction in neural models is used here

g(x) = tanh(αx) (4)

with α > 0. The internal field hi of the neuron Siis given by

hi(t) =N∑j=1

JijSj(t) + ξi(t) (5)

with ξi(t) as the external signal. Usually, the ex-ternal stimulus only affects part of the neural net-works. Therefore, assuming that only the first M(M ≤ N) neurons are stimulated by the inputsignal:

ξi(t) = ξ(t) , i = 1, . . . , M .

ξi(t) = 0 , i = M + 1, . . . , N .(6)

Without any stimulus, the neural dynamicsare typically chaotic because of the asymmetriccoupling [Molgedey et al., 1992]. Now consider aconstant driving signal with ξ(t) = FConst. Equa-tions (3)–(6) show that the trajectory of the system


driven by −F with the initial conditions {Si(0)} isthe same as that driven by F with {−Si(0)}. Sothe dynamical properties, including the Lyapunovexponents, are the same for the system driven by−F or by F .

If M = N , a large stimulus can always drivethe neural network to be periodic. This is because,with a large stimulus, the internal fields of all theneurons are mainly determined by the stimulus andso Si(t)→ 1 for all neurons with any different initialconditions. Thus a periodic attractor is obtained.Our simulation results also show that for a largeenough M , a large stimulus can drive the neuralnetwork to be periodic. The observation that thestimulus F0 can drive the network to be periodicjust indicates a fact that, if the states of the firstMneurons approach 1, the neural network intrinsicallybecomes periodic. A larger stimulus F0 only meansthat the states of the first M neurons are closer to 1.So, if a large F0 can drive the neural network to beperiodic, any larger stimulus F (> F0) can drive thenetwork to be periodic. There is a threshold stim-ulus FTr that the chaotic state only appears in theregion of C = (−FTr, FTr). The fact that a F largeonly means close to 1 for the states of the first Mneurons also suggests that the Lyapunov exponentsΛF are almost constant with large F for periodicattractors.

Now there is a chaotic region C with |F | < |FTr|and a periodic region P with |F | > |FTr|. One canthen use the approach discussed in Sec. 2 to gener-ate SNA. To do it, the input signal ξ in Eq. (1) isapplied. It is easy to see that the dynamical tra-jectory of the system for ξ(t) = A0 + A1 sin(θ(t))with initial conditions ({Si(0)}, θ0) is the same asthat for: ξ(t) = A0 − A1 sin(θ(t)) with ({Si(0)},θ0 + π); ξ(t) = −A0 −A1 sin(θ(t)) with initial con-ditions ({−Si(0)}, θ0); or ξ(t) = −A0 +A1 sin(θ(t))with ({−Si(0)}, θ0 + π). Thus, the attractors ofthe system are identical for the external signals of±A0 ± A1 sin(θ(t)). In other words, the dynami-cal properties, e.g. Lyapunov exponents and fractaldimension, are symmetric about the A0-axis andA1-axis in A0 − A1 plane. So in the following, weonly discuss the situation of the force in the firstquadrant in A0–A1 plane, i.e. A0, A1 > 0.

With a low frequency, the slowly changed stim-ulus ξ(t) can be approximated by the constant inputFt = A0 +A1 sin(θ(t)) within a small time interval.If A0 +A1 < FTr, i.e. the stimulus is always withinthe chaotic region C, the dynamics of the networksare typically expanding, resulting in a chaotic at-

tractor. If A0 − A1 > FTr, i.e. the stimulus is al-ways out of chaotic region C, the dynamics of thenetworks are typically contracting, resulting in atorus. If A0 + A1 > FTr > A0 − A1, the stimulusξ(t) goes in and out of the chaotic region C twiceduring each driving period. So its dynamics be-come expanding or contracting repeatedly. In thisregion, a strange but nonchaotic attractor may begenerated. In the following, a simulation exampleis discussed in detail to confirm this discussion.

4. Simulation Results of SNA

In the simulation, let N = 100 and M = 20. Aset of randomly selected coupling weights {Jij} uni-formly distributed from −1 to 1 is used in the sim-ulation. Without any extended signal, the neuraldynamics are chaotic with the maximum Lyapunovexponent Λ = 0.095. Figure 1(a) gives the depen-dence of the Lyapunov exponent ΛF as a functionof positive scalar stimulus F . The chaotic attractor

0 3 6 9 12 15-1.0

-0.5

0.0

0.5

1.0

FTr

S 100(t

)Λ

F

F

-0.10

-0.05

0.00

0.05

0.10

(b)

(a)

Fig. 1. (a) Lyapunov exponent ΛF and (b) the bifurcationdiagram of neuron 100 versus the constant signal F for theneural system driven by F . Here F is from 0 to 15.


undergoes a transition to periodic when F = FTr ≈5.56. The chaotic states only appear in a finite re-gion C = (−FTr, FTr) of the parameter space. Asexpected, ΛF is almost constant for F > 10. Fig-ure 1(b) gives its bifurcation diagram of neuron 100with F from 0 to 15.

In this paper, a quasiperiodical stimulus ξ(t)with ω = 0.001 ·

√5, i.e. the driving period T ≈ 448

is used to stimulate the network. Simulation results

show the following: (1) for A0 = 9 and A1 = 2, atorus attractor A is obtained with the nontrivialmaximum Lyapunov exponent Λ = −0.074; (2) forA0 = 2 and A1 = 9, a nonchaotic attractor B is ob-tained with Λ = −0.085; (3) for A0 = 2 and A1 = 8,a chaotic attractor C is obtained with Λ = 0.005;and (4) for A0 = 2 and A1 = 2, a chaotic attractorD is obtained with Λ = 0.078. The spatiotempo-ral patterns of these four attractors are shown in

10000 10250 10500 10750 11000

100

1

t(a)

10000 10250 10500 10750 11000

100

1

t(b)

10000 10250 10500 10750 11000

100

1

t(c)

10000 10250 10500 10750 11000

100

1

t(d)

Fig. 2. Oscillating states of the 100 neurons via time from 10 000 to 11 000 of the attractor (a) A, (b) B, (c) C and (d) D.


Fig. 2 which shows that both the nonchaotic attrac-tor B and the chaotic attractor C possess the in-termittency behavior between laminar and burstingstates. Their geometric structures are quite similar.These observations suggest that attractor B is anSNA.

SNAs also exhibit a singular continuouspower spectrum lying among the sharp peaks. InFigs. 3(A)–3(D), the power spectra versus fre-quency plots are given for the four trajectories ofthe 100th neuron which is not driven directly by thesine signal. The spectral shape of attractors B andC are quite similar as shown in Figs. 3(B) and 3(C).There is a broadband spectrum with the additionof some large peaks as expected from the combinedbehavior of periodic [Fig. 3(A)] and chaotic systems[Fig. 3(D)]. The broad band spectrum for attractorB implies that there is a finite-time expanding be-havior in a chaotic manner that yields a strangestructure, rather than a torus.

The finite-time nontrivial maximum Lyapunovexponents λτ0 (t) versus time t with the observing

0 200 400 600 800 10001E-111E-101E-91E-81E-71E-6

1E-121E-111E-101E-91E-81E-71E-61E-5

1E-121E-111E-101E-91E-81E-71E-61E-5

1E-151E-141E-131E-121E-111E-101E-91E-81E-7

Pow

er s

pect

ra

Frequency

D

C

B

A

Fig. 3. Power spectra versus frequency in the region of(0, 1000) for the trajectories of neuron 100 of the four at-tractors. Here 16384 points are calculated for each curve.The frequency is in arbitrary unit.

10000 11000 12000 13000 14000 15000

0.0

0.1

0.2

-0.15

0.00

0.15

-0.2

0.0

0.2

E

t

A0+

A1s

in(θ

(t))

λ τ 0(t)

λ τ 0(t)

λ τ 0(t)

λ τ 0(t)

D

C

B

A

-0.10

-0.08

-0.06

Fig. 4. Finite-time Lyapunov exponents λτ0(t) versus time

t from 10 000 to 15 000 with observing time window τ0 = 10for attractors A, B, C and D. Here (E) shows the appliedquasiperiodically driving force as a function of time.

time window τ0 = 10 are plotted in Fig. 4 for thefour attractors. For attractor A, λτ0 (t) is typi-cally negative, indicating an torus obtained. Forattractor D, λτ0 (t) is typically positive. Thereforea chaotic attractor occurs. For attractors B andC, driven by the quasiperiodical force, λτ0 (t) os-cillates between negative and positive values. Inparticular, their signs change from the negative tothe positive twice in a driving period. The intermit-tency between laminar and burst behavior is thena characteristic dynamic of such attractors.

As shown in Fig. 1, the dynamics of the net-works are expanding only when F falls into thechaotic region C = (−5.56, 5.56). For attractorA, stimulus 9 + 2 sin(2πωt) is always out of re-gion C, so λτ0 (t) is typically negative and its tra-jectory is a laminar state. For attractor D, stimu-lus 2 + 2 sin(2πωt) is always within region C andso λτ0 (t) is typically positive and the trajectory is


0 500 1000 1500 2000-0.05

0.00

0.05

0.10

τ=TB=430

τ=980

τ=1620

3T 4T

4T3T

τ=TA=90

τ=267, t=t1

2TT

2TT

τ=182, t=t2

τ=TA=90 (a)

(b)

λ τ(t2)

λ τ(t1)

τ

-0.05

0.00

0.05

0.10

Fig. 5. Finite-time Lyapunov exponents λτ (t) versus observing time window τ from 0 to 2000 for attractors B with thestarting time (a) t1 = 10 033 and (b) t2 = 10 215.

always bursting. For attractor B or C, stimulus2 + 9 sin(2πωt) or 2 + 8 sin(2πωt) goes in and outof the chaotic region C twice during each drivingperiod, so λτ0 (t) changes its sign twice in a driv-ing period. Simulation results also confirm that thechanges of the sign occur when F (t) ≈ ±5.56. Hereτ0 = 10. The subsequent time interval of expand-ing dynamics is about TA = 10τ0 = 100 for attrac-tor B, as shown in Fig. 4(b). Within the repeatedlyexpanding time intervals, a strange geometric struc-ture is yielded for nonchaotic attractor B.

Figure 5 gives the plot of λτ (t) versus observ-ing time window τ for attractor B with fixed start-ing time t1 = 10 033 and t2 = 10 215 which arein the same period of the driving force. At timet1, t2, the driving forces are about 5.56 and −5.56,respectively, and the trajectory is being driven tothe expanding region. In this case, the maximumtime interval for a positive finite-time Lyapunov ex-ponent can be obtained. Figure 5(a) shows thatthe exponents have large positive values as longas τ = TA = 90, within which the dynamics areexpanding. Then for 90 < τ < 182, the trajec-tory is driven to the contracting region althoughλτ (t) only decreases to a small positive level. Whenτ = 182 (i.e. t = t2), the dynamics become ex-panding again and then λτ (t1) reaches another pos-itive peak. Exponent λτ (t) remains positive untilτ = TB = 430. However, λτ (t1) is positive for τ as

long as 1620. Simulation results show that λτ (t1)asymptotically approach to −0.085 in an oscillatingmanner with longer τ . In Fig. 5(b), λτ (t) can bepositive when τ = 980. As discussed above, λτ (t)is always negative when τ = nT , which means thatthe contracting dynamics are stronger than the ex-panding dynamics in each driving period and there-fore its asymptotic Lyapunov exponent is negative.Simulations show that similar results can be ob-served with any starting time t = t1,2 + nT .

5. Phase Diagram

For a dynamic system, one can distinguish itschaotic phase from nonchaotic phase easily bychecking if the value of its maximum nontrivialLyapunov exponent Λ is positive. A nonchaotic at-tractor is by definition an SNA if it has a strangegeometric structure. In particular, we observe thegeometric structure of the nonchaotic attractor inthe S–θ phase space. The phase space of a neuronis (−1, 1). For a typical SNA, its trajectory repeat-edly bursts in a chaotic manner to a large phasespace (whose scale is more than 1) with a long timeperiod. In other words, a long positive finite timeLyapunov exponent can be observed for an SNA.Simulation results of phase diagram are given inFig. 6 in A0–A1 plane with 0 < A0, A1 < 25 for thepresent example.


0 4 8 12 16 200

4

8

12

16

20

trans

ition r

egion

Type

-II tor

us

D

C

B

A

A1

A0

Type-I torus

SNA

Chaos

Fig. 6. Phase diagram of the present neural networks in the first quadrant in A0–A1 plane. The four points correspondingto attractors A, B, C and D are also shown in the parameter space.

As shown in Fig. 6, the chaotic phase only oc-curs in a finite region near the origin in A0–A1

plane. This can be comprehended based on theapproximation of constant driving force which sug-gests that the Lyapunov exponent Λ of the sys-tem can be estimated from the average of a set ofLyapunov exponents ΛF of the autonomous systems[Shuai & Durand, 2000]. If the values of A0 or A1

of the driving force are quite larger than FTr, mostof the applied constant forces Ft must be largerthan FTr and so most of the finite-time Lyapunovexponents are negative. As a result, a negativeLyapunov exponent Λ can be expected, indicatinga nonchaotic attractor.

In Fig. 6, two types of torus (type-I and IItorus) are distinguished. The type-I torus is in theregion with A1 < A0 − FTr = A0 − 5.56. Becausethe instant driving force is always in the periodicregion, the dynamics of a type-I torus are typi-cally contracting and its finite-time Lyapunov ex-ponent is always negative. The dynamics of a type-II torus sometimes can become expanding, i.e. itsfinite-time Lyapunov exponent can be positive. Theexpanding time interval can sometimes be as longas 50. But the corresponding time intervals are sosmall that the resultant trajectory does not show a

large bursting state. Simulation results show thatin the nonchaotic region between A1 < A0 − 4 andA1 > A0−5.56 type-II torus is observed. By furtherincreasing the parameter A1, the scales of the burst-ing states are enlarged gradually and the attrac-tor translates from torus to SNA after a transitionregion.

SNAs can be found typically in the nonchaoticregion of A1 > A0 − 3.2. According to the ap-proximation of constant force, in order to obtain apiece of expanding dynamics, the region of drivingforce ξ(t), i.e. (A0−A1, A0 +A1), must overlap thechaotic region C = (−FTr, FTr). So, A0−A1 < FTr.Furthermore, to generate a strange structure, ex-panding dynamics must occur for long time in-tervals. Long-time expanding dynamics guaranteethe trajectory can burst in a chaotic manner to alarge phase space. Simulation results show that ifeach time interval of expanding dynamics is morethan 80, a largely bursting state can occur. Thus,A1 > A0 − 3.2 can be expected. Because the dy-namic properties (Lyapunov exponents and fractaldimension) are symmetric about the A0-axis andA1-axis in the A0–A1 plane, the phase diagram ofthe present neural network in A0 − A1 plane canbe easily obtained. The chaotic phase occurs only


within a small region around the origin in A0–A1

plane. The nonchaotic phase exists in a large region,more than half of which is the SNA phase. This isthe first example to observe an SNA region that islarger than chaotic region or torus region. Anotherinteresting observation is that, except near the A0-axis, the chaotic phase is typically surrounded bySNAs.

6. Discussion and Conclusion

A general method is presented in this paper forthe creation of SNA in spatiotemporal systems thatcan be chaotic or periodic depending on a constantdriving force. When a low-frequency quasiperiodicforce is applied, the dynamics of the system canbe driven to oscillate between expanding and con-verging states. Long-time expanding dynamics leadto a strange attractor. If the contracting dynam-ics dominate the expanding dynamics, the resul-tant attractor is strange but nonchaotic. With thismethod SNAs can occur in a large area in parameterspace. For such an SNA, the time interval of contin-uously positive finite-time Lyapunov exponent mustbe smaller than the period of the driving force, al-though it has a long-time positive tail.

A simple neural network driven by a low-frequency quasiperiodic signal is discussed as an ex-ample. Driven by the low-frequency quasiperiodicstimulus, the chaotic phase of the neural networkappears only in a small region around the originin A0–A1 plane and the SNA can occur in a largearea of the nonchaotic region. Because this simpleneural model contains some important features ofthe real neural system, it is capable of reproducingsome basic behavior of real neural networks. It hasbeen difficult to prove whether a biological signal ischaotic [Freeman, 1987; Lutzenberger et al., 1995].However, our results indicate that some complex bi-ological signals that have a strange structure maynot be chaotic. SNAs may be observed in some realneural systems. This new classification could haveimportant consequences in the analysis of the bi-ological signals, particularly for complex biologicalsignals such as those seen in epilepsy.

Strange nonchaotic dynamics could be applica-ble to the analysis of biological neural signals gen-erated by biological networks such as the electroen-cephalogram (EEG). It has been shown that thehuman EEG has a fractal dimension [Lutzenbergeret al., 1995]. The EEG has been studied in the view

of low-dimensional chaotic dynamics [Babloyantzet al., 1985; Freeman, 1987]. However, a numberof recent studies analyzing the Lyapunov exponent[Theiler, 1995] and surrogate data [Rombouts et al.,1995; Palus, 1996; Stam et al., 1997] suggest thatthe EEG is unlikely to be chaotic. It is also shownthat a positive Lyapunov exponent could be calcu-lated with time series of SNA [Shuai et al., 2001].Different mechanisms such as a noisy limit cycle,a noisy two-frequency oscillation and quasiperiodicoscillations, as well as the linearly filtered noise,have been proposed to explain the EEG dynamics[Theiler, 1995; Rombouts et al., 1995; Palus, 1996;Stam et al., 1997]. It has been shown that differ-ent neuronal population in the cortex can oscillateat different frequencies [Gray et al., 1989]. If thequasiperiodic dynamics can occur and the EEG isnot chaotic, it is then a reasonable hypothesis thatEEG dynamics may be due to strange nonchaoticbehavior.

Acknowledgments

We would like to thank W. C. Stacey for help-ful discussion. This work is supported by NSFgrant no. IBN 93-19599 and a Whitaker Devel-opment Award to the Department of BiomedicalEngineering.

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