Synchronization: Adaptive Mechanism Linking

Internal and External Dynamics

Alex Pitti Max Lungarella Yasuo KuniyoshiIntelligent Systems and Informatics Lab

Dept. of Mechano-InformaticsThe University of Tokyo, Japan

Email:{alex,maxl,kuniyosh}@isi.imi.i.u-tokyo.ac.jp

Abstract

Synchronization is the dynamic adjustmentof rhythms of oscillating systems. The ques-tion arises of whether the complex patternsemerging from the interaction of an embod-ied system’s internal and external dynamicscan be explained and quantified in terms ofsynchronization. Taking an information the-oretical stance, we make the assumption thatsynchronization between coupled dynamicalsystems provides a powerful adaptive mech-anism to receive, encode, and store externalevents into a system’s internal dynamics. Weillustrate our ideas with two simulation ex-periments: a system composed of two bidi-rectionally coupled chaotic map lattices, andan embodied system composed of a body andtwo coupled map lattices. We show how inboth model systems a framework based onsynchronization can smoothly integrate sens-ing and acting, and learn simple sensorimotorpatterns without the need to postulate anya priori learning scheme. In the light of ourresults, we discuss the potential link betweensynchronization and the emergence and devel-opment of communication and imitation.

1. Introduction

Synchronization is the adjustment of two or morecoupled systems to each other to give rise to somecommon dynamical behavior. Such behavior canresult from either coupling the systems or byforcing them through a common external signal.Synchronization is such a pervasive phenomenonthat it is studied in a wide range of researchfields (Pikovsky et al., 2001, Strogatz, 2003). Inbrain science, appropriate synchronization is hy-pothesized to play an important role as a supportingmechanism for various modes of reciprocal (shortand long-range) interaction between different brainregions, for perceptual and temporal binding, and asa necessary condition for the emergence of percep-

tual and conscious states (Roelfsema et al., 1997,Varela et al., 2001, Tsuda et al., 2004). In develop-mental psychology and psychophysics it is suggestedthat in infants the contingency (or, synchrony) be-tween different sensory and motor channels may bea crucial aspect of the development of social cogni-tion (Nadel et al., 2005, Prince and Hollich, 2005).Also, synchronization may be exploited to im-prove communication between humans androbots (Breazeal, 2002), or to enable a more efficientexploration of the sensorimotor space of embodiedsystems (Lungarella and Berthouze, 2002).

In this paper, we suggest to go one step furtherand use synchronization as a basic nonlinear mech-anism to drive the learning of sensorimotor patternsand skills, and the development of communicationand imitation. Synchronization may give a unitary,coherent, and hopefully formal view on how humanslearn from and gradually mature with their own ex-periences. We start by giving some background onthe notion of phase synchronization, a particularkind of synchronization which assesses the relationbetween the temporal structures of measured signals(typically, phases) regardless of their amplitudes. Wethen describe the methods used in this paper to de-tect synchronized states. By studying a system ofcoupled chaotic maps we show how while at synchro-nization such maps can exchange and store informa-tion, they are not able to do so when they are notsynchronized. We proceed by studying an embodiedsystem realized as a simulated humanoid robot, andshow how externally imposed dynamics can affect itsinternal dynamics. Finally, we address the questionof how to link synchronization and cognition, dis-cuss the possibility to formalize semantic higher-levelconcepts relevant to psychology (e.g. memory, com-munication, and imitation) with the tools of physics,and point to some avenues worth exploring.

2. Phase Synchronization

In this section we first give an overview of synchro-nization and then introduce the methods used.

2.1 Background

Quite a few relevant applications of coupled non-linear dynamical systems exists: excitable systemsused for pattern detection (Baier and Muller, 2004),chaotic communication channels employed to trans-fer and encrypt information (Hayes et al., 1994),master-slave (driver-response) systems which can actas memories or ”knowledge” predictors (Voss, 2000,Calvo et al., 2004). In all these applications, thecoupling of two nonlinear systems (continuous ordiscrete) creates a communication channel alongwhich information can flow (Hayes et al., 1994).It can be shown that the information flow-ing along the channel is proportional to thelevel of synchronization between the linked sys-tems (Baptista and Kurths, 2005). For particularvalues of the coupling between the systems, the com-munication channel behaves poorly, that is, someparts of the information are transferred while otherparts are lost or absorbed by the systems’ subcom-ponents. What may be a drawback for communi-cation can be of actual relevance from the point ofview of embodied systems. As we hope to show inthis paper, synchronization between coupled dynam-ical systems provides in fact a powerful mechanismto receive, encode, and embed external events intoa system’s internal dynamics. The balance betweenplasticity and stability of the system, and the abilityto embed or to ignore external events is directly re-lated to the amount of coupling between brain, body,and environment.

From a physical point of view, the coupling be-tween two or more nonlinear (but active) systems(e.g. self-sustained oscillators) can be realizedthrough modulation of the systems’ phases and syn-chronization (Pecora and Carroll, 1990). Phase syn-chrony (PS) represents the counter-intuitive situa-tion during which the phase of the coupled systemsare locked, but their amplitudes are uncorrelated. Inother words, the irregularities of the amplitudes canactually hide phase synchronization. While phasesynchronized, the system’s dynamics is integrated aswell as differentiated. The ”integrated” parts of thedynamics support the exchange of information byestablishing communication channels between them,while the ”differentiated” parts do not exchange anyinformation. In this particular situation (denotedby [γPS−, γPS+] in Fig. 1), both systems are flexiblycoupled to each other and sustain a common rhythm.

Note that synchronization is not equivalent to res-onance. Two coupled systems at resonance are notnecessarily synchronized. Resonance is a responseof a system that is non-active, i.e. demonstrates nooscillations without external forcing. Synchroniza-tion and resonance can be discerned by switchingoff or reducing the driving force and by observing ifthe rhythms disappear or not. In the case of reso-

Figure 1: Phase synchronization and modulation.

nance, some of the oscillations will decay after a tran-sient. Thus, synchronization relies on the presenceof weakly coupled active systems (e.g. self-sustainedoscillators) which can function mostly (but not com-pletely) independent from each other (cf. Fig.1).

2.2 Methods

Here we introduce the tools used to study synchro-nization, in particular, phase synchrony: a) phasesynchronization index (Ψ); b) spectral bifurcationdiagram (SBD); and c) wavelet bifurcation diagram(WBD).

a) Ψ: The method used to measure phasesynchronization between two coupled sys-tems is based on the concept of instantaneousphase (Rosenblum et al., 1996). The instantaneousphase can be calculated as φ = arctanxH(t)

x(t) wherexH(t) is the Hilbert transform of the narrow-bandsignal x(t). The condition of 1:1 phase synchroniza-tion between two signals x1(t) and x2(t) is that thedifference between their phases ∆φ(t) = φ1(t)−φ2(t)stays bounded, whereas for unsynchronized statesit corresponds to an unbounded growth of ∆φ(t).1

In other words, two systems are synchronized if|∆φ(t)| < C where C is a constant. Given thisdefinition of synchronization, we can define thephase synchronization index as:

Ψ =√

< cos∆φ(t) >2t + < sin∆φ(t) >2

t (1)

where < . >t denotes the temporal average. If thesignals x1(t) and x2(t) are phase synchronized Ψ = 1(which indicates a constant phase difference). For auniformly distributed phase difference (that is, nosynchronization) Ψ = 0.

b) SBD: In order to investigate qualita-tive changes of dynamics in high-dimensionalsystems, we use the spectral bifurcation dia-gram (Orrel and Smith, 2003). Essentially, SBDdisplays the power density spectrum of multiple

1A more general definition includes rational relationshipbetween the phases: |nφ1−mφ2| where n and m are arbitraryintegers.

system variables as a function of a system control pa-rameter (e.g. force, temperature, coupling strength)(the power spectra of the individual variables aresuperposed). This method allows identification ofresonant states characterized by sharp frequencycomponents, chaotic states having rather broadpower spectra, as well as bifurcations, that is,qualitative changes in the system’s behavior (phasetransitions from one state attractor to another).

c) WBD: Because we are interested in understand-ing the spatial correlations between coupled dynam-ical systems (e.g. neural units), as well as correla-tions at different spatio-temporal scales, we also usedthe wavelet bifurcation diagram (Pitti et al., 2006).This measure based on the wavelet transform spansfrequency, time, and the index of the time series an-alyzed, and allows visualization of spatio-temporalpatterns, bifurcations and chaotic itinerancy at dif-ferent scales for a high dimensional system. It canbe conceptualized as a multi-resolution tool for theanalysis of multi-variate time series.

3. Experimental Studies

To illustrate our ideas we investigate two model sys-tems: a system composed of two bidirectionally cou-pled chaotic map lattices (CML), and an embodiedsystem consisting of a body which is bidirectionallycoupled to two CMLs. A CML is a dynamical systemwith discrete time, discrete space, but a continuousstate. It consists of L coupled dynamical units whichtypically exhibit nonlinear dynamics. In this paperfor the dynamical units we choose the logistic map:

f(z) = 1− αz2. (2)

The behavior (chaoticity) of the chaotic map f(z) iscontrolled by the parameter α ∈ [0, 2] (identical forall units). The amplitude of the map varies between[−1, 1]. All units are connected to their two nearestneighbors and to an external input (e.g. sensor feed-back) whose amplitude is normalized between [−1, 1]by a coupling γ ∈ [0, 1] (external coupling).

3.1 Coupled chaotic maps

As a first example, we study phase synchronizationin two bidirectionally coupled CMLs xn(i) and yn(i)consisting of L = 32 chaotic units each. The coupledsystem’s behavior is expressed as:

xn(i + 1) = (1− γ) f(xn(i)) + γ2 (xint

n (i) + xextn (i))

yn(i + 1) = (1− γ) f(yn(i)) + γ2 (yint

n (i) + yextn (i))

with

xintn (i) = (xn−1(i) + xn+1(i))/2

yintn (i) = (yn−1(i) + yn+1(i))/2

xextn (i) = yn(i)

yextn (i) = xn(i)

(3)

Figure 2: Phase synchronization index CML 1 - CML 2.

Black continuous line: average of Ψ over 32 neural units;

gray dashed line: standard deviation.

where n is a discrete time step and i ∈ [1, 32] is theindex of the ith chaotic unit; xint, yint, and xext, yext

are the internal and external couplings respectively.Depending on the coupling parameter γ, the two

coupled CMLs may or may not produce sustainedoscillations. In this experiment we vary the cou-pling parameter γ between [0, 0.6] and observe thecoupled system’s behavior after perturbing the firstCML with an arbitrary signal sn(i) ∈ [−1, 1] (thesecond CML yn(i) is not externally perturbed). Ac-cordingly, the equation governing the dynamics ofthe first CML is modified as follows:

xextn (i) = (yn(i) + sn(i))/2. (4)

The experiments are conducted with N = 100000samples. By analyzing the outcome of the experi-ments in the spectral domain and in the phase space,we can better understand the structure of the inter-actions and their time evolution. The phase synchro-nization index Ψ between the two coupled CMLs isplotted in Fig. 2. The black continuous line denotesthe average of Ψ calculated over all 32 units, thegray dashed lines indicate the standard deviation.At phase synchronization (PS) the standard devia-tion is rather large indicating flexibility of the sys-tem to assume different states. The evolution overtime of the probability distribution of a representa-tive neural unit for different values of γ is displayedin Fig. 3. As evident from both figures, for this par-ticular system, PS occurs in two intervals for γ in[0.15, 0.18] and [0.27, 0.37] characterized by a highlevel of synchronization and a high variability. Inthe subsequent analysis, we focus our study on theinterval [0.15, 0.18].

Below a certain value of the external coupling con-stant γ < γPS− = 0.15, Ψ ≈ 0.5, and we do notobserve any synchronization between the two CMLs(Fig. 2). Both systems are uncoupled, and their dy-namics is unconstrained and independent from eachother (Fig. 3 a). The information fed to the systemthrough the external perturbation sn is absorbed by

the first CML’s dynamics and there is no exchangeof information between the two maps.

At PS, for γ = γPS− the two systems begin tointeract. Both systems are weakly coupled and syn-chronized and Ψ ≈ 0.5. The two systems form aweak communication channel and their phases as-sume values so that information can transit from onesystem to the other. External information can nowflow through the first CML and, at least partially,influence the state of the second CML (see Figs. 3 band c). Because the two systems reciprocally affecteach other, some information can be stored in thecoupled system’s dynamics.

For γ ∈ [γPS− , γPS+ ], the amount of informationexchanged is proportional to the coupling γ. As indi-cated by the probability distributions, the differencebetween parts of the dynamics of the two systemsincreases (differentiation) while at the same time theexternal information tends to intergrate other parts(integration). At PS the systems are able to trade-offplasticity and stability, that is, they are able to reactadaptively to external perturbations while maintain-ing their own intrinsic dynamics.

For γ > γPS+ = 0.23, the dynamics of both sys-tems are fully coupled and they behave as one uniquesystem. The two systems are completely synchro-nized and information is transmitted with a higheraccuracy and with less loss through the establishedchannel and less flexibility (Fig. 3 e).

These results suggest that phase synchronized sys-tems are characterized by stability and plasticity.They also explain how information is exchanged andstored independently of the internal structure of thecoupled systems and of the nature of the information.

3.2 Embodied system

Our second experimental system is a simulated hu-manoid robot realized with the Novodex graphicsengine of Aegia (”http://www.aegia.com”). We usefour of its mechanical degrees of freedom (two elbowsand two knees) all of which are connected to twocoupled CMLs using Eq. 3 (see Fig. 4). Both CMLsconsist of L = 32 coupled logistic maps (α = 2.0).The first lattice receives inputs from the angularsensors (chaotic units i = 5, 13, 21 and 29) form-ing the system ”body-CML 1”. The units with thesame index of the second lattice (with outputs nor-malized between [-1,1]) are connected to the actua-tors, forming the system ”CML 2-body”. To extractthe phase embedded in the chaotic signal the out-put is filtered with a mexican-hat wavelet expressedas: F body

i (n) = mexha(yi(n)), where mexha(x) =2√3a

π−1/4(1− x2)e−x2

2 , with a being the scale of thewavelet (a = 64 for our application). Our model sys-tem consists of two active oscillating systems (”body-

a)

b)

c)

d)

e)

Figure 3: Probability distribution as a function of time

for representative neural unit. Left column: CML 1; right

column: CML 2. Increasing the coupling γ increases the

influence of external input on systems’ internal dynamics.

The coupling constants are: γ = 0.10 (a); γ = γPS− =

0.15 (b); γ = 0.16 (c); γ = γPS+ = 0.18 (d); γ = 0.23

(e).

CML1” and ”CML 2-body”) and hence satisfies theminimal requirement for displaying phase synchro-nized states. The sampling time is 2.4 ms.

3.2.1 Varying the coupling parameter

As in the previous section, we analyze the synchro-nization level between the two CMLs by varying thecoupling parameter γ. We plot the phase synchro-nization index and the corresponding spectral bifur-cation diagram of the CMLs (Fig. 5) for γ ∈ [0, 0.60].For this model system, phase synchronization oc-curs for several intervals γ – specifically [0.15, 0.18],[0.23, 0.27] and [0.28, 0.41]. In the subsequent analy-sis, we focus on the interval [0.15, 0.18].

Figure 4: Outline of our embodied model system. ”Body-

CML 1” and ”CML2-body” form two self-sustained os-

cillators.

For γ < γPS− = 0.15, the chaotic units composingthe CMLs are characterized by broadband spectraldistribution with no well-pronounced peaks. We ob-serve rapid movements of arms and legs with smallamplitudes. When perturbed (e.g. through touch),the body’s limbs oppose only weak resistance; theexternal dynamics has no influence on the internaldynamics. There is some interaction but definitelyno synchronization between the body and the neuraldynamics, and each system acts independently.

For γ ∈ [0.15, 0.18], PS occurs between the bodyand the neural system. The spectral dynamics is nownarrow-band and characterized by a few significantpeaks (i.e. a dimension reduction of the global sys-tem dynamics occurs). Both dynamics are now cor-related and mutually integrated through the phase(condition of phase-locking). In analogy with the in-formation theoretic framework discussed previously,a communication channel between body and neuralsystem is established through which information canbe exchanged. In this situation, touching or ma-nipulating the limbs externally affects the internaldynamics of the system. After releasing the forceapplied to the limbs, the system is for some timeable to reproduce the previously imposed movement(see also the following subsection). Because the twoCMLs are synchronized, it is possible to link inter-nal and external dynamics, and embed externally im-posed sensorimotor patterns through phase modula-tion.

Beyond phase synchronization (0.23 > γ > 0.18 =γPS+), the level of synchronization between the twoCMLs increases and their dynamics becomes moreuniform. Information is exchanged between the twosystems but is not embedded.

Figure 5: Embodied setup – variable γ. Top: Phase syn-

chronization index vs. coupling γ for CML 1 - CML 2.

Center: Spectral bifurcation diagram for CML 1. Bot-

tom: Spectral bifurcation diagram for CML 2.

3.2.2 At phase synchronization

In this section, we analyze the behavior of the em-bodied system for coupling parameter γ = 0.15, thatis, body, CML1 and CML 2 are weakly coupled andphase synchronized.

For γ = 0.15 the system is active, sensitive to ex-ternal oscillations, and able to embed and sustainthem in its internal dynamics. By forcing the body’slimbs to follow particular movements, we can influ-ence the systems’s internal dynamics and store ex-ternally applied movements. As a result, the systemcan partially learn and reproduce the imposed move-ments. If the movement is blocked (e.g. by an obsta-cle or an external force), then the behavior flexiblyadapts to the new boundary condition. To analyzethe internal organization of the systems, we plot thephase synchronization index Ψ for the system com-posed of CML 1 - CML 2 (Fig. 6 top). The activity

of the joint angle sensor in the elbow is displayed inFig. 6 center (the dashed line is the imposed pattern,the continuous line is the pattern at the output of thesystem). To show that patterns are actually storedand reproduced, we also plot the output for the caseγ = 0.40 (strong synchronization; Fig. 6 bottom).

The large variance of Ψ around 0.5 suggests readi-ness of the system to react to different sensorimotorpatterns – it allows exchanges of information andthe mutual adaptation of rhythms of CML1 andCML2. The oscillatory fluctuations of Ψ point toperiods of synchronization (high values), synchro-nization breaks (low values), and resynchronization.To better understand the underlying dynamics, wecompute the wavelet bifurcation diagram for the twoCMLs (Fig. 7). Low scales (high frequencies) em-phasize the local variabilities while high scales (lowfrequencies) express the global structure of the sys-tem. We can also observe chaotic itinerancy in thetwo CMLs for time scales higher than 3 (Fig. 7 c).For these time scales the dynamics of the clusters aresimilar, that is, synchronization occurs but is ”itiner-ant” and characterized by instabilities which producea transitory behavior between attractor states, andthe appearance and disappearance of phase-relatedsynchronizations. The near-zero Lyapunov expo-nents of the time series indicate that the system iseffectively in a state of chaotic itinerancy and phasesynchronization (Fig. 8) (Tsuda et al., 2004).

The synchronization break (t = 6500 ms; Fig. 7) isexplained as follows: the externally imposed move-ment is ”forced” upon the internal systems andthe previous patterns are (at least partially) de-stroyed. When resynchronization occurs, patches ofsynchronous but decaying neural activity can be ob-served.

4. Discussion

Synchronization is the adaptation of rhythms of cou-pled oscillators. Using a disembodied and an embod-ied setup we have shown how at synchronization in-dividual (internal) and environmental (external) dy-namics can be coupled more efficiently, e.g. exter-nally imposed sensorimotor patterns are transferredand stored more easily internally. This result pointsto an important role of phase synchronization for se-lective attentional processes: selection of sensorimo-tor patterns is achieved dynamically through modu-lation of phases between the coupled systems. More-over, depending on the system’s current dynamicsand on the amount of coupling between internal andexternal dynamics, filtered external signals can per-sist for more or less time (patches of synchronous,persistent, but decaying neural activity are shownin Fig. 7) and the neural system can function aseither a short-term or a long-term memory. Suchpersistence is related to the idea of perceptual con-

Figure 6: Embodied setup – fixed γ. Top: Phase synchro-

nization index vs. time; coupling between CML1 and

CML 2 fixed to γ = 0.15. Center: Input-output joint an-

gle activity (elbow) at phase synchronization (γ = 0.15).

Bottom: Input-output angle activity (elbow) for highly

synchronized systems (γ = 0.40).

tinuity (Hock et al., 2003), that is, of coherent andstable perception despite disruption by environmen-tal perturbations or noise. The interaction betweeninternal and external dynamics is reciprocal. If thebrain and body-environment are adequately coupled,stable behavioral patterns such as walking or runningcould emerge despite perturbations or noisy signals(because (PS) occurs regardless of the amplitude ofthe signal), and yield complex emergent dynamics(because (PS) is a general mechanism independentof the movement patterns’s complexity).

Phase synchronization between the neural unitsand phase transitions are also exhibited by sev-eral models of memory dynamics (Tsuda et al., 2004,Raffone and van Leeuwen, 2003). Adaptive Reso-nance Theory is also related to the notion of synchro-

a)

b)

c)

d)

e)

Figure 7: Wavelet bifurcation diagrams. Left column:

CML 1; right column: CML 2. The scales are s=1 (a),

s=2 (b), s=3 (c), s=5 (d), s=10 (e).

Figure 8: Fluctuations of the largest Lyapunov exponent.

nization (Grossberg, 2005). Specifically, at phasesynchronization there is a trade-off between stabil-ity and plasticity, that is, the neural system learnsquickly novel patterns (plasticity) without forgettingcatastrophically past knowledge (stability or persis-

tence). Grossberg further suggests that all consciousstates in the brain are resonant states, and that theseresonant states trigger learning of sensory and cogni-tive representations. It might be worthwhile to gen-eralize these results in an embodied viewpoint andjustify them from a physical and information theo-retical point of view (as attempted in this paper).

As synchronization is the physically-based adap-tation of systems rhythms, it can also be concep-tualized as an adaptive learning mechanism whichneither makes any a priori assumptions nor usesany specific learning scheme. We hypothesize thatsynchronization may actually allow a reinterpreta-tion of Hebb’s law: ”Neurons that fire together,wire together” may be re-formulated as ”neuronsthat synchronize, wire together”. We suggest thattwo synchronized nonlinear oscillators, even if onlyweakly coupled (that is, potentially physically dis-connected), can actually ”wire” together, i.e. ex-change information.

In the context of social interaction, the no-tion of synchronization (or synchrony, resonance,social entrainment or regulation) has been al-ready introduced as a concept (e.g. (Breazeal, 2002,Nadel et al., 2005)). We hypothesize that phase syn-chronization may also shed some light on the mech-anisms underlying social interaction, and may helpformalize and explain high-level semantic conceptssuch as self-agency and resonance, coupling and au-tonomy (e.g. language, communication or their dys-functionment like autism).

It is also interesting to consider coupled systemsin the context of cause-effect relationships. Since thefirst observation of anticipation of synchronization ina physical system (Voss, 2000), it has been shownthat at PS, one of the two coupled systems can an-ticipate the state of the other (Calvo et al., 2004).As the systems act together, it follows that theeffect may actually precede its cause. From thisresult, we can imply that for a particular sys-tem it might be necessary to revise the notionof unidirectional relationship between causality andtime (Prigogine, 1980). It also follows that synchro-nization can stand for the design of a physically-based and dynamical predictive model and for amemory model.

Before concluding, we would like to point outthat phase as used here is a dubious conceptfor discrete time systems and can in generalonly be introduced for continuous dynamical sys-tems (Rosenblum et al., 2001). The reason is thatphase can be only defined for systems having zeroLyapunov exponents. As evident from Fig. 8,the results obtained with coupled logistic maps ex-hibited intermittent near-zero Lyapunov exponentswhich are characteristic of chaotic itinerancy butwhich weakly violate the strict definition of phase

given in (Rosenblum et al., 2001). Recent work onphase synchronization, however, seems to confirmour observations, and that is, phase can be also de-fined for discrete systems (at least in particular in-stances) (Koronovski et al., 2005). Our results seemto indicate that the framework described in this pa-per can be actually used for discrete time systems.

Acknowledgments

The authors would like to thank Shogo Yonekurafor having built the Novodex interface, Prof. IchiroTsuda, Prof. Rosenblum for his correspondenceand the Japan Society for the Promotion of Science(JSPS) for supporting this research.

References

Baier, G. and Muller, M. (2004). The nonlineardynamic conversion of analog signals into exci-tation patterns. Phys. Rev. E., 70(037201).

Baptista, M. S. and Kurths, J. (2005). Chaoticchannel. Phys. Rev. E, 72(045202).

Breazeal, C. (2002). Regulation and entrainment inhuman-robot interaction. Int. J. of Experimen-tal Robotics, 21(10/11):883–902.

Calvo, O., Chialvo, D. R., Eguiluz, V. R., Mirasso,C., and Toral, R. (2004). Anticipated synchro-nization: A metaphorical linear view. Chaos,14(1):7–13.

Grossberg, S. (2005). Linking attention to learning,expectation, competition, and consciousness.Neurobiology of Attention, Elsevier, 107:652–662.

Hayes, S., Grebogi, C., Ott, E., and Mark, A.(1994). Experimental control of chaos for com-munication. Phys. Rev. Lett., 13(73):1781–1784.

Hock, H. S., Schoner, G., and Giese, M. (2003).The dynamical foundations of motion patternformation: Stability, selective adaptation, andperceptual continuity. Percept. Psychophys.,65(3):429–457.

Koronovski, A. A., Hramov, A. E., and Khramova,A. E. (2005). Synchronous behavior of cou-pled systems with discrete time. JETP Letters,82(3):160–163.

Lungarella, M. and Berthouze, L. (2002). On theinterplay between morphological, neural, andenvironmental dynamics: a robotic case study.Adaptive Behavior, 10(3/4):223–241.

Nadel, J., Prepin, K., and Okanda, M. (2005). Ex-periencing contingency and agency: first step

towards self-understanding in making a mind?Interaction Studies, 6(3):447–462.

Orrel, D. and Smith, L. A. (2003). Visualizing bifur-cations in high dimensional systems: The spec-tral bifurcation diagram. Int. J. of Bifurcationand Chaos, 13(10):3015–3027.

Pecora, L. and Carroll, T. (1990). Synchronizationin chaotic systems. Phys. Rev. Lett., 64:821.

Pikovsky, A. S., Rosenblum, M. G., and Kurths, J.(2001). Synchronization. Cambridge Press.

Pitti, A., Lungarella, M., and Kuniyoshi, Y. (2006).Exploration of natural dynamics through reso-nance and chaos. Proc. of 9th Conf. on Intelli-gent Autonomous Systems, pages 558–565.

Prigogine, I. (1980). From Being To Becoming.Freeman.

Prince, C. G. and Hollich, G. J. (2005). Synchingmodels with infants: a perceptual-level modelof infant synchrony detection. J. of CognitiveSystems Research, 6(3):205–228.

Raffone, A. and van Leeuwen, C. (2003). Dy-namic synchronization and chaos in an associa-tive neural network with multiple active memo-ries. Chaos, 13:1090–1104.

Roelfsema, P. R., Engel, A. K., Konig, P., andSinger, W. (1997). Visuomotor integration isassociated with zero time-lag synchronizationamong cortical areas. Nature, 385:157–161.

Rosenblum, M. G., Pikovsky, A. S., and Kurths, J.(1996). Phase synchronization of chaotic oscil-lators. Phys. Rev. Lett., 76(11):1804–1807.

Rosenblum, M. G., Pikovsky, A. S., and Kurths,J. (2001). Comment on phase synchroniza-tion in discrete chaotic systems. Phys. Rev. E,63(058201):1–2.

Strogatz, S. (2003). Sync: The Emerging Scienceof Spontaneous Order. New York: Hyperion.

Tsuda, I., Fujii, H., Tadokoro, S., Yasuoka, T., andYamaguti, Y. (2004). Chaotic itinerancy as amechanism of irregular changes between syn-chronization and desynchronization in a neuralnetwork. J. of Integr. Neurosc., 3:159–182.

Varela, F., Lachaux, J., Rodriguez, E., and Mar-tinerie, J. (2001). The brain web: phase-synchronization and large-scale brain integra-tion. Nat. Rev. Neuroscience, 2:229–239.

Voss, H. U. (2000). Anticipating chaotic synchro-nization. Phys. Rev. E, 61:51155119.