Abstract. Composite stimulation techniques are present-ed here which are based on a soft (i.e., slow and mild)reset. They effectively desynchronize a cluster of globallycoupled phase oscillators in the presence of noise. Acomposite stimulus contains two qualitatively differentstimuli. The first stimulus is either a periodic pulse train ora smooth, sinusoidal periodic stimulus with an entrainingfrequency close to the cluster’s natural frequency. In thecourse of several periods of the entrainment, the cluster’sdynamics is reset (restarted), independently of its initialdynamic state. The second stimulus, a single pulse, isadministered with a fixed delay after the first stimulus inorder to desynchronize the cluster by hitting it in avulnerable state. The incoherent state is unstable, andthus the desynchronized cluster starts to resynchronize.Nevertheless, resynchronization can effectively beblocked by repeatedly delivering the same compositestimulus. Previously designed stimulation techniquesessentially rely on a hard (i.e., abrupt) reset. With thecomposite stimulation techniques based on a soft reset,an effective desynchronization can be achieved even ifstrong, quickly resetting stimuli are not available or nottolerated. Accordingly, the soft methods are very prom-ising for applications in biology and medicine requiringmild stimulation. In particular, it can be applied toeffectively maintain incoherency in a population ofoscillatory neurons which try to synchronize their firing.Accordingly, it is explained how to use the soft techniquesfor (i) an improved, milder, and demand-controlled deepbrain stimulation for patients with Parkinson’s disease oressential tremor, and for (ii) selectively blocking gammaactivity in order to manipulate visual binding.
1 Introduction
Synchronization processes abound in neuroscience(Eckhorn et al. 1988; Gray and Singer 1989) and
medicine (Llinas and Jahnsen 1982; Bergman et al.1994; Volkmann et al. 1996). Stimulation is a majorexperimental tool for studying and manipulating syn-chronization under physiological as well as pathologicalconditions. Let us consider two examples in the follow-ing.
Deep brain stimulation. Parkinsonian resting tremorappears to be caused by a cluster of neurons located inthe thalamus and the basal ganglia which fire synchro-nously at a frequency similar to that of the tremor(Llinas and Jahnsen 1982; Pare et al. 1990; Lenz et al.1994). By contrast, under physiological conditions theneurons in this cluster fire incoherently (Nini et al.1995). In patients with Parkinson’s disease (PD) thiscluster acts like a pacemaker and activates premotorareas (premotor cortex and supplementary motor area)and the motor cortex (Alberts et al. 1969; Lamarre et al.1971; Bergman et al. 1994; Nini et al. 1995; Volkmannet al. 1996), where the latter synchronize their oscillatoryactivity (Tass et al. 1998). Similarly, essential tremoralso appears to be caused by a central cluster ofsynchronously firing neurons, although they are locatedin different brain areas compared to PD (Elble andKoller 1990).In patients with advanced PD or with essential tremor
who do not respond to drug therapy any more, elec-trodes are chronically implanted within a particularneuronal cluster of the brain with millimeter precision(Benabid et al. 1991; Blond et al. 1992). Up to now, apermanent high-frequency stimulation with a high-fre-quency (>100 Hz) periodic pulse train has been admin-istered via the deep brain electrodes in order to suppressthe pathological synchronized activity of the pacemaker-like cluster which, in turn, suppresses the peripheraltremor (Benabid et al. 1991; Blond et al. 1992). Themethod presented here can be used to develop a mildand efficient, demand-controlled deep brain stimulationtechnique for patients with PD or essential tremor.
Sensory manipulation of visual binding. An importantand still unsolved issue in neuroscience is the visual
Correspondence to: P. A. Tass(e-mail: [emailprotected])
Biol. Cybern. 87, 102–115 (2002)DOI 10.1007/s00422-002-0322-5� Springer-Verlag 2002
Desynchronization of brain rhythmswith soft phase-resetting techniques
Peter A. Tass
Institute of Medicine, Research Centre Julich, 52425 Julich, Germany
Received: 3 July 2001 /Accepted in revised form: 7 February 2002
binding problem, which is the question of how infor-mation that is spatially distributed in the brain getsbound together to form a meaningful pattern ofperception. It was suggested that the mechanism whichrealizes the visual binding is the synchronization of thefiring in the gamma frequency range (30–80 Hz, espe-cially around 40 Hz) of neurons coding for specificfeatures belonging to an object in the visual field(Eckhorn et al. 1988; Gray and Singer 1989). Inanesthetized cat (Eckhorn et al. 1988; Gray and Singer1989) and awake monkey (Kreiter and Singer 1992;Eckhorn et al. 1993) it was observed that smoothlymoving stimuli induce sustained gamma oscillations indifferent visual cortical areas which are synchronized inphase. While rapidly changing stimuli evoke stimulus-locked fast and transient responses, the gamma oscilla-tions occur with longer and more variable latencies(Gray et al. 1992).The functional role of gamma oscillations in visual
binding is still a matter of debate (cf. Ghose andMaunsell1999; Riesenhuber and Poggio 1999). For instance, due tothe latencies of these oscillations their contribution tovisual binding in rapidly changing scenes is questionable(Ghose and Freeman 1992). To address this issue, in vi-sual cortical areas of anesthetized cat sustained gammaoscillations were produced by a slowly drifting visualgrating pattern and then perturbed by intermingled sud-den random accelerations of the grating (Kruse andEckhorn 1996). With increasing amplitude of the randomperturbations the corresponding evoked fast responsesincreased, whereas the amplitude of gamma oscillationsgradually decreased. Kruse and Eckhorn (1996) suggestthat this suppression of gamma oscillations is necessaryfor switching between different percepts.To manipulate gamma oscillations without addition-
al, perturbing stimuli, entraining and resonance effectsof visual stimuli were studied. It was shown that visualstimuli flickering at a frequency that is close to the res-onance frequency of 40 Hz strongly entrain neuronalpopulations in cat (Rager and Singer 1998) and human(Herrmann 2001) visual cortex. Moreover, Kanisza-likevisual stimuli flickering at the resonance frequency(40 Hz) are more rapidly processed by the brain thanstimuli flickering at other frequencies (Elliot and Muller1998), and they are connected with reduced latencies ofstimulus-evoked gamma responses as measured withelectroencephalography (Elliot et al. 2000). The novelsoft phase-resetting technique shown here may be usedfor visual stimulation to block gamma activity in orderto selectively study its role in visual binding processes.For this, the same visual stimulus has to be administeredwith an appropriate timing and intensity sequence. Inthis way it may be possible to study how the stability ofa single percept is manipulated by desynchronizinggamma oscillations without using additional stimuli re-lated to different percepts. In other words, the rela-tionship between a single percept and gammaoscillations can be studied without switching betweendifferent percepts.The present study is based on a stochastic phase-
resetting approach (Tass 1999). To investigate desyn-
chronizing effects of pulsatile stimuli, the concept ofphase resetting (Winfree 1980) was extended to popu-lations of noninteracting (Tass 1996a, b) and interacting(Tass 1999, 2000) oscillators in the presence of randomforces. With this aim in view, limit-cycle oscillators areapproximated by phase oscillators (Kuramoto 1984),and desynchronization is caused by stimuli that exclu-sively affect the oscillators’ phases. A fully synchronizedcluster of oscillators is desynchronized by a single pulseof the correct intensity and duration provided the pulsehits the cluster in a vulnerable phase range that corre-sponds to only a small fraction (5% or even less) of aperiod of the oscillation. Of course, this is tricky to re-alize under noisy experimental conditions typically en-countered in biological systems. Moreover, differentstimulation parameters have to be used to desynchronizea cluster which is not in its fully synchronized state (Tass1999, 2001a).By contrast, stimulation techniques were recently
presented which effectively desynchronize a cluster ofoscillators irrespective of the cluster’s dynamic state atthe beginning of the stimulation (Tass 2001a–c). Thesemethods are much more appropriate for medical or bi-ological applications and have one common feature: thestimulus contains two qualitatively different stimuli. Thefirst stimulus is either a strong single pulse (Tass 2001a,c) or a high-frequency pulse train (with a frequency thatexceeds the cluster’s natural frequency by a factor of 20or more; Tass 2001b). The strong first stimulus causes ahard reset (i.e., abrupt reset) during which the collectiveoscillation runs for less than one period. After this resetthe cluster restarts in a stereotypical way. The second,weaker stimulus is a single pulse that is administeredafter a constant time delay and desynchronizes byhitting the cluster in a vulnerable state.It has to be stressed that a hard reset requires that the
stimulus strongly affects the stimulated system withoutcausing any damage. In biology and medicine, however,there are many systems for which direct and strongstimuli are not available or that do not tolerate strongstimuli (Stoney et al. 1968; Winfree 1980). To overcomethis problem, a novel, effectively desynchronizing com-posite stimulus is presented here. It consists of twoqualitatively different stimuli. The first stimulus causes asoft reset, and is either a periodic pulse train or a smoothperiodic (e.g., sinusoidal) stimulus, where in both casesthe entraining frequency is close to the natural frequencyof the cluster (before stimulation). During this soft reset(i.e., slow reset), the collective oscillation is not quicklyabrupted. By contrast, in the course of this entrainmentthe influence of the initial dynamic state at the beginningof the stimulation fades away while the collective oscil-lation runs through several periods. The second stimulusis a single pulse which follows after a constant time delayand hits the cluster in a vulnerable state, in this waydesynchronizing it.Since such a combined stimulus desynchronizes a
cluster no matter at which initial dynamic state it isadministered, this method can be used to block thecluster’s resynchronization. To this end a combinedstimulus has to be administered repeatedly, whenever
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the cluster becomes synchronized again. This noveltechnique may find several significant applications inbiology and medicine. In this article I shall focus ondemand-controlled deep brain stimulation and on sen-sory manipulation of visual binding with desynchroniz-ing visual stimulation. A part of the analysis of thedynamics caused by a composite stimulus with pulsatileentrainment was presented in a rapid communication(Tass 2002).
2 Stochastic approach
2.1 Model
The dynamics of a neuronal population can be modeledby means of networks of phase oscillators. A detailedexplanation of this approach is presented in Kuramoto(1984), Ermentrout and Kopell (1991), Grannan et al.(1993), and Hansel et al. (1993). The dynamics of acluster of coupled phase oscillators subjected to astimulus S and to random forces is governed by theLangevin equation
_wwj ¼ X þ 1
N
XNk¼1
Cðwj � wkÞ þ X ðtÞSðwjÞ þ FjðtÞ ; ð1Þ
where wj denotes the phase of the jth phase oscillator,i.e., the jth model neuron (Tass 1999). For the sake ofsimplicity all oscillators have the same eigenfrequency:xj ¼ X. The global coupling is a 2p-periodic function.For the time being we consider a simple sine coupling ofthe form
Cðwj � wkÞ ¼ �K sinðwj � wkÞ ; ð2Þ
where K is a nonnegative coupling constant. This type ofcoupling is sufficient in this study, because we focus on acluster of oscillators synchronized in phase. The influ-ence of cosine couplings such as cosðwj � wkÞ andcos 2½ðwj � wkÞ� is discussed in Sect. 7. Sine couplingterms of second and higher order such as sin½2ðwj � wkÞ�and sin½3ðwj � wkÞ� give rise to noisy cluster states wherethe population consists of distinct phase-locked clusters.The stimulation techniques presented in this article alsoeffectively desynchronize cluster states. Actually, themechanism by which cluster states are desynchronized ispractically the same as the desynchronizing mechanismfor the in-phase synchronized cluster (see Sect. 7).The impact of an electrical stimulus on a single neu-
ron depends on the neuron’s phase at which the stimulusis administered (Best 1979; Guttman et al. 1980). Ac-cordingly, the stimulus is modelled by a 2p-periodic,time-independent function SðwjÞ ¼ Sðwj þ 2pÞ. We firstassume that the stimulus is of lowest order and definedby
SðwjÞ ¼ I cosðwjÞ ; ð3Þ
where I is a constant intensity parameter. The effect ofhigher-order terms of S is discussed in Sect. 7. Switchingthe stimulator on and off is taken into account by
X ðtÞ ¼ 1 : stimulus is on at time t0 : stimulus is off at time t .
nð4Þ
The random forces FjðtÞ are modelled by Gaussian whitenoise: hFjðtÞi ¼ 0 and hFjðtÞFkðt0Þi ¼ Ddjkdðt � t0Þ, withconstant noise amplitude D.To investigate the dynamics of (1) we first derive the
corresponding Fokker–Planck equation which is an evo-lution equation for the probability density f ðw; tÞ, wherew is the vector ðw1; . . . ;wN Þ. f ðw; tÞ dw1 � � � dwN gives usthe probability of finding the oscillators’ phases in theintervals wk; . . . ;wk þ dwk. In order to simplify the anal-ysis we consider the dynamics on a more macrospcopiclevel of description by introducing the average number
density nðw; tÞ ¼ h~nnðw;wÞit ¼R 2p0 � � �
R 2p0 dw1 � � � dwN
~nnðw;wÞf ðw; tÞ, where the number density is defined by~nnðw;wÞ ¼ 1
N
PNk¼1 dðw � wkÞ (Kuramoto 1984). The
probability density f ðw; tÞ provides us with informationconcerning the phase of each single oscillator. By con-trast, nðw; tÞ tells us how many oscillators of the wholepopulation most probably have phase w at time t.With a little calculation we finally obtain the evolu-
tion equation for the average number density
onðw; tÞot
¼� o
ownðw; tÞ
Z2p
dw0Cðw�w0Þnðw0; tÞ
8<:
9=;
� o
ownðw; tÞX ðtÞSðwÞ�X
o
ownðw; tÞþD
2
o2nðw; tÞow2
;
ð5Þ
which holds for large N (Tass 1999). For a detailedanalytical and numerical investigation of (5), please referto Tass (1999).
2.2 Spontaneously emerging synchrony
The time-dependent extent of synchronization is quan-tified by means
ZðtÞ ¼ RðtÞ exp½iuðtÞ� ¼Z2p
nðw; tÞ expðiwÞdw ; ð6Þ
where RðtÞ and uðtÞ are the real amplitude and the realphase of Z, respectively (Kuramoto 1984). Due to thenormalization condition
R 2p0 nðw; tÞdw ¼ 1, the ampli-
tude fulfills 0 � RðtÞ � 1 for all times t. Perfect in-phasesynchronization corresponds to R ¼ 1, whereas anincoherent state, given by nðw; tÞ ¼ 1=ð2pÞ, correspondsto R ¼ 0. ZðtÞ corresponds to the center of mass of thecircularly aligned density nðw; tÞ expðiwÞ in the Gaussianplane (Fig. 1).To study the stimulation-induced dynamics, first the
cluster’s behavior without stimulation (i.e., X ðtÞ ¼ 0 in4) has to be clarified. Let us assume that the coupling isgiven by (2). (The influence of higher-order couplingterms is explained in Sect. 7.) Noisy in-phase synchro-
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nization emerges out of the ncoherent state n ¼ 1=ð2pÞdue to a decrease of the noise amplitude D (Kuramoto1984) or, analogously, because of an increase of thecoupling strength (Tass 1999). When K exceeds its crit-ical value Kcrit ¼ D, Z from (6) becomes an order pa-rameter (Haken 1983) which governs the dynamics ofthe other, infinitely many stable modes (i.e., frequencycomponents) on the center manifold. In this way forK > D a stable limit cycle ZðtÞ ¼ Y exp½iðX þ �XXÞt�evolves, where Y is a complex constant, and �XX is a realfrequency shift term that depends on model parametersand vanishes if the coupling C contains no cosine terms,as in (2) (Tass 1999).The cluster’s collective dynamics will not only be vi-
sualized with the order parameter Z, but also by con-sidering the collective firing. A single firing/burstingmodel neuron fires/bursts whenever its phase vanishes(modulo 2). Accordingly, the cluster’s collective firingactivity is given by the firing density pðtÞ ¼ nð0; tÞ whichcorresponds to quantities registered in neurophysiolog-ical experiments such as multiunit activity, local fieldpotentials (LFP), and magnetic or electric fields mea-sured with magnetoencephalography or electroenceph-alography.
3 Desynchronizing soft phase resetting
3.1 Desynchronizing single pulse
During a single pulse, X ðtÞ ¼ 1 and S is constant in time.If the stimulus S is sufficiently strong with respect to thecoupling strength, nðw; tÞ tends to a stationary densitynstatðwÞ for t ! 1. The latter is the attractor of (5),irrespective of the initial state nðw; 0Þ at which thestimulation starts (Tass 1999). Correspondingly, theorder parameter is attracted by
Zstat ¼Z2p
nstatðwÞ expðiwÞdw ; ð7Þ
as shown in Fig. 1a, where the collective dynamics of thecluster is visualized by plotting the trajectory of Z in theGaussian plane. Desynchronization corresponds toZ ¼ 0. Hence, to desynchronize the synchronized cluster,the single pulse has to be administered at a critical
Fig. 1a–h. Trajectories of Z from (6) are plotted in the Gaussianplane. In a–e the unit circle indicates the maximal range of jZj. Singlepulse: a Series of identical stimuli SðwÞ ¼ I cosw (with I ¼ 7)administered at different initial phases uB in the stable synchronizedstate (open circles). Z approaches the attractor Zstat from (7) fort ! 1. Only the stimulus administered at the vulnerable initial phase(filled circles) moves Z through the origin. Trajectory of Z before andduring (b) and after (c) a desynchronizing single pulse (parameters asin a): b After running on its stable limit cycle (inner circle) incounterclockwise direction, Z is moved by the pulse into the origin(Z ¼ 0). Stimulation starts at the open circle and ends at the filledcircle. cAfter the stimulation the cluster spontaneously spirals back toits stable limit cycle. Composite stimulus with pulsatile entrainment:The periodic pulse train entrains the cluster so that Z performs aperiodic motion (d): During each pulse of the train Z is shifted fromZb ¼ limk!1 Zðtb;kÞ to Ze ¼ limk!1 Zðte;kÞ (upward arrow), whileduring each pause Z relaxes back from Ze to Zb (downward arrow). tb;kand te;k denote the begin and end of the kth pulse. e At the end of thepulse train Z is sufficiently close to Ze (star), and then relaxes back toits stable limit cycle. The desynchronizing pulse starts at the open circleand moves Z into the origin (filled circle). After the desynchronizationZ spirals back to its stable limit cycle (as in c). Composite stimulus withsmooth entrainment: f Before the stimulation Z is on its stable limitcycle (inner circle). The smooth stimulus starts at the triangle andrapidly moves Z to its entrained limit cycle (outer circle) where itoscillates with frequency xs. This transition is indicated by the arrow.gWhile Z is running on the entrained limit cycle, the second stimulus– a desynchronizing single pulse (with parameters as in b) – isadministered at the open circle and moves Z into the origin (filledcircle). h A periodic pulse train is applied to the fully synchronizedcluster at m ¼ 100 different initial phases equally spaced in ½0; 2p�.dðte;kÞ=dð0Þ (plus), i.e., the normalized mean mutual distance of theorder parameter Z at the end of the kth pulse during the pulse train,vanishes within a few periods where the pulse train starts at t ¼ 0.Likewise the smooth periodic stimulus (8) is administered to the fullysynchronized cluster at m ¼ 100 different initial phases equally spacedin ½0; 2p�. In this series of simulations dðtÞ=dð0Þ (solid line) vanishessimilarly. Model parameters were as follows. a–h: CðxÞ ¼ � sin x,D ¼ 0:4, X ¼ 2p. Single pulse in a, b, e, and g: SðwÞ ¼ I cosw withI ¼ 7, pulse duration T ¼ 0:31 in b, 0:33 in e, and 0:45 in g. Periodicpulse train (d): SðwÞ ¼ I cosw with I ¼ 21, pulse duration T1 ¼ 0:2,pause duration T2 ¼ 0:47, i.e., xp ¼ 3p, number of pulses M ¼ 10(same results for M > 100). Smooth periodic pulse (f):SðwÞ ¼ I cosðw � xstÞ with I ¼ 7, xs ¼ 3p
b
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(vulnerable) initial phase and it has to be turned off assoon as Z reaches the origin of the Gaussian plane(Fig. 1b). The desynchronized state is unstable. There-fore after the desynchronizing stimulation, Z spiralsback to its stable limit cycle, so that the cluster becomessynchronized again (Fig. 1c).
3.2 Pulsatile entrainment
A composite stimulus with pulsatile entrainment consistsof two qualitatively different stimuli. The first stimulus isa train of M identical pulses. The kth pulse begins at tb;kand ends at te;k. The frequency of the pulse train isxp ¼ 2p=ðT1 þ T2Þ, where T1 ¼ te;k � tb;k and T2 ¼ tb;kþ1�te;k are pulse duration and pause duration, respectively.The periodic pulse train resets the cluster by entrainmentat a rate similar to the natural frequency of the cluster(before stimulation). Accordingly, the entraining fre-quency xp is of the same order of magnitude as X from(1). In the fully entrained state Z moves fromZb ¼ limk!1 Zðtb;kÞ to Ze ¼ limk!1 Zðte;kÞ during eachpulse, whereas Z relaxes back from Ze to Zb during eachpause (Fig. 1d). The second stimulus follows after thepulse train with a constant time delay (corresponding toless than one period of the cluster’s spontaneousoscillation), hits the cluster in a vulnerable state, anddesynchronizes it by shifting Z into the origin (Fig. 1e).
3.3 Smooth Entrainment
A composite stimulus with smooth entrainment alsocontains two qualitatively different stimuli. The firststimulus is smooth, periodic, and explicitly timedepen-dent. Instead of (3), we now start from a general ansatzfor a first-order stimulus Sðwj; tÞ ¼ AðtÞ sinðwjÞ þ BðtÞcosðwjÞ, where A and B are smooth periodic functionswith period P : AðtÞ ¼ Aðt þ P Þ and BðtÞ ¼ Bðt þ P Þ. Letus consider a special case connected with a straightfor-ward dynamics that is sufficient to illustrate the mainprinciple of smooth entrainment. To this end we assumethat AðtÞ ¼ I sinðxstÞ and BðtÞ ¼ I cosðxstÞ, which can berewritten as
Sðwj; tÞ ¼ I cosðwj � xstÞ ; ð8Þ
where xs is the entraining frequency of the smoothentraining stimulus. Introducing a rotating coordinatesystem hjðtÞ ¼ wjðtÞ � xst for j ¼ 1; . . . ;N , the Langevinequation (1) in this case reads _hhj ¼ ~XX þ N�1PN
l¼1Cðhj � hlÞ þ X ðtÞ~SSðhjÞ þ FjðtÞ with ~XX ¼ X � xs and~SSðhjÞ ¼ I cos hj. The Langevin equation in the rotatingcoordinate system is of the same form as (1). Inparticular, ~SS is no longer explicitly timedependent.Hence, the dynamics caused by the smooth periodicstimulus (8) in the rotating coordinate systemðh1; . . . ; hN Þ corresponds to the dynamics caused by thesingle pulse studied above with SðwjÞ ¼ I coswj in theinitial coordinate system ðw1; . . . ;wNÞ.
Accordingly, during the smooth periodic stimulation,Z tends to a fixed point in the rotating coordinate systemprovided I is sufficiently large. This fixed point isequivalent to a limit cycle of frequency xs in the initialcoordinate system ðw1; . . . ;wN Þ (Fig. 1f). The secondstimulus, a single pulse, is administered at a vulnerablestate and causes a desynchronization by shifting Z intothe origin of the Gaussian plane (Fig. 1g).The very goal of both the pulsatile and the smooth
entrainment is that the cluster’s dynamics in the en-trained state no longer depend on the initial dynamicconditions at the beginning of the entraining stimula-tion. To show this, a periodic pulse train (i.e., the firststimulus of the composite stimulus with pulsatile en-trainment) is administered to the fully synchronizedcluster at m ¼ 100 different initial phases equally spacedin ½0; 2p�. Zð jÞðtÞ is the order parameter belonging to thesimulation starting at the jth initial phase ( j ¼ 1; . . . ;m).The mean mutual distance of the order parameter in thisseries of simulations is given by
dðtÞ ¼ 2
ðm� 1ÞmXmj¼1
Xk>j
jZðjÞðtÞ � ZðkÞðtÞj ; ð9Þ
where jxj denotes the absolute value of x, and thesummation runs over all ðm� 1Þm=2 pairs of differenttrajectories (Tass 2001b). Likewise, a smooth entrainingstimulus (i.e., the first stimulus of the composite stimuluswith smooth entrainment) is applied at m ¼ 100 differ-ent, equally spaced initial phases, and dðtÞ is determined.In the course of both the pulsatile and the smoothperiodic entrainment d vanishes (Fig. 1h), which meansthat the entraining stimuli reset the cluster during severalperiods of the entrained oscillation. The reset guaranteesthat the influence of the initial dynamic conditionsdisappears. Therefore the desynchronizing effect of acomposite stimulus is (practically) independent of thecluster’s initial dynamic conditions, as demonstrated inSect. 4.
4 Vulnerability to stimulation
4.1 Desynchronization
Let us compare the effect of a single pulse, a compositestimulus with pulsatile entrainment, and a compositestimulus with smooth entrainment on a cluster in thestable synchronized state, where /B ¼ uB=ð2pÞ mod 1,which is the phase of Z at the beginning of thestimulation, is varied within one cycle (Fig. 2). Thedynamics is visualized with the amplitude R of the orderparameter from (6) and with the firing densitypðtÞ ¼ nð0; tÞ; i.e., the number density of all firingneurons, where the single neuron fires whenever itsphase equals 0 mod 2p.Obviously, the single pulse causes a desynchroniza-
tion only provided it hits Z at or close to a vulnerablephase /critB � 0:38 (Fig. 2a, b). By contrast, the com-posite stimuli temporarily desynchronize the cluster no
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matter at which initial phase they are administered(Fig. 2c–f). To visualize the entrained firing in a pro-nounced way, in Fig. 2 the frequencies of the pulsatileand the smooth entrainment differ from the cluster’seigenfrequency X by 50% (xp ¼ xs ¼ 1:5X). Thesmaller the frequency mismatch, jX � xpj or jX � xsj,the more rapidly and easily a resetting entrainment isachieved. Thus, to save stimulation intensity (corre-sponding to the parameter I) and entrainment duration,
in applications the entraining frequency (xp or xs)should be chosen close to the cluster’s eigenfrequency X.
4.2 Block of resynchronization
The composite stimuli presented here desynchronize acluster irrespective of its initial dynamic state and, inparticular, irrespective of the extent of synchronization(i.e., the amplitude R from Eq. 6). For this reason thecomposite stimuli can be used to effectively block thecluster’s resynchronization. To this end the samecomposite stimulus is administered to the cluster when-ever it tends to resynchronize. This is illustrated in Fig. 3by composite stimuli with pulsatile entrainment, andworks analogously for composite stimuli with smoothentrainment. The larger the coupling strength K, themore often a composite stimulus has to be administeredto prevent the cluster from resynchronization.The amplitude of the synchronized firing in the en-
trained state crucially depends on the type of entrain-ment (pulsatile vs. smooth) and, furthermore, on thestimulation parameters. For the particular type ofsmooth entrainment considered here, the amplitude ofthe entrained firing is larger compared to the amplitudebefore stimulation (Fig. 2f). This would, of course, be adisadvantage in applications where the cluster has to beprevented from synchrony. By contrast, with a suitablechoice of the stimulation parameters of the pulsatile
Fig. 2a–f. Time course of R, the amplitude of the order parameterdefined by (6) (a, c, e), and the firing density pðtÞ ¼ nð0; tÞ (b, d, f)before, during, and after a single pulse (a, b), a composite stimulus withpulsatile entrainment (c, d), and a composite stimulus with smoothentrainment (e, f), where /B ¼ uB=ð2pÞ mod 1 is varied within onecycle. Stimulation starts at t ¼ 0; for t < 0 the cluster is in the stablesynchronized state. At the bottom of each plot single pulses areindicated by bars, whereas the time course of the strength of thesinusoidal entraining pulse is illustrated by plotting cosðxstÞ, where xsis the entraining frequency from (8). In a, b; c, d; and e, f theparameters were as in Fig. 1b; Fig. 1d, e; and Fig. 1f, g; respectively
Fig. 3a,b. Time course of the firing density p ¼ nð0; tÞ. a Twosuccessively administered composite stimuli with pulsatile entrainmentboth with identical parameters (from Fig. 1d, e). The pulses of thepulse train are modelled by SðwÞ ¼ I cosw with I ¼ 21. The firstcomposite stimulus desynchronizes the cluster, whereas the secondblocks the resynchronization. b Two successively administeredcomposite stimuli with pulsatile entrainment, where all parametersare as in a except for the intensity parameter I of the pulse train, whichnow has negative sign (I ¼ �21), and the (longer) pause betweenresetting pulse train and desynchronizing single pulse. Note thedifference in amplitude of the entrained firing. The begin and end ofthe composite stimuli are indicated by vertical lines; each single pulse isshown by a shaded region at the top of each panel
107
entrainment, the amplitude of the entrained firing isdamped. For illustration, Fig. 3a shows the effect of apulse train, where the pulses are modeled bySðwÞ ¼ I cosw with I ¼ 21, whereas Fig. 3b refers to apulse train with I ¼ �21. The amplitude of the entrainedfiring in Fig. 3b is clearly larger. To understand thisdifference we introduce a negative sign of the intensityparameter I in (1) and (3) by means of SðwjÞ ¼�I coswj ¼ I cosðwj þ cÞ with I > 0 and c ¼ p. Thetransformation /j ¼ wj þ c ð j ¼ 1; . . . ;NÞ yields theLangevin equation _//j ¼ X þ N�1PN
l¼1 Cð/j � /lÞþX ðtÞSð/jÞ þ FjðtÞ which equals (1) and, hence, producesthe same dynamics. The difference, however, is that nowthe single neuron fires whenever its phase equals/fire ¼ wfire þ c ¼ p, where wfire ¼ 0. As Z’s orbit in theentrained state is not radially symmetric (Fig. 1d),the amplitude of the entrained firing is small when theneurons fire at phase zero (Fig. 3a), whereas it is largewhen the neurons fire at phase p (Fig. 3b). In general, wecan minimize the amplitude of the entrained firingcaused by a first-order stimulus SðwÞ ¼ I cosðw þ cÞ(with I > 0) by means of an appropriate choice of c,which here is equal to 0.18.A resynchronization block cannot be achieved by
repeatedly stimulating with the same single pulse. Asingle pulse appropriate for desynchronizing the fullysynchronized cluster is too strong for a weakly syn-chronized cluster; instead of a desynchronization itcauses a synchronization (Fig. 4). Therefore it is notpossible to block the resynchronization by repeatedlyadministering the same single pulse used for desyn-chronizing the fully synchronized cluster. Even the fol-lowing modified single-pulse method would not work.First, a desynchronization of the fully synchronizedcluster is achieved with a stronger single pulse as shownin Fig. 4a. To block the resynchronization a weakersingle pulse is then administered whenever the recover-ing amplitude of the order parameter R grows back to asmall, fixed threshold R0. Note that for blocking the re-synchronization, the same pulse (with the same intensityand duration) is repeatedly applied. As shown inFig. 1b, desynchronizing the cluster means shifting Zinto the origin of the Gaussian plane, so that after thepulse Z has to be as close to 0 as possible. For Z ¼ 0
there is a phase singularity, and minimal variations ofthe stimulation parameters as well as noise – both in-evitable in an experiment – let the phase of Z vary within½0; 2p�. For this reason the recovering Z may run alonginfinitely many spirals towards its stable limit cycle.Consequently, R0 can be associated with infinitely manyvalues of /. Only one of these phase values is appro-priate for a desynchronization. Generically, however,the weaker single pulse hits the cluster at a wrong initialphase, in this way synchronizing the cluster similarly asin Fig. 4.
5 Deep brain stimulation
In PD the standard, permanent high-frequency(>100 Hz) deep brain stimulation aims at suppressingpathological, synchonized activity in particular targetareas such as the thalamic ventralis intermedius nucleusor the subthalamic nucleus (Benabid et al. 1991, 1994,2000; Blond et al. 1992). Based on the results presentedin this article I suggest a different therapy: demand-controlled deep brain stimulation with composite stim-uli. For this, the deep brain electrode is used for bothstimulation and registration of the feedback signal (thelocal field potential). A desynchronizing compositestimulus is administered only and whenever the pace-maker-like cluster becomes synchronized; put otherwise,whenever its LFP exceeds a critical value. The goal ofthis approach is to effectively block the resynchroniza-tion (Fig. 3a). As yet, no demand-controlled deep brainstimulation is used for the therapy of PD.Already in the late 1950s it was shown that parkin-
sonian tremor is entrained by periodic deep brain pul-satile electrical stimulation of the pallidum at ratessimilar to the peripheral tremor frequency (Hassler et al.1960). However, a desynchronizing composite stimulushas never been applied. There are two main reasons whydemand-controlled stimulation technique should be lessaggressive, and thus reduce side effects such as dysarthriaand dysesthesia: (i) reducing the stimulating currentreduces current spread and prevents stimulation ofneighboring areas; and (ii) the demand-controlledmethod does not simply suppress the pathological firing,
Fig. 4a,b. Two single pulses with identical pulse duration andintensity parameter I are successively administered at the same clusterphase u. As in Fig. 1a and b, the single pulses are modeled bySðwÞ ¼ I cosw with I ¼ 7. The first single pulse hits the cluster in itsfully synchronized state, whereas the second single pulse is adminis-tered to the weakly synchronized cluster. The firing density p is plottedin a, where the begin and end of pulses are indicated by vertical lines
connected by a shaded region. The corresponding trajectory of theorder parameter Z in the Gaussian plane between the end of the firstand the end of the second single pulse is shown in b. After the firstsingle pulse, Z starts spiraling towards its limit cycle. The second singlepulse is too strong for the weakly synchronized cluster, and hence Z isshifted halfway to the attractor Zstat from (7) (cf. Fig. 1a). Samenetwork parameters as in Fig. 1
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but rather it maintains an incoherent – in other words,nearly physiological (Nini et al. 1995) – firing, intersectedby periods of entrained residual synchronous firing.For this application it is important to choose appropri-ate stimulation parameters in order to minimize theduration of the entrained epochs and, in particular, theamplitude of the entrained firing (Fig. 3). Of course,the impact of these epochs of low-amplitude residualsynchrony on tremor generation has to be studiedexperimentally. For deep brain stimulation, a pulsatilesoft reset has to be chosen instead of a sinusoidal softreset, because extracellular sinusoidal stimulation at afrequency in the 5 Hz to 20 Hz range is not effective(Reilly 1998).Before a desynchronizing composite stimulation can
be performed, the critical stimulation parameters (theintensity of the single pulses, their duration, and theduration of the pauses in between) have to be deter-mined in a series of test stimuli with a calibration pro-cedure which corresponds to that of the double-pulsestimulation. Since the latter was explained in detail inTass (2001c), here I merely mention that by means ofphase-resetting curves, first, the quality of the reset hasto be tested and, second, the pause between first andsecond stimulus as well as the intensity of the secondstimulus are determined.After the calibration the demand-controlled deep
brain stimulation with composite stimuli starts. Alwaysthe same composite stimulus is applied to the samestimulation site as soon as the LFP exceeds its criticalvalue. No further time-consuming online phase or fre-quency estimation or calibration has to be performed,provided the network parameters remain sufficientlystable (see Sect. 7).
5.1 Comparison of different stimulation techniques
The mechanism by which the standard, permanenthigh-frequency deep brain stimulation suppressespathological rhythmic activity has not yet beenclarified experimentally (Ashby and Rothwell 2000;Benabid et al. 2000; Benazzouz and Hallett 2000).Likewise, up to now results of experimental tests of thenovel demand-controlled stimulation methods were notavailable. Therefore, in this section we compare thedifferent stimulation techniques by applying them tomodel (1).The demand-controlled methods aim at blocking the
resynchronization (see Sect. 4.2). In modeling studies,three stimulation techniques turned out to be mostsuitable for demand-controlled deep brain stimulation(see Sect. 1): repeated administration (i) of a doublepulse (Fig. 5a; Tass 2001a, c), (ii) of a high-frequencypulse train followed by a single pulse (Fig. 5b; Tass2001b), or (iii) of a composite stimulus with pulsatileentrainment (Fig. 3a). In contrast, permanent high-fre-quency pulse-train stimulation applied to the modelunder consideration completely stops the neurons firingdue to a high-frequency entrainment of the orderparameter (Tass 2001b; Fig. 5c). As soon as the high-
frequency stimulation ends, a particularly synchronousfiring occurs in a rebound-like manner. Thus, to sup-press the firing persistently, the periodic pulse-trainstimulation has to be applied permanently.We focus on the cumulative stimulation strength
necessary either to maintain an uncorrelated firing(Figs. 3a, 5a, b) or to suppress the firing (Fig. 5c). Thegoal of this comparison is to provide an estimate of theenergy consumption of the different demand-controlledstimulation techniques compared to the standard high-frequency stimulation. For the different stimulationtechniques shown in Figs. 3a and 5, the stimuli weremodeled by Sðw; tÞ ¼ IðtÞ cosw (cf. Eq. 3). IðtÞ is an in-tensity parameter which is constant during a single pulsebut may vary between pulses. Typically, the intensityparameter of a desynchronizing single pulse is smallercompared to that of a pulse belonging to a stronger,resetting stimulus. Since the effect of a stimulus on thesingle oscillator depends on the phase of the oscillator,we introduce
SmaxðtÞ ¼ maxfjSðw; tÞj;w 2 ½0; 2p�g ð10Þ
in order to determine the maximal stimulation strengthwithin a cycle ½0; 2p� at time t. For Sðw; tÞ ¼ IðtÞ cosw,we thus obtain SmaxðtÞ ¼ IðtÞ.The begin and end of a stimulus are denoted by sB
and sE, respectively. For a desynchronizing stimulus, sBdenotes the start of the resetting first single pulse(Fig. 5a) or the start of the resetting high- or low-frequency pulse train (Figs. 3a, 5b), whereas sE is theend of the directly following desynchronizing singlepulse. Analogously, a long high-frequency pulse trainstarts at sB and ends at sE (Fig. 5c). The mean cumulativestimulation strength during stimulation is given by
S ¼ 1
sE � sB
ZsE
sB
SmaxðtÞX ðtÞdt ð11Þ
with X ðtÞ from (4). In deep brain stimulation theintensity parameter IðtÞ corresponds to a current flowthrough the brain tissue. Hence, S corresponds to themean energy consumption during stimulation.The permanent high-frequency stimulation suppresses
the firing, so that the firing density vanishes: p ¼ 0(Fig. 5c). In contrast, the goal of the desynchronizing,demand-controlled stimulation techniques (Figs. 3a, 5a,b) is to maintain an uncorrelated firing by repeatedlyadministering a composite stimulus, so that the firingdensity p is kept close to the value belonging to a uniformdesynchronization; i.e., close to p ¼ 1=ð2pÞ. Denotingthe maximal value of the firing density in the stablesynchronized state before stimulation by pmax (seeFig. 5b), the value belonging to an 80% suppression ofpmax with respect to the uncorrelated firing is given byp80% ¼ 1=ð2pÞ þ ½pmax � 1=ð2pÞ�ð1� 80%Þ. With pmax ¼0:81, we obtain p80% ¼ 0:29. To describe how long thestimuli shown in Figs. 3a and 5 keep the firing densitybelow p80%, we introduce the following notations. Stim-ulation starts at time t0, whereas t80% is the timing pointwhen the firing density p again exceeds p80% (see Fig. 5b).
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The effective cumulative stimulation strength necessary foran 80% suppression of the firing density is
S80% ¼ 1
t80% � t0
Zt80%
t0
SmaxðtÞX ðtÞdt : ð12Þ
For the composite stimulus with soft pulsatile entrain-
ment (Fig. 3a), t80% � t0 (the denominator of Eq. 12)has to be replaced by the duration T80% during which p isactually below p80%. T80% is smaller than t80% � t0 if pexceeds p80% during the pulsatile entrainment.
S80% corresponds to the mean energy consumption
necessary to keep the firing density below p80%. In thesame way we determine Sx%, the effective stimulationstrength necessary for an x% suppression of the firingdensity. To compare the stimulation strength of thedemand-controlled methods with the stimulationstrength of the standard, high-frequency stimulation inmodel (1), we introduce the ratios of the mean cumulativestimulation strengths during stimulation as
MDP ¼SSTSDP
; MHF ¼ SSTSHF
; MSR ¼ SSTSSR
; ð13Þ
where DP stands for the double pulse from Fig. 5a, andHF denotes the high-frequency pulse train followed by asingle pulse from Fig. 5b. SR (‘‘soft reset’’) stands forthe composite stimulus with soft pulsatile entrainmentfrom Fig. 3a, and ST is the abbreviation for thestandard high-frequency stimulation from Fig. 5c. SDP,SHF, SSR, and SST are the mean cumulative stimulationstrength of DP, HF, SR, and ST, respectively. Inanalogy to (13) we introduce the ratios of the effectivecumulative stimulation strengths necessary for an 80%suppression of the firing density as
E80%DP ¼ S80%STS80%DP
; E80%HF ¼ S80%STS80%HF
; E80%SR ¼ S80%STS80%SR
: ð14Þ
Figure 5d and e displays the ratios defined by (13) and(14) which were determined for the simulations shown inFigs. 3a and 5a–c. A 100% suppression would not be arealistic goal for experimental applications. Accordingly,the comparison between the different stimulation tech-niques is performed for reasonable suppression levels. Itturns out that the demand-controlled methods areconsiderably more effective than the standard high-
frequency stimulation: E80%DP ¼ 8:63, E80%HF ¼ 7:97,E80%SR ¼ 3:38, and E70%DP ¼ 8:82, E70%HF ¼ 8:11,E70%SR ¼ 4:25 (Fig. 5e). The difference between the 70%
Fig. 5a–d. Comparison between different demand-controlled stimu-lation techniques and the standard method, where all stimuli areapplied to the same network as in Fig. 1. a Two successivelyadministered double pulses. The begin and end of a single pulse aredenoted by vertical lines connected by a shaded region. b The samehigh-frequency pulse train followed by a single pulse is applied twice.sB and sE denote the begin and end of a (composite) stimulus(downward arrows). pmax is the maximal value of the firing density inthe stable synchronized state, whereas p80% ¼ 1=ð2pÞþ½pmax � 1=ð2pÞ�ð1� 80%Þ is the 80% suppression level compared tothe uncorrelated firing p ¼ 1=ð2pÞ (horizontal arrows). Stimulationkeeps p below p80% between times t0 and t80% (upward arrows).Vertical lines connected by a shaded region indicate the begin and endof a high-frequency pulse train or of a single pulse. c A permanenthigh-frequency pulse train suppresses the collective firing. Directlyafter stimulation the cluster restarts in a rebound-like manner. Verticallines indicate the begin and end of the high-frequency stimulation. Theratios of the mean cumulative stimulation strengths during stimula-tion (MDP, MHF, and MSR) from (13) are shown in d. During stimulusadministration, our stimulation technique is milder than the standardhigh-frequency stimulation provided its value of M is greater than 1.The ratios of the effective cumulative stimulation strengths necessary
for a 70% suppression (E70%DP , E70%HF , and E70%SR , in black) and an 80%
suppression (E80%DP , E80%HF , and E80%SR , in gray) of the firing density as
defined by (14) are displayed in e (DP , double pulse; HR, high-frequency pulse train; SR soft reset). Our stimulation technique ismore efficient than the standard high-frequency stimulation providedits value of E is greater than 1. Stimulation parameters were asfollows: In a–c, all stimuli are modeled by SðwÞ ¼ I cosw. Intensityparameter I: a I ¼ 21 for the first and I ¼ 7 for the second pulse; bI ¼ 21 for the high-frequency pulse train and I ¼ 7 for the subsequentdesynchronizing pulse; c I ¼ 21 for the high-frequency pulse train.Pulse durations: a duration of the first pulse ¼ 0:5 and of the secondpulse ¼ 0:31; b duration of the pulses in the pulse train ¼ 0:02 withpauses of length ¼ 0:03 in between, duration of the desynchronizingsingle pulse ¼ 0:31; c same pulse train as in b, but without thesubsequent desynchronizing single pulse
b
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and 80% suppression levels is largest for the SR method,since during the pulsatile entrainment the peaks of p arelarger than p80% and smaller than p70%, so that T80% issmaller than t80% � t0 (see above).Especially for the demand-controlled techniques, the
effective cumulative stimulation strengths S70% and S80%
from (12) crucially depend on the time necessary toresynchronize, and thus on the ratio between couplingstrength and noise amplitude. The cluster of oscillatorsresynchronizes provided its coupling K exceeds thecritical value Kcrit ¼ D ¼ 0:4 (Tass 1999). No experi-mental data are available that would allow estimation ofappropriate values of K. Therefore, throughout thepresent study K was chosen to be clearly supercritical,namely equal to 1. This means that the cluster resyn-chronizes rather rapidly, which is more challenging forthe demand-controlled techniques. For K closer to Kcrit,the ratios from (14) are even larger than those obtainedfor K ¼ 1. In other words, the weaker the coupling thesmaller is the energy consumption of the demand-con-trolled methods.From Fig. 5e it follows that the double pulse is most
effective. However, from Fig. 5d we see that the doublepulse is the only stimulation technique with a meancumulative stimulation strength during stimulation thatis greater than that of the standard high-frequencystimulation (MDP ¼ 0:72, MHF ¼ 1:10, MSR ¼ 1:30;Fig. 5d). This means that the high-frequency pulse trainfollowed by a single pulse, and the composite stimuluswith soft pulsatile entrainment are the only methodswhich are both more efficient and milder compared tothe standard high-frequency stimulation. Concerningthe energy consumption, the high-frequency pulse trainfollowed by a single pulse is most advantageous. How-ever, avoiding a hard reset may be beneficial for clinicalapplications, as discussed in Sect. 7.
6 Sensory manipulation of visual binding
Neuronal oscillatory activity in cat (Rager and Singer1998) and human (Herrmann 2001) visual cortex can beentrained by means of a flickering visual stimulus; i.e., aperiodic train of visual single-pulse stimuli, which has afrequency xp (see Sect. 3.2) that is close to resonancefrequencies such as 10 Hz, 20 Hz, 40 Hz, and 80 Hz.This leads to a resonance-like increase of the amplitudeof the neuronal activity. Resonance phenomena of thiskind are not restricted to simple light flashes. They arefound even for complex visual stimuli such as Kanizsa-like patterns (i.e., arrangements of particularly formedangles mixed with randomly distributed angles): cerebralprocessing of Kanisza-like visual stimuli is much fasterwhen they flicker at the resonance frequency (40 Hz)than at other frequencies (Elliot and Muller 1998).Furthermore, a flicker frequency close to 40 Hz givesrise to reduced latencies of stimulus-evoked electroen-cephalographic gamma responses (Elliot et al. 2000).Instead of enhancing gamma activity by entrainment,
I suggest an opposite manipulation, which is to desyn-
chronize gamma activity by means of the combinedstimulation technique presented in this article, where xpis close to 40 Hz. In a first step this should be done withsimple flickering light flashes, where the duration of alight flash corresponds to the duration of a single pulseof the composite stimulus from Fig. 3, and the intensityof the light flash corresponds to the intensity parameterI . To investigate visual binding, more complex flickeringstimuli such as Kanisza patterns have to be used, where– similar to the light flashes – the timing and intensitysequence of the Kanisza patterns realizes a desynchro-nizing composite stimulus. One could also try to replacethe flickering soft reset (with a pulse train) by a smoothsoft reset using a sinusoidal visual stimulus with fre-quency xs from (8) close to 40 Hz. The motivation be-hind this approach is to try to block the gamma activityat least temporarily, as shown in Fig. 3, and to measurethe consequences both with respect to electrophysiologyand psychophysiology: the impact on brain activity canthen be assessed with electroencephalography andmagnetoencephalography in humans, or with electricalrecordings in animals, whereas psychophysical testingprovides estimates of the velocity of cerebral informa-tion processing.To compare the functional role of gamma activity
with that of other brain rhythms during binding, oneshould perform the desynchronizing visual stimulationwith visual patterns that both essentially require and donot require visual binding. To illustrate this approach letus consider an experiment where the composite visualstimulation is separately performed with a completetriangle (Fig. 6a) as well as with an incomplete triangle(Fig. 6b). To perceive the latter as a whole (i.e., asan unbroken triangle), all different short lines of thetriangular arrangement have to be linked up correctlyby the visual system (see Singer 1989). Accordingly,for recognizing Fig. 6b as a triangle, visual binding isessential. The experiment consists of two differentparts:
1. Entrainment by constant flickering. It has to be testedwhether a constantly flickering, periodic stimulationentrains activity in visual cortical areas. The entrain-ment has to be performed for both visual stimuli inFig. 6 separately. This part of the experiment corre-sponds to the entrainment experiments mentionedabove (i.e., Rager and Singer 1998; Elliot et al. 2000;
Fig. 6a,b. The complete (a) and incomplete (b) triangles used fordesynchronizing visual composite stimulation, as explained in the text.The incomplete triangle consists of an arrangement of short lineswhich have to be bound by the visual system in order to be perceivedas an intact triangle
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Herrmann 2001). For this, the flickering frequencyhas to be varied within a large range of frequencies,e.g., from 1 Hz to 100 Hz. The goal of this part is toidentify optimal entrainment frequencies for the twovisual stimuli from Fig. 6. A robust entrainment isnecessary for a soft reset.
2. Desynchronization with composite stimuli at optimalentrainment frequencies. The next step is to perform acomposite visual stimulation with the two differentstimuli in Fig. 6 separately. For both visual stimuli ithas to be tested whether a desynchronization can beachieved for all optimal entrainment frequencies. Tothis end the frequency of a soft reset has to be iden-tical to an optimal entrainment frequency. The goalof this part of the experiment is to selectively desyn-chronize oscillatory brain activity and to study thefunctional (i.e., electrophysiological and especiallypsychophysiological) consequences.
Comparing the entrainment and desynchronizationbehavior of the two different stimuli from Fig. 6 mayreveal brain rhythms which are exclusively necessary forvisual binding. For example, let us assume that activityin the 40-Hz range can be desynchronized by means ofcomposite stimulation with both stimuli in Fig. 6. If theperception of the complete triangle (Fig. 6a) would notbe affected by the desynchronization, whereas theperception of the incomplete triangle (Fig. 6b) wouldstrongly be retarded or weakened, the functionalrelevance of gamma activity for visual binding couldbe demonstrated.In this way it can be verified whether gamma activity
can be blocked with composite visual stimuli, and howthat correlates with function, e.g., in terms of an increaseof response latencies or an increase of error rates. In-stead of the visual patterns shown in Fig. 6, a differentpair of patterns can alternatively be chosen. The mainpoint is that perception of one of the stimuli requiresvisual binding, whereas perception of the other one doesnot require visual binding.The critical flicker fusion frequency (CFF) is the
minimal frequency at which a visual stimulus is per-ceived in a fused, steady way (Kelly 1972): for scotopicvision, maximal values of CFF lie in the 22–25 Hzrange, whereas for photopic vison the CFF cruciallydepends on stimulus parameters such as light intensityand size. The interruption of the visual stimulation oc-curring after each composite stimulus (Fig. 3) may leadto a discontinuous, nonfused perception of the stimulus.To compare effects of the visual patterns from Fig. 6presented either flickering at 40 Hz or as a compositestimulus (Fig. 3), one should exclude effects that arerelated to discontinuous perception caused by a stimuluspresented at a frequency which is repeatedly lower thanthe CFF. For this, one might alternatively modify thevisual stimulation by replacing the periods without anyvisual stimulation by a stimulation with the same visualstimulus periodically flickering at a different frequencyxf. This stimulation should be sufficiently detuned sothat it does not strongly entrain the gamma activity and,furthermore, xf should be greater than the CFF. The
timing sequence of the visual stimulation would then beas follows. A composite stimulus with a soft reset withfrequency xp is performed to desynchronize activity inthe frequency range related to xp. Directly after thecomposite stimulus, the same visual pattern is adminis-tered with a flicker frequency xf, where xf 6¼ xp. Assoon as the amplitude of the activity around xp in-creases again, the next composite stimulus is adminis-tered (again, with the same visual stimulus), which isthen followed by the xf flicker, and so on.
7 Discussion
Two composite stimulation techniques are presented inthis article which make it possible to effectively desyn-chronize a cluster of interacting phase oscillatorswithout making use of any strong stimulus. This isparticularly important for applications in biology andmedicine, since previously designed methods for effectivedesynchronization (Tass 1999, 2000, 2001a–c) essentiallyrely on a hard reset that requires a strong, abruptlyresetting stimulus. A hard reset is achieved within lessthan one period of the collective oscillation by means ofa strong single pulse (Tass 2001a, c) or a high-frequencypulse train (with an entraining frequency that isapproximately 20 times larger than the cluster’s eigen-frequency; Tass 2001b). Such a maneuver, however,might be too strong and might even injure neuronaltissue (or other biological systems). A way to avoid thisrisk is provided by the novel composite stimulationmethods which use a soft reset: during a pulsatile or asmooth periodic entrainment at a rate close to thecluster’s eigenfrequency, the influence of the initialdynamic state at the beginning of the periodic stimula-tion disappears, while the collective oscillation runsthrough several periods. After the soft reset a moderatesingle pulse follows with a constant time delay, whichdesynchronizes the cluster by hitting it in a vulnerablestate.While the desynchronizing effect of a composite
stimulus does not depend on the particular type of theentraining stimulus, the pattern – and especially theamplitude – of the entrained firing crucially do. Nomatter whether a pulsatile entrainment or a smoothentrainment is used, the desynchronization obtainedwith the composite stimulus is equally good (Fig. 2c–f).By contrast, the amplitude of the entrained firing may besmaller (Fig. 3a) or larger (Figs. 2f, 3b) compared to thefiring before stimulation. The choice of the most appro-priate type of entrainment depends on the applicationand, of course, on possible experimental restrictions. Forinstance, sinusoidal extracellular stimulation of neuraltissue at frequencies in the 5 Hz to 20 Hz range is noteffective (Reilly 1998). Thus, for extracellular deep brainstimulation, a pulsatile entrainment has to be usedinstead of a sinusoidal entrainment. Furthermore, it hasto be tested whether a variation of the stimulationparameter c in SðwÞ ¼ I cosðw þ cÞ can be performedexperimentally (see Sect. 4.2). If so, an appropriate valuefor c has to be chosen in order to minimize the amplitude
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of the entrained firing (Fig. 3). For applications it is alsoimportant that a soft reset is not restricted to a (pulsatileor smooth) 1:1 entrainment. Rather, one can also use a(pulsatile or smooth) n:m entrainment, where n and mare small integers, and n:m is the ratio between thestimulation frequency and the mean frequency of therhythm which is to be desynchronized. Hence, in anexperimental application where 1:1 entrainment is noteffective, different n:m entrainment ratios should betested.Since rhythmic activity abounds in physiology, the
presented stimulation techniques may find various ap-plications where rhythms have to be desynchronized forscientific or therapeutic purposes. Two applications aresuggested: demand-controlled deep brain stimulation(Sect. 5) and sensory manipulation of gamma activity(Sect. 6). Both applications are very promising becausepulsatile entrainment has already been demonstratedexperimentally for deep brain stimulation in PD (Hassleret al. 1960) as well as for visual stimulation of gammaactivity in cat (Rager and Singer 1998) and human(Herrmann 2001) visual cortex. However, up to nowcomposite stimulation for desynchronization has neverbeen applied. Likewise, the novel methods can also beapplied to desynchronize other brain rhythms such asalpha or beta rhythm which both can also be entrainedby flickering stimuli (Rager and Singer 1998; Herrmann2001).Smoothly moving stimuli produce sustained gamma
oscillations in different visual cortical areas that aresynchronized in phase, both in anesthetized cat (Eckhornet al. 1988; Gray and Singer 1989) and awake monkey(Kreiter and Singer 1992; Eckhorn et al. 1993). Per-turbing the smooth visual stimulation with qualitativelydifferent stimuli – namely with intermingled suddenrandom accelerations of the grating – suppresses thegamma oscillations, where with increasing amplitude ofthe random perturbations the related evoked fastresponses increase, whereas the amplitude of gammaoscillations gradually decreases (Kruse and Eckhorn1996). The suppression of gamma oscillations is, hence,intimately related to a switching between different per-cepts (Kruse and Eckhorn 1996). Compared to theapproach used by Kruse and Eckhorn (1996), thedesynchronizing composite visual stimulation techniquein Sect. 6 would enable investigation of the relationshipbetween a single percept and the extent of gamma oscil-lations without making use of additional stimuli relatedto different percepts, i.e., without switching betweendifferent percepts.For applications of the novel stimulation techniques,
it has to be taken into account that higher-order terms ofthe stimulus (3), such as SðwÞ ¼ I cosðmwÞ with m > 1,may cause an excitation of higher-order frequencycomponents with an average number density nðw; tÞ,which shows up as a higher-frequency, pronounced earlyresponse of the cluster’s firing directly after the stimulus.The mechanism behind this unwanted effect has beenstudied in detail in the context of single-pulse (Tass1999) and double-pulse stimulation (Tass 2001c). Thereit was explained how to avoid this phenomenon, namely
by suitably modifying the stimulation mechanism in away that it damps higher-order modes. The impact of astimulus on the different frequency components is reli-ably assessed by extracting these components out of theexperimetal data by means of band-pass filtering andHilbert transformation (Tass 1999).Instead of the suggested demand-controlled stimula-
tion mode (Sects. 5, 6), one can also choose a technicallymore straightforward type of stimulation, withoutfeedback control: by simply delivering a compositestimulus periodically, the unwanted synchronized firingcan also be kept down. In this case the period of stim-ulus administration has to be smaller compared to theexperimentally determined minimal resynchronizationtime. Since it is technically much easier to realize, theperiodic stimulation mode might be a relevant alterna-tive for the sensory stimulation of gamma activity(Sect. 6). For deep brain stimulation (Sect. 5), however,demand-controlled stimulation is clearly superior. Onthe one hand stimulation has to be avoided during silentperiods, which means when there is no pacemaker-likepathological rhythm. Keeping the stimulation current ata minimum improves the battery life, so that the surgicalreplacement of the generator and its battery could occurless frequently. Furthermore, minimizing the stimulationcurrent reduces the possibility of adaptive reactions ofthe stimulated network. Adaptation of the network and,in general, variations of network parameters may spoilthe stimulation outcome. Accordingly, monitoring theactivity of the target area is required to detect a dimi-nution of the stimulus action that might evolve on a longtimescale. In such a case, the stimulator has to be re-calibrated in order to maintain a strong desynchroni-zation.Based on model (1), the energy consumption of the
demand-controlled stimulation techniques was theoreti-cally estimated and compared to that of the standardhigh-frequency stimulation (Sect. 5.1). The demand-controlled techniques are clearly superior, even in thecase of rather strong coupling which gives rise to a rapidresynchronization. However, this comparison cannot bereduced to evaluating only one parameter, namely theenergy consumption. Since there are neuronal popula-tions in the target areas used for deep brain stimulationwhich are not primarily part of motor loops, other brainfunctions such as cognition are also affected by deepbrain stimulation (see Saint-Cyr et al. 2000). As yet,effects of this kind cannot be treated sufficiently bymeans of network simulations; rather, a clinical evalu-ation of the different demand-controlled stimulationtechniques is inevitable. This can only be achieved byapplying the novel techniques in patients and measuringthe benefit with respect to an attenuation of motor aswell as nonmotor symptoms.According to theoretical studies there are several ef-
fective desynchronizing stimulation techniques. Let metherefore sketch how an experimentalist selects the ap-propriate one for a given application. First, the experi-mentalist has to check whether a hard reset can, inprinciple, be obtained with a compatible stimulationintensity. To this end, in a series of test stimuli the ex-
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perimentalist has to study whether the response of thestimulated cluster is independent of its dynamic state atthe beginning of the stimulation. This criterion was ex-plained in Tass (2001a–c).If a hard reset can be applied, the experimentalist can
choose between three different variants of double-pulsestimulation (Tass 2001a, c) on the one hand, and acombined high-frequency pulse train, single-pulse stim-ulation (Tass 2001b) on the other hand. The applica-bility of the different double-pulse methods is comparedin Tass (2001c). The resetting effect of a strong singlepulse compared to that of a high-frequency pulse train(e.g., when using the same stimulating current) is prac-tically the same (Tass 2001b). Thus, for several appli-cations the combined high-frequency pulse trainsingle-pulse stimulation may be milder.If a hard reset is not feasible, the experimentalist has
to test whether a soft reset can be performed. To this endthe experimentalist measures phase and amplitude of theorder parameter Z from (6), and of higher-order fre-quency modes before and after entraining stimulation or– if there are no stimulus artifacts – continuously duringstimulation by means of the mean mutual distance dðtÞfrom (9). How to reconstruct the order parameter andhigher-order modes from the experimental data withband-pass filtering and Hilbert transformation is ex-plained in Tass (1999). At the end of the entraining,periodic stimulation phase and amplitude of the orderparameter and the higher-order modes have to beidentical, so that they no longer depend on the initialdynamic conditions. The intensity and number of en-trainment periods have to be large enough to fulfill thiscriterion.If both a hard and a soft reset are possible, the ex-
perimentalist can choose which to use whilst taking intoaccount that a soft reset will often be milder. However,the residual entrained firing during the soft reset mayspoil the functional outcome of the resynchronizationblock (Fig. 3). Furthermore, the shorter the reset theless probable fluctuations or unforeseen external influ-ences interfere with the desynchronizing effect of thestimulus.The composite stimulation techniques presented in
this article also work perfectly when applied to noisycluster states – the so-called noisy m-cluster states. Theseare complex synchronized states, where a large cluster ofoscillators breaks into m different subclusters, in each ofwhich all oscillators have (nearly) the same phase(m ¼ 2; 3; . . .; Golomb et al. 1992). Noisy clusterstates are caused by coupling terms of higher order suchas CðxÞ ¼ �Km sinðmxÞ (with Km > 0). An m-clusterstate emerges when Km exceeds its critical value mD(Tass 1999). For example, two clusters synchronized inantiphase form a two-cluster state. The order parameterof an m-cluster state is ZmðtÞ ¼
R 2p0 nðw; tÞ expðimwÞdw.
Zm runs on a limit cycle similar to Z’s limit cycle de-scribed in Sect. 2.2, and jZmj quantifies the extent ofsynchronization of the m-cluster state, where0 � jZmj � 1 for all times t. To desynchronize Zm mosteffectively (as shown in Fig. 1d–g), the stimulus shouldcontain terms of mth order such as SðwÞ ¼ I cosðmwÞ.
Note that composite stimuli are equally effective if thecoupling contains cosine terms.Although model (1) is rather simple, it nevertheless
reproduces some dynamical features observed in exper-iments with peripheral stimulation or repetitive deepbrain stimulation (for a detailed discussion, see Tass1999). Thus, one may consider the present model also asa suitable starting point for more microscopic modeling.Accordingly, the impact of bipolar pulses as well asspatially distributed synaptic coupling strengths, eigen-frequencies, and stimulation strengths on the stimula-tion effects is now being studied by us in networks ofphase oscillators and Hodgkin-Huxley neurons. Theresults will be presented in the near future. Concerningthe influence of the spatial pattern of the synaptic cou-pling strengths, it is important to stress that the basicdesynchronizing mechanism shown in Fig. 1b holds aswell for an ensemble of noninteracting phase oscillators(Tass 1996a, 1999). In other words, if the stimulationstrength is sufficiently large compared to the couplingstrength, the desynchronizing effect does not depend onthe coupling pattern (provided the stimulator is appro-priately calibrated). By contrast, the dynamics followingthe desynchronization crucially does. While the inter-acting cluster resynchronizes (Fig. 2), the noninteractingensemble remains incoherent (Tass 1996a, 1999).
Acknowledgements. I am grateful to Gereon R. Fink and PeterH. Weiss for our fruitful discussions. This study was supportedby the German–Israeli Foundation for Scientific Research andDevelopment (grant no. 667/00).
References
Alberts WW, Wright EJ, Feinstein B (1969) Cortical potentials andparkinsonian tremor. Nature 221: 670–672
Ashby P, Rothwell JC (2000) Neurophysiologic aspects of deepbrain stimulation. Neurology 55[Suppl 6]: S17–S20
Benabid AL, Pollak P, Gervason C, Hoffmann D, Gao DM,Hommel M, Perret JE, de Rougemont J (1991) Long-termsuppression of tremor by chronic stimulation of the ventralintermediate thalamic nucleus. Lancet 337: 403–406
Benabid AL, Pollak P, Gross C, Hoffmann D, Benazzouz A, GaoDM, Laurent A, Gentil M, Perret J (1994) Acute and long-termeffects of subthalamic nucleus stimulation in Parkinson’sdisease. Stereotact Funct Neurosurg 62: 76–84
Benabid AL, Koudsie A, Benazzouz A, Fraix V, Ashraf A, Le BasJF, Chabardes S, Pollak P (2000) Subthalamic stimulation forParkinson’s disease. Arch Med Res 31: 282–289
Benazzouz A, Hallett M (2000) Mechanism of action of deep brainstimulation. Neurology 55[Suppl 6]: S13–S16
Bergman H, Wichmann T, Karmon B, DeLong MR (1994) Theprimate subthalamic nucleus. II. Neuronal activity in theMPTP model of parkinsonism. J Neurophysiol 72: 507–520
Best EN (1979) Null space in the Hodgkin–Huxley equations: acritical test. Biophys J 27: 87–104
Blond S, Caparros-Lefebvre D, Parker F, Assaker R, Petit H,Guieu JD, Christiaens JL (1992) Control of tremor andinvoluntary movement disorders by chronic stereotacticstimulation of the ventral intermediate thalamic nucleus.J Neurosurg 77: 62–68
Eckhorn R, Bauer R, Jordan W, Brosch M, Kruse W, Munk M,Reitboeck HJ (1988) Coherent oscillations: a mechanism offeature linking in the visual cortex? Biol Cybern 60: 121–130
114
Eckhorn R, Frien A, Bauer R, Woelbern T, Kehr H (1993) Highfrequency 60–90 Hz oscillations in primary visual cortex ofawake monkey. Neuroreport 4: 243–246
Elble RJ, Koller WC (1990) Tremor. John Hopkins UniversityPress, Baltimore, Md.
Elliot MA, Muller HJ (1998) Synchronous information presentedin 40-Hz flicker enhances visual feature binding. Psychol Sci 9:277–283
ElliotMA,HerrmannCS,MecklingerA,MullerH (2000) The loci ofoscillatory visual-object priming: a combined electroencephalo-graphic and reaction-time study. Int J Psychophysiol 38: 225–242
Ermentrout GB, Kopell N (1991) Multiple pulse interactions andaveraging in systems of coupled neural oscillators. J Math Biol29: 195–217
Ghose GM, Freeman RD (1992) Oscillatory discharge in the visualsystem:does it have a functional role? J Neurophysiol 68: 1558–1574
Ghose GM, Maunsell J (1999) Specialized representations in visualcortex: a role for binding? Neuron 24: 79–85
Golomb D, Hansel D, Shraiman B, Sompolinsky H (1992) Clus-tering in globally coupled phase oscillators. Phys Rev A 45:3516–3530
Grannan ER, Kleinfeld D, Sompolinsky H (1993) Stimulus-dependent synchronization of neuronal assemblies. NeuralComput 4: 550–569
Gray CM, Singer W (1989) Stimulus-specific neuronal oscillationsin orientation columns of cat visual cortex. Proc Natl Acad SciUSA 86: 1698–1702
Gray CM, Engel AK, Konig P, Singer W (1992) Synchronizationof oscillatory neuronal responses in cat striate cortex: temporalproperties. Vis Neurosci 8: 337–347
Guttman R, Lewis S, Rinzel J (1980) Control of repetitive firing insquid axon membrane as a model for a neuroneoscillator.J Physiol (Lond) 305: 377–395
Haken H (1983) Advanced synergetics. Springer, Berlin HeidelbergNew York
Hansel D, Mato G, Meunier C (1993) Phase dynamics of weaklycoupled Hodgkin–Huxley neurons. Europhys Lett 23: 367–372
Hassler R, Riechert T, Mundinger F, Umbach W, Ganglberger JA(1960) Physiological observations in stereotaxic operations inextrapyramidal motor disturbances. Brain 83: 337–350
Herrmann CS (2001) Human EEG responses to 1–100 Hz flicker:resonance phenomena in visual cortex and their potential cor-relation to cognitive phenomena. Exp Brain Res 137: 346–353
Kelly DH (1972) Flicker. In: Jameson D, Hurvich LM (eds)Handbook of sensory physiology and visual psychophysics, volVII/4. Springer, Berlin Heidelberg New York, pp 271–302
Kreiter AK, Singer W (1992) Oscillatory neuronal responses in thevisual cortex of the awake macaque monkey. Eur J Neurosci 4:369–375
Kruse W, Eckhorn R (1996) Inhibition of sustained gamma oscil-lations (35–80 Hz) by fast transient responses in cat visualcortex. Proc Natl Acad Sci USA 93: 6112–6117
Kuramoto Y (1984) Chemical oscillations, waves, and turbulence.Springer, Berlin Heidelberg New York
Lamarre Y, de Montigny C, Dumont M, Weiss M (1971) Har-maline-induced rhythmic activity of cerebellar and lower brainstem neurons. Brain Res 32: 246–250
Lenz FA, Kwan HC, Martin RL, Tasker RR, Dostrovsky JO, LenzYE (1994) Single unit analysis of the human ventral thalamicnuclear group. Tremor-related activity in functionally identifiedcells. Brain 117: 531–543
Llinas R, Jahnsen H (1982) Electrophysiology of mammalianthalamic neurons in vitro. Nature 297: 406–408
Nini A, Feingold A, Slovin H, Bergman H (1995) Neurons in theglobus pallidus do not show correlated activity in the normalmonkey, but phase-locked oscillations appear in the MPTPmodel of parkinsonism. J Neurophysiol 74: 1800–1805
Pare D, Curro’Dossi R, Steriade M (1990) Neuronal basis of theparkinsonian resting tremor: a hypothesis and its implicationsfor treatment. Neuroscience 35: 217–226
Rager G, Singer W (1998) The response of cat visual cortex toflicker stimuli of variable frequency. Eur J Neurol 10: 1856–1877
Reilly JP (1998) Applied bioelectricity – from electrical stimulationto electropathology. Springer, Berlin Heidelberg New York
Riesenhuber M, Poggio T (1999) Are cortical models really boundby the ‘‘binding problem’’? Neuron 24: 87–93
Saint-Cyr JA, Trepanier LL, Kumar R, Lozano AM, Lang AE(2000) Neuropsychological consequences of chronic bilateralstimulation of the subthalamic nucleus in Parkinson’s disease.Brain 123: 2091–2108
Singer W (1989) Search for coherence: a basic principle of corticalself-organization. Concepts Neurosci 1: 1–26
Stoney SD Jr, Thompson WD, Asanuma H (1968) Excitationof pyramidal tract cells by intracortical microstimulation:effective treatment of stimulating current. J Neurophysiol 31:659–669
Tass PA (1996a) Resetting biological oscillators – a stochasticapproach. J Biol Phys 22: 27–64
Tass PA (1996b) Phase resetting associated with changes of burstshape. J Biol Phys 22: 125–155
Tass PA, Rosenblum MG, Weule J, Kurths J, Pikovsky A, Volk-mann J, Schnitzler A, Freund H-J (1998) Detection of n:mphase locking from noisy data: application to magnetoen-cephalography. Phys Rev Lett 81: 3291–3294
Tass PA (1999) Phase resetting in medicine and biology – stochasticmodelling and data analysis. Springer, Berlin Heidelberg NewYork
Tass PA (2000) Stochastic phase resetting: a theory for deep brainstimulation. Prog Theor Phys Suppl (Kyoto) 139: 301–313
Tass PA (2001a) Effective desynchronization by means of double-pulse phase resetting. Europhys Lett 53(1): 15–21
Tass PA (2001b) Effective desynchronization with a resetting pulse-train followed by a single pulse. Europhys Lett 55: 171–177
Tass PA (2001c) Desynchronizing double-pulse phase resetting andapplication to deep brain stimulation. Biol Cybern 85: 343–354
Tass PA (2002) Effective desynchronization with a stimulationtechnique based on soft phase resetting. Europhys Lett 57:164–170
Volkmann J, Joliot M, Mogilner A, Ioannides AA, Lado F, FazziniE, Ribary U, Llinas R (1996) Central motor loop oscillations inparkinsonian resting tremor revealed by magnetoencephalog-raphy. Neurology 46: 1359–1370
Winfree AT (1980) The geometry of biological time. Springer,Berlin Heidelberg New York
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