optimal number of stimulation contacts for coordinated reset … · 2017. 4. 12. · original...

ORIGINAL RESEARCH ARTICLEpublished: 22 July 2013

doi: 10.3389/fneng.2013.00005

Optimal number of stimulation contacts for coordinatedreset neuromodulationBorys Lysyansky1*, Oleksandr V. Popovych1 and Peter A. Tass1,2

1 Institute of Neuroscience and Medicine – Neuromodulation (INM-7), Research Center Juelich, Juelich, Germany2 Department of Neuromodulation, University of Cologne, Cologne, Germany

Edited by:

Laura Ballerini, University of Trieste,Italy

Reviewed by:

Alexander Dityatev, DeutschesZentrum für NeurodegenerativeErkrankungen, GermanyWalter Rocchia, Istituto Italiano diTecnologia, ItalyTjitske Heida, University of Twente,Netherlands

*Correspondence:

Borys Lysyansky, Institute ofNeuroscience and Medicine –Neuromodulation (INM-7), Research

In this computational study we investigate coordinated reset (CR) neuromodulationdesigned for an effective control of synchronization by multi-site stimulation of neuronaltarget populations. This method was suggested to effectively counteract pathologicalneuronal synchrony characteristic for several neurological disorders. We study how manystimulation sites are required for optimal CR-induced desynchronization. We found thata moderate increase of the number of stimulation sites may significantly prolong thepost-stimulation desynchronized transient after the stimulation is completely switchedoff. This can, in turn, reduce the amount of the administered stimulation current for theintermittent ON–OFF CR stimulation protocol, where time intervals with stimulation ONare recurrently followed by time intervals with stimulation OFF. In addition, we foundthat the optimal number of stimulation sites essentially depends on how strongly theadministered current decays within the neuronal tissue with increasing distance from thestimulation site. In particular, for a broad spatial stimulation profile, i.e., for a weak spatialdecay rate of the stimulation current, CR stimulation can optimally be delivered via a smallnumber of stimulation sites. Our findings may contribute to an optimization of therapeuticapplications of CR neuromodulation.

Keywords: coordinated reset stimulation, neuronal synchronization, electrical stimulation, parameter

optimization, synchronization control

1. INTRODUCTIONSynchronization plays a fundamental role in many interactingsystems (Winfree, 1980; Kuramoto, 1984; Tass, 1999; Pikovskyet al., 2001; Strogatz, 2003). However, pathological synchroniza-tion is a hallmark of some neurological disorders, e.g., Parkinson’sdisease (PD) or essential tremor (Lenz et al., 1994; Levy et al.,2002; Timmermann et al., 2003; Hammond et al., 2007; Amtageet al., 2008; Smirnov et al., 2008). Nowadays, high-frequency(HF, >100 Hz) electrical deep brain stimulation (DBS) is widelyapplied for the treatment of PD (Benabid et al., 1991; Blond et al.,1992) in patients who have inadequate therapeutic response tomedication or have intolerable side effects from it. Mechanismof HF DBS is not fully understood yet, it may significantly mod-ulate the neuronal firing by, e.g., suppressing or overacting it(Beurrier et al., 2001; Hashimoto et al., 2003; Filali et al., 2004;McIntyre et al., 2004). However, HF DBS may be ineffectiveor lead to side effects, and the clinical effect may decline withtime (Limousin et al., 1999; Kumar et al., 2003; Volkmann, 2004;Rodriguez-Oroz et al., 2005), which motivated the developmentof novel stimulation methods (Tass, 1999). They are aimed at thecontrol of undesirable neuronal synchronization, which is high-lighted by the finding that the physiological dynamics of neuronalpopulations is characterized by uncorrelated firing (Nini et al.,1995). During the last decade several desynchronizing techniquesfor DBS have been developed with the methods of non-lineardynamics (Tass, 1999, 2003a,b; Rosenblum and Pikovsky, 2004;Hauptmann et al., 2005; Popovych et al., 2005, 2006; Pyragas

et al., 2007), which provide mild, but, nevertheless, effectivemeans for the control of pathological neuronal synchronization(Tass et al., 2006). The main distinction of the novel methods incomparison to HF DBS is that coordinated reset (CR) stimulation(Tass, 2003a,b) as well as feedback techniques (Rosenblum andPikovsky, 2004; Hauptmann et al., 2005; Popovych et al., 2005;Pyragas et al., 2007) selectively counteract pathological synchro-nization of neuronal target populations and restore uncorrelatedneuronal firing.

In this paper we consider CR stimulation (Tass, 2003a,b). Itsmechanism of action is based on the phase reset of neuronaloscillations where electrical stimuli or synaptic input of sufficientstrength reset the phase of a neuron (Winfree, 1977; Best, 1979;Tass, 1999; Popovych and Tass, 2012). After the stimulation theneuronal oscillations restart from a preferred phase. Accordingto the CR stimulation protocol, the population of synchronizedneurons is stimulated via a few stimulation sites in a timelycoordinated manner. The entire neuronal population is dividedinto several sub-populations by CR stimulation where the phasesof the neuronal oscillations of the sub-populations get phase-shifted with respect to each other, and the total synchronization isreplaced by, e.g., a cluster state (Tass, 2003a,b; Lysyansky et al.,2011). Due to the pathologically strong synaptic connectivity,the entire target population runs from the cluster state througha transient characterized by pronounced desynchronization andfinally resynchronizes if left unperturbed. Accordingly, to keepthe neuronal ensemble in a desynchronized state, CR stimuli are

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Lysyansky et al. Optimal number of CR contacts

delivered intermittently (Tass, 2003a,b), for instance, by apply-ing CR in an m : n ON–OFF mode, where m cycles with CR arefollowed by n cycles without any stimulation (Lysyansky et al.,2011). Such a stimulation protocol has computationally beenfound to be effective in inducing transient desynchronizationin the stimulated neuronal ensembles (Tass, 2003a,b; Lysyanskyet al., 2011).

The desynchronizing effect of CR stimulation has beenanalyzed in several modeling papers (Tass, 2003a,b; Tass andMajtanik, 2006; Hauptmann and Tass, 2007, 2009; Tass andHauptmann, 2007, 2009; Popovych and Tass, 2012). The CR-induced long-lasting desynchronization has experimentally beenconfirmed in an in vitro study in rat hippocampal slices (Tasset al., 2009). In addition, CR neuromodulation was testedin the 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP)-treated macaque monkeys, the best characterized model ofexperimental parkinsonism, played an important role in thedevelopment of stereotactic treatments of PD (Bergman et al.,1990; Benazzouz et al., 1993; Nini et al., 1995; Hammond et al.,2007). CR neuromodulation of the subthalamic nucleus (STN)turned out to have sustained long-lasting after-effects on motorfunction in MPTP monkeys (Tass et al., 2012b). In contrast, long-lasting after-effects were not observed with classical HF DBS (Tasset al., 2012b). The clinical and preclinical studies performed so farhave not revealed adverse effects of CR neuromodulation (Tasset al., 2012a,b).

In clinical applications of CR neuromodulation optimal ther-apeutic effects should be achieved with a minimal amount ofstimulation current, which can prevent from unnecessarily strongperturbation of physiological neuronal activity in the target pop-ulation and spread of the stimulation current in the neuronal tis-sue and affecting, in such a way, neighboring regions. This can, inturn, prevent from possible side effects. In addition, high stimula-tion amplitudes lead to faster battery depletion of the stimulator,and prolonging battery life is a desirable outcome of parame-ter optimization, too. Apart from the stimulation strength, thenumber of stimulation contacts (for a given electrode topol-ogy, e.g., linear vs. circular alignment of stimulation contacts)is another stimulation parameter that is central to the outcomeof CR neuromodulation. Accordingly, in this study we considerthe impact of the number of stimulation sites on the desyn-chronizing effect of CR stimulation. This problem is stronglyconnected to the clinically important problem of the optimaldesign of DBS electrodes. For example, in a computational studyButson and McIntyre (2006) suggested improvements of the con-tact form by using finite-element models of the electrode andsurrounding medium. In this way the volume of the neuronal tis-sue activated by DBS stimulation can effectively be controlled,which depends on many factors such as electrode impedanceand capacitance, voltage drop at the electrode-tissue interface,and tissue properties (Butson et al., 2006; Chaturvedi et al.,2010).

In the present paper we show that for a linear arrangementof stimulation sites the optimal number of stimulation sitesfor CR stimulation essentially depends on the signal decay ratewith distance from the stimulation site. On the one hand, fora rapid decay of the stimulation signal in the neuronal tissue

the desynchronizing effect of CR stimulation can be improvedby additional stimulation sites. On the other hand, CR stimula-tion of a medium with a broad spatial signal spread is optimallydelivered via a small number of stimulation sites. The results areillustrated on three different oscillatory ensembles: the Kuramotosystem of coupled phase oscillators, a population of FitzHugh–Nagumo (FHN) spiking neurons coupled via excitatory chemicalsynapses, and a network of synaptically coupled adaptive expo-nential integrate-and-fire bursting neurons. We also discuss theapplication of our findings to the clinically used Medtronic leadsno. 3387 and no. 3389 DBS electrodes.

2. MATERIALS AND METHODS2.1. COUPLED PHASE OSCILLATORSIn this section we introduce a network of coupled phase oscilla-tors as a simple model of a neuronal ensemble subjected to CRstimulation. We consider the well-known Kuramoto system ofN all-to-all (or globally) coupled phase oscillators, where eachoscillator receives the coupling from the other N − 1 oscillators,and which reflects several main synchronization properties of anumber of oscillatory networks (Kuramoto, 1984; Acebrón et al.,2005),

θj = ωj + C

N

N∑k = 1

sin(θk − θj) + Sj(t), j = 1, 2, . . . , N, (1)

where θj(t) are the phases, ωj are the natural frequencies, C isthe coupling strength in the ensemble, and Sj(t) is the stim-ulation signal which will be defined below. We consider N =400 oscillators and coupling C = 0.1. The natural frequenciesωj are Gaussian distributed with mean ωmean = π and standarddeviation σω = 0.02. For numerical integration we use a Runge–Kutta method of order 5(4) with adaptive step size (Hairer et al.,1993).

Equation (1) governs the time-dependent dynamics of theoscillator phases θj(t), t ≥ 0, which, in the absence of stimula-tion (Sj(t) ≡ 0), have been found to spontaneously synchronizefor sufficiently strong coupling (Kuramoto, 1984; Strogatz, 2000).In that case, a large group of oscillators starts to oscillate at thesame frequency, and their phases get narrowly distributed byforming a single phase cluster, in this way constituting an in-phase synchronization. The onset of in-phase synchronization inensemble (Equation 1) is reflected by an increasing amplitudeof the mean field, which is given by the first order parameter

R1 defined as Rm =∣∣∣N−1 ∑N

j = 1 exp(imθj)

∣∣∣ for m = 1 (Haken,

1983; Kuramoto, 1984; Tass, 1999). Accordingly, large values ofR1, 0 ≤ R1 ≤ 1, close to 1 correspond to in-phase synchroniza-tion of oscillators (Equation 1), whereas a desynchronized state,where the phases θj are uniformly distributed on the unit circle,is indicated by small values of all order parameters Rm, m ≥ 1.For example, the time-averaged first order parameter 〈R1〉 ≈ 0.98for the considered coupling strength and distribution of the nat-ural frequencies. The above order parameters will thus be usedbelow to characterize the extent of synchronization in ensem-ble (Equation 1). The order parameters Rm can also reflect theformation of symmetric cluster states in ensemble (Equation 1),






where the phases θj (mod 2π) are divided into a few synchro-nized groups (clusters) of equal size being equidistantly spacedon a circle of length 2π. Such a cluster state comprising l clustersis characterized by large values of Rl combined with small valuesof Rm, 1 ≤ m < l (see, for example, the values of R1 and R4 inFigure 2 below for a CR-induced 4-cluster state).

We note here that since model (Equation 1) is dimensionless,all parameters are also considered dimensionless and will, thus, bemeasured and illustrated in figures below in arbitrary units. Weare, however, able to compare the obtained results with realisticsetups and provide a mapping between the parameter space of themodel and experimentally measured quantities (see section 4).

2.2. CR STIMULATIONCR stimulation is delivered to the phase ensemble (Equation 1)in the following setup: The oscillators are assumed to be equidis-tantly arranged on a 1D lattice of length L = 10 with lat-tice coordinates xj = (j − 1)L/(N − 1), j = 1, N. The stimula-tion signals are sequentially delivered via Ns stimulation sites,which are equidistantly spaced within the stimulated ensem-bles with coordinates ck = (k − 1

2 )L/Ns, k = 1, Ns. The controlof system (Equation 1) by CR stimulation of strength I ismodeled by the following stimulation term (Tass, 2003a,b) inEquation (1):

Sj(t) = INs∑

k = 1

D(xj, k)ρk(t)P(t) cos θj, (2)

where P(t) is a HF pulse train of unit amplitude, and ρk(t) arethe indicator functions such that ρk(t) = 1 if the kth stimulationsite is active and ρk(t) = 0 otherwise. Functions D(xj, k) definethe spatial spread of the stimulation signals in the target pop-ulation such that the impact of the stimulation signal deliveredvia the kth stimulating site on the jth oscillator depends on thedistance |xj − ck| between the oscillator and the stimulation site.Following Richardson et al. (2003) we consider a quadratic spatialprofile of the current spread

D(xj, k) = 1

1 + (xj − ck)2/σ2

, (3)

where σ defines the spatial decay rate of the stimulation current(Figure 1A).

The CR stimulation signals are short HF pulse trains(i.e., bursts with an intra-burst frequency as for standard HFDBS) sequentially delivered via different stimulation sites (Tass,2003a,b; Tass et al., 2012b). To define the stimulation signals inour models, we consider a long sequence of rectangular pulsesof unit amplitude P(t) with the pulse period Tp = 0.025 (i.e., 40pulses per time unit) and pulse width Tp/2 = 0.0125. Then thestimulation signal delivered to the neurons via the kth stimulationsite during its active period reads ρk(t)P(t), where ρk(t) controlsthe switching on and off of the kth stimulation site as definedabove. During each stimulation cycle of length T, all Ns stimu-lation sites are sequentially activated delivering an HF pulse trainof length T/Ns , respectively (Figure 1B). For the phase oscillators(Equation 1) the length of the CR stimulation cycle is considered

0 2.5 5 7.5 10

D(x,1) D(x,2) D(x,3) D(x,4)

c1

c2

c3

c4

A

B

0 2 4 6

T

Stim

ulat

ion

sign

als

Time

1

2

3

4

FIGURE 1 | (A) Schematic illustration of the spatial profiles D(x, k), k = 1, 4from Equations (2, 3) of the current spread in the stimulated neuronalpopulation for the case of Ns = 4 stimulation sites and σ = 0.5. Stimulationsites are depicted by filled circles. (B) Stimulation signals ρk (t)P(t),k = 1, 4, see Equation (2), for the 4-site CR stimulation protocol are shownby the same color as the corresponding stimulation sites from plot (A).

T = 2 which equals the mean period of the stimulation-freesynchronized phase ensemble.

2.3. SPIKING NEURONAL OSCILLATORSIn the previous section 2.1 we introduced the phase ensemble(Equation 1) controlled by CR stimulation. For applications, it isimportant to reveal whether the results for this model presentedin sections 3.1 and 3.2 are robust and generic enough to be validfor more realistic neuronal models. Here we present a networkof FHN (FitzHugh, 1961; Nagumo et al., 1962) spiking neuronsinteracting via excitatory chemical synapses,

vj = vj − 1

3v3

j − wj + 1 + I(syn)

j + I(stim)j ,

wj = εj(vj + 0.7 − 0.8wj),

sj = 2(1 − sj)

1 + exp(−10vj)− sj, j = 1, 2, . . . , N. (4)

The variable vj models the membrane potential of a single neu-ron, wj is a recovery variable, and the parameter εj determinesthe natural spiking frequency of a neuron (number of spikes pertime unit). The synaptic coupling among the neurons is real-ized via post-synaptic potentials sj triggered by spikes of neuronj (Terman, 2005; Izhikevich, 2007), which are modeled by addi-tional differential equations for sj, see Terman et al. (2002) and

Terman (2005). Then the synaptic current I(syn)

j in Equation (4)reads

I(syn)

j = C(V − vj)1

N

N∑k = 1

sk, (5)

where V is a reversal potential, taken as V = 2 to realize an exci-tatory coupling, and C defines the coupling strength. We consider






N = 400 neurons, coupling strength C = 0.11, and parameters εj

are Gaussian distributed with mean 0.08 and standard deviation0.002.

In order to quantify the extent of synchronization in thisnetwork, we use the order parameters Rm, m ≥ 1, defined forthe phase ensemble, see section 2.1, where phase θj of neu-ron j is approximated based on its spike timings tj, k: θj(t) =2π(t − tj, k)/(tj, k + 1 − tj, k) + 2πk, tj, k ≤ t < tj, k + 1, k = 0,

1, 2, . . . (Pikovsky et al., 2001). For example, the time-averagedfirst order parameter 〈R1〉 ≈ 0.96 for the considered couplingstrength and distribution of parameters εj.

The stimulation current I(stim)j in Equation (4) of CR stimula-

tion is given by the following expression:

I(stim)j = I

Ns∑k = 1

D(xj, k)ρk(t)P(t), (6)

where the functions D, ρk, and P are the same, as defined insection 2.2 for the ensemble of phase oscillators. We use thestimulation period T = 38 of CR stimulation, which approx-imates the period of the intrinsic spiking dynamics of theneuronal ensemble (Equations 4, 5). The period of the high-frequency pulse train P(t) is taken Tp = 0.5, and the pulse widthTp/2 = 0.25.

2.4. BURSTING NEURONAL OSCILLATORSSynchronized firing of bursting neurons has beenreported in basal ganglia of MPTP-treated monkeys(Bergman et al., 1994, 1998). In this section we present amodel network of coupled adaptive exponential integrate-and-fire (aEIF) bursting neurons (Brette and Gerstner, 2005; Naudet al., 2008; Touboul and Brette, 2008). The model is given bythe following equations for the membrane potentials Vj andadaptation currents wj:

CVj = −gL(Vj − EL) + gL�T exp

(Vj − VT

�T

)− wj + I

(syn)

j

+ I(stim)j + Ij,

wj = (a(Vj − EL) − wj)/τw, j = 1, 2, . . . , N. (7)

The jth oscillator emits a spike if its membrane potential Vj

overcomes some threshold, here we set it to −25 (mV). At thismoment the variables (Vj, wj) are instantaneously reset to thevalues:

Vj → Vr,

wj → wj + b.

The parameters of the model are gL = 30 nS, EL = −70.6 mV,VT = −50.4 mV, �T = 2 mV, τw = 40 ms, a = 4 nS, b = 80 pA,Vr = −47.2 mV, and C = 281 pF. The values Ij are randomlychosen according to a Gaussian distribution with mean 780 andσI = 1.0 pA. These parameters imply a bursting mode of theoscillators (Touboul and Brette, 2008) with period ≈ 70 ms. The

synaptic coupling I(syn)

j is given by

I(syn)

j = K(Vr.p. − Vj)1

N

N∑k = 1

α(t − t(sp)

k ),

α(x) = 4x exp(−4x),

where K is the coupling strength, t(sp)

k is the last spike of the kthoscillator, and Vr.p. = −20 mV is a reversal potential. We con-sider K = 12 nS such that the time-averaged first order parameter〈R1〉 ≈ 0.92 calculated as for the FHN ensemble except that thetimings tj, k stand for the onsets of bursts. The stimulation current

I(stim)

j of CR stimulation is considered in the form (Equation 6)

with parameter Tp = 2 ms. The stimulation period T = 70 ms,and the number of neurons in the ensemble is taken N = 200.

3. RESULTS3.1. EFFECTS OF CONTINUOUS CR STIMULATION OF PHASE

OSCILLATORSThe desynchronizing impact of CR stimulation on the controlledoscillators depends on the stimulation parameters mentionedabove, see Lysyansky et al. (2011) where, in particular, the role ofthe stimulation strength I and the current decay rate σ was investi-gated. In the present work the main attention is paid to the impactof the number of stimulating sites Ns on desynchronizing effect.CR stimulation can be administered either continuously, wherethe stimulation cycles described above of length T are applied oneafter another (without interruption), or in an intermittent way,where a few stimulation cycles are repeatedly followed by a fewcycles without stimulation. In this section we consider the formerstimulation protocol.

Examples of the time courses of the order parameters R1 andR4 before, during, and after 4-site continuous CR stimulationare illustrated in Figure 2A. CR stimulation is administered toan ensemble of strongly synchronized oscillators (Equation 1)(switched on at t = 400) where, because of the strong enoughcoupling, the phases are narrowly distributed (Figure 2C), andthe first order parameter attains a large value R1 ≈ 0.98 inthe pre-stimulus interval (Figure 2A for t < 400). After a shorttransient the stimulation suppresses the in-phase synchroniza-tion marked by small values of R1 (Figure 2A, black curve for400 < t < 700). We found that during the stimulation the orderparameters slightly fluctuate around their mean values 〈R1〉 ≈0.07, 〈R2〉 ≈ 0.13, 〈R3〉 ≈ 0.17, and 〈R4〉 ≈ 0.55 (Figure 2A, onlyR1 and R4 are depicted by black and red curves, respectively).Therefore, continuous CR stimulation administered via 4 stim-ulation sites can reliably induce a 4-cluster state such that thephases are grouped into four clusters equidistantly spaced onthe unit circle (Figure 2D). The emergence of the cluster stateis reflected by large values of R4 combined with small valuesof R1, R2, and R3, see section 2.1 and Lysyansky et al. (2011).After the stimulation is switched off (at t = 700) the oscillatoryensemble transiently relaxes from the stimulation-induced clus-ter state to a desynchronized state where the phases get nearlyuniformly distributed (Figure 2E), and which is characterized bysmall values of both order parameters R1 and R4 (Figure 2A).






Then the desynchronization is followed by a resynchroniza-tion due to the persistent strong coupling. The phases of thestimulation-free oscillators form a single cluster (Figure 2F) andapproach the original strongly synchronized state (Figure 2C),and the order parameters increase (Figure 2A for t > 800). Thediscussed cluster state induced by the continuous CR stimulationis a robust phenomenon (Lysyansky et al., 2011). For example,for the considered 4-site CR stimulation there exists a rangeof the stimulation strength I where the time-averaged first andfourth order parameters 〈R1〉 and 〈R4〉 attain small and large val-ues, respectively, which is characteristic for a four-cluster state,see Figure 2B. The minimal value of 〈R1〉 obtained for the opti-mal stimulation strength Iopt is depicted by the blue circle inFigure 2B.

300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1

R1

R4

Time

Ord

er p

aram

eter

s CR stimulation

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

<R1>

<R4>

I

Ord

er p

aram

eter

s

Iopt

A

B

DC

FE

FIGURE 2 | (A) Time courses of the order parameters R1 and R4 for thephase oscillators (Equation 1) controlled by 4-site CR stimulation(Equation 2) administered to synchronized oscillators in the time intervalt ∈ [400,700]. (B) Time-averaged (over 400 stimulation periods T ) orderparameters 〈R1〉 and 〈R4〉 vs. stimulation strength I. Blue circle depicts theminimal value of 〈R1〉 achieved for the optimal stimulation strengthIopt = 6.25, which is taken for the simulation in plot (A). (C–F) Snapshots ofthe phase distribution density histograms at the times indicated in theplots. Current decay rate σ = 0.5 in Equation (2).

As mentioned in the Introduction, in this study we estimatethe desynchronizing impact of CR stimulation for different num-bers of stimulation sites Ns. For this the stimulation strength Iis varied in some interval [0, Imax] whereas the other parametersremain fixed. The minimal (optimal) value 〈R1〉opt of the time-averaged first order parameter 〈R1〉 achieved in the consideredinterval of I (see Figure 2B) is then used to quantify the opti-mal effect of CR stimulation for a given number of stimulationsites Ns. As Ns increases, the optimal values 〈R1〉opt may demon-strate different behaviors depending on the spatial decay rate σ ofthe stimulation current, see Figure 3. 〈R1〉opt saturates for largeNs, and additional stimulation sites do not further improve theoptimal effect of CR stimulation. The saturation levels of 〈R1〉opt

decisively depend on the values of σ, where small (large) σ impliessmall (large) values of 〈R1〉opt. Moreover, small σ (�0.5) ensuresthat the increasing number of stimulation sites Ns up to ≈5–10sites improves the stimulation outcome as compared to the stim-ulation delivered via Ns ≈2–3 sites. In contrast, large σ (�1.25)implies that Ns = 2 sites is the most preferable choice, and largerNs leads to larger values of 〈R1〉opt and to a suboptimal desyn-chronizing effect of continuous CR stimulation. For intermediatevalues of σ (≈0.75–1.00) 〈R1〉opt does not significantly changewith varying number of stimulation sites.

In summary, if the properties of the neuronal tissue imply aspatially selective stimulation profile, corresponding to a smallσ, additional stimulation sites can improve the desynchronizingeffect of CR stimulation. In contrast, a small number of stimula-tion sites is the best choice for a broad profile of the spatial currentdecay (i.e., for large σ).

3.2. EFFECTS OF INTERMITTENT CR STIMULATION OF PHASEOSCILLATORS

The suggested protocol of CR stimulation (Tass, 2003a,b) impliesan intermittent (ON–OFF) administration of the stimulation,where several cycles of CR stimulation (ON-cycles) are followed

2 3 4 5 7 10 12 15 17 20 22 250

0.1

0.2

0.3σ=0.25σ=0.50σ=0.75σ=1.00σ=1.25σ=1.50σ=2.00σ=2.50

Ns

<R1>

opt

FIGURE 3 | Minimal values 〈R1〉opt of the time-averaged first order

parameter 〈R1〉 of the phase ensemble (Equation 1) controlled by

continuous CR stimulation vs. the number of stimulation sites Ns for

different values of the parameter σ in Equation (3) as indicated in the

legend. The stimulation strength I was varied in the interval [0, 60], i.e.,Imax = 60. Other parameters as in Figure 2.






by a few rest cycles without stimulation (OFF-cycles). Such anintermittent stimulation allows the oscillators to freely evolveduring the OFF-cycles according to their intrinsic unperturbeddynamics. During the post-stimulus transient (OFF cycles) theoscillatory ensemble (Equation 1) relaxes into a desynchronizedstate followed by resynchronization, both phenomena (transientdesynchronization and subsequent resynchronization) being aconsequence of the persisting strong coupling in the network(Tass, 2003a,b). This effect is illustrated in Figure 2A, where thepost-stimulus desynchronization is reflected by small values of theorder parameters. Intermittent stimulation can also decrease theamount of current delivered to the target tissue. This, in turn, canminimize the spread of the stimulation to the neighboring areasand, hence, prevent from possible side effects. In networks withspike timing-dependent plasticity (STDP) (Markram et al., 1997;Bi and Poo, 1998) intermittent CR stimulation causes an anti-kindling effect (Tass and Majtanik, 2006; Hauptmann and Tass,2007). Anti-kindling is a desynchronization-induced unlearningof pathologically upregulated synaptic connectivity and, in turn,synchrony, i.e., a stabilization of a desynchronized activity ina network, which persists even if the stimulation is completelyswitched off (Tass and Majtanik, 2006; Hauptmann and Tass,2009). The time scale of STDP is essentially larger than the m :n timing of the intermittent stimulation and has an order oftens or hundreds periods of the oscillation (Bi and Poo, 1998;Hauptmann and Tass, 2009). In this paper we investigate theimpact of CR stimulation on oscillatory networks without STDP.

To quantify the desynchronizing effect of intermittent CRstimulation, we use an approach suggested in Lysyansky et al.(2011). In each rest interval Ik we evaluate the maximal value

0 10 20 30 40Time

Stim

ulat

ion

sign

als

rest

inte

rval

rest

inte

rval

rest

inte

rval

rest

inte

rval

0 10 20 30 400

0.1

0.2

r1r2

r3 r4

Time

B

A

R1

FIGURE 4 | (A) Schematic illustration of the intermittent m : n ON–OFF CRstimulation with m = 2 and n = 3 stimulation and rest cycles of length T ,respectively. Number of stimulation sites Ns = 4. (B) Dynamics of the firstorder parameter R1 for the phase ensemble (Equation 1) subjected to theintermittent CR stimulation. Red circles depict the maximal values rk of theorder parameter achieved in the corresponding rest intervals Ik.Stimulation strength I = 10 and σ = 0.5. Other parameters as in Figure 2.

of the first order parameter such that a set of the maximalvalues rk = max

t ∈Ik

R1(t), k = 1, 2, . . . is calculated. For example,

during the rest intervals the first order parameter R1(t) canincrease and reach its maximal values rk at the end of the OFFcycles (Figure 4B, red circles). The mean value of rk, i.e., 〈r〉 =N−1

rp

∑Nrp

k = 1 rk, where Nrp is the number of rest periods, is thenused to estimate the effect of the intermittent ON–OFF CR stim-ulation. In what follows we denote by m and n the number of theON and OFF cycles of length T of the interchanging stimulationand rest time intervals, respectively, where T is the stimulationperiod (Figure 4A). If m and n can attain non-integer values, aninteresting anti-resonance-like behavior of 〈r〉 vs. m and n can beobserved (Lysyansky et al., 2011). In the present paper we con-sider integer values of m and n only and analyze the impact of thenumber of stimulation sites Ns on the optimal value of 〈r〉. Wealso vary the stimulation strength I as well as the stimulation tim-ing, i.e., the values of m and n in order to find optimal parametersfor the intermittent CR stimulation.

In Figure 5A we illustrate the dynamics of 〈r〉 vs. the stim-ulation intensity I for fixed number m = 3 of the ON cyclesand different numbers n of the OFF cycles. As in the case ofthe continuous stimulation protocol (Figure 2B), there exists an

0 10 20 300

0.2

0.4

0.6

0.8

1m:n=3:1m:n=3:2m:n=3:4m:n=3:6m:n=3:10m:n=3:15

I

<r>

A

BIopt

0 2 3 4 5 7 10 12 15 17 20 22 250

0.25

0.5

0.75 m:n=3:1m:n=3:2m:n=3:4m:n=3:6

Ns

<r>opt

m:n=3:15

m:n=3:10

FIGURE 5 | (A)


Behavior of the order parameter 〈r〉 [averaged maxima ofR1(t) of the phase ensemble (Equation 1) during the rest intervals, see textfor definition] vs. the stimulation strength I of the intermittent CRstimulation for fixed m = 3 and different numbers n of the stimulation andrest cycles of length T, as indicated in the legend. Red circles indicate theminimal values 〈r〉opt of the order parameter 〈r〉 obtained for thecorresponding optimal stimulation intensities Iopt. (B) 〈r〉opt vs. the numberof stimulation sites Ns for different n indicated in the legend. The dashedhorizontal line depicts the threshold 〈r〉opt = 0.5, and the dashed boxindicates the number of sites Ns = 4 used in plot (A). σ = 1.0, the numberof the rest periods used for calculations Nrp = 400, and other parametersas in Figure 3.





optimal stimulation strength Iopt = Iopt(m, n), where 〈r〉 attainsa minimal value 〈r〉opt (Figure 5A, red circles). Increasing nresults in larger values of 〈r〉 and 〈r〉opt since longer rest inter-vals imply more time for resynchronization and, hence, largermaximal values of R1 occurring during the rest intervals. Wefollow the optimal values 〈r〉opt as the number of stimula-tion sites Ns increases and find that 〈r〉opt saturates if Ns getslarge enough (Figure 5B). This phenomenon is tightly relatedto the saturation observed for the continuous CR stimulation(Figure 3).

Since we are interested in the longest possible post-stimulusdesynchronized transient as discussed above, we are lookingfor the stimulation parameters leading to the maximal admis-sible length nmax of the rest intervals. For this we first inves-tigate the dynamics of 〈r〉opt induced by the intermittent CRstimulation vs. the length of the rest intervals n for differentnumbers of stimulation sites Ns, see Figure 6. As expected (seealso Figure 5A), 〈r〉opt = 〈r〉opt(n) increases as the rest intervalsget longer. At some n = nmax + 1, 〈r〉opt exceeds the predefinedthreshold 〈r〉opt = 0.5 (Figure 6, dashed line). We thus considerthe value n = nmax (Figure 6, empty circles) as the maximaladmissible length of the rest intervals for the stimulation strengthI ∈ [0, Imax]. nmax can thus serve as a criterion for an optimalparameter choice for the intermittent CR stimulation since largenmax guarantees long stimulation-free OFF intervals. We foundthat an increasing number of stimulation sites Ns can prolong theadmissible rest periods, i.e., nmax increases, and thus improve thedesynchronizing impact of the intermittent CR stimulation if thelatter is spatially rather selective, i.e., the stimulation current isnarrowly distributed around the stimulation site within the stim-ulated population (Figure 6A for σ = 0.5). For a broad spatial

0 5 10 15 200

0.25

0.5

0.75N

s=2

Ns=3

Ns=4

Ns=15

<r>opt

A

B

nmax

nmax

nmax

n (off cycles)

0 5 10 15 200

0.25

0.5

0.75

Ns=2

Ns=3

Ns=4

Ns=15

n (off cycles)

<r>opt

nmax

nmax

FIGURE 6 | Behavior of 〈r〉opt of the phase ensemble (Equation 1)

stimulated by the intermittent CR stimulation vs. the number n of

OFF cycles in the rest intervals for (A) σ = 0.5 and (B) σ = 2.0 in

Equation (3) and for different numbers of stimulation sites Ns as

indicated in the legends. Empty circles indicate the maximal admissiblelength nmax of the rest interval for a given Ns for which 〈r〉opt stillremains below or equal to 0.5. Number of the ON cycles m = 3, andother parameters as in Figure 3.

spread of the stimulation current the situation, however, is exactlyopposite, i.e., nmax decreases for larger Ns (Figure 6B for σ = 2.0)such that the longest rest periods can be achieved for the smallestnumber of stimulation sites Ns = 2.

The minimal values 〈r〉opt of the first order parameter illus-trated in Figure 6 are attained at the optimal stimulationstrengths I = Iopt ∈ [0, Imax] (Figure 5A) shown in Figure 7 vs.the number n of the OFF cycles in the rest intervals. Accordingly,one has to increase the stimulation strength in order to reachthe possible minimal value of the order parameter 〈r〉opt duringthe longer post-stimulus rest periods if the number of stimu-lation sites Ns is fixed. Nevertheless, the optimal intermittentCR stimulation with the maximal admissible length nmax ofthe rest periods (Figure 6) can be realized for smaller optimalstimulation strength Iopt if the number of stimulation sites isincreased (Figure 7, empty circles). Note, this effect is well pro-nounced for small σ (Figure 7A) in contrast to the case of large σ

(Figure 7B).We have shown above that the desynchronizing impact of the

continuous as well as the intermittent CR stimulation essentiallydepends on the spatial decay rate of the stimulation current σ asthe number of stimulation sites varies (Figures 3, 6). We summa-rize our findings in Figure 8. If the number of stimulation sitesNs increases, the maximal length nmax of the admissible rest peri-ods can either increase or decrease depending on the values ofσ, see Figure 8A. The spatially selective stimulation with smallσ (Figure 8A, black curve for σ = 0.5) implies relatively shortrest intervals for the stimulation delivered via small number ofsites. Additional stimulation sites strongly elongate the admissiblerest intervals, which, however, saturate for large Ns. In contrast,

0 5 10 15 200

20

40

60 Ns=2

Ns=3

Ns=4

Ns=15I

opt

A

n (off cycles)

0 1 3 5 7 9 11 131

3

5

7

9

Ns=2

Ns=3

Ns=4

Ns=15

n (off cycles)

Iopt

B

FIGURE 7 | Behavior of the optimal stimulation strength Iopt ∈ [0, 60]providing the minimal values 〈r〉opt shown in Figure 6 vs. the

number n of OFF cycles in the rest intervals for (A) σ = 0.5 and (B)

σ = 2.0 in Equation (3) and for different numbers of the stimulation

sites Ns as indicated in the legend. Empty circles indicate the optimalstimulation strength Iopt at which the maximal admissible length nmax ofthe rest interval can be achieved for a given Ns. Other parameters as inFigure 6.






the intermittent CR stimulation with a broad spatial spread ofthe stimulation signal (large σ) admits the longest rest periodsfor a small number of stimulation sites (Figure 8A, red curvefor σ = 2). Adding further stimulation sites can only worsen thestimulation outcome in this case.

As follows from Figures 5A, 7 the optimal value 〈r〉opt isattained for larger optimal stimulation strength Iopt if the numbern of the OFF cycles increases. It is thus important to clarify howmuch stimulation an individual oscillator receives for the optimalintermittent CR stimulation as in Figure 8A. We therefore calcu-late the effective amount of the administered stimulation Ieff =0.5I

m

m + n

1

Ns × N

Ns∑k = 1

N∑j = 1

D(xj,k) which is the time and ensem-

ble average of the stimulation signal (Equation 2). In Figure 8Bthe values of Ieff are plotted vs. the number of stimulation sitesNs for the maximal admissible length of the rest interval nmax

from Figure 8A. For small σ (spatially selective stimulation) thedesynchronizing effect of the intermittent CR stimulation is sig-nificantly improved by increasing Ns, where the length of theadmissible rest periods is increased (Figure 8A, black curve), andthe amount of required stimulation is decreased (Figure 8B, blackcurve). For the large σ the situation is opposite, see Figures 8A,B(red curves). However, due to the saturation effect, the improve-ment of the stimulation impact for the considered small σ = 0.5becomes less effective if the number of stimulation sites getslarger than Ns ≈ 5 − 7 sites (Figures 8A,B, black curves). It isthus unreasonable to strongly increase the number of stimulationsites, which can either induce no significant improvements of thedesynchronizing impact of CR stimulation, or even worsen it ifthe spatial stimulation profile is broad enough.

2 3 4 5 6 7 1010 12 15 20

5

10

15

20σ=0.5σ=2.0

nmax

A

Ns

2 3 4 5 6 7 10 12 15 200

0.2

0.4

0.6

0.8σ=0.5σ=2.0

Ns

3:11

3:14

3:183:19 3:19

3:19 3:20 3:203:20

Ieff 3:9

3:6

3:6

3:5 3:53:5 3:5 3:5

3:5

B

3:5

3:20

FIGURE 8 | (A) Maximal admissible number nmax of OFF cycles and (B)

effective amount of the stimulation Ieff received on average by a singleoscillator of the phase ensemble (Equation 1) during the intermittent CRstimulation vs. the number of stimulation sites Ns for σ = 0.5 (black curves)and σ = 2 (red curves). Other parameters as in Figure 6.

Since the best intermittent CR stimulation is leading to thelongest OFF periods achieved for a small stimulation current, weintroduce a measure Q of the quality of the intermittent CR stim-ulation, which is proportional to the maximal admissible lengthof the rest intervals given by nmax and inversely proportional tothe effective amount of the stimulation Ieff and the number ofstimulation sites Ns, Q = nmax/ (Ieff × Ns). This function esti-mates the efficacy of CR stimulation “per each stimulation site”such that for similar values of nmax/Ieff more preferable is thestimulation setup with a smaller number of sites. We found thatfor fixed spatial decay rate of the stimulation signal from a rangeσ ∈ [0.25, 2.5], Q achieves a clear maximum Qmax = Qmax(σ)

for an optimal numbers of stimulation sites Ns, opt depicted by ablack circle in Figure 9, respectively. Ranges of Ns are indicatedby colored stripes where the quality Q of the stimulation doesnot deviate for more than 30% from its best value Qmax. In con-trast, the white regions correspond to parameter values of clearlysub-optimal CR stimulation and should be avoided. This diagramsupports our conclusion that an optimal CR stimulation has tobe administered via a small number of sites if the stimulationcurrent is broadly spread in the tissue, i.e., if σ is large, whereasfor small σ (spatially selective stimulation) a larger number ofstimulation sites leads to a better desynchronizing impact of CRstimulation.

3.3. RESULTS FOR SPIKING NEURONAL MODELFor the FHN model (Equations 4, 5) controlled by CR stimulation(Equation 6) we use the same evaluation techniques applied to thephase oscillators (Equation 1) and described in section 3.1. Wethus calculate the optimal (minimal) values 〈R1〉opt of the time-averaged first order parameter 〈R1〉 obtained under variation ofthe stimulation strength I ∈ [0, 10]. The effect of continuous CRstimulation on the FHN network is illustrated in Figure 10 vs. thenumber of stimulation sites Ns. Different curves in the plot rep-resent the stimulation-induced minimal values of the first order

FIGURE 9 | Quality Q = nmax/(Ieff × Ns) of the intermittent CR

stimulation depicted in color ranging from blue (small values) to red

(large values), see the color bar. The optimal numbers of stimulation sitesNs,opt, where the maximal values Qmax(σ) = max

NsQ(Ns, σ) are achieved for

fixed σ, are indicated by black circles. The corresponding parameters(Iopt, nmax) of the stimulation are given in the round brackets. The values ofQ smaller than 0.7Qmax are not shown (white regions). Other parametersas in Figure 6.






parameter 〈R1〉opt for different values of the spatial decay rate σ.This diagram is very similar to that in Figure 3 obtained for thephase oscillators. Indeed, 〈R1〉opt saturates for large number ofstimulation sites. The saturation levels of 〈R1〉opt depend on thevalues of σ as for the phase oscillators, such that for small (large)σ CR stimulation leads to a small (large) order parameter whichsaturates for large Ns.

Also for the intermittent ON–OFF CR stimulation of the FHNensemble the impact of the stimulation can be quantified in thesame way as described in section 3.2 for the network of phaseoscillators. In particular, Figure 11A illustrates the influence ofthe number of stimulation sites Ns on the maximal admissi-ble length nmax of the rest intervals. As for the phase oscillators(Figure 8A), a spatially selective (i.e., with small σ) intermittentCR stimulation is less effective for small number of stimulationsites Ns (nmax is small), whereas the length of the rest intervalscan significantly be prolonged if Ns increases (Figure 11A, blackcurve for σ = 0.5). Simultaneously, the amount of the stimulationcurrent Ieff received on average by a single neuron in the networkdecays with increasing Ns (Figure 11B, black curve). Therefore,a selective intermittent CR stimulation optimally delivered to aneuronal network via a large number of stimulation sites enableslong stimulation-free desynchronization time intervals with aminimal amount of the administered stimulation current. In con-trast, for a broad spatial spread of the stimulation current in theneuronal tissue (large σ) the intermittent CR stimulation has to beoptimally administered via a small number on stimulation sites,e.g., Ns = 2. Only for such a stimulation protocol CR stimulationcan lead to the longest possible rest periods achieved for the small-est possible delivered stimulation current (Figure 11, red curvesfor σ = 2). In this case, a larger number of stimulation sites canworsen the desired desynchronizing impact of CR stimulation.

Note, that for the FHN neuronal ensemble (Equations 4, 5)controlled by the intermittent CR stimulation (Equation 6) wehave observed the same behavior of the order parameter 〈r〉opt

2 3 4 5 7 10 12 15 17 20 22 250

0.1

0.2

0.3

0.4

0.5

σ=0.25σ=0.50σ=0.75σ=1.00σ=1.25σ=1.50σ=2.00σ=2.50

Ns

<R1>

opt

FIGURE 10 | Minimal values 〈R1〉opt of the time-averaged first order

parameter 〈R1〉 of the FHN neuronal ensemble (Equations 4–6)

controlled by continuous CR stimulation vs. the number of stimulation

sites Ns for different values of the parameter σ in Equation (3) as

indicated in the legend. The stimulation strength I has been varied in theinterval [0, 10].

and the optimal stimulation strength Iopt vs. the number ofthe OFF cycles n in the rest intervals (figures are not shown)as for the phase ensemble, see Figures 6, 7. Together with theresults illustrated in the above Figures 10, 11 this indicates therobustness of the reported phenomena, which can equally befound for phase oscillators and for spiking neurons interactingvia excitatory synapses.

3.4. RESULTS FOR THE BURSTING NEURONAL MODELWhen continuous CR stimulation is administered to synchro-nized aEIF bursting neurons (Equation 7), the in-phase syn-chronization is replaced by a cluster state (Figures 12A,B). Thebursting neuronal discharges get organized in a few sub-groups(clusters) corresponding to the number of stimulation sites, andthe neurons within the same cluster fire simultaneously withequidistant time shift between neighboring clusters. The samedynamics is observed for the phase oscillators (Figure 2) andFHN neurons. The synchronization order parameter of the aEIFneuronal ensemble is suppressed from 〈R1〉 ≈ 0.92 in the in-phase synchronized regime to 〈R1〉 ≈ 0.014 in the cluster state forthe parameter values used in Figures 12A,B. In the latter regimethe other order parameters are 〈R2〉 ≈ 0.063, 〈R3〉 ≈ 0.088, and〈R4〉 ≈ 0.766, which is a clear indication of the 4-cluster state thatcan be seen in Figures 12A,B.

The spatial decay rate σ of the stimulation current and thenumber Ns of stimulation sites can significantly influence thedesynchronizing impact of permanent CR stimulation as mea-sured by the values of the first order parameter. In particular,for large σ the time-averaged first order parameter 〈R1〉 grows

2 3 4 5 6 7 12 1510 15 200

4

8

12

σ=0.5σ=2.0n

max

A

Ns

2 3 4 5 6 7 10 12 15 200

0.04

0.08

0.12

σ=0.5σ=2.0

Ns

Ieff

3:5

3:63:8

3:8 3:10

3:93:12

3:11 3:10 3:11

3:7

3:4 3:3

3:2

3:2

3:1 3:2 3:23:1

3:2

B

FIGURE 11 | (A) Maximal admissible number nmax of OFF cycles and (B)

effective amount of the stimulation Ieff received on average by a singleneuron of the FHN ensemble (Equations 4–6) during the intermittent CRstimulation vs. the number of stimulation sites Ns for σ = 0.5 (black curves)and σ = 2 (red curves). Number of ON cycles m = 3, and other parametersas in Figure 10.






as Ns increases, whereas the opposite is true for small σ, seeFigure 12C. This effect revealed for the considered bursting neu-rons is again in perfect agreement with the results obtainedfor the phase oscillators (Figure 3) and FHN spiking neurons(Figure 10).

Also the intermittent ON–OFF CR stimulation of the aEIFneuronal ensemble can become more effective for a large num-ber of stimulation sites. However, this requires spatially selec-tive stimulation, i.e., small σ (Figure 13A, black diamonds). Inthis case the maximally admissible number of OFF intervalsnmax increases with the number of stimulation sites, which per-mits longer time intervals with desynchronized dynamics of thestimulation-free neurons. We illustrate this effect on the synchro-nized dynamics for σ = 0.5 and different numbers of stimulationsites in Figures 13B,C. The suboptimal number of stimulationsites Ns = 2 (Figure 13C) implies shorter OFF-intervals and afaster re-increase of both order parameter R1 and amplitude ofthe ensemble mean field 〈V〉 during the OFF-intervals in com-parison to the stimulation delivered via a larger number Ns = 5

FIGURE 12 | Impact of continuous CR stimulation on the ensemble of

aEIF bursting neurons (Equation 7). (A) Time courses of the membranepotentials Vj of four selected neurons with indices j indicated in the plot.The neurons are assigned to each of the four stimulation sites located atthe same lattice coordinates. Stimulation begins at t = 700 ms withstimulation intensity I = 1550 pA, and spatial current decay rate σ = 0.5.The vertical solid lines comprise the CR stimulation cycles of length T = 70ms, and the dashed lines indicate the time intervals, where thecorresponding stimulation site is active. (B) The corresponding raster plotof the burst onsets. (C) Minimal values 〈R1〉opt of the time-averaged firstorder parameter of the neuronal ensemble controlled by continuous CRstimulation vs. the number of stimulation sites Ns for different values ofparameter σ in Equation (3) as indicated in the legend. The stimulationstrength I was varied in the interval [0, 2000] pA.

of stimulation sites (Figure 13B). The situation is opposite for abroad spatial spread of the stimulation current in the neuronaltissue (large σ), where the intermittent CR stimulation can opti-mally be administered via a small number of stimulation sitesonly (Figure 13A, red circles). In this case, a larger number ofstimulation sites can significantly shorten the admissible lengthof the rest intervals. The above effects of the intermittent CRstimulation for the aEIF bursting neurons (Equation 7) are againin good agreement with those observed for the phase oscillators(Figure 8) and FHN spiking neurons (Figure 11), which furtherconfirms the generality of the presented results.

4. DISCUSSIONAn effective desynchronization of pathological neuronal synchro-nization, present in several neurological disorders like Parkinson’sdisease or essential tremor, is an important clinical challenge(Tass, 1999). The considered CR stimulation technique providesan effective means for the desynchronization of neuronal net-works (Tass, 2003b). The problem of an appropriate calibrationof the stimulation parameters like stimulation strength or timing

2 3 5 7 10N

s

0

10

20

30

40

nmax σ=0.5

σ=4.0

A

60 60.5 61 61.5 62 62.5 63 63.5time (s)

-51

-50

-49

-48

<V

> (

mV

)

60 60.5 61 61.5 62 62.5 63 63.50

0.1

0.2

0.3

R1

B

60 60.5 61 61.5 62 62.5 63 63.5time (s)

-51

-50

-49

-48

<V

> (

mV

)

60 60.5 61 61.5 62 62.5 63 63.50

0.1

0.2

0.3

R1

C

FIGURE 13 | Impact of intermittent ON–OFF CR stimulation on the

ensemble of aEIF bursting neurons (Equation 7). (A) Maximaladmissible number nmax of OFF cycles vs. the number of stimulation sitesNs for σ = 0.5 (black diamonds) and σ = 4 (red circles). The synchronizationthreshold is considered 〈r〉opt = 0.25, see section 3.2 for definition. (B)

Time courses of the ensemble mean filed 〈V 〉 = N−1 ∑Nj = 1 Vj (black curve)

and the first order parameter R1 (red curve, scale on the right vertical axis)between two successive ON epochs (indicated by dashed vertical lines andbars on the top of the plot). Parameters σ = 0.5, n = nmax = 40,I = Iopt = 392 pA, and Ns = 5. (C) The same as in (B) for the parametersσ = 0.5, n = nmax = 14, I = Iopt = 460 pA, and Ns = 2. Number of ONcycles m = 3, and other parameters as in Figure 12.






parameters of the intermittent CR stimulation has been addressedin our previous study (Lysyansky et al., 2011). However, theimpact of the number of stimulation sites has not been studiedso far. This problem is computationally addressed in the presentpaper.

The optimization problem of CR stimulation can be consid-ered from different points of view. First of all, the stimulationshould have a significant desynchronizing effect. On the otherhand, the stimulation should be mild, i.e., the total amount ofthe delivered stimulation current should be minimized. The lat-ter quantity depends on the stimulation strength as well as on thetiming parameters of the intermittent stimulation, i.e., the lengthsof the ON- and OFF-periods of the stimulation protocol. We,thus, analyze the impact of the number of stimulation sites Ns

on these optimality criteria. The obtained results are illustratedon three different models of a neuronal network subjected to CRstimulation: the Kuramoto model of phase oscillators, a networkof spiking FHN neurons, and a network of adaptive exponen-tial integrate-and-fire bursting neurons. We show that all threemodels demonstrate a striking similarity in their responses to CRstimulation, which indicates the robustness of the demonstratedphenomena.

We found that the impact of the number of stimulation sitescrucially depends on the spatial spread of the stimulation signal inthe neuronal tissue as given by the parameter σ. If the stimulationsignal is narrowly spread (small σ) we can speak about spatiallyselective CR stimulation where each stimulation site essentiallyaffects neurons in its nearest proximity only. For a broad spatialsignal spread (large σ) the stimulation sites perturb much largersubpopulations which may significantly overlap with each other.In our calculations the stimulation strength I is varied in the inter-val [0, Imax] in order to find an optimal value Iopt leading to thestrongest desynchronization characterized by minimal values ofthe time-averaged order parameter 〈R1〉opt for continuous CRstimulation or 〈r〉opt for intermittent CR stimulation. For the for-mer stimulation protocol we found that an increase of the numberof stimulation sites Ns results in a saturation of the optimal (min-imal) value of the order parameter 〈R1〉opt. The saturation leveldepends on σ such that small (large) σ implies small (large) valuesof 〈R1〉opt (Figures 3, 10). Therefore, the desynchronizing impactof the continuous CR stimulation can be improved by a moder-ate increase of the number of stimulation sites Ns for small σ. Incontrast, for large σ, Ns has to be kept small.

For the intermittent ON–OFF CR stimulation the same con-clusion can also be drawn. More precisely, for small σ a moderateincrease of the number of stimulation sites Ns can improveall optimality criteria mentioned above (Figures 8, 11). In thiscase the intermittent CR stimulation admits long rest periods ofthe post-stimulus desynchronizing transient which are achievedfor a minimal amount of the administered stimulation current.However, much larger Ns does not lead to any further signifi-cant improvement, and a saturation effect is observed. Moreover,for large σ the best stimulation outcome is observed for a smallnumber of stimulation sites, e.g., Ns = 2, and any additional stim-ulation site can only worsen the desynchronizing impact of theintermittent CR stimulation.

To relate the obtained results to possible pre-clinical or clini-cal applications, the following values are necessary: the size Lexp

of the stimulated target region where all sites are placed and thereal value of the signal decay rate in the neuronal tissue σexp.The latter quantity can be estimated by an analysis of the vol-ume of tissue activated (VTA) and the corresponding voltagesnecessary to activate neurons at a distance d to the stimulationsite (Chaturvedi et al., 2010). For example, the threshold acti-vating stimulation voltage is reported to be Vth = kd2, wherek ranges from 0.42 to 0.68 in the subthalamic nucleus (STN)(Chaturvedi et al., 2010). Therefore, σexp = 1√

kcan be estimated

as 1.2 ≤ σexp ≤ 1.5. Rescaling the real domain to the linear modelsegment of length L = 10 considered in the present study, weobtain the corresponding value of σ = σexp

LLexp

. If the length of

the domain under stimulation is, e.g., 10mm, the value of σ liesbetween 1.2 and 1.5. Thus, the optimal number of stimulationsites Ns, opt = 2, but also for slightly larger Ns = 3 or 4 we expecta comparable outcome, as follows from Figure 9.

In a first approximation, we can apply our results to realisticDBS electrodes. Medtronic leads no. 3387 and no. 3389 com-prise four stimulation sites, each of length 1.5 mm, and thecontact spacings are 1.5 and 0.5 mm, respectively. As mentionedabove, σ depends on the realistic decay rate in the neuronaltissue as well as on the size of the stimulated domain. In theframework of our approach, i.e., approximating the realistic, spa-tially extended stimulation contacts by points through which amonopolar stimulation is delivered (Figure 1), the Medtroniclead no. 3387 covers a target domain of length Lexp = 12 mm,whereas for the Medtronic no. 3389 lead the corresponding lengthLexp = 8 mm. Assuming that the target region is large enoughto embrace all stimulation sites, the values of σ range from 1.0to 1.25 and from 1.5 to 1.9 for the former and latter electrode,respectively. Therefore, for the Medtronic no. 3387 lead fromtwo to four sites and for the Medtronic no. 3389 lead only tworemote sites enable an optimal desynchronizing effect of CRneuromodulation.

CR neuromodulation can be realized by means of a numberof different stimulation modalities including electrical deep brainstimulation and sensory, e.g., acoustic stimulation for the treat-ment of tinnitus (Popovych and Tass, 2012; Tass and Popovych,2012; Tass et al., 2012a,b; Adamchic et al., in press; Silchenkoet al., 2013). The neuronal target regions for the former stim-ulation setup can be, for example, the STN or globus pallidus,whereas the non-invasive acoustic CR neuromodulation aims ata reduction of pathological synchronization in tinnitus-relatedauditory and non-auditory brain areas. For a more sophisticatedmodeling of CR deep brain stimulation one has to account forimportant details of the interface between the DBS electrode andthe neuronal tissue and their properties as well as for the geome-try and arrangement of fibers in the vicinity of the DBS electrode(Butson and McIntyre, 2006; Butson et al., 2006; Chaturvediet al., 2010). For the acoustic CR neuromodulation, however, aprecise tonotopic organization of the auditory cortex and audi-tory pathway has to be considered (Ehret and Romand, 1997).Furthermore, one can use models based on phase response curves(PRC) (Winfree, 1980; Ermentrout, 1996; Lücken et al., 2013) and






incorporate the PRC measured either experimentally or obtainedby detailed modeling of the STN, globus pallidus or corticalregions (Netoff et al., 2005; Tateno and Robinson, 2007; Tsuboet al., 2007; Stiefel et al., 2008; Schultheiss et al., 2010; Farries andWilson, 2012a,b). Detailed neuronal models, although reflect-ing the richness and complexity of neuronal dynamics, are, onthe other hand, so complicated and specialized that they mayundermine the generality of their predictions, in particular, forother stimulation modalities and target regions. In this studywe considered relatively simple models of neuronal networksand stimulation which approximate a particular realistic stimu-lation setup only to some extend. Nevertheless, the conclusionsof the present investigation are based on fundamental proper-ties of neuronal ensembles such as synchronization and phaseresetting which are universal phenomena and can be observedunder a variety of conditions, see, e.g., Winfree (1977); Best(1979), and Pikovsky et al. (2001). Ongoing oscillatory neuronalactivity can be reset by both electrical and sensory stimulation(Brandt, 1997; Meissner et al., 2005; Jahangiri and Durand, 2011;Thorne et al., 2011), which serves as a basis for using differentstimulation modalities for CR neuromodulation (Popovych andTass, 2012). This supports a broad applicability of the reportedresults.

As mentioned in the Introduction, CR stimulation leads tosustained long-lasting after-effects on motor function in MPTPmonkeys in contrast to classical HF DBS (Tass et al., 2012b).Remarkably, these sustained CR after-effects were obtained ata stimulation strength (i.e., amplitude of the single electricalpulses) equal to a third of the stimulation strength used for clas-sical HF DBS. In contrast, delivering CR neuromodulation at alarger stimulation strength equal to that of the classical HF DBSled to only weak and considerably shorter CR after-effects. Thisfinding is in accordance with a previous computational studywhere we investigated the influence of the stimulation strengthand the lengths of the ON and OFF intervals (i.e., intervals withand without CR stimulation) for the intermittent CR stimulationprotocol (Lysyansky et al., 2011). We found that there exists anoptimal stimulation strength for CR stimulation, whereas a largerstimulation intensity can lead to a sub-optimal outcome of thestimulation. These results have now been confirmed by large-scalesimulations of a precise neuronal network model of the STN andglobus pallidus (Ebert, 2012) as well as experimentally in MPTPmonkeys (Tass et al., 2012b). In the present study we use a similarapproach and anticipate that the drawn conclusions concerningthe optimal number of stimulation sites for CR stimulation maybe of relevance for therapeutic effects of CR stimulation.

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Conflict of Interest Statement: Theauthors declare that the researchwas conducted in the absence of anycommercial or financial relationshipsthat could be construed as a potentialconflict of interest.

Received: 09 January 2013; accepted: 01July 2013; published online: 22 July 2013.Citation: Lysyansky B, Popovych OV andTass PA (2013) Optimal number of stim-ulation contacts for coordinated resetneuromodulation. Front. Neuroeng. 6:5.doi: 10.3389/fneng.2013.00005Copyright © 2013 Lysyansky, Popovychand Tass. This is an open-access arti-cle distributed under the terms of theCreative Commons Attribution License,which permits use, distribution andreproduction in other forums, providedthe original authors and source are cred-ited and subject to any copyright noticesconcerning any third-party graphics etc.


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