Brain Pacemaker

Glossary

Coordinated reset stimulation:

Coordinated reset (CR) stimulation is an effectively desynchronizing control technique, where a population of synchronized oscillators is stimulated via several stimulation sites in such a way that spatially and timely coordinated phase reset is achieved in subpopulations assigned to each of the stimulation sites. This method is suggested for the counteraction of abnormal neuronal synchronization characteristic for several neurological diseases and amelioration of their symptoms. It has successively been verified in a number of experimental and clinical studies.

Deep brain stimulation:

Electrical deep brain stimulation (DBS) is the standard therapy for medically refractory movements disorders, e.g., Parkinson’s disease and essential tremor. It requires a surgical treatment, where depth electrodes are chronically implanted in target areas like the thalamic ventralis intermedius nucleus or the subthalamic nucleus. For standard DBS electrical high-frequency (>100 Hz) stimulation is permanently delivered via depth electrodes. More sophisticated deep brain stimulation techniques are in the process of being established for clinical use.

Delayed feedback:

Delayed feedback is a method for the creation of a closed-loop forcing, where a portion of the measured output signal of a system is time delayed, linearly or non-linearly processed, and fed back into the system. This approach is often used to control the dynamic behavior of complex systems. In this article delayed feedback is used to control synchronization in ensembles of coupled oscillators, e.g., neurons.

Order parameter:

The order parameter is a quantity characterizing a phase transition or phase change in the transformation of a complex system from one phase (state) to another. The order parameter is convenient for characterizing the onset and extent of synchronization in larger ensembles: Perfect phase synchronization corresponds to a large value of the order parameter, whereas an incoherent (desynchronized) state is associated with a small value of the order parameter. In synergetics it has been shown that the dynamics of complex systems may be governed by only a few order parameters.

Synchronization:

Synchronization (from Greek syn = the same, common and chronos = time) means the adjustment of rhythms of self-sustained oscillators due to their weak interaction. The interacting oscillators can be regular (periodic) or chaotic. There are several different forms of synchronization including phase, complete, generalized, and lag synchronization, etc. In this article we focus on phase synchronization. In the simplest form, the oscillators, rotating with the same frequency, become phase synchronized (phase locked) to each other, if they tend to oscillate with the same repeating sequence of relative phase angles. Put otherwise, the oscillators adjust their rhythms, while their amplitude dynamics need not be correlated.

Definition of the Subject

A brain pacemaker is a medical device that is implanted into the brain with the purpose to stimulate nervous tissue with electrical signals. Brain pacemakers are used for the therapy of patients suffering, for example, from Parkinson’s disease, epilepsy or mental disorders. Brain stimulation is either called deep brain stimulation (DBS) if structures deeply inside the brain are targeted or cortical stimulation (intracortical or epicortical), if the electrical contacts of the stimulator are positioned within the cortex or on its surface. Apart from direct brain stimulation, other targets may also be used, such as spinal cord (e.g., for the treatment of pain) or the vagus nerve (for the treatment of epilepsy). The electrical stimulation of the nervous system has a long history which goes back to the nineteenth century where first tests with cortical stimulation were documented (Gildenberg 2005). The first intraoperative deep brain stimulation was performed by Spiegel et al. in 1947 in a patient suffering from Huntington’s chorea, and in the eighties DBS was introduced as a treatment for motor disorders (Brice and McLellan 1980; Benabid et al. 1987). DBS was approved by the Food and Drug Administration (FDA) as a treatment for essential tremor in 1997, for Parkinson’s disease in 2002, and dystonia in 2003. The treatment of severe neurological and psychiatric diseases with brain pacemakers is a rapidly growing and promising field. Novel, model-based approaches, which use methods from synergetics, nonlinear dynamics, and statistical physics, to specifically restore brain function and connectivity, demonstrate how insights into the dynamics of complex systems contribute to the development of novel therapies.

Introduction

Self-organization processes are abundant in numerous fields of the natural sciences (Haken 1977, 1983). For instance, the nervous system elegantly utilizes self-organization principles for motor control purposes (Haken et al. 1985; Schöner et al. 1986; Haken 1996; Kelso 1995). A classical example of a self-organization process is synchronization of populations of interacting oscillators, which is widely observed in physics (Haken 1970, 1983; Pikovsky et al. 2001; Strogatz 2003), chemistry (Kuramoto 1984), biology (Winfree 1980), neuroscience (Steriade et al. 1990; Haken 2002), and medicine (Elble and Koller 1990; Milton and Jung 2003; Tass 1999). In the nervous system synchronization processes are important, e.g., in the context of information processing (Singer 1989) and motor control (Andres and Gerloff 1999). However, pathological, excessive synchronization strongly impairs brain function (Elble and Koller 1990; Milton and Jung 2003). In fact, pathological synchronization processes are the hallmark of several neurological diseases like Parkinson’s disease (PD) or essential tremor (Alberts et al. 1969; Nini et al. 1995). For example, Parkinsonian resting tremor appears to be caused by a pacemaker-like population of neurons which fires in a synchronized and periodical manner (Alberts et al. 1969; Smirnov et al. 2008). In contrast, under healthy conditions these neurons fire in an uncorrelated, i.e. desynchronized manner (Nini et al. 1995).

Permanent deep brain stimulation (DBS) at high frequencies (>100 Hz) is the standard therapy for medically refractory patients suffering from Parkinson’s disease and essential tremor (Benabid et al. 1991, 2002; Blond et al. 1992), see Fig. 1. High-frequency (HF) DBS has been developed empirically, mainly based on experimental results and clinical observations. The mechanism of HF DBS is still a matter of debate (Benabid et al. 2005). Clinical studies showed that HF DBS essentially has similar effects as observed after tissue lesioning. Compared to lesioning, DBS is reversible and has a lower rate of side effects (Tasker 1998; Schuurman et al. 2000). However, in spite of many beneficial effects, in some patients DBS may not help, or may cause side effects, or the therapeutic effects may disappear over time (Tasker 1998; Volkmann 2004; Rodriguez-Oroz et al. 2005; Deuschl et al. 2006). With the objective of finding better tolerated and more effective DBS techniques, a model-based development of novel stimulation methods has been initiated (Tass 1999, 2002a, c, 2003b; Hauptmann et al. 2005a, b, c, 2007b; Tass et al. 2006; Popovych et al. 2005, 2006a, b), see Fig. 2. In these studies, relevant neuronal target populations were modeled mathematically and stimulation techniques have been developed utilizing principles from nonlinear dynamics and statistical physics (Tass 1999).

Fig. 1

Standard DBS setup. A depth electrode is implanted into the target structure (e.g., the subthalamic nucleus). The electrode is subcutaneously connected with the generator of the high-frequency stimulation signal (not shown in this image). The stimulation signal is delivered through one or more of the four stimulation contacts labeled from 0 to 3

Fig. 2

Model-based development of novel deep brain stimulation techniques: Along the lines of a top-down approach target areas for deep brain stimulation are modeled by means of oscillator networks and physiology- and anatomy-based neural networks. Methods from nonlinear dynamics, statistical physics, and synergetics are employed to develop stimulation techniques which specifically utilize dynamic self-organization principles and plasticity rules. Experimental feedback from both animal experiments and clinical evaluation serves to validate, falsify or modify theoretical assumptions and predictions. This iterative approach aims at steadily improving the mathematically designed stimulation techniques and, hence, at establishing superior therapies

One goal of this approach is to control the pathological neural dynamics appropriately in order to achieve a mild and efficient relief of symptoms (Tass 1999). The second, more ambitious goal is to stimulate in a way that the formerly affected neuronal populations unlearn their pathological connectivity and, hence, their tendency to produce pathological synchronization (Tass and Majtanik 2006). Put otherwise, the very goal of this approach is to induce long-lasting therapeutic effects which outlast the cessation of stimulation (Tass and Majtanik 2006; Tass and Hauptmann 2007; Hauptmann and Tass 2007). To this end, stimulation algorithms have been developed and optimized to exploit dynamic self-organization principles and plasticity rules (Tass and Majtanik 2006; Tass and Hauptmann 2006, 2007; Hauptmann and Tass 2007; Hauptmann et al. 2007b).

Several novel stimulation techniques have computationally been developed in the past. In this article four of these control methods will be presented in detail: coordinated reset (CR) stimulation (Tass 1999, 2002a, c, 2003b), multisite linear delayed feedback (MLDF) stimulation (Hauptmann et al. 2005a, b, c), nonlinear delayed feedback (NDF) stimulation (Popovych et al. 2005, 2006a, b; Popovych and Tass 2010), and proportional–integro–differential feedback (PIDF) stimulation (Pyragas et al. 2007). These techniques have the common objective of reducing the synchronized activity of the target population by reestablishing a normal desynchronized physiological activity in a highly synchronized population of neurons. For other stimulation methods we refer to Rosenblum and Pikovsky (2004a), Tukhlina et al. (2007), Kiss et al. (2007), Luo et al. (2009), Danzl et al. (2009), and Nabi and Moehlis (2011).

CR stimulation, in its original realization, uses short electrical pulse trains to subsequently reset sub-populations of the neuronal network, which induces a desynchronized state (Tass 1999, 2002a, c, 2003b). The stimulation is applied through a small number of stimulation sites which are equally spaced within the neuronal population. CR stimulation induced desynchronization is achieved by utilizing self-organization principles, in particular, the slaving principle induced by the pathological neuronal interactions (i.e., interactions which have the potential to induce a pathological synchronization) (Tass 2002a, c, 2003b). MLDF (Hauptmann et al. 2005a, b, c) and NDF (Popovych et al. 2005, 2006a, b; Popovych and Tass 2010) stimulation use delayed feedback for stabilizing a desynchronized state which is intended to be as close to the physiological mode of action as possible. Here, the local field potential (LFP) of the target population is measured, amplified, delayed, and fed back into the ensemble. The PIDF feedback (Pyragas et al. 2007) utilizes an instantaneous LFP and is designed for a particularly difficult situation characterized by a separate registration and stimulation setup.

It has been shown experimentally, that synaptic plasticity enhances pathological synchronization (Nowotny et al. 2003). From the kindling phenomenon in the context of epilepsy it is well known that neural networks may learn pathological strong interactions (Speckmann and Elger 1991; Morimoto et al. 2004). The novel desynchronizing stimulation protocols are designed to invert this pathological process, so that the affected neuronal populations unlearn their pathological connectivity, and physiological neuronal activity is re-established on a long-term basis. In a nutshell, the novel stimulation techniques aim at a well-directed employment of fundamental principles of dynamic brain action to induce long-lasting therapeutic effects.

Standard High-Frequency Stimulation

High-frequency (HF) deep brain stimulation (DBS) is the standard therapy for patients suffering from medically refractory PD or essential tremor (Benabid et al. 1991, 2002). To this end, depth electrodes are chronically implanted in the thalamic ventralis intermedius nucleus or the subthalamic nucleus (Benabid et al. 1991, 2002) and a permanent HF (>100 Hz) periodic pulse train stimulation is applied (Fig. 1). HF DBS has been developed empirically, mainly based on intraoperative observations. HF DBS strongly alters the neuronal firing and mimics the effect of tissue lesioning, e.g., by suppressing neuronal firing, which, in turn, suppresses the peripheral tremor (Benabid et al. 2002; Filali et al. 2004; McIntyre et al. 2004b; Volkmann 2004). However, as yet, the mechanism of HF DBS is not sufficiently understood (McIntyre et al. 2004b).

During stimulation HF DBS seems to induce a regular bursting mode (Beurrier et al. 2002). After a reduction of stimulation artifacts, robust bursting activity in subthalamic nucleus (STN) neurons was observed in slices from naive or resperine-treated rats. After offset of stimulation, blockade of activity, i.e., a non-specific suppression of the neuronal activity in the target structures through a depolarization blockade which did not require synaptic transmission was observed (Beurrier et al. 2001).

Other hypotheses are that HF DBS applied to a PD patient mimics the effect of tissue lesioning and appears to block neuronal activity in relevant target areas during stimulation (Benabid et al. 2002). In single-compartment conductance-based biophysical models of isolated STN neurons the HF stimulation may cause a suppression of neuronal activity on an elementary membrane level, where a neuron’s resting state or low-amplitude subthreshold oscillations can get stabilized (Pyragas et al. 2013). The obtained theoretical results resemble the clinically observed relations between stimulation amplitude and stimulation frequency required to suppress Parkinsonian tremor (Benabid et al. 1991). The most probable hypothesis was offered by Benabid et al. (2005), in which a mixture of different mechanisms was discussed. The contributing mechanisms resulting in the observed effects of HF DBS might be membrane inhibition, jamming, excitation of excitatory and inhibitory afferents, excitation of efferents and plasticity (Benabid et al. 2005). In particular, HF stimulation of afferent axons projecting to STN can account for a therapeutic effect of HF DBS within STN (Gradinaru et al. 2009).

To precisely evaluate the contribution of these different mechanisms, spatially extended multicompartment neuron models were used to demonstrate the effects of extracellular stimulation on the different structures of the stimulated neuronal population (Grill and McIntyre 2001). Depending on the stimulation amplitude and the shape of the stimulation pulses, either the cells were activated directly or fibers mediating excitatory or strong inhibitory action were activated (Grill and McIntyre 2001). Modeling studies indicate that already at the level of single neurons, the activation of a larger number of structures can take place with different and possibly conflicting impacts on the single neuron dynamics (Grill and McIntyre 2001). The collective dynamics of neuronal populations further adds aspects which are important for the creation of synchronized activity: cells responding differently to external inputs like somatosensory stimulation or stimulation due to active movements are present in the target tissue together with so called no-response cells (Lenz et al. 1994). HF stimulation has a complex impact on these structures (Benabid et al. 2002; Shen et al. 2003).

Experimental and modeling studies also indicate that the globus pallidum interior (GPi) – one structure of the basal ganglia – might be strongly involved in the mechanisms of DBS (Hashimoto et al. 2003; Garcia et al. 2005; McIntyre et al. 2004a; Rubin and Terman 2004; Miocinovic et al. 2006). The results of modeling studies indicate that under parkinsonian conditions the rhythmic inhibition from GPi to the thalamus compromises the ability of thalamocortical relay cells to respond to depolarizing inputs, such as sensorimotor signals. HF stimulation of STN regularizes GPi firing, and this restores the responsiveness of the thalamus (Rubin and Terman 2004). In such a way, one may distinguish between local and non-local effects of HF DBS. Locally, in the vicinity of the stimulation electrode, the axons rather than cell bodies (somas) get activated (McIntyre et al. 2004a), while the latter can even be effectively inhibited by the HF stimulation (Beurrier et al. 2001; Benabid et al. 2002; Welter et al. 2004; Meissner et al. 2005). The stimulation-induced axonal activity propagates antidromically and ortodromically (Hammond et al. 2008) and can change the firing in the output structures downstream to the neuronal target population. The pathological discharge patterns there can be replaced by a HF spiking or suppressed depending on whether the efferent fibers of the stimulated nucleus are excitatory or inhibitory, respectively (Hashimoto et al. 2003; Anderson et al. 2003; McIntyre et al. 2004b). In other words, local and non-local effects of HF DBS may differ considerably.

HF DBS is reversible and has a much lower rate of side effects than lesioning with thermocoagulation (Schuurman et al. 2000). Although HF DBS is the golden standard for the therapy of medically refractory movement disorders, there are still limitations of HF DBS: On the one hand HF DBS may cause adverse effects like dysarthria, dysesthesia, cerebellar ataxia, and memory decline (Volkmann 2004; Rodriguez-Oroz et al. 2005; Freund 2005). On the other hand HF DBS may be ineffective or its therapeutic effect may wear off over time (Kumar et al. 2003; Rodriguez-Oroz et al. 2005). For instance, 11–15% of PD patients have unsatisfactory outcomes although their depth electrodes are properly placed (Limousin et al. 1999).

Coordinated Reset Stimulation

To study the impact of pulsatile stimuli on single oscillators and, in particular, populations of oscillators in a biologically more realistic setting, it was necessary to take into account random forces (Tass 1996a, b, 1999). To this end, a stochastic concept of phase resetting has been developed for populations of non-interacting (Tass 1996a, b) as well as interacting (Tass 1999) oscillators in the presence of noise. In this approach limit cycle oscillators were approximated by phase oscillators (Hansel et al. 1993b), so that the pulsatile stimulation only affects the oscillators’ phases. If a single pulse of the right intensity and duration is delivered to the population in the stable synchronized state, it causes an at least temporary desynchronization provided it hits the population at a vulnerable phase. Theoretically, single pulse stimulation has also been studied in more complex networks, for instance, networks of coupled phase oscillators with inertia, modeling dendritic dynamics (Dolan et al. 2005; Majtanik et al. 2006). Based on the stochastic phase resetting theory and utilizing a phase oscillator as a model for a single neuron (Hansel et al. 1993b), demand-controlled single pulse deep brain stimulation has been suggested for the therapy of movement disorders like Parkinson’s disease or essential tremor (Tass 1999, 2000).

However, there are drawbacks to single-pulse stimulation which decisively limit its applicability (Tass 2001b; Zhai et al. 2005): First, if the mutual coupling is not weak, the vulnerable phase range we have to hit in order to cause an effective desynchronization is only a small fraction (e.g., 5%) of a period of the collective oscillation. Second, the critical stimulation parameters required to achieve a good desynchronization depend on the initial dynamical state of the population. Thus, different stimulation parameters have to be used if the cluster is not in its stable synchronized state.

To overcome the limitations of single pulse stimulation, double pulse (Tass 2001a, b) stimulation has been proposed: Two qualitatively different stimuli are successively delivered. The first, stronger pulse resets (restarts) the collective oscillation irrespective of the initial state of the population. The second, weaker pulse is applied after a fixed time delay, where it hits the cluster in its vulnerable state and, hence, causes a desynchronization. There are different variants of double pulse stimulation, depending on the type of stimuli used to achieve a reset or a desynchronization (Tass 2001c, 2002a, b, c). For instance, the first resetting pulse can be replaced by a brief high-frequency pulse train (Tass 2001c) or by a softly resetting low-frequency pulse train (Tass 2002a, b).

Although double pulse stimulation causes a desynchronization irrespective of the initial dynamical state at which this stimulus is delivered, there are still limitations which may hinder an application to a biological system (Tass 2003a, b): On the one hand, double pulse stimulation requires a calibration. On the other hand, double pulse stimulation is only effective if the system parameters are reasonably stable. The required quasi-stationarity of the system parameters combined with the possibly time consuming calibration may cause problems when applied to a real biological system, where fluctuations of model parameters are inevitable.

To provide a stimulation technique which is robust with respect to system parameters and which does not require calibration, coordinated reset (CR) stimulation has been developed (Tass 2003a, b). The idea behind this approach is to abstain from achieving a perfect desynchronization by a well-calibrated stimulus. Rather, by means of a robust and comparably mild stimulus the stimulated population is shifted into a dynamical state which is not the desired desynchronized state, but sufficiently close to it. Close in the sense that due to the pathologically strong coupling the population automatically relaxes into the desired desynchronized state. This approach essentially exploits the pathological tendency of the neuronal population to establish a synchronized state. Accordingly, CR stimulation is in a way comparable to Asian martial arts, where ideally a minimal amount of energy (i.e., CR stimulation) is invested to control the adversary by utilizing the adversary’s own energy (i.e., the neurons’ pathological strong coupling).

The scheme of the stimulation setup is presented in Fig. 3a. Several stimulation sites are placed within the target network and weak resetting stimulation signals are administered via these stimulation sites. In this way the oscillatory population is divided into several sub-populations, where each of them is assigned to the corresponding stimulation site and receiving the stimulation signal mostly from that stimulation site. CR stimulation means that a synchronized population of neurons is stimulated with a sequence of brief resetting stimuli (typically brief HF stimulus trains) via the different sites. The delay between the subsequent resetting stimuli can be chosen as τ = T/n with respect to that at the preceding site, where n is the number of stimulation sites, and T approximates the mean period of the collective dynamics of synchronized oscillators (Tass 2003a, b).

Fig. 3

Stimulation setup of CR stimulation method. (a) Brief and mild resetting stimuli are administered at different sites at subsequent times and effectively divide the stimulated population into several sub-populations such that (b) their phases ψ_j form phase clusters equidistantly (or close to that) distributed over the unit circle. (c) Nearly uniform distribution of the oscillator phases during the post-stimulation transient

The subsequent reset of the different sub-populations induces a so-called cluster state, i.e., the whole population is divided into n sub-populations which differ with respect to their mean phase. This effect is illustrated in Fig. 3b where a snapshot of the distribution of the phases ψ_j of stimulated oscillators is shown in the (cos(ψ_j),sin(ψ_j))-plane after CR stimulation. The phases of the population of oscillators stimulated via, e.g., four sites form four phase clusters distributed equidistantly (or close to that) over the unit circle. To estimate the extent and type of synchronization of the whole population of N oscillators, the cluster variables

$$ {Z}_m(t)={R}_m(t)\;{\mathrm{e}}^{\mathrm{i}{\varPsi}_m(t)}=\frac{1}{N}\sum \limits_{j=1}^N{\mathrm{e}}^{\mathrm{i}\;m{\psi}_j(t)}, $$

(1)

can be used. R_m(t) and Ψ_m(t) are the corresponding real amplitude and real mean phase, where 0 ≤ R_m(t) ≤ 1 for all time t (Daido 1992; Tass 1999). Cluster variables are convenient for characterizing synchronized states of different types: Perfect in-phase synchronization corresponds to R₁ = 1, whereas an incoherent state, with uniformly distributed phases, is associated with R_m = 0, m = 1, 2, 3, … Small values of R₁ combined with large values of R_m are indicative of an m-cluster state consisting of m distinct and equally spaced clusters, where all oscillators within the same cluster have similar phases. In Fig. 3b, for instance, R₁ ≈ 0.02, whereas R₄ ≈ 0.87, indicating a four-cluster state induced by CR stimulation administered via four stimulation sites as in Fig. 3a.

From the cluster state the neurons typically relax to a uniformly desynchronized state (Fig. 3c) before they revert back to the in-phase synchronized state, if left unperturbed. To understand how a stimulus-induced clustering leads to an effective desynchronization, the dynamics of the leading modes Z₁, Z₂, …, can be considered. When the coupling among oscillators becomes sufficiently large, e.g., it exceeds a certain critical value, Z₁ from (Eq. 1) becomes an order parameter (Kuramoto 1984), which according to the slaving principle (Haken 1983) governs the dynamics of the other stable modes Z_m (m = 2, 3, …) on the center manifold (Pliss 1964): The order parameter Z₁ acts on a slow time scale, whereas the stable modes Z_m act on a fast time scale and relax to values given by the order parameter Z₁ (Wunderlin and Haken 1975; Haken 1983). In a system with a large number of oscillators this relationship reads (Tass 1999):

$$ {R}_m\propto {R}_1^{\nu}\quad \mathrm{with}\quad \nu \ge 2,\quad \quad m=2,3,4,\dots $$

(2)

Hence, to maintain a desynchronized neuronal firing, CR stimuli have to be administered repetitively.

CR stimulation exploits transient responses which are due to the oscillators’ (pathologically strong) interactions. The general stimulation protocol of the intermittent CR stimulation is illustrated in Fig. 4. Here, the collective dynamics is visualized by considering the collective firing of the neurons. A single firing/bursting model neuron fires/bursts whenever its phase is close to zero (modulo 2π) (Kuramoto 1984; Ermentrout and Kopell 1991; Grannan et al. 1993; Hansel et al. 1993a; Tass 1999). The collective firing can be illustrated with the relative number of neurons producing an action potential or burst at time t given by

$$ {n}_{\mathrm{fire}}(t)=\frac{\mathrm{number}\ \mathrm{of}\ \mathrm{neurons}\ \mathrm{with}\cos {\psi}_j>0.99}{N}. $$

(3)

0 ≤ n_fire(t) ≤ 1 for all t.  n_fire(t) = 0 means that no neuron fires/bursts, while all neurons fire/burst at time t if n_fire(t) = 1. Varying the threshold parameter 0.99 in a reasonable range does not change the results. As shown in Fig. 4, stimulation starts when the neurons are synchronized and the collective firing demonstrates high-amplitude rhythmic oscillations (upper-right insert in Fig. 4). After a few periods of stimulation the oscillatory population is set to a cluster state (bottom-right insert in Fig. 4). Then the stimulation is switched off and the ensemble returns to a synchronized state, on this way running through a uniformly desynchronized state (bottom-left insert in Fig. 4). And the procedure is repeated such that the ensemble is kept in a transient desynchronized state. The relaxation to a clustered state is due to the system being attracted by the center manifold as characterized by Eq. 2. By imposing a cluster state, the stimulation does only half of the desynchronizing work. The rest, namely approaching a uniformly desynchronized state, is done by the system itself. In this way the coupling, which causes the synchronization, is used for improving the desynchronizing effect. In the course of the post-stimulus transient R₁ and according to Eq. 2 also R₂, R₃, … recover again. The system finally reaches its stable in-phase synchronized state again. In summary, by shifting the system into an unstable cluster state, the system reacts by automatically running through a desynchronized state. Finally, the system reverts back to the synchronized state, if left unperturbed.

Fig. 4

A general scheme of the intermittent CR stimulation. Desynchronized firing of neurons is maintained by repetitive administration of CR stimuli intermingled with epochs of no stimulation

The effectively desynchronizing intermittent CR stimulation can be used to block the resynchronization. For this, the repetitive stimulus administration can be organized either regardless of the state of the stimulated ensemble (open-loop control) or in a demand-controlled way (closed-loop control), where the following three different control strategies can be utilized:

• (i) Periodic administration of CR stimuli: The most simple, open-loop type of stimulation is a periodic administration of CR stimuli. Here the time intervals of fixed length of CR stimulation (ON cycles) alternate with time intervals of fixed length where the stimulation is switched off (OFF cycles).

• (ii) Demand-controlled timing of the administration of identical stimuli: Whenever the population tends to resynchronize, the same stimulus is administered (Fig. 5). The stronger synchronization among the neurons is, the more often a stimulus has to be administered to maintain an uncorrelated firing. In addition, for an ideal performance in an experimental application one has to observe the synchronized oscillation during a sufficiently long period of time in order to perform a frequency analysis which yields the period T of the population in the absence of stimulation and, thus, the critical stimulation parameter τ (the time delay between the two successive HF pulse trains administered via different stimulation sites, see Fig. 3). Moreover, instead of performing such a calibration of, pre-set values of can be used by adapting the latter to the typical frequency range of the pathological oscillation (see, e.g., Tass et al. 2012b).

• (iii) Periodically administered HF pulse trains of demand-controlled length: The stimuli are periodically administered with offset times t_k = kνT, where k = 0, 1, 2, 3, … is the index labeling the different stimuli, $T=\tilde{T}+\varepsilon$ is a time interval in the range of the period $\tilde{T}$ of the population without stimulation, and ν is a small integer such as 2 or 3. This means that a 1:ν entrainment of the four sub-populations is performed, where the spontaneous frequency of the neurons is approximately ν times larger compared to the frequency of stimulus administration. The smaller |ε|, the smaller is the stimulation strength necessary to achieve an entrainment.

Fig. 5

Desynchronizing effect of the demand-controlled intermittent CR stimulation. Time course of R₁ from Eq. 1 (a and c) and of n_fire from Eq. 3 (b and d) during different types of stimulation. Demand-controlled timing of stimulus administration (a and b): As soon as the amplitude R₁ of the recovering order parameter reaches the value of 0.5, the stimulus is administered again. Periodical stimulation with demand-controlled length of HF pulse train (c and d): The stimulus is administered periodically, where the length of the HF pulse trains is adapted to R₁ according to Eq. 4 with M_max = 15 and M_min = 0. First published in Tass (2003b)

The closed-loop variants (ii) and (iii) require that the ensemble’s activity or at least a quantity representing the extent of synchronization can be measured appropriately. Here, either the start times of identical CR stimuli or the length of periodically administered stimuli are calculated from the values of R₁. For example, in the case (ii) the stimulation is started if R₁ becomes larger than a certain threshold (Fig. 5, upper two plots), whereas in the case (iii) the stimulation period is longer for larger values of R₁ measured at the onset of the stimulation (Fig. 5, bottom two plots). In the latter case the length of the HF pulse trains increases linearly between a minimal value M_min and a maximal value M_max of single-pulses (except for rounding), where the latter is initially used for desynchronizing the fully synchronized population. R₁ is measured at (or, in practice, close to) times ${t}_k^{\prime }={t}_k-{t}_{\mathrm{max}}$ where t_max the maximal duration of a HF pulse train (containing M_max single-pulses). ${R}_1\left({t}_k^{\prime}\right)$ determines the number of pulses of the HF pulse trains administered via each of the stimulation sites of the kth stimulus according to

$$ {M}_k=\min \left\{{\left[\frac{R_1\left({t}_k^{\prime}\right)\left({M}_{\mathrm{max}}-{M}_{\mathrm{min}}\right)}{R_1\left({t}_0\right)}\right]}_{\mathrm{\mathbb{Z}}}+{M}_{\min, }{M}_{\mathrm{max}}\right\}, $$

(4)

where k = 0, 1, 2, 3, …, [x]_ℤ stands for rounding x to the nearest integer, and min{x₁, x₂} stands for the minimum of {x₁, x₂}. The kth stimulus ends precisely at time t_k = kνT, whereas it starts somewhere between ${t}_k^{\prime }$ (for M_k = M_max) and t_k (for M_k = M_min = 0), depending on its duration. If the suppression of R₁ is not sufficient one may (i) choose a larger intensity of stimulation, (ii) increase M_min, (iii) administer the stimuli at a higher rate, i.e. decrease ν, so that the inter-stimulus interval t_k + 1 − t_k = νT gets smaller, (iv) increase the duration of each single pulse of the pulse trains and/or increase the intra-burst frequency of the pulse trains (i.e., bursts). The feedback value of R₁ can also be evaluated before time ${t}_k^{\prime }$, especially in case of a slow order parameter dynamics (i.e., when the synchronization is weak with respect to the noise). One could also use the mean of R₁ in a period of evaluation.

Applying the standard, permanent HF stimulation (Benabid et al. 1991; Blond et al. 1992) (in a first approximation) corresponds to stimulating each neuron with the same HF pulse train. During a permanent HF stimulation a high-frequency entrainment of the order parameter Z₁ captures Z₁ in a small portion of the complex plane (Tass 2001c), so that the individual neurons’ firing is stopped, but no desynchronization occurs. In contrast, during stimulation R₁ can be even larger compared to its pre-stimulus level, and after stimulation the synchronous firing continues immediately. To suppress the firing with such a simple pulse train persistently, it has to be administered permanently. The number of single pulses used to suppress the firing in the case of the standard permanent HF pulse train stimulation is about five to eight times larger than that used for blocking the resynchronization in Fig. 5a–d, respectively. This illustrates the effectiveness of the demand-controlled CR stimulation. The latter can effectively desynchronize stimulated oscillators with a significantly smaller amount of stimulation current compared to the standard permanent HF pulse-train stimulation.

The efficacy of CR stimulation can further be improved by an optimal choice of stimulation parameters. Several computational studies on neuronal models of different complexity have addressed this problem and showed that the intermittent m: n ON–OFF CR stimulation, where m cycles with stimulation ON are recurrently followed by n cycles with stimulation OFF, is most effective for a weak stimulation intensity and short ON intervals (Lysyansky et al. 2011a). The stimulation-induced cluster state leads to the longest desynchronizing post-stimulation transient which can further be prolonged for non-uniform timing of the stimuli onsets (Luecken et al. 2013). The number of stimulation sites is another important stimulation parameter, and its optimal choice essentially depends on the properties of the neuronal tissue (Lysyansky et al. 2013). For a weak (strong) spatial decay rate of the stimulation current with distance to the stimulation site, CR stimulation can optimally be delivered via small (large) number of stimulation sites.

The theoretical findings on the properties of CR stimulation have been verified experimentally. The resetting impact and the induced transient desynchronization of an electrical short-pulse stimulation, on which the CR technique is based, have been reported in vivo for coupled neuronal bursters in paddle fish (Neiman et al. 2007). Taking into account the spike timing-dependent synaptic plasticity (see section “Plasticity”), the long-lasting desynchronizing effects of CR stimulation have been investigated in detail in theoretical studies (Tass and Majtanik 2006; Tass and Hauptmann 2006, 2007; Hauptmann and Tass 2007), and the results have been confirmed experimentally in vitro in rat hippocampal slice (Tass et al. 2009). The beneficial therapeutic long-lasting aftereffects of weak CR stimulation have been observed in the 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP)-treated macaque monkeys in contrast to a stronger CR stimulation and to the standard HF DBS (Tass et al. 2012b).

Modeling shows that CR stimulation can be effective for a number of stimulation setups and demonstrate a great applicability. Based on a computational study, CR stimulation has been suggested for counteraction of cerebral hypo-activity, in particular, to activate hypo-active or inactive neuronal populations found in a number of diseases without promoting pathological synchronization by a multi-frequency and phase-shifted activation of the stimulated neuronal networks (Lysyansky et al. 2011b). Other computational studies showed that CR stimulation can be effective in inducing desynchronization for direct somatic stimulation and as well as for excitatory or inhibitory synaptically meditated stimulation (Popovych and Tass 2012). The latter stimulation setup might correspond to stimulation of afferent or efferent fibers or sensory stimulation where the stimulation signals arrive at the neural target population as post-synaptic potentials. Sensory CR stimulation has been suggested for suppression of the neural synchrony underlying tinnitus (Tass and Popovych 2012) and successively verified in a clinical proof of concept study in tinnitus patients treated with non-invasive acoustic CR stimulation (Tass et al. 2012a; Silchenko et al. 2013; Adamchic et al. 2013).

Multisite Linear Delayed Feedback

Similarly as in the case of CR stimulation, multisite linear delayed feedback (MLDF) (Hauptmann et al. 2005a, b, c, 2007a) is administered via several stimulation sites, e.g., via four sites as illustrated in Fig. 3a. The individual stimulation signals S_m(t) of each of the stimulation sites are however derived from the delayed mean field Z(t) of the stimulated ensemble using different time delays for different stimulation signals. The mean field characterizes the collective macroscopic dynamics of the oscillators and can be viewed as the ensemble average of the signals z_j(t),  j = 1, …, N, of individual oscillators, $Z(t)={N}^{-1}\sum \limits_{j=1}^N{z}_j(t)$.

For n stimulation sites, the stimulation signals are calculated as S_m(t) = KZ(t − τ_m), m = 1, …, n, where K is the amplification parameter, and the values of delay τ_m, for example, for n = 4 are calculated from the following relation:

$$ {\tau}_m=\frac{11-2\left(m-1\right)}{8}\tau, \quad m=1,2,3,4. $$

(5)

The delays τ_m are symmetrically distributed with respect to the main delay τ, where the smallest time delay between neighboring stimulation sites is chosen as τ/4. In the case τ = T (mean period of the ensemble), the delays τ_m are uniformly distributed over the mean period T. In another realization, instead of four delays τ_m, m = 1, …, 4 one can use only two of them, e.g., τ₁ and τ₂. One can put τ₃ = τ₁ and τ₄ = τ₂, where the polarity of the stimulation signals S₃(t) and S₄(t) is reversed: S₃(t) =  − S₁(t) and S₄(t) =  − S₂(t). Assuming that the mean field of the ensemble uniformly oscillates with period T = τ, the alternating polarity of the signal corresponds to a shift in time by half a period. Therefore, under this condition the stimulation signal S₃(t) =  − S₁(t) =  − KZ(t − τ₁) approximates the stimulation signal S₁(t + τ/2) which is shifted in time by half of the period, which, in turn, is equal to KZ(t − τ₃), where τ₃ is calculated according to Eq. 5. Analogous arguments are applicable to the stimulation signal S₄(t) =  − S₂(t) =  − KZ(t − τ₂).

If the phase Ψ(t) of the mean field Z(t) (see also Z₁ from Eq. 1) uniformly rotates with a constant frequency Ω = 2π/τ, the phases Φ_m(t) = Ψ(t − τ_m) of the stimulation signals S_m(t) are distributed uniformly over the unit circle as illustrated in Fig. 6a. Then the phases ψ_j(t) of the stimulated neuronal sub-population assigned to the stimulation site m are attracted to the phase Ψ(t − τ_m) of the corresponding stimulation signal. Hence, the phases of all oscillators stimulated with MLDF become symmetrically redistributed on the circle (0, 2π) in a cluster state. The order parameter R₁(t) thus gets minimized. Depending on the value of delay τ, the stimulation can induce different clustered states in the stimulated ensemble, where the corresponding order parameter R_m attains large values.

Fig. 6

Control of synchronization by multisite linear delayed feedback (MLDF) stimulation. (a) Distribution of the phases Φ_m(t) of the stimulation signals S_m(t) administered via four stimulation sites (as in Fig. 3a), which are the delayed mean phase, Φ_m(t) = Ψ(t − τ_m), with delays τ_m from Eq. 5 for τ = T. (b and c) Time courses of the amplitudes of the cluster variables (Eq. 1), the order parameters R₁, R₂ and R₄. In the subplots four trajectories from each of four stimulated sub-populations assigned to each of four different stimulation sites are shown for t ∈ (320,340). Parameter τ = T in (b) and τ = 2T in (c)

As shown in Fig. 6b, c, the in-phase synchronization in the stimulated ensemble is effectively suppressed (for time t > 200, where both coupling and stimulation are switched on), where the order parameter R₁ =  ∣ Z₁(t)∣ from Eq. 1 attains small values (Fig. 6b, c, red curve). This indicates a symmetrical redistribution of the oscillator phases ψ_j(t) over the unit circle. For the parameter τ close to the mean period T of the stimulation-free ensemble a four cluster state is induced by the stimulation, where the order parameters R₁ and R₂ are small, whereas R₄ is relatively large (Fig. 6b). In the subplot, where four trajectories from each of the stimulated sub-populations are depicted, the emerging four-cluster state induced by MLDF is illustrated. For τ closer to, for example, 2T the stimulation induces a two-cluster state, where R₁ is small, whereas R₂ and R₄ are large (Fig. 6c). The oscillators thus split into two clusters, which is also illustrated in the subplot in Fig. 6c.

MLDF robustly suppresses the in-phase synchronization as shown in Fig. 7a, where the time-averaged order parameter 〈R₁〉 attains small values for a broad range of parameters τ and K. On the other hand, depending on system and stimulation parameters, MLDF can induce either a two-cluster state, where the second order parameter R₂ attains relatively large values (e.g., for τ  ≈  2T, see Fig. 7b), or a four-cluster state, where R₂ becomes small and the fourth order parameter R₄ increases (e.g., for τ  ≈  T, see Fig. 7c). Therefore, the whole stimulated population is divided into two or four distinct sub-populations. Within the phase clusters the individual oscillators have phases close to each other, while the different phase clusters are equidistantly distributed within the cycle. Hence, depending on the values of the parameters τ and K, MLDF with four stimulation sites may cause either a two-cluster state, where R₁ is close to zero and R₂ is large, or a four-cluster state, where both R₁ and R₂ are small, but R₄ is large. The cluster states become less pronounced and the phases redistribute on the circle even more uniformly if a local coupling as well as spatially decaying profile of the current spread is taken into account (Hauptmann et al. 2005a).

Fig. 7

Impact of the MLDF stimulation versus parameters τ and stimulus amplification K. The time-averaged order parameters 〈R₁〉, 〈R₂〉, and 〈R₄〉 are depicted in plots (a–c), respectively, and encoded in color ranging from 0 (blue) to 1 (red). In plot (d) the impact of the single-site linear delayed feedback (SLDF) on the oscillatory population is illustrated, where the values of the order parameter 〈R₁〉 are depicted in color versus parameters τ and K (First published in Popovych et al. (2006a))

In Fig. 7d a similar two-parameter diagram for the averaged order parameter ⟨R₁(t)⟩ is presented for a single-site linear delayed feedback (SLDF) suggested for synchronization control in references (Rosenblum and Pikovsky 2004a, b). The stimulation is performed via one stimulation electrode in such a way that all oscillators of the ensemble (in the first approximation) receive the same stimulation signal S(t). In this case the stimulation signal S(t) attains the form S(t) = KZ(t − τ). For the stimulation with SLDF, in the corresponding two-parameter diagram (Fig. 7d) islands of perfect desynchronization are complemented by areas of stimulation-enhanced synchronization. In the limit N→∞ the order parameter R₁ = 0 in the desynchronization regions, where the phases are uniformly distributed on the circle (0,  2π) (Rosenblum and Pikovsky 2004a, b). This is the state of complete desynchronization, where the stimulated oscillators rotate with different frequencies indicating an absence of any clustered state whatsoever. The island-like structure of desynchronization regions in parameter space of SLDF (Fig. 7d) was also experimentally confirmed for arrays of coupled electrochemical oscillators (Zhai et al. 2008).

The important property of the stimulation with the multi- and single-site linear delayed feedback is the inherit demand-controlled character of the methods. As soon as the desired desynchronized state is achieved, the values of the order parameter R₁(t), i.e., the amplitude of the mean field become small. Along with the order parameter, in the desynchronized state the amplitude of the stimulation signal S(t) vanishes as well. The stimulation with multi- and single-site linear delayed feedback thus represents noninvasive control methods for desynchronization of coupled oscillators. The stimulated ensemble is then subjected to a highly effective control at a minimal amount of stimulation force.

Nonlinear Delayed Feedback

As for the case of the single-site linear delayed feedback, for the stimulation with nonlinear delayed feedback (NDF) only one registering and one stimulating site is required, see Fig. 8a. All stimulated oscillators receive the same stimulation signal S(t) which is constructed from the measured mean field of the ensemble. It is assumed that the measured mean field Z(t) of the ensemble has the form of a complex analytic signal Z(t) = X(t) + iY(t), where X(t) and Y(t) are the real and imaginary parts of Z(t), respectively. If only a real part X(t) of the mean field is measured, the imaginary part can be calculated, e.g., with the help of the Hilbert transform (Pikovsky et al. 2001). The stimulation signal is then constructed by a nonlinear combination of a delayed complex conjugate mean field with the instantaneous mean field (Popovych et al. 2005, 2006a, b; Tass et al. 2006),

$$ S(t)={KZ}^2(t)Z\left(t-\tau \right), $$

(6)

where K is a stimulus amplification parameter, τ is a time delay, and the asterisk denotes complex conjugacy.

Fig. 8

Control of synchronization by nonlinear delayed feedback (NDF) stimulation. (a) The macroscopic activity (mean field) of the controlled population is measured, delayed, nonlinearly combined with the instantaneous mean field, amplified, and fed back via a single stimulation site. (b and c) Desynchronization of strongly synchronized oscillators by NDF. Time courses of the order parameter R₁(t) (red curves) and the amplitude of the stimulation signal ∣S(t)∣ (blue curves) are plotted for delay (b) τ = T/2 and (c) τ = T, where T is the mean period of the stimulation-free ensemble. In the subplots trajectories of two selected oscillators are depicted in the stimulated regime (First published in Popovych et al. (2008))

The desynchronizing effect of the stimulation with NDF is illustrated in Fig. 8b, c. The onset of stimulation at t = 550 results in desynchronization of the stimulated oscillators and the order parameter R₁(t) reaches the values of approximately the same order of magnitude as in the uncoupled regime (t < 400). This indicates a high level of desynchronization. The stimulation does not destroy the normal oscillatory activity of the individual oscillators. In the insets in Fig. 8b, c individual trajectories of two selected oscillators of stimulated ensemble are plotted. The stimulated oscillators rotate with different individual frequencies just as in the coupling- and stimulation-free regime.

As soon as a desynchronized state is achieved, the stimulation force declines and the stimulated system is subjected to a highly effective control with a minimal amount of stimulation force. Also, as soon as a resynchronization occurs, the mean field starts to exhibit large-amplitude oscillations and the stimulation signal increases its amplitude and brings the ensemble back to a desynchronized state. This demand-controlled character of the nonlinear delayed feedback is illustrated in Fig. 8c where the onsets of resynchronization (increase of R₁(t), red curve) at times around t ≈ 850, 1,050, and 1,200 lead to an increase of the amplitude of the stimulation signal ∣S(t)∣ (blue curve), which in turn results in a suppression of the resynchronization.

The impact of the nonlinear delayed feedback on the stimulated oscillators is twofold. On one hand, the stimulation can effectively desynchronize even strongly interacting oscillators for a large range of the stimulus amplification K, see Fig. 9a. This effect is very robust with respect to the variation of the delay τ and, as a result, with respect to the variation of the mean frequency Ω of the stimulated ensemble. On the other hand, in a weakly coupled ensemble the stimulation can induce synchronization in island-like regions of small values of the stimulus amplification K complemented by domains of desynchronization, see Fig. 9b.

Fig. 9

Robustness of the NDF effects. (a) Stimulation-induced desynchronization and (b) stimulation-induced synchronization in ensembles of (a) strongly coupled and (b) weakly coupled oscillators. The time-averaged values of the order parameter ⟨R₁⟩ are encoded in color ranging from red (synchronization) to blue (desynchronization) versus delay τ and stimulus amplification K

An increase of the stimulus amplification parameter K results in a gradual decay of the order parameter R₁ for both strongly and weakly coupled oscillators, which indicates an onset of desynchronization in the stimulated ensemble. Simultaneously, the amplitude of the stimulation signal ∣S(t)∣ decays as well, indicating the demand-controlled character of the nonlinear delayed feedback stimulation. For a fixed delay τ > 0 the order parameter and the amplitude of the stimulation signal decay as ∣K∣ increases according to the following power law (Fig. 10a):

$$ {R}_1\sim {\left|K\right|}^{-1/2},\quad \mid S\mid \sim {\left|K\right|}^{-1/2}. $$

(7)

Fig. 10

Impact of NDF as the stimulus amplification K increases. (a) Log–log plot of the time-averaged order parameter ⟨R₁⟩ and amplitude of the stimulation signal ⟨|S(t)|⟩ versus K. The dashed line has the slope −0.5 and is given for comparison. (b) The observed individual frequencies ${\overline{\omega}}_j$ of the stimulated oscillators versus K. Blue diamonds at the right vertical axis depict values of the natural frequencies of the oscillators (First published in Popovych et al. (2006a))

The desynchronization transition for increasing K also manifests itself in a sequence of frequency-splitting bifurcations, where the observed individual frequencies ${\overline{\omega}}_j=\left\langle {\dot{\psi}}_j\right\rangle$ of the stimulated oscillators split, one after another from the mean frequency Ω as K increases (Fig. 10b) and approach the natural frequencies of the unperturbed oscillators (Fig. 10b, blue diamonds). For large values of K all stimulated oscillators rotate with different frequencies close to the natural frequencies ω_j. The oscillators thus exhibit a uniform desynchronous dynamics without any kind of cluster states. In addition, depending on the values of the delay τ, the nonlinear delayed feedback can significantly change the mean frequency Ω, i.e., the frequency of the mean field Z(t) of the stimulated ensemble (Popovych et al. 2005, 2006a). The macroscopic dynamics can thus be either accelerated or slowed down, whereas the individual dynamics remains close to the original one. This opens an approach for the frequency control of the oscillatory population stimulated with the nonlinear delayed feedback.

Mixed Nonlinear Delayed Feedback

The NDF method can also be applied for desynchronization and decoupling of two (or more) interacting oscillator populations. For this, the mixed NDF can be used (Popovych and Tass 2010), see Fig. 11a. For a drive-response coupling scheme, the coupling within population 2 is assumed to be weak, so that, being isolated from population 1, no synchronization emerges in population 2. In contrast, the coupling in population 1 is strong enough to cause synchronization within population 1. It then drives the second population, which synchronizes because of the driving and sends a response signal back to population 1. The second ensemble is stimulated with signal S(t), which is constructed from the mixed mean field Z_ε according to the rule of NDF from Eq. 6. The mixed mean field Z_ε = εW₁ + (1 − ε)W₂ is a linear combination of the mean fields W₁ and W₂ of populations 1 and 2, respectively.

Fig. 11

Desynchronization and decoupling of interacting populations by the mixed NDF. (a) Stimulation setup: The measured mean fields of populations 1 and 2 is linearly combined into a mixed mean field, processed by the NDF algorithm (Eq. 6), and fed back to the target population 2. (b and c) Time-averaged order parameter ⟨R₁⟩ of (b) intrinsically synchronized, drive population 1 and (c) stimulated population 2 driven to synchronization by population 1 versus time delay τ and mixing parameter ε. The color coding as in Fig. 9 (First published in Popovych and Tass (2010). Copyright (2010) by the American Physical Society)

The level of mixing of the mean fields W₁ and W₂ within the stimulation signal is given by the parameter ε. Depending on it the mixed NDF can have different desynchronizing effects on populations 1 and 2.

• Small ε: mostly population 2 contributes to the stimulation signal. The mixed NDF desynchronizes the driven and stimulated population 2 (Fig. 11b), but the driving ensemble 1 remains unaffected and exhibits strongly synchronized dynamics (Fig. 11c). The populations get effectively decoupled from each other.

• Intermediate ε: both populations equally contribute to the stimulation signal. Both ensembles remain synchronized (Fig. 11b, c).

• Large ε: mostly population 1 contributes to the stimulation signal. Both ensembles are effectively desynchronized by the mixed NDF (Fig. 11b, c).

In the latter case the desynchronization induced by the mixed NDF in the driven and stimulated population 2 propagates to the drive population 1 which is not directly stimulated. This is indicative of an indirect control of synchronization by the mixed NDF.

Proportional–Integro–Differential Feedback

For a particularly difficult situation, where the measurement and stimulation are not possible at the same time and at the same place, there is another control method which is based on a proportional–integro–differential feedback (PIDF). The scheme of this stimulation protocol is sketched in Fig. 12a, see also Fig. 11a for ε = 1 except for the measured signal being processed by a PIDF algorithm. The controlled ensemble of N coupled oscillators is divided into two separate sub-populations of N₁ and N₂ = N − N₁ oscillators, one being exclusively measured and the other being exclusively stimulated. In this way a separate stimulation-registration setup is realized, where the recording and stimulating sites are spatially separated and the measured signal is not corrupted by stimulation artifacts. The observed signal is considered to be the mean field W₁ of the measured sub-population. Below the main attention will be paid to the proportional-differential (PD) feedback only (for more details, see Pyragas et al. (2007)). Then, the stimulation signal S(t) administered to the second, stimulated sub-population is constructed as

$$ S(t)={PW}_1(t)+D{\dot{W}}_1(t), $$

(8)

where the parameters P and D define the strength of the proportional and differential feedback, respectively. The effect of the stimulation with PD feedback is illustrated in Fig. 12b. As the strength of the feedback (parameters P and D) increases the stimulation results in a complete desynchronization of the whole ensemble. The threshold of the onset of desynchronization depends on the relative splitting N₁ : N₂ of the oscillators between sub-populations and on the mean frequency Ω: The threshold is larger for smaller number of oscillators N₂ in the stimulated populations or for larger frequency Ω. The later dependence can be eliminated if an integral component is included in the stimulation signal, see Pyragas et al. (2007). Moreover, if the coupling in the ensemble is rather weak, the desynchronization can be achieved by applying the proportional feedback only. In contrast, in the case of strong coupling the stimulation signal additionally requires the differential feedback for robust desynchronization. As illustrated in the two-parameter diagrams in Fig. 12c, d, there exists a certain threshold in parameters P and D values, where the stimulation with PIDF desynchronizes both sub-populations in the target ensemble: stimulated sub-population (Fig. 12c) and also measured, non-stimulated sub-population (Fig. 12d). In this sense the PIDF stimulation method appears to be very effective even for a complicated stimulation protocol with a separate stimulation-registration setup.

Fig. 12

PIDF control: (a) The mean field is measured in one part of the controlled ensemble and, after processing according to proportional–integro–differential feedback (PIDF) algorithm, is administered to the other part of the ensemble. (b) The time-averaged order parameter ⟨R₁⟩ of the whole ensemble versus the strength of the PD feedback (with P = D) for different splitting N₁ : N₂ and different mean frequencies Ω. (c and d) The time-averaged order parameters ⟨R₁⟩ (encoded in color) of (c) the measured sub-population and (d) stimulated sub-population versus stimulation parameters P and D. The white curve is the parameter threshold for the onset of desynchronization in the sub-populations (First published in Pyragas et al. (2007). Used with permission from EPL)

Plasticity

Plasticity is a fundamental property of the nervous system: In order to learn and to adapt to sensory inputs, neurons continuously adapt the strength of their synaptic connections in relation to the mutual timing properties of their firing or bursting (Hebb 1949; Gerstner et al. 1996; Markram et al. 1997; Debanne et al. 1998; Kilgard and Merzenich 1998; Abbott and Nelson 2000; Feldman 2000; Song et al. 2000; van Hemmen 2001; Zhou et al. 2003). However, plasticity may not only lead to desired learning and optimization processes. Rather neuronal populations can learn pathologically strong interactions which may lead, e.g., to the emergence of epilepsies (Morimoto et al. 2004; Speckmann and Elger 1991). This is well-known from the so-called kindling phenomenon (Goddar 1967), where preparatory stimulation induces the spontaneous production of epileptic seizures without gross morphological changes (Morimoto et al. 2004).

The impact of plasticity on synaptic weights and collective neuronal dynamics has been accounted for by several theoretical studies on desynchronizing stimulation methods (Tass and Majtanik 2006; Hauptmann and Tass 2007; Tass and Hauptmann 2007; Maistrenko et al. 2007). They have initiated an approach which aims at unlearning pathologically strong synaptic interactions by desynchronizing brain stimulation and which has further been developed in latter papers (Hauptmann and Tass 2009, 2010; Tass and Popovych 2012; Popovych and Tass 2012). This approach exploits plasticity in two different ways: On the one hand, due to plasticity desynchronizing stimulation may decrease the strength of the neurons’ synapses by decreasing the rate of coincidences. On the other hand, neuronal networks with synaptic plasticity may exhibit bi- or multistability (Seliger et al. 2002; Tass and Majtanik 2006; Hauptmann and Tass 2007; Tass and Hauptmann 2007; Maistrenko et al. 2007). Accordingly, by decreasing the mean synaptic weight, desynchronizing stimulation may shift a neuronal population from a stable synchronized (pathological) state to a stable desynchronized (healthy) state, where the neuronal population remains thereafter, if left unperturbed. In Fig. 13 an exemplary simulation of a model neural network is displayed, for further details concerning the mathematical model we refer to references (Hauptmann and Tass 2007; Tass and Hauptmann 2007). Induced by appropriate stimulation protocols a switching between the different stable states is realizable. Starting from a desynchronized state, associated with a physiological model dynamics, low-frequency stimulation can induce a kindling of the synaptic connectivity and causes a stabilization of the synchronized state. After stimulation offset, the system remains in the pathological state (Fig. 13). In contrast, desynchronizing CR stimulation (see section “Coordinated Reset Stimulation”) results in an anti-kindling of the pathological connectivity and, finally, the physiological weakly coupled and desynchronized state is reestablished (Fig. 13).

Fig. 13

Effects of kindling and anti-kindling stimulation on a population of model neurons. Low frequency stimulation is applied between 50 and 200 s, and CR stimulation (see section “Coordinated Reset Stimulation”) is applied between 300 and 550 s. The local field potential, the level of synchronization within the network and the mean connectivity is plotted (from top to bottom). Five patterns representing the coupling topology of the network at different times are plotted below. Blue (red) colors represent low (high) values of the interneuronal connectivity

From a mathematical point of view, in a first approximation this situation may be illustrated by considering a double well potential, where each minimum corresponds to a stable attractor, surrounded by a basin of attraction (Fig. 14). The strongly synchronized state (Pathol. in Fig. 14) serves as a model for a disease state, whereas the uncorrelated or weakly synchronized state (Healthy in Fig. 14) is used as a model for a healthy state. As soon as the system, i.e., the neuronal population (illustrated by the ball in Fig. 14), enters a particular basin of attraction, it gets attracted by the corresponding attractor, so that it relaxes towards the corresponding minimum of the potential.

Fig. 14

Kindling and anti-kindling stimulation can move the neuronal population from one attractor to another: Schematic plot of the attractors symbolizing pathological (Pathol.) or healthy (Healthy) dynamical model states. (a) Periodic, kindling stimulation (red trajectory) shifts the population from a healthy, desynchronized state (Healthy) to a pathological, synchronized state (Pathol.). (b) Conversely, desynchronizing stimulation shifts the population from a pathological state (Pathol.) to the healthy uncorrelated state (Healthy). This anti-kindling is achieved by a desynchronizing stimulation of sufficient duration, so that after stimulus offset the population is close to the healthy state (red trajectory). (c) Alternatively, the same long-term anti-kindling effect can be achieved with a brief desynchronizing stimulation, which shifts the population to an intermediate state (red trajectory), which may still be connected with pronounced synchrony. However, since the intermediate state (blue ball) lies within the basin of attraction of a healthy state (Healthy) the population spontaneously relaxes to the healthy state without any further intervention (blue trajectory)

Appropriate stimulation protocols may shift the neuronal population from one state to another. Kindling stimulation of appropriate duration shifts the neuronal population from a desynchronized state close to a strongly synchronized state or at least into the basin of attraction of such a state (Fig. 14a, red trajectory from Healthy to Pathol.). Conversely, anti-kindling can be achieved by means of a desynchronizing stimulation which shifts the neuronal population close to the desynchronized state (Tass and Majtanik 2006; Hauptmann and Tass 2007, 2009, 2010; Tass and Hauptmann 2007; Tass and Popovych 2012; Popovych and Tass 2012) (Fig. 14b, red trajectory from Pathol. to Healthy). However, with respect to the long-term anti-kindling outcome, it is even sufficient to move the neuronal population from the synchronized state just into the basin of attraction of the desynchronized state (Fig. 14c, red trajectory from Pathol. to the intermediate state). After stimulus offset there may still be pronounced synchrony, but being captured within the basin of attraction of the desynchronized state, without further intervention the neuronal population spontaneously relaxes to the desynchronized state (Fig. 14c, blue trajectory from the intermediate state to Healthy). Note the short as well as the long desynchronizing stimulation in this schematic illustration have the same long-term anti-kindling outcome.

In PD neuronal populations of the basal ganglia are strongly synchronized (Beurrier et al. 2002; Schnitzler et al. 2006; Timmermann et al. 2007) and synaptic plasticity results in a further amplification of the synchronized activity by a strengthening of the synaptic connections (Nowotny et al. 2003). Properly designed electrical stimulation may be used to break this vicious circle and to induce an anti-kindling (Tass and Majtanik 2006; Hauptmann and Tass 2007, 2009, 2010; Tass and Hauptmann 2007; Tass and Popovych 2012; Popovych and Tass 2012), which finally might reestablish the normal level of connectivity, associated with a mostly uncorrelated neuronal activity. In this way a sustained long-lasting desynchronization can be achieved, and therapeutic after-effects can be expected after the cessation of desynchronizing stimulation as predicted computationally (see references above). In parkinsonian MPTP monkeys it was shown that unilateral CR stimulation delivered to the subthalamic nucleus (STN) for only 2 h per day during 5 days leads to significant and sustained therapeutic aftereffects for at least 30 days, while standard 130 Hz DBS has no aftereffects (Tass et al. 2012b).

Closed-Loop DBS

The standard setup of HF DBS assumes an open-loop stimulation protocol, where, after the corresponding parameter calibration, a permanent HF electrical pulse train is administered to the target nucleus without relation to the ongoing neuronal activity (Benabid et al. 1991; Volkmann 2004). Both, clinical studies and modeling studies systematically investigated the influence of stimulation parameters and focused on the optimization of the standard HF DBS via an appropriate parameter calibration (Rizzone et al. 2001; Moro et al. 2002; Rubin and Terman 2004) including a closed-loop optimization setup (Feng et al. 2007a, b).

In monkeys rendered parkinsonian with the neurotoxin MPTP a closed-loop DBS has been tested under acute conditions (Rosin et al. 2011). To this end, a short train (comprising seven pulses at 130 Hz) was delivered through a pair of electrodes located in the GPi at a predetermined, fixed latency (80 ms) following each action potential recorded through an electrode placed in the primary motor cortex (M1). This type of stimulation caused a strong decrease of the firing rate of the pallidal neurons together with a pronounced decrease of the oscillatory neuronal activity at tremor frequency (4–7 Hz) and at double tremor frequency (9–15 Hz) along with an amelioration of the MPTP-induced akinesia. After cessation of this type of closed-loop DBS the initial firing pattern reverted back, i.e., pallidal firing rate and pallidal oscillatory activity attained pre-stimulus levels (Rosin et al. 2011). In contrast, standard continuous 130 Hz DBS caused a less pronounced decrease of the pallidal firing rate, the oscillatory neuronal activity and the amelioration of the akinesia (Rosin et al. 2011).

Another study (Little et al. 2013) confirmed the efficacy of the closed-loop adaptive DBS (aDBS) in PD patients, where the onsets and offsets of HF stimulation were triggered by a threshold crossing by LFP in beta band measured via the same stimulation electrode implanted in STN. The stimulation trigger threshold for the LFP amplitude was heuristically determined in such a way that a reduction of the stimulation time of approximately 50% was achieved while maintaining clinical effect. The onset of HF stimulation was delayed by 30–40 ms after the crossing of the threshold by LFP, and the stimulation was sustained until beta amplitude fell below the threshold again (Little et al. 2013). For the same stimulation intensity and stimulation frequency (130 Hz), the aDBS can be about 30% more effective than standard continuous HF DBS, while less than 50% of the total electrical energy is delivered in the aDBS mode as compared to continuous HF DBS. Moreover, despite of the used fixed beta threshold, the triggered stimulation duration (per 10-s block) progressively drops over time during stimulation in the aDBS mode, which suggests that aDBS may lead to positive adaptive effects in pathological Parkinsonian networks (Little et al. 2013).

Summary

High-frequency deep brain stimulation is the golden standard for the treatment of medically refractory movement disorders (Benabid et al. 1991; Volkmann 2004).

Apart from the empirically developed standard DBS protocols (Benabid et al. 1991; Volkmann 2004), new stimulation approaches were successfully tested in pre-clinical and early clinical settings. For instance, a closed-loop neurostimulation controlled by beta-band activity showed a better performance than classical DBS in reducing motor signs as well as pallidal firing rate and oscillatory activity in parkinsonian nonhuman primates and PD patients (Rosin et al. 2011; Little et al. 2013).

Another approach, i.e., the model-based development of novel deep brain stimulation techniques, especially targets the pathological neuronal synchrony associated with Parkinson’s disease (Tass 1999). This approach bases on dynamic neuronal self-organization principles and fundamental plasticity rules of the nervous system (Fig. 15) (Tass and Majtanik 2006; Tass and Hauptmann 2006, 2007; Hauptmann and Tass 2007, 2009, 2010; Tass and Popovych 2012; Popovych and Tass 2012).

Fig. 15

Schematic illustration of unlearning of pathological connectivity by desynchronizing stimulation. The latter reduces the overall rate of coincidences in the neuronal population. This is effectively achieved by using dynamic self-organization principles. The reduction of the rate of coincidences, in turn, reduces the synaptic weights and shifts the stimulated network in a weakly coupled state. Because of the multistability, not only strongly coupled and synchronized states, but also weakly coupled and weakly synchronized states are stable. Accordingly, the neuronal population stably remains in a desynchronized or weakly synchronized state, after having been shifted into the basin of attraction of that state by means of desynchronizing stimulation. Hence, a suitable control of the dynamics of the network can lead to long-lasting changes of its connectivity and dynamics (First published in Tass and Hauptmann (2007). Used with permission from the International Journal of Psychophysiology)

The control methods discussed in this article differ to each other with respect to the stimulation setup and stimulation effects as well as other properties such as robustness and applicability. For example, CR stimulation in an open-loop protocol uses standard stimulation pulses (as used for HF stimulation) applied in a dedicated pattern and its technical realization was proven to be feasible (Hauptmann et al. 2009; Tass et al. 2012b). CR does not require sophisticated calibration and effectively causes transient desynchronization of the stimulated oscillators via a stimulation-induced cluster state. As mentioned in section “Coordinated Reset Stimulation,” a number of computational, experimental and pre-clinical studies confirmed the applicability and efficacy of CR stimulation under different stimulation modalities.

Other smart feedback methods are more difficult to realize conceptually and technically and await experimental proof of concept, see sections “Multisite Linear Delayed Feedback,” “Nonlinear Delayed Feedback,” and “Proportional–Integro–Differential Feedback.” In modeling studies they effectively result in a sustained desynchronized regime. The feedback methods can be applied under a variety of conditions, and possess an intrinsic demand-controlled character, where the stimulation signal is significantly reduced or even vanishes as soon as desynchronization is achieved. The experimental and clinical realization of these methods is a challenging task, first of all, from the technical side, since stimulation signals have to fulfill all safety aspects like charge density limits. These limits strongly affect the applicability of the slow feedback signals, i.e., slow compared to the timescales of HF and CR pulses. In this way, the application conditions of the methods have to be handled with care, and stimulation setups and effects have to clearly be distinguished for different feedback methods. Otherwise, one can come up with misleading conclusions and erroneous interpretations of the efficacy of feedback methods, see the computational study (Dovzhenok et al. 2013) where nonlinear and linear techniques were not properly distinguished. To this end, only an experimental proof of concept can finally assess the applicability and performance of the control methods.

In forthcoming studies the mathematical modeling needs to be refined to incorporate further anatomical and physiological details, for instance, the contributions of glial cells (Silchenko and Tass 2008). By the same token, control techniques have to be optimized and further developed. As currently done for CR stimulation (Tass et al. 2012b), clinical studies are necessary to evaluate the therapeutic effects of the novel stimulation techniques under real conditions. This interdisciplinary endeavor might finally provide superior therapies for patients with neurological or psychiatric diseases.