Desynchronization (computational neuroscience)

Desynchronization is a process inverse to synchronization, where initially synchronized oscillating systems desynchronize as parameters change or they do so under the influence of an external force or feedback. Desynchronization is important, for example, in neuroscience and medicine, where pathologically strong synchronization of neurons may severely impair brain function as, e.g., in Parkinson's disease or epilepsy.

This article briefly overviews several methods for the control of (de)synchronization in oscillatory networks.

[edit] Ensemble of coupled oscillators

Figure 1: Synchronization in the Kuramoto model (1): order parameter \(R\) versus coupling strength (bottom plot) and the corresponding distributions (snapshots) of the phases in the plane \((\cos(\psi_j), \sin(\psi_{j}))\ .\)

The widely accepted model capturing fundamental properties of the collective dynamics of interacting oscillators like oscillatory neurons is the Kuramoto model of globally coupled phase oscillators (Kuramoto 1984, Strogatz 2000, Acebron et al. 2005) \[\tag{1} \dot{\psi}_j = \omega_j +\frac{C}{N} \sum_{k=1}^N \sin(\psi_k-\psi_j), \quad j = 1, 2, \dots, N. \]

The phases \(\psi_{j}\) characterize the oscillatory dynamics of the elements of the ensemble and increase by \(2\pi\) after each completed cycle. For oscillating neurons, for instance, the cycle can be defined as the time period between two successive spikes or bursts.

For the inhomogeneous ensemble (1), i.e., if the natural frequencies \(\omega_j\) are different, the phase oscillators remain desynchronized and oscillate with different frequencies if the coupling among the oscillators (parameter \(C\)) is sufficiently weak. The desynchronization-synchronization transition takes place in system (1) if the coupling among the oscillators increases. In the limit \(N \to \infty\) and if the natural frequencies \(\omega_j\) are randomly chosen from a unimodal symmetric probability density \(g(\omega)\ ,\) \(g(\Omega+\omega)=g(\Omega-\omega)\ ,\) where \(\Omega\) is the mean frequency, the critical coupling of spontaneous synchronization is given by (Kuramoto 1984, Strogatz 2000) \[\tag{2} C_{cr} = \frac{2}{\pi g(\Omega)}. \]

For \( C < C_{cr} \ ,\) the system relaxes to an incoherent state, where all oscillators are not synchronized, but for \(C > C_{cr}\ ,\) mutual synchronization occurs in a group of oscillators. This transition can be characterized by values of the order parameter \(R(t)\) calculated as \[\tag{3} R(t)\exp(i\Psi(t)) = \displaystyle{\frac{1}{N}} \sum_{j = 1}^{N}\exp(i \psi_j(t)), \]

where \(\Psi(t)\) is the mean phase. The state of in-phase synchronization, were all phases are close to each other \(\psi_{j} \approx \psi_{k}\ ,\) is characterized by large values of \(R \approx 1\) ( Figure 1). For a desynchronized state, where the phases are uniformly distributed on the circle \((0, 2\pi)\ ,\) the order parameter is small, \(R \approx 0\ ,\) and scales as \(R \sim 1/\sqrt{N}\) with the number of elements \(N\) -- so-called finite-size effect (Pikovsky et al. 2001).

The problem of desynchronization consists in designing a method which can effectively desynchronize an ensemble of strongly synchronized oscillators by delivering a stimulation signal \(S(t)\) to the oscillators. The efficacy and the main properties of several desynchronization methods will be illustrated below on exemplary phase ensemble (1). However, the discussed methods have successfully been applied to more complicated and realistic models including neuronal ensembles. Some of them have been tested experimentally as well, see the bibliography for further reading at the end of the corresponding sections.

[edit] Single-pulse stimulation

Figure 2: Two identical single pulses are delivered at the ensemble's vulnerable phase (Tass 2001b). The first single pulse hits the ensemble in a stably synchronized state and causes a desynchronization. The second single pulse is delivered to the ensemble in a weakly synchronized state. Instead of causing a desynchronization, the second single pulse strongly synchronizes the ensemble.

A single pulse of appropriate strength delivered at a vulnerable phase of a fully synchronized population of phase oscillators desynchronizes the population both in the absence (Winfree 1980) and in the presence (Tass 1999) of noise, see Figure 2, where the time course of the firing density \(p(t)\ ,\) the average number density of the oscillators which have zero phase at time \(t\) is shown. The vulnerable phase denotes the critical value of the mean phase \(\Psi_{cr}(t)\) of synchronized ensemble at which the population is most capable to exhibit stimulation-induced desynchronization (Tass 1999, see also Winfree 1977). The strength of an effectively desynchronizing pulse is typically weaker than that of a resetting single pulse. Effective desynchronization, however, requires that the vulnerable phase and the stimulus strength are thoroughly calibrated. Hence, variations of system parameters require a recalibration. There is another limitation of this stimulation technique that is relevant, in particular, for possible clinical applications to deep brain stimulation (Tass 2000). The effect of a single pulse crucially depends on the initial conditions of the population. For desynchronization a single pulse has to hit the fully synchronized population at the vulnerable phase ( Figure 2). Delivered to an only partially or weakly synchronized population, such a pulse will cause a synchronization ( Figure 2) (Tass 2001b). Hence, such a stimulus cannot be used to reliably prevent from resynchronization.

Bibliography for further reading: (Zhai et al. 2005).

[edit] Double-pulse stimulation

Figure 3: Two identical double pulse stimuli are delivered to the initially stably synchronized population (Tass 2001b). Both double pulse stimuli desynchronize the population -- irrespective of the population's initial state at stimulus onset. Accordingly, double-pulse stimulation enables to block the resynchronization.

Motivated by possible clinical applications, stimulation techniques have been developed which robustly cause a desynchronization -- irrespective of the initial state of the neuronal population at stimulus onset (Tass 2001b). For this purpose, the double-pulse method uses two qualitatively different stimuli: The first, stronger pulse resets the collective dynamics, irrespective of its initial state at stimulus onset. The second, weaker pulse follows after a calibrated delay; it hits the population in the vulnerable phase and causes a desynchronization (Tass 2001a, Tass 2001b). The pause between the first and the second pulse as well as the strength of, in particular, the second pulse have to be calibrated thoroughly. Instead of a strong first pulse which causes a hard reset, i.e., a rapid reset achieved within at most a cycle of the synchronized oscillation, alternatively, one can also utilize a soft reset (Tass 2002a, Tass 2002b). The latter is typically achieved by a brief low-frequency entraining pulse train, which causes a reset within several cycles of the synchronized oscillation. A soft reset may be superior, e.g., whenever strong stimuli may cause tissue damage (Tass 2002a, Tass 2002b). However, with double-pulse stimulation one still has to face the problem that biologically inevitable variations of system parameters require frequent recalibration.

[edit] Coordinated reset stimulation

Figure 4: Mechanism of action of CR: Brief and mild resetting stimuli are administered at different sites at subsequent times and cause an effective transient desynchronization. Desynchronized firing of neurons is maintained by repetitive administration of CR stimuli.

The motivation behind the development of coordinated reset stimulation was to design a robustly desynchronizing stimulation technique which is considerably more robust with respect to variations of system parameters as opposed to single- and double-pulse stimulation.

The mechanism of action of coordinated reset (CR) stimulation is schematically shown in Figure 4. The main idea behind CR stimulation is to achieve a desynchronization indirectly, by shifting the network into an unstable state, from where it transiently relaxes into a desynchronized state. For this, weak phase resetting stimuli are sequentially delivered at different sites (Tass 2003a, Tass 2003b). In this way the oscillatory population is split into distinct clusters related to the different stimulation sites, respectively. In the case of electrical stimulation of brain tissue the resetting stimuli are brief high-frequency pulse trains. The latter are sequentially delivered with a delay of \(T/n\ ,\) where \(n\) is the number of the stimulation sites, and \(T\) approximates the mean period of the ensemble. \(T = 2\pi /\Omega\ ,\) where \(\Omega\) is the mean frequency of the ensemble.

Figure 5: Phase clusters induced by CR stimulation delivered via four stimulation sites. Snapshot of the phases with coordinates \((\cos(\psi_j), \sin(\psi_{j}))\) aligned on the (black) unit circle.

Modelling the effect of electrical pulse stimulation on the phase dynamics of a neuron, the stimulation signal for the phase oscillator \(j\) from the ensemble (1) attains the form \(S_{j}(t) = I X(t)\cos(\psi_{j})\ ,\) where \(I\) is the intensity of the stimulation and \(X(t) = 1\) if the oscillator \(j\) is stimulated at time \(t\) and \(X(t) = 0\) otherwise (Tass 1999, Tass 2003b, Tass 2003a). The term which has to be added to the right-hand side of Eq. (1) then takes the form \[\tag{4} \sum_{m = 1}^{n} I_{m} \rho_{m,j} X_{m}(t) \cos(\psi_{j}), \]

where \(X_{m}(t)\) is the pulse train administered to the \(m\)th stimulation site and parameters \(\rho_{m,j}\) model the spatial profile of the current spread.

CR stimulation splits the whole population into an \(n\)-cluster state, i.e., into \(n\) different phase clusters ( Figure 5). An \(n\)-cluster state can be detected with the cluster variables \[\tag{5} Z_{n} = R_{n}\exp(i\Psi_{n}) = \frac{1}{N} \sum_{j = 1}^{N} \exp(i n \psi_{j}), \]

where \(R_n(t)\) and \(\Psi_n(t)\) are the corresponding real amplitude (order parameter of the degree \(n\)) and real phase, where \(0 \le R_n(t) \le 1\) for all times \(t\ .\) Cluster variables are convenient for characterizing synchronized states of different types: Perfect in-phase synchronization corresponds to \(R_1 = 1\) (\(R_{1}\) equals the order parameter \(R\) from Eq. (3)), whereas an incoherent state with uniformly distributed phases is associated with \(R_n = 0\) (\(n = 1, 2, 3, \dots\)). Small values of \(R_1\) combined with large values of \(R_n\) are indicative of an \(n\)-cluster states consisting of \(n\) distinct and equally spaced clusters, where all oscillators have similar phase within each cluster, see Figure 5, where \(R_{1} \approx 0.02\) and \(R_{4} \approx 0.87\ .\)

Figure 6: Demand-controlled CR stimulation via four stimulation sites. Upper plot: periodical application of CR stimuli of demand-controlled duration (based on values of \(R_1\)); bottom plot: demand-controlled timing (based on values of \(R_1\)) of the administration of identical CR stimuli. Horizontal bars symbolize CR stimuli.

The stimulation mechanism of CR stimulation is as follows ( Figure 4): After a few periods of stimulation the oscillatory population is shifted into a cluster state. After switching off the stimulation, the ensemble returns to a synchronized state, on this way transiently passing through a uniformly desynchronized state. This procedure is repeated, so that the ensemble is kept in a desynchronized state. The repetitive stimulus administration can be performed either regardless of the state of the stimulated ensemble (open loop control), or in a demand-controlled way (closed loop control). In the later case, e.g., either identical stimuli are delivered whenever \(R_1\) exceeds a critical threshold ( Figure 6, lower plot), or CR stimuli are periodically delivered with a stimulus duration that is adapted to \(R_{1}\) ( Figure 6, upper plot).

Bibliography for further reading: (Tass 1999, Tass 2003a, Tass 2003b, Tass et al. 2006, Tass et al. 2009, Lysyansky et al. 2011, Tass et al. 2011).

[edit] Linear multisite delayed feedback

Figure 7: The macroscopic activity (mean field) of the controlled population is measured, delayed, amplified and fed back in a spatially coordinated way via several stimulation sites using different delays for different stimulation sites.

Similarly to coordinated reset stimulation, the linear multisite delayed feedback is delivered via several stimulation sites, e.g. via four sites as illustrated in Figure 7, where separate stimulation signals are applied via each site. These separate stimulation signals are derived from the delayed mean field of the ensemble by using different time delays for the 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 of individual oscillators \(Z(t) = N^{-1}\sum_{j=1}^{N} z_j(t)\ ,\) where \(z_j(t)\ ,\) \(j = 1, \dots , N\) are the signals of the individual elements of the ensemble. In the phase representation, \(Z(t)\) attains the form of \(Z_{1}(t)\) from Eq. (5).

For \(n\) stimulation sites, the stimulation signals are calculated as \(S_{m}(t) =KZ(t - \tau_{m})\ ,\) \(m = 1,\dots,n\ ,\) where the values of delay, for example, for \(n = 4\) are calculated from the following relations \[\tag{6} \tau_m = \frac{11-2(m-1)}{8}\tau, \qquad m = 1,2,3,4. \]

The delays \(\tau_m\) are symmetrically distributed with respect to the main delay \(\tau\ ,\) where the smallest time distance between neighboring stimulation sites is chosen as \(\tau/4\ .\) In the case \(\tau = T\) (mean period of the ensemble), the delays \(\tau_m\) are uniformly distributed over the mean period \(T\ .\)

Figure 8: Upper-left plot: Distribution of the delayed phases \(\Psi(t-\tau_m)\) of the stimulation signals \(S_{m}(t)\) of the multisite delayed feedback for four stimulation sites and for \(\tau = T\) in Eq. (6). The other plots illustrate the stimulation-induced clustered states indicated by the time-averaged order parameters \(\langle R_{1} \rangle\ ,\) \(\langle R_{2} \rangle\ ,\) and \(\langle R_{4} \rangle\) (encoded in color) versus delay \(\tau\) and stimulus amplification \(K\ .\)

In the framework of the phase ensemble (1) the stimulation signals attain the form \(S_{m}(t) = K R(t - \tau_{m}) \sin(\Psi(t - \tau_{m}) - \psi_{j}(t))\) and the term added to the right-hand side of Eq. (1) reads \[\tag{7} \sum_{m = 1}^{n} \rho_{m,j} S_{m}(t), \]

where the parameters \(\rho_{m,j}\) play the same role as for the coordinated reset stimulation (see Eq. (4)). Assuming that the mean phase \(\Psi(t)\) rotates with a constant frequency \(\Omega\ ,\) the delayed mean phases \(\Psi(t-\tau_{m})\) are uniformly distributed on the unit circle (see Figure 8, upper left plot). Then the phases \(\psi_{j}(t)\) of the sub-population assigned to the stimulation site \(m\) are attracted to the phase \(\Psi(t-\tau_{m})\) of the corresponding stimulation signals. Hence, the phases of all oscillators stimulated with the multisite delayed feedback are redistributed symmetrically on the circle \((0,2\pi)\) in a cluster state. The order parameter \(R_{1}(t)\) gets thus minimized. Depending on the values of delay \(\tau\) in Eq. (6), the stimulation can induce phase clusters in the stimulated ensemble, where the corresponding order parameter \(R_{m}\) attains large values, see Figure 8. For example, for the optimal value of the delay \(\tau = T\) the stimulation induces a four-cluster state (for four stimulation sites), where \(R_{1}\) and \(R_{2}\) are small and \(R_{4}\) is larger. For other values of \(\tau\ ,\) for instance, for \(\tau = 2T\) the stimulated ensemble exhibits a two-cluster dynamics, where \(R_{1}\) is small and \(R_{2}\) is large. The cluster states become less pronounced, and the phases redistribute on the circle even more uniformly if in the model ensemble a local coupling as well as spatially decaying profile of the current spread is taken into account (Hauptmann et al. 2005a).

An important property of the linear multisite delayed feedback stimulation is its inherent demand-controlled character. As soon as the desired desynchronized state is achieved, the values of the order parameter \(R(t)\) become small and, thus, the amplitude of the stimulation signals \(S_{m}(t)\) becomes small as well. The stimulated ensemble is then subjected to a highly effective control at a minimal amount of stimulation force.

Bibliography for further reading: (Hauptmann et al. 2005a, Hauptmann et al. 2005c, Hauptmann et al. 2005b, Tass et al. 2006).

[edit] Linear single-site delayed feedback

Figure 9: Impact of the single-site linear delayed feedback on the collective dynamics of the stimulated oscillators (1) versus delay \(\tau\) and stimulus amplification \(K\ .\) The values of the time-averaged order parameter \(\langle R \rangle\) (see Eq. (3)) are encoded in color ranging from red (synchronization) to blue (desynchronization).

Stimulation with linear single-site delayed feedback utilizes only one recording site and, in contrast to the above methods, only one stimulation site. In a first approximation, all oscillators of the ensemble are stimulated with the same stimulation signal \(S(t)\) constructed from the delayed mean field of the ensemble \(S(t) = KZ(t-\tau)\ .\) For the coupled phase oscillators (1) the stimulation signal \(S(t) = KR(t-\tau)\sin(\Psi(t-\tau) - \psi_{j}(t))\) is added to the right-hand side of Eq. (1).

Depending of the values of the stimulation parameters \(\tau\) and \(K\) the stimulation can result either in an enhancement or in a suppression of synchronization in the stimulated ensemble. The impact of the stimulation is illustrated in Figure 9. Red is the region of parameters \((\tau,K)\ ,\) where the extent of synchronization is enhanced (large values of \(R\)). Blue indicates regions of desynchronization (small values of \(R\)). In the limit \(N \to \infty\) the order parameter \(R = 0\) in the desynchronization regions, where the phases are uniformly distributed on the circle \((0,2\pi)\) (Rosenblum & Pikovsky 2004a, Rosenblum & Pikovsky 2004b). This is the state of complete desynchronization, where the stimulated oscillators rotate with different frequencies indicating an absence of any ordered state whatsoever. Along with the order parameter, in the desynchronized state the amplitude of the stimulation signal \(S(t)\) vanishes as well. In this sense, the stimulation with linear single-site delayed feedback represents a noninvasive control method for desynchronization of coupled oscillators.

Bibliography for further reading: (Rosenblum & Pikovsky 2004a, Rosenblum & Pikovsky 2004b, Rosenblum et al. 2006, Zhai et al. 2008).

[edit] Nonlinear delayed feedback

Figure 10: 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.

As in the case of the linear single-site delayed feedback, for the stimulation with nonlinear delayed feedback (NDF) only one registration and one stimulation site are required, see Figure 10. All stimulated oscillators receive the same stimulation signal \(S(t)\) which is constructed from the 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. In the case if only a real part \(X(t)\) of the mean filed is measured, the imaginary part can be constructed, e.g., with the help of the Hilbert transform.

The stimulation signal is then constructed by a nonlinear combination of a delayed complex conjugate mean field with the instantaneous mean field \[\tag{8} S(t) = K Z^2(t)Z^{\ast}(t - \tau), \]

where \(K\) is a stimulus amplification parameter, \(\tau\) is a time delay, and the asterisk denotes complex conjugacy. For the phase oscillators (1) the stimulation signal added to the right-hand side of Eq. (1) attains the form \[\tag{9} S(t) = KR^2(t)R(t-\tau) \sin \left ( 2\Psi(t) - \Psi(t-\tau)-\psi_{j}(t) \right ). \]

Figure 11: Nonlinear delayed feedback: stimulation-induced desynchronization and synchronization in ensembles of strongly (left plot) or weakly (right plot) coupled oscillators (1). The time-averaged values of the order parameter \(\langle R \rangle\) are encoded in color ranging from red (synchronization) to blue (desynchronization) versus delay \(\tau\) and stimulus amplification \(K\) (Popovych et al. 2005, Popovych et al. 2006a).

The impact of the NDF on the stimulated oscillators is twofold. On one hand, the stimulation can effectively desynchronize even strongly interacting oscillators for a broad range of values of the stimulus amplification \(K\ ,\) see Figure 11, left plot. This effect is very robust with respect to the variation of the delay \(\tau\) and, as a result, with respect to the variation of the mean frequency \(\Omega\) of the stimulated ensemble. On the other hand, in a weakly coupled ensemble the stimulation can induce synchronization in small, island-like parameter regions complemented by wide domains of desynchronization, see Figure 11, right plot.

An increase of the stimulus amplification parameter \(K\) results in a gradual decay of the order parameter \(R\) ( Figure 12, upper plot). This enables a precise control of the extent of desynchronization in the stimulated ensemble. Simultaneously, the amplitude of the stimulation signal \(\vert S(t) \vert\) decays as well, indicating a demand-controlled character of the NDF stimulation. For a fixed delay \(\tau > 0\) the order parameter and the amplitude of the stimulation signal decay according to the following power law as \(\vert K \vert\) increases: \[\tag{10} R \sim \vert K \vert ^{-1/2}, \quad \vert S \vert \sim \vert K \vert^{-1/2}, \]

see Figure 12, upper plot. The desynchronization transition for increasing \(K\) also manifests itself in a sequence of frequency-splitting bifurcations ( Figure 12, bottom plot), where the observed individual frequencies \(\overline{\omega}_{j} = \langle \dot{\psi}_{j} \rangle\) of the stimulated oscillators split, one after another away from the mean frequency \(\Omega\) and approach the natural frequencies \(\omega_j\) of the oscillators (black diamonds in Figure 12, bottom plot) as \(K\) increases.

Figure 12: Time-averaged order parameter \(\langle R \rangle\) and amplitude of the stimulation signal \(\langle \vert S(t) \vert \rangle\) (upper plot, log-log scale) and the observed individual frequencies \(\overline{\omega}_{j}\) of the stimulated oscillators (1) (bottom plot) versus the stimulus amplification parameter \(K\) (Popovych et al. 2005, Popovych et al. 2006a).

For large values of \(K\) all stimulated oscillators thus rotate with the frequencies close to the natural ones and exhibit a uniform desynchronous dynamics without any kind of cluster states. Simultaneously, depending on the values of the delay \(\tau\ ,\) the NDF can significantly change the mean frequency \(\Omega\ ,\) i.e., the frequency of the mean field \(Z(t)\) of the stimulated ensemble (Popovych et al. 2005, Popovych et al. 2006a). The macroscopic dynamics can thus be either accelerated or slowed down, whereas the individual dynamics remains close to the original one (without stimulation and coupling). This opens up an approach for a frequency control of the oscillatory population stimulated with the NDF.

The above properties, namely, the robustness with respect to parameter variation (compare Figure 11 to Figure 9), precise control of the extent of synchronization, and the frequency control distinguish the NDF desynchronizing method from the linear single-site delayed feedback. Another peculiarity of NDF is an indirect control of synchronization as discussed below.

In the case of two (or more) oscillator populations interacting according to a drive-response coupling scheme, a particular stimulation setup, called mixed nonlinear delayed feedback, can be applied (Popovych & Tass 2010) ( Figure 13, upper plot). The coupling within population \(2\) is assumed to be weak, so that -- 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, driven ensemble is stimulated with signal \(S(t)\ ,\) which is constructed from a linear combination \(S_{\varepsilon}\) of the mean fields \(Z\) and \(W\) of populations \(1\) and \(2\ ,\) respectively, according to the rule of the NDF, see Eq. (8) \[\tag{11} S(t) = K S_{\varepsilon}^2(t) S^{\ast}_{\varepsilon}(t - \tau), \quad S_{\varepsilon}(t) = \varepsilon Z(t) + (1-\varepsilon)W(t). \]

The level of mixing of the mean fields \(Z\) and \(W\) within the stimulation signal is given by the parameter \(\varepsilon\ .\) If \(\varepsilon = 0\ ,\) only the driven and stimulated population \(2\) is measured, and if \(\varepsilon = 1\ ,\) only the drive population \(1\) contributes to the stimulation signal. For intermediate values of the mixing \(\varepsilon \in (0,1)\) the mixed dynamics of both ensembles is used as stimulation signal. Such a mixing signal can be used for stimulation in the case where only different linear combinations of the mean fields of the drive and response ensembles can be measured, e.g., due to electrophysiological conditions.

Figure 13: Stimulation setup of mixed NDF (upper plot) and the time-averaged order parameter \(\langle R \rangle\) of the not stimulated, drive population (bottom left) and the stimulated, response population (bottom right) encoded in color versus time delay \(\tau\) and mixing parameter \(\varepsilon\) (Popovych & Tass 2010).

Depending on the level of mixing \(\varepsilon\) different regimes can be observed ( Figure 13, bottom plots):

• small values of \(\varepsilon\ :\)

The mixed NDF desynchronizes the driven and stimulated population \(2\) ( Figure 13, bottom right plot), but the driving ensemble \(1\) remains unaffected and exhibits strongly synchronized dynamics ( Figure 13, bottom left plot). The populations are effectively decoupled from each other.

• intermediate values of \(\varepsilon\ :\)

Both ensembles are synchronized.

• large values of \(\varepsilon\ :\)

Both ensembles are effectively desynchronized by the stimulation, which is indicative of an indirect control of synchronization by the mixed NDF.

Bibliography for further reading: (Popovych et al. 2005, Popovych et al. 2006a, Popovych et al. 2006b, Tass et al. 2006, Majtanik 2007, Popovych & Tass 2010).

[edit] Proportional-integro-differential feedback

Figure 14: PID Control: The mean field is measured in one part of the controlled ensemble and, after processing according to the proportional-integro-differential algorithm, it is administered to the other part of the ensemble.

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 (PID) feedback. The scheme of this stimulation protocol is sketched in Figure 14. The controlled ensemble of \(N\) coupled oscillators is split into two separate sub-populations of \(N_{1}\) and \(N_{2} = N - N_{1}\) 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 \(Z_{1}\) of the measured sub-population. Below, only the proportional-differential (PD) feedback is illustrated (for more details, see (Pyragas et al. 2007)). The stimulation signal \(S(t)\) which is delivered to the second, stimulated sub-population is constructed as

\[\tag{12} S(t) = P Z_{1}(t) + D\dot{Z_{1}}(t), \]

where the parameters \(P\) and \(D\) define the strength of the proportional and differential feedback, respectively. In the framework of the phase oscillators (1) one arrives to the following equation for the phases \(\psi_{j}\ :\)

\[\tag{13} \dot{\psi}_j = \omega_j +\frac{C}{N} \sum_{k=1}^N \sin(\psi_k-\psi_j) - H(j-N_1)F_j, \]

\[\tag{14} F_j = \frac{P}{N_1}\sum_{k=1}^{N_1}\sin(\psi_k-\psi_j)+ \frac{D}{N_1}\sum_{k=1}^{N_1} \dot\psi_k \cos(\psi_k-\psi_j), \]

Figure 15: Upper plot: The time averaged order parameter \(\langle R \rangle\) of the whole population (13), (14) versus the strength of the PD feedback (with \(P = D\)) for different splitting ratios \(N_{1}:N_{2}\) and different mean frequencies \(\Omega\ .\) Bottom plot: The values of the order parameter \(\langle R \rangle\) (encoded in color) of the measured sub-population (left) and stimulated sub-population (right) versus stimulation parameters \(P\) and \(D\ .\) The white curve is the parameter threshold of the onset of desynchronization in both sub-populations obtained in (Pyragas et al. 2007).

where \(F_j\) is the corresponding phase representation of the above stimulation signal \(S\) and \(H(\cdot)\) is the Heaviside function defined as \(H(k)=0\) if \(k\leq0\) and \(H(k)=1\) if \(k>0\ .\)

The effect of the stimulation with PD feedback is illustrated in Figure 15. As the strength of the feedback (parameters \(P\) and \(D\)) increases the stimulation results in a complete desynchronization of both, measured and stimulated sub-populations. The threshold of the onset of desynchronization depends on the splitting ratio \(N_{1}:N_{2}\) of the size of the sub-populations and on the mean frequency \(\Omega\ :\) The threshold is larger for a smaller number of oscillators \(N_2\) in the stimulated population or for larger frequency \(\Omega\ .\) 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 must also contain the differential feedback for robust desynchronization.

[edit] Effects of desynchronizing stimulation in the presence of synaptic plasticity

The nervous system is able to learn and to adapt to sensory inputs (Hebb 1949). Neurons continuously adapt the strength of their synaptic connections in relation to the mutual timing properties of their firing or bursting by means of the spike timing-dependent plasticity (STDP) (Herz et al. 1989, Gerstner et al. 1996, Markram et al. 1997, Debanne et al. 1998, Kilgard & Merzenich 1998, Abbott & Nelson 2000, Feldman 2000, Song et al. 2000, van Hemmen 2001, Zhou et al. 2003). STDP is fundamental for neuronal information processing and memory (Abbott & Nelson 2000, Dan & Poo 2004, Morrison et al. 2008). However, it may also contribute to pathological process. Neuronal populations may learn to pathologically upregulate their interactions, which may lead, e.g., to the emergence of epileptic activity. This mechanism is well known from the so-called kindling phenomenon (Morimoto et al. 2004, Speckmann & Elger 1991, Goddar 1967).

Externally applied desynchronizing brain stimulation was recently proposed as a potential candidate for reversing the kindling process, i.e., for unlearning pathologically strong synaptic interactions (Tass & Majtanik 2006, Hauptmann & Tass 2007, Tass & Hauptmann 2007).

Two basic ingredients form the basis of this concept:

• Desynchronizing stimulation may decrease the strength of the neurons' synapses by decreasing the rate of coincidences.
• Neuronal networks with STDP may exhibit bi- or multistability.

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.

Figure 16: Illustration of the effects of kindling and anti-kindling stimulation on a population of neuronal bursters: the local field potential \(LFP\) (top), the mean synaptic connectivity (middle) and the synchronization measure (bottom) are plotted as a function of time. Periods ON stimulation are indicated by red bars and vertical lines. Anti-kindling coordinated reset stimulation is performed in a demand-controlled manner here. The same results are, however, obtained with open-loop coordinated reset stimulation. For details of the model and its parameters see (Hauptmann & Tass 2007, Tass & Hauptmann 2007).

The kindling and anti-kindling processes are illustrated in Figure 16. A kindling stimulation, i.e., spatially homogeneous periodic low-frequency stimulation, is first administered to the initially desynchronized population of bursting neurons. The synaptic connectivities are modified following a simplified synaptic plasticity rule with symmetric spike timing characteristics (Debanne et al. 1998, Magee & Johnston 1997, Abbott & Nelson 2000, Kepecs et al. 2002, Wittenberg & Wang 2006, Pfister & Gerstner 2006). The stimulation induces an increase of the rate of coincident bursts, which, in turn, results in an increase of the corresponding synaptic connections, see Figure 16 (middle plot). Induced by the kindling stimulation a synchronized state is established, which stably persists if left unperturbed. The stimulation thus shifts the population from a stable desynchronized state (modeling a healthy state) characterized by rather weak coupling coefficients to a stable synchronized states (modeling disease states) of strongly coupled oscillators. In a second step, an anti-kindling stimulation, i.e., desynchronizing coordinated reset stimulation, is applied to the kindled population (Hauptmann & Tass 2007, Tass & Hauptmann 2007). It results in a reduction of the rate of coinciding bursts and leads to a reduction of the synaptic connections, which, finally, ends up in a stabilization of a desynchronized state which also persists thereafter if left unperturbed, see Figure 16.

Desynchronizing stimuli (Tass & Majtanik 2006, Hauptmann & Tass 2007, Tass & Hauptmann 2007) have the potential to shift the population from the basin of attraction of a stable synchronized state into the basin of attraction of a stable desynchronized state. This concept might have substantial impact on novel therapeutic stimulation strategies for the therapy of neurological and psychiatric diseases characterized by abnormal synchrony. In fact, long-lasting desynchronizing effects of coordinated reset stimulation have been observed in rat hippocampal slice rendered epileptic by magnesium withdrawal (Tass et al. 2009). Another aspect which is relevant from the neuroscientific and, especially, clinical standpoint is the stimulation effect on physiological synaptic connections. In particular, do physiological (e.g., sensory input-related) synaptic connections survive the desynchronization-induced anti-kindling, or are they erased together with pathological (i.e., up-regulated) connections? In a first computational study (Hauptmann & Tass 2010) CR stimulation turned out to restore physiological synaptic connections during the anti-kindling process.

Bibliography for further reading: (Tass & Majtanik 2006, Hauptmann & Tass 2007, Tass & Hauptmann 2007, Tass & Hauptmann 2009, Tass et al. 2009, Hauptmann & Tass 2009, Hauptmann & Tass 2010).

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