Failure of Delayed Feedback Deep Brain Stimulation for Intermittent Pathological Synchronization in Parkinson's Disease

Suppression of excessively synchronous beta-band oscillatory activity in the brain is believed to suppress hypokinetic motor symptoms of Parkinson's disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize the synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson's disease exhibits complex intermittent synchronous patterns, far from the idealized synchronous dynamics used to study the delayed feedback stimulation. This study explores the action of delayed feedback stimulation on partially synchronized oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a model of the basal ganglia networks which reproduces experimentally observed fine temporal structure of the synchronous dynamics. When the parameters of our model are such that the synchrony is unphysiologically strong, the feedback exerts a desynchronizing action. However, when the network is tuned to reproduce the highly variable temporal patterns observed experimentally, the same kind of delayed feedback may actually increase the synchrony. As network parameters are changed from the range which produces complete synchrony to those favoring less synchronous dynamics, desynchronizing delayed feedback may gradually turn into synchronizing stimulation. This suggests that delayed feedback DBS in Parkinson's disease may boost rather than suppress synchronization and is unlikely to be clinically successful. The study also indicates that delayed feedback stimulation may not necessarily exhibit a desynchronization effect when acting on a physiologically realistic partially synchronous dynamics, and provides an example of how to estimate the stimulation effect.Comment: 19 pages, 8 figure

Complex partial synchronization patterns in networks of delay-coupled neurons

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

Uncertainty Principle for Control of Ensembles of Oscillators Driven by Common Noise

We discuss control techniques for noisy self-sustained oscillators with a focus on reliability, stability of the response to noisy driving, and oscillation coherence understood in the sense of constancy of oscillation frequency. For any kind of linear feedback control--single and multiple delay feedback, linear frequency filter, etc.--the phase diffusion constant, quantifying coherence, and the Lyapunov exponent, quantifying reliability, can be efficiently controlled but their ratio remains constant. Thus, an "uncertainty principle" can be formulated: the loss of reliability occurs when coherence is enhanced and, vice versa, coherence is weakened when reliability is enhanced. Treatment of this principle for ensembles of oscillators synchronized by common noise or global coupling reveals a substantial difference between the cases of slightly non-identical oscillators and identical ones with intrinsic noise.Comment: 10 pages, 5 figure

Mechanisms of Zero-Lag Synchronization in Cortical Motifs

Zero-lag synchronization between distant cortical areas has been observed in a diversity of experimental data sets and between many different regions of the brain. Several computational mechanisms have been proposed to account for such isochronous synchronization in the presence of long conduction delays: Of these, the phenomenon of "dynamical relaying" - a mechanism that relies on a specific network motif - has proven to be the most robust with respect to parameter mismatch and system noise. Surprisingly, despite a contrary belief in the community, the common driving motif is an unreliable means of establishing zero-lag synchrony. Although dynamical relaying has been validated in empirical and computational studies, the deeper dynamical mechanisms and comparison to dynamics on other motifs is lacking. By systematically comparing synchronization on a variety of small motifs, we establish that the presence of a single reciprocally connected pair - a "resonance pair" - plays a crucial role in disambiguating those motifs that foster zero-lag synchrony in the presence of conduction delays (such as dynamical relaying) from those that do not (such as the common driving triad). Remarkably, minor structural changes to the common driving motif that incorporate a reciprocal pair recover robust zero-lag synchrony. The findings are observed in computational models of spiking neurons, populations of spiking neurons and neural mass models, and arise whether the oscillatory systems are periodic, chaotic, noise-free or driven by stochastic inputs. The influence of the resonance pair is also robust to parameter mismatch and asymmetrical time delays amongst the elements of the motif. We call this manner of facilitating zero-lag synchrony resonance-induced synchronization, outline the conditions for its occurrence, and propose that it may be a general mechanism to promote zero-lag synchrony in the brain.Comment: 41 pages, 12 figures, and 11 supplementary figure

Synchronization of coupled neural oscillators with heterogeneous delays

We investigate the effects of heterogeneous delays in the coupling of two excitable neural systems. Depending upon the coupling strengths and the time delays in the mutual and self-coupling, the compound system exhibits different types of synchronized oscillations of variable period. We analyze this synchronization based on the interplay of the different time delays and support the numerical results by analytical findings. In addition, we elaborate on bursting-like dynamics with two competing timescales on the basis of the autocorrelation function.Comment: 18 pages, 14 figure

Desynchronization of systems of coupled Hindmarsh-Rose oscillators

It is widely assumed that neural activity related to synchronous rhythms of large portions of neurons in specific locations of the brain is responsible for the pathology manifested in patients' uncontrolled tremor and other similar diseases. To model such systems Hindmarsh-Rose (HR) oscillators are considered as appropriate as they mimic the qualitative behaviour of neuronal firing. Here we consider a large number of identical HR-oscillators interacting through the mean field created by the corresponding components of all oscillators. Introducing additional coupling by feedback of Pyragas type, proportional to the difference between the current value of the mean-field and its value some time in the past, Rosenblum and Pikovsky (Phys. Rev. E 70, 041904, 2004) demonstrated that the desirable desynchronization could be achieved with appropriate set of parameters for the system. Following our experience with stabilization of unstable steady states in dynamical systems, we show that by introducing a variable delay, desynchronization is obtainable for much wider range of parameters and that at the same time it becomes more pronounced.Comment: 5 pages, 2 figures, RevTe

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback