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For a feedforward loop of oscillatory Hodgkin-Huxley neurons interacting via excitatory chemical synapses, we show that a great variety of spatiotemporal periodic firing patterns can be encoded by properly chosen communication delays and synaptic weights, which contributes to the concept of temporal coding by spikes. These patterns can be obtained by a modulation of the multiple coexisting stable in-phase synchronized states or traveling waves propagating along or against the direction of coupling. We derive explicit conditions for the network parameters allowing us to achieve a desired pattern. Interestingly, whereas the delays directly affect the time differences between spikes of interacting neurons, the synaptic weights control the phase differences. Our results show that already such a simple neural circuit may unfold an impressive spike coding capability. [ABSTRACT FROM AUTHOR]

Spike timing-dependent plasticity is a fundamental adaptation mechanism of the nervous system. It induces structural changes of synaptic connectivity by regulation of coupling strengths between individual cells depending on their spiking behavior. As a biophysical process its functioning is constantly subjected to natural fluctuations. We study theoretically the influence of noise on a microscopic level by considering only two coupled neurons. Adopting a phase description for the neurons we derive a two-dimensional system which describes the averaged dynamics of the coupling strengths. We show that a multistability of several coupling configurations is possible, where some configurations are not found in systems without noise. Intriguingly, it is possible that a strong bidirectional coupling, which is not present in the noise-free situation, can be stabilized by the noise. This means that increased noise, which is normally expected to desynchronize the neurons, can be the reason for an antagonistic response of the system, which organizes itself into a state of stronger coupling and counteracts the impact of noise. This mechanism, as well as a high potential for multistability, is also demonstrated numerically for a coupled pair of Hodgkin-Huxley neurons. [ABSTRACT FROM AUTHOR]

Periodic pulse train stimulation is generically used to study the function of the nervous system and to counteract disease-related neuronal activity, e.g., collective periodic neuronal oscillations. The efficient control of neuronal dynamics without compromising brain tissue is key to research and clinical purposes. We here adapt the minimum charge control theory, recently developed for a single neuron, to a network of interacting neurons exhibiting collective periodic oscillations. We present a general expression for the optimal waveform, which provides an entrainment of a neural network to the stimulation frequency with a minimum absolute value of the stimulating current. As in the case of a single neuron, the optimal waveform is of bang-off-bang type, but its parameters are now determined by the parameters of the effective phase response curve of the entire network, rather than of a single neuron. The theoretical results are confirmed by three specific examples: two small-scale networks of FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a large-scale network of synaptically coupled quadratic integrate-and-fire neurons. [ABSTRACT FROM AUTHOR]

We derive an algorithm for constructing an optimal waveform that ensures an entrainment of a spiking neuron to an external periodic stimulation with the minimum mean absolute value of the stimulating current. Such an optimization problem is relevant for the design of stimulation waveforms employed by implantable pulse generators for the treatment of brain disorders. Specifically, it is motivated by the desire to minimize tissue damage. The optimization problem is solved by involving the phase reduction theory and Pontriagyn's maximum principle. We show that the optimal waveform is of bang-off-bang type. The theoretical results are confirmed by three specific examples: an analytically solvable Stuart-Landau oscillator, a randomly generated phase response curve, and the classical Hodgkin-Huxley neuron model. [ABSTRACT FROM AUTHOR]

We study the collective response of small random tree networks of diffusively coupled excitable elements to stimuli applied to leaf nodes. Such networks model the morphology of certain sensory neurons that possess branched myelinated dendrites with excitable nodes of Ranvier at every branch point and at leaf nodes. Leaf nodes receive random inputs along with a stimulus and initiate action potentials that propagate through the tree. We quantify the collective response registered at the central node using mutual information. We show that in the strong-coupling limit, the statistics of the number of nodes and leaves determines the mutual information. At the same time, the collective response is insensitive to particular node connectivity and distribution of stimulus over leaf nodes. However, for intermediate coupling, the mutual information may strongly depend on the stimulus distribution among leaf nodes. We identify a mechanism behind the competition of leaf nodes that leads to nonmonotonous dependence of mutual information on coupling strength. We show that a localized stimulus given to a tree branch can be occluded by the background firing of unstimulated branches, thus suppressing mutual information. Nonetheless, the mutual information can be enhanced by a proper stimulus localization and tuning of coupling strength.

Recently, we have presented spatially modulated delayed feedback as a novel mechanism, which generically generates chimera states, remarkable spatiotemporal patterns in which coherence coexists with incoherence [O. E. Omel'chenko, Phys. Rev. Lett. 100, 044105 (2008)]. Remarkably, such chimera states serve as a natural link between completely coherent states and completely incoherent states. So far, we have studied this mechanism with a self-consistency-based numerical analysis only. In contrast, in this paper we perform a thorough dynamical description and, in particular, a stability analysis of the emerging chimera states. For this, we apply a recently developed reduction procedure [A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. 101, 264103 (2008)]. By combining analytical and numerical approaches, we systematically describe the relationship between the parameters of the delayed feedback on one hand and the properties of the chimera states on the other hand. We provide the general rules for an effective control and manipulation of the chimera states.

We show that a nonlinear coupling with delayed feedback between two limit-cycle oscillators can lead to phase-locked, periodically modulated, and chaotic phase synchronization as well as to desynchronization. Parameter regions with stable phase-locked states attain the well-known form of the swallows or shrimps found and studied for nonlinear maps. We demonstrate that the swallow regions can be accompanied by a different bifurcation scenario where the periodic orbits of the phase-locked states undergo a torus bifurcation instead of a previously reported period-doubling bifurcation. This property has an impact on the spatial organization of the swallows in the parameter space. The swallow regions contribute to the synchronization domain of the considered system, and we analytically approximate the parameter synchronization threshold.

We propose a method for the control of synchronization in two oscillator populations interacting according to a drive-response coupling scheme. The response ensemble of oscillators, which gets synchronized because of a strong forcing by the intrinsically synchronized driving ensemble, is controlled by mixed nonlinear delayed feedback. The stimulation signal is constructed from the mixed macroscopic activities of both ensembles. We show that the suggested method can effectively decouple the interacting ensembles from each other, where the natural desynchronous dynamics can be recovered in a demand-controlled way either in the stimulated ensemble, or, intriguingly, in both stimulated and not stimulated populations. We discuss possible therapeutic applications in the context of the control of abnormal brain synchrony in loops of affected neuronal populations.

Chimera states are remarkable spatiotemporal patterns in which coherence coexists with incoherence. As yet, chimera states have been considered as nongeneric, since they emerge only for particular initial conditions. In contrast, we show here that in a network of globally coupled oscillators delayed feedback stimulation with realistic (i.e., spatially decaying) stimulation profile generically induces chimera states. Intriguingly, a bifurcation analysis reveals that these chimera states are the natural link between the coherent and the incoherent states.

We studied the transient dynamics of synchronized coupled neuronal bursters subjected to repeatedly applied stimuli, using a hybrid neuroelectronic system of paddlefish electroreceptors. We show experimentally that the system characteristically undergoes poststimulus transients, in which the relative phases of the oscillators may be grouped in several clusters, traversing alternate phase trajectories. These signature transient dynamics can be detected and characterized quantitatively using specific statistical measures based on a stochastic approach to transient oscillator responses.

We present a simplified phase model for neuronal dynamics with spike timing-dependent plasticity (STDP). For asymmetric, experimentally observed STDP we find multistability: a coexistence of a fully synchronized, a fully desynchronized, and a variety of cluster states in a wide enough range of the parameter space. We show that multistability can occur only for asymmetric STDP, and we study how the coexistence of synchronization and desynchronization and clustering depends on the distribution of the eigenfrequencies. We test the efficacy of the proposed method on the Kuramoto model which is, de facto, one of the sample models for a description of the phase dynamics in neuronal ensembles.

For a system of two phase oscillators coupled with delayed self-feedback we study the impact of pulsatile stimulation administered to both oscillators. This system models the dynamics of two coupled phase-locked loops (PLLs) with a finite internal delay within each loop. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. Remarkably, for large coupling strength the two PLLs are completely desynchronized. We study stimulus-locked responses emerging in the different dynamical regimes. Simple phase resets may be followed by a response clustering, which is intimately connected with long poststimulus resynchronization. Intriguingly, a maximal perturbation (i.e., maximal response clustering and maximal resynchronization time) occurs, if the system gets trapped at a stable manifold of an unstable saddle fixed point due to appropriately calibrated stimulus. Also, single stimuli with suitable parameters can shift the system from a stable synchronized state to a stable desynchronized state or vice versa. Our result show that appropriately calibrated single pulse stimuli may cause pronounced transient and/or long-lasting changes of the oscillators' dynamics. Pulse stimulation may, hence, constitute an effective approach for the control of coupled oscillators, which might be relevant to both physical and medical applications.

Various stimulation desynchronization techniques are explored in a laboratory experiment on electrochemical oscillators, a system that exhibits transient dynamics, heterogeneities, and inherent noise. Stimulation with a short, single pulse applied at a vulnerable phase can effectively desynchronize a cluster. A double pulse method, that can be applied at any phase, can be improved either by adding an extra weak pulse between the original two pulses or by adding noise to the first pulse.

A complex high-dimensional chaotic behavior, phase chaos, is found in the finite-dimensional Kuramoto model of coupled phase oscillators. This type of chaos is characterized by half of the spectrum of Lyapunov exponents being positive and the Lyapunov dimension equaling almost the total system dimension. Intriguingly, the strongest phase chaos occurs for intermediate-size ensembles. Phase chaos is a common property of networks of oscillators of very different natures, such as phase oscillators, limit-cycle oscillators, and chaotic oscillators, e.g., Rössler systems.

We show that nonlinear delayed feedback opens up novel means for the control of synchronization. In particular, we propose a demand-controlled method for powerful desynchronization, which does not require any time-consuming calibration. Our technique distinguishes itself by its robustness against variations of system parameters, even in strongly coupled ensembles of oscillators. We suggest our method for mild and effective deep brain stimulation in neurological diseases characterized by pathological cerebral synchronization.

A model of two n:m coupled phase oscillators is studied, where both oscillators are subject to random forces, but only one oscillator is repetitively stimulated with a pulsatile stimulus. The focus of the paper is on transmission of transient responses as well as transient synchronization and desynchronization, which are stimulus locked, i.e., tightly time locked to the stimulus. A bistability or multistability of stable synchronized states of the two-phase oscillators (modulo 2pi ) occurs due to the n:m coupling. Accordingly, after stimulation the two oscillators may tend to qualitatively different stable states, which leads to a cross-trial (CT) response clustering (i.e., a switching between qualitatively different poststimulus responses across trials) of either one of the oscillators or both. A stochastic CT phase resetting analysis allows one to detect such transient responses and provides a reliable estimation of the transmission time. In contrast, CT averaging (averaging over an ensemble of responses), CT standard deviation, and CT cross correlation fail in studying the transmission of such stimulus-locked responses, even in the simpler case of 1:1 coupling. In particular, even though being used as golden standard for the analysis of evoked responses in medicine and neuroscience, CT averaging typically causes severe artifacts and misinterpretations.

A model of two coupled phase oscillators is presented, where the oscillators are subject to random forces and are stimulated at different times. Transient phase dynamics, synchronization, and desynchronization, which are stimulus locked (i.e., tightly time locked to a repetitively administered stimulus), are investigated. Complex coordinated responses, in terms of a noise-induced switching across trials between qualitatively different responses, may occur when the two oscillators are reset close to an unstable fixed point of their relative phases. This can be achieved with an appropriately chosen delay between the two stimuli. The switching of the responses shows up as a coordinated cross-trial (CT) response clustering of the oscillators, where the two oscillators produce two different pairs of responses. By varying noise amplitude and coupling strength we observe a stochastic resonance and a coupling-mediated resonance of the CT response clustering, respectively. The presented data analysis method makes it possible to detect such processes in numerical and experimental signals. Its time resolution is enormous, since it is only restricted by the time resolution of the preprocessing necessary for extracting the phases from experimental data. In contrast, standard data analysis tools applied across trials relative to stimulus onset, such as CT averaging (where an ensemble of poststimulus responses is simply averaged), CT standard deviation, and CT cross correlation, fail in detecting complex coordinated responses and lead to severe misinterpretations and artifacts. The consequences for the analysis of evoked responses in medicine and neuroscience are significant and are discussed in detail.

This paper is devoted to the desynchronizing effects of bipolar stimuli on a synchronized cluster of globally coupled phase oscillators. The bipolar pulses considered here are symmetrical and consist of a positive and a negative monopolar pulse. A bipolar single pulse with the right intensity and duration desynchronizes a synchronized cluster provided the stimulus is administered at a vulnerable initial phase of the cluster's order parameter. A considerably more effective desynchronization is achieved with a bipolar double pulse consisting of two qualitatively different bipolar pulses. The first bipolar pulse is stronger and resets the cluster, so that the second bipolar pulse, which follows after a constant delay, hits the cluster in a vulnerable state and desynchronizes it. A bipolar double pulse desynchronizes the cluster independently of the cluster's dynamical state at the beginning of the stimulation. The dynamics of the order parameter during a bipolar single pulse or a bipolar double pulse is different from the dynamics during a monopolar single pulse or a monopolar double pulse. Nevertheless, concerning their desynchronizing effects the monopolar and the bipolar stimuli are comparable, respectively. This is significant for applications where bipolar stimulation is required. For example, in medicine and physiology charge-balanced stimulation is typically necessary in order to avoid tissue damage. Based on the results presented here, demand-controlled bipolar double-pulse stimulation is suggested as a milder and more efficient therapy compared to the standard permanent high-frequency deep brain stimulation in neurological patients.