We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill-defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine the isophases for experimentally observed irregular oscillations, providing a basis for a refined phase description of observed oscillatory dynamics.

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control.

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We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.

Neural circuit motifs producing coexistent rhythmic patterns are treated as building blocks of multifunctional neuronal networks. We study the robustness of such a motif of inhibitory model neurons to reliably sustain bursting polyrhythms under random perturbations. Without noise, the exponential stability of each of the coexisting rhythms increases with strengthened synaptic coupling, thus indicating an increased robustness. Conversely, after adding noise we find that noise-induced rhythm switching intensifies if the coupling strength is increased beyond a critical value, indicating a decreased robustness. We analyze this stochastic arrhythmia and develop a generic description of its dynamic mechanism. Based on our mechanistic insight, we show how physiological parameters of neuronal dynamics and network coupling can be balanced to enhance rhythm robustness against noise. Our findings are applicable to a broad class of relaxation-oscillator networks, including Fitzhugh-Nagumo and other Hodgkin-Huxley-type networks.

We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincaré surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled of the amplitude dynamics, and provides a proper description of phase resetting of chaotic oscillations. The method is illustrated with the Rössler and Lorenz systems.

The dynamics of neurons is characterized by a variety of different spiking patterns in response to external stimuli. One of the most important transitions in neuronal response patterns is the transition from tonic firing to burst discharges, i.e., when the neuronal activity changes from single spikes to the grouping of spikes. An increased number of interspike-interval sequences of specific temporal correlations was detected in anticipation of temperature induced tonic-to-bursting transitions in both, experimental impulse recordings from hypothalamic brain slices and numerical simulations of a stochastic model. Analysis of the modelling data elucidates that the appearance of such patterns can be related to particular system dynamics in the vicinity of the period-doubling bifurcation. It leads to a nonlinear response on de- and hyperpolarizing perturbations introduced by noise. This explains why such particular patterns can be found as reliable precursors of the neurons' transition to burst discharges.

SUMMARYAppearances of alpha waves in the sleep electrencephalogram indicate physiological, brief states of awakening that lie in between wakefulness and sleep. These microstates may also cause the loss in sleep quality experienced by individuals suffering from insomnia. To distinguish such pathological awakenings from physiological ones, differences in alphawave characteristics between transient awakening and wakefulness observed before the onset of sleep were studied. In polysomnographic datasets of sleep-healthy participants (n = 18) and patients with insomnia (n = 10), alpha waves were extracted from the relaxed, wake state before sleep onset, wake after sleep-onset periods and arousals of sleep. In these, alpha frequency and variability were determined as the median and standard deviation of inverse peak-to-peak intervals. Before sleep onset, patients with insomnia showed a decreased alpha variability compared with healthy participants (P < 0.05). After sleep onset, both groups showed patterns of decreased alpha frequency that was lower for wake after sleep-onset periods of shorter duration. For patients with insomnia, alpha variability increased for short wake after sleep-onset periods. Major differences between the two groups were encountered during arousal. In particular, the alpha frequency in patients with insomnia rebounded to wake levels, while the frequency in healthy participants remained at the reduced level of short wake after sleeponset periods. Reductions in alpha frequency during wake after sleeponset periods may be related to the microstate between sleep and wakefulness that was described for such brief awakenings. Reduced alpha variability before sleep may indicate a dysfunction of the alpha generation mechanism in insomnia. Alpha characteristics may also prove valuable in the study of other sleep and attention disorders.