Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which \textit{unstable attractors} arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.Comment: 14 pages, 12 figure

Possible Origin of Stagnation and Variability of Earth's Biodiversity

The magnitude and variability of Earth's biodiversity have puzzled scientists ever since paleontologic fossil databases became available. We identify and study a model of interdependent species where both endogenous and exogenous impacts determine the nonstationary extinction dynamics. The framework provides an explanation for the qualitative difference of marine and continental biodiversity growth. In particular, the stagnation of marine biodiversity may result from a global transition from an imbalanced to a balanced state of the species dependency network. The predictions of our framework are in agreement with paleontologic databases.Comment: 5 pages, 6 pages supplemen

Stochastic Resonance in Neuron Models: Endogenous Stimulation Revisited

The paradigm of stochastic resonance (SR)---the idea that signal detection and transmission may benefit from noise---has met with great interest in both physics and the neurosciences. We investigate here the consequences of reducing the dynamics of a periodically driven neuron to a renewal process (stimulation with reset or endogenous stimulation). This greatly simplifies the mathematical analysis, but we show that stochastic resonance as reported earlier occurs in this model only as a consequence of the reduced dynamics.Comment: Some typos fixed, esp. Eq. 15. Results and conclusions are not affecte

Statistics of Extreme Waves in Random Media

Waves traveling through random media exhibit random focusing that leads to extremely high wave intensities even in the absence of nonlinearities. Although such extreme events are present in a wide variety of physical systems and the statistics of the highest waves is important for their analysis and forecast, it remains poorly understood in particular in the regime where the waves are highest. We suggest a new approach that greatly simplifies the mathematical analysis and calculate the scaling and the distribution of the highest waves valid for a wide range of parameters

Long Chaotic Transients in Complex Networks

We show that long chaotic transients dominate the dynamics of randomly diluted networks of pulse-coupled oscillators. This contrasts with the rapid convergence towards limit cycle attractors found in networks of globally coupled units. The lengths of the transients strongly depend on the network connectivity and varies by several orders of magnitude, with maximum transient lengths at intermediate connectivities. The dynamics of the transient exhibits a novel form of robust synchronization. An approximation to the largest Lyapunov exponent characterizing the chaotic nature of the transient dynamics is calculated analytically.Comment: 4 pages; 5 figure

Quantum Reversibility: Is there an Echo?

We study the possibility to undo the quantum mechanical evolution in a time reversal experiment. The naive expectation, as reflected in the common terminology ("Loschmidt echo"), is that maximum compensation results if the reversed dynamics extends to the same time as the forward evolution. We challenge this belief, and demonstrate that the time $t_r$ for maximum return probability is in general shorter. We find that $t_r$ depends on $lambda = eps_evol/eps_prep$, being the ratio of the error in setting the parameters (fields) for the time reversed evolution to the perturbation which is involved in the preparation process. Our results should be observable in spin-echo experiments where the dynamical irreversibility of quantum phases is measured.Comment: 4 pages, 4 figures, to be published in Phys. Rev. Let

Correlations and Synchrony in Threshold Neuron Models

We study how threshold model neurons transfer temporal and interneuronal input correlations to correlations of spikes. We find that the low common input regime is governed by firing rate dependent spike correlations which are sensitive to the detailed structure of input correlation functions. In the high common input regime the spike correlations are insensitive to the firing rate and exhibit a universal peak shape independent of input correlations. Rate heterogeneous pairs driven by common inputs in general exhibit asymmetric spike correlations. All predictions are confirmed in in vitro experiments with cortical neurons driven by synthesized fluctuating input currents.Comment: 5 pages, 10 figure