Levy Flights in Inhomogeneous Media

We investigate the impact of external periodic potentials on superdiffusive random walks known as Levy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Levy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Levy index $\mu$. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdiffusion is caused by topological complexity, such as diffusion on folded polymers and scale-free networks.Comment: 4 pages, 4 figure

Sequential Desynchronization in Networks of Spiking Neurons with Partial Reset

The response of a neuron to synaptic input strongly depends on whether or not it has just emitted a spike. We propose a neuron model that after spike emission exhibits a partial response to residual input charges and study its collective network dynamics analytically. We uncover a novel desynchronization mechanism that causes a sequential desynchronization transition: In globally coupled neurons an increase in the strength of the partial response induces a sequence of bifurcations from states with large clusters of synchronously firing neurons, through states with smaller clusters to completely asynchronous spiking. We briefly discuss key consequences of this mechanism for more general networks of biophysical neurons

Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao

Metal-insulator transitions in cyclotron resonance of periodic nanostructures due to avoided band crossings

A recently found metal-insulator transition in a model for cyclotron resonance in a two-dimensional periodic potential is investigated by means of spectral properties of the time evolution operator. The previously found dynamical signatures of the transition are explained in terms of avoided band crossings due to the change of the external electric field. The occurrence of a cross-like transport is predicted and numerically confirmed

Scaling detection in time series: diffusion entropy analysis

The methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance. This means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved. We illustrate a method of statistical analysis based on the Shannon entropy of the diffusion process generated by the time series, called Diffusion Entropy Analysis (DEA). We adopt artificial Gauss and L\'{e}vy time series, as prototypes of ordinary and anomalus statistics, respectively, and we analyse them with the DEA and four ordinary methods of analysis, some of which are very popular. We show that the DEA determines the correct scaling exponent even when the statistical properties, as well as the dynamic properties, are anomalous. The other four methods produce correct results in the Gauss case but fail to detect the correct scaling in the case of L\'{e}vy statistics.Comment: 21 pages,10 figures, 1 tabl

What determines the spreading of a wave packet?

The multifractal dimensions D2^mu and D2^psi of the energy spectrum and eigenfunctions, resp., are shown to determine the asymptotic scaling of the width of a spreading wave packet. For systems where the shape of the wave packet is preserved the k-th moment increases as t^(k*beta) with beta=D2^mu/D2^psi, while in general t^(k*beta) is an optimal lower bound. Furthermore, we show that in d dimensions asymptotically in time the center of any wave packet decreases spatially as a power law with exponent D_2^psi - d and present numerical support for these results.Comment: Physical Review Letters to appear, 4 pages postscript with figure

Experimental evidence for the role of cantori as barriers in a quantum system

We investigate the effect of cantori on momentum diffusion in a quantum system. Ultracold caesium atoms are subjected to a specifically designed periodically pulsed standing wave. A cantorus separates two chaotic regions of the classical phase space. Diffusion through the cantorus is classically predicted. Quantum diffusion is only significant when the classical phase-space area escaping through the cantorus per period greatly exceeds Planck's constant. Experimental data and a quantum analysis confirm that the cantori act as barriers.Comment: 19 pages including 9 figures, Accepted for publication in Physical Review E in March 199

Fractal Conductance Fluctuations in a Soft Wall Stadium and a Sinai Billiard

Conductance fluctuations have been studied in a soft wall stadium and a Sinai billiard defined by electrostatic gates on a high mobility semiconductor heterojunction. These reproducible magnetoconductance fluctuations are found to be fractal confirming recent theoretical predictions of quantum signatures in classically mixed (regular and chaotic) systems. The fractal character of the fluctuations provides direct evidence for a hierarchical phase space structure at the boundary between regular and chaotic motion.Comment: 4 pages, 4 figures, data on Sinai geometry added to Fig.1, minor change

Statistics of resonances and of delay times in quasiperiodic Schr"odinger equations

We study the statistical distributions of the resonance widths ${\cal P} (\Gamma)$, and of delay times ${\cal P} (\tau)$ in one dimensional quasi-periodic tight-binding systems with one open channel. Both quantities are found to decay algebraically as $\Gamma^{-\alpha}$, and $\tau^{-\gamma}$ on small and large scales respectively. The exponents $\alpha$, and $\gamma$ are related to the fractal dimension $D_0^E$ of the spectrum of the closed system as $\alpha=1+D_0^E$ and $\gamma=2-D_0^E$. Our results are verified for the Harper model at the metal-insulator transition and for Fibonacci lattices.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let