Asymptotic analysis of first passage time in complex networks

The first passage time (FPT) distribution for random walk in complex networks is calculated through an asymptotic analysis. For network with size $N$ and short relaxation time $\tau\ll N$, the computed mean first passage time (MFPT), which is inverse of the decay rate of FPT distribution, is inversely proportional to the degree of the destination. These results are verified numerically for the paradigmatic networks with excellent agreement. We show that the range of validity of the analytical results covers networks that have short relaxation time and high mean degree, which turn out to be valid to many real networks.Comment: 6 pages, 4 figures, 1 tabl

On a Case of Spontaneous Thrombosis in the Left Femoral and Saphena Vein

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The fate of bezafibrate, carbamazepine, ciprofloxacin and clarithromycin in the wastewater treatment process

The progress of four pharmaceuticals (bezafibrate, carbamazepine, clarithromycin and ciprofloxacin) is followed through the treatment stages (screened sewage, settled sewage and final effluent) of a large urban wastewater treatment plant (WWTP) employing activated sludge treatment. Concentrations at the inlet to the WWTP are generally higher than those predicted from consideration of local pharmaceutical consumption and typical excretion data. Percentage removal efficiencies are variable (22.5 – 94.3%) with carbamazepine being the most resistant to elimination

Sources and pathways for pharmaceuticals in the urban water environment

The progress of five pharmaceutical compounds (bezafibrate, carbamazepine, diclofenac, ibuprofen and sulfasalazine) and one antibacterial agent (triclosan) were monitored through the treatment stages of a large sewage treatment works (STW) using activated sludge as well as in the receiving water both upstream and downstream of the effluent discharge. All except sulfasalazine were detected in the influent at concentrations ranging from 1.44-3.75 µg/L. The analysis of prescription data has been used as a tool to predict the amount of pharmaceuticals potentially released into the catchment of the investigated sewage treatment works and the results compared with the measured influent concentrations. A reduction in concentration between influent and final effluent samples (51-97%) indicates the variable removal of these compounds and therefore their potential to be discharged into receiving surface waters. The analysis of primary and final effluents highlight the important processes involved in the removal of pharmaceuticals and indicate that sorption processes are important for bezafibrate, carbamazepine and diclofenac. These three PPCPs were observed at higher concentrations (0.07-0.35 µg/L) downstream of the discharged effluent compared to upstream (0.02-0.04 µg/L) although the risks that these compounds pose in the environment are not yet fully understood

Crossover between Levy and Gaussian regimes in first passage processes

We propose a new approach to the problem of the first passage time. Our method is applicable not only to the Wiener process but also to the non--Gaussian L$\acute{\rm e}$vy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first passage time problems in the truncated L$\acute{\rm e}$vy flights (the so-called KoBoL processes), in which the arbitrarily large tail of the L$\acute{\rm e}$vy distribution is cut off. We find that the asymptotic scaling law of the first passage time $t$ distribution changes from $t^{-(\alpha +1)/\alpha}$-law (non-Gaussian L$\acute{\rm e}$vy regime) to $t^{-3/2}$-law (Gaussian regime) at the crossover point. This result means that an ultra-slow convergence from the non-Gaussian L$\acute{\rm e}$vy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated L$\acute{\rm e}$vy flight but also in the first passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cut-off length of the L$\acute{\rm e}$vy distribution are discussed.Comment: 18pages, 7figures, using revtex4, to appear in Phys.Rev.

Desynchronization in diluted neural networks

The dynamical behaviour of a weakly diluted fully-inhibitory network of pulse-coupled spiking neurons is investigated. Upon increasing the coupling strength, a transition from regular to stochastic-like regime is observed. In the weak-coupling phase, a periodic dynamics is rapidly approached, with all neurons firing with the same rate and mutually phase-locked. The strong-coupling phase is characterized by an irregular pattern, even though the maximum Lyapunov exponent is negative. The paradox is solved by drawing an analogy with the phenomenon of ``stable chaos'', i.e. by observing that the stochastic-like behaviour is "limited" to a an exponentially long (with the system size) transient. Remarkably, the transient dynamics turns out to be stationary.Comment: 11 pages, 13 figures, submitted to Phys. Rev.

Noise Induced Complexity: From Subthreshold Oscillations to Spiking in Coupled Excitable Systems

We study stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. Diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed in the course of this transition. In order to understand details and mechanisms of noise-induced dynamics we consider a thermodynamic limit $N\to\infty$ of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In the Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good agreement with dynamical scenarios observed in the stochastic simulations of large ensembles

Generalized Rate-Code Model for Neuron Ensembles with Finite Populations

We have proposed a generalized Langevin-type rate-code model subjected to multiplicative noise, in order to study stationary and dynamical properties of an ensemble containing {\it finite} $N$ neurons. Calculations using the Fokker-Planck equation (FPE) have shown that owing to the multiplicative noise, our rate model yields various kinds of stationary non-Gaussian distributions such as gamma, inverse-Gaussian-like and log-normal-like distributions, which have been experimentally observed. Dynamical properties of the rate model have been studied with the use of the augmented moment method (AMM), which was previously proposed by the author with a macroscopic point of view for finite-unit stochastic systems. In the AMM, original $N$-dimensional stochastic differential equations (DEs) are transformed into three-dimensional deterministic DEs for means and fluctuations of local and global variables. Dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the AMM are in good agreement with those obtained by direct simulation. The synchronization in the neuronal ensemble is discussed. Variabilities of the firing rate and of the interspike interval (ISI) are shown to increase with increasing the magnitude of multiplicative noise, which may be a conceivable origin of the observed large variability in cortical neurons.Comment: 19 pages, 9 figures, accepted in Phys. Rev. E after minor modification

First Passage Time Densities in Non-Markovian Models with Subthreshold Oscillations

Motivated by the dynamics of resonant neurons we consider a differentiable, non-Markovian random process $x(t)$ and particularly the time after which it will reach a certain level $x_b$. The probability density of this first passage time is expressed as infinite series of integrals over joint probability densities of $x$ and its velocity $\dot{x}$. Approximating higher order terms of this series through the lower order ones leads to closed expressions in the cases of vanishing and moderate correlations between subsequent crossings of $x_b$. For a linear oscillator driven by white or coloured Gaussian noise, which models a resonant neuron, we show that these approximations reproduce the complex structures of the first passage time densities characteristic for the underdamped dynamics, where Markovian approximations (giving monotonous first passage time distribution) fail

First Passage Time Densities in Resonate-and-Fire Models

Motivated by the dynamics of resonant neurons we discuss the properties of the first passage time (FPT) densities for nonmarkovian differentiable random processes. We start from an exact expression for the FPT density in terms of an infinite series of integrals over joint densities of level crossings, and consider different approximations based on truncation or on approximate summation of this series. Thus, the first few terms of the series give good approximations for the FPT density on short times. For rapidly decaying correlations the decoupling approximations perform well in the whole time domain. As an example we consider resonate-and-fire neurons representing stochastic underdamped or moderately damped harmonic oscillators driven by white Gaussian or by Ornstein-Uhlenbeck noise. We show, that approximations reproduce all qualitatively different structures of the FPT densities: from monomodal to multimodal densities with decaying peaks. The approximations work for the systems of whatever dimension and are especially effective for the processes with narrow spectral density, exactly when markovian approximations fail.Comment: 11 pages, 8 figure