Search

We discuss effects of pairwise aligning interactions in an ensemble of central place foragers or of searchers that are connected to a common home. In a wider sense, we also consider self moving entities that are attracted to a central place such as, for instance, the zooplankton Daphnia being attracted to a beam of light. Single foragers move with constant speed due to some propulsive mechanism. They explore at random loops the space around and return rhytmically to their home. In the ensemble, the direction of the velocity of a searcher is aligned to the motion of its neighbors. At first, we perform simulations of this ensemble and find a cooperative behavior of the entities. Above an over-critical interaction strength the trajectories of the searcher qualitatively changes and searchers start to move along circles around the home position. Thereby, all searchers rotate either clockwise or anticlockwise around the central home position as it was reported for the zooplankton Daphnia. At second, the computational findings are analytically explained by the formulation of transport equations outgoing from the nonlinear mean field Fokker-Planck equation of the considered situation. In the asymptotic stationary limit, we find expressions for the critical interaction strength, the mean radial and orbital velocities of the searchers and their velocity variances. We also obtain the marginal spatial and angular densities in the under-critical regime where the foragers behave like individuals as well as in the over-critical regime where they rotate collectively around the considered home. We additionally elaborate the overdamped Smoluchowski-limit for the ensemble.

We develop a prototypical stochastic model for local search around a given home. The stochastic dynamic model is motivated by experimental findings of the motion of a fruit fly around a given spot of food but shall generally describe local search behavior. The local search consists of a sequence of two epochs. In the first the searcher explores new space around the home whereas it returns to the home during the second epoch. In the proposed two dimensional model both tasks are described by the same stochastic dynamics. The searcher moves with constant speed and its angular dynamics is driven by a symmetric {\alpha}-stable noise source. The latter stands for the uncertainty to decide the new direction of motion. The main ingredient of the model is the nonlinear interaction dynamics of the searcher with its home. In order to determine the new heading direction, the angles of its position to the home and of the heading vector need to be known. A bound state to the home is realized by a permanent switch of a repulsive and attractive forcing of the heading direction from the position direction corresponding to search and return epochs. Our investigation elucidates the analytic tractability of the deterministic and stochastic dynamics. The noise enables a faster finding of a target distinct from the home with optimal intensity. This optimal situation is related to the noise dependent relaxation time. It is uniquely defined for all {\alpha} and distinguishes between the stochastic dynamics before and after its value. For times large compared to this we derive the corresponding Smoluchowski equation and find diffusive spreading of searcher in the space. As result of its simplicity the model aims to reproduce the local search behavior of simple units during their exploration of surrounding space and their quasi-periodic return to a home.

We extend a recently introduced prototypical stochastic model describing uniformly the search and return of objects looking for new food sources around a given home. The model describes the kinematic motion of the object with constant speed in two dimensions. The angular dynamics is driven by noise and describes a "pursuit" and "escape" behavior of the heading and the position vectors. Pursuit behavior ensures the return to the home and the escaping between the two vectors realizes exploration of space in the vicinity of the given home. Noise is originated by environmental influences and during decision making of the object. We take symmetric {\alpha}-stable noise since such noise is observed in experiments. We now investigate for the simplest possible case, the consequences of limited knowledge of the position angle of the home. We find that both noise type and noise strength can significantly increase the probability of returning to the home. First, we review shortly main findings of the model presented in the former manuscript. These are the stationary distance distribution of the noise driven conservative dynamics and the observation of an optimal noise for finding new food sources. Afterwards, we generalize the model by adding a constant shift {\gamma} within the interaction rule between the two vectors. The latter might be created by a permanent uncertainty of the correct home position. Non vanishing shifts transform the kinematics of the searcher to a dissipative dynamics. For the latter we discuss the novel deterministic properties and calculate the stationary spatial distribution around the home.

We are concerned with the dynamical description of the motion of a stochastic micro-swimmer with constant speed and fluctuating orientation in the long time limit by adiabatic elimination of the orientational variable. Starting with the corresponding full set of Langevin equations, we eliminate the memory in the stochastic orientation and obtain a stochastic equation for the position alone in the overdamped limit. An equivalent procedure based on the Fokker-Planck equation is presented as well.

The physics of activated escape of objects out of a metastable state plays a key role in diverse scientific areas involving chemical kinetics, diffusion and dislocation motion in solids, nucleation, electrical transport, motion of flux lines superconductors, charge density waves, and transport processes of macromolecules, to name but a few. The underlying activated processes present the multidimensional extension of the Kramers problem of a single Brownian particle. In comparison to the latter case, however, the dynamics ensuing from the interactions of many coupled units can lead to intriguing novel phenomena that are not present when only a single degree of freedom is involved. In this review we report on a variety of such phenomena that are exhibited by systems consisting of chains of interacting units in the presence of potential barriers. In the first part we consider recent developments in the case of a deterministic dynamics driving cooperative escape processes of coupled nonlinear units out of metastable states. The ability of chains of coupled units to undergo spontaneous conformational transitions can lead to a self-organised escape. The mechanism at work is that the energies of the units become re-arranged, while keeping the total energy conserved, in forming localised energy modes that in turn trigger the cooperative escape. We present scenarios of significantly enhanced noise-free escape rates if compared to the noise-assisted case. The second part deals with the collective directed transport of systems of interacting particles overcoming energetic barriers in periodic potential landscapes. Escape processes in both time-homogeneous and time-dependent driven systems are considered for the emergence of directed motion. It is shown that ballistic channels immersed in the associated high-dimensional phase space are the source for the directed long-range transport.

Using our generalized Fick-Jacobs approach [Martens et al., PRL 110, 010601 (2013); Martens et al., Eur. Phys. J. Spec. Topics 222, 2453-2463 (2013)] and extensive Brownian dynamics simulations, we study particle transport through three-dimensional periodic channels of different height. Directed motion is caused by the interplay of constant bias acting along the channel axis and a pressure-driven flow. The tremendous change of the flow profile shape in channel direction with the channel height is reflected in a crucial dependence of the mean particle velocity and the effective diffusion coefficient on the channel height. In particular, we observe a giant suppression of the effective diffusivity in thin channels; four orders of magnitude compared to the bulk value., Comment: 16 pages, 8 figures

We study the escape of a chain of coupled units over the barrier of a metastable potential. It is demonstrated that a very weak external driving field with suitably chosen frequency suffices to accomplish speedy escape. The latter requires the passage through a transition state the formation of which is triggered by permanent feeding of energy from a phonon background into humps of localised energy and elastic interaction of the arising breather solutions. In fact, cooperativity between the units of the chain entailing coordinated energy transfer is shown to be crucial for enhancing the rate of escape in an extremely effective and low-energy cost way where the effect of entropic localisation and breather coalescence conspire.

Using the analytical Fick-Jacobs approximation formalism and extensive Brownian dynamics simulations we study particle transport through two-dimensional periodic channels with triangularly shaped walls. Directed motion is caused by the interplay of constant bias acting along the channel axis and a pressure-driven flow. In particular, we analyze the particle mobility and the effective diffusion coefficient. The mechanisms of entropic rectification is revealed in channels with a broken spatial reflection symmetry in presence of hydrodynamically enforced entropic trapping. Due to the combined action of the forcing and the pressure-driven flow field, efficient rectification with a drastically reduced diffusivity is achieved., Comment: 11 pages, 7 figures

We study the transport of Brownian particles through a corrugated channel caused by a force field containing curl-free (scalar potential) and divergence-free (vector potential) parts. We develop a generalized Fick-Jacobs approach leading to an effective one-dimensional description involving the potential of mean force. As an application, the interplay of a pressure-driven flow and an oppositely oriented constant bias is considered. We show that for certain parameters, the particle diffusion is significantly suppressed via the property of hyrodynamically enforced entropic particle trapping., Comment: 5 pages, 4 figures, in press with Physical Review Letters

We numerically investigate the transport of a suspended overdamped Brownian particle which is driven through a two-dimensional rectangular array of circular obstacles with finite radius. Two limiting cases are considered in detail, namely, when the constant drive is parallel to the principal or the diagonal array axes. This corresponds to studying the Brownian transport in periodic channels with reflecting walls of different topologies. The mobility and diffusivity of the transported particles in such channels are determined as functions of the drive and the array geometric parameters. Prominent transport features, like negative differential mobilities, excess diffusion peaks, and unconventional asymptotic behaviors, are explained in terms of two distinct lengths, the size of single obstacles (trapping length) and the lattice constant of the array (local correlation length). Local correlation effects are further analyzed by continuously rotating the drive between the two limiting orientations., Comment: 10 pages 13 figures

According to the conventional approach to model neural ensembles the extracellular environment has fixed ionic concentrations. However, in many cases the extracellular concentration of potassium ions can not be regarded as constant. That represents specific chemical pathway for neurons to interact and can influence strongly the behavior of a single neuron as well as large ensembles. The released chemical agent follows a diffusive dynamics in the external medium, that lowers the threshold of individual excitable units. We address this problem by studying simplified excitable units given by a modified FitzHugh-Nagumo dynamics. In our model neurons interact only chemically via the released and diffusing potassium in the surrounding non-active medium. We study the dynamics of a single excitable unit embedded in the extracellular matter. That leads to a number of noise-induced effects, like self-modulation of firing rate in an individual neuron. In the spatially extended situation various patterns appear ranging from spirals and traveling waves to oscillons and inverted structures depending on the parameters of the medium., Comment: 27 pages, 14 figures

We consider motion of an underdamped Brownian particle in a washboard potential that is subjected to an unbiased time-periodic external field. While in the limiting deterministic system in dependence of the strength and phase of the external field directed net motion can exist, for a finite temperature the net motion averages to zero. Strikingly, with the application of an additional time-delayed feedback term directed particle motion can be accomplished persisting up to fairly high levels of the thermal noise. In detail, there exist values of the feedback strength and delay time for which the feedback term performs oscillations that are phase locked to the time-periodic external field. This yields an effective biasing rocking force promoting periods of forward and backward motion of distinct duration, and thus directed motion. In terms of phase space dynamics we demonstrate that with applied feedback desymmetrization of coexisting attractors takes place leaving the ones supporting either positive or negative velocities as the only surviving ones. Moreover, we found parameter ranges for which in the presence of thermal noise the directed transport is enhanced compared to the noise-less case.

Granular flows through pipes show interesting phenomena, e.g. clogging and density waves, 1/f-noise. These things are fairly good studied by computer-experiments, but there is a lack in theoretical and analytical consideration. We introduce a simple "minimal" model describing such a flow of granular particles and examine the stability of an initially homogeneous system against perturbations. In order to define the collisions between the granular particles we use two different approaches. For both, the simple and the more advanced collision definition, the model shows the qualitative same behaviour., Comment: Workshop on Traffic and Granular Flow: HLRZ, Forschungszentrum, Julich, Germany, Ooctober 9-11, 1995

We study the Langevin dynamics of a two-dimensional discrete oscillator chain absorbed on a periodic substrate and subjected to an external localized point force. Going beyond the commonly used harmonic bead-spring model, we consider a nonlinear Morse interaction between the next-nearest-neighbors. We focus interest on the activation of directed motion instigated by thermal fluctuations and the localized point force. In this context the local transition states are identified and the corresponding activation energies are calculated. As a novel feature it is found that the transport of the chain in point force direction is determined by stepwise escapes of a single unit or segments of the chain due to the existence of multiple locally stable attractors. The non-vanishing net current of the chain is quantitatively assessed by the value of the mobility of the center of mass. It turns out that the latter as a function of the ratio of the competing length scales of the system, that is the period of the substrate potential and the equilibrium distance between two chain units, shows a resonance behavior. More precisely there exist a set of optimal parameter values maximizing the mobility. Interestingly, the phenomenon of negative resistance is found, i.e. the mobility possesses a minimum at a finite value of the strength of the thermal fluctuations for a given overcritical external driving force., Comment: 25 pages, 12 figures

We study the collective escape dynamics of a chain of coupled, weakly damped nonlinear oscillators from a metastable state over a barrier when driven by a thermal heat bath in combination with a weak, globally acting periodic perturbation. Optimal parameter choices are identified that lead to a drastic enhancement of escape rates as compared to a pure noise-assisted situation. We elucidate the speed-up of escape in the driven Langevin dynamics by showing that the time-periodic external field in combination with the thermal fluctuations triggers an instability mechanism of the stationary homogeneous lattice state of the system. Perturbations of the latter provided by incoherent thermal fluctuations grow because of a parametric resonance, leading to the formation of spatially localized modes (LMs). Remarkably, the LMs persist in spite of continuously impacting thermal noise. The average escape time assumes a distinct minimum by either tuning the coupling strength and/or the driving frequency. This weak ac-driven assisted escape in turn implies a giant speed of the activation rate of such thermally driven coupled nonlinear oscillator chains.

The nonintegrable Hamiltonian dynamics of particles placed in a symmetric, spatially periodic potential and subjected to a periodically varying field is explored. Such systems can exhibit a rich diversity of unusual transport features. In particular, depending on the setting of the initial phase of the drive, the possibility of a giant transient directed transport in a symmetric, space-periodic potential when driven with an adiabatically varying field arises. Here, we study the escape scenario and corresponding mean escape times of particles from a trapping region with the subsequent generation of a transient directed flow of an ensemble of particles. It is shown that for adiabatically slow inclination modulations the unidirectional flow proceeds over giant distances. The direction of escape and, hence, of the flow is entirely governed whether the periodic force, modulating the inclination of the potential, starts out initially positive or negative. In the phase space, this transient directed flow is associated with a long-lasting motion taking place within ballistic channels contained in the non-uniform chaotic layer. We demonstrate that for adiabatic modulations all escaping particles move ballistically into the same direction, leading to a giant directed current.

We present the noise free escape of a chain of linearly interacting units from a metastable state over a cubic on-site potential barrier. The underlying dynamics is conservative and purely deterministic. The mutual interplay between nonlinearity and harmonic interactions causes an initially uniform lattice state to become unstable, leading to an energy redistribution with strong localization. As a result a spontaneously emerging localized mode grows into a critical nucleus. By surpassing this transition state, the nonlinear chain manages a self-organized, deterministic barrier crossing. Most strikingly, these noise-free, collective nonlinear escape events proceed generally by far faster than transitions assisted by thermal noise when the ratio between the average energy supplied per unit in the chain and the potential barrier energy assumes small values.

A spatially extended Lotka-Volterra system of two competing species in the presence of two correlated noise sources is analyzed: (i) an external multiplicative time correlated noise, which mimics the interaction between the system and the environment; (ii) a dichotomous stochastic process, whose jump rate is a periodic function, which represents the interaction parameter between the species. The moment equations for the species densities are derived in Gaussian approximation, using a mean field approach. Within this formalism we study the effect of the external time correlated noise on the ecosystem dynamics. We find that the time behavior of the $1^{st}$ order moments are independent on the multiplicative noise source. However the behavior of the $2^{nd}$ order moments is strongly affected both by the intensity and the correlation time of the multiplicative noise. Finally we compare our results with those obtained studying the system dynamics by a coupled map lattice model., Comment: 12 pages, 7 figures, to appear in Acta Phys. Pol. B

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 figures

The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulations of the time discrete version of LV equations, that is coupled map lattice (CML) model, we conclude that the anticorrelated oscillations of the species densities are strictly related to non-overlapping spatial patterns., Comment: 10 pages, 3 figures

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.

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.

We develop a theory to calculate the effective phase diffusion coefficient and the mean phase velocity in periodically driven stochastic models with two discrete states. This theory is applied to a dichotomically driven Markovian two state system. Explicit expressions for the mean phase velocity, the effective phase diffusion coefficient and the P\'eclet number are analytically calculated. The latter shows as a measure of phase-coherence forced synchronization of the stochastic system with respect to the periodic driving. In a second step the theory is applied to a non Markovian two state model modeling excitable systems. The results prove again stochastic synchronization to the periodic driving and are in good agreement with simulations of a stochastic FitzHugh-Nagumo system., Comment: 11 pages, 7 figures

The response of a neural cell to an external stimulus can follow one of the two patterns: Nonresonant neurons monotonously relax to the resting state after excitation while resonant ones show subthreshold oscillations. We investigate how do these subthreshold properties of neurons affect their suprathreshold response. Vice versa we ask: Can we distinguish between both types of neuronal dynamics using suprathreshold spike trains? The dynamics of neurons is given by stochastic FitzHugh-Nagumo and Morris-Lecar models with either having a focus or a node as the stable fixpoint. We determine numerically the spectral power density as well as the interspike interval density in response to a random (noise-like) signals. We show that the information about the type of dynamics obtained from power spectra is of limited validity. In contrast, the interspike interval density gives a very sensitive instrument for the diagnostics of whether the dynamics has resonant or nonresonant properties. For the latter value we formulate a fit formula and use it to reconstruct theoretically the spectral power density, which coincides with the numerically obtained spectra. We underline that the renewal theory is applicable to analysis of suprathreshold responses even of resonant neurons., Comment: 7 pages, 8 figures

Bistability generated via a pure noise-induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics. It allows an analytical study of the phase diagram in more general cases than previous methods. In addition using this approach we investigate patially-extended systems with two degrees of freedom per site. For this system, the analytic solution of the stationary Fokker-Planck equation is not available and a standard mean field approach cannot be used to find noise induced phase transitions. A new approach based on cumulant dynamics predicts a noise-induced phase transition through a Hopf bifurcation leading to a macroscopic limit cycle motion, which is confirmed by numerical simulation., Comment: 8 pages, 8 figures

We study a non Markovian three state model, subjected to an external periodic signal. This model is intended to describe an excitable systems with periodical driving. In the limit of a small amplitude of the external signal we derive expressions for the spectral power amplification and the signal to noise ratio as well as for the inter-spike interval distribution., Comment: 5 pages, 3 figures

We investigate globally coupled stochastic three-state oscillators, which we consider as general models of stochastic excitable systems. We compare two situations:in the first case the transitions between the three states of each unit 1->2->3->1 are determined by Poissonian waiting time distributions. In the second case only transition 1->2 is Poissonian whereas the others are deterministic with a fixed delay. When coupled the second system shows coherent oscillations whereas the first remains in a stable stationary state. We show that the coherent oscillations are due to a Hopf-bifurcation in the dynamics of the occupation probabilities of the discrete states and discuss the bifurcation diagram., Comment: 10 pages, 4 figures, submitted to Physica A

We consider networks of coupled stochastic oscillators. When coupled we find strong collective oscillations, while each unit remains stochastic. In the limit (N\to \infty) we derive a system of integro-delay equations and show analytically that the collective oscillations persist in a large region in parameter. For a regular topology with \emph{few} connections between the oscillators, islands of coherent oscillations are formed, which merge as the amount of topological disorder increases. We link this transition to typical network quantities in the framework of small-world networks., Comment: 4 pages, 8 figures. Submitted to PRL

A model of globally coupled phase oscillators under equilibrium (driven by Gaussian white noise) and nonequilibrium (driven by symmetric dichotomic fluctuations) is studied. For the equilibrium system, the mean-field state equation takes a simple form and the stability of its solution is examined in the full space of order parameters. For the nonequilbrium system, various asymptotic regimes are obtained in a closed analytical form. In a general case, the corresponding master equations are solved numerically. Moreover, the Monte-Carlo simulations of the coupled set of Langevin equations of motion is performed. The phase diagram of the nonequilibrium system is presented. For the long time limit, we have found four regimes. Three of them can be obtained from the mean-field theory. One of them, the oscillating regime, cannot be predicted by the mean-field method and has been detected in the Monte-Carlo numerical experiments., Comment: 9 pages 8 figures