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We discuss and analyze the driving a polar active particle with a head-tail asymmetry based on the dynamics of an internal motor variable driven by an energy depot and a broken symmetry of friction with respect to the internal degree of freedom. We show that such a driving may be advantageous for driving large masses with small energy uptake from the environment and exhibits interesting properties such as resonance-driven optimal propulsion.

The stochastic dynamics of an active particle undergoing a constant speed and additionally driven by an overall fluctuating torque is investigated. The random torque forces are expressed by a stochastic differential equation for the angular dynamics of the particle determining the orientation of motion. In addition to a constant torque, the particle is supplemented by random torques, which are modeled as an Ornstein-Uhlenbeck process with given correlation time τ(c). These nonvanishing correlations cause a persistence of the particles' trajectories and a change of the effective spatial diffusion coefficient. We discuss the mean square displacement as a function of the correlation time and the noise intensity and detect a nonmonotonic dependence of the effective diffusion coefficient with respect to both correlation time and noise strength. A maximal diffusion behavior is obtained if the correlated angular noise straightens the curved trajectories, interrupted by small pirouettes, whereby the correlated noise amplifies a straightening of the curved trajectories caused by the constant torque.

We study the effect of different types of fluctuation on the motion of self-propelled particles in two spatial dimensions. We distinguish between passive and active fluctuations. Passive fluctuations (e.g., thermal fluctuations) are independent of the orientation of the particle. In contrast, active ones point parallel or perpendicular to the time dependent orientation of the particle. We derive analytical expressions for the speed and velocity probability density for a generic model of active Brownian particles, which yields an increased probability of low speeds in the presence of active fluctuations in comparison to the case of purely passive fluctuations. As a consequence, we predict sharply peaked Cartesian velocity probability densities at the origin. Finally, we show that such a behavior may also occur in non-Gaussian active fluctuations and discuss briefly correlations of the fluctuating stochastic forces.

We study time-delayed coupling in one-dimensional arrays of discrete excitable FitzHugh-Nagumo elements. For small time delays in an excitatory coupling, noise-triggered spikes undergo a slight prolongation. For larger delay times, the coupling to adjacent elements leads to a splitting of spikes and an "echo" effect that produces dampened series of spikes. For very large delay times, a transition to a stationary regime appears, as the spikes are sustained by mutual excitations, resulting in antiphase synchronization of all elements. We characterize several features exhibited by the system, such as speed and duration of pulses as well as bifurcations leading to the stationary regime.

In order to move effectively in unpredictable or heterogeneous environments animals must make appropriate decisions in response to internal and external cues. Identifying the link between these components remains a challenge for movement ecology and is important in understanding the mechanisms driving both individual and collective motion. One accessible way of examining how internal state influences an individual's motion is to consider the nutritional state of an animal. Our experimental results reveal that nutritional state exerts a relatively minor influence on the motion of isolated individuals, but large group-level differences emerge from diet affecting inter-individual interactions. This supports the idea that mass movement in locusts may be driven by cannibalism. To estimate how these findings are likely to impact collective migration of locust hopper bands, we create an experimentally parametrized model of locust interactions and motion. Our model supports our hypothesis that nutrient-dependent social interactions can lead to the collective motion seen in our experiments and predicts a transition in the mean speed and the degree of coordination of bands with increasing insect density. Furthermore, increasing the interaction strength (representing greater protein deprivation) dramatically reduces the critical density at which this transition occurs, demonstrating that individuals' nutritional state could have a major impact on large-scale migration.

The effect of intrinsic channel noise is investigated for the dynamic response of a neuronal cell with a delayed feedback loop. The loop is based on the so-called autapse phenomenon in which dendrites establish connections not only to neighboring cells but also to its own axon. The biophysical modeling is achieved in terms of a stochastic Hodgkin-Huxley model containing such a built in delayed feedback. The fluctuations stem from intrinsic channel noise, being caused by the stochastic nature of the gating dynamics of ion channels. The influence of the delayed stimulus is systematically analyzed with respect to the coupling parameter and the delay time in terms of the interspike interval histograms and the average interspike interval. The delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution, exhibiting a delay-induced reduction in the spontaneous spiking activity at characteristic frequencies. Moreover, a specific frequency-locking mechanism is detected for the mean interspike interval.

A two-state unit is considered as an abstract modification for an excitable system. Each state is characterized by a different waiting time distribution. This non-Markovian approach allows for a renewal process description of the system dynamics. Exact formulas for the interspike interval distribution and power spectral density are found. In the limit of an infinity ensemble of globally coupled units the mean-field equations for the populations of both states are derived. Depending on the coupling strength and on the noise intensity the ensemble undergoes saddle-node bifurcations and demonstrates bistability, while a pitchfork bifurcation emerges on a cusp point. The ensemble undergoes Hopf bifurcations and bulk oscillations emerge, in the onset of coherent activation events, only when the feedback affects individual units with a certain time delay.

We consider overdamped Brownian dynamics in a periodic potential with temporally oscillating amplitude. We analyze the transport which shows effective diffusion enhanced by the oscillations and derive approximate expressions for the diffusion coefficient. Furthermore we analyze the effect of the oscillating potential on the transport if additionally a constant force is applied. We show the existence of synchronization regimes at which the deterministic dynamics is in resonance with the potential oscillations, giving rise to transport with extremely low dispersion. We distinguish slow and fast oscillatory driving and give analytical expressions for the mean velocity and effective diffusion.

We study the effect of noisy oscillatory input on the signal discrimination by spontaneously firing neurons. Using analytically tractable model, we contrast signal detection in two situations: (i) when the neuron is driven by coherent oscillations and (ii) when the coherence of oscillations is destroyed. Analytical calculations revealed a region in the parameter space of the model where oscillations act to reduce the variability of neuronal firing and to enhance the discriminability of weak signals. These analytical results are employed to unveil a possible role of coherent oscillations in peripheral electrosensory system of paddlefish in improvement of detection of weak stimuli. The proposed mechanism may be relevant to a wide range of phenomena involving coherently driven oscillators.

Motivated by recent experiments on intracellular calcium release we study the effects of different types of coupling on the dynamics of arrays of excitable elements. We intend to find a mechanism that produces a sustained activity of the elements following a spike. While instantaneous diffusive coupling does not exhibit this property, we show that, for a coupling term with temporal delay, signals from adjacent elements can serve as mutual excitations and thus prolong the duration of the signal. We propose that time delayed coupling is generated by diffusion between isolated clusters of calcium channels. Our model could thus provide an explanation for two different release modes observed in the Ca2+ system.

Recent studies suggest that noncooperative behavior such as cannibalism may be a driving mechanism of collective motion. Motivated by these novel results we introduce a simple model of Brownian particles interacting by biologically motivated pursuit and escape interactions. We show the onset of collective motion for both interaction types and analyze their impact on the global dynamics. We demonstrate a strong dependence of experimentally accessible macroscopic observables on the relative strength of escape and pursuit and determine the scaling of the migration speed with model parameters.

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.

We consider a three-dimensional model of active Brownian particles interacting via a Morse-type potential. The system exhibits two modes of motion: translation and a coherent rotation in a torus-shape structure. We observe noise-induced transitions in both directions between the two states. These occur at different noise intensities, thus leading to a hysteresis curve. For certain parameter regions, the system switches persistently between the states such that the center of mass alternates between ballistic and diffusive motion. The coherent rotation leads to a pronounced mean angular momentum that changes its direction diffusively. We derive an analytic expression for the diffusion of the angular momentum of one particle in an external harmonic potential and show that it is always faster than the stochastic switching of the direction of motion in the two-dimensional case.

The present work derives the spatiotemporal field equation of neural populations considering two types of neurons. The model considers pyramidal cells, which may excite or inhibit other neurons, and GABAergic interneurons inhibiting terminal neurons. Additionally, taking into account excitatory and inhibitory synapses, the neural population obeys a vector-field equation involving nonlocal spatial interactions. The work studies the effect of the anesthetic agent propofol, which increases the decay time of inhibitory synapses. In addition, it explains the bifurcation mechanism in some detail and finds a saddle-node bifurcation subject to the propofol concentration. This bifurcation may model the transition between consciousness and nonconsciousness and vice versa during the administration of general anesthetics in medicine.

We analytically investigate the interspike interval (ISI) density, the Fano factor, and the coefficient of variation of a leaky integrate-and-fire neuron model driven by exponentially correlated Gaussian noise with a large correlation time tau . We find a burstinglike behavior of the spike train, which is revealed by a dominant peak of the ISI density at small intraburst intervals and a slow power-law decay of long interburst intervals. The large, power-law distributed ISIs give rise to a coefficient of variation which diverges as square root [tau] . This leads to the paradoxical effect that ISI correlations, as expressed by the serial correlation coefficient, vanish for large correlation times. This is in contrast to findings of previous works on a simpler neuron model where the effect of noise correlations appeared in higher-order statistical measures.

We apply a stochastic center manifold method to the calculation of noise-induced phase transitions in the stochastic Swift-Hohenberg equation. This analysis is applied to the reduced mode equations that result from Fourier decomposition of the field variable and of the temporal noise. The method shows a pitchfork bifurcation at lower perturbation order, but reveals a novel additive-noise-induced postponement of the Turing bifurcation at higher order. Good agreement is found between the theory and the numerics for both the reduced and the full system. The results are generalizable to a broad class of nonlinear spatial systems.

The motions of many physical particles as well as living creatures are mediated by random influences or 'noise'. One might expect that over evolutionary time scales internal random processes found in living systems display characteristics that maximize fitness. Here we focus on animal random search strategies [G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. Da Luz, E.P. Raposo, H.E. Stanley, Optimizing the success of random searches, Nature 401 (1999) 911-914; F. Bartumeus, J. Catalan, U.L. Fulco, M.L. Lyra, G.M. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus Brownian stratagies, Phys. Rev. Lett. 88 (2002) 097901 and 89 (2002) 109902], and we describe experiments with the following Daphnia species: D. magna, D. galeata, D. lumholtzi, D. pulicaria, and D. pulex. We observe that the animals, while foraging for food, choose turning angles from distributions that can be described by exponential functions with a range of widths. This observation leads us to speculate and test the notion that this characteristic distribution of turning angles evolved in order to enhance survival. In the case of theoretical agents, some form of randomness is often introduced into search algorithms, especially when information regarding the sought object(s) is incomplete or even misleading. In the case of living animals, many studies have focused on search strategies that involve randomness [H.C. Berg, Random Walks in Biology, Princeton University, Princeton, New Jersey, 1993; A. Okubo, S.A. Levin (Eds.), Diffusion and Ecological Problems: Modern Perspectives, second ed., Springer, New York, 2001]. A simple theory based on stochastic differential equations of the motion backed up by a simulation shows that the collection of material (information, energy, food, supplies, etc.) by an agent executing Brownian-type hopping motions is optimized while foraging for a finite time in a supply patch of limited spatial size if the agent chooses turning angles taken from an exponential distribution with a specific stochastic intensity or 'noise width'. Search strategies that lead to optimization is a topic of high current interest across many disciplines [D. Wolpert, W. MacReady, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1 (1997) 67].

In this paper we develop a theory to describe stochastic influences on the fate of new species with non-linear growth rates in evolutionary processes. We develop a theoretical framework based on notions of species, network, innovation, competition, survival and fitness. We introduce a stochastic picture describing the role of fluctuations in the survival of new species in non-linear systems. In particular we consider the fate of new species with non-linear growth. As an application of the general model framework we consider the fate of 'rare species' in early biological evolution. We show that hypercycle systems do not represent the end of the evolutionary process as they may evolve further in small niches. This has implications for different types of applications ranging from biological systems on one level to socio-technological systems on a more metaphoric level.

Processing of external stimuli by sensory neurons often involves bursting, when epochs of fast firing alternate with intervals of quiescence. In particular, sensory neurons of electroreceptors in paddlefish (Polyodon spathula) undergo bursting when stimulated externally with broad-band noise, but otherwise fire spontaneously in a quasiperiodic tonic manner. We use a simple phenomenological model for noise-induced bursting to quantify analytically, by means of the Kullback entropy and Fisher information, the gain in information transfer and electroreceptor sensitivity for external noisy stimuli. A good agreement between theoretical predictions, numerical simulations and experimental data is shown.

We investigate stochastic resonance in the nonlinear, one-dimensional Fisher-Eigen model (FEM), which represents an archetypal model for biological evolution based on a global coupling scheme. In doing so we consider different periodically driven fitness functions which govern the evolution of a biological phenotype population. For the case of a simple harmonic fitness function we are able to derive the exact analytic solution for the asymptotic probability density. A distinct feature of this solution is a phase lag between the driving signal and the linear response of the system. Furthermore, for more complex systems a general perturbation theory (linear response approximation) is put forward. Using the latter approach, we investigate stochastic resonance in terms of the spectral amplification measure for a quadratic, a quartic single-peaked, and for a bistable fitness function. Our analytical results are also compared with those of detailed numerical simulations. Our findings vindicate that stochastic resonance does occur in these nonlinear, globally coupled biological systems.

In this paper two different evolutionary strategies are tested by means of harmonic landscapes. Both strategies are based on ensembles of searchers, spreading over the search space according to laws inspired by nature. The main difference between the two prototypes is given by the underlying selection mechanism, governing the increase or decrease of the local population of searchers in certain regions of the search space. More precisely, we compare a thermodynamic strategy, which is based on a physically motivated local selection criterion, with a biologically motivated strategy, which features a global selection scheme (i.e., global coupling of the searchers). Confining ourselves to a special class of initial conditions, we show that, in the simple case of harmonic test potentials, both strategies possess particular analytical solutions of the same type. By means of these special solutions, the velocities of the two strategies can be compared exactly. In the last part of the paper, we extend the scope of our discussion to a mixed strategy, combining local and global selection.

The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.

We study the transport of overdamped Brownian particles in periodic potentials subject to a spatially modulated Gaussian white noise. We derive an analytical expression for the diffusion coefficient of particles. By means of velocity, diffusion coefficient, and their ratio (Péclet number) we discuss (a) symmetric potential and modulation of noise intensity and (b) a ratchet profile with strong noise modulation. It is shown that state dependent fluctuations may not only induce directed transport, but also a pronounced coherence of transport if the potential exhibits a strong asymmetry.

Calcium release is an important tool for cellular signaling processes where chemical signals are converted into spatio-temporal variations of intracellular calcium concentration. We investigated the temporal behavior of a single cluster of inositol-(1,4,5)-triphosphate receptor (IP3R)-I channels and will present an analytic approach to obtain the spectrum of the calcium signal within the cluster. We compare these results with stochastic simulations and obtain an intermediate number of channels per cluster for optimal signaling periodicity.

We study noise-induced resonance effects in the leaky integrate-and-fire neuron model with absolute refractory period, driven by a Gaussian white noise. It is demonstrated that a finite noise level may either maximize or minimize the regularity of the spike train. We also partition the parameter space into regimes where either or both of these effects occur. It is shown that the coherence minimization at moderate noise results in a flat spectral response with respect to periodic stimulation in contrast to sharp resonances that are observed for both small and large noise intensities.

We consider a dilute or semidilute polymer solution with localized attracting centers near a flat phase boundary and assume it driven by both stochastic and periodic forces. The attracting inhomogeneities restrict the free motion of macromolecules and play the role of fixed pinning centers. The flat boundary is modeled by a bistable potential whose minima attract the movable polymer segments between neighboring pinning points. We study the motion of these segments. The stochastic forces lead to stochastic oscillations of the polymer parts between the two potential wells near the phase boundary. Application of a small temporal periodic force can synchronize these oscillations and leads to the phenomenon of stochastic resonance for a nonvanishing noise intensity. As an outcome of our theory in agreement with numerical simulations, the resonance is stronger for wider and/or less deep potentials and observed at smaller values of the noise intensity. Additionally, we discuss under what conditions doubly stochastic resonance of the macromolecular motion occurs, that is, if bistability of the potential near the boundary originates in the action of multiplicative noise.

The phenomenon of effective phase synchronization in stochastic oscillatory systems can be quantified by an average frequency and a phase diffusion coefficient. A different approach to compute the noise-averaged frequency is put forward. The method is based on a threshold crossing rate pioneered by Rice. After the introduction of the Rice frequency for noisy systems we compare this quantifier with those obtained in the context of other phase concepts, such as the natural and the Hilbert phase, respectively. It is demonstrated that the average Rice frequency R typically supersedes the Hilbert frequency H, i.e. R > or = H. We investigate next the Rice frequency for the harmonic and the damped, bistable Kramers oscillator, both without and with external periodic driving. Exact and approximative analytic results are corroborated by numerical simulation results. Our results complement and extend previous findings for the case of noise-driven inertial systems.

We show that external fluctuations are able to induce propagation of harmonic signals through monostable media. This property is based on the phenomenon of doubly stochastic resonance, where the joint action of multiplicative noise and spatial coupling induces bistability in an otherwise monostable extended medium, and additive noise resonantly enhances the response of the system to a harmonic forcing. Under these conditions, propagation of the harmonic signal through the unforced medium is observed for optimal intensities of the two noises. This noise-induced propagation is studied and quantified in a simple model of coupled nonlinear electronic circuits.

Zooplankton emit weak electric fields into the surrounding water that originate from their own muscular activities associated with swimming and feeding. Juvenile paddlefish prey upon single zooplankton by detecting and tracking these weak electric signatures. The passive electric sense in this fish is provided by an elaborate array of electroreceptors, Ampullae of Lorenzini, spread over the surface of an elongated rostrum. We have previously shown that the fish use stochastic resonance to enhance prey capture near the detection threshold of their sensory system. However, stochastic resonance requires an external source of electrical noise in order to function. A swarm of plankton, for example Daphnia, can provide the required noise. We hypothesize that juvenile paddlefish can detect and attack single Daphnia as outliers in the vicinity of the swarm by using noise from the swarm itself. From the power spectral density of the noise plus the weak signal from a single Daphnia, we calculate the signal-to-noise ratio, Fisher information and discriminability at the surface of the paddlefish's rostrum. The results predict a specific attack pattern for the paddlefish that appears to be experimentally testable., (Copyright 2002 Academic Press.)

We show that external fluctuations induce excitable behavior in a bistable spatially extended system with activator-inhibitor dynamics of the FitzHugh-Nagumo type. This can be understood as a mechanism for sustained signal propagation in bistable media. The phase diagram of the stochastic system is analytically obtained and numerically verified. For small-noise intensities, front propagation becomes unstable, and excitable pulses arise as the only possible spatiotemporal behavior of the system. For large-noise intensities, on the other hand, the system enters an effective regime of oscillatory behavior, where it exhibits spontaneous nucleation of pulses and synchronized firing.

We study the nonlinear response of a stochastic bistable system driven by both a weak periodic signal and a dichotomic noise in terms of stochastic phase synchronization. We show that the effect of noise-induced phase synchronization can be significantly enhanced by the addition of a dichotomic noise.

Neuronal populations receive signals through temporally inhomogeneous spike trains which can be approximated by an input consisting of a time dependent mean value (additive signal) and noise with a time dependent intensity (noise coded signal). We compare the linear response of an ensemble of model neurons to these signals. Our analytical solution for the mean activity demonstrates the high efficiency of the transmission of a noise coded signal in a broad frequency band. For both kinds of signal we show that the transmission by the ensemble reveals stochastic resonance as well as a nonmonotonous dependence on the driving frequency.

Juvenile paddlefish prey upon single zooplankton by detecting a weak electric signature resulting from its feeding and swimming motions. Moreover, it has recently been shown that paddlefish make use of stochastic resonance near the threshold for prey detection: a process termed behavioral stochastic resonance. But this process depends upon an external source of electric noise. A swarm of plankton, for example, Daphnia, can provide this noise. Assuming that juvenile paddlefish attack single Daphnia as outliers in the vicinity of the swarm, making use of noise from the swarm, we calculate the spatial distribution of the average phase locking period for the subthreshold signals acting at the paddlefish rostrum. Numeric evaluation of analytic formulas supports the notion of a noise-induced widening of the capture area quantitatively.

We study linear responses of a stochastic bistable system driven by dichotomic noise to a weak periodic signal. We show that the effect of stochastic resonance can be greatly enhanced in comparison with the conventional case when dichotomic forcing is absent, that is, both the signal-to-noise ratio and the spectral power amplification reach much greater values than in the standard stochastic resonance setup.

The subject of our study is a two-state dynamics driven by Gaussian white noise and a weak harmonic signal. The system resulting from a piecewise linear FizHugh-Nagumo model in the case of perfect time scale separation between fast and slow variables shows either bistable, excitable, or oscillatory behavior. Its output spectra as well as the spectral power amplification of the signal can be calculated for arbitrary noise strength and frequency, allowing characterization of the coherence resonance in the bistable and excitable regimes as well as quantification of nonadiabatic resonances with respect to the external signal in all regimes.

The phenomenon of noise-induced transport in ratchet devices offers an explanation for directed motion on the molecular scale observed in many biological systems. Net transport through a series of discrete states, occurring in cyclic processes or reactions, can be related to widely investigated continuous ratchet models in the context of thermally activated transitions. The transport process can be described effectively in terms of two characteristic coefficients: velocity and diffusion. Their relation to model parameters and limitations for the ratchet mechanism are discussed in this paper. As an application we consider a four-state model for uphill transmembrane transport and compare theoretical results with existing data from a related experiment.

We study, in terms of synchronization, the nonlinear response of noisy bistable systems to a stochastic external signal, represented by Markovian dichotomic noise. We propose a general kinetic model which allows us to conduct a full analytical study of the nonlinear response, including the calculation of cross-correlation measures, the mean switching frequency, and synchronization regions. Theoretical results are compared with numerical simulations of a noisy overdamped bistable oscillator. We show that dichotomic noise can instantaneously synchronize the switching process of the system. We also show that synchronization is most pronounced at an optimal noise level-this effect connects this phenomenon with aperiodic stochastic resonance. Similar synchronization effects are observed for a stochastic neuron model stimulated by a stochastic spike train.