Stochastic chaos: An analog of quantum chaos

Some intriging connections between the properties of nonlinear noise driven systems and the nonlinear dynamics of a particular set of Hamilton's equation are discussed. A large class of Fokker-Planck Equations, like the Schr\"odinger equation, can exhibit a transition in their spectral statistics as a coupling parameter is varied. This transition is connected to the transition to non-integrability in the Hamilton's equations.Comment: Uuencoded compressed postscript file, 4 pages, 3 fig

Temporally Asymmetric Fluctuations are Sufficient for the Operation of a Correlation Ratchet

It has been shown that the combination of a broken spatial symmetry in the potential (or ratchet potential) and time correlations in the driving are crucial, and enough to allow transformation of the fluctuations into work. The required broken spatial symmetry implies a specific molecular arrangement of the proteins involved. Here we show that a broken spatial symmetry is not required, and that temporally asymmetric fluctuations (with mean zero) can be used to do work, even when the ratchet potential is completely symmetric. Temporal asymmetry, defined as a lack of invariance of the statistical properties under the operation to temporal inversion, is a generic property of nonequilibrium fluctuation, and should therefore be expected to be quite common in biological systems.Comment: 17 pages, ps figures on request, LaTeX Article Forma

Recovering ‘lost’ information in the presence of noise: application to rodent–predator dynamics.

A Hamiltonian approach is introduced for the reconstruction of trajectories and models of complex stochastic dynamics from noisy measurements. The method converges even when entire trajectory components are unobservable and the parameters are unknown. It is applied to reconstruct nonlinear models of rodent–predator oscillations in Finnish Lapland and high-Arctic tundra. The projected character of noisy incomplete measurements is revealed and shown to result in a degeneracy of the likelihood function within certain null-spaces. The performance of the method is compared with that of the conventional Markov chain Monte Carlo (MCMC) technique

Trans-membrane Signal Transduction and Biochemical Turing Pattern Formation

The Turing mechanism for the production of a broken spatial symmetry in an initially homogeneous system of reacting and diffusing substances has attracted much interest as a potential model for certain aspects of morphogenesis such as pre-patterning in the embryo, and has also served as a model for self-organization in more generic systems. The two features necessary for the formation of Turing patterns are short-range autocatalysis and long-range inhibition which usually only occur when the diffusion rate of the inhibitor is significantly greater than that of the activator. This observation has sometimes been used to cast doubt on applicability of the Turing mechanism to cellular patterning since many messenger molecules that diffuse between cells do so at more-or-less similar rates. Here we show that stationary, symmetry-breaking Turing patterns can form in physiologically realistic systems even when the extracellular diffusion coefficients are equal; the kinetic properties of the 'receiver' and 'transmitter' proteins responsible for signal transduction will be primary factors governing this process

Observable and hidden singular features of large fluctuations in nonequilibrium systems

We study local features, and provide a topological insight into the global structure of the probability density distribution and of the pattern of the optimal paths for large rare fluctuations away from a stable state. In contrast to extremal paths in quantum mechanics, the optimal paths do {\it not} encounter caustics. We show how this occurs, and what, instead of caustics, are the experimentally observable singularities of the pattern. We reveal the possibility for a caustic and a switching line to start at a saddle point, and discuss the consequences.Comment: 10 pages, 3 ps figures by request, LaTeX Article Format (In press, Phys. Lett. A

Applications of dynamical inference to the analysis of noisy biological time series with hidden dynamical variables.

We present a Bayesian framework for parameter inference in noisy, non-stationary, nonlinear, dynamical systems. The technique is implemented in two distinct ways: (i) Lightweight implementation: to be used for on-line analysis, allowing multiple parameter estimation, optimal compensation for dynamical noise, and reconstruction by integration of the hidden dynamical variables, but with some limitations on how the noise appears in the dynamics; (ii) Full scale implementation: of the technique with extensive numerical simulations (MCMC), allowing for more sophisticated reconstruction of hidden dynamical trajectories and dealing better with sources of noise external to the dynamics (measurements noise)

Realistic Models of Biological Motion

The origin of biological motion can be traced back to the function of molecular motor proteins. Cytoplasmic dynein and kinesin transport organelles within our cells moving along a polymeric filament, the microtubule. The motion of the myosin molecules along the actin filaments is responsible for the contraction of our muscles. Recent experiments have been able to reveal some important features of the motion of individual motor proteins, and a new statistical physical description - often referred to as ``thermal ratchets'' - has been developed for the description of motion of these molecules. In this approach the motors are considered as Brownian particles moving along one-dimensional periodic structures due to the effect of nonequilibrium fluctuations. Assuming specific types of interaction between the particles the models can be made more realistic. We have been able to give analytic solutions for our model of kinesin with elastically coupled Brownian heads and for the motion of the myosin filament where the motors are connected through a rigid backbone. Our theoretical predictions are in a very good agreement with the various experimental results. In addition, we have considered the effects arising as a result of interaction among a large number of molecular motors, leading to a number of novel cooperative transport phenomena.Comment: 12 pages (5 figures). submitted to Elsevier Preprin

Swarm field dynamics and functional morphogenesis

A class of models with application to swarm behavior as well as many other types of complex systems is studied with an emphasis on analytic techniques and results. Special attention is given to the role played by fluctuations in determining the behavior of such systems. In particular it is suggested that such fluctuations may play an active role, and not just the usual passive one, in the organization of structure in the vicinity of a non-equilibrium phase transition. One model, that of an ant swarm, is analyzed in more detail as an illustration of these ideas

Parameter and Structure Inference for Nonlinear Dynamical Systems

A great many systems can be modeled in the non-linear dynamical systems framework, as x = f(x) + xi(t), where f() is the potential function for the system, and xi is the excitation noise. Modeling the potential using a set of basis functions, we derive the posterior for the basis coefficients. A more challenging problem is to determine the set of basis functions that are required to model a particular system. We show that using the Bayesian Information Criteria (BIC) to rank models, and the beam search technique, that we can accurately determine the structure of simple non-linear dynamical system models, and the structure of the coupling between non-linear dynamical systems where the individual systems are known. This last case has important ecological applications