Biomedical subjects

I M Sokolov

Publications and source records attributed to I M Sokolov.

At least 19 recordsLinked to original sources

PubMed2006

Field-induced dispersion in subdiffusion.

We discuss the response of continuous-time random walks to an oscillating external field within the generalized master equation approach. We concentrate on the time dependence of the two first moments of the walker's displacement. We show that for power-law waiting-time distributions with 0<alpha<1 corresponding to a semi-Markovian situation showing nonstationarity, the mean particle position tends to a constant; namely, the response to the external perturbation dies out. On the other hand, the oscillating field leads to a new additional contribution to the dispersion of the particle position, proportional to the square of its amplitude and growing with time. These new effects, amenable to experimental observation, result directly from the nonstationary property of the system.

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PubMed2006

Linear response to perturbation of nonexponential renewal process: A generalized master equation approach.

The work by Barbi, Bologna, and Grigolini [Phys. Rev. Lett. 95, 220601 (2005)] discusses a response to alternating external field of a non-Markovian two-state system, where the waiting time between the two attempted changes of state follows a power law. It introduced a new instrument for description of such situations based on a stochastic master equation with reset. In the present Brief Report we provide an alternative description of the situation within the framework of a generalized master equation. The results of our analytical approach are corroborated by direct numerical simulations of the system.

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PubMed2006

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 non-Markovian 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.

Action Potentials↗

PubMed2006

Reaction-subdiffusion equations.

To analyze possible generalizations of reaction-diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion A --> B. We derive the corresponding kinetic equations for the local and concentrations. Their form is rather unusual: The parameters of the reaction influence the diffusion term in the equation for a component A, a consequence of the non-Markovian nature of subdiffusion. The equation for the product contains a term which depends on the concentration of A at all previous times. Our discussion shows that reaction-subdiffusion equations may not resemble the corresponding reaction-diffusion ones and are not obtained by a trivial change of the diffusion operator for a subdiffusion one.

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PubMed2006

Blowing DNA bubbles.

We report here experimental observations which indicate that topologically or covalently formed polymer loops embedded in an ultrathin liquid film on a solid substrate can be "blown" into circular "bubbles" during scanning force microscopy (SFM) imaging. In particular, supercoiled vector DNA has been unraveled, moved, stretched, and overstretched to two times its B-form length and then torn apart. We attribute the blowing of the DNA bubbles to the interaction of the tapping SFM tip with the ultrathin liquid film.

DNA↗

PubMed2005

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion.

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

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PubMed2004

Reshuffling scale-free networks: from random to assortative.

To analyze the role of assortativity in networks we introduce an algorithm which produces assortative mixing to a desired degree. This degree is governed by one parameter p . Changing this parameter one can construct networks ranging from fully random (p=0) to totally assortative (p=1) . We apply the algorithm to a Barabási-Albert scale-free network and show that the degree of assortativity is an important parameter governing the geometrical and transport properties of networks. Thus, the average path length of the network increases dramatically with the degree of assortativity. Moreover, the concentration dependences of the size of the giant component in the node percolation problem for uncorrelated and assortative networks are strongly different. The behavior of the clustering coefficient is also discussed.

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PubMed2004

From subdiffusion to superdiffusion of particles on solid surfaces.

We present a numerical and partially analytical study of classical particles obeying a Langevin equation that describes diffusion on a surface modeled by a two-dimensional potential. The potential may be either periodic or random. Depending on the potential and the damping, we observe superdiffusion, large-step diffusion, diffusion, and subdiffusion. Superdiffusive behavior is associated with low damping and is in most cases transient, albeit often long. Subdiffusive behavior is associated with highly damped particles in random potentials. In some cases subdiffusive behavior persists over our entire simulation and may be characterized as metastable. In any case, we stress that this rich variety of behaviors emerges naturally from an ordinary Langevin equation for a system described by ordinary canonical Maxwell-Boltzmann statistics.

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PubMed2004

Spectra and waiting-time densities in firing resonant and nonresonant neurons.

The response of a neural cell to an external stimulus can follow one of two patterns: nonresonant neurons monotonically relax to the resting state after excitation while resonant ones show subthreshold oscillations. We investigate how these subthreshold properties of neurons affect their suprathreshold response. Conversely we ask the following: 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 either having a focus or a node as the stable fixed point. We determine numerically the spectral power density as well as the interspike interval density in response to random (noiselike) 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 provides 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.

Action Potentials↗

PubMed2004

Diffusion on a solid surface: anomalous is normal.

We present a numerical study of classical particles diffusing on a solid surface. The particles' motion is modeled by an underdamped Langevin equation with ordinary thermal noise. The particle-surface interaction is described by a periodic or a random two-dimensional potential. The model leads to a rich variety of different transport regimes, some of which correspond to anomalous diffusion such as has recently been observed in experiments and Monte Carlo simulations. We show that this anomalous behavior is controlled by the friction coefficient and stress that it emerges naturally in a system described by ordinary canonical Maxwell-Boltzmann statistics.

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PubMed2004

SIS epidemics with household structure: the self-consistent field method.

We consider a stochastic SIS infection model for a population partitioned into m households assuming random mixing. We solve the model in the limit m --> infinity by using the self-consistent field method of statistical physics. We derive a number of explicit results, and give numerical illustrations. We then do numerical simulations of the model for finite m and without random mixing. We find in many of these cases that the self-consistent field method is a very good approximation.

Endemic Diseases↗

PubMed2003

Correlations in scale-free networks: tomography and percolation.

We discuss three related models of scale-free networks with the same degree distribution but different correlation properties. Starting from the Barabási-Albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes. We point out that the Barabási-Albert model displays dissortative behavior with respect to the nodes' degrees, while the node-randomized network shows assortative mixing. These kinds of correlations are visualized by discussing the shell structure of the networks around an arbitrary node. In spite of different correlation behaviors, all three constructions exhibit similar percolation properties. This result for percolation is also detected for a network with finite second moment and its corresponding randomized models.

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PubMed2003

Cyclization of a polymer: first-passage problem for a non-Markovian process.

We discuss a closure of a polymer chain released from a state with given end-to-end distance. Considering the situation as a first passage problem for a non-Markovian process, we express the survival probability through the three-time distribution of the end-to-end distance and evaluate it numerically by solving a Volterra equation. We also show how the Wilemski-Fixmann approximation emerges within this scheme and discuss its geometrical nature.

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PubMed2002

Geography in a scale-free network model.

We offer an example of a network model with a power-law degree distribution, P(k) approximately k(-alpha), for nodes, but which nevertheless has a well-defined geography and a nonzero threshold percolation probability for alpha>2, the range of real-world contact networks. This is different from p(c)=0 for alpha<3 results for the original well-mixed scale-free networks. In our lattice-based scale-free network, individuals link to nearby neighbors on a lattice. Even considerable additional small-world links do not change our conclusion of nonzero thresholds. When applied to disease propagation, these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering.

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PubMed2002

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).

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PubMed2002

Solutions of a class of non-Markovian Fokker-Planck equations.

We show that a formal solution of a rather general non-Markovian Fokker-Planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the Markovian equation with the same Fokker-Planck operator. This allows us to classify memory kernels into safe ones, for which the solution is always a probability density, and dangerous ones, when this is not guaranteed. The first situation describes random processes subordinated to a Wiener process, while the second one typically corresponds to random processes showing a strong ballistic component. In this case the non-Markovian Fokker-Planck equation is only valid in a restricted range of parameters, initial and boundary conditions.

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PubMed2002

Evolving networks with disadvantaged long-range connections.

We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability of getting connected to a site at distance d is proportional to d(-alpha), where alpha is a tunable parameter of the model. We show that the properties of the networks grown with alpha<1 are close to those of the genuine scale-free construction, while for alpha>1 the structure of the network is quite different. Thus, in this regime, the node degree distribution is no longer a power law, and it is well represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of alpha.

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PubMed2002

Relative dispersion in fully developed turbulence: the Richardson's law and intermittency corrections.

Relative dispersion in fully developed turbulence is investigated by means of direct numerical simulations. Lagrangian statistics is found to be compatible with Richardson description although small systematic deviations are found. The value of the Richardson constant is estimated as C2 approximately equal to 0.55, in a close agreement with recent experimental findings [S. Ott and J. Mann, J. Fluid Mech. 422, 207 (2000)]. By means of exit-time statistics it is shown that the deviations from Richardson's law are a consequence of Eulerian intermittency. The measured Lagrangian scaling exponents require a set of Eulerian structure function exponents zeta(p) which are remarkably close to standard ones known for fully developed turbulence.

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