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The SOS/ERK cascades are key signaling pathways that regulate cellular processes ranging from cellular proliferation, differentiation and apoptosis to tumor formation. However, the properties of these signaling pathways are not well understood. More importantly, how stochastic perturbations of internal and external cellular environment affect these pathways remains unanswered. To answer these questions, we, in this paper, propose a stochastic model according to the biochemical reaction processes of the SOS/ERK pathways, and, respectively, research the dynamical behaviors of this model under the four kinds of noises: Gaussian noise, colored noise, Lévy noise and fraction Brown noise. Some important results are found that Gaussian and colored noises have less effect on the stability of the system when the strength of the noise is small; Lévy and fractional Brownian noises significantly change the trajectories of the system. Power spectrum analysis shows that Lévy noise induces a system with quasi-periodic trajectories. Our results not only provide an understanding of the SOS/ERK pathway, but also show generalized rules for stochastic dynamical systems.

Models of quantum mechanical anharmonic lattice systems ("anharmonic crystals") are described. Temperature quantum Gibbs states are represented by classical Gibbs measures for lattice systems of loop-valued spin variables. These Gibbs measures are also obtained as invariant (equilibrium) measures of a system of stochastic differential equations ("stochastic dynamics", "stochastic quantization"). Existence and uniqueness results for these equations are established and a construction of the solution via a finite volume approximation is given. The Markov property of this solution is also exhibited and properties of the Gibbs distributions (existence, a prioiri estimates, regularity of support) are characterized in terms of the stochastic dynamics. Ergodicity and uniqueness of the Gibbs distributions are also discussed.

Hopfield neural network (HNN) is a nonlinear computational model successfully applied in finding near-optimal solutions of several difficult combinatorial problems. In many cases, the network energy function is obtained through a learning procedure so that its minima are states falling into a proper subspace (feasible region) of the search space. However, because of the network nonlinearity, a number of undesirable local energy minima emerge from the learning procedure, significantly effecting the network performance.

In the neural model analyzed here, we combine both a penalty and a stochastic process in order to enhance the performance of a binary HNN. The penalty strategy allows us to gradually lead the search towards states representing feasible solutions, so avoiding oscillatory behaviors or asymptotically instable convergence. Presence of stochastic dynamics potentially prevents the network to fall into shallow local minima of the energy function, i.e., quite far from global optimum. Hence, for a given fixed network topology, the desired final distribution on the states can be reached by carefully modulating such process.

The model uses pseudo-Boolean functions both to express problem constraints and cost function; a combination of these two functions is then interpreted as energy of the neural network. A wide variety of NP-hard problems fall in the class of problems that can be solved by the model at hand, particularly those having a monotonic quadratic pseudo-Boolean function as constraint function. That is, functions easily derived by closed algebraic expressions representing the constraint structure and easy (polynomial time) to maximize.

We show the asymptotic convergence properties of this model characterizing its state space distribution at thermal equilibrium in terms of Markov chain and give evidence of its ability to find high quality solutions on benchmarks and randomly generated instances of two specific problems taken from the computational graph theory.

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Lagrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dX_t = b(X_t)dt+ϵdW_t, where W_t is a Brownian motion. In the limit of vanishingly small ϵ, the solution to the stochastic differential equation other than are all rare events. However, conditioned on an occurrence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with and Hamiltonian equations with H(p, q) = ‖p‖²+b(q)⋅p. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for X_t as f(x, t) = e^{-u(x, t)/ϵ}, where u(x, t) is called a large-deviation rate function which satisfies the corresponding Hamilton–Jacobi equation. An irreversible diffusion process with ∇×b≠0 corresponds to a Newtonian system with a Lorentz force . The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions and integrable systems.

In order to study the stochastic Markov processes conditioned on a specific value of a time-integrated observable, the concept of ensembles of trajectories has been recently used extensively. In this paper, we consider a generic reaction–diffusion process consisting of classical particles with nearest-neighbor interactions on a one-dimensional lattice with periodic boundary conditions. By introducing a time-integrated current as a physical observable, we have found certain constraints on the microscopic transition rates of the process under which the effective process contains local interactions; however, with rescaled transition rates comparing to the original process. A generalization of the linear Glauber model is then introduced and studied in detail as an example. Associated effective dynamics of this model is investigated and constants of motion are obtained.

We study the overdamped Brownian dynamics of a particle in a sawtooth potential along with a temporal asymmetric driving force. We observe that in the deterministic limit, the transport coherence which is determined by a dimensionless quantity, called Peclet number Pe, is quite high under certain circumstances. For all the regime of parameter space of this model, Pe in our model shows similar features of current like Stokes efficiency. Diffusion as a function of driving amplitude shows a nonmonotonic behavior and results a minimum exactly at which the current shows a maximum. Unlike the previously studied models, Pe in our model shows a peaking behavior with temperature. Moreover, the diffusion shows a nonlinear dependence of temperature in the long-time limit and it is sensitive to the potential asymmetry parameter.

Induced pluripotent stem cells (iPSCs) provide a great model to study the process of stem cell reprogramming and differentiation. Single-cell RNA sequencing (scRNA-seq) enables us to investigate the reprogramming process at single-cell level. Here, we introduce single-cell entropy (scEntropy) as a macroscopic variable to quantify the cellular transcriptome from scRNA-seq data during reprogramming and differentiation of iPSCs. scEntropy measures the relative order parameter of genomic transcriptions at single cell level during the process of cell fate changes, which show increase tendency during differentiation, and decrease upon reprogramming. Hence, scEntropy provides an intrinsic measurement of the cell state, and can be served as a pseudo-time of the stem cell differentiation process. Moreover, based on the evolutionary dynamics of scEntropy, we construct a phenomenological Fokker-Planck equation model and the corresponding stochastic differential equation for the process of cell state transitions during pluripotent stem cell differentiation. These equations provide further insights to infer the processes of cell fates changes and stem cell differentiation. This study is the first to introduce the novel concept of scEntropy to quantify the biological process of iPSC, and suggests that the scEntropy can provide a suitable macroscopic variable for single cells to describe cell fate transition during differentiation and reprogramming of stem cells.

It is a great pleasure to be part of the celebration of Nitant Kenkre’s scientific career. In this paper, we review mathematical models for studying evolutionary branching and diversification in biology. We highlight insights provided by Nitant Kenkre’s work in the early 2000s as part of the Consortium of the Americas for Interdisciplinary Science. We focus on the use of these models for understanding the conditions that promote phenotypic diversification and innovation in biological systems, understood as the branching of stationary solutions, and some of the ideas that emerged as part of the research Consortium. We highlight recent developments and applications of stochastic, nonlocal partial derivative formalisms.

The concept of deterministic dynamical chaos has a long history and is well established by now. Nevertheless, its field theoretic essence and its stochastic generalization have been revealed only very recently. Within the newly found supersymmetric theory of stochastics (STS), all stochastic differential equations (SDEs) possess topological or de Rahm supersymmetry and stochastic chaos is the phenomenon of its spontaneous breakdown. Even though the STS is free of approximations and thus is technically solid, it is still missing a firm interpretational basis in order to be physically sound. Here, we make a few important steps toward the construction of the interpretational foundation for the STS. In particular, we discuss that one way to understand why the ground states of chaotic SDEs are conditional (not total) probability distributions, is that some of the variables have infinite memory of initial conditions and thus are not “thermalized”, i.e., cannot be described by the initial-conditions-independent probability distributions. As a result, the definitive assumption of physical statistics that the ground state is a steady-state total probability distribution is not valid for chaotic SDEs.

The stability of portfolio investment in stock market crashes with Markowitz portfolio is investigated by the method of theoretical and empirical simulation. From numerical simulation of the mean escape time (MET), we conclude that: (i) The increasing number $(Np)$ of stocks in Markowitz portfolio induces a maximum in the curve of MET versus the initial position; (ii) A critical value of $Np$ in the behavior of MET versus the long-run variance or amplitude of volatility fluctuations maximumlly enhances the stability of portfolio investment. When $Np$ takes value below the critical value, the increasing $Np$ enhances the stability of portfolio investment, but restrains it when $Np$ takes value above the critical value. In addition, a good agreement of both the MET and probability density functions of returns is found between real data and theoretical results.

Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares, etc., that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity, etc. Within the recently proposed supersymmetric theory of stochastic dynamics, the N-phase can be roughly interpreted as the noise-induced “overlap” between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or aperiodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

A stochastic dynamic model of a Kaplan turbine is established in this paper during the transient process. When the Kaplan turbine operates with fluctuating load, the synergistic relationship between the guide vanes and blades experiences random fluctuation resulting from the mechanical, hydraulic and signal factors. To study the effect of stochastic fluctuations of the synergistic relationship, Chebyshev polynomial approximation method is adopted to analyze the stochastic dynamic characteristics of the Kaplan turbine during the transient process. Using Chebyshev polynomial approximation, the stochastic model of the Kaplan turbine is simplified to its equivalent deterministic model, and the stochastic dynamic characteristics of the model are investigated in the transient process. The effects of stochastic intensity on the dynamic behaviors of the Kaplan turbine are analyzed by means of numerical simulation. Moreover, the influences of PID parameters on the stochastic dynamic characteristics of the Kaplan turbine are studied through bifurcation diagrams. Analysis of stochastic characteristics and dynamic behaviors suggests that transient performance improvement can be obtained by controlling the synergistic stochastic intensity and PID parameters.

Some chaotic and a series of stochastic neural firings are multimodal. Stochastic multimodal firing patterns are of special importance because they indicate a possible utility of noise. A number of previous studies confused the dynamics of chaotic and stochastic multimodal firing patterns. The confusion resulted partly from inappropriate interpretations of estimations of nonlinear time series measures. With deliberately chosen examples the present paper introduces strategies and methods of identification of stochastic firing patterns from chaotic ones. Aided by theoretical simulation we show that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractor or two limit cycle attractors. In contrast, the multimodal chaotic firing trains are generated by the dynamics of a specific strange attractor. Three systems were carefully chosen to elucidate these two mechanisms. An experimental neural pacemaker model and the Chay mathematical model were used to show the stochastic dynamics, while the deterministic Wang model was used to show the deterministic dynamics. The usage and interpretation of nonlinear time series measures were systematically tested by applying them to firing trains generated by the three systems. We successfully identified the distinct differences between stochastic and chaotic multimodal firing patterns and showed the dynamics underlying two categories of stochastic firing patterns. The first category results from the effects of noise on the neuronal system near a Hopf bifurcation. The second category results from the effects of noise on the period-adding bifurcation between two limit cycles. Although direct application of nonlinear measures to interspike interval series of these firing trains misleadingly implies chaotic properties, definition of eigen events based on more appropriate judgments of the underlying dynamics leads to accurate identifications of the stochastic properties.

We have studied the effect of a particular kind of non-Gaussian noise (NGN), mainly of its deviation q from Gaussian noise, on the intercellular calcium (Ca²⁺) oscillations in an array of bidirectionally coupled cells. It is found that as q is increased, the Ca²⁺ oscillation becomes the most regular at an intermediate optimal q value, representing the occurrence of coherence resonance (CR). This deviation-induced CR behavior shows that the intercellular Ca²⁺ oscillations of the coupled cells can be enhanced and even optimized by the appropriate NGN. This result provides a new insight into the constructive role of the NGN on the transmission of Ca²⁺ signaling in coupled cells.

Synchronization is a common phenomenon whereby a dynamical system follows the pacemaker provided by an external forcing. Often, such systems have multiple synchronization modes, which are equivalent solutions. We investigate the specific case of two to one synchronization produced by the periodic forcing of a van der Pol oscillator where two possible modes, shifted by one period of the modulation, exist. By studying the flow and the local Lyapunov exponents along the orbit we give an explanation of the noise induced jumps observed in a stochastic forced oscillator. While this investigation gives results that are specific to this system, the framework presented is more general and can be applied to any system showing similar jumping dynamics.

The paper deals with a mathematical description of human feelings (such as hostility, or indifference or love) toward other human beings. The discussion is carried out in a somewhat simplified scheme, by describing the psychological state of each individual by only one state variable, conceived as a kind of "measure" of its feelings. This is, however, to be meant as simply a first step towards a more comprehensive description of human psychology, taking into account character, tastes and possibly past experiences of each individual involved in a social relation. For the simplified description here developed, we propose a system of nonlinear integro-differential stochastic equations, aiming at giving at least a probabilistic forecast of the evolution of reciprocal feeling of two relating individuals. Thus, the solution of the system will be a couple of probability density functions, for which we present a short preliminary discussion of existence and uniqueness as well as of stability and the possible occurrence of strong instability effects. A number of numerical simulations are presented to test the significance of the model and some ways to extend the scheme, by introducing a number of additional external as well as internal parameters that can reasonably be assumed to influence the reciprocal feelings, are suggested and discussed, with special attention to the concept of stochastic "constitutive laws" for the feelings.

In this paper, we provide an analytical framework for investigating the efficiency of a consensus-based model for tackling global optimization problems. This work justifies the optimization algorithm in the mean-field sense showing the convergence to the global minimizer for a large class of functions. Theoretical results on consensus estimates are then illustrated by numerical simulations where variants of the method including nonlinear diffusion are introduced.

During the inflationary regime, the expansion of the Universe is driven by a scalar field ϕ(t) which may be in thermal contact with the radiation fluid. In this work, we study the influence of the thermal bath assuming that it is responsible for the stochastic evolution of the inflaton field. Assuming that the fluctuation dynamics is described by a Langevin-type equation of motion, a large set of analytical solutions including white and colored noises are derived. It is found that even in the case of white noise the field experience an anomalous diffusion. Such results may be important for studying thermally induced initial density perturbations in inflationary cosmologies, mainly in the framework of warm inflation.

In the present paper, a modeling in the complex space is combined with complex-valued fractional Brownian motion to get some new results in biological systems. The rational of this approach is as follows. Biological dynamics which evolve continuously in time but are not time differentiable, necessarily exhibit random properties. These random features appear also as a result of the randomness of the proper time of biological systems. Usually, this is taken into account by using white noises that is to say fractals of order two. Fractals of order n larger than two are more suitable for increments with large amplitudes, and they may be introduced by using either real-valued fractal noises with long range memory or Brownian motions with independent increments, which are necessarily complex-valued. In the later case, we are then led to describe biological systems in the complex plane. After some background on the complex-valued fractional Brownian motion, we shall deal successively with population growth, information thermodynamics of order n, nonequilibrium phase transition via fractal noises and complexity of Markovian processes via the concept of informational divergence.

We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for simple special cases are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.