In this paper, we develop a Bayesian framework for the empirical estimation of the parameters of one of the best known
nonlinear models of the business cycle: The Marx-inspired model of a growth cycle introduced by R. M. Goodwin. The
model predicts a series of closed cycles representing the dynamics of labor's share and the employment rate in the
capitalist economy. The Bayesian framework is used to empirically estimate a modified Goodwin model. The original
model is extended in two ways. First, we allow for exogenous periodic variations of the otherwise steady growth rates of
the labor force and productivity per worker. Second, we allow for stochastic variations of those parameters. The resultant
modified Goodwin model is a stochastic predator-prey model with periodic forcing. The model is then estimated using a
newly developed Bayesian estimation method on data sets representing growth cycles in France and Italy during the
years 1960-2005. Results show that inference of the parameters of the stochastic Goodwin model can be achieved. The
comparison of the dynamics of the Goodwin model with the inferred values of parameters demonstrates quantitative
agreement with the growth cycle empirical data.
A novel conceptual model is introduced in which ion permeation is coupled to the protein wall vibration and the
later in turn modulates exponentially strongly the permeation via radial oscillations of the potential of mean
force. In the framework of this model of ion-wall-water interaction we discuss problems of selectivity between
alike ions and coupling of ion permeation to gating.
Ionic motion in the bulk solution away from the mouth of a biological ion channel, and inside the channel, is
analyzed using Poisson-Nernst-Planck (PNP) equation. The one-dimensional method allows us to connect in
a self-consistent way ion dynamics in the bulk solution and inside the channel by taking into account access
resistance to the channel. In order to glue the PNP solution in the bulk to that inside the channel, a continuity
condition is used for the concentration and the current near the channel mouth at the surface of the hemisphere.
The resulting one dimensional (1D) current-voltage characteristics are compared with the Kurnikova16 results
which are in good agreement with experimental measurement on the channel, by using a filling factor as the
only fitting parameter. The filling factor compensates the fact that the radial charge distribution is non-uniform
in a real channel as compared to the cylindrically symmetrical channel used in the 1D approximation.
A direct comparison between continuous and discrete forms of
analysis of control and stability is investigated theoretically
and numerically. We demonstrate that the continuous method provides
a more energy-efficient means of controlling the switching of a
periodically-driven class-B laser between its stable and unstable
pulsing regimes. We provide insight into this result using the
close correspondence that exists between the problems of
energy-optimal control and the stability of a steady state.
We consider the following general problem of applied stochastic
nonlinear dynamics. We observe a time series of signals y(t) = y(t0+hn) corrupted by noise. The actual state and the nonlinear vector field of the dynamical system is not known. The question is how and with what accuracy can we determine x(t) and functional form of f(x). In this talk we discuss a novel approach to the solution of this problem based on the application of the path-integral approach to the full Bayesian inference. We demonstrate a reconstruction of a dynamical state of a system from corrupted by noise measurements. Next we reconstruct the corresponding nonlinear vector field. The emphasis are on the theoretical analysis. The results are compared with the results of earlier research.
The polarization dynamics of a vertical cavity surface emitting laser is investigated as a nonlinear stochastic dynamical system.
The polarization switches in the device are considered as activation processes in a two dimensional system with a saddle cycle; the optimal way of switching is determined as the solution of a boundary value problem. The theoretical results are in good agreement with the numerical simulations.
The response of a noisy FitzHugh-Nagumo (FHN) neuron-like model to
weak periodic forcing is analyzed. The mean activation time is
investigated as a function of noise intensity and of the parameters of the external signal. It is shown by numerical simulation that there exists a frequency range within which the phenomenon of resonant activation occurs; resonant activation is also observed in coupled FHN elements. The mean activation time with small noise intensity is compared with the theoretical results.
An application of the path-integral approach to an analysis of the
fluctuations in complex dynamical systems is discussed. It is
shown that essentially the same ideas underly recent progress in
the solutions of a number of long-standing problems in complex
dynamics. In particular, we consider the problems of prediction,
control and inference of chaotic dynamics perturbed by noise in
the framework of path-integral approach.
Ionic motion through an open ion channel is analyzed within the framework of self-consistent Brownian dynamics formalism. A novel conceptual model of coupling of the ion's motion to the vibrations of the pore walls is introduced. The model allows one to include into simulations an important additional mechanism of energy dissipation and the effects of self-induced strong modulation of the channel conductivity.
A new method of inferencing of coupled stochastic nonlinear
oscillators is described. The technique does not require extensive
global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range of dynamical models. We illustrate the main ideas of the technique by inferencing a model of five globally and locally coupled noisy oscillators. Specific modifications of the technique for inferencing hidden degrees of freedom of coupled nonlinear oscillators is discussed in the context of physiological applications.
The vertical-cavity surface-emitting laser (VCSEL) is a novel type of semiconductor laser that is having a dramatic influence on many optical applications, including computing, communication, and sensing. Unfortunately, VCSELs suffer from a randomly fluctuating polarization whose dynamics are not fully understood. The model of stochastic polarization dynamics developed by San Miguel is rather complicated, and comparisons between experiment and simulations are quite difficult. One of the approaches to solve this problem was suggested in, where a simplified spin-eliminated linearized model (i.e., with dynamics reduced to that of a class-A laser) is used to analyze experimentally measured fluctuations and fluctuational switchings. In this talk, we present a recently developed technique that estimates the parameters of a nonlinear stochastic dynamical model by Bayesian inference, and demonstrate its application to the characterization of VCSELs. We start by considering the problem of diffusion of polarization in a potential well, onto which the dynamics of a class-A laser are usually mapped. We demonstrate the ability to infer laser parameters in numerical and analogue simulations, with the emphasis being placed on the role of large deviations. We specifically show that, contrary to one's intuition, the quality of inference can be improved by neglecting those data points in experimental time series that correspond to the rising part of large deviations. We then extend our technique to the full set of equations describing the polarization dynamics of a VCSEL in terms of
the motion of its Stokes vector on the Poincare sphere. Application of this technique to other standard problems encountered in characterizing semiconductor lasers, such as the identification of laser parameters from measurements of relaxational oscillations, is also discussed.
We study fluctuational transitions in a discrete dynamical system
between two co-existing chaotic attractors separated by a
fractal basin boundary. It is shown that there is a generic
mechanism of fluctuational transition through a fractal boundary
determined by a hierarchy of homoclinic original saddles. The most
probable escape path from a chaotic attractors to the fractal boundary is found using both statistical analysis of fluctuational trajectories and Hamiltonian theory of fluctuations.
Fluctuational escape via an unstable limit cycle is investigated
in stochastic flows and maps. A new topological method is
suggested for analysis of the corresponding boundary value
problems when the action functional has multiple local minima
along the escape trajectories and the search for the global
minimum is otherwise impossible. The method is applied to the
analysis of the escape problem in the inverted Van der Pol
oscillator and in the Henon map. An application of this technique
to solution of the escape problem in chaotic maps with fractal
boundaries, and in maps with chaotic saddles embedded within the
basin of attraction, is discussed.
The non-equilibrium distribution in dissipative dynamical systems with unstable limit cycle is analyzed in the next-to-leading order of the small-noise approximation of the Fokker-Planck equation. The noise-induced variations of the non-equilibrium distribution are described in terms of topological changes in the pattern of optimal paths. It is predicted that singularities in the pattern of optimal paths are shifted and cross the basin boundary in the presence of finite noise. As a result the probability distribution oscillates at the basin boundary. Theoretical predictions are in good agreement with the results of numerical solution of the Fokker-Planck equation and Monte Carlo simulations.
We investigate theoretically and numerically the activation process in a single-out and coupled FitzHugh-Nagumo elements. Two qualitatively different types of the dependence of the mean activation time and of the mean cycling time on the coupling strength monotonic and non-monotonic have been found for identical elements. The influence of coupling strength, noise intensity and firing threshold on the synchronization regimes and its characteristics is analyzed
We suggest a fresh approach to the modeling of the human cardiovascular system. Taking advantage of a new Bayesian inference technique, able to deal with stochastic nonlinear systems, we show that one can estimate parameters for models of the cardiovascular system directly from measured time series. We present preliminary results of inference of parameters of a model of coupled oscillators from measured cardiovascular data addressing cardiorespiratory interaction. We argue that the inference technique offers a very promising tool for the modeling, able to contribute significantly towards the solution of a long standing challenge -- development of new diagnostic techniques based on noninvasive measurements.