We study a system consisting of two coupled phase oscillators in the presence of noise. This system is
used as a model for the cardiorespiratory interaction in wakefulness and anaesthesia. We show that longrange
correlated noise produces transitions between epochs with different n:m synchronisation ratios, as
observed in the cardiovascular system. Also, we see that, the smaller the noise (specially the one acting
on the slower oscillator), the bigger the synchronisation time, exactly as happens in anaesthesia compared
with wakefulness. The dependence of the synchronisation time on the couplings, in the presence of noise,
is studied; such dependence is softened by low-frequency noise. We show that the coupling from the slow
oscillator to the fast one (respiration to heart) plays a more important role in synchronisation. Finally, we
see that the isolines with same synchronisation time seem to be a linear combination of the two couplings.
We present a model of the cardiovascular system (CVS) based on a system of coupled oscillators. Using this
approach we can describe several complex physiological phenomena that can have a range of applications. For
instance, heart rate variability (HRV), can have a new deterministic explanation. The intrinsic dynamics of the
HRV is controlled by deterministic couplings between the physiological oscillators in our model and without
the need to introduce external noise as is commonly done. This new result provides potential applications not
only for physiological systems but also for the design of very precise electronic generators where the frequency
stability is crucial. Another important phenomenon is that of oscillation death. We show that in our CVS
model the mechanism leading to the quenching of the oscillations can be controlled, not only by the coupling
parameter, but by a more general scheme. In fact, we propose that a change in the relative current state
of the cardiovascular oscillators can lead to a cease of the oscillations without actually changing the strength
of the coupling among them. We performed real experiments using electronic oscillators and show them to
match the theoretical and numerical predictions. We discuss the relevance of the studied phenomena to real
cardiovascular systems regimes, including the explanation of certain pathologies, and the possible applications
in medical practice.
We address the problem of interactions between the phase of cardiac and respiration oscillatory components.
The coupling between these two quantities is experimentally investigated by the theory of stochastic Markovian
processes. The so-called Markov analysis allows us to derive nonlinear stochastic equations for the reconstruction
of the cardiorespiratory signals. The properties of these equations provide interesting new insights into the
strength and direction of coupling which enable us to divide the couplings to two parts: deterministic and
stochastic. It is shown that the synchronization behaviors of the reconstructed signals are statistically identical
with original one.
KEYWORDS: Oscillators, Systems modeling, Solids, Stochastic processes, Data modeling, Signal generators, Neurons, Control systems, Diagnostics, Diffusion
A Bayesian framework for parameter inference in non-stationary, nonlinear, stochastic, dynamical systems is
introduced. It is applied to decode time variation of control parameters from time-series data modelling physiological
signals. In this context a system of FitzHugh-Nagumo (FHN) oscillators is considered, for which synthetically
generated signals are mixed via a measurement matrix. For each oscillator only one of the dynamical
variables is assumed to be measured, while another variable remains hidden (unobservable). The control parameter
for each FHN oscillator is varying in time. It is shown that the proposed approach allows one: (i) to
reconstruct both unmeasured (hidden) variables of the FHN oscillators and the model parameters, (ii) to detect
stepwise changes of control parameters for each oscillator, and (iii) to follow a continuous evolution of the control
parameters in the quasi-adiabatic limit.
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 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.
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.
Preliminary results are reported from a research project analysing
congestive heart failure in terms a stochastic coupled-oscillator
model of the cardiovascular system. Measurements of blood flow by
laser Doppler flowmetry (LDF) have been processed by use of the
wavelet transform to separate its oscillatory components, which
number at least five. Particular attention was concentrated on the
frequency content near 0.01 Hz, which is known to be associated
with endothelial function. The LDF was carried out in conjunction
with iontophoretically administered acetylcholine (ACh) and sodium
nitroprusside (SNP) in order to evaluate endothelial reactivity.
Measurements were made on 17 congestive heart failure (CHF)
patients (a) on first diagnosis, and (b) again several weeks later
after their treatment with a β-blocker had been stabilised.
The results of these two sets of measurements are being compared
with each other, and with data from an age and sex-matched group
of healthy controls. It is confirmed that endothelial reactivity
is reduced in CHF patients, as compared to healthy controls, and
it is found that one effect of the Beta-blocker is to ameliorate the loss of endothelial function in CHF. The implications of these results are discussed.
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.
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.
A numerical approach based on dynamic importance sampling (DIMS) is introduced to investigate the activation problem in two-dimensional nonequilibrium systems. DIMS accelerates the simulations and allows the investigation to access noise intensities that were previously forbidden. A shift in the position of the escape path compared to a heteroclinic trajectory calculated in the limit of zero noise intensity is directly observed. A theory to account for such shifts is presented and shown to agree with the simulations for a wide range of noise intensities.
The human cardiovascular system (CVS), responsible for the delivery of nutrients and removal of waste products to/from the entire body, is a highly complex system involving many control mechanisms. Signals derived from the CVS are inherently difficult to analyse because they are noisy, time-varying, and of necessarily limited duration. The application of techniques drawn from nonlinear science has, however, yielded many insights into the nature of the CVS, and has provided strong evidence for a large degree of determinism in the way it functions. Yet there is compelling evidence that random fluctuations (noise) also play an essential role. There are at least five oscillatory processes of widely differing frequency involved in the blood distribution. The evidence for them, and their probable physiological origins, are discussed. Interactions between some of the processes can give rise to modulation and synchronization phenomena, very similar to those observed in classical oscillators in many areas of physics. The extent to which the CVS can be modelled as a stochastic nonlinear dynamical system is reviewed, and future research directions and possible applications based on this perception are considered.
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.
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.
Experiments show that the amplitude of turbulent pulsation in
submerged jets rises with increasing distance from the nozzle, at
first slowly and then, after a certain distance, rapidly. This
dependence on distance from the nozzle closely resembles the
dependence of an order parameter on temperature in the case of a
second-order phase transition. Following an idea introduced by
Landa and Zaikin in 1996, it is suggested that the onset of
turbulence is a noise-induced phase transition similar to that in
a pendulum with a randomly vibrated suspension axis. The Krylov-Bogolyubov asymptotic method is used to provide an approximate description of the transition. Results obtained in this way are shown to coincide closely with experimental data. Such an approach is appropriate because the convective character of the instability means that turbulence in nonclosed flows cannot be a self-oscillatory process, as is often assumed. Rather, it must originate in the external random disturbances that are always present in real flows.
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.
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
KEYWORDS: Fractal analysis, Statistical analysis, Probability theory, Dynamical systems, Complex systems, Solids, Physics, Stochastic processes, Gas lasers, Control systems
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.
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.
The responses of an all-optical bistable system and an analog model of the Brownian motion in the symmetric Duffing potential to a weak periodic force in the presence of noise are investigated. The appearance of a stochastic resonance in both cases is explained in the theory of a linear response.
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