scholarly journals Methods of Stochastic Analysis of Complex Regimes in the 3D Hindmarsh–Rose Neuron Model

2018 ◽  
Vol 17 (01) ◽  
pp. 1850008 ◽  
Author(s):  
Irina Bashkirtseva ◽  
Lev Ryashko ◽  
Evdokia Slepukhina
Keyword(s):  
Stochastic Dynamics ◽  
Random Noise ◽  
Dynamic Regime ◽  
Sensitivity Function ◽  
Interspike Intervals ◽  
Hr Model ◽  
Random States ◽  
Parametric Region ◽  
Good Agreement ◽  
Qualitative Changes

A problem of the stochastic nonlinear analysis of neuronal activity is studied by the example of the Hindmarsh–Rose (HR) model. For the parametric region of tonic spiking oscillations, it is shown that random noise transforms the spiking dynamic regime into the bursting one. This stochastic phenomenon is specified by qualitative changes in distributions of random trajectories and interspike intervals (ISIs). For a quantitative analysis of the noise-induced bursting, we suggest a constructive semi-analytical approach based on the stochastic sensitivity function (SSF) technique and the method of confidence domains that allows us to describe geometrically a distribution of random states around the deterministic attractors. Using this approach, we develop a new algorithm for estimation of critical values for the noise intensity corresponding to the qualitative changes in stochastic dynamics. We show that the obtained estimations are in good agreement with the numerical results. An interplay between noise-induced bursting and transitions from order to chaos is discussed.

scholarly journals Stochastic Bifurcations and Noise-Induced Chaos in 3D Neuron Model

2016 ◽  
Vol 26 (12) ◽  
pp. 1630032 ◽  
Author(s):  
Irina Bashkirtseva ◽  
Sergei Fedotov ◽  
Lev Ryashko ◽  
Evdokia Slepukhina
Keyword(s):  
Stochastic Dynamics ◽  
Stable Equilibrium ◽  
Deterministic Model ◽  
Three Dimensional ◽  
Random Disturbances ◽  
Stable Equilibria ◽  
Relationship Of ◽  
Stochastic Bifurcations ◽  
Good Agreement ◽  
Qualitative Changes

The stochastically forced three-dimensional Hindmarsh–Rose model of neural activity is considered. We study the effect of random disturbances in parametric zones where the deterministic model exhibits mono- and bistable dynamic regimes with period-adding bifurcations of oscillatory modes. It is shown that in both cases the phenomenon of noise-induced bursting is observed. In the monostable zone, where the only attractor of the system is a stable equilibrium, this effect is connected with a stochastic generation of large-amplitude oscillations due to the high excitability of the model. In a parametric zone of coexisting stable equilibria and limit cycles, bursts appear due to noise-induced transitions between the attractors. For a quantitative analysis of the noise-induced bursting and corresponding stochastic bifurcations, an approach based on the stochastic sensitivity function (SSF) technique is applied. Our estimations of the strength of noise that generates such qualitative changes in stochastic dynamics are in a good agreement with the direct numerical simulation. A relationship of the noise-induced generation of bursts with transitions from order to chaos is discussed.

The Exploitation of Variability and Noise in Micro Devices

2007 ◽  
Author(s):  
Kiyohisa Nishiyama ◽  
M. C. L. Ward
Keyword(s):  
Statistical Theory ◽  
Brownian Motion ◽  
Random Noise ◽  
Mean Value ◽  
Measurement Tool ◽  
Electronic Circuits ◽  
Johnson Noise ◽  
Micro Systems ◽  
Switch Group ◽  
Good Agreement

The strength of Micro Systems Technology (MST) is the ability to fabricate a large number of small devices economically. However such devices tend to have errors caused by the variations of fabrication and inherent noise signals such as Brownian motion or Johnson noise. This paper develops the understanding of Micro Switch Group Sensors (MSGS), which works by exploiting this random noise. In the experimental work reported here, an MSGS comprising of 20 switches has been built using electronic circuits and tested to verify the performance. The output of the device is simply the number of switches turned on, this is then transformed into the expected mean value of the input signal using standard statistical theory. The performance of the devices was shown to be in good agreement with the theoretical prediction. The linearity and the standard deviation of the output signal of MSGS are investigated and it is concluded that an MSGS may be successfully applied as a measurement tool.

Bifurcations Induced in a Bistable Oscillator via Joint Noises and Time Delay

2016 ◽  
Vol 26 (06) ◽  
pp. 1650102 ◽  
Author(s):  
Jin Fu ◽  
Zhongkui Sun ◽  
Yuzhu Xiao ◽  
Wei Xu
Keyword(s):  
Time Delay ◽  
Qualitative Behavior ◽  
Deterministic Case ◽  
Van Der Pol ◽  
Stochastic Case ◽  
Stationary Probability Density Function ◽  
Theoretical Analyses ◽  
Probability Density Function Pdf ◽  
Good Agreement ◽  
Qualitative Changes

In this paper, noise-induced and delay-induced bifurcations in a bistable Duffing–van der Pol (DVP) oscillator under time delay and joint noises are discussed theoretically and numerically. Based on the qualitative changes of the plane phase, delay-induced bifurcations are investigated in the deterministic case. However, in the stochastic case, the response of the system is a stochastic non-Markovian process owing to the existence of noise and time delay. Then, methods have been employed to derive the stationary probability density function (PDF) of the amplitude of the response. Accordingly, stochastic P-bifurcations can be observed with the variations in the qualitative behavior of the stationary PDF for amplitude. Furthermore, results from both theoretical analyses and numerical simulations best demonstrate the appearance of noise-induced and delay-induced bifurcations, which are in good agreement.

Approximating Chaotic Attractors by Period-Three Cycles in Discrete Stochastic Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550138 ◽  
Author(s):  
Irina Bashkirtseva ◽  
Lev Ryashko
Keyword(s):  
Stochastic Systems ◽  
Original System ◽  
Sensitivity Function ◽  
Function Technique ◽  
Chaotic Attractors ◽  
Suggested Approach ◽  
One Dimensional ◽  
Discrete Time Systems ◽  
Random States ◽  
Time Systems

We study a distribution of random states forced outwards of the general deterministic attractor (regular or chaotic) for one-dimensional discrete-time systems with unimodal map. To approximate this distribution, we replace the original system by the appropriate modeling system with stable 3-cycle and use the stochastic sensitivity function technique. Constructive abilities of the suggested approach are demonstrated in the analysis of noise-induced transitions from chaos to order.

scholarly journals Evolution and Extinction Dynamics in Rugged Fitness Landscapes

1998 ◽  
Vol 12 (04) ◽  
pp. 361-391 ◽  
Author(s):  
Paolo Sibani ◽  
Michael Brandt ◽  
Preben Alstrøm
Keyword(s):  
Time Distribution ◽  
Stochastic Dynamics ◽  
Mean Field ◽  
Life Time ◽  
Added Mass ◽  
Fitness Landscapes ◽  
Mass Extinctions ◽  
And Behavior ◽  
Average Fitness ◽  
Good Agreement

After an introductory section summarizing the paleontological data and some of their theoretical descriptions, we describe the "reset" model and its (in part analytically soluble) mean field version, which have been briefly introduced in Letters.1,2 Macroevolution is considered as a problem of stochastic dynamics in a system with many competing agents. Evolutionary events (speciations and extinctions) are triggered by fitness records found by random exploration of the agents' fitness landscapes. As a consequence, the average fitness in the system increases logarithmically with time, while the rate of extinction steadily decreases. This non-stationary dynamics is studied by numerical simulations and, in a simpler mean field version, analytically. We also consider the effect of externally added "mass" extinctions. The predictions for various quantities of paleontological interest (life-time distribution, distribution of event sizes and behavior of the rate of extinction) are robust and in good agreement with available data.

Stochastic Sensitivity Analysis and Noise-Induced Chaos in 2D Logistic-Type Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650053 ◽  
Author(s):  
Irina Bashkirtseva ◽  
Ekaterina Ekaterinchuk ◽  
Lev Ryashko
Keyword(s):  
Random Noise ◽  
Type Model ◽  
Invariant Curves ◽  
Form Complex ◽  
Suggested Approach ◽  
Sensitivity Functions ◽  
Stochastic Sensitivity ◽  
Random Disturbances ◽  
Stochastic Sensitivity Analysis ◽  
Random States

We study the dynamics of stochastically forced 2D logistic-type discrete model. Under random disturbances, stochastic trajectories leaving deterministic attractors can form complex dynamic regimes that have no analogue in the deterministic case. In this paper, we analyze an impact of the random noise on 2D logistic-type model in the bistability zones with coexisting attractors (equilibria, closed invariant curves, discrete cycles). For the constructive probabilistic analysis of the random states distribution around such attractors, a stochastic sensitivity functions technique and method of confidence domains are used. For the considered model, on the base of the suggested approach, a phenomenon of noise-induced transitions between attractors and the generation of chaos are analyzed.

scholarly journals The determination of the activation energy for a Vibro-impact system under multiple excitations

2021 ◽  
Author(s):  
jianlong wang ◽  
Xiaolei Leng ◽  
Xianbin Liu
Keyword(s):  
Activation Energy ◽  
Stochastic Stability ◽  
Large Deviation ◽  
Random Noise ◽  
Steady States ◽  
Impact System ◽  
Working Out ◽  
Probability Density Function Pdf ◽  
Good Agreement

Abstract In this paper, the stochastic stability of a Vibro-impact system with multiple excitation forces is studied. Due to the multiple external excitations, the probability density function (PDF) of the system is extremely difficult to solve. In addition, the existence of coexisting steady states is very common for the Vibro-impact system, and the perturbation of random noise will cause the transitions between the steady states. In this case, we are more interested in working out each attractor’s activation energy, which is specifically used to characterize the attractor’s stochastic stability, rather than the solution of the PDF. Based on the large deviation theory, the asymptotic analysis is carried out, and a time-varying Hamilton’s equation for the quasi-potential is derived. To verify the effectiveness of the theoretical analysis, two detailed examples, where an impact attractor and a non-impactor coexist in the system, are conducted. By the application of the action plot method, the activation energies and the most probable exit paths (MPEP) for each attractor are derived. Compared with the numerical simulation, it shows very good agreement. Moreover, it is found that the existence of transient chaos near the attractor could seriously deteriorate the attractor’s stability.

Stochastic Dynamics of Sea Surface Height Variability

2010 ◽  
Vol 40 (7) ◽  
pp. 1582-1596 ◽  
Author(s):  
Philip Sura ◽  
Sarah T. Gille
Keyword(s):  
Stochastic Dynamics ◽  
Kuroshio Extension ◽  
Random Noise ◽  
Equations Of Motion ◽  
Sea Surface Height ◽  
Ocean Dynamics ◽  
Sea Surface ◽  
Double Exponential ◽  
Ocean Topography ◽  
Non Gaussian

Abstract Sea surface height anomalies measured by the Ocean Topography Experiment (TOPEX)/Poseidon satellite altimeter indicate high values of skewness and kurtosis. Except in a few regions, including the Gulf Stream, the Kuroshio Extension, and the Agulhas Retroflection, that display bimodal patterns of sea surface height variability, kurtosis is uniformly greater than 1.5 times the squared skewness minus an adjustment constant. This relationship differs substantially from what standard Gaussian or double-exponential noise would produce. However, it can be explained by a simple theory in which the noise is assumed to be multiplicative, meaning that a larger background state implies larger random noise elements. The existence of multiplicative noise can be anticipated from the equations of motion, if ocean dynamics are split into a slowly decorrelating deterministic component and a rapidly decorrelating contribution that is approximated as noise. Such a model raises the possibility of predicting the probabilities of extreme sea surface height anomalies from first physical principles and may provide a useful null hypothesis for non-Gaussian sea surface height variability.

scholarly journals Improving LADCP Velocity with External Heading, Pitch, and Roll

2017 ◽  
Vol 34 (8) ◽  
pp. 1713-1721 ◽  
Author(s):  
A. M. Thurnherr ◽  
I. Goszczko ◽  
F. Bahr
Keyword(s):  
Random Noise ◽  
The Arctic ◽  
Accelerometer Data ◽  
Velocity Measurements ◽  
Beam Velocity ◽  
Attitude Measurements ◽  
Magnetic Poles ◽  
Acoustic Doppler Current Profilers ◽  
Magnetic Disturbances ◽  
Good Agreement

AbstractData collected with acoustic Doppler current profilers installed on CTD rosettes and lowered through the water column [lowered ADCP (LADCP) systems] are routinely used to derive full-depth profiles of ocean velocity. In addition to the uncertainties arising from random noise in the along-beam velocity measurements, LADCP-derived velocities are commonly contaminated by bias errors due to imperfectly measured instrument attitude (heading, pitch, and roll). Of particular concern are the heading measurements, because it is not usually feasible to calibrate the internal ADCP compasses with the instruments installed on a CTD rosette, away from the magnetic disturbances of the ship. Heading data from dual-headed LADCP systems, which consist of upward- and downward-pointing ADCPs installed on the same rosette, commonly indicate heading-dependent compass errors with amplitudes exceeding 10°. In an attempt to reduce LADCP velocity errors, several dozen profiles of simultaneous LADCP and magnetometer/accelerometer data were collected in the Gulf of Mexico. Agreement between the LADCP profiles and simultaneous shipboard velocity measurements improves significantly when the former are processed with external attitude measurements. Another set of LADCP profiles with external attitude data was collected in a region of the Arctic Ocean where the horizontal geomagnetic field is too weak for the ADCP compasses to work reliably. Good agreement between shipboard velocity measurements and Arctic LADCP profiles collected at magnetic dip angles exceeding and processed with external attitude measurements indicate that high-quality velocity profiles can be obtained close to the magnetic poles.

MODELING A SIMPLE CHOICE TASK: STOCHASTIC DYNAMICS OF MUTUALLY INHIBITORY NEURAL GROUPS

2001 ◽  
Vol 01 (02) ◽  
pp. 159-191 ◽  
Author(s):  
ERIC BROWN ◽  
PHILIP HOLMES
Keyword(s):  
Model Building ◽  
Stochastic Dynamics ◽  
Random Noise ◽  
Numerical Data ◽  
Cognitive Test ◽  
Stochastic Neural Networks ◽  
One Dimensional ◽  
Probability Densities ◽  
Low Dimensional ◽  
And Control

We describe the dynamical and bifurcational behavior of two mutually inhibitory, leaky, neural units subject to external stimulus, random noise, and "priming biases". The model describes a simple forced choice experiment and accounts for varying levels of expectation and control. By projecting the model's dynamics onto slow manifolds, using judicious linear approximations, and solving for one-dimensional (reduced) probability densities, analytical estimates are developed for reaction time distributions and shown to compare satisfactorily with "full" numerical data. A sensitivity analysis is performed and the effects of parameters assessed. The predictions are also compared with behavioral data. These results may help correlate low-dimensional models of stochastic neural networks with cognitive test data, and hence assist in parameter choices and model building.

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