P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience

This study aims to determine the phenomenological bifurcation(P-bifurcation)occur-ring in the van der Pol(VDP)neuronal model of burst neurons with a random signal.We observe the P-bifurcation under an intense noise stimulus which would become chaotic transitions.Bursting and spiking simulations are used to describe the causes of chaotic transitions between two periodic phases that are the reason for the neuronal activities.Ran-domness plays a crucial role in detecting the P-bifurcation.To determine the equilibrium points,stability or instability of the stochastic VDP equation,and bifurcation,we use the stochastic averaging method and some related theorems.Apart from theoretical methods,we also examine numerical simulations in the particular case of that stochastic equation that illustrates the outcome of theorems for various noise types.The most striking part of our theoretical findings is that these results are also valid for the Izhikevich-FitzHugh model,Bonhoeffer-van der Pol oscillator in dynamical systems of neuroscience.Finally,we will discuss some applications of the VDP equation in neuronal activity.

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

This paper investigates the stochastic P-bifurcation(SPB)of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise.Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function(PDF)of the system response.Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function.Numerical results are given to show the effectiveness of the proposed approach.Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions.

Stationary response of stochastic viscoelastic system with the right unilateral nonzero offset barrier impacts

The stationary response of viscoelastic dynamical system with the right unilateral nonzero offset barrier impacts subjected to stochastic excitations is investigated. First, the viscoelastic force is approximately treated as equivalent terms associated with effects. Then, the free vibro-impact(VI) system is absorbed to describe the periodic motion without impacts and quasi-periodic motion with impacts based upon the level of system energy. The stochastic averaging of energy envelope(SAEE) is adopted to seek the stationary probability density functions(PDFs). The detailed theoretical results for Van der Pol viscoelastic VI system with the right unilateral nonzero offset barrier are solved to demonstrate the important effects of the viscoelastic damping and nonzero rigid barrier impacts condition. Monte Carlo(MC) simulation is also performed to verify the reliability of the suggested approach. The stochastic P-bifurcation caused by certain system parameters is further explored. The variation of elastic modulus from negative to zero and then to positive witnesses the evolution process of stochastic P-bifurcation. From the vicinity of the common value to a wider range, the relaxation time induces the stochastic P-bifurcation in the two interval schemes.

Method for extracting geometrical characteristics of joint probability density based on contour lines

This paper presents a method for extracting geometrical features of the joint probability density function(PDF)of two-dimensional systems based on its contour lines,with particular interests given to the number and position of peaks and craters.In order to detect those two types of structures,a series of horizontal planes are applied to truncate the joint PDF with contour lines generated.Starting with the analysis of contour lines in a single plane,shape characteristics of the peak and the crater can be reflected on the contour lines in the aspects of gradient direction and inclusion relationship.Aided by the properties of PDF,the information about gradient direction and inclusion relationship of contour lines can be obtained simultaneously if the contour tree is built.According to the contour tree,the contour lines can be classified as two groups.Then the corresponding relation between contour lines in different planes is discussed.Based on the corresponding relation,clustering analysis about contour lines belonging to the same group but having different heights is performed.Two sets of contour lines are finally obtained as the simplest expression of geometrical characteristics of a joint PDF.They can be used to obtain the number and position of each peak and crater.Three oscillators of different types are chosen to test this method,which shows that this method can pave the way for numerical calculation about the stochastic P-bifurcation of multi-dimensional systems.