Development of spiral wave in a regular network of excitatory neurons due to stochastic poisoning of ion channels

Some experimental evidences show that spiral wave could be observed in the cortex of brain, and the propagation of this spiral wave plays an important role in signal communication as a pacemaker. The profile of spiral wave generated in a numerical way is often perfect while the observed profile in experiments is not perfect and smooth. In this paper, formation and development of spiral wave in a regular network of Morris–Lecar neurons, which neurons are placed on nodes uniformly in a two-dimensional array and each node is coupled with nearest-neighbor type, are investigated by considering the effect of stochastic ion channels poisoning and channel noise. The formation and selection of spiral wave could be detected as follows. (1) External forcing currents with diversity are imposed on neurons in the network of excitatory neurons with nearest-neighbor connection, a target-like wave emerges and its potential mechanism is discussed; (2) artificial defects and local poisoned area are selected in the network to induce new wave to interact with the target wave; (3) spiral wave can be induced to occupy the network when the target wave is blocked by the artificial defects or poisoned area with regular border lines; (4) the stochastic poisoning effect is introduced by randomly modifying the border lines (areas) of specific regions in the network. It is found that spiral wave can be also developed to occupy the network under appropriate poisoning ratio. The process of growth for the poisoned area of ion channels poisoning is measured, the effect of channels noise is also investigated. It is confirmed that perfect spiral wave emerges in the network under gradient poisoning even if the channel noise is considered.     

Transition from spiral wave to target wave and other coherent structures in the networks of Hodgkin-Huxley neurons

Transition of spiral wave in the regular networks of Hodgkin-Huxley (H-H) neurons is simulated and discussed in detail when the effect of membrane temperature and forcing current is considered. Neurons are distributed in the sites of two-dimensional array, neurons are connected with complete nearest-neighbor connections, no-flux boundary conditions, appropriate initial values and physiological parameters are used to develop a stable rotating spiral wave. A statistic factor of synchronization is defined to discuss the transition and development of spiral wave in the two parameters space (membrane temperature T and forcing current I), and it is found that spiral wave keeps alive due to positive current forcing and the spiral wave can be removed completely when the temperature is increased to a threshold about T = 22.3 °C at a fixed current intensity. Periodical forcing current is imposed on the networks of neurons globally and locally, respectively. It is found that spiral wave could be suppressed by the new generated traveling wave or target wave when periodical forcing current is imposed on the border of networks of neurons, and the most effective frequency of the external forcing current is close to the intrinsic frequency of the spiral wave of the networks.     

Dynamics of chemical wave segments with free ends

A quite simple but useful approach is performed for the analysis of chemical wave segments with free ends. By integrating a reaction–diffusion system we can obtain an analytical expression to understand the dynamics of the wave segments. This integration can yield qualitative information regarding wave development under an external forcing having feedback or noise effects. We conclude that this wave development is influenced not only by medium excitability but also by wave size.     

Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin-Huxley neurons

An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction-diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin-Huxley neurons is measured by imposing an additive term V_E on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 × 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive.     

Enhanced logical stochastic resonance under periodic forcing

It was demonstrated recently that noise in an optimal window allows a bistable system to operate reliably as reconfigurable logic gates (Murali et al., 2009) [1], as well as a memory device (Kohar and Sinha, 2012) [11]. Namely, in a range of moderate noise, the system can operate flexibly, both as a NAND/AND/OR/NOR gate and a Set Reset latch. Here we demonstrate how the width of the optimal noise window can be increased by utilizing the constructive interplay of noise and periodic forcing, namely noise in conjunction with a periodic drive yields consistent logic outputs for all noise strengths below a certain threshold. Thus we establish that in scenarios where noise level is below the minimum threshold required for logical stochastic resonance (or stochastic resonance in general), we can add a periodic forcing to obtain the desired effects. Lastly, we also show how periodic forcing reduces the switching time, leading to faster operation of devices and lower latency effects.     

Dynamics of spiral waves driven by a rotating electric field

We study the behavior of spiral wave under the driving of a rotating electric field. The rotating electric field can drive a spiral wave to be synchronous, depending on four factors: its frequency and amplitude, chirality, and polarized modes. Rotation-synchronization characterized by the rotating direction is focused on. We discuss the behavior of synchronization, such as the dependence of angle-differences between the spiral tip and the electric field on ratio of frequency, the influences of different polarized modes of the electric field, the radius of synchronous spiral wave, and so on. A circularly polarized electric fields (CPEF) can suppress meandering spiral to rigid one and prevent breakup of spiral in medium with low excitability. The phase diagram describing the controllable region in excitability-period plane is presented. The influences of polarized modes of electric field on minimum excitability of medium are also studied.     

The drift of spirals under competitive illumination in an excitable medium

We study numerically the resonant drift of spirals induced by periodic illuminations in excitable media for the Oregonator model. Differential phase illumination of a competitive system with an interface causes interesting spiral drift behavior which can be related to the phase difference. It is found that the drift directions and velocities have been controlled by changing their phase differences, and the spiral has been forbidden to drift back to the initial region due to the interface. Furthermore, the simulation result seems to be reliable as it is also consistent perfectly with the theoretical analysis based on the weak deformation approximation.     

Multiplicity of rotating spirals under curvature flows with normal tip motion

We study the global dynamics of a singular nonlinear ordinary differential equation, which is autonomous of second order. This equation arises from a model for steadily rotating spiral waves in excitable media. The sharply located spiral wave fronts are modeled as planar curves. Their normal velocity is assumed to depend affine linearly on curvature. The spiral tip rotates along a circle with a constant rotation frequency. It neither grows nor retracts tangentially to the curve. With rotation frequency as a parameter, we derive the global structure of solutions of the associated initial value problem for this ODE, by an analytical approach. In particular, the number of solutions for each given rotation frequency can be computed. The multiplicity of coexisting rotating spiral curves can be any positive integer.     

Mather theory for piecewise smooth Lagrangian systems

We establish the Mather theory for a type of piecewise smooth and positive definite Lagrangian systems.It models a mechanical system subject to external impulsive forcing.We show the existence of the minimal measure and the Lipschitz property of Aubry set.In addition,the weak KAM solution to this kind of piecewise smooth Lagrangian is also established.     

Additive noise driven phase transitions in a predator-prey system

We explore the impact of additive noise on phase transitions in a predator-prey system, which is formulated by stochastic partial differential equations (SPDEs). The system is observed to experience twice phase transitions under certain level of additive noise. We extend the multiple scaling approach from single SPDE to multiple SPDEs and find a necessary and sufficient condition to excite the occurrence of the first transition from a spatially homogeneous state to a spatially regular spiral wave. Numerical experiments show the second phase transition from the regular spatial pattern to spiral turbulence. We conclude that additive noise has a destabilising effect on population dynamics by triggering the onset of Hopf bifurcation.     

Numerical resolution of stochastic focusing NLS equations

In this note, we numerically investigate a stochastic nonlinear Schrödinger equation derived as a perturbation of the deterministic NLS equation. The classical NLS equation with focusing nonlinearity of power law type is perturbed by a random term; it is a strong perturbation since we consider a space-time white noise. It acts either as a forcing term (additive noise) or as a potential (multiplicative noise). For simulations made on a uniform grid, we see that all trajectories blow-up in finite time, no matter how the initial data are chosen. Such a grid cannot represent a noise with zero correlation lengths, so that in these experiments, the noise is, in fact, spatially smooth. On the contrary, we simulate a noise with arbitrarily small scales using local refinement and show that in the multiplicative case, blow-up is prevented by a space-time white noise. We also present results on noise induced soliton diffusion.     

A nonlinear fracture differential kinetic model to depict chaotic atom motions at a fatigue crack tip based on the differentiable manifold methodology

A nonlinear differential kinetic model describing dynamical behaviours of an atom at a fatigue crack tip is developed in this paper. It is assumed that the forces acted on this atom by its surrounding atoms consist of the following three components: (1) an elastic restoring force governed by Leonard-Jones potential, which describes the elastic interaction between atoms; (2) a nonlinear damping force proportional to its velocity through a linear function of its displacement as a coefficient that empirically simulates the energy loss from the crack tip to its surroundings; (3) an external remote driving force to represent thermally activated energy supplied to the crack tip from the surroundings. Based on these assumptions of the interaction forces between the atoms around the crack tip, a nonlinear dynamic equation describing the motion of the atom at a crack tip using the Newton’s second principle is derived. For a periodic external force and a random one influenced by parameters omitted, deterministic and a stochastic analyses on the dynamic equation obtained above are completed. Based on the theories of the Hopf bifurcation, global bifurcation and stochastic bifurcation, the extent and some possible implications of the existence of atomic-scale chaotic and stochastic bifurcative motions involving the fracture behaviour of actual materials are systematically and qualitatively discussed and the extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. As demonstrated in the paper, chaotic behaviour may be observed in the case of a larger amplitude of the driving force and a smaller damping constant. The white noise introduced in the atomistic motion process may leads to a drift of the divergence point of the nonlinear stochastic differential kinetic system in contrast to the homoclinic divergence of the nonlinear deterministic differential kinetic system.     

Mean-square stability properties of an adaptive time-stepping SDE solver

We consider stability properties of a class of adaptive time-stepping schemes based upon the Milstein method for stochastic differential equations with a single scalar forcing. In particular, we focus upon mean-square stability for a class of linear test problems with multiplicative noise. We demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls, one for the drift term and one for the diffusion.     

Oscillatory reaction-diffusion equations with temporally varying parameters

Periodic wave trains are the generic one-dimensional solution form for reaction-diffusion equations with a limit cycle in the kinetics. Such systems are widely used as models for oscillatory phenomena in chemistry, ecology, and cell biology. In this paper, we study the way in which periodic wave solutions of such systems are modified by periodic forcing of kinetic parameters. Such forcing will occur in many ecological applications due to seasonal variations. We study temporal forcing in detail for systems of two reaction diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. In this case, the kinetics can be approximated by the Hopf normal form, giving reaction-diffusion equations of λ-ω type. Numerical simulations show that a temporal variation in the kinetic parameters causes the wave train amplitude to oscillate in time, whereas in the absence of any temporal forcing, this wave train amplitude is constant. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the amplitude of the wave train oscillations with small forcing. We show that the amplitude of these oscillations depends crucially on the period of forcing.     

完整Coriolis力作用下带有外源强迫的非线性ZK方程

在正压流体中,从包含完整Coriolis参数的准地转位涡方程出发,在弱非线性长波近似下,采用多时空尺度和摄动方法,推导出大气非线性Rossby波振幅演变满足带有外源强迫的二维Zakharov-Kuznetsov(ZK)方程.然后利用Jacobi椭圆函数展开法,求解了ZK方程的椭圆正弦波解和孤立波解.分析结果表明,Coriolis参数的水平分量将影响二维Rossby波传播的频率特征,而外源不仅对二维Rossby波动的传播的频率有影响,对振幅也产生一个调制作用.     

Stochastic resonance with different periodic forces in overdamped two coupled anharmonic oscillators

We study the stochastic resonance phenomenon in the overdamped two coupled anharmonic oscillators with Gaussian noise and driven by different external periodic forces. We consider (i) sine, (ii) square, (iii) symmetric saw-tooth, (iv) asymmetric saw-tooth, (v) modulus of sine and (vi) rectified sinusoidal forces. The external periodic forces and Gaussian noise term are added to one of the two state variables of the system. The effect of each force is studied separately. In the absence of noise term, when the amplitude f of the applied periodic force is varied cross-well motion is realized above a critical value (f_c) of f. This is found for all the forces except the modulus of sine and rectified sinusoidal forces. For fixed values of angular frequency ω of the periodic forces, f_c is minimum for square wave and maximum for asymmetric saw-tooth wave. f_c is found to scale as Ae^0.75ω + B where A and B are constants. Stochastic resonance is observed in the presence of noise and periodic forces. The effect of different forces is compared. The stochastic resonance behaviour is quantized using power spectrum, signal-to-noise ratio, mean residence time and distribution of normalized residence times. The logarithmic plot of mean residence time τ_MR against 1/(D − D_c) where D is the intensity of the noise and D_c is the value of D at which cross-well motion is initiated shows a sharp knee-like structure for all the forces. Signal-to-noise ratio is found to be maximum at the noise intensity D = D_max at which mean residence time is half of the period of the driving force for the forces such as sine, square, symmetric saw-tooth and asymmetric saw-tooth waves. With modulus of sine wave and rectified sine wave, the SNR peaks at a value of D for which sum of τ_MR in two wells of the potential of the system is half of the period of the driving force. For the chosen values of f and ω, signal-to-noise ratio is found to be maximum for square wave while it is minimum for modulus of sine and rectified sinusoidal waves. The values of D_c at which cross-well behaviour is initiated and D_max are found to depend on the shape of the periodic forces.     

Vibrational resonance in a time-delayed genetic toggle switch

Biological oscillators can respond in a surprising way when they are perturbed by two external periodic forcing signals of very different frequencies. The response of the system to a low-frequency signal can be enhanced or depressed when a high-frequency signal is acting. This is what is known as vibrational resonance (VR). Here we study this phenomenon in a simple time-delayed genetic toggle switch, which is a synthetic gene-regulatory network. We have found out how the low-frequency signal changes the range of the response, while the high-frequency signal influences the amplitude at which the resonance occurs. The delay of the toggle switch has also a strong effect on the resonance since it can also induce autonomous oscillations.     

Ergodicity of Truncated Stochastic Navier Stokes with Deterministic Forcing and Dispersion

Turbulence in idealized geophysical flows is a very rich and important topic. The anisotropic effects of explicit deterministic forcing, dispersive effects from rotation due to the \(\beta \)-plane and F-plane, and topography together with random forcing all combine to produce a remarkable number of realistic phenomena. These effects have been studied through careful numerical experiments in the truncated geophysical models. These important results include transitions between coherent jets and vortices, and direct and inverse turbulence cascades as parameters are varied, and it is a contemporary challenge to explain these diverse statistical predictions. Here we contribute to these issues by proving with full mathematical rigor that for any values of the deterministic forcing, the \(\beta \)- and F-plane effects and topography, with minimal stochastic forcing, there is geometric ergodicity for any finite Galerkin truncation. This means that there is a unique smooth invariant measure which attracts all statistical initial data at an exponential rate. In particular, this rigorous statistical theory guarantees that there are no bifurcations to multiple stable and unstable statistical steady states as geophysical parameters are varied in contrast to claims in the applied literature. The proof utilizes a new statistical Lyapunov function to account for enstrophy exchanges between the statistical mean and the variance fluctuations due to the deterministic forcing. It also requires careful proofs of hypoellipticity with geophysical effects and uses geometric control theory to establish reachability. To illustrate the necessity of these conditions, a two-dimensional example is developed which has the square of the Euclidean norm as the Lyapunov function and is hypoelliptic with nonzero noise forcing, yet fails to be reachable or ergodic.     

Optimal tax depreciation with loss carry-forward and backward options

The choice of depreciation method from among straight-line and accelerated methods can have a significant impact on the present value of expected tax payments. This is a problem that has been studied for decades, with most results indicating the optimality of accelerated methods. Recent research questions this claim by relaxing the assumption of positive taxable income. The situation where net-operating losses may be carried-forward and backward in time is the subject of this paper. We model this situation and establish conditions that allow straight-line depreciation to be preferred over accelerated methods. The results are focused around a threshold number of periods of consecutive losses, which are determined by the allowable periods to carry a loss forward. For consecutive losses beyond this threshold, straight-line will always be optimal. When the cumulative depreciation charges up to and including the window are guaranteed to be applied on or before the threshold period, then straight-line will never be optimal.