The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circ...The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circuit theory,biology and the area of population genetics.For this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a tool.The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule.By the help of this feature which is not provided by other popular fractional derivatives,nonlinear fractional partial differential equation is turned into an integer order differential equation.The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method.This comparison is made both with the help of three-dimensional graphical representations and tables for different values of theγ.展开更多
We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expafunction and extended sinh-Gor...We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expafunction and extended sinh-Gordon equation expansion(ESh GEE) schemes are utilized. The solutions obtained include dark, bright, dark-bright, periodic and other kinds of solitons. These analytical wave solutions are gained and verified with the use of Mathematica software. These solutions do not exist in literature. Some of the solutions are demonstrated by 2D, 3D and contour graphs. This model is mostly used in circuit theory, transmission of nerve impulses, and population genetics. Finally, both the schemes are more applicable, reliable and significant to deal with the fractional nonlinear partial differential equations.展开更多
In this paper, a spatial perturbation scheme is proposed to suppress the spiral wave in the modified Orengonator model, which is used to describe the chemical reaction in the light-sensitive media. The controllable ex...In this paper, a spatial perturbation scheme is proposed to suppress the spiral wave in the modified Orengonator model, which is used to describe the chemical reaction in the light-sensitive media. The controllable external illumination Ф is perturbed with a spatial linear function. In our numerical simulation, the scheme is investigated by imposing the external controllable illumination on the space continuously and/or intermittently. The numerical simulation results confirm that the stable rotating spiral wave still can be removed with the scheme proposed in this paper even if the controllable Ф changed vs. time and space synchronously. Then the scheme is also used to control the spiral wave and turbulence in the modified Fitzhugh-Nagumo model. It is found that the scheme is effective to remove the sable rotating and meandering spiral wave but it costs long transient period and intensity of the gradient parameter to eliminate the soiral turbulence.展开更多
It is well known that the strong coupling can synchronize a network of nonlinear oscillators. Synchronization provides the basis of the remarkable computational performance of the brain. In this paper the FitzHugh-Nag...It is well known that the strong coupling can synchronize a network of nonlinear oscillators. Synchronization provides the basis of the remarkable computational performance of the brain. In this paper the FitzHugh-Nagumo neuron network is constructed. The dependence of the synchronization on the coupling strength, the noise intensity and the size of the neuron network has been discussed. The results indicate that the coupling among neurons works to improve the synchronization, and noise increases the neuron random dynamics and the local fluctuations; the larger the size of network, the worse the synchronization. The dependence of the synchronization on the strength of the electric synapse coupling and chemical synapse coupling has also been discussed, which proves that electric synapse coupling can enhance the synchronization of the neuron network largely.展开更多
It has been proved recently that the spike timing can play an important role in information transmission, so in this paper we develop a network with N-unlt FitzHugh-Nagumo neurons coupled by gap junctions and discuss ...It has been proved recently that the spike timing can play an important role in information transmission, so in this paper we develop a network with N-unlt FitzHugh-Nagumo neurons coupled by gap junctions and discuss the dependence of the spike timing precision on synaptic coupling strength, the noise intensity and the size of the neuron ensemble. The calculated results show that the spike timing precision decreases as the noise intensity increases; and the ensemble spike timing precision increases with coupling strength increasing. The electric synapse coupling has a more important effect on the spike timing precision than the chemical synapse coupling.展开更多
This paper aims to investigate the stochastic resonance （SR） in an FitzHugh-Nagumo （FHN） model with an additive LEvy noise numerically. The non-Gaussian LEvy noise is a kind of general random noise which is differ...This paper aims to investigate the stochastic resonance （SR） in an FitzHugh-Nagumo （FHN） model with an additive LEvy noise numerically. The non-Gaussian LEvy noise is a kind of general random noise which is different from the usual Gaussian noise, and it has small fluctuations with the unpredictable jumps to describe the random fluctuations in an FHN model. SR is determined by the signal-to-noise ratio （SNR）, and the numerical simulation results show the occurrence of the SR phenomena in the given FHN system. The influence of various parameters of the LEvy noise and the FHN model on the SR will be exam- ined, and some mechanisms of the LEvy noise-induced SR are presented which are different from those of the Gaussian noise.展开更多
We numerically study the dynamics of spiral waves in the excitable system with the excitability modulated by a rectangle wave. The tip trajectories and their variations with the modulation period T are explained by th...We numerically study the dynamics of spiral waves in the excitable system with the excitability modulated by a rectangle wave. The tip trajectories and their variations with the modulation period T are explained by the corre- sponding spectrum analysis. For a large T, the external modulation leads to the occurrence of more frequency peaks and these frequencies change with the modulation period according to their specific rules, respectively. Some of the frequencies and a primary frequency fl determine the corresponding curvature periods, which are locked into rational multiplies of the modulation period. These frequency-locking behaviours and the limited life-span of the frequencies in their variations with the modulation period constitute many resonant entrainment bands in the T axis. In the main bands, which follow the relation T/T12 = m/n, the size variable Rx of the tip trajectory is a monotonic increasing function of T. The rest of the frequencies are linear combinations of the two ones. Due to the complex dynamics, many unique tip trajectories appear at some certain T. We find also that spiral waves are eliminated when T is chosen from the end of the main resonant bands. This offers a useful method of controling the spiral wave.展开更多
文摘The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circuit theory,biology and the area of population genetics.For this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a tool.The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule.By the help of this feature which is not provided by other popular fractional derivatives,nonlinear fractional partial differential equation is turned into an integer order differential equation.The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method.This comparison is made both with the help of three-dimensional graphical representations and tables for different values of theγ.
文摘We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expafunction and extended sinh-Gordon equation expansion(ESh GEE) schemes are utilized. The solutions obtained include dark, bright, dark-bright, periodic and other kinds of solitons. These analytical wave solutions are gained and verified with the use of Mathematica software. These solutions do not exist in literature. Some of the solutions are demonstrated by 2D, 3D and contour graphs. This model is mostly used in circuit theory, transmission of nerve impulses, and population genetics. Finally, both the schemes are more applicable, reliable and significant to deal with the fractional nonlinear partial differential equations.
基金supported by National Natural Science Foundation of China under Grant Nos.10572056 and 30670529Natural Science Foundation of Lanzhou University under Grant No.Q200706
文摘In this paper, a spatial perturbation scheme is proposed to suppress the spiral wave in the modified Orengonator model, which is used to describe the chemical reaction in the light-sensitive media. The controllable external illumination Ф is perturbed with a spatial linear function. In our numerical simulation, the scheme is investigated by imposing the external controllable illumination on the space continuously and/or intermittently. The numerical simulation results confirm that the stable rotating spiral wave still can be removed with the scheme proposed in this paper even if the controllable Ф changed vs. time and space synchronously. Then the scheme is also used to control the spiral wave and turbulence in the modified Fitzhugh-Nagumo model. It is found that the scheme is effective to remove the sable rotating and meandering spiral wave but it costs long transient period and intensity of the gradient parameter to eliminate the soiral turbulence.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10645006 and 10747123)the Natural Science Foundation of Hebei Province (Grant Nos C2005000011and C2007000026)the Key Subject Construction Project of Hebei Provincial University
文摘It is well known that the strong coupling can synchronize a network of nonlinear oscillators. Synchronization provides the basis of the remarkable computational performance of the brain. In this paper the FitzHugh-Nagumo neuron network is constructed. The dependence of the synchronization on the coupling strength, the noise intensity and the size of the neuron network has been discussed. The results indicate that the coupling among neurons works to improve the synchronization, and noise increases the neuron random dynamics and the local fluctuations; the larger the size of network, the worse the synchronization. The dependence of the synchronization on the strength of the electric synapse coupling and chemical synapse coupling has also been discussed, which proves that electric synapse coupling can enhance the synchronization of the neuron network largely.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10375016 and 10474018） and the Natural Science Foundation of Hebei Province （Grant Nos C2005000011and A2004000005） and the Key Subject Construction Project of Hebei Provincial University.
文摘It has been proved recently that the spike timing can play an important role in information transmission, so in this paper we develop a network with N-unlt FitzHugh-Nagumo neurons coupled by gap junctions and discuss the dependence of the spike timing precision on synaptic coupling strength, the noise intensity and the size of the neuron ensemble. The calculated results show that the spike timing precision decreases as the noise intensity increases; and the ensemble spike timing precision increases with coupling strength increasing. The electric synapse coupling has a more important effect on the spike timing precision than the chemical synapse coupling.
基金supported by the the National Natural Science Foundation of China(Grant Nos.11372247&11472224)the NPU Foundation for Undergraduate Graduation Design
文摘This paper aims to investigate the stochastic resonance （SR） in an FitzHugh-Nagumo （FHN） model with an additive LEvy noise numerically. The non-Gaussian LEvy noise is a kind of general random noise which is different from the usual Gaussian noise, and it has small fluctuations with the unpredictable jumps to describe the random fluctuations in an FHN model. SR is determined by the signal-to-noise ratio （SNR）, and the numerical simulation results show the occurrence of the SR phenomena in the given FHN system. The influence of various parameters of the LEvy noise and the FHN model on the SR will be exam- ined, and some mechanisms of the LEvy noise-induced SR are presented which are different from those of the Gaussian noise.
基金supported by the National Natural Science Foundation of China (Grant No. 11005030)the Science Foundation of Hebei Education Department,China (Grant No. 2009135)+1 种基金the Science Foundation of Inner Mongolia Education Department,China(Grant No. NJ09178)the Science Foundation of Hebei Normal University,China
文摘We numerically study the dynamics of spiral waves in the excitable system with the excitability modulated by a rectangle wave. The tip trajectories and their variations with the modulation period T are explained by the corre- sponding spectrum analysis. For a large T, the external modulation leads to the occurrence of more frequency peaks and these frequencies change with the modulation period according to their specific rules, respectively. Some of the frequencies and a primary frequency fl determine the corresponding curvature periods, which are locked into rational multiplies of the modulation period. These frequency-locking behaviours and the limited life-span of the frequencies in their variations with the modulation period constitute many resonant entrainment bands in the T axis. In the main bands, which follow the relation T/T12 = m/n, the size variable Rx of the tip trajectory is a monotonic increasing function of T. The rest of the frequencies are linear combinations of the two ones. Due to the complex dynamics, many unique tip trajectories appear at some certain T. We find also that spiral waves are eliminated when T is chosen from the end of the main resonant bands. This offers a useful method of controling the spiral wave.
