Bixiang Wang
The existence of a random attractor for the stochastic FitzHugh-Nagumo system defined on an unbounded domain is established. The pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.
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This paper is devoted to pulse solutions in FitzHugh--Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh--Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.
The existence of a pullback attractor is established for the singularly perturbed FitzHugh-Nagumo system defined on the entire space $R^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.
In this paper, we prove the existences of pullback attractors in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ for stochastic Fitzhugh-Nagumo system driven by both additive noises and deterministic non-autonomous forcings. The nonlinearity is polynomial like growth with exponent $p-1$. The asymptotic compactness for the cocycle in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ is proved by using asymptotic a priori method, where the plus and minus signs of the nonlinearity at large value are not required.
We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between kinetic and reaction-diffusion models.
Nonlinear Stability of Large-Period Traveling Waves Bifurcating from the Heteroclinic Loop in the FitzHugh-Nagumo Equation
本文研究FitzHugh-Nagumo方程中由异宿环分岔产生的大周期行波，通过谱分析和相位调制方法证明了其非线性稳定性。
In this work a new mathematical model for complex neural networks is presented by the partly diffusive FitzHugh-Nagumo equations with ensemble boundary coupling. We analyze the dissipative dynamics and boundary coupling dynamics of the solution semiflow with sharp estimates. The exponential synchronization of this kind complex neural networks is proved under the condition that synaptic stimulation signal strength reaches a threshold quantitatively expressed.
We consider a spatially extended mesoscopic FitzHugh-Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of Dirac distribution concentrated to an averaged membrane potential. Our approach is based on techniques widely developed in kinetic theory (Wasserstein distance, relative entropy method), where macroscopic quantities…
In this paper we focus on a spatially extended FitzHugh-Nagumo model with interactions. In the regime where strong and local interactions dominate, we quantify how the probability density of neurons concentrates into a Dirac distribution. Previous work investigating this question have provided relative bounds in integrability spaces. Using a Hopf-Cole framework, we derive precise $L^\infty$ estimates using subtle explicit sub- and super- solutions which prove, with rates of convergence, that the blow up profile is Gaussian.
A Differential Harnack Inequality for the FitzHugh-Nagumo Equation
本文针对ℝⁿ上FitzHugh-Nagumo方程的正解建立了Li-Yau-Hamilton型微分Harnack估计，并借此证明了该方程正解的经典Harnack不等式及行波解速度下界等性质。
We introduce a geometrical extension of the FitzHugh-Nagumo equations describing propagation of electrical impulses in nerve axons. In this extension, the axon is modeled as a warped cylinder, rather than a straight line, as is usually done, while pulses propagate on its surface, as is the case with real axons. We prove the stability of electrical impulses for a standard cylinder and existence and stability of pulse-like solutions for warped cylinders whose radii are small and vary slowly along their lengths.
Stability of Traveling Fronts of the FitzHugh-Nagumo Equations on Cylindrical Surfaces
本文研究FitzHugh-Nagumo方程在柱面上的行波前稳定性，证明标准柱面上非线性稳定，并扩展至缓慢变化半径的扭曲柱面，数值模拟验证理论结果。
Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.
The FitzHugh-Nagumo system on undulated cylinders: spontaneous symmetrization and effective system
本文研究波纹圆柱上的FitzHugh-Nagumo系统，证明小半径下解趋于径向平均，且可由含半径函数的有效系统近似，解释实验吻合现象。
Instability of the Standing Pulse in Skew-Gradient Systems and Its Application to FitzHugh-Nagumo Type Systems
本文研究了非自共轭斜梯度系统中稳定脉冲的不稳定性，并将其应用于FitzHugh-Nagumo型系统，通过拓扑方法分析非线性抑制对脉冲稳定性的影响。
Consider the bidomain equations subject to ionic transport described by the models of FitzHugh-Nagumo, Aliev-Panfilov, or Rogers-McCulloch. It is proved that this set of equations admits a unique, strong T-periodic solution provided it is innervated by T-periodic intra- and extracellular currents. The approach relies on a new periodic version of the classical Da Prato-Grisvard theorem on maximal L^p-regularity in real interpolation spaces.
We consider a spatially extended mean-field model of a FitzHugh-Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its asymptotic limit in the regime of strong local interactions converges toward a system of reaction-diffusion equations taking account for theaverage quantities of the network. Our approach is based on a modulated energy argument, to compare the macroscopic quantities computed from the solution of the transport equation, and the solution of the limit system. The main difficulty, compared to the literature, lies in the need of regularity …
Large coupling in a FitzHugh-Nagumo neural network: quantitative and strong convergence results
针对空间扩展FitzHugh-Nagumo神经网络介观模型，证明短程作用主导区电位概率密度集中为质心满足经典非局部反应扩散系统的狄拉克分布，通过相对熵技术和正则性传播推导两个定量强收敛估计，证实集中爆破剖面为高斯型。
Patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems. A common theme in investigating such systems is to identify the intrinsic factors responsible for such self-organization. The $Γ$-convergence is a well-known technique applicable to variational formulation in studying the concentration phenomena of stable patterns. A geometric variational functional associated with the $Γ$-limit of standing waves of FitzHugh-Nagumo system has recently been built. This article studies the $Γ$-limit of traveling waves. To the best of our knowledge, t…
The initial value problem P0, in all of the space, for the spatio - temporal FitzHugh - Nagumo equations is analyzed. When the reaction kinetics of the model can be outlined by means of piecewise linear approximations, then the solution of P0 is explicitly obtained. For periodic initial data are possible damped travelling waves and their speed of propagation is evaluated. The results imply applications also to the non linear case.
The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension $d\leq 3$, by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria parallel t…
A theory on bi-spatial random attractors developed recently by Li \emph{et al.} is extended to study stochastic Fitzhugh-Nagumo system driven by a non-autonomous term as well as a general multiplicative noise. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, it is showed that every generated random cocycle has a pullback attractor in $L^l(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ with $l\in(2,p]$, and the family of obtained attractors is upper semi-continuous at any intensity of noise. Moreover, if some additional conditions are added, then the s…
We study invasion fronts in the FitzHugh--Nagumo equation in the oscillatory regime using singular perturbation techniques. Phenomenologically, localized perturbations of the unstable steady-state grow and spread, creating temporal oscillations whose phase is modulated spatially. The phase modulation appears to be selected by an invasion front that describes the behavior in the leading edge of the spreading process. We construct these invasion fronts for large regions of parameter space using singular perturbation techniques. Key ingredients are the construction of periodic orbits, their unsta…
Synchronization of Memristive FitzHugh-Nagumo Neural Networks
本文研究带忆阻器和线性突触耦合的扩散FitzHugh-Nagumo神经网络，证明其全局耗散性，并给出耦合强度阈值条件下网络指数同步的显式判据。
We investigate existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh-Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we show exponential nonlinear stability in the small connectivity regime.
Global Strong Well-Posedness of the stochastic bidomain equations with FitzHugh-Nagumo transport
研究受电流噪声影响的FitzHugh-Nagumo输运双域方程，证明其在临界空间中存在唯一全局强路径解，方法结合随机与确定性极大正则性及外推空间技术。
Propagation failure in heterogeneous FitzHugh--Nagumo systems via coupled upper and lower solutions
本文通过构造耦合上下解，为异质FitzHugh--Nagumo系统提供传播失败的充分条件，并建立方向性信号阻断，适用于神经元模型。
Analysis and stability of the FitzHugh-Nagumo system with rapidly oscillating Neumann boundary condition
本文利用平均化方法和改进的收缩矩形法，分析了半直线上受高频正弦Neumann边界条件驱动的FitzHugh-Nagumo系统，证明了高频下系统可近似分解，并建立了时变边界条件下的稳定性结果。
Chaotic motion and singularity structures of front solutions in multi-component FitzHugh-Nagumo-type systems
本文研究多分量反应扩散系统中前沿解的动力学，利用几何奇异摄动、Evans函数和中心流形约化证明可通过耦合函数与慢部分维数N控制前沿运动复杂性，并揭示N≥3时前沿速度的混沌行为。
We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special c…
In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall be mainly concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure $ν$ associated with the transition semigroup $P_t$; further, we identify its infinitesimal generator in the space $L^2(H;ν)$.
Diffusive synchronization of phase waves in the FitzHugh-Nagumo system
研究通过新颖方法揭示FitzHugh-Nagumo系统中相波的扩散同步机制及潜在不稳定性。
Continuous Data Assimilation for Memristive FitzHugh--Nagumo Neural Networks
本文研究忆阻 FitzHugh--Nagumo 神经网络的连续数据同化，提出仅向膜电位方程添加反馈的 nudging 算法，证明指数收敛和能量类唯一性，并给出噪声误差界。
Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth
本文研究二维区域上具有临界指数增长的FitzHugh-Nagumo系统，通过约化方法转化为非局部标量问题，利用山路论证和Nehari水平比较证明基态解存在，并借助谱分解的鞍点连接论证在无单调性条件下获得非平凡解。
We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary …
Multiple front standing waves in the FitzHugh-Nagumo equations
本文以两个稳定驻波前沿为构件，通过非局部Lyapunov-Schmidt约化和临界点变分刻画，构造并证明鞍—焦平衡点情形下FitzHugh-Nagumo方程的多重前沿稳定及不稳定驻波。
We consider the long-time behavior of a population of mean-field oscillators modeling the activity of interacting excitable neurons in large population. Each neuron is represented by its voltage and recovery variables, which are solution to a FitzHugh-Nagumo system, and interacts with the rest of the population through a mean-field linear coupling, in the presence of noise. The aim of the paper is to study the emergence of collective oscillatory behaviors induced by noise and interaction on such a system. The main difficulty of the present analysis is that we consider the kinetic case, where i…
Approximation and stability results for the parabolic FitzHugh-Nagumo system with combined rapidly oscillating sources
本文提出了一种新的平均方法，用于研究带有结合快速振荡源的抛物型FitzHugh-Nagumo系统的近似和稳定性问题。
Nonlinear stability of periodic wave trains in the FitzHugh-Nagumo system against fully nonlocalized perturbations
本文针对FitzHugh-Nagumo系统中周期波列对完全非局域扰动的稳定性问题，提出一种适用于一般耗散半线性问题的新方案，通过逆Laplace表示揭示高频阻尼，并将非线性阻尼估计扩展到非局域扰动，从而克服了缺乏抛物性和局域化带来的困难。
Selection of pushed pattern-forming fronts in the FitzHugh-Nagumo system
本文研究了推动前沿的非线性稳定性，揭示了其在半轴初始数据下的吸引性，确定了传播速度和所选波数，验证了模式形成前沿的边缘稳定性猜想，适用于一般耗散PDE模型。
We investigate the dynamics of a limit of interacting FitzHugh-Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a nonlinear evolution partial differential equation representing the probability distribution of one given neuron in a large network. The case of weak connectivity previously studied displays a unique, exponentially stable, stationary solution. Here, we consider the case of strong connectivities, and exhibit the presence of possibly non-unique stationary behaviors or non-stationary behaviors. To this end…
Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients
针对空间非均匀FitzHugh-Nagumo型系统，将现有常系数理论扩展到变号变系数情形，证明了山路解的存在性，并在系数不变号时建立了分量正负解的存在性，得到稳态解的存在性和多重性结果。
We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüg…
Stability of coherent pattern formation through invasion in the FitzHugh-Nagumo system
本文研究了FitzHugh-Nagumo系统中通过入侵形成周期性图案的前沿稳定性，提出了一种非线性稳定性分析方法，解决了边缘扩散与尾部模式相互作用的问题，为生长过程中的波数选择定律提供了理论基础。
This paper studies the solutions of a reaction--diffusion system with nonlinearities that generalise the Lengyel--Epstein and FitzHugh--Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the system's solutions and confirmed through numerical Examples.
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