We consider a class of three-dimensional systems having an equilibrium point at the origin, whose principal part is of the form (-Dy h(x, y), Dx h(x,y), f(x,y))^T . This principal part, which has zero divergence and does not depend on the third variable z, is the coupling of a planar Hamiltonian vector field Xh(x,y)=(-Dy h(x, y), Dx h(x,y))^T with a one-dimensional system. We analyze the quasi-homogeneous orbital normal forms for this kind of systems, by introducing a new splitting for quasi-homogeneous three-dimensional vector fields. The obtained results are applied to the nondegenerate Hopf-zero singularity that falls into this kind of systems. Beyond the Hopf-zero normal form, a parametric normal form is obtained, and the analytic expressions for the normal form coefficients are provided. Finally, the results are applied to a case of the three-dimensional Fitzhugh-Nagumo system.
For a coupled slow--fast FitzHugh--Nagumo(FHN) equation derived from a reaction-diffusion-mechanics (RDM) model, Holzer, Doelman and Kaper in 2013 studied existence and stability of the travelling pulse, which consists of two fast orbit arcs and two slow ones, where one fast segment passes the unique fold point with algebraic decreasing and two slow ones follow normally hyperbolic critical curve segments. Shen and Zhang in 2019 obtained existence of the travelling pulse, whose two fast orbit arcs both exponentially decrease, and one of the slow orbit arcs could be normally hyperbolic or not at…
FitzHugh-Nagumo equation: bifurcations, slow-fast system and dynamics near infinity
Lisbon School of Economics and Management, Lisbon University（里斯本经济与管理学院，里斯本大学）；School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran（德黑兰大学理学院数学、统计与计算机科学系）；School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）
In this paper, we use geometric singular perturbation theory and blowup, as our main technical tool, to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. In particular, we demonstrate that the MMOs in this model are not due to generic folded singularities, but rather due to singularities at a cusp -- not a fold -- of the critical manifold. Using blowup, we determine the number of SAOs analytically, showing -- as for the folded nodes -- that they are determined by the Weber equation and the ratio of eigenvalues. We al…
From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model
研究了多项式动力系统满足常微分方程的Pu i n x级数与不可约不变代数曲线的次数之间的关系。得到了一类多项式动力系统不可约不变代数曲线的次数的一个界。证明了无穷远点附近的Pu i n x级数可以用来显式地求出不可约代数曲线。作为一个例子，我们得到了著名的FitzHugh-Nagumo系统的所有不可约不变代数曲线。
We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of traveling $1$-pulse solutions with undetermined width and propagating speed. Next, we compute the associated action functional for this profile from which we derive the conditions for existence and a saddle-node bifurcation as the zeros of the action functional and its derivatives. We obtain the same conditions by using a different analytical approach that exploits the singular limit of the …
We investigate the impact of spatial-temporal discretisation schemes on the dynamics of a class of reaction-diffusion equations that includes the FitzHugh-Nagumo system. For the temporal discretisation we consider the family of six backward differential formula (BDF) methods, which includes the well-known backward-Euler scheme. The spatial discretisations can feature infinite-range interactions, allowing us to consider neural field models. We construct travelling wave solutions to these fully discrete systems in the small time-step limit by viewing them as singular perturbations of the corresp…
This article communicates results on regular depolarization cascades in periodically-kicked feedforward chains of excitable two-dimensional FitzHugh-Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and…
Since Noble adapted in 1962 the model of Hodgkin and Huxley to fit Purkinje fibres the refinement of models for cardiomyocytes has continued. Most of these models are high-dimensional systems of coupled equations so that the possible mathematical analysis is quite limited, even numerically. This has inspired the development of reduced, phenomenological models that preserve qualitatively the main featuture of cardiomyocyte's dynamics. In this paper we present a systematic comparison of the dynamics between two notable low-dimensional models, the FitzHugh-Nagumo model \cite{FitzHugh55, FitzHugh6…
Machine learning algorithms have been successfully used to approximate nonlinear maps under weak assumptions on the structure and properties of the maps. We present deep neural networks using dense and convolutional layers to solve an inverse problem, where we seek to estimate parameters of a FitzHugh-Nagumo model, which consists of a nonlinear system of ordinary differential equations (ODEs). We employ the neural networks to approximate reconstruction maps for model parameter estimation from observational data, where the data comes from the solution of the ODE and takes the form of a time ser…
In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequenci…
In this work we propose a new model of 2D cellular neural networks (CNN) in terms of the lattice FitzHugh-Nagumo equations with boundary feedback and prove a threshold condition for the exponential synchronization of the entire neural network through the \emph{a priori} uniform estimates of solutions and the analysis of dissipative dynamics. The threshold to be satisfied by the gap signals between pairwise boundary cells of the network is expressed by the structural parameters and adjustable. The new result and method of this paper can also be generalized to 3D and higher dimensional FitzHugh-…
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…
We present a computer-assisted approach to prove the existence of Hopf bubbles and degenerate Hopf bifurcations in ordinary and delay differential equations. We apply the method to rigorously investigate these nonlocal bifurcation structures in the FitzHugh- Nagumo equation, the extended Lorenz-84 model and a time-delay SI model.
