Nagumo System Research Articles

КОМПАКТНЫЕ РАЗНОСТНЫЕ СХЕМЫ ДЛЯ СЛАБО НЕЛИНЕЙНЫХ ЗАДАЧ И ГРАНИЧНЫЕ УСЛОВИЯ, ИМИТИРУЮЩИЕ ЗАДАЧУ КОШИ

Compact finite-difference schemes are well known and provide high accuracy order for differential equation with constant coefficients. Algorithms for constructing compact schemes of the 4-th order for boundary value problems with variable (smooth or jump) coefficient are developed. For the diffusion equations with a smooth variable coefficient and the Levin – Leontovich equation, compact finite-difference schemes are also constructed and their 4-th order is experimentally confirmed. The method of constructing compact schemes of the 4-th order can be generalized to partial differential equations and systems with weak nonlinearity, for example, for the Fisher – Kolmogorov – Petrovsky – Piskunov equation, for the nonlinear Schrödinger equation or for the Fitzhugh – Nagumo system. For such nonlinear problems, a combination of simple explicit schemes and relaxation is used. Richardson’s extrapolation increases the order of the circuits to the 6-th. To approximate multidimensional problems with discontinuous coefficients, for example, the two-dimensional stationary diffusion equation in inhomogeneous media, it is necessary to estimate the possible asymptotics of solutions in the vicinity of the boundary line’s breaks. To do this, we use generalized eigen-functions in the angle, which can be used as a set of test functions and build compact difference schemes approximating the problem on triangular grids with high order of accuracy. The asymptotics along the radius of these generalized eigen-functions (in polar coordinates in the vicinity of the vertex of the angle) have irrational indices which can be found from a special dispersion equation and which determine the indices of the corresponding Bessel functions along the radius. For a number of difference schemes approximating the most important evolutionary equations of mathematical physics, it is possible to construct special boundary conditions imitating the Cauchy problem (ICP) on the whole space. These conditions depend not only on the original equation, but also on the type of the difference scheme, and even on the coefficients of the corresponding differential equation. The ICP conditions are determined with accuracy to a gauge. But the choice of this gauge turns out to be essential with numerical implementation. The role of rational approximations of the Pade – Hermite type of the symbol of the corresponding pseudo-differential operator is important. Examples of movie solutions of problems with ICP conditions for various finite-difference schemes approximating the basic mathematical physics equations, see https://cs.hse.ru/mmsg/transbounds. The study was realized within the framework of the Academic Fund Program at the National Research University – Higher School of Economics (HSE) in 2016–2017 (grant No. 16-05-0069) and by the Russian Academic Excellence Project «5–100».

Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators

We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol–Duffing oscillators and the neuron-like FitzHugh–Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.

Minimal lamellar structures in a periodic FitzHugh–Nagumo system

A singular limit of a FitzHugh–Nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions. We study the stationary lamellar set and give a criterion to select out the one with the lowest energy. Such an optimal structure is called a minimal lamella. While the empty set or the full torus is a global minimizer for appropriate parameter regimes, the minimal lamellae beat both in other circumstances. The concept of minimal lamella points out that a preferred 1D mesh size is universal.

Rigorous Derivation of the Nonlocal Reaction-Diffusion Fitzhugh--Nagumo System

We introduce a spatially extended transport kinetic FitzHugh--Nagumo model with forced local interactions and prove that its hydrodynamic limit converges toward 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 us to make the rigorous link between kinetic and reaction-diffusion models.

Traveling Waves for Spatially Discrete Systems of FitzHugh--Nagumo Type with Periodic Coefficients

We establish the existence and nonlinear stability of traveling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh--Nagumo system with alternati...

Fronts in subdiffusive FitzHugh–Nagumo systems

Front solutions of subdiffusive FitzHugh–Nagumo equations are studied for a piece-wise linear nonlinearity. Multiple solutions of the problem and the dependence of the propagation velocity on the parameters are discussed.

On the microscopic bidomain problem with FitzHugh–Nagumo ionic transport

The microscopic bidomain problem with FitzHugh–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 $$d\le 3$$ , 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 those of the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained by combining recent results on Dirichlet-to-Neumann operators (Pruss in J Integral Equ Appl, 2018; Pruss and Simonett in Moving interfaces and quasilinear parabolic evolution equations, monographs in mathematics, vol 105. Birkhauser, Basel, 2016), on critical spaces for parabolic evolution equations (Pruss et al. in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.

An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes.

Periodicity induced by noise and interaction in the kinetic mean-field FitzHugh–Nagumo model

We consider the long-time behavior of a population of mean-field oscillators modeling the activity of interacting excitable neurons in a large population. Each neuron is represented by its voltage and recovery variables, which are the 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 interaction and noise are only imposed on the voltage variable. We prove the existence of a stable cycle for the infinite population system, in a regime where the local dynamics is small.

Spiral wave chimera in two-dimensional nonlocally coupled Fitzhugh–Nagumo systems

Spiral wave chimeras with spatially extended unsynchronized cores have been reported in nonlocally coupled phase oscillators and chaotic oscillators. Here, we investigate the two-dimensional nonlocally coupled FitzHugh–Nagumo oscillators with open boundary conditions. Rich dynamics of spiral wave chimeras is discovered numerically. Besides outwardly propagating and inwardly propagating single-spiral wave chimeras, we find the double-spiral wave chimera with two spiral chimeras rotating with each other and the breakup of spiral wave chimeras. We also find a transition between a normal spiral wave with a phase singularity at its center and spiral wave chimera.

An approach to periodic, time-varying parameter estimation using nonlinear filtering

Many systems arising in biological applications are subject to periodic forcing. In these systems the forcing parameter is not only time-varying but also known to have a periodic structure. We present an approach to estimating periodic, time-varying parameters that imposes periodic structure by treating the time-varying parameter as a piecewise function with unknown coefficients. This method allows the resulting parameter estimate more flexibility in shape than prescribing a specific functional form (e.g. sinusoidal) to model its behavior, while still maintaining periodicity. We employ nonlinear filtering, more specifically, a version of the augmented ensemble Kalman filter (EnKF), to estimate the unknown coefficients comprising the piecewise approximation of the periodic, time-varying parameter. This allows for straightforward comparison of the proposed method with an EnKF-based parameter tracking algorithm, where periodicity is not guaranteed.We demonstrate the effectiveness of the proposed approach on two bio- logical examples: a synthetic example with data generated from the nonlinear FitzHugh–Nagumo system, modeling the excitability of a nerve cell, to estimate the external voltage parameter, and a case study using reported measles incidence data from three locations during the pre-vaccine era to estimate the seasonal transmission parameter. The formulation of the proposed approach also allows for simultaneous estimation of initial conditions and other static system parameters, such as the reporting probability of measles cases, which is vital for predicting under-reported incidence data.

Bubbles and droplets in a singular limit of the FitzHugh–Nagumo system

The FitzHugh–Nagumo system gives rise to a geometric variational problem, when its parameters take values in a particular range. A stationary set of the variational problem satisfies an Euler–Lagrange equation that involves the curvature of the boundary of the set and a nonlocal term that inhibits unlimited growth and spreading. The nonlocal term is the solution of a nonhomogeneous modified Helmholtz equation with the characteristic function of the stationary set as the inhomogeneous term. Two types of stationary sets are studied: disc shaped stationary sets in the plane, termed bubbles, and unions of perturbed small discs, termed droplet assemblies, in bounded domains. A complete description of the existence and the stability of bubbles is established. Depending on the parameters, there may be zero, one, two, or even three bubbles. Droplet assemblies are constructed by a reduction argument. Each droplet in a stationary assembly is close in size and shape to the corresponding bubble in the plane. The locations of the droplets in the assembly are determined by the Green’s function of the modified Helmholtz equation.

On the bidomain problem with FitzHugh–Nagumo transport

The bidomain problem with FitzHugh–Nagumo transport is studied in the $$L_p\!-\!L_q$$ -framework. Reformulating the problem as a semilinear evolution equation, 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 $$d\le 4$$ , 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 are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Pruss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Pruss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.

Synchronization control for reaction–diffusion FitzHugh–Nagumo systems with spatial sampled-data

To gain a better understanding the synchronization mechanism of networked neurons, this paper studies the synchronization control for a class of reaction–diffusion FitzHugh–Nagumo systems, associated with a digraph containing at least one directed spanning tree. A novel control method adopting spatial sampling strategies is proposed, in which the control inputs are constructed directly on the spatial means of system state variables on sampling subsets. After discussing the existence and uniqueness of classical solutions, we show analytically that the synchronization of the controlled systems is equivalent to that of the corresponding FitzHugh–Nagumo systems under the corresponding control inputs. Based on the study on general algebraic connectivity, a sufficient condition for the system synchronization is given, together with case simulations to illustrate the effectiveness and potential of the new control method.

Continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system on [formula omitted

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in L2(RN)×L2(RN) by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most p-times integrability where p is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in (Lp(RN)∩H1(RN))×L2(RN) with respect to the initial values belonging to L2(RN)×L2(RN). As an application of the result on the continuity, the existence of a pullback attractor in (Lp(RN)∩H1(RN))×L2(RN) is proved for arbitrary N≥1 and p>2.

Meromorphic solutions in the FitzHugh–Nagumo model

The FitzHugh–Nagumo model is studied in the framework of analytic theory of differential equations. The Nevanlinna theory is used to find all meromorphic solutions of a second-order ordinary differential equation related to the FitzHugh–Nagumo model. As a consequence new exact solutions of the FitzHugh–Nagumo system are obtained in explicit form.

Pulses in FitzHugh--Nagumo Systems with Rapidly Oscillating Coefficients

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, ther...

On the Existence and Stability of Fast Traveling Waves in a Doubly Diffusive FitzHugh--Nagumo System

The FitzHugh--Nagumo equation, which was derived as a simplification of the Hodgkin--Huxley model for nerve impulse propagation, has been extensively studied as a paradigmatic activator-inhibitor system. We consider the version of this system in which two agents diffuse at an equal rate. Using geometric singular perturbation theory, we prove the existence and stability of fast traveling pulses. The stability proof makes use of the Maslov index---an invariant of symplectic geometry---to count unstable eigenvalues for the linearization about the wave. The calculation of the Maslov index is carried out by tracking the evolution of the unstable manifold of the rest state using the timescale separation. This entails a careful consideration of how the transition from fast to slow dynamics occurs in the tangent bundle over the wave. Finally, we observe in the calculation that the Maslov index lacks monotonicity in the spatial parameter, which distinguishes this application of the Maslov index from similar analyses of Hamiltonian systems.

Higher Dimensional Bubble Profiles in a Sharp Interface Limit of the FitzHugh--Nagumo System

The FitzHugh--Nagumo system gives rise to a nonlocal geometric variational problem defined on subsets of a domain. The energy of a subset contains three terms: its perimeter, its volume, and a long...

ЗАДАЧА СТАРТОВОГО УПРАВЛЕНИЯ И ФИНАЛЬНОГО НАБЛЮДЕНИЯ ДЛЯ СИСТЕМЫ УРАВНЕНИЙ ФИТЦ ХЬЮ–НАГУМО С УСЛОВИЕМ ДИРИХЛЕ–ШОУОЛТЕРА–СИДОРОВА

The paper discusses the start control and the final observation of solutions of the Dirichlet–Showalter–Sidorov problem for the degenerate Fitz Hugh–Nagumo system of equations. This system refers to the class of reaction-diffusion equations and describes the propagation of waves in active biological media, such as the heart muscle, or brain tissue. On the one hand, the Fitz Hugh–Nagumo system of equations is the development of more familiar model of Kolmogorov–Petrovsky–Piskunov, and on the other hand, the simplification of the Hodgins–Huxley model. When constructing a mathematical model taking into account that the speed of one desired function of the Fitz Hugh–Nagumo system of equations significantly exceeds the speed of the other, it has been suggested to investigate the degenerate case. The studied task of the start control and final observation simulates the situation when, after a short-time control action, the expected result for a certain period of time, i.e. at the initial moment of time sends a high-power pulse is sent to a nerve system and the required state of the system is expected after a certain set time. On the basis of Galerkin's methods and compactness, the existence theorem of the problem of starting control and final observation in a weak generalized case is proved.