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.     

Control of space-time chaos in a system of equations of the FitzHugh–Nagumo type

We perform an analytic and numerical study of a system of partial differential equations that describes the propagation of nerve impulses in the heart muscle. We show that, for fixed parameter values, the system has infinitely many distinct stable wave solutions running along the spatial axis at arbitrary velocities and infinitely many distinct modes of space-time chaos, where the bifurcation parameter is the velocity of running wave propagation along the spatial axis, which does not explicitly occur in the original system of equations. We suggest an algorithm for controlling the space-time chaos in the system, which permits one to stabilize any of its unstable periodic running waves.     

Two-frequency self-oscillations in a FitzHugh–Nagumo neural network

A new mathematical model of a one-dimensional array of FitzHugh–Nagumo neurons with resistive-inductive coupling between neighboring elements is proposed. The model relies on a chain of diffusively coupled three-dimensional systems of ordinary differential equations. It is shown that any finite number of coexisting stable invariant two-dimensional tori can be obtained in this chain by suitably increasing the number of its elements.     

Pattern selection in the 2D FitzHugh–Nagumo model

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

     

Controlling bifurcations and chaos in TCP–UDP–RED

We study the bifurcation and chaotic behavior of the Transmission Control Protocol (TCP) and User Datagram Protocol (UDP) network with Random Early Detection (RED) queue management. These bifurcation and chaotic behaviors may cause heavy oscillation of an average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the internet congestion control system. The theoretical analysis and the simulation experiments show that this method can obtain the stable average queue length without sacrificing the other advantages of RED.     

Periodic solutions in an array of coupled FitzHugh–Nagumo cells

We analyse the dynamics of an array of N²$N^2$ identical cells coupled in the shape of a torus. Each cell is a 2-dimensional ordinary differential equation of FitzHugh–Nagumo type and the total system is _ZN×_ZN${\mathbb{Z}}_N\times {\mathbb{Z}}_N$-symmetric. The possible patterns of oscillation, compatible with the symmetry, are described. The types of patterns that effectively arise through Hopf bifurcation are shown to depend on the signs of the coupling constants, under conditions ensuring that the equations have only one equilibrium state.     

Weak Solutions for Stochastic FitzHugh–Nagumo Equations

This article studies the existence of weak solutions for a stochastic version of the FitzHugh–Nagumo equations. The random elements are introduced through initial values and forcing terms of associated Cauchy problem, which may be white noise in the time. Moreover there is a dependence of a stochastic parameter.     

Standing Waves Joining with Turing Patterns in FitzHugh–Nagumo Type Systems

An article by Kondo and Asai demonstrated that the pattern formation and change on the skin of tropical fishes can be predicted well by reaction-diffusion models of Turing type. As being observed, a common pattern structure is the rearrangement of stripe pattern, and defect like heteroclinic solution appeared between the patterns with different number of stripes. We consider FitzHugh–Nagumo type reaction-diffusion systems with anisotropic diffusion. Under a sufficient condition in diffusivity, we apply variational arguments to show the existence of standing waves joining with Turing patterns.     

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 Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.     

Robust synchronization of FitzHugh–Nagumo network with parameter disturbances by sliding mode control

In this paper, based on the sliding mode control method, the robust synchronization for a coupled FitzHugh–Nagumo (FHN) neurobiological network with parameter disturbances is investigated. Some theoretical criteria are derived to realize the robust synchronization of the FHN network with disturbed parameters, and the synchronization occurs without dependence on the type and magnitude of the noise, which greatly extend some existing results for two or three coupled FHN neurons. Finally, a numerical example is given to illustrate the effectiveness of the proposed theoretical results.     

Stability of Traveling Pulses with Oscillatory Tails in the FitzHugh–Nagumo System

The FitzHugh–Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter \(\varepsilon \) goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh–Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with \(\varepsilon \), while the relevant scaling in the oscillatory case is \(\varepsilon ^{2/3}\).     

Experimental and numerical study of noise effects in a FitzHugh–Nagumo system driven by a biharmonic signal

Using a nonlinear circuit ruled by the FitzHugh–Nagumo equations, we experimentally investigate the combined effect of noise and a biharmonic driving of respective high and low frequency F and f. Without noise, we show that the response of the circuit to the low frequency can be maximized for a critical amplitude B¹ of the high frequency via the effect of Vibrational Resonance (V.R.). We report that under certain conditions on the biharmonic stimulus, white noise can induce V.R. The effects of colored noise on V.R. are also discussed by considering an Ornstein–Uhlenbeck process. All experimental results are confirmed by numerical analysis of the system response.     

Singleton sets random attractor for stochastic FitzHugh–Nagumo lattice equations driven by fractional Brownian motions

The paper is devoted to the study of the dynamical behavior of the solutions of stochastic FitzHugh–Nagumo lattice equations, driven by fractional Brownian motions, with Hurst parameter greater than 1/2. Under some usual dissipativity conditions, the system considered here features different dynamics from the same one perturbed by Brownian motion. In our case, the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.     

Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations

This paper studies standing pulse solutions to the FitzHugh–Nagumo equations. Since the reaction terms are coupled in a skew-gradient structure, a standing pulse solution is a homoclinic orbit of a second order Hamiltonian system. In this work, an index theory for the Hamiltonian system is employed to study the stability of standing pulses for the FitzHugh–Nagumo equations. Related results for more general skew-gradient systems are also obtained.     

Hyperbolic Chaos in Systems Based on FitzHugh – Nagumo Model Neurons

In the present paper we consider and study numerically two systems based on model FitzHugh–Nagumo neurons, where in the presence of periodic modulation of parameters it is possible to implement chaotic dynamics on the attractor in the form of a Smale–Williams solenoid in the stroboscopic Poincaré map. In particular, hyperbolic chaos characterized by structural stability occurs in a single neuron supplemented by a time-delay feedback loop with a quadratic nonlinear element.     

Strong convergence rate in averaging principle for stochastic FitzHugh–Nagumo system with two time-scales

This article deals with averaging principle for stochastic FitzHugh–Nagumo system with different time-scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved, and as a consequence, the system can be reduced to a single stochastic ordinary equation with a modified coefficient. Moreover, the rate of convergence for the slow component towards the solution of the averaging equation is of order 1/2.