Signal propagation and failure in one-dimensional FitzHugh-Nagumo equations with periodic stimuli

Non-Trivial Dynamics in the FizHugh-Rinzel Model and Non-Homogeneous Oscillatory-Excitable Reaction-Diffusions Systems

This article focuses on the qualitative analysis of complex dynamics arising in a few mathematical models in neuroscience context. We first discuss the dynamics arising in the three-dimensional FitzHugh-Rinzel (FHR) model and then illustrate those arising in a class of non-homogeneous FitzHugh-Nagumo (Nh-FHN) reaction-diffusion systems. FHR and Nh-FHN models can be used to generate relevant complex dynamics and wave-propagation phenomena in neuroscience context. Such complex dynamics include canards, mixed-mode oscillations (MMOs), Hopf-bifurcations and their spatially extended counterpart. Our article highlights original methods to characterize these complex dynamics and how they emerge in ordinary differential equations and spatially extended models.

Pulse propagation and failure in the discrete FitzHugh-Nagumo model subject to high-frequency stimulation

We investigate the effect of a homogeneous high-frequency stimulation (HFS) on a one-dimensional chain of coupled excitable elements governed by the FitzHugh-Nagumo equations. We eliminate the high-frequency term by the method of averaging and show that the averaged dynamics depends on the parameter A=a/ω equal to the ratio of the amplitude a to the frequency ω of the stimulating signal, so that for large frequencies an appreciable effect from the HFS is attained only at sufficiently large amplitudes. The averaged equations are analyzed by an asymptotic theory based on the different time scales of the recovery and excitable variables. As a result, we obtain the main characteristics of a propagating pulse as functions of the parameter A and derive an analytical criterion for the propagation failure. We show that depending on the parameter A, the HFS can either enhance or suppress pulse propagation and reveal the mechanism underlying these effects. The theoretical results are confirmed by numerical simulations of the original system with and without noise.

Propagation of bursting oscillations

We investigate a system of partial differential equations of reaction-diffusion type which displays propagation of bursting oscillations. This system represents the time evolution of an assembly of cells constituted by a small nucleus of bursting cells near the origin immersed in the middle of excitable cells. We show that this system displays a global attractor in an appropriated functional space. Numerical simulations show the existence in this attractor of recurrent solutions which are waves propagating from the central source. The propagation seems possible if the excitability of the neighbouring cells is above some threshold.

Self-sustained collective oscillation generated in an array of nonoscillatory cells

Oscillations are ubiquitous phenomena in biological systems. Conventional models of biological periodic oscillations usually invoke interconnecting transcriptional feedback loops. Some specific proteins function as transcription factors, which in turn negatively regulate the expression of the genes that encode these "clock proteins." These loops may lead to rhythmic changes in gene expression in a cell. In the case of multicellular tissue, collective oscillation is often due to the synchronization of these cells, which manifest themselves as autonomous oscillators. In contrast, we propose here a different scenario for the occurrence of collective oscillation in a group of nonoscillatory cells. Neither periodic external stimulation nor pacemaker cells with intrinsically oscillator are included in the present system. By adopting a spatially inhomogeneous active factor, we observe and analyze a coupling-induced oscillation, inherent to the phenomenon of wave propagation due to intracellular communication.

Bifurcation analysis of solitary and synchronized pulses and formation of reentrant waves in laterally coupled excitable fibers

We study the dynamics of a reaction-diffusion system comprising two mutually coupled excitable fibers. We consider a case in which the dynamical properties of the two fibers are nonidentical due to the parameter mismatch between them. By using the spatially one-dimensional FitzHugh-Nagumo equations as a model of a single excitable fiber, synchronized pulses are found to be stable in some parameter regime. Furthermore, there exists a critical coupling strength beyond which the synchronized pulses are stable for any amount of parameter mismatch. We show the bifurcation structures of the synchronized and solitary pulses and identify a codimension-2 cusp singularity as the source of the destabilization of synchronized pulses. When stable solitary pulses in both fibers disappear via a saddle-node bifurcation on increasing the coupling strength, a reentrant wave is formed. The parameter region, where a stable reentrant wave is observed in direct numerical simulation, is consistent with that obtained by bifurcation analysis.

Density-matrix renormalization for model reduction in nonlinear dynamics

We present an approach for model reduction of nonlinear dynamical systems based on proper orthogonal decomposition (POD). Our method, derived from the density-matrix renormalization group, provides a significant reduction in computational effort for the calculation of the reduced system, compared to a POD. The efficiency of the algorithm is tested on the one-dimensional Burgers equations and a one-dimensional equation of the Fisher type as nonlinear model systems.

Input-output relation of FitzHugh-Nagumo elements arranged in a trifurcated structure

In this study, the propagation of an action potential in a network of excitable elements is studied numerically. The network we consider consists of excitable elements arranged in the shape of a trifurcated structure having three cables. The system has a branch point, a Y junction, at which the three cables are joined. Two types of external stimulations are considered: a single impulsive stimulation at one of the cable terminals, and a pair of stimuli applied to two different terminals. We have found three basic phases depending on the excitability of the elements for a single external stimulus as follows: (1) signal distributor--as the excitability gets higher, the pulse generated by a stimulus splits into two at the branch point, and two pulses are transmitted to the opposite terminals, (2) propagation block--the pulse in the lower excitable chain is blocked at the branch point, and (3) transient propagation--as the excitability is decreased further, we see that the pulse vanishes before reaching the branch point. By the interaction between the pulses that originate from different sources, signal transmission is recovered if the pulses arrive at the branch point nearly synchronously or after a specific delay time. The effects of the repetition of these two types of stimulation are also investigated. Complex spatiotemporal patterns occur due to pulse-pulse interaction and collisions at the branch point. The input-output relationship, which depends crucially on the repetition period and the time lag between the pair of stimuli, is characterized by the stimulus-response ratio and the interspike interval. We also show the effects of noise on the distribution of the interspike interval.

Synchronization and firing death in the dynamics of two interacting excitable units with heterogeneous signals

We study the response of two coupled FitzHugh-Nagumo systems to heterogeneous external inputs. The latter, modeled by periodic parametric stimuli, force the uncoupled excitable systems into a regime of chaotic firing. Due to parameter dispersion involved in randomly distributed amplitudes and/or phases of the external forces the units are nonidentical and their firing events will be asynchronous. Interest is focused on mutually synchronized spikings arising through the coupling. It is demonstrated that the phase difference of the two external forces crucially affects the onset of spike synchronization as well as the resulting degree of synchrony. For large phase differences the degree of spike synchrony is constricted to a maximal possible value and cannot be enhanced upon increasing the coupling strength. We even found that overcritically strong couplings lead to suppression of firing so that the units perform synchronous subthreshold oscillations. This effect, which we call "firing death," is due to a coupling-induced modification of the excitation threshold impeding spiking of the units. In clear contrast, when only the amplitudes of the forces are distributed perfect spike synchrony is achieved for sufficiently strong coupling.

Boundary-induced patterns in excitable systems: the structure of oscillatory domain

The present work extends our recently published study [Phys. Rev. E 73, 066224 (2006)] on a mechanism of pattern formation in excitable media due to inhomogeneous boundary conditions (BC). To that end, we analyze a pair of coupled excitable and oscillatory cells, a distributed FitzHugh Nagumo model, and a distributed five-variable model that describes CO catalytic oxidation. For the three systems we determine the structure of the oscillatory domains, composed of bands of complex firing solutions with period-adding bifurcations, and show the commonality of the structures. The obtained results account for the recently reported experimental observations of mixed-mode oscillations showing a period-adding bifurcation during CO oxidation on a disk-shaped catalytic cloth with imposed cold temperature BC.

Emergent reaction-diffusion phenomena in capillary tubes

Pattern formation in the Belousov-Zhabotinsky reaction experiments carried out by filling capillary glass tubes with catalyst-immobilized gel for the reaction is reported. Under unperturbed and oscillatory conditions, helicoidal waves appear spontaneously. Quantitative structural data of those helices are obtained by devising an optical tomography technique for extracting rotationally symmetric structures from time-lapse data. Space-time representation of the catalyst oxidation reveals wave transmission phenomenon that is studied further by numerical simulations of a reduced spatial model.

Pair of excitable FitzHugh-Nagumo elements: synchronization, multistability, and chaos

We analyze a pair of excitable FitzHugh-Nagumo elements, each of which is coupled repulsively. While the rest state for each element is globally stable for a phase-attractive coupling, various firing patterns, including cyclic and chaotic firing patterns, exist in an phase-repulsive coupling region. Although the rest state becomes linearly unstable via a Hopf bifurcation, periodic solutions associated to the firing patterns is not connected to the Hopf bifurcation. This means that the solution branch emanating from the Hopf bifurcation is subcritical and unstable for any coupling strength. Various types of cyclic firing patterns emerge suddenly through saddle-node bifurcations. The parameter region in which different periodic solutions coexist is also found.