Related papers: Spatial localization in the FitzHugh-Nagumo model
The FitzHugh-Nagumo equation, originally conceived in neuroscience during the 1960s, became a key model providing a simplified view of excitable neuron cell behavior. Its applicability, however, extends beyond neuroscience into fields like…
Spatiotemporal localized and extended structures associated with a subcritical finite wavenumber Hopf bifurcation are studied in the Purwins model (a three-variable FitzHugh-Nagumo version). Steady and time-dependent numerical continuation…
Nonlinear dynamical stochastic models are ubiquitous in different areas. Excitable media models are typical examples with large state dimensions. Their statistical properties are often of great interest but are also very challenging to…
We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the…
Hysteresis dynamics has been described in a vast number of biological experimental studies. Many such studies are phenomenological and a mathematical appreciation has not attracted enough attention. In the paper, we explore the nature of…
We report how strategic evolution can stabilize topological states in a network of FitzHugh-Nagumo systems. The evolution follows a repeated process of adding or deleting of links between two nodes that is decided based on a threshold set…
Neurons are the central biological objects in understanding how the brain works. The famous Hodgkin-Huxley model, which describes how action potentials of a neuron are initiated and propagated, consists of four coupled nonlinear…
Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics…
The FitzHugh-Nagumo equation has been investigated with a wide array of different methods in the last three decades. Recently a version of the equations with an applied current was analyzed by Champneys, Kirk, Knobloch, Oldeman and Sneyd…
Two-dimensional spatially localized structures in the complex Ginzburg-Landau equation with 1:1 resonance are studied near the simultaneous occurrence of a steady front between two spatially homogeneous equilibria and a supercritical Turing…
In this article, we study the FitzHugh-Nagumo $(1,1)$--fast-slow system where the vector fields associated to the slow/fast equations come from the reduction of the Hodgin-Huxley model for the nerve impulse. After deriving dynamical…
We study the nonlinear dynamics of two delay-coupled neural systems each modelled by excitable dynamics of FitzHugh-Nagumo type and demonstrate that bistability between the stable fixed point and limit cycle oscillations occurs for…
Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial…
Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie--Gower type. %in the vicinity of a Turing-Hopf interaction. Two regimes are studied in detail. In the first, the homogeneous…
We focus on the qualitative analysis of the phase portraits arising in the three-parameter FitzHugh-Nagumo system and its compactified form. The investigation is split into three parameter-dependent cases. In one of these cases, the system…
The passive conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) corresponds to a gradient dynamics for a single order parameter field related to density. It provides a simple microscopic description of the thermodynamic…
We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity…
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a…
This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with quadratic term and delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try…
We study the interplay between traveling action potentials and spatial inhomogeneities in the FitzHugh-Nagumo model to investigate possible mechanisms for the occurrence of fibrillatory states in the atria of the heart. Different dynamical…