This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh–Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a ...

The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. We demonstrate how two different codimension-one bifurcations of a saddle-node periodic orbit with homoclinic orbits can explain transitions between tonic spiking and burst...

Drifting pattern domains (DPDs), i.e., moving localized patches of traveling waves embedded in a stationary (Turing) pattern background and vice versa, are observed in simulations of a reaction-diffusion model with nonlocal coupling. Within this model, a region of bistability between Turing patterns and traveling waves arises from a codimension-2 Turing-wave bifurcation (TWB). DPDs are found wi...

Abstract. In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov Theory. Secondly, we give exponentially small upper bounds in some cases in w...

We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius α. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter α. We relate this pattern to the Morse index of the static solution with maximal energy. Using a h...

Exploiting the specific structure of neuron conductance-based models, the paper investigates the mathematical modeling of neuronal bursting. The proposed approach combines singularity theory and geometric singular perturbations to unfold the geometry of multiple time-scales attractors in the neighborhood of high-codimension singularities. We detect a three-time scale bursting attractor in the u...

Pattern formation in semiconductor heterostructures is studied on the basis of a spatially two{dimensional model of reaction{diiusion type. In particular, we investigate the neighbourhood of a codimension{two Turing{Hopf instability by analytical methods. Amplitude equations are derived which predict the absence of mixed modes but extended ranges of bistability between homogeneous oscillatory s...

Novak and Tyson have proposed a realistic mathematical model of the biochemical mechanism that regulates M-phase promoting factor (MPF), the major enzymatic activity controlling mitotic cycles in frog eggs, early embryos, and cell-free egg extracts. We use bifurcation theory and numerical methods (AUTO) to characterize the codimension-one and -two bifurcation sets in this model. Our primary bif...

We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli. Recalling and extending results from the literature, we compute phase response curves (PRCs) valid near bifurcations to periodic firing for Hindmarsh-Rose, Hodgkin-Huxley, FitzHugh-Nagumo, and Morris-Lecar models, encompassing the four generic (codimension one) bifurcatio...

Parametric excitations are capable of stabilizing an unstable state, but they can also destabilize modes that are otherwise stable. Resonances come into play when the periodically forced base state undergoes a Hopf bifurcation from a limit cycle to a torus. A Floquet analysis of the basic state will identify such an event, but in order to characterize the resonances, one requires precise knowle...