Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between distinct large and small amplitudes. MMOs have been observed studied for over thirty years chemical, physical, biological systems. Few attempts made thus far to classify different patterns MMOs, contrast the classification related phenomena bursting oscillations. This paper gives survey types concentrating its analysis on whose small-amplitude produced by local, multiple-time-scale...

We give a geometric analysis of canards folded node type in singularly perturbed systems with two-dimensional (2D) critical manifold using the blow-up technique. The existence two primary is known provided nonresonance condition $\mu\notin\mathbb{N}$ satisfied, where $\mu=\lambda_1/\lambda_2$ denotes ratio eigenvalues associated singularity reduced flow. show that, due to resonances, bifurcation secondary occurs. detailed explanation this phenomenon an extension Melnikov theory prove...

We extend canard theory of singularly perturbed systems to the general case $k$ slow and $m$ fast dimensions, with $k\ge 2$ $m\ge 1$ arbitrary. A folded critical manifold a system, generic requirement for canards exist, implies that there exists local $(k+1)$-dimensional center spanned by variables eigendirection variables. If one further assumes $m-1$ nonzero eigenvalues $m\times m$ Jacobian matrix equation have all negative real part, then $(k+m)$-dimensional problem is locally governed...

In recent work [J. Rubin and M. Wechselberger, Biol. Cybern. 97, 5 (2007)], we explained the appearance of remarkably slow oscillations in classical Hodgkin-Huxley (HH) equations, modified by scaling a time constant, using recently developed theory about mixed-mode (MMOs). This is only rigorously valid, however, for ε sufficiently small, where parameter that arises from nondimensionalization HH system. Here, illustrate how regime over which MMOs exist, features MMO patterns within this...

Cell volume changes are ubiquitous in normal and pathological activity of the brain. Nevertheless, we know little how cell affects neuronal dynamics. We here performed first detailed study effects on By incorporating swelling together with dynamic ion concentrations oxygen supply into Hodgkin-Huxley type spiking dynamics, demonstrate spontaneous transition between epileptic seizure spreading depression states as swells contracts response to osmotic pressure. Our use an order parameter...

Stellate cells (SCs) of the medial entorhinal cortex (layer II) display mixed-mode oscillatory activity, subthreshold oscillations (small-amplitude) interspersed with spikes (large amplitude), at theta frequencies (8–12 Hz). In this paper we study mechanism generation such patterns in an SC biophysical (conductance-based) model. particular, show that is based on three-dimensional canard phenomenon and intrinsically nonlinear, involving participation both components (fast slow) a...

Applying a low current to recent biophysical model for inhibitory neurons in the cortex leads mixed mode oscillations (MMOs), mixture of spikes and subthreshold oscillations. At higher currents, fire regularly. We show that specific slow potassium underlies this behavior. Next we reduce five-dimensional an neuron three dimensions with slow/fast time-scale structure. then there is range parameters under which reduced shows MMOs can be explained by canard phenomenon. Many interneurons are...

Pseudo-plateau bursting is a type of oscillatory waveform associated with mixed mode dynamics in slow/fast systems and commonly found neural models. In recent model for the electrical activity calcium signaling pituitary lactotroph, two types pseudo-plateau bursts were discovered: one which drives another simply follows them. Multiple methods from dynamical theory have been used to understand bursting. The classic 2-timescale approach treats concentration as slowly varying parameter...

We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The analyzed investigates the phenomenon post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674–1697, [2012]). Propofol modulates decay time-scale inhibitory GABAa synaptic current. Interestingly, this system gives rise within specific range doses. Using techniques geometric singular perturbation theory, we identify structures,...

A network of excitatory neurons within the pre-Bötzinger complex ({\mbox{pre-BötC}}) mammalian brain stem has been found experimentally to generate robust, synchronized population bursts activity. An calibrated model for {\mbox{pre-BötC}} cells yields typical square-wave bursting behavior in absence coupling, over a certain parameter range, with quiescence or tonic spiking outside this range. Previous simulations showed that introduction synaptic coupling extends range significantly and...

We develop a general technique for proving the existence of chaotic attractors three-dimensional vector fields with two time scales. Our results connect important areas dynamical systems: theory discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two-dimensional are diffeomorphisms that limit on non-invertible one-dimensional maps. Wang Young formulated hypotheses suffice to prove in these families. Three-dimensional singularly perturbed have return also...

Thermoregulatory responses are partially controlled by the preoptic area and anterior hypothalamus (PO/AH), which contains a mixed population of temperature-sensitive insensitive neurons. Immunohistochemical procedures identified extent various ionic channels in rat PO/AH These included pacemaker current [i.e., hyperpolarization-activated cyclic nucleotide-gated (HCN)], background potassium leak (TASK-1 TRAAK), transient receptor potential channel (TRP) TRPV4. neurons showed dense TASK-1...

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns electrical activity. Several types produce short bursts activity which are more effective than single spikes evoking hormone release. These bursts, called pseudo-plateau unlike studied mathematically neurons (plateau bursting) and standard fast-slow analysis used for plateau bursting is limited use. Using an alternative analysis, with one fast two slow variables, we show that a canard-induced mixed mode...

We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point view. focus on inherent nature and define via bifurcations. In particular, we show that type I is associated with novel Bogdanov-Takens/SNIC bifurcation while II Andronov-Hopf bifurcation. both cases, canards play an important role understanding unfolding these structures. also explain transition between two types highlight all bifurcations involved, thus providing complete...

The existence of traveling wave solutions to a haptotaxis dominated model is analyzed. A version this has been derived in Perumpanani et al. [Phys. D, 126 (1999), pp. 145--159] describe tumor invasion, where diffusion neglected as it assumed play only small role the cell migration. By instead allowing be small, we reformulate singular perturbation problem, which can then analyzed using geometric theory. We prove three types physically realistic case diffusion. These reduce no-diffusion limit...