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In recent work [J. Rubin and M. Wechselberger, Biol. Cybern. 97, 5 (2007)], we explained the appearance of remarkably slow oscillations in the classical Hodgkin-Huxley (HH) equations, modified by scaling a time constant, using recently developed theory about mixed-mode oscillations (MMOs). This theory is only rigorously valid, however, for epsilon… (More)

A network of excitatory neurons within the pre-Bötzinger complex (pre-BötC) of the mammalian brain stem has been found experimentally to generate robust, synchronized population bursts of activity. An experimentally calibrated model for pre-BötC cells yields typical square-wave bursting behavior in the absence of coupling, over a certain parameter range,… (More)

The electrical dynamics in the heart is modeled by a two-component PDE. Using geometric singular perturbation theory, it is shown that a traveling pulse solution, which corresponds to a single heartbeat, exists. One key aspect of the proof involves tracking the solution near a point on the slow manifold that is not normally hyperbolic. This is achieved by… (More)

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast… (More)

This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin-Huxley model, depending on parameters tau( h ), tau( n ) that scale the rates of change of the gating variables. After reducing the model to an appropriate nondimensionalized form featuring one fast and two slow variables, we use geometric singular perturbation… (More)

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 the mechanism of generation of such patterns in a SC biophysical (conductance-based) model. In particular, we… (More)

Applying a low current to a recent biophysical model for inhibitory neurons in the cortex leads to mixed mode oscillations (MMOs), a mixture of spikes and subthreshold oscillations. At higher currents, the neurons fire regularly. We show that a specific slow potassium current underlies this behavior. Next we reduce this five-dimensional biophysical model… (More)

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with… (More)

We develop a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Our results connect two important areas of dynamical systems: the theory of chaotic attractors for discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two-dimensional Henon-like maps are… (More)