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- Martin Wechselberger
- SIAM J. Applied Dynamical Systems
- 2005

- Mathieu Desroches, John Guckenheimer, Bernd Krauskopf, Christian Kuehn, Hinke M. Osinga, Martin Wechselberger
- SIAM Review
- 2012

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)

- Janet Best, Alla Borisyuk, Jonathan E. Rubin, David Terman, Martin Wechselberger
- SIAM J. Applied Dynamical Systems
- 2005

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)

- Jonathan E. Rubin, Martin Wechselberger
- Biological Cybernetics
- 2007

This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin–Huxley model, depending on parameters τ h , τ 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 theory to… (More)

- Margaret Beck, Christopher K. R. T. Jones, David Schaeffer, Martin Wechselberger
- SIAM J. Applied Dynamical Systems
- 2008

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)

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)

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory –… (More)

- Martin Wechselberger, Chadwick L. Wright, Georgia A. Bishop, Jack A. Boulant
- American journal of physiology. Regulatory…
- 2006

Thermoregulatory responses are partially controlled by the preoptic area and anterior hypothalamus (PO/AH), which contains a mixed population of temperature-sensitive and insensitive neurons. Immunohistochemical procedures identified the extent of various ionic channels in rat PO/AH neurons. These included pacemaker current channels [i.e.,… (More)

- Jonathan E. Rubin, Martin Wechselberger
- Chaos
- 2008

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)

- Horacio G. Rotstein, Martin Wechselberger, Nancy J Kopell
- SIAM J. Applied Dynamical Systems
- 2008

Abstract 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… (More)