caa a22 4500 463244842 CHVBK 20180405153322.0 cr unu---uuuuu 170326e20070701xx s 000 0 eng 10.1007/s00422-007-0153-5 doi (NATIONALLICENCE)springer-10.1007/s00422-007-0153-5 Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model [Elektronische Daten] [Jonathan Rubin, Martin Wechselberger] 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 analyze the model's dynamics under systematic variation of the parameters τ h , τ n , and applied current I. As expected, we find that for fixed (τ h , τ n ), the model undergoes a transition from excitable, with a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO's), featuring slow action potential generation, arise for an intermediate range of I values, if τ h or τ n is sufficiently large. Our analysis explains in detail the geometric mechanisms underlying these results, which depend crucially on the presence of two slow variables, and allows for the quantitative estimation of transitional parameter values, in the singular limit. In particular, we show that the subthreshold oscillations in the observed MMO patterns arise through a generalized canard phenomenon. Finally, we discuss the relation of results obtained in the singular limit to the behavior observed away from, but near, this limit. Springer-Verlag, 2007 Rubin Jonathan Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, PA, USA aut Wechselberger Martin School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia aut Biological Cybernetics Springer-Verlag 97/1(2007-07-01), 5-32 0340-1200 97:1<5 2007 97 422 https://doi.org/10.1007/s00422-007-0153-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00422-007-0153-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Rubin Jonathan Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, PA, USA aut NATIONALLICENCE 700 1- Wechselberger Martin School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia aut NATIONALLICENCE 773 0- Biological Cybernetics Springer-Verlag 97/1(2007-07-01), 5-32 0340-1200 97:1<5 2007 97 422 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer