The potassium Acurrent, low firing rates and rebound excitation in HodgkinHuxley models
 Maureen E. Rush,
 Dr. John Rinzel
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It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium Acurrent (I _{A}). Using a reduction of the classical HodgkinHuxley model, we study the effects ofI _{A} on steady firing rate, especially in the nearthreshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steadystate currentvoltage relation be Nshaped rather than monotonic. For experimentally based genericI _{A} parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI _{A} parameters given by Connor and Stevens for the crab axon. When theI _{A} inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg _{A} decreases and as inactivation rate decreases. We also study howI _{A} affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.
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 Title
 The potassium Acurrent, low firing rates and rebound excitation in HodgkinHuxley models
 Journal

Bulletin of Mathematical Biology
Volume 57, Issue 6 , pp 899929
 Cover Date
 19951101
 DOI
 10.1007/BF02458299
 Print ISSN
 00928240
 Online ISSN
 15229602
 Publisher
 Kluwer Academic Publishers
 Additional Links
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 Industry Sectors
 Authors

 Maureen E. Rush ^{(1)}
 Dr. John Rinzel ^{(2)}
 Author Affiliations

 1. Department of Mathematics, California State University, 93311, Bakersfield, California, USA
 2. Mathematical Research Branch, NIDDK, National Institutes of Health, BSA Bldg., Suite 350, 20892, Bethesda, Maryland, USA