Abstract
Buchholtz et al. constructed a detailed conductance-based model of the LP cell of the stomatogastric ganglion of crustacea based upon the experimental work of Golowasch. Their model incorporated seven ionic currents and had 13 dynamical variables. We have produced a simplification of this model that has a seven-dimensional phase space by using the method of equivalent potentials, suggested by Abbott and Kepler, to combine several dynamical variables with similar time scales. Analysis of the dynamics of the reference and reduced model reveals similar bifurcation diagrams and similar dynamical behavior of the individual ionic currents.
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References
Abbott LF, Kepler TB (1990) Model neurons: from Hodgkin-Huxley to Hopfield. In: Garridol (ed) Statistical mechanics of neural networks. Springer, Berlin Heidelberg New York
Back A, Guckenheimer J, Myers M, Wicklin F, Worfolk P (1992) dstool: computer assisted exploration of dynamical systems. Not Am Math Soc 39:303–309
Buchholtz F, Golowasch J, Epstein IR, Marder E (1992) Mathematical model of an identified stomatogastric ganglion neuron. J Neurophysiol 67:332–340
Fitzhugh R (1961) Impulses and physiological stages in theoretical models of nerve membrane. Biophys J 1:445–466
Golowasch J (1990) Characterization of a stomatogastric ganglion neuron. A biophysical and a mathematical description. PhD dissertation, Brandeis University, Waltham, Mass
Golowasch J, Marder E (1992) Ionic currents of the lateral pyloric neuron of the stomatogastric ganglion of the crab. J Neurophysiol 67:318–331
Guckenheimer J, Holmes P (1983)Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, Berlin Heidelberg New York
Guckenheimer J, Gueron S, Harris-Warrick RM (1993) The dynamics of conditionally bursting neuron. Proc R Soc Lond (Biol) in press
Hodgkin AL, Huxley AF (1952) Measurement of current-voltage relations in the membrane of the giant axon of the Loligo. J Physiol (Lond) 117:500–544
Johnson BR, Peck JH, Harris-Warrick RM (1991) Temperature sensitivity of graded synaptic transmission in the lobster stomatogastric ganglion. J exp Biol 156:267–285
Kepler TB, Abbott LF, Marder E (1992) Reduction of conductancebased neuron models. Biol Cybern 66:381–387
Krinskii VI and Kokoz YM (1973) Analysis of equations of excitable membranes. 1. Reduction of the Hodgkin-Huxley equations to a second-order system. Biofizika 18:506–511
Meunier C (1992) Two and three dimensional reductions of the Hodgkin-Huxley system: separation of time scales and bifurcation schemes. Biol Cybern 67:461–468
Nagumo JS, Arimito S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070
Rinzel J (1985) Excitation dynamics: insights from simplified membrane models. Fed Proc 44:2944–2946
Rose RM, Hindmarsh JL (1989) The assembly of ionic currents in a thalamic neuron. I. three-dimensional model. Proc R Soc Lond (Biol) 237:267–288
Tuckwell HC (1988) Introduction to theoretical biology. Cambridge University Press, Cambridge
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Golomb, D., Guckenheimer, J. & Gueron, S. Reduction of a channel-based model for a stomatogastric ganglion LP neuron. Biol. Cybern. 69, 129–137 (1993). https://doi.org/10.1007/BF00226196
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DOI: https://doi.org/10.1007/BF00226196