By Luigi Ricciardi and Alwyn Scott (Eds.)
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Additional resources for Biomathematics in 1980Papers presented at a workshop on Biomathematics: Current Status and Future Perspectives, Salerno, April 1980
V) w + 9. (t) ( 9) 27 THE MATHEMA TICS OF EXCITATION where y!. (x,t) is a 3-dimensional vector function which determines ionic conductances. 9. are linear in V but nonlinear in:d. The features of interest are due to the vector~. The forms taken by the nonlinear functions I, £ and a are empirical and mathematically idiosyncratic: their choice is determined by the fact they fit the data, and fit in with some physical ideas about the control of ion movement. There is no particular mathematical interest in the H-H equations as such, and so most of the analysis of excitation has not been on the H-H system but on simpler analogs.
C. is increase; (d) response to sinusoidal excitation. 200 action potentials rate /51 100 o 50 100 150 I /IJAcm-2 Figure 5. Computed rate of period solutions of the H-H membrane equations for constant injected current densities. 23 THE MATHEMA TICS OF EXCITA TlON may be approximated, for 50 < r < 125, by r = 27 ln (I + 1). For currents above 70 ilA/cm2 the membrane current is not regenerat i ve, and for 1arge currents the response consists of oscillations rather than action potentials. The relation between rate and injected current is plotted in Figure 5.
Hastings (1975) has shown that there are single pulse travelling wave so 1ut ions of the system of (11) for b smalL Such sol ut ions have homoc 1in i c orbits in the 3-dimensional phase-space: they return to the stable equilibrium point at t = ±oo, and there are two possible values of conduction velocity. Green and Sleeman (1974) obtain the upper and lower bounds for these conduction velocities. Hastings (1974) obtains periodic, travell ing wave solutions for an open set of parameters, confirming a conjecture by Conley (unpublished, referred to in Conley and Smoller, 1976).