Hodgkin-Huxley

The Hodgkin-Huxley (HH) equations model action potentials in membranes. Variants of these equations are used in most models for electrical activity of excitable membranes and have been studied by many authors. 

FIG. 22 The Hodgkin-Huxley equivalent for an axon (a) and the modified HH circuit for sieve tubes in phloem (b). 

A model, based on a perturbation analysis of the highly successful empirical formulation of Hodgkin and Huxley (1), has been developed which makes predictions of the effect of oscillating fields on a particular nerve membrane system (2). In the present paper, a theoretical model will be presented along with some of the predicted effects of AC electric fields on the Hodgkin-Huxley (HH) model of squid axon membranes. 

Modified Hodgkin-Huxley Model. In the HH model, the membrane current I, written as a function of V is expressed by the system of coupled equations given in Table I. In these equations V is the displacement of membrane potential from the resting value (depolarization negative). Constants Cm, g, gjga, 8i VK VNa and V] are explained in detail in ( 1). In Table I, l n and g m h are the potassium and sodium conductances, respectively. The dimensionless dynamical quantities m, n, and h are solutions of the given first order differential equations and vary between zero and unity after a change in membrane potential. The a and 3 rate constants are assumed to depend only on the instantaneous value of membrane potential. 

TABLE I. The Hodgkin-Huxley equations which describe the relationship between total membrane current 1 and transmembrane potential V in the squid giant axon [see (1)]. 

R. E. Pant and M. Kim Mathematical descriptions of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys.J. 1976,16 227-244. 

Zhou, C., and Kurths, J. Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons. Chaos 2003,13 401— 409. 

Noble, D. A modification of the Hodgkin Huxley equations applicable to Purkinje... 

The P s are now permeability coefficients and are related to the mobilities of the ions as in the original Nernst theory. The subscripts in and out refer to the concentrations of the ions inside and outside the membrane and the P s describe diffusion coefficients, mobilities, and the membrane thickness, but, in the Hodgkin-Huxley theory, were used as adjustable parameters. 

The demise of the famous Hodgkin-Huxley theory of nerve conductance brings to mind other Nobel prizes in electrochemically related areas. In 1959 Heyrovsky was recognized for a new analytical method, and this polarography has been the origin of many modem methods of electroanalysis. The award for Nobel Prize to Mitchell in 1978 (for a chemiosmotic model of membrane function) and metabolism seems to have been based on a lack of awareness of a simpler, clearer (prior) model by Williams for interpreting the same functions. The award to Marcus in 1992 for the theory of redox reactions (1956) seems to have lacked awareness of an earlier publication by Weiss that described similar ideas. 

The best-known theory of the spike potential produced in an impulse down a nerve is that due to Hodgkin, Huxley, and Katz (1952). Although this theory is still current among electrophysiologists, it is now regarded with skepticism by a number of physical scientists who have examined it in the light of modem... 

In the Goldman equation use the diffusion coefficients in question 1 in place of permeation coefficients. The rate of permeation is -D(dc/dx). The dx is essentially the membrane thickness and its neglect will cancel out of the equation (cf. the Goldman equation), (c) Is this flux consistent with actual radiotracer measurements of the movement concerned (In the Hodgkin-Huxley and Katz work, arbitrary values were used for the P s to ensure that the equation replicated the experiment. This, of course, makes it difficult to check its validity. Assume the starting concentration of ions on either side of the membrane is that shown in the text. The average internal diameter of a squid axon is about 1 mm.)... 

The 1952 Hodgkin-Huxley model for membrane electrical potential is perhaps the oldest and the best known cellular kinetic model that exhibits temporal oscillations. The phenomenon of the nerve action potential, also known as excitability, has grown into a large interdisciplinary area between biophysics and neurophysiology, with quite sophisticated mathematical modeling. See [103] for a recent treatise. 

Derived from Hodgkin-Huxley s celebrated theory and inspired by the experimental observations, cellular calcium dynamics, either stimulated via inositol 1,4,5-trisphosphate (IP3) receptor in many non-muscle cells [69,139], or via the ryanodine receptor in muscle cells [108], is another extensively studied oscillatory system. Both receptors are themselves Ca2+ channels, and both can be activated by Ca2+, leading to calcium-induced calcium release from endoplasmic reticulum. 

The Hodgkin-Huxley model involves three membrane currents due to potassium, sodium, and a leak current of charge through other pathways. The model assumes linear current-voltage relationships ... 

The voltage v in the Hodgkin-Huxley model is the membrane potential measured relative to the equilibrium voltage Veq v = AT — Veq, where Veq is the potential when no current is applied. The experimentally determined equilibrium potentials (which depend on the ion gradients across the membrane) for the model are... 

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