Hodgkin - Huxley Equations

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. 

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

If a semiconductor element with negative differential conductance is operated in a reactive circuit, oscillatory instabilities may be induced by these reactive components, even if the relaxation time of the semiconductor is much smaller than that of the external circuit so that the semiconductor can be described by its stationary I U) characteristic and simply acts as a nonlinear resistor. Self-sustained semiconductor oscillations, where the semiconductor itself introduces an internal unstable temporal degree of freedom, must be distinguished from those circuit-induced oscillations. The self-sustained oscillations under time-independent external bias will be discussed in the following. Examples for internal degrees of freedom are the charge carrier density, or the electron temperature, or a junction capacitance within the device. Eq.(5.3) is then supplemented by a dynamic equation for this internal variable. It should be noted that the same class of models is also applicable to describe neural dynamics in the framework of the Hodgkin-Huxley equations [16]. 

Oscillations of the bursting type have been modelled mainly in neurobiology, whieh is the field where they are most frequently observed. These models consist in modifications of the Hodgkin Huxley equations, which generally take into account the effect of Ca "" and the slow, cyclical variation of some ionic conductances (Plant Kim, 1976 Plant, 1978 Carpenter, 1979 Hindmarsh Rose, 1984 Chay Rinzel, 1985 Ermentrout Koppell, 1986 Rinzel Lee, 1986 Canavier etal, 1991 Smolen Keizer, 1992 Destexhe, Babloyantz Sejnowski, 1993 Bertram, 1994). 

Best, E. 1979. Null space in the Hodgkin-Huxley equations. A critical test. Biophys. J. 27 87-104. 

Troy, W.C. 1978. The bifurcation of periodic solutions in the Hodgkin-Huxley equations. Quart. Appl. Math. 36 73-83. 

PitzHugh-Nagumo The FitzHugh-Nagumo equations are also called the Bonhoeffer-Van der Pol equations and have been used as a generic system that shows excitability and oscillatory activity. FitzHugh [1969] showed that much of the behavior of the Hodgkin-Huxley equations can be reproduced by a system of two differential equations ... 

Noble, D. (1960). A description of cardiac pacemaker potentials based on the Hodgkin-Huxley equations. J. Physiol. (Land.) 154, 64P-65P. 

Noble, D. (1962). A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. J. Physiol. (Lond.) 160,317-352. 

To conclude this section we would note that, despite the fact that physically the two models are totally different, the stimulus propagation mechanisms in both are very similar in both potential equations it is the second space derivative which is the highest, and if one compares specifically Eqs. (17) and (24) one will also notice the absence in both of the time derivative of potential. Further, in both cases there is another dynamic variable in the potential equation which determines the evolution of the process. In the next section we will show that to a good approximation the same situation occurs in a giant axon which is described by the Hodgkin-Huxley equations. 

Consider the problem of the passage of a single nervous impulse along a fiber it is described by the Hodgkin-Huxley equations. " Our aim is not so much to obtain exact relationships between the speed and various other parameters (Huxley has already done this ), as to find out the most essential features of these equations. On the qualitative level an explanation of the nervous impulse looks very simple. 

Therefore, the existence of such a thing as a gating current may be regarded as an established fact now, although its precise identification is a matter of future studies. Furthermore modern techniques of measurement have demonstrated that a simplistic interpretation of the Hodgkin-Huxley equations is imperfect. 

Although there now exists considerably more knowledge concerning membrane structure, and correlations between membrane structure and ionic permeability changes under a host of different conditions have been documented, there still does not exist a molecular theory explaining excitable membrane phenomena in the usual sense where the application of statistical mechanics yields empirically observed and derived thermodynamic relationships, e.g., the Hodgkin-Huxley equations. 

In spite of extensive research the molecular mechanisms underlying the excitability of biological membranes are still by and large unresolved. The success of the Hodgkin-Huxley equations in describing the action potential... 

Noble D, Gamy A, Noble PJ (2012). How the Hodgkin-Huxley equations inspired the cardiac physiome project. J Physiol 590 2613-2628. 

Electrical noise in biological membrane systems is explained in terms of opening and closing of ionic channels (for two useful review see De Felice (1981) and Frehland (1982)). The qualitative behaviour of membrane activity strongly depends on the density of membrane channels (Holden, 1981 Holden Yoda 1981). Integrating numerically the celebrated Hodgkin-Huxley equations (Hodgkin Huxley, 1952) it was demonstrated that the number of channels is a bifurcation parameter. The fact that channel... 

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