The Hodgkin-Huxley model

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 

The variables m, n, and h are phenomenological variables that describe the observed gating of the sodium and potassium channels in response to changes in 

A Matlab computer code for the Hodgkin-Huxley model is given below. The code to compute the time derivatives of the state variables (the right-hand side of Equation (7.30)) is  

The model may be simulated and the predicted voltage transient plotted using the following commands  

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Figure 7.7 Functions m< >, n00, and /too predicted by Equations (7.31) and (7.32) for the Hodgkin-Huxley model. Figure adapted from [108],...

Use computer simulation to determine if the solution to the Hodgkin-Huxley model of Section 7.3.3 is periodic at Iapp = 6.2 pA-cm 2. What happens when the applied current is lowered to 6.0 pA-citr2 ... 

As was recalled by Keynes (1994a), it was first observed by Chandler Meves (1970) that in squid axons perfused with NaF a small flow of Na ions persisted in the inactivated steady state. Their tentative conclusion based on the Hodgkin-Huxley model was to suppose that the inactivation parameter h was the sum of two components h, and Zij, where A, predominated at negative potentials, and Zij predominated at positive ones. This proposition predicted the existence of one type of Na+ conductance that increased transiently with depolarization as in the Hodgkin—Huxley system, and a second type that persisted with depolarization to give a steady low level of conductance. Combined with m kinetics the idea fitted well with the experimental data. 

The psychologist Maslow wrote that if the only tool you have is a hammer, you tend to treat everything as if it were a nail (4). Markov processes based on the Hodgkin-Huxley model had been widely used to describe ionic currents measured in many different experiments. However, in 1986, we began to use a new tool to analyze the patch clamp data. The insight gained from this new analysis has changed our ideas about the processes that open and close the ion channel. The new tool is based on fractals. 

The Markov description of ion channel kinetics, originally derived from the Hodgkin-Huxley model, implies that the ion channel protein has certain physical properties. 

Hassard, B. 1978. Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon. J. Theor. Biol. 71 401-20. 

FIGURE 19.3 Nerve action potential. Computed with the Hodgkin-Huxley model for nerve membrane (membrane patch). B, baseline E, excitation E, plateau E, recovery A, afterpotential. 

Figure 4. Effect of potential on the parametric functions and in the Hodgkin-Huxley model (u = m,n, h).

A concrete example of a model for neuronal bursting (Plant, 1981) has been analyzed in considerable detail by Rinzel and Lee (1987). The model utihzes the Hodgkin-Huxley model, eqs. (13.3) and (13.4), as the mechanism for action-potential generation and introduces two additional conductances, a calcium channel and a calcium-activated potassium channel, to produce the bursting behavior. The membrane potential is given by... 

US to quite different excitation models, namely, a molecular model in the following section and a dissipative one in Section 6.2. We will come back to the Hodgkin-Huxley model in Section 5.6 in context with a generalized network language. 

R. N. Miller and J. Rinzel The Dependence of Impulse Propagation Speed on Firing Frequency, Dispersion, for the Hodgkin-Huxley Model , Biophys. J. 3 i 227-259, 1981. 

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