Properties of the Hodgkin Huxley Model

This section reviews some of the results obtained using the Simnon program HODHUX.t given in the previous section. Using Hodgkin and Huxley's original sign convention, currents (positive ion flow) entering the axon from outside, such as JNa, are positive. JK leaving the axon lumen is negative. Vm is shown as it would be measured; the resting Vmr = -70 mV. Jfa < 0 depolarizes the membrane, i.e., drives Vm positive. (Jfa is a - ion current density.)

Figure 1.4-4 illustrates an action potential produced when Jfa = -2 |A for 5 ms. Also plotted are the scaled membrane capacitance current density, potassium ion current density and the sodium ion current density. Scaling details are in the figure caption. Figure 1.4-5 plots the auxiliary parameters n, m, and h when the same action potential (AP) is generated. The AP is scaled by 1/70 in order that the same scale can be used as for n, m, and h. Note that the sodium activation parameter, m, falls rapidly after reaching its peak near unity. Finally, in Figure 1.4-6, the specific ionic conductances, gK, gNa, and gnet are plotted with (Vm(t) + 70) mV. Note that gK dominates the recovery phase of Vm following the spike.

FIGURE 1.4-4 Results of Simnon simulation of the HH model run with Euler integration; 8t = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (2) Vm mV; (3) Jin = -2 |A/cm2 from t = 1 to 6 ms, else 0; (4) JCsc = 10 JC (scaled capacitive current density, |A/cm2); (5) JKsc = JK/10 (scaled potassium ion current density, |A/cm2); (6) JNasc = JNa /10 (scaled sodium ion current density, |A/cm2).

FIGURE 1.4-4 Results of Simnon simulation of the HH model run with Euler integration; 8t = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (2) Vm mV; (3) Jin = -2 |A/cm2 from t = 1 to 6 ms, else 0; (4) JCsc = 10 JC (scaled capacitive current density, |A/cm2); (5) JKsc = JK/10 (scaled potassium ion current density, |A/cm2); (6) JNasc = JNa /10 (scaled sodium ion current density, |A/cm2).

A little-appreciated property of the HH model is its propensity to fire on rebound from a prolonged hyperpolarization of Vm. Hodgkin and Huxley (1952) called this phenomenon anode break excitation. Figure 1.4-7 illustrates what happens when Jin is positive, forcing Vm more negative, hyperpolarizing the membrane. Rebound firing occurs when Jfa ^ 0; Vm overshoots Vmr enough to cause a spike to occur for Jin = 2, 3, and 5 |A. There is not enough rebound in Vm for Jin = 1 or 1.5 |A to cause firing, however.

Another property of the HH model is its nonlinear behavior as a current-to-frequency converter. In this case, a prolonged, negative Jin is applied. For the model parameters given, and Cm = 0.003 mF, it is found that for Jin 3 -7 |A, the model

FIGURE 1.4-5 Results of Simnon simulation of the HH model run with Euler integration; 8t = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (1) Vm(t)/70 mV; (2) n(t); (3) m(t); (4) h(t); (5) zero.
FIGURE 1.4-6 Results of Simnon simulation of the HH model run with Euler integration; 8t = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (1) Jin |A/cm2; (2) [Vm(t) + 70] mV; (3) gK (t, Vm) mS/cm2; (4)gNa(t, Vm) mS/cm2; (5) gnet = gK + gNa + gL mS/cm2.
Tendon Reflex Stimulation
FIGURE 1.4-7 Simulation of anode break excitation. HH model run with Euler integration; 8t = 0.00001, Cm = 0.001. Horizontal axis, time in milliseconds. Top trace: Jin (hyperpolarizing current densities of 1, 1.5, 2, 3, 5 |A/cm2; no spike for Jin = 1 and 1.5).

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