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where Vm is the membrane resting potential in volts, -0.070 V for squid at 20°C (note Vm depends on temperature); R is the gas constant, 8.314 J/mol K; F is the Faraday constant, 9.65 x 104 C/mol; T is the Kelvin temperature; [Na+]o and [Na+]i are the concentrations of sodium ions outside (460 mM) and inside (46 mM) the membrane, respectively, in mol/l for squid axon. ENa < 0 means the net force on Na+ is inward. That is, a sodium ion outside the cell membrane has a total potential energy of ENa electronvolts acting to force it inward. For squid axon at 293 K (20°C), ENa can be calculated:

ENa = - 0.070 - (8.314 x 293/9.65 x 104) ln (460/46) = -0.128 eV 1.4-2

Similarly, one can calculate the potential energy, EK, acting to force a K+ ion outward through the resting membrane:

Ek = - 0.070 - (8.314 x 293/9.65 x 104) ln(10/410) = + 0.024 eV 1.4-3

Na+ ions generally enter the axon through special voltage-gated Na+ channels. With the membrane unexcited and in its steady-state condition, there is a very low, random leakage of Na+ ions inward through the membrane. Potassium ions also leak out because the net potassium potential energy is dominated by the concentration gradient (the inside having a much higher potassium concentration than the outside, i.e., [K+]i > [K+]o). Other small ions leak as well, being driven through the membrane by the potential energy difference between the membrane resting potential and the ion Nernst potential.

At steady-state equilibrium, a condition known as electroneutrality exists in unit volumes inside and immediately outside the axon. That is, in each volume there are equal numbers of positive and negatively charged ions and molecules. Inside the axon, a significant percentage of negative charges are bound to large protein molecules that cannot pass through the membrane because of their sizes.

Not specifically pertinent to the short-term solution to the HH model is the fact that nerve membrane (on soma, axon, and dendrites), and those of nearly all other types of cells, contains molecular "pumps" driven from the energy in ATP molecules which can eject Na+ from the interior of the cell against the potential energy barrier, ENa, often as an exchange operation with K+ being pumped in. Ionic pumps are ubiquitous in nature, and are responsible for the maintenance of the steady-state intracellular resting potentials of nerves, muscle cells, and all other cells.

To try to describe and summarize the electrical events associated with nerve membrane, a simple, parallel, electrical circuit was developed (Hodgkin and Huxley, 1952; Katz, 1966) that includes specific ionic conductances for Na+, K+, and "other ions" (leakage) for a 1-cm2 area of axon membrane. Figure 1.4-3 illustrates the circuit. VNa and VK are the Nernst potentials for sodium and potassium ions, respectively, and VL is the equivalent Nernst potential for all leakage ions, including chloride. A specific ionic conductance is defined by the specific ionic current density divided by the difference between the actual transmembrane potential and the Nernst potential. Thus,

Outside axon

Outside axon

FIGURE 1.4-3 Details of the HH membrane patch. Batteries represent the Nernst potentials for the Na+, K+, and "leakage" ions. Leakage includes Cl- and all other non-voltage-sensitive ions. Currents are given as current densities (units, A/cm2). Also shown are the metabolically driven ion pumps as current sources. The pumps slowly restore steady-state internal ion concentrations that maintain the Nernst potentials.

FIGURE 1.4-3 Details of the HH membrane patch. Batteries represent the Nernst potentials for the Na+, K+, and "leakage" ions. Leakage includes Cl- and all other non-voltage-sensitive ions. Currents are given as current densities (units, A/cm2). Also shown are the metabolically driven ion pumps as current sources. The pumps slowly restore steady-state internal ion concentrations that maintain the Nernst potentials.

For example, in resting squid axon, gNa = JNa/(-0.070 - 0.058). JNa is negative inward by definition, so gNa is positive as it should be.

The mathematical model devised by Hodgkin and Huxley (1952) to describe the generation of the nerve action potential begins by writing a node equation for the simplified equivalent circuit of Figure 1.4-3:

Cmv = Jin JK JNa JL 1.4-5B i v = (Jin JK JNa JL )/Cm ^^

By definition, v = (Vmr - Vm ). v is in millivolts. If v < 0, Vm is depolarizing (going positive from the resting Vmr = -70 mV). The leakage current density is assumed to obey Ohm's law:

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