The electrical behavior of a UM can be modeled electrically by a lumped-parameter, RC model. This model can be used to describe the behavior of dendrites with passive membrane (see Section 4.2) and also, with some modification, can be used to describe the generation of the nerve impulse (see Section 1.4).

A very important property of nerve UM is its electrical capacitance. Because the hydrophobic lipid center of the UM acts like an insulator (neglecting ions leaking through the membrane-bound glycoproteins), and the internal and external liquids surrounding the membrane generally have relatively high conductances because of the dissolved ions, the membrane behaves like a parallel-plate capacitor. The plates are the inside surface of the UM in contact with the conductive axoplasm on the inside of the cell, and the outside surface of the UM in contact with the extracellular fluid. A patch of UM from a squid giant axon, for example, has a measured capacitance of about 1 |aF/cm2. In MKS units, this is 10-2 F/m2. If one assumes that a UM patch is modeled by a parallel-plate capacitor with capacitance, then

d where C is the capacitance in farads, A is the plate area in m2, d is the plate separation in m, eo is the permittivity of free space in F/m, and k is the dielectric constant. By substituting C = 10-6 F, A = 10-4 m2, d = 7.5 x 10-9 m, and eo = 8.85 x 10-12 F/m, the equivalent dielectric constant for the UM is k = 8.47, which is a bit high for long-chain lipids, but not so for hydrated proteins (Plonsey, 1969, Ch. 3). Assuming the resting potential across the nerve cell membrane is 65 mV, the electric field across the dielectric is approximately E = 0.065/(7.5 x 10-9) = 8.67 x 106 V/m, or 8.67 x 104 V/cm, quite high. If one takes the dielectric thickness as 5 nm (50 A), then k = 5.65, and E = 1.30 x 105 V/cm. Because of this high electric field in the membrane, it would not be surprising to find that membrane capacitance is in fact a function of the transmembrane potential, Vm.

Membrane capacitance is important because any potential change across the neuron UM cannot occur unless the capacitance is supplied with a current density according to the relation: Jc(t) = Cm Vm (t) amp/m2 (Cm is the capacitance in F/m2). Jc is generally supplied by ions flowing through gated-ion channel proteins, and axial ion currents outside and within the axon or dendrite.

In formulating mathematical models of how passive dendrites behave electrically when subjected to transient changes of input voltage, it is more convenient to describe cylindrical dendrites in terms of per-unit-length parameters, including capacitance. It is easy to show that for a dendrite or axon, cm = Cm n D F/m 1.2-2

where Cm is the membrane capacitance in F/m2, and D is the dendrite diameter in meters. When cm is multiplied by the length, L, of a cylindrical section of dendrite, one again obtains Cm F.

There are three constant resistive parameters used along with cm needed to describe the electrical behavior of a dendrite with a passive membrane. (Passive nerve membrane has transmembrane conductances that are constant over the range of membrane voltage of interest.)

First, an expression for ri, the internal longitudinal or axial resistance of the dendrite in ohm s/m will be derived. Assume that the axoplasm inside the dendrite has a net resistivity of pi ohm cm. (Resistivities are commonly given in ohm cm.) The net internal resistance of a tube of axoplasm L cm long and D cm in diameter is

Changing the length units to meters and dividing by L, ri = 0.04 pi/(n D2) ohm/m 1.2-4

(pi is still in ohm cm.) Thus (ri Ax) is a resistance in series with (internal) axial current flow in the lumped parameter model.

The external longitudinal (spreading) resistance per unit length, ro, can also be found. ro is also in series with the external axial current near the dendrite. ro is generally small compared with ri, and its exact value depends on the tissue structures and extracellular fluid composition surrounding the dendrite. (In the case of the squid giant axon, the external medium is seawater which has a low resistivity, po).

The passive membrane has an equivalent constant radial leakage conductance for the various ions found in the axoplasm and extracellular fluid. The total net leakage conductance is gm S/m in parallel with the membrane capacitance per unit length, cm F/m. In general, the leakage conductance for K+ is higher than that for Na+.

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