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FIGURE 4.2-2 The infinitely long dendrite example used in the derivation of the space constant, X.

Equations 4.2-4 and 4.2-5 are called the "telegrapher's equations," or "telephone equations" (Lathi 1965); they describe the spatial and temporal distribution of transmembrane voltage and longitudinal current on an idealized dendrite (RC transmission line). To see how they can be used, examine the simplest case of steady-state (in time) behavior of an infinitely long dendrite (0 3 x 3 x) given a dc voltage source at one end (see Figure 4.2-2). The voltage distribution in x is given by

Laplace transforming Equation 4.2-6 yields

Equation 4.2-7 has roots at s = ± ^(ro + ri)gm , so the general solution is of the form:

vm (x) = Aexp^-^ (r0 + r )gm + Bexp[+^ (r0 + r )gm ] 4.2-8

Considering the boundary conditions, vm(0) = V0, vm(x) = 0, B must = 0, and A = V0. Therefore:

The parameter X is called the dendrite space constant (analogous to the time constant in an RC circuit):

Example 4.2-1

As an example, calculate the cable parameters and space constant of a "typical" dendrite of length L in which C = 1 |F/cm2, D = 0.6 |im or 6 x 10-5 cm (diameter), axoplasm conductivity ai = 1.333 x 10-2 S/cm, assume ri > ro so neglect ro ^ 0, Gm = 4 x 10-4 S/cm2, line length L = 103 |im. Am/L is the dendrite membrane area /length in cm2/cm. First, calculate ri:

The shunt conductance /length, gm, is next considered:

gm = Gm (Am/L) = Gm (nDL/L) = 4 x 10-4 (n 6 x 10-5) = 7.540 x 10-8 S/cm 4.2-12 Thus rm = 1.326 x 107 Q cm. Similarly, cm = C (Am/L) = 10-6 (n 6 x 10-5) = 1.885 x 10-10 F/cm 4.2-13 Thus, the dendrite space constant is

1 = l/^/g^r" = ^7.540 x 10-8 S/cm x 2.653 x 1010 ohm/cm = 2.236 x 10

2 cm

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