Dendritic Cable

The current distribution shown in Fig. 2(b) may be written

In both terms, the factors 5(y)5(z) ensure that the source lies on the x-axis. In the first term, the factor S(x) puts the sink at x = 0. In the second term, the transmembrane current per unit length im(x) is given by (2.2). Inserting (4.8) into (4.3) gives

J V • JS(r) d3r = J [I0 S(x) - im(x)] dx = 0 (4.9)

where the last equality follows from direct integration of (2.2). Thus the monopole contribution vanishes by current conservation, i.e., the total of sources and sinks equals zero. Inserting (4.8) into (4.5) gives p = J r [Io S(x) - im(x)] S(y)S(z) d3r (4.10)

The three vector components may be evaluated separately. Because of the factor r = (x, y, z), integration over y and z gives py = 0 and pz = 0, respectively. Similarly for px, integrating over x causes the first term involving S(x) to vanish, leaving

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