As a simple example, assume a dendrite of diameter D stretching from x = 0 to x = x. At x = 0, a voltage source holds vm(0) at Vm0 V. vm(x) = 0, Ie - 0. Thus it is clear that B = K = 0, and A = Vm0. The dc voltage along the dendrite decays exponentially as x ^ x.
The space constant, X, of a dendrite is an important parameter because it is a measure of the steady-state electrotonic spread of epsps and ipsps. It can be shown that the bulk parameters of the UM and the axoplasm are related to X by
where D is the diameter of the dendrite in cm, GM is the leakage conductivity of the membrane in S/cm2, and pi is the axoplasm resistivity in ohm cm. Note that X increases with the square root of the diameter of the dendrite, other bulk parameters being held constant.
The rate of spread of a transient voltage induced on a passive dendrite is inversely proportional to the product, (ro + ri)cm (Kandel et al., 1991). If one assumes that ri > ro, and expresses (ri cm)-1 in terms of the bulk parameters of the membrane, then
where D is the dendrite diameter in cm, pi is the axoplasm resistivity in ohm cm, and Cm is the bulk membrane capacitance in F/cm2. Thus, the larger the dendrite diameter, the faster the passive spread velocity.
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