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91

5.4.3 Discrete cable

The previously presented equations describe the potential distribution in a continuous cable.
However, unlike an axon, the rod network does not have continuously distributed capac-
itance and resistance, but consists of discrete photoreceptors interconnected via gap junc-
tions. Each photoreceptor can be considered isopotential, with an associated membrane
capacitance and resistance. The continuous cable equations give a reasonable estimate of
the voltage distribution in networks of discretely coupled cells when the length constant λ
is large compared to the distance between cells (λ > D).

The analytic form of the impulse response function (Greens function) for a discrete two-
dimensional cable is given by:

τr. /.Λ lτ ∕2i∖τ ∕2t∖    /-(7 + 4)t

(5.9)


Vmln( ) ~ ^fylm I I ɪm ! I θXp I

C ∖'yτ∕   ∖ 7τ /    ∖ 7τ
τ = RmC
~ Rd Rm

where Im and In are modified Bessel functions of the first kind. Photoreceptors are on a
grid centered at (m=0, n=0). This solution, derived by Alan Hodgkin, is given in Lamb and
Simon, 1976 [69].

5.4.4 Numerical Methods

Although equation 5.9 can be convolved over space and time to give the response of the
two-dimensional network, this is very inconvenient, and an analytic solution may not be



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