90
time with the current input to yield the solution to a space and time-dependent input. These
Greens functions are identical to the concept of an impulse response from engineering.
5.4.2 Cable equation in two dimensions
While the one-dimensional cable equation is useful for computing the response of the rod
network to a bar stimulus, where current flow is restricted to the dimension perpendicular
to the bar (see figure 5.1), to understand how coupling in the rod network reduces noise in
photoreceptors, the entire network must be considered.
The cable equation has been studied extensively for one-dimensional problems relating
to the voltage distribution in a neuron, but a two-dimensional electrical cable is a much less
common topology. One example of a two-dimensional cable an the array of rod photore-
ceptors connected to one another in a grid, as shown in figures 4.3 and 5.8.
The steady-state response for a current i injected into a continuous two-dimensional
cable is:
τz∕ ʌ .Rcv (r∖
(5.7)
(5.8)
Where K is a modified Bessel function of the second kind with a = 0 [69]. The solution
is radially symmetric, so r = y∕x2 + y2.
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