87
of neurons.
Figure 5.6 A shows a diagram of a neuronal cable, with axial specific resistance Ra, dis-
tributed radial resistance through the membrane Rm, and distributed capacitance C. This
cable is analogous to a one-dimensional section of rod photoreceptors, shown in figure 5.6
B, with rod membrane resistance Rm, membrane capacitance C, and Rc the coupling resis-
tance to an adjacent rod.
The current flow through an infinitely long section of cable of the form given in figure
5.6 gives rise to the following differential equation:
V ∂v 1 ∂2υ .
~s~ + ɑnɪ ~ ~d~^R~2 ~ г (ʒ-^)
Rm ∂t Ra Oxz
where u is a function of x and t. The general solution↑ for the voltage in a one-dimensional
infinite cable as a function of time and space is:
z-,* / ,ʌ / ,∖ ! Ra 1 /,∖ ( t ʌ / CRaX2 ∖ .
G {x, t) = v(x, t) = ∖ --=u(t) exp —— exp --—-- (5.3)
V c y∕4τrt ∖Orr∕ ∖ 4i /
for a point source of current that is impulsive in time (z(æ, t) = d(ʃ)d(t)), where u(t)
is the Heaviside step function. The derivation for this solution is given in section A.4 of the
appendix.
By solving for an impulse (point) of current in space and an impulse (flash) in time, the
solution can be solved for any time vary current by superimposing a scaled version of this
response for each point in a known stimulus distribution through time and space. The time
↑This solution is derived in section A.4