The name is absent

87

of neurons.

Figure 5.6 A shows a diagram of a neuronal cable, with axial specific resistance R_a, dis-
tributed radial resistance through the membrane R_m, 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 R_m, membrane capacitance C, and R_c 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-^)

R_m    ∂t  R_a Ox^z

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 / CR_aX² ∖          .

G {x, t) = v(x, t) = ∖ --=u(t) exp —— exp --—-- (5.3)

V c y∕4τrt ∖Or_r∕ ∖ 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

<<   93   94   95   96   97   98   99   >>

More intriguing information