AC Length Constant
Along a neurite
V x
= e
B x ⁄λ
ω
where
λ
ω
=
Z m
⁄ r a
, Z m
= membrane impedance of a unit
V
0 length of neurite, and r a
= resistance of a unit length of cytoplasm.
Assume f is sufficiently high that transmembrane current can be regarded as entirely capacitive
( f
> 5 ⁄ 2 πτ m
, i.e. for a neuron with
τ m
= 30 ms, f > 25 Hz). Then Z m
≈ 1 ⁄ j ω C where C = capacitance of a unit length of neurite.
Thus at high frequencies
λ
ω
≈ 1 ⁄ j ω C r a
and
V x
≈ e
B x j ω C r a = e
B x 1 A j
ω C r a
2 . The
V
0 real part of the exponent is the signal attenuation, and the imaginary part is the phase shift, so
V x
≈ e
B x
ω C r a
2 .
V
0
Substituting r a
= R i
⁄π a
2
and C = 2 π a C m
, where a = radius of neurite, R i
= cytoplasmic resistivity in
Ω
cm, and C m
= specific membrane capacitance in µf/cm
2
, we have r a
2
C
=
1
⋅
2
R i
π a
2
⋅ 2 π a C m
=
R i
C a m
. Therefore
V x
V
0
≈ e
B x
ω R i
C m a and the AC space constant is
λ
ω
≈
2
=
R i a
C m
ω
=
2 π a f R i
C m
. If a is in µm, f in Hz, R r a
C ω
C m
in µf/cm
2
, the numerical result must be multiplied by 10
5
to convert it to µm.
i
in
Ω
cm, and
Example: consider a neurite with radius of 1 µm, R m
= 50,000
Ω
cm
2
, R i
= 100
Ω
cm, and C m
=
1 µf/cm
2
. The membrane time constant is 50 ms, so the frequency at which membrane resistive and capacitive current are equal is ~ 3.2 Hz. The DC length constant is
λ
DC
= r m
⁄ r a
, which turns out to be ~ 1500 microns. The AC length constant at 100 Hz is only 400 microns, roughly
4 times shorter.
Addendum: NEURON uses R a
to signify R i
, and neurite diameter is specified rather than radius.
Thus in the context of NEURON models it is more convenient to rewrite the AC length constant formula as
λ f
≈
1
2 d
π f R a
C m
.
N.T. Carnevale Page 1 of 1