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The internal resistance and membrane resistance both
contribute to voltage attenuation in space
II.C. How does voltage change in a
neuron
as we get further away from the injection point? (cont’d)
II.C.3
.. neurons
II.C.3.a.
The model
Segment of free nerve ending
Δ
V(x) = I*R
in
[e
(x/
λ
)
] =
Δ
V
ss(inject)
[e
(x/
λ
)
]
Membrane
resistance of a unit
length of neuron
(in
Ω
cm)
Internal resistance
of a unit length of
neuron (in
Ω
/cm)
Magnitude of
injected current
Input resistance of
the cell
where
λ
=(r
m
/r
i
)
1/2
is the patch of
neuron’s
space or
length constant
and
Δ
V
ss
= I*R
in
is the steady state
voltage change at
the injection point
(note in the graph
Δ
Vss(x) is not
limited to the
injection point)
is the distance it takes
for the voltage change to
drop by 63%
If we increase the membrane resistance, r
m
, will the cell's voltage
attenuate more or less as we get farther away from the injection
point? ________
If we increase r
m
, we _______
λ
.
Therefore, by
________
λ
,
voltage will attenuate less as we get further from the injection point.
r
i
Voltage attenuates less
r
m
λ
r
m
λ
Why? _____________________________________________
As an exercise, you might want to fill in the rest of the chart above.
In general, increasing
λ
decreases attenuation as shown
on the chart below.
The value
λ
is useful because it gives us a single
parameter with which to determine how rapidly the
cell's voltage will attenuate as a function of distance.
r
i
Voltage attenuates more
II.C.3.c
Properties of
λ
internal axoplasmic resistance
II.C.3.b.
The space constant equation
λ
2
λ
2
We can model the system as a resistor network
Cell with larger
λ
less
Fewer ions leak out of the CM per unit distance
increase
increasing
Note looking at Steady state voltage only
For simplicity assume R
in
and r
m
not related
Δ
V
ss
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View Full DocumentV(x,t) =
λ
2
(
δ
2
V/
δ
x
2
)
τ
(
δ
v/
δ
t)
The truth
II.D.2 The simplified unified equation
II.D.1 The cable equation
The time and space equations can be simplified & combined into
a unified equation by making certain assumptions that are
acceptable for this class.
Δ
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 Fall '08
 Chapman
 Neurobiology

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