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BN102: Passive Membrane Properties
C. Aizenman, 2/13/07
What are passive electrical properties?
1. Resting membrane resistance (Rm).
2. Membrane capacitance (Cm).
3. Intracellular (axial) resistance (Ri).
Passive electrical properties are the membrane
properties that allow neurons to conduct electricalimpulses without using voltage-gated ion channels.
What is affected by passive properties?
1. The magnitude of
change in membranepotential
after current entry.
2. The time course of
change in membranepotential
after current entry.
3. The distance over
which the change involtage
travels.
4. Speed of action
potentialpropagation.
IV
1. Membrane Resistance (Input Resistance): Rinput
Determines how much the
membrane potential willchange in response to a
current.
2
IV
1. Membrane Resistance (Input Resistance):
V=IR
Rinput: Overall resistance of a cell.
!V=IRInput
Rin= 10 M" (10x106 ")
I= 1 nA (1x10 -9A) !V= (1x10-9)(10x106)=10 mV
1. The density of the channels open at rest.
2. The size of the cell.
What does Rin depend on? !V=IRInput
The specific membrane resistance, Rm, describes the
resistance of a unit area (how leaky the membrane is).
Rm Rm
Rin Rin
Rm Rm
RinRin
>
>
=
>
Surface area of a sphere = 4!a2
Rin"1/a2
a=radius
!
Rinput =Rm
4"a2
20 #m
Purkinje Neuron (rat) Tectal Neuron (tadpole)
Neurons vary in size and structure - passive properties differ
Rinput Rinput<Water
Pump
Pressure difference
R
Plumber’s version of a membrane: R
3
Circuit model of a membrane: Rm
Voltage change through a resistor is instantaneous.
IV
2. Membrane Capacitance: Cinput
Voltage change across a
membrane is notinstantaneous due to
membrane capacitance.
A capacitor is a device that stores energy in the
electric field created between a pair of conductors,
separated by an insulating layer, on which equal
but opposite electric charges have been placed.
The lipid bilayer in a cell’s membrane acts as a capacitor.
!
C(F) =Q
V
Water
Pump
Pressure difference = 0
C
t=0
Plumber’s version of a capacitor
4
Water
Pump
Pressure difference = Vt
C
t=x
Circuit model of a membrane: Cm
• Voltage change through a capacitor is gradual and
proportional to the current.
• Specific membrane cap. (Cm) has a fixed value =1#F/cm2 due
to the uniform thickness (4nm) of the cell membrane.• Larger cells have larger capacitance.
!
Cinput = Cm(4"a2)
20 #m
Purkinje Neuron (rat) Tectal Neuron (tadpole)
Neurons vary in size and structure - passive properties differ
CinputCinput >
Rinput Rinput<WaterPump
No flow
C
t=0
Capacitor
charging R
Plumber’s version of an RC circuit
Total flow =
Flow at R
+
Flow at C
5
WaterPump
Pressure difference
C
t=1
RCapacitor
charged
Water
Pump
Pressure difference
C
t=2
RCapacitor
discharging
Circuit model of a membrane: Rin. + Cin.
Since Im = Ir + Ic , as the capacitor gets charged the amount
of current flowing through the resistor gradually increases,
gradually increasing Vm until the voltage reaches a steady
state. The membrane time constant (#) determines the rate
of change in Vm.# = CinRin
• # describes how fast
the voltage changes.
• The greater # is, the
longer it will take to
reach maximal voltage
change (ImRin).
• Also the greater # is,
the slower is the decay
of the voltage.
#= CinRinCin= Cm (4!a2)Rin= Rm/4!a2
#= (Cm (4!a2)) (Rm/4!a2)#= Cm Rm
# does not depend on the size of the cell
# depends on Rm
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Rising phase Falling phase
!Vm(t)= ImRin (1$e$t/#)
!Vm(t) = Change in voltage at time t.ImRin = maximal change in voltage
t = time t=0, t>> #
# = time constant (sec)
#= The time it takes to reach (1-1/e) (~63%) of maximal change in voltage
#= The time it takes to drop to 1/e (~37%) of maximal change in voltage
Rising phase
!Vm(t)= ImRin (1$e$t/#)
After # seconds, t= #
!Vm(#)= ImRin (1$e$ # /#)
!Vm(#)= ImRin (1$e$ 1) = ImRin (1$1/e) e=2.7
63% ImRin
Falling phase!Vm(t)= ImRin e$t/#
t= #
!Vm(t)= ImRin e$1 = ImRin (1/e) = ImRin (1/2.7)
!Vm(#)=0.37 ImRin
= ImRin (1$1/2.7)
= 0.63 ImRin
3. Internal Resistance (Axial Resistance): Ra
Determines how far and how fast an impulse will travel.
Determined by the specific resistance of the
cytoplasm and the diameter of the central core.
The length constant (%) describes the change in Vm at distance (x)
!
" =rm
ra
!Vm(x)= V0 e-x/%
Voltage drops off exponentially:
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Cross section through an axon
ions
High ra Low ra
a
channels
High rm Low rm
!
" =rm
ra
Fatter axons have longer length constants
rm=Rm/2!a
ra=&/2!a2
& = specific resistance of cytoplasm
larger circumference = more channels
larger area = more ions
As a increases, ra decreases
faster than rm
, thus % gets bigger.
!
" = aRm#
Why are passive membrane properties important?
% will affect the speed of
AP propagation.
Fatter axons will
conduct faster.
Myelination effectively increases rm making % greater.
It also decreases Cm (#=rmcm), allowing Vm to change faster.
!
" =rm
ra
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Time constants and
length constants will
affect temporal and
spatial summation,
determining whethera subthreshold input
will elicit an action
potential.
A cell receives a synaptic input 100 !m away from the axon hillock.
Activation of the input results in voltage change of +45 mV at the site of
input. You have experimentally determined that % is 100 !m and that
action-potential threshold at the axon hillock is -50 mV. Does the cell fire
an action potential as a result of the synaptic input? Assume a restingmembrane potential of -65mV.
% = 100 #m
!Vm(x)= V0 e-x/%
X= 100 #m
!Vm(100)= 45 x e-1/1
-65 + 16.7 = -48.3 mV
= 45 x 1/e = 45 x 0.37= 16.7 mV
100#m
Problem
Giga = x109
Mega = x106
Kilo = x103
UnitMilli = x10-3
Micro = x10-6
Nano = x10-9
Pico = x10-12
Useful Tidbits
V=IR
g=1/R
C=Q/V
Voltage = volts (V)
Current = amps (A)
Resistance = ohms (")
Capacitance = Farads (F)Conductance = Siemens (S)
Charge = Coulombs