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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