BME 502 / handout #2
BME 502 -- Handout #2
Passive Membrane Properties
all of the ligand, voltage, and ion-sensitive conductances flow through the cell membrane, which has
two important properties that determine the consequences of the current flow: membrane resistance
and capacitance
consider resistance first
membrane resistance
i.
membrane resistance consists of
both rm (membrane resistance) and
ri (internal resistance)
ii. ri increases with distance: for all
patches in membrane, rm is
equivalent but for any one patch of
membrane, ri 's sum with distance
iii. thus, will be less current to flow through rm for patches of membrane farther from a point of
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BME 502 / handout #2
current injection
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BME 502 / handout #2
precise relation between amplitude and distance given by:
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where V0 is the membrane potential at point zero, Vx is the membrane potential at point x,
and λ = space constant, or distance at which Vx decrements to 1/3 of original value
units: λ is a distance, in cm; x is a distance, in cm
Vx and V0 both in volts
functional consequence: spatial summation
V0 is proportional to the size of the injected current, because
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rinput is the input resistance, or the average resistance presented by the membrane to the
flow of current along the axoplasm and back out through the membrane
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BME 502 / handout #2
so the two parameters, rinput and λ determine the shape of the curve of membrane potential
produced in response to an injection of current: r input determines the height of the potential at
the point of injection, and λ determines the lateral spread, i.e., the rate of decay as a function
of distance
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BME 502 / handout #2
rinput and λ depend on both rm and ri
relation between λ and membrane resistance:
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interpretation:
λ, the distance over which the potential change spreads increases with increases in
membrane resistance (which prevents loss of current across the membrane)
and decreases with increasing internal resistance (which resists current flow along the core
of the fiber)
 these parameters are influenced by morphology of neuron:
rm inversely proportional to surface area (# of channels)
surface area of a cylinder = 2 π a l
 as surface area of cell  , (diameter  ),
rm  , so λ also 
ri inversely proportional to cell volume
volume of cylinder = π a2 l
 as cell size  , volume  in proportion to square of diameter, so
ri  disproportionately more than rm , so that λ  as cell size increases
bottom line: larger cells will have a higher length constant than smaller cells
likewise, rinput depends on both rm and ri:
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BME 502 / handout #2
rm and ri are always specified for a cylindrical segment of axon, 1-cm in length
the dimensions of ri are ohm/cm
because the resistance along the membrane increases as length increases
the dimensions of rm are Ω-cm
because the resistance through the membrane decreases as length increases
(more channels for current to flow through); so rm must be inversely proportional
to length, so Ω-cm / cm = Ω
these parameters provide information about the resistive characteristics of a cylindrical segment of
axon, 1-cm in length
do not provide information about the specific resistivities of the membrane or the cytoplasm because
they depend on the size of the neural process (dendrite or axon)
for example, all other things being equal, would expect a 1-cm length of small diameter fiber
to have a higher membrane resistance than a larger diameter fiber of the same constituent
membrane, because the smaller diameter fiber has less membrane surface, and thus would
have fewer channels
if the radius, a, is known, then can determine the specific resistance of the membrane and the
cytoplasm
the specific membrane resistance, Rm, is defined as the resistance of 1 cm2 of membrane
is independent of geometry, and thus allows comparison of membranes
membrane resistance  as circumference 
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Install Equation Editor and doubleclick here to view equation.
where a is the radius
units: rm in ohm-cm, a in cm, Rm in ohm-cm2
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BME 502 / handout #2
specific resistance of the axoplasm, R i, is the internal longitudinal resistance of a 1-cm
length of axon 1 cm2 in cross-sectional area
also independent of geometry, and gives a measure of how freely ions move through the
intracellular space
because ri increases as cross-sectional area decreases:
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Install Equation Editor and doubleclick here to view equation.
units: ohm-cm
thus, input resistance also depends on geometry:
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BME 502 / handout #2
membrane capacitance
cell membrane also can store charge, so acts as capacitor
capacitor consists of two conducting surfaces separated by a layer of non-conducting, or
insulating, material
capacitance, C, is defined as the amount of charge, Q, it will accumulate for each volt of
potential, V, applied to it:
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Install Equation Editor and doubleclick here to view equation.
capacitance has units of coulombs per volt, or farads, F
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capacitance proportional to membrane surface area, because greater change in Ic
required for a given change in voltage
and inversely proportional to membrane thickness (decreased capability for storing
charge)
because cell membranes are on the order of 7nm, capable of storing relatively large amount
of charge; cell membranes typically have a capacitance on the order of 1μF/cm2
functional consequence of capacitance:
considering original RC membrane circuit,
a.
b.
c.
d.
driving force = battery (polarity in figure is arbitrary)
membrane resistance, rm = reciprocal of conductance
ri, internal resistance
Cm, membrane capacitance
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BME 502 / handout #2
current injected into cell initially flows to charge the capacitance of the
membrane (because sees the capacitor as a zero resistance)
changes Vm at that local part of membrane
any change in Vm for the capacitor must be matched by a change in
Vm for the resistive part of the membrane
as ΔVm increases, ionic current across the resistance also increases
since Im = Ii + Ic , Ic decreases as ΔVm and Ii increase
relationship expressed by:
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or,
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where
V0= the voltage at time 0, at the beginning of the injection of current
Vt = the voltage at time t
t = the time from the beginning of the injection of current
τ = membrane time constant
where τ = RC where R = rm
and represents time to increase charge to 2/3 of asymptotic Vm
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BME 502 / handout #2
time to decrease to 1/e from initial voltage, V0 :
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consequence: alters the time course of the transmembrane voltage
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BME 502 / handout #2
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BME 502 / handout #2
consequence: temporal summation
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BME 502 / handout #2
because of multiple capacitances in successive patches of membrane, time constant is not
constant, but varies according to distance:
at point of current injection, Vm rises to 84% of
maximum within 1 time constant
at 1 length constant distance, V m rises to 63% of
maximum, i.e., τ accurate
at 2 length constants distance, V m rises to 35% of
maximum within 1 time constant
so how to measure τ accurately ?
must measure Vm at a site other than the site of
current injection
must know the distance between site of current
injection and site of measurement
the three parameters of rin, λ, and τ provide all of the information necessary to predict the
evolution of membrane potential changes in response to a current injection
provided the geometrical properties of the axon / dendrite / soma are known
assuming only membrane exhibits only passive properties, i.e., no voltage-dependent or
ion-sensitive conductances
same principles apply to case when hyperpolarizing current is injected
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