Passive Electrical
Properties of the Neuron
Reference: Eric R. Kandel: Essentials of
neural Science and Behavior. P149 - 159
I. Equivalent Circuit of the
Membrane and
Passive Electrical Properties
Equivalent Circuit of the Membrane and
Passive Electrical Properties
• Equivalent Circuit of the Membrane
– What Gives Rise to C, R, and V?
– Model of the Resting Membrane
• Passive Electrical Properties
– Time Constant and Length Constant
– Effects on Synaptic Integration
Ions Cannot Diffuse Across the Hydrophobic
Barrier of the Lipid Bilayer
The Lipid Bilayer Acts Like a Capacitor
The voltage (Vm)across a capacitor is
proportional to the charge (Q) stored
on the capacitor:
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Vm = Q/C
∆Vm = ∆Q/C
∆Q must change before
∆Vm can change
Capacitance is Proportional to Membrane Area
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Vm = Q/C
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The Bulk Solution Remains Electroneutral
Electrical Signaling in the Nervous System is
Caused by the
Opening or Closing of Ion Channels
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The Resultant Flow of Charge into the Cell
Drives the Membrane Potential Away From its Resting Value
Each K+ Channel Acts as a Conductor
(Resistance)
γ conductance; r resistance
Ion Channel Selectivity and Ionic Concentration
Gradient Result in an Electromotive Force
An Ion Channel Acts Both as a
Conductor and as a Battery
γk , conductance of one k+
channel
EK =
RT
zF
•ln
[K+]o
[K+]i
All the K+ Channels Can be Lumped into
One Equivalent Structure
An Ionic Battery Contributes to VM in Proportion to the
Membrane Conductance for That Ion
When gK is Very High, gK•EK Predominates
The K+ Battery Predominates at Resting Potential
≈
gK
The K+ Battery Predominates at Resting Potential
≈
gK
+10
Experimental points
Membrane
potential
(millivolts)
-60
-70
(Red line shows values
according to Nernst equation)
-130
1
5
10
100
Extracellular potassium concentration (millimoles)
[K+]o = 4 mmol.l-1
Equivalent Circuit of the Membrane and
Passive Electrical Properties
• Equivalent Circuit of the Membrane
– What Gives Rise to C, R, and V?
– Model of the Resting Membrane
• Passive Electrical Properties
– Time Constant and Length Constant
– Effects on Synaptic Integration
Passive Properties Affect Synaptic Integration
Experimental Set-up for
Injecting Current into a Neuron
Equivalent Circuit for Injecting Current into Cell
Im total membrane current
Ii Ionic membrane current
Ic Capacitive membrane current
If the Cell Had Only Resistive Properties
If the Cell Had Only Resistive Properties
∆Vm = I x Rin
If the Cell Had Only Capacitive Properties
PNS, Fig 8-2
If the Cell Had Only Capacitive Properties
∆Vm = ∆Q/C
The rate of change in the membrane potential is
slowed by the membrane capacitance
t = Rin x Cin
t
Time constant (τ): The time taken to reach 63% of
the final voltage .
The time constants of different neurons typically
range from 1 to 20 ms
The Vm Across C is Always Equal to
Vm Across the R
∆Vm = IxRin
Out
In
∆Vm = ∆Q/C
Synaptic potentials that originate in dendrites are
conducted along the dendrite toward the cell body and
the trigger zone.
The cytoplastic core of a dendrite offers significant
resistance to the longitudinal flow of current because it
has a relatively small cross-sectional area and ions
flowing down the dendrite collide with other molecules.
The greater the length of the cytoplastic core, the
greater the resistance since the ions experience more
collisions the further they travel.
The larger the diameter of the cytoplasmic core, the
lower the resistance will be in a given length due to the
greater number of charges carriers at any point.
Spread of Injected Current is Affected by ra and rm
A neuronal process, either an axon or dendrite, can be divided
into unit lengths, which can be represented in an electrical
equivalent circuit.
Each unit length of the process is a circuit with its own membrane
resistance (rm) and capacitance(cm).
All the circuits are connected by resistors(ra), which represent the
axial resistance of segments of cytoplasm.
ra and rm
ra: The axial resistance of a unit
length (1 cm) of the cytoplasmic
core, expressed in Ω /cm.
Axial resistance depends on both the specific resistivity of the
cytoplasm, p, measured in Ω.cm, and the cross-sectional area of a
dendrite with radius a: ra = p/(πa2)
rm, the membrane resistance per unit length of cylinder is
expressed in Ω.cm. Membrane resistance depends on both the
specific resistance of a unit area of membrane, Rm, measured in Ω
cm2, and the circumference of the dendrite rm = Rm/2πa
For a dendrite of a uniform diameter, rm is the same for equal
lengths of membrane cylinder.
Length Constant
The current that is injected
flows out through several
pathways across successive
membrane cylinders along
the length of the process.
Each of these current
pathways is made up of two
resistive components in
series: the total axial
resistance rx, and the
membrane resistance rm, of
the unit membrane cylinder.
rx = ra x
The membrane component,
rm, has the same value at
each outflow pathway along
the cell process.
More current flows across a membrane
cylinder near the site of injection than at
more distant regions because current
always follows the path of least
resistance, and the total axial resistance,
rx, increase with distance form the site of
injection
Because ΔVm = Imrm, the change in membrane potential, ΔVm (x),
produced by the current across a membrane cylinder becomes
smaller as one moves down the dendrite away from the current
electrode. This decay with distance has an exponential shape,
expressed by the equation: Δvm (x) = ΔVoe-x/λ
λ is the membrane length constant, x is the distance from the site
of current injection, and V0 is the change in membrane potential
produced by the current flow at the site of the current electrode
(x=0)
The length constant is the distance along the dendrite form the
site of current injection to the site where Vm has decayed to 1/e,
or 37% of its initial value, and is determined as follows:
Length Constant l = √rm/ra
ra = p/(πa2)
rm = Rm/2πa
l = √Rma/2p
Large-diameter axon will have a longer
length constant than narrower axons.
Typical values of the length constant
fall in the range 0.1 to 1.0 mm.
Such passive spread of voltage changes along
the neuron is called electrotonic conduction.
The efficiency of this process, which is measured
by the length constant, has two important effects
on neuronal function.
First, it influences spatial summation, the process
by which synaptic potentials generated in
different regions of the neuron are added together
at the trigger zone, the decision-making
component of the neuron.
A second important feature of electrotonic
conduction is its role in the propagation of the
action potential.
Once the membrane at any point along an axon
has been depolarized beyond threshold, an axon
has been depolarized beyond threshold, an
action potential potential is generated in that
region in response to the opening of voltagegated Na+ channels.
This local depolarization then spreads
electronically along the axon, causing the
threshold for generating an action potential.
II. Propagation of the action
potential.
Why does action potential, once
initiated, run the length of the
axon?
Passive electrical properties of a plasma membrane
can be thought of as a simple electrical circuit.
Cable properties
of an axon.
The change in Vm
passively spreads in
both directions
along the axon
Amplitude of the
change decays
exponentially as it
moves away from
its source
Length constant:
- distance over
which the
potential falls by
1-(1/e) or 63%
from its original
value.
- depends on the
rm (resistance of
the membrane)
and the ra
(longitudinal
resistance).
Unlike the passive local current, action potentials
travel down the length of the axon without decrement
How is an action potential propagated along
the length of the axon without any decline in
amplitude?
Hodgkin's undergraduate research project
Hypothesis: The inactive membrane ahead of the
action potential becomes depolarized by
the electronically conducted local current.
Hodgkin's undergraduate research project
Conclusion: the passive cable
properties of the axon permit
the electronic spread of local
currents from areas undergoing
an action potential to inactive
membrane areas ahead of the
action potential.
How does the passive local current bring about an
action potential in membrane areas that are
inactive?
Summary Propagation of an action potential depends on:
1. Passive cable properties of the axons
- Local currents spread electrotonically.
- Distance conducted depends on the resistance of
the membrane and the cytosol.
2. Presence of voltage-sensitive Na+ channels that
respond to the passive depolarization due to the
electrotonically spreading local current.
- this is what is meant by an excitable membrane
- the opening of the Na+ channels with positive
feedback regulation regenerates the action
potential in the inactive membrane area.
Passive Membrane Properties and Axon
Diameter Affect the Velocity of Action
Potential Propagation
1. According to Ohm’s law, I =V/R, the larger the
axoplasmic resistance, the smaller the current flow
around the loop, and thus the longer it takes to
changes the charge on the membrane of the adjacent
segment.
2. Since ΔV = Q/C, the larger the membrane
capacitance, the more charge must be deposited on
the membrane to change the potential across the
membrane, so the current must flow for a longer
time to produce a given depolarization.
Therefore, the time takes for depolarization to
spread along the axon is determined by both the
axial resistance and the capacitance per unit
length of the axon (ra and cm).
The rate of passive spread varies inversely with
the produce racm.
If this product is reduced, the rate of passive
spread of a given depolarization will increase and
the action potential will propagate faster
Rapid propagation of the action potential is
functionally important, and two distinct
mechanisms have evolved to increase it.
One adaptive strategy is to increase
conduction velocity by increasing the
diameter of the axon core. ra decrease in
proportional to the square of axon diameter.
The second mechanism for increasing conduction
velocity by reducing racm is myelination, the
wrapping of glial cell membrane around an axon.
This process is functional equivalent to increasing
the thickness of the axonal membrane by as
much as 100 times.
Because the capacitane of a parallel-plate
capacitor such as the membrane is inversely
proportional to the thickness of the insulatin,
myelination decrease cm and thus racm.
Schwann cell surrounding an individual
axon in a nerve fiber
In a neuron with a myelinated axon the action potential
is triggered at the nonmyelinated membrane of the axon
hillock (Action potential could not be initiated at the
myelinated membrane.)
The inward current that flows through this region of
membrane is then available to discharge the
capacitance of the myelinated axon ahead of it.
Even though the thickness of myelin makes the
capacitance of the axon quite small, the amount of
current flowing down the core of the axon from the
trigger zone is not enough to discharge the capacitance
along the entire length of the myelinated axon.
Myelinated neuron of the central nervous system
The myelin sheath is interrupted every 1 to 2 mm by the
nodes of Ranvier.
The bare patches of axon membrane at the nodes are
only about 2 µm in length.
Each nodal membrane contains a relatively high density
of voltage-gated Na+ channels and thus can generate an
intense depolarizing inward Na+ current in response to
the passive spread of depolarization from the axon
upstream.
These regularly distributed nodes thus boost the
amplitude of the action potential periodically, preventing
it from dying out.
Saltatory conduction:
The action potential jumps from node to node
The action potentials, which spread quite rapidly
between nodes because of the low capacitance of the
myelin sheath, slows down as it cross the highcapacitance region of each bare node.
Consequently, as the action potential moves down the
axon, it seems to jump quickly from node to node. For
this reason, the action potential in a myelinated axon is
said to to move by saltatory conduction.
Because ionic membrane current flows only at the nodes
in myelinated fibers, saltatory conduction is also
favorable from a metabolic standpoint.
Several diseases of the nervous system,such as multiple
sclerosis and Guillain-barre syndrome, cause demyelination.
As an action potential goes from a myelinated region to a
bare stretch of axon, it encounters a region of relatively high
cm and low rm.
For this unmyelinated segment of membrane to reach the
threshold for an action potential, the inward current
generated at the node just before this area has to flow for a
long time.
In addition, this local-circuit current does not spread as far as
normal because it is flowing into a segment of axon that ,
because of its low rm, has a short length constant.
These two factors can combine to slow, and in some cases
actually block, the conduction of action potential.
III Synaptic Integration
Signaling between central neurons is more
complex than that at the neuromuscular
junction:
1) Most muscle fibers are innervated by only one
motor neuron, a central nerve cell such as the
motor neuron in the spinal cord receives
connections from hundreds of neurons.
2) The muscle fiber receives only excitatory input
(there are no inhibitory synapses onto vertebrate
skeletal muscle). Central neuron, on the other
hand, receive both excitatory and inhibitory
inputs.
3) All the excitatory connections on muscle fibers are
mediated by a single neurotransmitter, acetylcholine,
which activates the same kind of receptor-channel;
In the central nervous system the inputs to a single cell
are mediated by a variety of transmitters and any given
transmitter can control different types of ion channels,
some of which are directly gated and some indirectly
gated by second messengers.
As a result, unlike muscle fibers, central neurons must
integrate diverse sets of inputs into a coordinated
response.
4) The synapse of a motor neuron at a muscle is highly
effective – each action potential in a single motor neuron
produces a synaptic potential that is invariable
suprathreshold and always produces an action potential
in the muscle.
In contrast, the synaptic connections made by a single
presynaptic neuron onto the motor neuron are only
modestly effective,and perhaps 50 to 100 excitatory
presynaptic potential must fire together to produce a
synaptic potential large enough to trigger an action
potential.
1. A Central Neuron Receives Both
Excitatory and Inhibitory Signals
1) The excitatory postsynaptic potential (EPSP) produced by the
one sensory cell depolarizes the motor neuron by less than 1
mV, often only 0.2 to 0.4 mV – far below the threshold required
for generating an action potential.
2) The convergence of many excitatory synaptic potentials from
many afferent fibers can be integrated by the neuron to initiate
an action potential
3) Inhibitory synaptic potential, if strong enough, can prevent the
membrane potential from reaching threshold
4) Sculpturing role of inhibition: synaptic inhibition exert control
over spontaneously active nerve cells.
2. Excitatory and Iinhibitory Signals Are
Intergrated into a Single Response by the Cell
1) Concept of neuronal integration
Each neuron in the central nervous system is constantly
bombarded by synaptic input from other neurons.
These competing inputs are integrated in the
postsynaptic neuron by a process called neuronal
integration.
Neuronal integration, the decision to fire an action
potential, reflects at the level of the cell the task that
confronts the nervous system as a whole: decision
making.
2) Axon hillock: readout for the
integrative action of a neuron
In motor neurons and most interneurons the
decision to initiate an action potential is made at
the initial segment of the axon, the axon hillock.
This region of cell membrane has a lower
threshold than in the cell body or dendrites
because it has a higher density of voltagedependent Na+ channel.
The depolarization increment required to reach
the threshold at the axon hillock is only 10 mV
(from –65 to –55mV). In contrast, the membrane
of the cell body has to be depolarized by 30mg
before its threshold (-35 mV) is reached.
Synaptic excitation will therefore first discharge
the region of membrane at the axon hillock.
Membrane Potential (mV)
Spatial Summation
Excitatory
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Excitatory
b
Inhibitory
c
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Spatial
Summation
Time
Spatial
Summation
Spatial Summation
The process by which many presynaptic neurons
acting at different sites on the postsynaptic
neuron are added together.
The length constant of the cell determines the
degree to which a depolarizing current decreases
as it spreads passively.
In cells with a larger length constant the signals
spread to the trigger zone with minimal
decrement
Membrane Potential (mV)
Temporal Summation
Excitatory
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Excitatory
b
Inhibitory
c
d
a
b
c
d
Temporal
Summation
Time
Temporal & Spatial
Summation
the process by which consecutive synaptic
actions at the same site are added together in the
postsynaptic cell.
The time constant of the cell determines the time
course of the synaptic potential and thereby
affects temporal summation.
Neurons with a large time constant have a
greater capability for temporal summation
Time constant (τ): The time taken to reach 63% of the final
voltage .
The time constants of different neurons typically range from 1
to 20 ms
Synaptic Integration
PNS, Fig 12-13
Receptor Potentials and Synaptic Potentials
Convey Signals over Short Distances
Action Potentials Convey Signals over Long
Distances
PNS, Fig 2-11
The Action Potential
1) Has a threshold, is all-or-none, and is conducted without decrement
2) Carries information from one end of the neuron to the other in a pulse-code
PNS, Fig 2-10
Equivalent Circuit of the Membrane and
Passive Electrical Properties
• Equivalent Circuit of the Membrane
– What Gives Rise to C, R, and V?
– Model of the Resting Membrane
• Passive Electrical Properties
– Time Constant and Length Constant
– Effects on Synaptic Integration
•
Voltage-Clamp Analysis of the Action Potential
Sequential Opening of Na + and K+ Channels
Generate the Action Potential
Rest
Rising Phase of
Action Potential
Falling Phase of
Action Potential
Voltage-Gated
Channels Closed
Na + Channels
Open
Na + Channels Close;
K+ Channels Open
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A Positive Feedback Cycle Generates the
Rising Phase of the Action Potential
Open Na+
Channels
Depolarization
Inward INa
Voltage Clamp Circuit
Voltage Clamp:
1) Steps
2) Clamps
PNS, Fig 9-2
The Voltage Clamp Generates a Depolarizing Step by
Injecting Positive Charge into the Axon
Command
PNS, Fig 9-2
Opening of Na + Channels Gives Rise to Na +
Influx That Tends to Cause Vm to
Deviate from Its Commanded Value
Command
PNS, Fig 9-2
Electronically Generated Current
Counterbalances the Na + Membrane Current
Command
g = I/V
PNS, Fig 9-2
Where Does the Voltage Clamp
Interrupt the Positive Feedback Cycle?
Open Na+
Channels
Depolarization
Inward INa
The Voltage Clamp Interrupts the
Positive Feedback Cycle Here
Open Na+
Channels
Inward INa
Depolarization
X
Length constant of the passive local current can be
increased by:
1. Increasing the diameter of the neuron.
- reduces the internal longitudinal resistance
2. Increasing the resistance of the axonal membrane.
- insulating the axon - myelination