Passive Electrical Properties of the Neuron

Reference: Eric R. Kandel: Essentials of neural Science and Behavior. P149 - 159

I. Equivalent Circuit of the Membrane andPassive 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

Equivalent Circuit of the Membrane andPassive Electrical Properties

Ions Cannot Diffuse Across the Hydrophobic Barrier of the Lipid Bilayer

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- - - -

Vm = Q/C

∆ Vm = ∆Q/C

The Lipid Bilayer Acts Like a Capacitor

∆ Q must change before∆Vm can change

The voltage (Vm)across a capacitor is

proportional to the charge (Q) stored on the capacitor:

Capacitance is Proportional to Membrane Area

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Vm = Q/C

The Bulk Solution Remains Electroneutral

Electrical Signaling in the Nervous System isCaused by the

Opening or Closing of Ion Channels

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The Resultant Flow of Charge into the CellDrives 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

RT [K+]o

zF [K+]i

• lnEK =

γk , conductance of one k+ channel

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

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The K+ Battery Predominates at Resting Potential

gK

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1 10 100Extracellular potassium concentration (millimoles)

Membranepotential

(millivolts)

-130

-60

+10

(Red line shows valuesaccording to Nernst equation)

Experimental points

-70

5

[K+]o = 4 mmol.l-1

• 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

Equivalent Circuit of the Membrane andPassive Electrical Properties

Passive Properties Affect Synaptic Integration

Experimental Set-up forInjecting Current into a Neuron

Equivalent Circuit for Injecting Current into Cell

Im total membrane currentIi Ionic membrane currentIc 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

Rin x Cin

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 toVm Across the R

∆ Vm = ∆Q/C∆ Vm = IxRin

In

Out

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 = rax

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 = √rm/ra

ra = p/(πa2) rm = Rm/2πa

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.

√Rma/2p =

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 voltage-gated 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

- 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).

Length constant:

Unlike the passive local current, action potentials travel down the length of the axon without decrement

Hodgkin's undergraduate research project

Hypothesis: The inactive membrane ahead of the action potential becomes depolarized by the electronically conducted local current.

How is an action potential propagated along the length of the axon without any decline in amplitude?

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 high-capacitance 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 voltage-dependent 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.

Excitatory

Excitatory

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Spatial Summation

Spatial Summation

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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

Excitatory

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Temporal & Spatial Summation

Temporal Summation

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Temporal 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

1) Has a threshold, is all-or-none, and is conducted without decrement2) Carries information from one end of the neuron to the other in a pulse-code

The Action Potential

PNS, Fig 2-10

• 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

Equivalent Circuit of the Membrane andPassive Electrical Properties

Sequential Opening of Na + and K+ Channels Generate the Action Potential

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Rising Phase ofAction PotentialRest

Falling Phase ofAction Potential

Na + ChannelsOpen

Na + Channels Close;K+ Channels Open

Voltage-Gated Channels Closed

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A Positive Feedback Cycle Generates theRising Phase of the Action Potential

Depolarization

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Voltage Clamp Circuit

Voltage Clamp:1) Steps2) 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 ClampInterrupt the Positive Feedback Cycle?

Depolarization

Open Na+

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Inward INa

The Voltage Clamp Interrupts thePositive Feedback Cycle Here

Depolarization

Open Na+

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Inward INa

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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