I. Length constant:

So far we've been talking about how the concentration gradients and membrane permeabilities of different ions can establish a potential difference across the cell membrane and how changing the relative permeabilities of different ions can give rise to the sequence of potential changes that underlies the action potential. But we've been essentially looking at things as though all potential changes took place simultaneously everywhere along the neuronal membrane

Furthermore, we haven't considered the question of time: how long does it take the potential difference across the cell membrane to change in response to the movement of a current. So today's discussion concerns the two questions: how far does a localized change in membrane potential spread, and how faithfully does the change in transmembrane voltage follow a pulse of current?

The properties that determine this are often called the cable properties of nerve. That's because it occurred to the people who were trying to figure out the passive electrical properties of nerves that an axon was analogous to an undersea cable: both consist of a conducting core, surrounded by an insulator surrounded by a conducting fluid. Since the equations for current spread in undersea cables had been worked out in the 19th century, it made sense to apply them to current flow in axons as well.

 There are two reasons why these passive cable properties of nerve are important physiologically:

1. The first has to do with summation at the synapse. A neuron has many thousands of synaptic inputs, which it it integrates, and if the sum of all the inputs is such that the membrane potential at the axon hillock (trigger zone/initial segment) reaches threshold the neuron will fire an action potential. Given that the inputs are on the dendrites and the cell body, the distance that the potential can spread is critical to determining whether or not the neuron will fire. Similarly, if the voltage change disappeared immediately after the current disappeared, there would be no possibility of temporal summation. However, if the signal outlasts the initiating current, two different synaptic potentials slightly displaced in time can still sum with each other.

2. The cable properties of nerve also play a major role in determining the speed of action potential propagation.

length constant, l: determines how far a passive PD across the cell membrane will spread down an axon (or dendrite, or what have you).

If you have a neuron, and you pass depolarizing current or hyperpolarizing current, i, into it, you can measure a change in membrane potential. And as long as you stay in the range of membrane potentials where you aren't turning on any voltage dependent channels, the voltage change right at the site of current injection will be proportional to amount of current you injected:

The relationship between current injected and the voltage change produced defines the input resistance of the cell. The input resistance depends upon the density of open channels in the membrane of the cell at rest, and on the geometry of the cell.

If you measure the voltage change a little distance away from the site at which the current was injected, you'll find that it's smaller. To use a hydraulic analogy, this is sort of like water flowing out of a leaky hose: lets say you have a hose and you start pumping water in at point 0. Then you punch a bunch of holes down the length of the hose. The pressure differential (and hence the flow of water) will be greatest right near the site at which you're pumping in the water and will get progressively smaller the further you get from the source.

In a cylindrical axon, the decrease in voltage with distance is given by the formula:
 
 

For any particular neuron, l is defined by the ratio of the resistance to ionic current flow across the cell membrane compared to the resistance to ionic current flow down the axon:

where r_m is defined as the resistance to current flow across the membrane of a one cm length of axon (Wcm)

 and r_i is the resistance to current flow down the axoplasm of a one cm length of axon (W/cm)

So the greater the resistance to current leakage across the membrane, and the easier it is for current to flow down the axon, the longer the length constant, and the slower the drop off in voltage.

Both r_m and r_i are influenced by the diameter of the axon: r_m will decrease in proportion to the the circumference of the axon (2pr); while r_i will decrease in proportion to the cross-sectional area.(pr²).

THEREFORE, IF YOU HAVE A LARGER DIAMETER AXON, l WILL INCREASE

Squid: diameter ~ 1 mm; l ~ 13 mm

Frog muscle: diameter ~ 100 mm; l ~ 1.5 mm

Small mammalian nerve fiber: diameter ~ 1 mm; l ~ 0.2 mm

II. Time constant:

It's because of the time constant, t, that

The capacitance of a capacitor is defined by how much charge it will hold for each volt of potential applied to it:
 
 

Capacitance is directly proportional to the surface area of the capacitor, and inversely proportional to its thickness. Because the cell membrane is very thin, it makes a pretty good capacitor.

You can think of a patch of cell membrane as a resistor in parallel to a capacitor, with the resistor part being equivalent to the ion channels and the capacitor part being equivalent to the lipid bilayer:

Under these conditions, it takes time to build up or discharge the charge on the capacitor; the existence of a combined RC circuit adds an element of time to an electrical circuit.

The voltage change across a purely resistive circuit in response to an applied current is instantaneous:

while voltage builds up at a constant rate in response to constant current across a capacitor:

The time constant, t, is defined as the time it takes the potential to rise to (1-1/e) (~63%) of its final value and it is determined by both the resistance and capacitance of the circuit:

With a brief pulse of current across the cell membrane, the pulse may end before the membrane current is fully charged so that the change in potential won't get all the way to its final value. This will result in a shorter effective length constant. So the length constant and the time constant are both important in that they determine whether or not synaptic potentials will be able to sum with each other and bring the initial segment to threshold.

They also play critical roles in affecting the speed of action potential propagation.

III. Cable properties and action potential propagation

Lets go back to l , the length constant and the local spread and decay of passive depolarizing signals. If we inject a little bit of positive current at one end of an axon, we measure the biggest depolarization right near the site of injection, and then the voltage decreases as we move away from the site of current injection, as determined by l, until several mm away, we can't measure any depolarization at all.

But if we depolarize the nerve to threshold and generate an action potential at one end, we find that we can measure an action potential of the same amplitude at the other end of the neuron, and that's what we mean when we say that the action potential is propagated.

Just as the potential difference you can measure across the cell membrane spreads out from a small subthreshold depolarization, it spreads out from a big depolarization like the action potential as well. Only the depolarization from the action potential is so big, -- an action potential can be 90 or 100 mV in amplitude -- that even when the peak voltage across the membrane has dropped off to 37% of maximum (remember, thats how much the voltage decreases in one length constant) the depolarization is big enough for that patch of membrane to reach threshold.

Just how far away the depolarization is able to spread and still remain above threshold -- and therefore how fast the action potential is able to propagate -- depends upon the length constant. In a nerve with a longer length constant, the depolarization can spread farther, the neuron will reach threshold further away and the action potential can propagate faster.

So when you have an action potential, positively charged Na rushes into the neuron through the open Na channels and depolarizes the cell, and a little bit of that + current flows longitudinally down the axon and depolarizes the area of the cell membrane adjacent to the site where the action potential is taking place. So that patch of membrane becomes depolarized to threshold and develops a regenerative increase in the Na permeability and the action potential moves along a little bit.

Why doesn't the depolarization spread in both directions so that the action potential starts propagating itself backward as well?

Myelin:

The action potential is very good at propagating itself and traveling for long distances. But once the current has spread it takes a little time to depolarize the next little patch of membrane, and then it takes a little time for the Na channels to activate in that little patch of membrane; and, with a length constant of only 1/10 of a mm or so, the action potential travels relatively slowly. It moves at a speed of about 1 - 10 meters/sec and that really isn't fast enough to be efficient. In order to increase the speed of action potential propagation you need to increase l, since the farther away the neuron can be excited to threshold, the faster the action potential can travel down the axon.

How could you increasel?

Myelin, the insulating sheath that surround the axon speeds action potential propagation by increasing l (because of the increase in rm); the decrease in capacitance produced by increasing the thickness of the insulating layer around the cell is at least as important in speeding action potential propagation. By limiting the sites at which ions can flow out across the cell membrane, myelin also increases metabolic efficiency.

Saltatory conduction: even myelinating the nerve wouldn't be sufficient to extend l indefinitely; high-Na channel density internodes provide periodic "boost".

For a nifty simulation of action potential generation and propagation, check out this interactive site.

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