So far, we have learned

that the cell membrane contains ion channels of selective permeability

for specific ions. Ionic currents flowing across

the cell membrane can then drive membrane potential changes that are filtered, in time,

by the membrane capacitance. We’ve considered the cell

to be small and round, and therefore,

isopotential — that is, all parts of the cell membrane have

the same electrical potential. And, indeed, many types

of cells in our body are small, round, and isopotential. However, this is very far

from the situation when it comes to considering neurons. Neurons, in fact, have

very extensive arborisations. These outgrowths from the cell body are filled with cytoplasm

and covered by a plasma membrane, just like the cell body,

but much thinner. Whereas the cell body typically has

a diameter of 10 microns, your neuronal arborisations may have diameters of the order

of one micrometer or less. Typically, these branches of the neuron

extend for hundreds of microns. In the largest arborisations,

the process can extend several meters. Think of neurons in the giraffe brain

that need to communicate along the length of the spinal cord. It turns out that the neuronal arbors

can be considered as leaky electrical cables

with capacitance. These cables transmit and transform

the electrical signals generated in one part of the neuron

to other parts of the neuron. The most important point to realize is

that the membrane potential at different locations

across the neuronal arborisation will be different. Membrane potential can fluctuate strongly

in one part of a neuron and might make little impact

upon other regions of the neuron. To understand why, in this lesson

we’ll study the basic principles of leaky electrical cables

with capacitance. The simplest analogy to think about

in terms of neuronal arborisations and ionic current flow

in these arborisations is to think of a watering hose

as you might use in your garden. Water influx across this leaky cable

will be high at the input end, and if we make small holes

in this garden hose, water will leak out

along the length of this tube. The pressure at different points

in the tube will differ. Here we’ll have a high water pressure, and as the water leaks out,

at the far end of the water hose you’ll have a lower pressure. We can think of the same thing

in terms of the arborisation. We have intracellular cytoplasm, we have axial current flow,

ionic flow of electricity down the length

of the center of the arborisation. This is flanked by

lipid bi-layer membranes, and part of that current flow

will leak out across the ion channels

that are present in the plasma membrane. The current flow that enters here

will therefore decrease in each part of the cable by the amount of current that leaks out

across the plasma membrane. In addition, of course,

we must remember that the lipid bilayer is

an important capacitor, and we therefore need to think

of capacitance of the axial current in part goes to flow

across the ionic channels, in part charges the membrane,

and in part drives the additional current flow

down the length of the arborisation. We can draw the electrical equivalent

of any small length of this cable through thinking about the capacitance

across that unit length provided by the surface area

of the lipid bilayer. There’s the membrane conductance provided by the numbers

and types of ion channels present in that part of the membrane, and of course,

we have the axial current flow that flows down the length

of the arborisation, changing potential depending

upon the axial resistance that’s offered to that current. In general, we can write cable equations that describe this electrical situation. The simplest one is

the steady-state cable equation where we forget about

time-dependent changes and if we forget about

time-dependent changes, we don’t need to consider capacitance because current only flows

into a capacitor during charging, when which is obviously not

at steady state. The membrane potential

at any given point in this cable is therefore given by

the trans-membrane current multiplied by

the trans-membrane resistance. Ohm’s Law: V=IR. We can also see that the drop in potential

between adjacent points in the cable depends upon the axial resistance, and the change in potential across

arbitrarily small areas of the cable depend upon the current

and the resistance, axially. Finally, we can see that

the change in axial current depends upon how much of that current

leaves across the plasma membrane, and so, the change in axial resistance is

equal to minus the membrane current. We can take these three equations

and solve it for the cable equation in one dimension at steady state. We begin by Ohm’s Law. From this equation we can see that Im can also be written as

minus dI Axial over dx. And so, the membrane potential is

therefore equal to: minus Rm, dI axial by dx. The axial current we can rewrite here as being one over R axial, dv by dx. Differentiating this one more time gives us the second derivative here

with respect to space. We can then put this equation in here, and finally we solve for Voltage

as being the ratio of the membrane resistance

to the axial resistance times the second derivative

with respect to space. This second order

differential equation has a solution with an exponential form with respect to space,

length down the cable, and we introduce now the length constant which is equal to the square root

of the membrane resistance divided by the axial resistance. We can plot the exponential drop-off

of voltage across the length of the cable here as the exponential decay

of the membrane potential as the current traverses

this leaky cable at steady state. The distance where

the potential has dropped to 63% — that is, 1/e — is equal to the length constant

of the cell, Lambda. Lambda defined as the square root of membrane resistance

divided by axial resistance has some obvious intuitive properties. If we were to increase

the membrane resistance, there would be less leak of current

across the length of the cable, and the voltage would therefore

drop less across space. Conversely, if we had

a very high axial resistance, then the amount of current

that would leak out would increase and it would be more difficult

for current to flow down here, and we would therefore get

a more sharp drop in the membrane potential. Lambda would be low. The steady-state cable equation

can be rewritten to include the time-dependent charging

of the membrane capacitance. As we saw last week, there’s a time constant involved

with charging the membrane that’s equal to the product

of the membrane resistance and the membrane capacitance, and that can then be included in

this larger partial differential equation that describes the one-dimensional,

time-dependent spread of voltage across space. We have two important constants: we have a length constant that indicates the length scale over which

the membrane potential decreases, and we have another important constant, the time constant, that tells us over what time scale

the membrane potential is filtered as it traverses the neuronal arborisation

in space and time. It’s worth noting that these constants are

actually variable in time and space. The membrane resistance depends upon

how many ion channels are open

at any given time, and in general, the open probability of ion channels is

something that’s highly regulated, and so, the membrane conductance

and membrane resistance varies considerably over time; in fact, that’s what drives

the membrane potential changes in a cell. The axial resistance may not

change over time, but it certainly changes over space, and so, very thick arbors of the neuron

which have large diameters have low axial resistance, whereas the very thin arborisations

that might be far from a neuron might have much higher axial resistances and so the membrane time constants

are not constant across

the neuronal arborisation in space or time, leading to a great deal of complexity. In general, it’s fair to say

that there are no analytical solutions to the cable equation

for real neuronal structures, and therefore,

numerical computer simulations are therefore typically used, and one that’s particularly used often is the one called Neuron, which is freely available for download

at this World Wide Web address, and you can download it

and play with these simulations and see for yourself

how complicated and interesting the spread of neuronal membrane potential

is within individual neurons. So in terms of

membrane potential distributions, neurons are rather complicated. The membrane potential at one point

in a neuron differs from that in a different point of a neuron. The membrane potential is filtered as current flows down

the neuronal arborisations because there’s some leak

across the plasma membrane. Equally, membrane potential changes are

highly filtered in time. A membrane potential change

that occurs rapidly at a distal part

of a neuronal arborisation may have very little impact

upon the cell soma. In order to get a better feel

for how this works, let’s have a look at some example neurons

and how membrane potential varies across the neuronal arborisation. Let’s first think about membrane area. In the distal processes of a neuron, the diameter might be 0.1 micrometers or one micrometer,

or somewhere in between. The largest diameters are

present at the cell body, where the nucleus

and the genetic material is also located, and there we have diameters

of around ten micrometers. So there’s an approximately

hundredfold difference in the diameter of the structures that are located

at different points across a neuron. The surface area of a cable is approximately proportional

to the diameter of that structure. and the surface area,

of course, also for a capacitor, tells us how much capacitance there is, and so, the local capacitance at the soma might be a hundred times higher than the local capacitance

sitting here in a dendrite. A given amount of current or charge

flowing into a small dendrite here with a small amount of capacitance

might then give rise to a very large voltage change, whereas the same charge

flowing into the cell body will give rise to a smaller change

in membrane potential. One might, therefore, imagine

very large potential differences across time and space here

as conductances open here, compared to slower and smaller

membrane potential changes at the cell body, where there’s

considerably larger filtering due to the large amount of surface area

immediately available to this region. Let’s consider a situation where we have

three recording electrodes placed across the arborisation

of a given neuron. At each electrode we could record

the membrane potential at that location, and let’s see what happens

if we inject current into Location One. Let’s take a given point in time

where we begin to inject current. At recording electrode one,

location one on the dendritic arborisation we’re very close

to where the current is being injected. There’s only a small surface area here. The membrane is rapidly charged

and reaches steady state. Current then begins to flow

and charge the capacitors along the way and gradually, we charge location two and location three. And each time, there’s some delay that increases as the current

needs to flow down here, and the rate of charging slows down

because the overall amount of surface area that needs to be charged has increased. The steady-state voltage that’s reached

at these different locations is described by the steady-state

cable equation as we saw before, where we have our length constant Lambda that tells us how much of a drop

in membrane potential we might expect. Now, in real neurons,

the sizes of this dendritic arborisation might be hundreds of micrometers, and that’s exactly the length scale that

also the membrane length constant has. And so if we imagine

that there’s 500 microns between recording electrode one

and recording electrode three, we can then say that this change

in membrane potential will just be 37% of the steady state membrane potential

reached here at this point in the neuronal arborisation. The length constant of neurons, then being on the same scale

as the arborisation of a neuron, we realize that there’s

a considerable impact upon the steady-state membrane potential

across this neuronal arborisation. The situation, however,

becomes even more extreme when we think about brief current pulses. Again, we consider injecting current

here into electrode one, but now just a brief current pulse. We charge the membrane,

and then it relaxes, because of the current spread

rapidly flowing away from that location. What we see at electrode two, however, is a highly filtered version of this

which is much slower, and electrode three,

if we see anything at all, is again slower and smaller. The half-time of these membrane

potential changes is very different comparing these different locations. The amplitude might be forty times larger at distal dendrites compared to what we see at the soma, and equally, the time course

here, which is fast, will be much more extensive at these

small membrane potential fluctuations. Here at the soma,

we may have around ten milliseconds in terms of the half-width

of membrane potential changes, whereas here, at the distal dendrites,

we probably have time constants that are much closer to one millisecond. When currents and conductances

are opening at different times across different locations

of a neuronal arborisation, you can see that there will be

very complex spatiotemporal dynamics where the fast, large signals

taking place here are strongly filtered, giving rise to smaller amplitude,

longer duration potentials at the soma. Membrane potential is, therefore,

clearly very complex in real neurons. So in this lesson we’ve learned

about the complexity of neuronal membrane potential. Neurons have extensive arborisations

that electrically can be considered as leaky cables

with significant capacitance. Membrane potential, therefore, has

complex spatiotemporal dynamics within single neurons,

with significant attenuation and filtering of the signals

across the arbors. In general, this leads

to a complicated situation in terms of information processing, and next week we’ll see

that the nervous system has come across at least one solution

as to how to make reliable signaling across large distances in the form

of the axon potential. That, we’ll discuss next week.