Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.
Equations for alphas and betas
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Bioelectricity: A Quantitative Approach
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Hodgkin-Huxley Membrane Models
This week we will examine the Hodgkin-Huxley model, the Nobel-prize winning set of ideas describing how membranes generate action potentials by sequentially allowing ions of sodium and potassium to flow. The learning objectives for this week are: (1) Describe the purpose of each of the 4 model levels 1. alpha/beta, 2. probabilities, 3. ionic currents and 4. trans-membrane voltage; (2) Estimate changes in each probability over a small interval $$\Delta t$$; (3) Compute the ionic current of potassium, sodium, and chloride from the state variables; (4) Estimate the change in trans-membrane potential over a short interval $$\Delta t$$; (5) State which ionic current is dominant during different phases of the action potential -- excitation, plateau, recovery.
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Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics
Biomedical Engineering, Pediatrics
So hello, once again, this is Roger Coke-Barr talking for the Bioelectricity
course, in Week four Topic six. In this topic we talk about equations for
the alphas and the betas, but let's back up just a moment and say where we are.
We are trying to explain the big deflection from active tissue into action
potential. We saw that Hodgkin and Huxley performed a
series of experiments in the giant axon of the squid.
They expressed the result for individual ion channels in terms of probabilities.
And the probabilities increased or decreased as time went by.
And in particular they increased and decreased according to rate constants that
were called alphas and betas. The alphas and betas were determined
experimentally from the squid and that's what we're talking about in this segment.
Before talking about the alphas and betas specifically, we need to go back and
review what is the difference between Vm with a capital and vm with lowercase.
This is important, because it comes into the way that alphas and betas are given to
us by Hodgkin and Huxley, and how they expressed it in their experimental
results. It's not very hard, but it is confusing if
you don't understand the difference. Vm with a lowercase.
This vm is measured against the baseline of the recording.
So this much is vm. So at point A, vm is equal to 40.
Now on the other hand, capital VM is measured with respect to a voltage of
zero. So at point A, this much would be capital
Vm. So at point A, lower case vm the diff, the
voltage off the base line is plus 40. While capital Vm the voltage reference to
zero is minus 30. Now obviously one is related to the other.
So that we have as a matter of mathematical fact, VM relative to the
baseline is capital VM. The absolute VM minus the voltage of the
signal when the waveform is at rest, VR. Here's a few comments on while there is
both of, both capital VM and lower VM. Vm with the lower case V is much easier to
measure experimentally, because if you have the trace of voltage versus time, you
can take the actual recording, mark the base line, mark another point and just
find the voltage difference. Whereas, on the choice of voltage versus
time recorded from the squid, there's not going to be any line for zero.
The reason there is not any line for zero is that the base line zero, or that is to
say the zero value, is affected by the drift in the equipment.
In measuring DC voltages. In preparations such as the squid is
notoriously difficult, so when you do try to measure capital VM, it is often the
case that it is affected the measurement by what is called DC drift.
Or, in other words, for the value of voltage equal to zero to move up and down
the page a little bit as time goes by. Then I should call your attention to the
fact, that here I am trying as much as possible to be consistent, and call
voltages. That are measured relative to the va-,
baseline with a lower case vm. And the voltages that are absolute, to
call those uppercase VM. I'm trying to maintain that difference
consistently. And I hope I do it without making too many
mistakes. If you take the world around us, the big
wide world. I would say everyone recognizes everyone
in electro-physiology, recognizes that sometimes measurements were given with
respect to the base line and sometimes are given absolute, but the way they are
denoted changes from an author to another. So they may or may not follow this lower
case capital VMI notation, notational difference.
Now, in a minute, we're gonna look at the actual alpha and beta functions.
But let me give you a few preliminaries. As we said, the basic idea of the alpha
and beta functions are to measure how fast, for the alpha function is to measure
how fast the n, m, and h particles are opening.
Or, putting that a little bit differently, to measure the rate at which the
probability is increasing. If you go to Washington D.C., and ride on
the metro train in Washington D.C., one hears the phrase, doors are opening, each
time the doors open. And in a way that's a corollary.
Alpha is telling us, doors are opening, and here's the rate at which that's
happening. In contrast, the Beta function.
We're telling about closing, and they're knowledge to the subway is saying doors
are closing. How fast is the probability that n, m, and
h are moving into the closed position. In both cases both Alpha and Beta these
are quantities that have units, and as rates the units that are used there are
per millisecond. Now they could be expressed in some other
units, of course. But as Hodgkin and Huxley wrote them down,
they are per millisecond. And most people have just copied the
Hodgkin and Huxley expressions, so those are per millisecond too.
The only thing is, when they copy the Hodgkin and Huxley expression, they don't
always remember to copy the units. So you see these equations out there
sometimes, as just free standing entities, as if they did not have units.
But in fact, they do. It's worth making a note that although the
alpha and beta functions are expressed as equations.
Even though they are presented as mathematical functions, they are, in fact,
experimental results. You might say a part of the genius of
Hodkin and Huxley is they took their experimental results and presented them to
us as mathematical functions which allows us all to use them again very easily by
copying those experimental functions. It is also true that these values of alpha
and beta are ones that are measured for squid axon.
That is, they're not the right opening and closing functions for say, electrically
active muscle in the human heart. Those may have comparable functions of
alpha and beta, but they won't be the same.
Saying once again, the alpha is the turn on function and their beta is the turn off
function. There's an alpha and beta for each of the
three probabilities N, M and H.The fact that there is an alpha and beta for each
of them leads people to think in their minds that these actual functions are
probably pretty similar. Well that's just not true at all.
They're notably different, one from the other.
Even when the expression looks somewhat similar, they really are quite different
functions. That is to say the values of the
functions, at different values of Vm, are, changes.
Quite remarkably from one alpha to another alpha to another alpha.
Since there's a turn on function for each of the particles and M and H and also a
turn off function, then all together there have to be six functions in all and then
indeed that's what's given. So we'll be looking at the expressions for
all six functions putting them on slides, but then looking at them rather quickly.
Now here's another little detail, which has ensnared many people in the modern
era. It is important to realize that when
Hodgkin and Huxley did numerical calculations, they did their numerical
calculations manually. And that meant that when they did the
calculation, the calculations. Have some singular points.
Some points where the functions are undefined and they just fix that up.
To them, it was not a big deal because they knew the values that should go there
and they just put them in because they were doing it manually.
They rarely even mention it. On the other hand, when one codes these
functions for computer evaluation, as is now done most of the time.
Most alphas and betas nowadays come out of a computer program.
You have to watch out for these singular points.
And you have to be, in particular, sure that you code something in there to take
care of the case where the denominator of the expression goes to zero, which, from
the time to time, it does. Finally, you have to keep in mind that the
alpha and beta expressions as written, are done in terms of vm, little vm, the value
off of the baseline. They are not done in terms of capital VM,
the value absolute. The reason for this historically, is that
Hodgkin and Huxley could not measure capital VM.
Their equipment was not stable enough to do that measurement.
The ability to do that measurement reliably has, been developed in the 50
years since they did that original work. Therefore they did what they could and
that is to say what they could most recently and most readily.
They wrote the expressions based on the data they had which were VM displacements
while I take to the base one. We talked about earlier what the
difference was in capital VM and lower VM. So you know what that is.
Okay, so now let's look for the specific equations for alpha N and beta N.
As you know, N is the probability that a sodium channel is open, or out a large
number of sodium channels, a fraction of sodium channels that are open.
And the alpha function. It's given, right here.
The beta function, right there. If we look at the alpha function in the
sense that we were just describing it, we see that there is a point at which the
denominator is going to go to zero and in particular is going to go to zero when VM
is equal to ten. So when VM is equal to ten, the numerator
there is zero, the, the e to the power goes to one, we subtract one so the
denominator goes to zero, and all these other if statements, here.
Or intended to take care of that fact. The beta function, which does not help
that issue, is right there. For alpha M and beta M, the expression is
shown here. Here for our alpha.
Here for beta, as given to us by Hodgkin and Huxley with this other code intended
to take care of this special case, if present.
And for alpha H and beta H. We are very happy to see, that there's no
special case that has to be taken into account.
Should say no. Special case.
So here we have the alpha expression, and here we have the beta expression.
If you look at the actual expressions here for H, then you remember the VM is a,
always a positive number during an action potential.
Well, almost always a positive number. The voltage rises out of the baseline.
Then you say, what do these expressions mean, in terms of the channel?
You'd say, well, what alpha H is telling us, looking at this argument, is that as H
goes up. Alpha h goes down.
So it turns on less and less the higher the VMS, and conversely for beta, we'd
say, if you look at the way that expression is going to operate, as Vm goes
up the denominator's going to get smaller and smaller, so as VM goes up, beta H is
gonna go up. So the turn off function is going to get
to be higher and higher with increasing VM.
So you can see what the H channel is going to do.
As the voltage grows higher, turning on grows less and turning off grows more.
So more and more of the H channels are gonna turn off as VM gets bigger and
bigger. And indeed, that is what happens.
While I am in this section of showing specific experimental details I thought it
might also be useful to put in a slide that shows the parameters that normally
are present. In Hodgkin-Huxley membrane, by normally
are present, I put these in as values that Hodgkin and Huxley associated with normal
squid axon function, so you can use them as beginning points.
Here are the conductivities, and here are the Nernst potentials.
And the resting potential is usually about 60 millivolts absolute.
In the slide, also, I've written in some other values which are interesting to
think about in the context of various applications.
If you ask in this tissue what is it that changes dynamically?
Well what changes dynamically are the parameters VM, MM and H.
Here are some sample values for those. Done under the conditions.
The stimulus currents are equal to zero. Here's another set of sample values.
These are just put out as specific examples of two situations.
And they are referred to later on in the sections of the course.
Thank you for watching. We move on now, and talk about how this
plan comes together. I look forward to talking to you again in
the next segment.
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