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Â Hello.

Â We are now onto the 5th lecture on mathematical models of action potentials.

Â And this lecture is a little bit more conceptual.

Â Rather than rather than getting into details of data.

Â Or rather than getting into details of how you make

Â a mathematical model or how you simulate with a mathematical model?

Â But, even though it's, it's conceptual rather than

Â focus on the details, I think it's really important.

Â And the theme of this is going to be something that

Â we covered a little bit with the cell cycle model.

Â The idea of phenomenology versus mechanism.

Â We talked about the difference

Â between phenomenological representations and processes

Â versus mechanistic representations of processes back in the cell cycle lectures.

Â Now we're going to talk about this some more.

Â 0:54

And you know, the question we want to ask in this case is,

Â is the Hodgkin-Huxley model a mechanistic model, or is it a phenomenological model?

Â And when we discuss this, we'll see that

Â the some aspects of the Hodgkin-Huxley model are,

Â are clearly mechanistic, but other parts of it

Â may appear to be more phenomenological rather than mechanistic.

Â And I think that's common of a lot of models, is, is that the answer is both.

Â There are mechanistic aspects and phenomenological aspects.

Â 1:20

And then we're going to ask this question.

Â If you know what the mechanism is, can a phenomenological model still be useful?

Â What we discussed with the cell cycle lectures is that models often evolve in a

Â way that they start in a phenomenological

Â and then when more biological details are obtained.

Â Then more mechanistic representations of those processes are used so then

Â you, you might think that it never goes in the other direction.

Â And we're going to discuss an example where it went

Â the other direction and that, that's called The Fitzhugh-Nagumo model.

Â 1:52

And what I want to argue is that, usually, it goes in one direction.

Â Usually, phenomenological representations are, you know,

Â begin when you don't know the

Â details of the processes and then those usually become more mechanistic over time.

Â But, I think that moving in the other

Â direction can, in some cases, also be useful.

Â So that sometimes phenomenological models are extremely important

Â and extremely useful, even when the mechanism is well

Â known and that's the argument I want to make,

Â when I discuss the Fitzhugh-Nagumo model in this context.

Â 2:25

When we talk about phenomenology versus mechanism, one

Â questions we can ask in the context of

Â everything we just learned is, is the Hodgkin-Huxley

Â model a mechanistic model or a phenomenological model?

Â 2:37

Personally I think the answer is both.

Â This may be somewhat in the eye of

Â the beholder, other people may have a different opinion.

Â I'll explain to you why I think.

Â The Hodgkin-Huxley model is both.

Â 2:49

There is clearly a mechanistic aspect of this and that

Â Hodgkin-Huxley separated the, the total ion current across the membrane,

Â into a sodium current, and, and into a potassium and

Â in the previous lectures, we discussed how they did this.

Â And this aspect of the, the Hodgkin-Huxley is clearly mechanistic.

Â 3:07

But then other parts of the Hodgkin-Huxley model were more phenomenological.

Â And one example of this are the functions describing alpha as

Â a function of voltage, and beta as a function of voltage.

Â We talked about how these are derived directly from the data, but we didn't talk

Â very much about what the actual equations look

Â like, and the equations look something like this.

Â Beta and of voltage is 4 times e raised to the minus v plus 60 divided by 20.

Â There's no physical basis in Hodgkin-Huxley model

Â for using an exponential function for these.

Â And a numbers such as four or 60 or, or 20 are chosen simply to fit the data.

Â So what we discussed was that you know the the infinity values.

Â And the time constants came directly from the data, and

Â when you got the infinity values of the time constants, then

Â you're able to come up with good estimates of beta as

Â a function of voltage, and alpha as a function of voltage.

Â That's in terms of what data show, but then in terms

Â of what actual function you choose to, to fit those data points.

Â That parts more phenomenal logical and Hodgkin-Huxley

Â just sort of picked whatever worked the fit

Â their alphas and betas and in this aspect,

Â I think that the model is phenomenal logical.

Â Al though as I just discussed and it's clearly mechanistic in the

Â way that it separates ionic current into a sodium component and potassium component.

Â 4:30

we can think about with the Hodkin-Huxley model.

Â And this is when phenomenal begets mechanism.

Â Or this is what discussed in the context of the cell cycle model.

Â Sometimes you start with a phenomenal cycle representation then

Â over time when you learn more it becomes more mechanistic.

Â And a great example of this with the

Â Hodgkin-Huxley model, is with this four particle model.

Â Why do you have n raised to the 4th power here

Â in describing the potassium conductance and the, the, the potassium current?

Â 4:57

As we discussed in some of the previous lectures,

Â it chooses four part, particle model based on curve fitting.

Â But, now we know that I channels have most [INAUDIBLE] ion channels have a

Â tetrameric ion channel structure in particular, these

Â potassium channels are made up of tetramers.

Â So now we have a rigorous physical basis for the

Â fact that this is 10 raised to the 4th power.

Â At the time, this was something that they chose just based on curvenage.

Â Of when a representation in a mathematical method can start phenomenological

Â and then over time when more details learned about the biology.

Â 5:32

you, we, one, one can see that that

Â phenomenological representation actually has a mechanism bases of it.

Â So this is a, a, great example of when phenomenology begets this mechanism.

Â [BLANK_AUDIO]

Â What are the examples we've seen, when phenomenology begets mechanism?

Â First in the Hodgkin-Huxley model as we just discussed the fact that they

Â chose these four particles to describe

Â the Changes in potassium conductance were based

Â on curve fitting, they observed that the rising phase of potassium conductance as

Â a function of time had a delay, whereas the falling phase did not.

Â And this is how they hypothesized that this rising phase

Â looked like an exponential, raised to the 4th power, and

Â when you have, when you're closing the channel, if it's

Â only first particle out of four that has to close.

Â To to get the channel to close, that can explain why the rising

Â phase might have a delay because all four particles need to shift directions.

Â And the following phase, where only one out of four

Â needs to shift, that's why you wouldn't have a delay there.

Â So, this is based on curve fitting, but know

Â we know that this relates from tetrameric ion channel structure.

Â Where the potassium channel here will be a tetramer, one, two, three, four.

Â Each of these will have one alpha helix here the S4 segment that has charges

Â in it and these four S4 segments have to move to get the channel to open.

Â So this is a case where you go

Â from kerfitting or phenomenological representation in the days of

Â Hodgkin and Huxley 60 years ago Now we

Â know that there is a physical basis for this.

Â 7:07

The other example we saw of phenomenology begeting

Â mechanism was with the Novak and Tyson model.

Â Remember that intermediate enzyme here was included in the model.

Â Just so that they could account for the delay.

Â Between the activation of mpf here.

Â 7:20

And the activation of anaphase promoting complex here.

Â So, intermediate enzyme was something that they hypothesised.

Â And they put it into the model.

Â They didn't know what it was.

Â And now we know what the gene is.

Â Or what the protein is that intermediate enzyme corresponds to.

Â And so, I think these are great successes of models.

Â When something is put into the model, this is

Â a way to explain the data in quantitative terms.

Â And then, later, that becomes something that's known to have a biological basis.

Â And this is one of the strengths of models.

Â This is something that, when a model is, is well

Â built and well designed can be one of its great strengths.

Â 8:02

Getting back now to the theme of

Â phenomenology versus mechanism, we just discussed how

Â models can sometimes start out as phenomenological

Â and then become mechanistic with more biological details.

Â Are, are acquired.

Â Hm.

Â What about the other direction?

Â What about when when models start out making this and become phenomenological?

Â And, we're going to discuss a somewhat extreme case here.

Â The Fitzhugh-Nagumo model.

Â This model shows an actual potential here.

Â Voltage is a function of time.

Â And this is a sub threshold stimulus it fails to induce actual potential.

Â This model came about, just with these two equations here.

Â Changing voltage with respect to time.

Â And changing w with respect to time.

Â Where v is a voltage like variable not explicitly voltage.

Â And w is a recovery variable.

Â And this was published in a paper by Doctor Richard Fitzhugh in 1961.

Â 8:58

In a biophysical journal.

Â There are many different manifestations of the Fitzhugh-Nagumo model so if you

Â look up the original paper the equations will look slightly different than this.

Â I just picked these particular equations, because

Â they're very easy to implement and they

Â illustrate the same the same points as any of the other representations we can do.

Â Nagumo model.

Â And what you notice about this is that, this paper was published in 1961.

Â The Hodgkin-Huxley model was published in 1952,

Â so Fitzhugh certainly knew all about the Hodgkin-Huxley

Â model, other investigators knew all about the Hodgkin-Huxley

Â model, over the course of those nine years.

Â 9:31

What I want to argue is that?

Â Even though the Hodgkin-Huxley model was already in

Â the in the public domain and even though

Â it was already in the literature, the Fitzhugh-Nagumo

Â model was also extremely important and extremely valuable.

Â Let's

Â 9:46

think about that, and let's think about, how and why

Â the Fitzhugh-Nagumo model could have been could have been valuable?

Â This is an abstract and very clearly, a phenomenological model.

Â When we discuss Hodgkin-Huxley we had to argue that

Â parts of that were mechanistic and other parts were phenomenological.

Â This one, I think, we can conclude is entirely phenomenological.

Â These are the two equations, dV/dt and dW/dt.

Â It only has two variables.

Â What we discussed with Hodgkin-Huxley in

Â terms of what's actually going on mechanistically.

Â In terms of potassium conductance and sodium conductance.

Â You're certainly going to need more than two variables to describe that.

Â 10:22

This doesn't include any ionic currents, this includes this

Â minus I here as a, as a stimulus current

Â but other than that, in terms of how to

Â 10:29

membrane evolves autonomously after you give the stimulus current.

Â There's nothing in here that's called ionic current.

Â And this recovery variable W.

Â >> Is not related to any specific biological

Â process when we talked about recovery in the

Â Hudgen-Huxley model, we discussed how the refractory period

Â 10:46

result primarily from the recovery of the sodium current.

Â From the inactivation process recovery of the H jig in context to the model

Â here you have a recovery variable but

Â its not linked to any particular biological process.

Â 11:13

now you, can you back to a, a really abstract phenomenal representation?

Â Can it, can this have any value?

Â whatsoever.

Â And what I want to argue is that in this case it did have a lot of value.

Â The Fitzhugh-Nagumo model is a very famous model.

Â The original paper by, by Fitzhugh has been cited something like 18,000 times.

Â It's a.

Â It's a very, very, well respected, well know model.

Â And it has proven to be very valuable.

Â In terms of the simulation and in terms of our understand.

Â And how can this be, when is this coming, after the more mechanistic representation.

Â 11:51

So, this is the question that we want to address.

Â Why would anyone care about a two variable phenomenological model when a

Â more mechanistic, four variable model, in some ways a much better model.

Â When this other model already exists, why would

Â want to, someone want to work with a more abstract version.

Â 12:09

Well, one answer here is that, this occurred in the pre-digital era.

Â In these days, as we've seen and as you've seen yourself doing your homework.

Â It's really easy to write down a set of differential equations and

Â implement them and, and run simulations using a program language such as MATLAB.

Â That's because we have very powerful computers.

Â Here in the 21st century.

Â Working back here in the 1950's and in

Â the 60's, they didn't have such powerful computers.

Â 12:35

In fact, Hodgkin-Huxley, solved a lot of there equations using a calculator,

Â and it took them a really, really long time to do this.

Â So back in those days, back in the 50s and 60s.

Â There was value in simplifying.

Â There was value in going from a four variable model to a two variable model.

Â The reason this is called a

Â Fitzhugh-Nagumo model, it's not because Fitzhugh-Nagumo published

Â a paper together where they described this model, that was the case in Hodge-Huxley.

Â What happened in this case for the historical lesson is that

Â Fitzhugh published a paper that we just discussed in the last slide.

Â And then a year later, Nagumo and

Â coworkers, published another paper, here, but, so,

Â the two models were very similar, and

Â therefore they're described as the two Nagumo model.

Â And with Nagumo, Nagumo's model, in fact, it wasn't

Â just a set of equations that could be written down.

Â And then solved on a computer.

Â He actually implemented this in a box he used this device called

Â the tunnel diode and a tunnel diode has a cubic current voltage

Â relation similar to you may remember it in the Tunic Gilmo equations

Â there was a voltage to the three to the third power term.

Â 13:47

And so these tunnel diodes have a very unusual relationship between how

Â they pass current related to the voltage that you apply across them.

Â And, so in these days Nagumo's

Â representation was actually a box where you

Â could turn the dials and you could get your, your output like that.

Â And in those days when it was really

Â difficult to implement these on digital computers, and you

Â had these super computers there was great value in

Â having something that you could solve in a box.

Â So, this was one of the reasons why the Fitzhugh-Nagumo model was very valuable.

Â 14:35

And I think that had to do with

Â the fact that they went specifically with two variables.

Â I think if Fitzhugh-Nagumo, had gone from a

Â 20 variable model to an eight variable model.

Â It would have helped a lot in terms of the computing

Â power and computing speed, but it wouldn't have had the same impact.

Â 14:52

So what it, what's so special about going

Â specifically to a two variable model in this case?

Â We'll, we've already seen this in our lecture's name dynamical systems.

Â When you have two variables, you can plot things in the face plane.

Â You can plot one variable on one axis.

Â The other variable on the other axis, and you can visualize how the system moves,

Â visualize trajectories, and you can look at things

Â like, like fixed points, and analyze them graphically.

Â That's exactly what Fitzhugh did.

Â He applied it nullclines.

Â Calculate the voltage nullcline.

Â W equals voltage minus voltage to the third minus I.

Â And you can calculate the W nullcline, with

Â this equation here, and then you can plot that.

Â Black line here represents the voltage nullcline,

Â and the red line represents the W nullcline.

Â 15:39

And then, once you have the your nullclines plotted in the phase

Â plane, as we discussed in the lecture on, lectures on dynamical systems is.

Â You can, you can apply arrows you can say

Â in this quadrant the system is moving this way,

Â up here it's moving up and to the left, here it's moving down and to the left etc.

Â And those are the things that Fitzhugh did as he

Â said, okay where is my fixed point going to be stable?

Â When is it going to be unstable?

Â Which way is the system going to move etc?

Â 16:06

And i think this even more than the computing speed issue.

Â Is why the Fitzhugh-Nagumo model proved to be so influental.

Â It's because it was specifically two variables.

Â And because things could be plotted face plain.

Â This allowed for, for more conceptual insight using these graphical methods.

Â And conceptual insight into process such as [INAUDIBLE] sub threshold stimuli.

Â Super threshold stimuli etcetera.

Â 16:41

And how potting mill times and looking at things in the face

Â plan, and the idea of diamical systems can help us to understand these.

Â First let's look at what happens with the an electrical stimulus.

Â And we're going to represent an electrical stimulus here

Â as an instantaneous increase in the variable V.

Â So, there are are nullcline where we have voltage.

Â Voltage nullcline.

Â These are all the points where dvdt equals 0

Â and these are all the points where dwdt equals 0.

Â And if we

Â 17:09

have a small increase in V.

Â Then what we move from this point here from our fixed point, to this point to

Â the V-W nullcline, and then If we run a simulation we see that it goes back.

Â It goes back to the stable fixed point.

Â 17:41

And now what happens is our system moves like this, and now we have that.

Â Instead of both voltage decreasing afterwards, voltage increases.

Â And how do we know voltage is going to decrease?

Â Rather than increase after the stimulus,

Â well that's because we've crossed the nullcline

Â remember the blank nullcline here is a set of points where dbdt equals 0.

Â So, clearly at this point here, after we

Â release the stimulus, we'll just decrease, at this

Â point because we've cross the nullcline, welshes is

Â going to increase, so this so plotting nullclines.

Â Helped Fitzhugh get insight into where your threshold occurs, and why,

Â with a small increase in v, you return to the fixed point.

Â You return to, again, in physiological terms,

Â you return to around minus 60 millivolts.

Â Whereas in this case, voltage continues to increase.

Â Eventually w increases as well, and then you come

Â back to the fixed point, after the voltage decreases.

Â And in this case, when Fizhugh ran a simulation with

Â his equations, he saw something that looked like actual potential.

Â So, by reducing from four variables in the case of Hodgkin-Huxley to two

Â variables in the case of Fitzhugh-Nagumo, we're able to represent things like.

Â This is a sub-threshold stimulus, and this is

Â a super-threshold stimulus that induces an action potential.

Â 19:03

One of the other analyses that Fitzhugh did that was really cool

Â was he said, what happens when we, when we have constant current injection?

Â How does that change the stability of our fixed points?

Â Constant current injection in this case

Â corresponds is going to correspond to negative

Â value of I, and if we look at the equation for our v

Â nullcline over here we see w equals v minus v cubed minus I,

Â so if I is a negative number then you subtract the negative number.

Â That's going to be like adding to it, so your voltage

Â nullcline is going to occur at higher values of w.

Â In other words it's going to shift the voltage nullcline up.

Â 19:37

And this is what we see when we have constant current injection.

Â In this case, I equals minus 7.

Â The black line here, the nullcline has shifted up compared

Â to where, where it was before with our previous example.

Â 19:57

And when this fixed point becomes unstable, what do you have?

Â Instead, instead of a stable fixed point you

Â have a stable limit cycle, and what this model

Â is going to do is it's going to to continually

Â go around this trajectory, again and again and again.

Â 20:10

And if you taught voltage is a function of time, rather

Â than voltage and w together you'd see something that looks like this.

Â One [UNKNOWN], another one, another one, another one.

Â And that occurs because our fixed point.

Â Does now, become unstable.

Â So, in other words, what Fitzhugh was able

Â to conclude in the context of dynamical systems,

Â is that repetitive action potentials with constant current

Â injection, and this is something they haven't seen experimentally.

Â This is equivalent in dynamical systems terms

Â to conversion from a stable fixed point.

Â To an unstable fixed point and when you had

Â an unstable fixed point, you had a stable limit cycle.

Â So this is analogous to what we saw

Â before, with the beer model of yeast glycolysis,

Â where we could either have a stable fixed

Â point or we could have a stable limit cycle.

Â The is a stable limit cycle.

Â 21:12

To summarize then, what we seen is that the

Â Hodgkin-Huxley model and this is true of most mathematics model.

Â It contains a mixture of some mechanistic elements.

Â Some phenomenological elements.

Â 21:27

We've also seen that when a phenomenological

Â representation, is later found to have a rigorous

Â mechanistic basis, this is something that can be

Â considered a, a great success of the model.

Â We saw this in the case of the cell cycle model.

Â With intermediate enzyme and we saw this in the case of the Hodgkin-Huxley model.

Â Where they had four particles describing

Â the, changes in the [INAUDIBLE] endocrines.

Â And those four particles were later shown to, have an actual physical basis.

Â 21:54

However, I don't want to leave you with

Â the impression that things always go in that direction.

Â They usually start off as phenomenal logical and then become

Â more mechanistic over time, but even when mechanism is known phenomenal

Â logical representations can none the less be useful, because they

Â often time provide very general, very abstract insight into, into phenomena.

Â And one prominent example of this.

Â Which was the Fitzhugh-Nagumo model.

Â So remember that even the, though the Fitzhugh-Nagumo model

Â came after the Hodgkin-Huxley model and even though it was

Â an abstract and phenomenological representation,

Â it was nonetheless, very significant

Â and very important, for the general insight that it provided.

Â