Building a virtual neuron â part 2
Image credit: Ionut Stefan
Itâs been a tad longer than I intended since our intro on differential equations came out, but hopefully that means you had some extra time for memory consolidation. Otherwise, you can refresh your memory here. Today itâs finally time to tackle the long-awaited virtual neuron. But before we jump in, we need to have a quick housekeeping chat. As you can already glimpse from the list below, we mean business this time, so I strongly recommend that you read this article in chunks. Then again, Iâm just a disembodied voice on the Internet and I canât tell you what to do.
- Defining the goal
- The plurality of virtual neurons
- The foundation model
- From cats to neurons
- Small detour # 1: hypothesized, but not modeled
- Small detour # 2: positive, yet negative?
- From neurons to circuits
- From measurements to interventions
- Virtual neuron 1 â linear, boring, and instructive
- Virtual neuron 2 â to the moon and beyond
- Small detour # 3: that Nernst guy
- Back to virtual neuron 2
- Virtual neuron 3 â other ions have joined the party
- The full model â everybody gets a function
- Still alive?
Defining the goal
First, we need to understand what we want to do. âBuilding a virtual neuronâ sounds cool (well, about as cool as math can ever sound), but it tells us surprisingly little about the task. We need to define the level at which we build this neuron. Do we want to simulate every protein and ion, and all their interactions? I mean, maybe. I admit that does sound pretty cool, but would we be able to interpret the results? My computational neuroscience professor used to say: âIf you build a simulation as complex as the system youâre studying, you now have two systems you donât understand.â And leaving that aside, could we even construct such a simulation right now? Well, no, not really. So instead we need to define three things:
- what we want to do;
- what we can and want to get out of it;
- what we can realistically accomplish.
For today, we want to build a model capable of producing action potentials, just like real neurons (1). We want to use this model to understand how neurons produce these potentials and how they are affected by both external stimuli and ion channel properties (2). And we can realistically accomplish this with a run-of-the-mill laptop and our own brains (3).
The plurality of virtual neurons
There isnât just one single way to simulate a neuron. In fact, there are a lot of options. If you donât believe me, have a look here. Choosing a computational model is an act of balance between complexity and efficiency. On the one hand, we want something complex enough to capture what weâre interested in: for example, if we want to know what happens to a neuron when we mess with its calcium channels, we need a model that includes them. On the other hand, this model needs to run on the available hardware and we should be able to make some sense of its results. So if we only care about calcium channels, itâs not such a good idea to include 300 other types of ion channels.
The foundation model
For today, Iâve chosen the Hodgkin-Huxley (HH) model. As some of you might already know, this is kind of the bedrock of modern computational modeling, and often the first boss you will encounter if you ever attend such a course.
While arguably not the first computational model, the HH was pioneering as a quantitative, dynamic, biologically grounded one, and it remains remarkably elegant to this day. Of course, now itâs quite easy to look at it and think âwell, big whoop, we already know how action potentials workâ. But given the limited amount of information Hodgkin and Huxley had available at the time, itâs nothing short of fascinating how well the model reproduced empirical data and what predictions they were able to derive from it.
At the same time, coming from the biology side, I always had a bunch of questions about action potentials that remained largely unanswered until I made my way through the math jungle. For example, why do sodium (Na+) channels open slowly at first, then all at once? Why does the threshold for spike generation have that value and not another one? Why do potassium (K+) channels take so long to open? And why is it that we donât always get one spike after another?
As we work our way through the model, we will be answering these questions and more. But similar to the previous article, weâll start with a series of small, made-up examples (the code to follow along is here) and work our way up to the main beast. I hope that these examples bring clarity, but if they have the opposite effect, please let me know in the comments. That way, I can improve this guide (and future ones).
Throughout, Iâll try to highlight the underlying biology, as well as what Hodgkin and Huxley actually knew at the time. If youâd like a refresher on neuron structure and function, we do have this older post covering the basics, but Iâll try to weave those concepts in as we go.
From cats to neurons
Abstracting the movements of a cat to math is somewhat straightforward. If we get stuck in the equation, we have something tangible to go back to. So before we start with the math, letâs try to build the same kind of concreteness for neurons and action potentials.
We can begin from the same information Hodgkin and Huxley had available at the time. Neurons are enclosed by membranes, which usually block the movement of ions. Since the membrane is typically sealed, we can have different concentrations of ions on both sides: more Na+ outside, more K+ inside. While they didnât yet know how these concentration gradients were maintained, HH recognized their importance.
They also observed that, if one were to place an electrode outside of the neuron and stick another one inside, a potential difference in voltage of about -65 mV could be measured (by the way, these days itâs also known the exact voltage difference varies by neuron type). In other words, the inside of the cell is more negative compared to the outside. Importantly, the value and its sign donât matter that much, at least not for understanding the general principles. What matters is that there is a measurable difference and that sometimes there is a change in this difference.
If the membrane were forever sealed to the passage of any and all ions, then that would be the end of the story. Weâd have no action potential to talk about (and we couldnât anyway, because no intelligence, language, movement, nothing). But sometimes, the membrane allows ions to flow through it. You can imagine the ion concentrations we mentioned above as water stored in a tank. Thereâs much more Na+ outside the neuron than inside, so when the Na+ âtapâ (i.e., ion channels) opens, Na+ rushes into the neuron, like water gushing into an empty chamber. This happens very fast and leads to a temporary reversal of the voltage difference sign: the inside becomes more positive than the outside. Then the Naâș tap closes and the K+ tap opens, allowing K+ to flow out and bring things back to normal.
This information is pretty much all we need for the HH model, although Iâm sure you still have some questions.
Small detour # 1: hypothesized, but not modeled
We mentioned above that Hodgkin and Huxley didnât know how the Na+ and K+ gradients were maintained. However, they hypothesized there must be some active mechanism that pushes Na+ out and brings K+ into the neuron, thus working to maintain the concentration gradients. Otherwise, each neuron would only have a few action potentials to fire before the ion concentration on both sides of the membrane equalizes.
And they were right. Years later, we found out that there are proteins embedded in the membrane, called ion pumps, that are open only on one side of the neuron at a time. They act kind of like a shuttle bus that only allows Na+ to board from the inside going out and K+ from the outside going in.
Small detour # 2: positive, yet negative?
Iâm sure itâs not lost on any of you that: 1) both Na+ and K+ are positive ions, and 2) cells, including neurons, arenât electrically charged. So how can we talk about a voltage difference?
There are a few key points here:
- overall, the amount of positive and negative charges is equal both inside and outside the membrane, but itâs the distribution of these charges close to the membrane that makes a difference;
- inside the neuron, there are also large negatively charged proteins which canât leave the cell and tend to cluster close to the membrane;
- even though both Na+ and K+ each carry a +1 charge, the concentration of Na+ outside the cell is larger than that of K+ inside the cell (around 150 mM for Na+ vs 100 mM for K+, depending on neuron type). Additionally, the pump we mentioned earlier throws out 3 Na+ ions for every 2 K+ brought in, thus maintaining the imbalance;
- there are also some K+ channels that remain open at rest. Due to the K+ concentration gradient, some of it flows out of the neuron, which means that some positive charge trickles outward, leaving the inside slightly more negative relative to the outside.
The combination of these factors generates the voltage difference measured by Hodgkin and Huxley.
From neurons to circuits
Coming back to our neuron model, now that we have the biology basics, we can begin to abstract. But instead of inventing an entirely new mathematical framework to describe how neurons behave, Hodgkin and Huxley realized that it was easier to repurpose what was already in the physics of electric circuits.
All the elements we described above have an equivalent in a circuit:
- since the membrane stores charge, it behaves like something called a capacitor, i.e. a device which stores charge by accumulating it on two closely spaced surfaces insulated from each other;
- the only way for ions to passively go through the membrane is through ion channels, which are typically closed. In other words, the channels provide resistance to the flow of ions, so we can represent them through resistors;
- we also explained that there are differences in the concentration of ions between the inside and the outside of the neuron and that these differences drive the ion flow, so the ion concentration differences are our voltage sources or batteries;
- and finally, although not explicitly included in the HH model, the ion pumps which restore the concentration gradients represent the current sources in our circuit, pushing ions in a specific direction to keep the system going.
As I said, even at the time, there was already a lot of math for how to work with electrical circuits. And thatâs the key for cracking our simulations.
From measurements to interventions
In the circuit above, we could measure the voltage difference of the inside compared to the outside of the membrane. In fact, thatâs what Hodgkin and Huxley did at first. They used giant axons from squids and silver electrodes to measure the so-called membrane or resting potential, which we said sits at around -65 mV.
But measurements alone arenât enough. And by itself, the neuron and its membrane potential at rest arenât that exciting. We want actionâŠpotentials. Those happen when neurons receive stimuli or input. One could try to do these measurements in vivo, that is when the one neuron we measure receives input naturally, either from other neurons or from the environment. But in this particular situation, Hodgkin and Huxley wanted to have precise control over the neuronâs input and they wanted to use the circuit framework from above. So instead, they used another set of electrodes to directly inject current into the axon of an isolated neuron.
Now, looking at the circuit diagram, physics tells us that if we inject some external current (weâll call it ) into this system before the point where the individual elements (capacitor and resistors) are branching out, this current will split to flow through each available path. So weâll have a capacitive current and, for each type of channel, ionic currents, which for now weâll lump under a generic . As nothing is lost in this idealized circuit, our original will be the sum of the currents flowing through the individual elements, so: .
Cool, but we actually care about voltage, right? Thatâs what the action potential is, a change in voltage difference between the inside and the outside of the neuron over time. Yes, and hereâs how physics helps us again: it tells us that â our capacitive or membrane current, can be expressed in terms of the rate of change of the voltage, i.e. our old friend . Since weâre talking about membrane voltage, weâll just rename x to . And the full formula is , where represents something called the membrane capacitance, and itâs just a constant, a number that we normally determine experimentally or read from a paper that already measured it. In this case, Hodgkin and Huxley measured and found it equal to 1 (, but donât stress about the units yet; by the way, what youâve just heard is the collective shudder of all the worldâs physicists at the idea of not stressing about units).
With that, we can rewrite , and shifting the terms, we get . Since is a constant, you will often see it written on the same side as (basically, constant = we donât care much about it), but to make it clearer, we can also isolate . This will be our stepping stone for the full model. The lefthand side of the equation wonât change anymore. Thatâs the potential weâve been wanting to simulate for a while now. The righthand side will gradually expand in complexity until it allows us to get something looking like the image below:
Virtual neuron 1 â linear, boring, and instructive
In the equation , we already know that is a constant equal to 1 . is what we pump into the system and we have full control over it. For now, we will try out three values: 0, 1, and 2 mA/. tells us about how ions, like Na+ and K+, behave, but for now, we will completely ignore it by setting it to zero. So our equation reduces to or 0, 1 or 2 (mV), depending on which we pick. This is very similar to the first cat example from last time, except that our starting point, , is -65 mV.
But just because this example is so simple, it doesnât mean we canât extract any information from it. We observe that the higher the input current is, the faster our membrane voltage increases. And of course, if there is no input whatsoever, nothing happens.
We can also check what happens if we start from different values at (in this case, -100 mV, -65 mV, and 10 mV). And weâll look at just one external input value, = 1 mA/. As you see below, not much. The line looks exactly the same, except that it starts from different values of . Weâll check this again in the more complex model and see if it holds.
Virtual neuron 2 â to the moon and beyond
Now itâs time to tackle . Instead of zero, we could give it another random value, like 3. But no matter what fixed value we give it, the only thing that would change in our equation would be how fast the membrane voltage increases. More importantly, we know this is unrealistic in neurons because when Na+ and K+ channels open and the ions travel from one side of the membrane to the other, the ionic currents also change.
That means needs to be not a constant, but a function. More specifically, a function which changes over time (and later, over voltage too). One such example would be â at every time step, our ionic current would be equal to the negative value of that time step. Our base equation would then transform into . For mA/, we would get the following:
We see that the membrane voltage now rises much faster, up to very unrealistic values (in practice, if we actually injected the current necessary for reaching such voltages, weâd fry the neuron long before getting there). And if we were to slightly vary either or as we did above, there would be barely any noticeable difference in the result.
But remember how we represented our ion channels through resistors? Similar to capacitors, there is also a formula that relates current and voltage for these elements: . is our membrane voltage, the one weâve been plotting so far. So our base equation now expands into (Iâve moved to the lefthand side to avoid using too many brackets). is the conductance for that ion. Conductance is a measure of how easily electric current flows through a material. In our case, this means how easily the ions pass through their respective channels. For now, we will pretend that is a constant, like 0.1 (mS/).
And is our battery from the circuit above. It represents the equilibrium potential of each ion, what they aspire to, and the voltage at which the membrane would settle if there were no other ions around and if the membrane were permeable all the time. In this case, we donât need to pretend: is always constant for a given ion type. For example, for Na+, is about +45 mV. If the membrane potential, , were equal to +45 mV, we would say that Na+ is at equilibrium and there would be no movement of Na+ ions across the membrane. In real neurons, this is never reached, since other ions have different equilibrium potentials (for example, K+ sits at around -82 mV), but weâll learn more about that later.
Small detour # 3: that Nernst guy
But hold up: what does ion concentration have to do with voltage? And where do ion equilibrium potentials actually come from? Well, in practice, from neat little tables.
But conceptually, we need to make something clear, using Na+ as an example: we said that there are more Na+ ions outside than inside the neuron, so there is a higher concentration of Na+ on the outside of the membrane. If we open the tap, this concentration difference will push Na+ inside. But when does the pushing stop? Is it when the Na+ concentration is exactly equal on both sides of the membrane? It would be, if only Na+ were the only one around and there were no voltage difference between the two sides of the membrane.
But letâs imagine that we also have those negatively charged proteins from earlier. This changes the game, because even though the concentration of Na+ ions might equalize at some point, there would be another force pulling it in: the negative charge of the proteins, or the electrical gradient. Because these two forces compete, the actual voltage at which no Na+ moves around anymore is the one given above.
We can calculate this number from yet another equation that some guy named Nernst came up with: . R, T, z, and F are constants, so we again ignore them. What matters is that this formula allows us to relate the ion concentrations (outside) and (inside) the neuron to voltage, thus giving us the equilibrium potential of each ion.
Bonus: this nifty formula tells us why sudden influxes of K+ can kill you. When the concentration of K+ outside the neuron increases a lot, the equilibrium potential of K+ ends up being much higher than -82 mV. In turn, this messes with the generation of action potentials, thus impairing communication between neurons. Once we have the full HH model, weâll be able to check exactly how this happens.
Back to virtual neuron 2
For now, we see that if we were to model just Na+ currents and assume a constant conductance (in this example, mS/), the membrane potential would eventually settle to the equilibrium potential of Na+.
This time, if we change our starting point , we observe a different behavior compared to the first virtual neuron: here, the membrane potential always settles at the Na+ equilibrium, regardless of whether we start from a value above or below that.
But what happens if we keep the resting state voltage the same and change the conductance ? A higher conductance means that Na+ ions barrel through channels quicker (because more channels are open, not because the ions move any faster). That translates into the equilibrium potential being reached sooner.
I want to stress here that conductance isnât just an abstract thing that makes the graph sharper. In real life, alterations in Na+ channel conductance can have devastating effects. For example, tetrodotoxin, a powerful toxin derived from pufferfish, effectively decreases Na+ conductance to zero by blocking Na+ channels and preventing its influx into the cell. This is deadly. And in different types of epilepsy, Na+ conductance is again affected: either too high or too low, depending on the type of epilepsy. As weâll see later, changes in conductance affect the properties of action potentials, such as shape and timing. At the level of the whole brain, this results in abnormal communication between neurons and can lead to the symptoms observed in epilepsy.
Moving on to varying the external input current , we see that the membrane potential no longer settles at the ionâs equilibrium potential, but at another value that changes with the strength of the external input . Looking again at our equation , we see that when is zero, the membrane voltage is only governed by . But once we inject a steady flow of current into this system, the balance point shifts higher or lower, depending on the sign of . This will be important for action potential generation later on.
Virtual neuron 3 â other ions have joined the party
Alright, but we know Na+ doesnât act alone. There is at least a K+ current. There are other ions as well, but Hodgkin and Huxley lumped everything else that might act in a neuron under a so-called âleakâ current that is modeled as an additional resistor.
Once we add the K+ and leak currents in our model ( and ), we now have a slightly longer differential equation for the membrane voltage:
.
Simulating this allows us to see that, like before, the membrane voltage settles at an equilibrium point. But this point is no longer equal to the equilibrium voltage of any single ion. Instead, it sits somewhere in-between. This in-between value is nothing more than the weighted average of the contributions of all ions to the membrane potential. The contribution of an ion is given by the product between its equilibrium potential and its conductance, so the full equation reads like this: .
We saw above that changing the Na+ conductance when only Na+ is present allows us to manipulate how fast we reach the equilibrium potential. But the equilibrium potential itself remains unchanged. But now we have more than one ion, each with their own conductance, and we see in the equation above that the membrane equilibrium potential takes into account conductances as well. So what happens if we change each ionic conductance individually?
We should be able to deduce this from the equation, but weâll check it against the simulation results below. The blue line represents our original case from above. Since the K+ equilibrium potential is more negative than our original resting state potential , increasing the K+ conductance while keeping the Na+ conductance the same means that the membrane will settle at a new, more negative potential (orange line). In contrast, since the Na+ equilibrium potential is positive, increasing the ionic conductance while keeping the same means that our neuronâs equilibrium potential goes up and we also reach it faster (green line). Now, if we increase while maintaining this higher , our membrane resting potential comes down, closer to that of K+. But we still get there fast, since the Na+ conductance is so high (red line).
In principle, we could also play around with the leak conductance . However, as we will see later, in the HH model, the leak conductance is always assumed to be static, whereas and do change under certain conditions.
The full model â everybody gets a function
Weâve already added quite a few details to our model, but thereâs still a bit to go on. So far, we have a simulation of the membrane potential which includes multiple ion channels. This model is capable of settling at an equilibrium point, the resting state potential, but it still doesnât produce spikes yet. So letâs fix that. Fair warning, this next part is the trickiest (I know! As if the novel before was soooo easy!), so go slowly, pause often, and donât worry if things take a few reads to click.
Key takeaway # 1: conductances are voltage-dependent
Letâs bridge biology and math now: we said that when the Na+ conductance increases (i.e. Na+ channels open), the membrane voltage also increases. But we also know from experiments that when the membrane voltage increases, K+ channels open. In other words, the K+ conductance increases. In math terms, that suggests conductance (for both Na+ and K+) is voltage-dependent.
Key takeaway # 2: there is a maximum conductance
Imagine all Na+ channels are open. Even then, there is still a limit to how much Na+ can pass through the membrane at every time step, because the ions need to wait for their turn to go through the channels, just like cars have to wait to pass through a crowded tunnel. That means conductance has a maximum value, which we can call . When all channels are open, for Na+ and similarly, for K+.
Key takeaway # 3: we can work directly with proportions of open channels
But what if only 50% of the channels were open? Well, the limit would be half of the maximum: . Why is it this relevant? Because instead of directly relating conductance to voltage, we can relate proportion of open (or closed) channels to voltage. The math is easier and itâs a bit more intuitive.
Putting it all together
First of all, since conductances are voltage-dependent and the membrane voltage changes over time, we actually have voltage- and time-dependent conductances. Important to note, only for Na+ and K+; we assume the leak conductance to be fixed.
Secondly, we work with the proportion of open channels, not with conductances directly. Letâs pause for a moment and think about what we want to model. We basically want a sort of push-pull mechanism, such that when the voltage goes up, the proportion of open Na+ channels goes up, and when the voltage decreases, the proportion of closed channels increases. And the same way for K+.
Letâs start with K+. We can denote the proportion of open K+ channels with n. The proportion of closed channels will be simply 1 â n (total minus how many are open). Since weâre interested in how this evolves over time, we need to bring back our differential equation friend, in this case . The push-pull mechanism we want can be written in the following form: or following the Hodgkin-Huxley convention: . There are two parts that matter here:
- the two functions A and B act like weights for the proportion of closed, respectively open, channels. A controls how fast closed channels open and B controls how fast open channels close;
- the above is not enough. A and B are voltage-dependent functions themselves and they need to be chosen in such a way that, when the voltage goes up, A goes up and B goes down, and vice-versa when the voltage goes down.
But how to choose them? Well, the equation above is called a first-order differential equation and has a known solution. Without going further into mathematical detail, Hodgkin and Huxley used that solution together with experimental measurements of K+ currents to derive specific formulas for and . I am including them here for completeness and because you will see them in the code, but there is no reason to stress over them. In practice, unless you use them on a daily basis, youâre just going to look them up when needed (and by the way, depending on the neuron type, the actual numerical values in these formulas will change): and .
(Side note: the sign convention. One thing to notice above is that we use both and V. Thatâs not a typo. Normally, we define the membrane voltage , so the membrane voltage is negative at rest. In the HH model, however, V is defined as . That means the voltage is shifted such that at rest, mV. And because all s and s were fitted to these shifted values, we need to take that into account when working with the original HH model.)
For Na+, they modeled the Na+ channel activation in a similar manner, except they called the proportion of activated channels m. Again, for completeness, the respective equations were and .
Now we almost have the full functioning HH model, but there are just a couple of minor tweaks left. Because Hodgkin and Huxley fitted their model to experimental data, they observed two interesting tidbits:
- the model fit better when the variable n was raised to the power of 4 and when m was raised to the power of 3. At the time, they didnât know why that was the case, but in the meantime, weâve found out that the K+ ion channel is made up of 4 subunits, each of which needs to be activated for the channel to allow the passage of K+. In that case, you can think of n as the proportion of channels where subunit 1 is activated (or the probability for this subunit to be activated). The proportion of channels where 2 subunits are activated is , and so on. Similarly, Na+ channels have 3 activation domains that need to be opened for Na+ to pass through the channel;
- when the membrane voltage was held constant at a high value, K+ kept flowing out of the cell until the voltage was allowed to return to normal. But for Na+, Hodgkin and Huxley observed a different behavior: Na+ flowed into the cell at first, then it stopped. The Na+ current sharply decreased and regardless of how long the voltage was kept high, the Na+ current didnât increase anymore. To model this behavior, they introduced a second variable for Na+, called h, which they used to model the proportion of inactivated Na+ channels. This needed the same and functions, with and . Again, nowadays, we know that Na+ has an inactivation domain that rapidly blocks Na+ channels at high voltages and only unlocks them when the voltage goes down again. Thatâs also why action potentials cannot spread backwards from where they came from.
And thatâs it, we now have a full HH model. Put all together, it looks like this:
,
,
,
.
Importantly, by itself, the model doesnât really do anything. If the external input is zero and we start the model from an initial membrane voltage below a certain threshold (in this case, -60 mV), it quickly decays back to the resting state potential (which you can calculate yourself using the formula given above and the maximum conductances and ionic equilibrium potentials given in the code here.)
If we start the model above a certain threshold (for example, -50 mV), it will fire a single spike before going silent forever.
To get more than one spike, we need to drive it with external input current. So far, weâve used constant current, and weâll stick with that for today (in the next part, weâll also try out time-varying currents). For a high enough current, we see that the model fires one action potential after the next. You can try it out for yourself to see what happens for different values of , and next time weâll try a more systematic analysis as well.
Finally, we can inspect our gating variables m, h, and n, to see how they evolve over time. In the plot below, you see that the Na+ channel activation variable m (in blue), goes up really quickly â Na+ channels open fast; but it goes down just as quickly â they also close fast. The Na+ inactivation variable, h, quickly decreases during the spike â Na+ channels are blocked and cannot open again for some time. In the meantime, the K+ activation variable n goes up, lagging a bit behind m â K+ channels open more slowly and the membrane voltage goes back down.
Still alive?
I donât know about you, but Iâm tired. The good news is that now we have a functional HH model. Also good news is that we can do a lot of things with it, but unfortunately, that requires additional explanations, and I think we could all use a break. So Iâll see you for the next part. Until then, feel free to toy with the model parameters.
P.S.: If someone knows a better solution for displaying LaTeX equations in WordPress, do let me know. The current method is hurting my soul.
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References
Goaillard, J.-M., & Marder, E. (2021). Ion Channel Degeneracy, Variability, and Covariation in Neuron and Circuit Resilience. Annual Review of Neuroscience, 44(1), 335â357. https://doi.org/10.1146/annurev-neuro-092920-121538
Hodgkin, A. L., Huxley, A. F., & Katz, B. (1952). Measurement of currentâvoltage relations in the membrane of the giant axon of Loligo. The Journal of Physiology, 116(4), 424â448. Portico. https://doi.org/10.1113/jphysiol.1952.sp004716
Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500â544. Portico. https://doi.org/10.1113/jphysiol.1952.sp004764
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