Sirs Hodgkin and Huxley

A spike is an "all-or-none" phenomenon, a digital one, a 0/1. Along with synaptic potential, spikes (action potentials) form the 2 types of potential in the nervous system: the action potential, located at the axon of the pre-synaptic cell and the synaptic, at the dendritic spine of the post-synaptic cell (a reminder: synaptic potential is either EPSP or IPSP). We say spikes "overshoot" when momentarily they go above zero voltage (remember, resting potenital is -70mV) and they "undershoot" when momentarily the dive even below their resting potential, upon attenuation phase. This is also called "after-hyperpolarization) (lesson 4, video 1, 09:30-).

So this is a transient phenomenon, a very brief one, on the order of a millisecond or less, depending on temperature. By these experiments on squids (

*please go to the end of this post and watch the 2*), Hodgkin and Huxley did not only want to record what happens but also to**amazing YouTube videos**showing in-lab the actual experiments and theory behind!!**explain**the underlying mechanism that enables this kind of phenomena to occur. The "spiking activity" is a universal phenomenon applicable to almost all nerve cells, it's a universal "patent" of neurons to carry information (actually, to generate electrical signals that represent external information (visual, auditory, etc) or internal information (depression, love, anger, etc)). Their findings are today all integrated into the 4 famous H&H equations (check at the end of the post) that consist the mathematical representation of the process involved in the generation and propagation of an action potential.**The action potential (spike) as an "all-or-none" phenomenon**

So we have a sub-threshold and supra-threshold voltage level. Current injected below the sub-threshold voltage regime, it goes up and then decays to resting state. Contrary, when you inject current at the supra-threshold regime, there is spike generation. After a specific magnitude of depolarizing current, the spike won't grow any bigger (battery prevents it from doing so as well as the "h" variable, check some sections below) but it will occur earlier.

**What are the membrane currents underlying the spike?**

Or else, what makes the membrane of the axon excitable? Or, what enables the membrane of the axon to create spiking activity, after a certain voltage threshold is reached? Why doesn't this happen to the dendrite? What is so unique in axon that enables it to "fire" such bursts of electricity (or, these boosts of action potential)? To answer these questions, Hodgkin and Huxley developed two techniques: the space clamp and the voltage clamp.

**Space-clamp**

This technique means that you take a long axon and make it electrically isopotential, meaning that when you place inside the axon an axial low resistance (an axial electrode), all the points along the axon become isopotential, meaning in turn that there is no voltage drop (Prof. Segev definition: making the (long) axon effectively isopotential via the insertion of an axial conductive wire). So the space becomes "clamped" or shrinked because of the insertion of the low conductance axial electrode.

**Voltage clamp**

This is a most sophisticated method than the previous one. With this, you want to clamp/fix the voltage between the inside and the outside of the membrane of the axon. So you don't want the membrane to behave independently, as it wants normally to behave and generate a spike but you want to fix the voltage to a specific

**preset voltage**and clamp it there. And why do I want to do this? Because I don't want the action potential to interfere. The voltage clamp technique, which is a fast-feedback system, enables the experimenter to dictate the desired voltage difference between the inside and the outside of the membrane. This electronic feedback system injects current exactly to counter-balance the excitatory, voltage-dependent membrane current that the membrane wants to generate. You "feel", using this feedback system, the current the axon wants to generate to blow a spike, then you inject through this system a counter-current, of the same amount but in the reverse direction so you fix the voltage. And because now you can fix the voltage between the two sides of the membrane, you can ask the question: "what is the current that flows between the two sides of the axonal membrane for this particular, fixed, clamped voltage"? Lesson slides, here.

**Membrane currents underlying the spikes**

So first let's clamp the voltage to a value, say -50mV (check lessons slides). Then we record the current needed to fix this voltage. By doing so, we first get a capacitative current, since we have a change in voltage (remember: C=dV/dt, we have a change in V). We keep having a constant

**voltage step**and we see that we next have a resistive current so the system basically works as a passive RC circuit. Notice that we're talking about sub-threshold depolarizing voltage here. So, for sub-threshold depolarizing voltage clamp, the recorded membrane current is the current that flows via the "leak", or the passive conductance plus a small capacitative current at the start and at the end of the circuit (again, pls check the slides, they're quite self-explanatory). But what happens when we depolarize the cell further, to the supra-threshold regime?

Something very interesting happens. For supra-threshold depolarizing voltage clamp, the recorded membrane current (after the fast capacitative current) flows

**first inwards**(into the axon) and**later outward**(from inside to the outside). This was considered to be a surprise or revelation for the neuroscience studies because it was completely new. Let's explain it.At first, you see the capacitative current increase a lot (the dv/dt) because we have a big change in voltage, +60mV (plus, not a typo)

**relative to the resting state**. Then, when you hold the voltage clamped at zero, the capacitative current decays ("empties") very fast and very early on there is an**inward current, the fast inactivating current**(directed to inside the axon). You continue to clamp the voltage and this current gets smaller and smaller (it goes up in the diagram, it gets**inactive**) then it reverses and becomes an**outward current**, the**slow non-inactivating current**(from inside to outside) which remains activated (does not decay) as long as the voltage remains clamped. Until that day, nobody knew that the underlying currents of the squid's giant axon membrane during a voltage change actually consist of two different, say, "faces". But HH also found something else:They found that you can block these two currents separately. The inwards one by applying a very potent and dangerous drug found in some fish, called Tetrodotoxin (

**TTX**), on top of the axon and the outwards one by applying Tetraethylammonium (**TEA**) at the same place. So using pharmacology agents (drugs) you start to separate the two currents. So it seems that the two currents are**probably different**since you can get rid of one or another by applying a specific drug on top of the axon.They also found by changing the concentration of Sodium (Na+) and Potassium (K+) that the inward current is a Na+ (fast-inactivating) current while the outward current is a K+ one (slow-non inactivating).

So, the total current in the membrane is the capacitative current (Cm to the left), plus the voltage dependent, excitable Sodium current plus the voltage-dependent Potassium current and the passive (leaky) R conductance. This circuit is unique because it exists mostly in axons. When there is enough depolarization the two channels (Na+ and K+) open. The first one opens fast but it deactivates fast as well and the other opens after the first and it does not deactivate.

Hodgkin and Huxley were not content just with this finding. They wanted to understand exactly the "kinetics" of these Na+ and K+ ions. So they voltage-clamped the membrane to different values, step-by-step and measured the conductance of both potassium and sodium for the different voltages. One sees (check below) that potassium gets stronger and stronger as depolarization grows more and more, so the larger the voltage clamp the bigger the potassium conductance, which responds slowly to the increase of clamp but it reacts faster while the voltage clamp increases. The K conductance rises slower than it decays at the end of voltage clamp (slow-growing, fast-attenuating).

This is voltage-dependent conductance, these ion channels are sensitive to

**voltage**. So the amount of conductance depends on depolarization. Now let's see what goes on with sodium. In this case, sodium conductance responds fast, very early on. It's a fast conductance change for sodium. When the depolarization increases, also the conductance increases, as well. But after some time, the conductance decays, even if voltage is still clamped.**The mathematics of K current during VC**

So, HH represented mathematically the slow response (upstroke) with (1-exp(t)) to the power of 4 and the downstroke (fast-attenuation) as exp (-4t). Then they formed the following equation: the actual potassium conductance equals to the maximal conductance (conductance of total, absolute number of potassium channels) multiplied by the factor "n" (a voltage dependent parameter). The claim was that for a particular voltage clamp, "n" gets a value between 0 and 1. So at a very big voltage clamp, "n" gets near to 1. "n" depends on both voltage and time. It represents the proportion of K ion channels in an open stage. If n=1 then all the conductance (the maximal one) is available. They also tried to explain the power of 4 (check image below and quote at the bottom of it).

Now we will try to graphically represent the potassium channel. So we know that with depolarization the 4 n gates start to open (check below).

We can think of "n" the probability of a gate to open. So we need all n-gates to open to let potassium flow. They open and close all together in the probability of n to get the value of 1. This "n" parameter relies according to HH to another parameter, the a parameter and the b parameter (1-n=closed state, n=open state). The dn/dt equation describes the dependance of n to voltage.

In case of Sodium (Na+) the equation is a bit more complicated because there is both activation and inactivation for the same voltage. So here we have the "m" variable for activation phase and "h" variable for inactivation phase.

In Sodium we have 3 gates which open with depolarization as well but there is also the "h" gate.

Equation #1: It says that the action potential current (spike) depends on capacitance (dv/dt) plus sodium conductance plus potassium conductance plus the passive conductance (leak). The next three equations describe variables m, n and h.

**The refractory period**

It's the period between 2 spikes, the gap or delay between the first and second. So the frequency is limited in axon.

So the clock frequency of axon is 100Hz (100 spikes per second, it's a slow clock).

## related reading

- The original paper of Hodgkin & Huxley

- Conductance-based models

- The HH model

- Interactive Java applet of the HH mode

- Model DB

- HH model in NEURON

- HH model

- Conductance-based models

- The HH model

- Interactive Java applet of the HH mode

- Model DB

- HH model in NEURON

- HH model

## related videos

- The actual experiment of HH! Amazing video!!

- The squids' Giant Axon video!

- Great animation of voltage clamp technique!

- The Squid and its giant nerve fiber

- The squids' Giant Axon video!

- Great animation of voltage clamp technique!

- The Squid and its giant nerve fiber