Notes

Lecturer: Valerio Mante

An Action Potential is a depolarization that starts driving down the axon, which is then followed by hyperpolarization.

• Occurs in axons: travels away from soma.
• Lasts 1-2 ms.
• All-or-none, i.e., stereotyped, if $I_e$ increases it will lead to an AP initiated sooner, but with the same shape. It is non-linear, unlike passive membrane.
• Stimulus intensity encoded as AP rate ($r_{AP}$).
• Several phases:
  • No input: $V = V_{rest}$
  • Current $I_{e} \rightarrow V - V_{\infty}\sim exp\left( - \frac{t}{\tau} \right)$
  • V rapid increase in V
  • Peak at V > 0
  • Rapid decrease in V
  • Undershoot: $V < V_{rest}$

How to explain these properties of the AP

• Why only in axon?
• Why all-or-none?
• Why that shape?

Answer: g = g(V, t): voltage-dependent channels in the axon. They open and close dependent on the voltage that they experience on the membrane. Hodgkin-Huxley: Nobel Prize Medicine Physiology in 1963. Everything they did was before the existence of ion-channels (membrane channels) were known.

During an AP we see channels opening and pulling V towards E. The hypotheses is that in the rising phase of AP the sodium and calcium conductances increase (gNa and gCa), and in the decaying phase of AP the sodium and calcium conductances decrease or potassium and chloride conductances increase. All as function of V. For testing these hypotheses, we need to measure $g_{Na}$, $G_K$, etc... We can use the IV-relation: measuring $I_{Na}$, $I_K$, etc... for different V then infer $g_{Na}$, $G_K$, etc... To do this, we can use a voltage clamp.

Squid Giant Axon

Voltage Clamp A new technique invented by Hodgkin and Huxley. Previously, current $I_e$ was injected and voltage V was measured, now we set V and measure $I_e$ required to keep $V_{measured}$ = $V_{set}$. It measures the current required to clamp the membrane voltage. Fast feedback system to fix V and measure I. $I_e$ has the opposite sign, i.e., is positive if from outside to inside. But to keep $\mathrm{\Delta}V$ constant, it is necessary to inject a current opposite to the ionic current. In the end, the current injected can be read as the ionic current (in the ionic current convention).

Space Clamp

It makes the axon isopotential, do not have an AP but it is the same mechanism. The giant axon in squid has approximately 1mm of diameter and it is like a long wire, making the axon isopotential.

Voltage Clamp Experiment

• Command voltage is set by the experimenter, the feedback circuit holds the voltage constant.
• The voltage clamp allows the membrane voltage to be manipulated independently of ionic currents, allowing the current-voltage relationships of membrane channels to be studied.
• With negative feedback circuit, the Na+ current is auto-catalytic. An increase in the voltage increases conductance, which increases the Na+ current, which increases the voltage again.
• The threshold for action potential initiation is where the inward Na+ current exactly balances the outward K+ current.

Identifying the Currents

• Hodgkin & Huxley approach: remove Na concentration gradients (Na+ free seawater eliminates $I_{Na}$).

• Later: pharmacological blockade of specific channels
  • TTX: poison in pufferfish, it eliminates $I_{Na}$.
  • TEA: Tetraethylammonium eliminates $I_{K}$.

Voltage and time-dependent conductances for $g_{Na}$, $g_K$: $g_{Na}$ increases quickly (fast activation), but then inactivation kicks in and it decreases again (fast inactivation). $g_K$ increases more slowly (slow activation), and only decreases once the voltage has decreased (no inactivation).

Towards a Mechanistic Model They proposed an hypothesis of what might be causing voltage and time-dependence, which is going to be formalized in the lines of the previous image. The white dots is what they measured and the models estimate the lines. How to explain voltage and time-dependence in $g_{Na}$ and $g_K$?

Two possibilities:

• Single channels have variable (continuous) permeability (analog).
• Single channels are either open or closed whereby p(open) = f(V) (digital).

Today we know that the second possibility is correct:

• Patch clamp: record $I_{singlechannel}$ (Nobel 1991, Erwin Neher & Bart Sakmann)
  • It allows the study of currents across single channels of membranes.
  • Hodgkin & Huxley: inferred the patch clamp from their voltage clamp data.
  • Individual channels are probabilistic devices that are opened or closed. The conductance is measured by the average of all channels.

Single Channel Current

There are two types of voltage-dependent conductances:

• Persistent Conductance Type, it has two stages: deactivated (closed) and activated (opened). The channel opens and stay opened when the cell is depolarized. For example, $g_K$ in AP.
• Transient Conductance Type, it has three stages: deactivated, activated and inactivated. Here we have two gating variables that describe the opening and closing of the channel. Activation and Inactivation are two processes that work in opposite directions. The channel opens but then it closes while the cell is stil depolarized. For example, $g_{Na}$ in AP.

Hodgkin & Huxley formalism is used for active conductances in general: $g_{i} = \overline{g_{i}} \bullet P_{i}$ where $g_{i}$ is the overall conductance of channels of type i; $\overline{g_{i}}$ is the maximal conductance (if all channels were open); and $P_{i}$ is the probability of the channel to be open (or the fraction of channels that are open).

Persistent Conductances Assuming that k events (independents and identical) are necessary to open a single channel, then $P_{\ } = n^{k}$. n is a gating/activation variable: the probability of a subunit gate to be open, and it is voltage and time-dependent. k is the number of subunits necessary to open each channel. According to Hodgkin & Huxley $g_{K} = \overline{g_{K}} \bullet P_{K} \rightarrow P_{K} = n^{4} = n \bullet n \bullet n \bullet n$ (it is necessary 4 subunits to open the channel). When k was fitted to data it leaded to corrected predictions for K+ channels.

Gating-Variables: Time-Dependence

Persistent and Transient Conductances Transient conductance includes inactivation:

$$P_{Na} = m^{3} \bullet h$$

Where $m^{3}$ is the activation variable and $h$ is the inactivation variable, which also represents the probability that the channel is not blocked by the inactivation gate.

Gating-Variables: Voltage-Dependence

The Hodgkin and Huxley Model It is the model that describes how action potentials in neurons are initiated and propagated.

• The model parameters are fit to $g_{Na}(V,t)$ and $g_{K}(V,t)$ from voltage clamp.
• n and m are probabilities for a gate to be open.
• h is the probability that an open channel is not blocked.
• The gating variable have a voltage dependence.
• $\overline{g_{\ }}$ values are the maximum conductance possible.
• There is no inactivation for potassium, only for sodium.
• The membrane does not get locked at positive values.
• $\overline{g_{L}}$ stands for some generic leak.
• The functions $n_{\infty}(V)$, $m_{\infty}(V)$ and $h_{\infty}(V)$ determine whether gates serve to activate channels (with depolarization) or inactivate the channel (close with depolarization). $\tau_{m}$, $\tau_{h}$ and $\tau_{n}$ are time constants.

Fitting the Hodgkin and Huxley Model

{width="3.884027777777778in" height="0.4263888888888889in"}

Model Predictions

• Potassium Channel: the number of subunits (4) in K channel was verified much later with structural studies.
• Action Potential Shape
• Action Potential Threshold

• Refractory Period: it is harder (requires larger current injection) to generate AP immediately after an AP. The reason is that $g_K$ is still activated and $g_{Na}$ is still inactivated.
• Action Potential Propagation in Unmyelinated axon
• Action Potential Propagation in Myelinated axon: in the myelinated part of the axon we have passive AP propagation (small capacitance and large resistance), but in the nodes of ranvier, we have active AP regeneration. Compared to unmyelinated:
  • Faster AP propagation.
  • Smaller current.
  • Faster $V_{AP}$ increase with axon radius.
• Action Potential Collision: Action Potential propagates in one direction along axon. Reason: refractory period, $g_{Na}$ still inactivated in the wake of AP. Either direction is possible in principle. From soma to axon terminal: orthodromic. In the opposite direction: antidromic. In the brain we do not have usually antidromic AP. Antidromic AP can be generated artificially also during collision experiments, i.e., both antidromic and orthodromic AP are initiated, none achieve the other end, they annihilate each other in the center.
• Action Potential Not Reflected at Axon Terminal: at the end of the cable, there is no AP reflected because of refractory period.
• Why Action Potential Only in the Axon: AP does not usually propagate in dendrites because $g_{Na}$ & $g_K$ are missing. However, in a few cell types $g_{Na}$ & $g_K$ are present also in dendrites. It is not sufficient to generate an AP, but can propagate AP from soma into dendrite to some extent: axon backpropagation.

Single Neuron Computations