Reading action potential 2

Current and Voltage Clamps

• In the studies of the passive membrane in the previous unit, and the studies you did in this unit of the action potential, we controlled the membrane's voltage by injecting different amounts of current.
• Implicitly, the simulations assumed that if we requested 1 nA of current, the simulated electrode would continue to inject a steady 1 nA of current, even if conditions changed.
• In real electronic devices, a good deal of engineering effort has to be expended to ensure that the current injected remains steady even as conditions change. For example, an electrode's resistance may change, or the resistance of the cell membrane may change.
• Devices that can inject steady currents, even when voltages change, are referred to as current clamps, i.e., they keep the current "clamped", that is, fixed.
• This technique is very valuable for exploring how one neuron in a circuit affects the activity of another neuron.
• However, if one wishes to analyze the underlying biophysics of a single neuron, current clamp is not very helpful.
• To characterize the sodium conductance, one needs to understand how its value varies over time for a fixed voltage.
• Doing this using current clamp is not possible. The reason is that if you use the current clamp to set the membrane to a voltage at which the positive feedback loop we have described is activated, the sodium current flows in, and the membrane voltage changes.
• We need a technique that will break the connection between changes in voltage and changes in current.
• This technique was invented by K. C. Cole, and is referred to as voltage clamp. This technique makes it possible for the experimenter to hold the membrane at a fixed voltage, and to measure the currents that flow through the membrane, which are prevented from changing the membrane's voltage.
• Still more recently, a technique has been invented, patch clamping, that makes it possible to voltage clamp small regions of the nerve cell membrane, and look at the responses of individual ion channels to changes in voltage across the membrane. We will spend more time describing the patch clamp technique in the next unit.
• So to summarize:

Name	Controls	Measures	Purpose
Current clamp	Current	Voltage	Circuitry analysis
Voltage clamp	Voltage	Current	Conductance analysis
Patch clamp	Voltage	Current	Ion channel analysis

How Voltage Clamp Works

$Image: Amplifier.png|200px|thumb|Figure 4: A schematic diagram of an amplifier, a key element for many electronic instruments]]

• How does the voltage clamp work?
• To answer this question, we need to understand how a differential amplifier works.
• As shown in Figure 4, a differential amplifier has two inputs, $V^+$ and $V^-$, and a gain (amplification factor) $A$. The amplifier takes the difference between the two inputs, multiplies them by the gain, and this is its output. In symbols, $V_o = A \left(V^+ - V^-\right)$, where $V_o$ is the final output voltage of the amplifier.
• Using a differential amplifier, we can measure the membrane voltage by placing a wire within the axon, and connecting it to input $V^+$. We can place another wire in the bath, where it is connected to the electronic ground, and attach it to the other input of the amplifier, $V^-$. The difference between these inputs is amplified to determine the membrane voltage, $V_m$.
• One of the major reasons that the squid giant axon was the first nerve cell to be studied exhaustively was the large diameter of the axon made it feasible to place an electrode wire along its length.

$Image: MembranePotentialMeasure.png|400px|thumb|Figure 5: How membrane potential is measured in the voltage clamp]]

• How can we both control the value of the voltage and at the same time measure the current that flows through the membrane?
• To do this, we must use another amplifier, and a negative feedback loop which counteracts the positive feedback loop that generates the action potential, as shown in Figure 6.

$Image: CommandVoltage.png|400px|thumb|Figure 6: How command voltages are set in the voltage clamp]]

• In this figure, we see a second wire that is threaded into the axon; the voltage output of the amplifier goes through a resistor that converts the voltage into a current, and this current is therefore injected into the axon, changing the axon's voltage.
• The command voltage is sent into the $V^+$ input of the amplifier, and the actual membrane voltage, $V_m$, which is measured by the other amplifier, is sent into the $V^-$.
• Here is how the negative feedback loop works: if the membrane voltage is slightly different from the command voltage, the difference is amplified, and the output voltage is converted into a current that is injected into the axon to keep the membrane voltage steady. The current that is injected can be measured, and is the opposite of the current that would have flowed through the membrane if the clamp were not operating.
• So, the negative feedback loop keeps the membrane voltage steady, and the voltage clamp also measures the current that is flowing through the membrane due to that voltage.
• It is important to stress the importance of being able to thread the control wire through the axon, which is made possible by the large diameter of the axon.
• Having the wire run throughout the length of the axon ensures that all parts of the membrane are kept at a fixed voltage. This is referred to as space clamp, and is very important for obtaining meaningful results from voltage clamp experiments. If the membrane is not properly clamped, uncontrolled currents may flow, and voltage changes may occur, badly distorting the results.
• Using the voltage clamp, one can step the membrane to different voltages, and measure the current flows through the membrane over time at that fixed voltage.
• There are some important conventions that you need to know to be able to read voltage clamp records:
  • Inward currents are shown as negative, and are depolarizing. For example, at the resting potential, sodium currents flow into the nerve cell, and act to depolarize the nerve cell.
  • Outward currents are shown as positive, and are hyperpolarizing. For example, at the resting potential, potassium currents flow out of the nerve cell, and act to hyperpolarize it.
  • For historical reasons, these conventions are the opposite of those used for current clamp, in which depolarizing currents are shown as upward deflections, and hyperpolarizing currents are shown as downward deflections.
• Make sure to look carefully at the graphs, to see which variable is being controlled and which is being measured; this will clarify whether you are looking at current clamp or voltage clamp data.

Video Describing How Voltage Clamp Works

How Voltage Clamp Works

Analysis of the Action Potential of the Squid Giant Axon

$Image: ElectricalEquivalentCircuitActionPotential.png|300 px|thumb|Figure 7: The electrical equivalent circuit for the action potential]]

• It is now possible to understand how Hodgkin and Huxley dissected out the different components of the current, and then were able to deduce properties of the underlying sodium and potassium ion channel conductances.
• The overall task that they had to solve is illustrated schematically in Figure 7.
• As shown in Figure 7, there are two components of the current that are due to the passive membrane properties, which we studied in the previous unit.
• One of these components is the capacitative current, which flows across the capacitor whose value is $C_m$, and which we will discuss further in the next section.
• A second component is referred to as the leak current, which corresponds to the flow of ions through the ungated sodium and potassium ion channels, labeled $g_{L}$, is in series with a battery representing the leak potential, labeled $E_L$, and will be discussed at greater length in the next section. Together, these two properties are referred to as the passive properties of a nerve cell.
• A third component corresponds to the potassium current, which flows through a variable resistor (indicated by an arrow across the resistor symbol) labeled $g_{\text{K}^+}$, and is in series with a battery representing the potassium ion Nernst or equilibrium potential, labeled $E_{\text{K}^+}$.
• A fourth component corresponds to the sodium current, which also flows through a variable resistor (again indicated by an arrow across the resistor symbol), labeled $g_{\text{Na}^+}$, and is in series with a battery representing the sodium ion Nernst or equilibrium potential, labeled $E_{\text{Na}^+}$.
• The task before Hodgkin and Huxley was to characterize the different current components as a function of voltage and time.
• They took their analysis one step further by positing the existence of ion channels, and used this hypothesis as the basis of a model that they used to quantitatively reconstruct the action potential.
• In the remainder of this unit, we will describe how they analyzed the passive currents (i.e., the currents through the capacitor and the leak conductance), and the potassium current. In the next unit, we will discuss their analysis of the sodium channel. We will also introduce the modern technique, patch clamp, that has been used to work out the detailed biophysics of ion channels. In a subsequent unit, we will describe their quantitative model, and subsequent extensions of the model that have been made as additional ionic conductances have been discovered and characterized.
• Even before the full characterization of the ionic conductances, it is still possible to write down the equation describing the electrical equivalent circuit for the membrane, which states that the membrane current is the sum of the capacitative current, the leak current, and the ionic currents potassium and sodium: $

   I_m = I_C + I_L + I_{\text{K}^+} + I_{\text{Na}^+}$ which can be expanded using the equations for ionic currents that we defined in the passive membrane unit as $
   I_m = C_m \frac{dV_m}{dt} + g_L\left(V_m - E_L\right) + g_{\text{K}^+}\left(V_m - E_{\text{K}^+}\right) + g_{\text{Na}^+}\left(V_m - E_{\text{Na}^+}\right). \quad\text{(Equation 1)}$

• The goals of this and the next unit will be to fill in the details of these terms.

A Video on the Electrical Equivalent Circuit for the Squid Giant Axon

Electrical Equivalent Circuit for the Squid Giant Axon

Capacitance and Leak Currents

$Image: LeakCurrentsUnderClamp.png|300px|thumb|Figure 8: Change in leak conductance currents in response to a voltage step from the resting potential (-61 mv) to 0 mV]]

• Before analyzing the active currents, the two passive components of the squid giant axon needed to be characterized: the capacitative current and the leak current.
• The capacitative current only flows when the membrane potential is changing, and this happens for only a very brief time when the membrane voltage is changed by the voltage clamp, and so can be subtracted out based on its very short time course.
• In the previous unit, we described the passive ungated channels that allowed potassium, sodium and chloride ions to flow through them, generating the resting potential of the membrane.
• Because the permeability of these channels are not affected by voltage, Hodgkin and Huxley chose to lump together as a single equivalent conductance, which they referred to as the leak conductance, symbolized as $g_L$.
• By stepping the membrane to a series of positive and negative voltages, Hodgkin and Huxley established that the leak currents increased linearly with voltage, as illustrated in Figure 8, and then did not change any further with time.
• By stepping to hyperpolarized potentials, the voltage-gated channels were not activated, and the conductance of the leak channels, $g_L$, could be computed by dividing the current by the driving force, which is the difference between the command voltage, and the equilibrium potential for the leak channels, the voltage at which no net current flowed through the channels.
• Once this was done, the leak current, $I_L$ could be predicted from the equation $

   I_L = g_L\left(V_m - E_L\right). \quad\text{(Equation 2)}$

Video on Analyzing the Capacitative and Leak Currents

Analyzing the Capacitative and Leak Currents in the Squid Giant Axon

Evidence for Potassium and Sodium Currents

$Image: TotalIonicCurrentsUnderClamp.png|300px|thumb|Figure 9: Total ionic currents in response to a voltage step from the resting potential (-61 mV) to 0 mV. Capacitative and leak currents have been subtracted]]

• Once the capacitative and leak currents were subtracted from the record, what remained were the active, voltage-gated ionic currents.
• Figure 9 illustrates what these currents look like when the membrane is stepped from the resting potential to 0 mV, a value well above the threshold for generating an action potential.
• Note that there is an initial fairly rapid inward (depolarizing) current, followed by a slower and larger outward (hyperpolarizing) current.
• Hodgkin and Huxley hypothesized that the inward current was due to the inward flux of sodium ions, and the larger, slower outward current was due to the outward flux of potassium ions. How could this be proved directly?
• What Hodgkin and Huxley did to separate the currents was very ingenious. They knew that choline had a single positive charge, and could be substituted for sodium in the medium bathing the squid giant axon, but the choline molecule was much larger than the sodium ions, and would not be able to permeate the sodium ion channels. This would block any sodium ion flow, while maintaining osmotic balance and electroneutrality. They could then step the membrane to different potentials, and measure the current that resulted, which presumably would be due solely to the flux of the potassium ions. Subtracting the total current from these currents containing only potassium, they could estimate the sodium currents.
• Subsequent to their work, specific drugs were identified that could block specific ion channels. In particular, tetrodotoxin (TTX), a venom produced by the puffer fish, was found to selectively block voltage-gated sodium channels, and tetra-ethyl ammonium (TEA) was found to block voltage-gated potassium channels.

$Image: IndividualIonicCurrentsUnderClamp.png|300px|thumb|Figure 10: Sodium (purple) and potassium (blue) currents in response to a voltage step from the resting potential (-61 mV) to 0 mV. Capacitative and leak currents have been subtracted]]

• Using these drugs, subsequent investigators confirmed their results. Figure 10 shows the separate sodium and potassium ionic currents in response to stepping the membrane voltage from the resting potential to 0 mV, well above the threshold for generating an action potential.
• The inward sodium current develops quickly, and then collapses even though the membrane is still depolarized.
• The outward potassium current develops after a delay, and then remains on for as long as the membrane is depolarized.

Analysis of the Potassium Current

• Having isolated the potassium and sodium currents, Hodgkin and Huxley proceeded to analyze the time and voltage dependence of the underlying conductances that generated these currents.
• In this unit, we will focus on their analysis of the potassium current; in the next unit, we will conclude with their analysis of the sodium conductance, which was more complex.

A Video Describing How the Potassium Current was Analyzed

Analyzing the Potassium Current in the Squid Giant Axon

Analyzing the Potassium Current Using Voltage Clamp

• You now have enough information to explore the role of the potassium current in the generation of the action potential.

• Question 10: Press the Reset button. First, set the Leak potential to -70 mV, and set the number of pulses to 1. Set the simulation duration to 60 ms. Run the simulation, and observe the action potential.To explore the role of the potassium conductance in shaping the action potential, now set the conductance of potassium to one tenth of its initial value (i.e., change it from 36 to 3.6 $\mu$S). This is effectively equivalent to shutting down nine-tenths of the voltage gated potassium ion channels, which could be done pharmacologically with a drug like TEA (tetraethylammonimum). To answer these questions correctly, please make sure to actually measure values.
  • a. What happens to the height of the action potential as compared to the one with normal potassium conductance? Please report values. Explain.
  • b. What happens to the width of the action potential (measure it at half the height) as compared to the one with the normal potassium conductance? Please report values. Explain.
  • c. What happens to the after-hyperpolarization as compared to the one with the normal potassium conductance? Please report values. Explain.
• Question 11: Let us explore what happens if the potassium conductance is entirely absent, i.e., set to zero, and whether it is possible to generate a train of action potentials at all.
  • a. Press the Reset button, set the simulation duration to 150 ms, set the number of pulses to 10, and set the Leak potential to -70 mV. What do you observe? Explain briefly.
  • b. Now set the potassium conductance to 0, and re-run the simulation. What do you observe? Explain in terms of the gates; which one is critical?

• Using the voltage clamp, Hodgkin and Huxley were able to examine how the current changed over time at different fixed voltages.

$Image: PotassiumCurrents.png|300px|thumb|Figure 11: Potassium currents in response to three different voltage steps from the resting potential (-61 mV) to 0 mV, 50 mV, and 100 mV. Capacitative, leak and sodium currents have been subtracted]]

• Figure 11 shows how the potassium currents vary over time as the membrane voltage is stepped to different potentials.
• By stepping the membrane to different voltages, Hodgkin and Huxley were able to find the voltage at which no current flowed through the membrane; this allowed them to measure the value of $E_{\text{K}^+}$, the equilibrium or Nernst potential for potassium, since (by definition), when the membrane is at this potential, and one only focuses on potassium ions, there should be no net flux of potassium ions of the membrane (as we established in units 1 and 2).
• As they stepped the membrane potential to increasingly positive potentials, they observed that, after a delay, the potassium current increased, and it grew larger with increasingly positive voltage steps.
• Having measured a whole family of potassium currents as a function of different voltages, Hodgkin and Huxley could then estimate the potassium conductance using the equation $

   g_{\text{K}^+} = \frac{I_{\text{K}^+}}{V_m - E_{\text{K}^+}}, \quad\text{(Equation 3)}$ since they had determined the reversal potential and had measured the potassium currents.

$Image: PotassiumConductances.png|300px|thumb|Figure 12: Potassium conductances in response to three different voltage steps from the resting potential (-61 mV) to 0 mV, 50 mV, and 100 mV. Capacitative, leak and sodium currents have been subtracted]]

• Figure 12 shows some of typical potassium conductance curves that they found.
• Given this data, Hodgkin and Huxley could just have done a curve fit to the data.
• Instead, based on their best guesses about the underlying biophysics of the membrane (which was not fully understood at that time) they hypothesized that there must be pores in the membrane that could allow ions to flow through them rapidly, and that their probability of being open to a given ion was dependent on voltage.
• They assumed that there might be "gating particles" that needed to translocate due to changes in membrane potential, and when this translocation occurred, the ion channel would allow ions to flow.
• The simplest model would be to assume that there is some probability that one of these gates would be conducting. They chose to call this probability n.
• Examining the potassium conductance curves, they observed that the conductance did not turn on immediately, but did so after a delay.
• They could obtain a better fit to the curves by assuming that there were four independent gates.
• To determine the probability of independent events, one multiplies the probability of each event by the probability of each of the other events.
• For example, if the probability of heads occurring in a fair coin is $\frac{1}{2}$, then the probability that 4 fair coins, tossed independently, will all show heads, is $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16}$.
• Similarly, if four independent gates had to translocate to allow the channel to conduct, and the probability of one of them translocating was n, then the overall probability that the channel would become conducting was $

   n \times n \times n \times n = n^4. \quad\text{(Equation 4)}$

• To capture the change in the probabilities that the gates would translocate, Hodgkin and Huxley assumed that it would update according to a simple first order differential equation, with the rate of "turning on" or "turning off" controlled by rate constants that depended on voltage. We will explore these equations in a subsequent unit.
• You now have the information to explore the potassium current, conductances and gates using the following simulation:

• Question 12: Set the Total Duration to 100 ms. Set the Step Delay to 35 ms. Please change the step duration from 4 ms to 20 ms for this and the next question. Please focus only on the red lines for this exercise. In the top graph, the red line represents the potassium current; in the next graph, the red line represents the potassium conductance. In the third graph, the red line represents the n gate. Note that the neuron is held at the Holding Potential of -70 mV. You can step it to a different voltage after the initial delay, by inserting values into the First Step Potential box in the simulation. As you look at your results, please keep in mind that if current were allowed to flow across the membrane, the membrane voltage would change; so think carefully about what changes first when the membrane is stepped to a new potential, what happens next, and so on.
  • a. What is the effect on the n gate of stepping the membrane voltage from the initial Holding Potential of -70 mV to First Step Potentials of -90 mV, -120 mV, and -140 mV, i.e., to different hyperpolarized potentials? Explain.
  • b. What is the effect on the potassium conductance? Explain.
• Question 13: a. What is the effect on the potassium n gate and potassium conductance of stepping the membrane voltage from rest to -60 mV, to -30 mV, and to 0 mV, i.e., to different depolarized potentials?
  • b. What is the effect on the potassium current? Explain.
• Question 14: Now shorten the duration of the voltage step by changing Pulse Duration from 20 ms to 1 ms, and again step the membrane voltage from rest to -60 mV, to -30 mV, and to 0 mV, i.e., to different depolarized potentials.
  • a. What is the effect on the n gate and the potassium conductance?
  • b. What is the effect on the potassium current? Explain, and relate this to the delay in onset and offset of the potassium current during the action potential. In particular, focus on how the individual gate and the overall conductance respond to short pulses (as in this problem) versus long pulses (as in problem 13). Explain. Pay very close attention to the y-axis scales.
• Question 15. Press the Reset button. Again, change the pulse duration to 20 ms. Change the First step potential to -90 mV. Look closely at the initial response of the potassium current to the onset of the voltage step.
  • a. What is the reason that the current is negative? Modify Equation 2, above, to work for the potassium current, and reference it in your explanation.
  • b. To what potential must you change the First step potential so that the current goes to zero? Note that this is the way in which one can determine the reversal potential of a current using voltage clamp, which will be very helpful in the future.