Action Potential IV: Hodgkin-Huxley Equations and Other Conductances

Introduction

• In the unit before last, we found that the Hodgkin Huxley axon acted largely as an on/off switch.

• For rapid conduction of signals over long distances, this is extremely useful.

• However, neurons need to communicate other, more complex forms of information.

• First, they may need to respond proportionally to an input over some range. For example, it is more useful for a sensor for touch to respond proportionally to the strength of the pressure on an area of the skin, rather than simply turning on or off.

• Second, it may be important to be sensitive to whether a signal is changing or not, and so a neuron may become more or less responsive to a constant stimulus to capture this important feature of the environment.

• Third, it may be important for the nervous system to generate activity in the absence of sensory inputs. Animals do not generally wait for food to come to them, for example; they actively go out and seek it, even if there is no immediate sign of food in their vicinity. The crucial feature of neurons that allow them to initiate behavior is spontaneous activity.

• Fourth, it may be important for a neuron to activate the body in rhythmic patterns, for example, activating a muscle that swings a leg forward or pulls it backwards. Neurons that can rhythmically burst play an important role in generating these behaviors.

• Understanding these more complex dynamical properties of nerve cells requires an understanding of additional conductances that go beyond the two we have previously studied.

• To understand how these other conductances work, we need to more fully develop the Hodgkin Huxley equations, so it is clear how they can be generalized.

• Once we've done this, we will explore a few conductances, and then we will briefly describe some of the most recent developments in understanding action potentials and the role of other conductances in their generation.

The Hodgkin Huxley Equations

• We have many of the components of the Hodgkin Huxley equations; we only need to take a few more steps to see the whole set of equations, which will then naturally suggest how they may be generalized.

• In the absence of external current injection, the only currents present are those that charge or discharge the capacitance of the membrane due to the opening or closing of the voltage-dependent ion channels. Thus, the total membrane current, $I_{tot}$, is equal to the sum of the capacitative current ($I_{cap}$) and the sum of the ionic currents, $I_{ionic}$, or $$I_{tot} = I_{cap} + I_{ionic}. \quad\text{(Equation 1)}$$

• Since the capacitative current is a function of the rate of change of voltage, $$I_{cap} = C \frac{dV_{m}}{dt}. \quad\text{(Equation 2)}$$

• For the Hodgkin and Huxley model, the sum of the ionic currents is the sum of the leak, sodium and potassium ionic currents: $$I_{ionic} = I_{L} + I_{\text{K}^+} + I_{\text{Na}^+}. \quad\text{(Equation 3)}$$

• Combining Equations 1, 2 and 3, and assuming that the total current sums to zero, since none is being added or subtracted by an external electrode, we obtain $$C \frac{dV_{m}}{dt} = -I_{L} - I_{\text{K}^+} - I_{\text{Na}^+}. \quad\text{(Equation 4)}$$

• Recall that each ionic current can be re-written in terms of the product of its conductance and driving force: $$I_{ion} = g_{ion}(V_{m} - E_{ion}), \quad\text{(Equation 5)}$$ where $V_{m}$ is the membrane voltage and $E_{\text{ion}}$ is the Nernst or equilibrium potential for the ionic species.

• So, we can re-write Equation 4 as follows, using Equation 5: $$C \frac{dV_{m}}{dt} = -g_{L}(V_{m} - E_{L}) - g_{\text{K}^+}(V_{m} - E_{\text{K}^+}) - g_{\text{Na}^+}(V_{m} - E_{\text{Na}^+}), \quad\text{(Equation 6)}$$ an equation we saw in Action Potentials II when we looked at the electrical equivalent circuit for the membrane.

• Based on the analyses that Hodgkin and Huxley did, we can go further, and fill in the properties of the conductances for the leak, the voltage-dependent potassium channels, and the voltage-dependent sodium channels.

• First, recall that the leak current was just a linear function of the driving force; this is the same as saying that $g_{L}$ is just a constant. To emphasize that it is a fixed, maximal value, it is often written with a bar over the top, as $\bar{g}_{L}$.

• Second, recall that the potassium current, as analyzed in the unit before last, depended on the simultaneous activation of four gating elements, whose probability of moving into a conducting state was represented by the symbol n. Hodgkin and Huxley found the best fit for the data by assuming that there were four of these gates. Since they were assumed to act independently of one another, the overall probability of a potassium channel opening was the probability that all four of the gates had switched into a conducting state, or $n \times n \times n \times n = n^4$. Thus, if the maximum possible conductance for the potassium channels was $\bar{g}_{\text{K}^+}$, the overall conductance for the potassium channels was $$g_{\text{K}^+} = \bar{g}_{\text{K}^+} n^4. \quad\text{(Equation 7)}$$

• Third, recall that the sodium current, as analyzed in the previous unit, is dependent on the simultaneous activation of three gating elements, whose probability of moving into a conducting state was represented by the symbol m, and another gating element, which was initially open at rest, but then closed with a probability h. Thus, by the same reasoning, if the maximum possible conductance for the sodium channels was $\bar{g}_{\text{Na}^+}$, the overall conductance for the sodium channels was $$g_{\text{Na}^+} = \bar{g}_{\text{Na}^+} m^3 h. \quad\text{(Equation 8)}$$

• With these additional definitions, we can now rewrite Equation 6 as follows: $$C \frac{dV_{m}}{dt} = -\bar{g}_{L}(V_{m} - E_{L}) - \bar{g}_{\text{K}^+}n^4(V_{m} - E_{\text{K}^+}) - \bar{g}_{\text{Na}^+}m^3 h(V_{m} - E_{\text{Na}^+}). \quad\text{(Equation 9)}$$

• We now need to address a question that we only looked at qualitatively in the previous two units: how do the gating variables, m, n and h, change with time and with voltage?

• Hodgkin and Huxley made the simplest assumption for how these gating variable would change with time. Let us focus on the n variable to make the argument more concrete. Conceptually, one can think of it this way: if the probability that the gating variable will begin to conduct is n, where n ranges from 0 to 1, then the probability that it will stop conducting is 1 - n. For example, assume that we are at voltage where the gating variable is highly unlikely to move into the conducting configuration. Say that the probability of its moving into the conducting configuration was 0.1. Then the probability of its not moving into the conducting configuration is 1 - 0.1, or 0.9, since at any instant the gating particle is either in its conducting configuration, or it is not.

• How would this probability update with time? They assumed that there would be some rate (call it $\alpha_{n}$) at which a gate that is currently not conducting would become conducting. Thus, in the next time instant, $\Delta t$, the change in the probability, $\Delta n$, would be the rate at which the gate that is not conducting becomes conducting, multiplied by the proportion of gates that are not yet conducting, which we said was 1 - n, so that the total increase in the probability that new gates would move into the conducting configuration would be $\alpha_{n} (1 - n)$.

• By the same logic, if there was some rate (call it $\beta_{n}$) at which a gate that is currently conducting would move into a non-conducting configuration, then the probability that gates would move into the non-conducting configuration would be that rate multiplied by the proportion of gates that are currently conducting, which is n, so that the change in probability that new gates would move into the non-conducting configuration would be $\beta_{n} n$.

• So, the total rate of change in the probability of gates conducting would be the probability that some gates became conducting, minus the probability that some became non-conducting, or $$\frac{dn}{dt} = \alpha_{n} (1 - n) - \beta_{n} n. \quad\text{(Equation 10)}$$

• This equation can equivalently be written in a different form, which you will often see. The derivation for the alternative form is here. $$\frac{dn}{dt} = \frac{(n_{\infty} - n)}{\tau_{n}}. \quad\text{(Equation 11)}$$

• Equation 11 is mathematically identical to Equation 10, but can be thought of conceptually as saying that, at any given voltage, there is a value of the probability of the gating variable n which will be reached if the voltage is held fixed for a very long time (i.e., $n_{\infty}$), and the rate of change of the probability n will go to zero once this value is reached. The rate at which n reaches this steady-state value depends on a time constant, $\tau_{n}$. If the time constant is large, it will take the gating variable a long time to reach its steady-state value; if the time constant is small, the gating variable will reach its steady-state value quickly.

• We are almost done. We need a way of describing how the time constants change with voltage.

• The key idea is that the probabilities of opening or closing are related to the voltage across the membrane.

• Hodgkin and Huxley assumed that a change of a gating element is like a chemical reaction, and that the rate relationships could be thought of as illustrated in the following equation, using $k_{1}$ to represent that rate at which the gating elements move from the closed to the open configuration, and $k_{-1}$ to represent the rate at which the gating elements move from the open to the closed configuration: $$\text{Closed} \overset{k_{1}}{\underset{k_{-1}}{\rightleftharpoons}} \text{Open}. \quad\text{(Equation 12)}$$

• At equilibrium, the rate of opening will become equal to the rate of closing, so that the equilibrium constant $K$ will be $$K = \frac{k_{1}}{k_{-1}}. \quad\text{(Equation 13)}$$

• At constant temperature and pressure, which is a reasonable set of conditions to assume for channel gating processes, one can define the energy available for a reaction to occur, also known as the Gibbs energy. At equilibrium, thermodynamic arguments show that $$\Delta G^\circ = - RT \ln \ K, \quad\text{(Equation 14)}$$ where $\Delta G^\circ$ is the Gibbs energy under standard conditions (i.e., fixed temperature and pressure), $R$ is the universal gas constant, $T$ is the absolute temperature, and $K$ is the equilibrium constant we defined in Equation 13.

• Now the Gibbs energy has two components: that due to the chemical reaction that allows the ion channel to shift from an open to a closed configuration, and an electrical component due to the voltage field across the membrane, and the charges on the part of the channel that induce it to respond to that field. Thus, $$\Delta G^\circ = \Delta G_\text{chem} + \Delta G_\text{elec} = \Delta G_\text{chem} + zFV_{m}, \quad\text{(Equation 15)}$$ where $z$ is the charge on the ion channel, $V_{m}$ is the voltage across the membrane, and $F$ is Faraday's constant. Combining Equations 14 and 15, and solving for K, we obtain $$K = e^{-\tfrac{\Delta G^\circ}{RT}} = e^{-\tfrac{\Delta G_\text{chem} + \Delta G_\text{elec}}{RT}} = e^{-\frac{\Delta G_\text{chem} + zFV_{m}}{RT}} = e^{-\tfrac{\Delta G_\text{chem}}{RT}} e^{-\tfrac{zFV_{m}}{RT}}$$

• If we define $K_{o}$ as $e^{-\tfrac{\Delta G_\text{chem}}{RT}}$, then this simplifies to $$K = K_{o} e^{\tfrac{-zFV_{m}}{RT}}. \quad\text{(Equation 16)}$$

• These considerations were the basis for the actual equations that Hodgkin and Huxley used to fit the data they had obtained for the voltage dependence of the rate constants. Thus, for example, for the rate constant $\alpha_{n}$ describing the rate at which the n gating variable went from the closed to the open configuration as a function of voltage, they used the equation $$\alpha_{n}(V_{m}) = \frac {-(V_{m} - V_{2})} {V_{1} \left(e^\tfrac{-(V_{m} - V_{2})}{V_{3}} - 1\right)}, \quad\text{(Equation 17)}$$ where $V_{1}$, $V_{2}$, and $V_{3}$ were parameters that allowed them to shift the center of the curve and alter its sharpness, which they fit to their data.

• There was another equation for the rate constant $\beta_{n}$, describing the rate at which the n gating variable went from the open to the closed configuration as a function of voltage: $$\beta_{n}(V_{m}) = V_{4} e^\tfrac{-(V_{m} - V_{5})}{V_{6}}, \quad\text{(Equation 18)}$$ where $V_{4}$, $V_{5}$, and $V_{6}$ were again parameters fit from their data.

• The same logic led to two additional differential equations for the m and h gating variables, $$\frac{dm}{dt} = \alpha_{m} (1 - m) - \beta_{m} m, \quad\text{(Equation 19)}$$ and $$\frac{dh}{dt} = \alpha_{h} (1 - h) - \beta_{h} h. \quad\text{(Equation 20)}$$

• Of course, each of the four rate constants for Equations 19 and 20 were also voltage dependent, leading to these four equations: $$\alpha_{m}(V_{m}) = \frac {-(V_{m} - V_{8})} {V_{7} \left(e^\tfrac{-(V_{m} - V_{8})}{V_{9}} - 1\right)}, \quad\text{(Equation 21)}$$ $$\beta_{m}(V_{m}) = V_{10} e^\tfrac{-(V_{m} - V_{11})}{V_{12}}, \quad\text{(Equation 22)}$$ $$\alpha_{h}(V_{m}) = V_{13} e^\tfrac{-(V_{m} - V_{14})}{V_{15}}, \quad\text{(Equation 23)}$$ $$\beta_{h}(V_{m}) = \frac {1} {V_{16} \left(e^\tfrac{-(V_{m} - V_{17})}{V_{18}} + 1\right)}. \quad\text{(Equation 24)}$$

• So the key equations for the Hodgkin and Huxley model are Equations 9 and 10, and 17 through 24. A list of the parameters for the model is here.

• Given this information, you could type in the Hodgkin and Huxley equations, and their parameters, into your favorite differential equation solver (e.g., MATLAB, Mathematica), and obtain action potentials.

An Informal Introduction to the Hodgkin Huxley Equations

Understanding the Hodgkin Huxley Equations Informally

An Informal Introduction to Other Ionic Conductances

Understanding Other Ionic Conductances, especially the Fast Potassium (or A) Current

Other Conductances and their Functional Consequences

• Thanks to studies on molluscan nerve cells other than the squid giant axon, and to studies on mammalian nerve cells using patch clamp techniques, it has become clear that there are many other ionic conductances beyond the two characterized by Hodgkin and Huxley.

• Because you will see this in the literature, you should know that the voltage-dependent potassium conductance that Hodgkin and Huxley studied is often referred to as the delayed rectifier current. This name comes from the delay prior to the conductance turning on, and because it acts as a rectifier, that is, it tends to shunt current injected into the nerve cell.

• Although many other conductances were discovered, the tools, methodology, and models pioneered by Hodgkin and Huxley have continued to set the agenda for how biophysicists analyze excitable membranes. The general approach is to find pharmacological agents that block the channel; use voltage clamp to characterize its activation and inactivation when other channels are blocked; and use ion substitution to determine which ions are permeated by the channel.

• More recently, investigators have sequenced the genes expressing different ion channels, cloned them, and expressed them in systems that are much more easily studied than the original nerve cells in which they were found (e.g., in the oocytes of the frog Xenopus, which are large and relatively easy to maintain in culture).

• To give you a feeling for the other conductances that have been discovered, and some of the functional consequences that are associated with them, we will allow you to do some experiments with simulations of them. The simulations below are based on actual values measured in hypoglossal motor neurons and used in a model published by Purvis and Butera in 2005.

• Here is a simulation that incorporates additional conductances, and allows you to study them using current clamp:

• Once again, you may want to open up the simulation in a different web page as you work through the questions.

• Question 1: If you click on the simulation, you will see that it lists multiple potassium currents, sodium currents, a nonspecific current, and multiple calcium currents. Because looking at all these currents at once is confusing, the next few questions will allow you to examine them one at a time. The first conductance that we will examine is known as the sag or $I_{H}$ conductance (in the simulation, it is listed under the Nonspecific Currents as H-current). The reason for the name will become clear as we look at its effects. Note that the $I_{H}$ conductance's reversal potential is -38.8 mV. What does this imply about the ions that permeate this channel? Could it only permeate potassium ions, or only sodium ions? Explain.

• Question 2: Please click on the button labeled "Sag current". Please set the Stimulus current first pulse to -1 nA, the Stimulus delay to 50 ms the Pulse duration to 200 ms, and the Total duration to 300 ms. Under Potassium Currents, set the Delayed rectifier potassium conductance to 0 and under Sodium Currents, set the Fast transient sodium conductance to 0. Under Nonspecific Currents, set the H-current conductance to 0, and run the simulation. Take a screenshot of the Membrane Potential, and Stimulation Current plots. Note the final voltage reached during the hyperpolarizing pulse, and also note its shape. What features of the membrane are responsible for this voltage change? Recall the electrical equivalent circuit equation that you worked with in the Passive Membranes unit.

• Question 3:
  • Now set the H-current conductance back to 0.005 $\mu S$. Note the scale of the y-axis. How does the voltage change during the initial part of the hyperpolarizing pulse? What is the maximum negative value that the voltage reaches after the pulse has been on for 200 ms? Take a screenshot of the Membrane Potential, and Stimulation Current plots (2 plots).
  • The change in conductance and current due to the $I_{H}$ current is plotted in gray in the Sag Current, Conductance and Gate plots. What is the $I_{H}$ current doing? Explain, including the reason it is also known as the "sag" current. What has happened to the value of the resting potential after the current pulse stops? Explain in terms of the electrical equivalent circuit equation that you worked with in the Passive Membranes unit (Equation 7 in that unit). Take a screenshot of the Sag Current, Conductance, and Gates (3 plots).

• Question 4:
  • Press the Sag current button again to restore the Hodgkin Huxley currents. To see what happens after a hyperpolarizing current pulse when no sag current is present, set the H-current conductance to zero, set the Stimulus current first pulse to -0.5 nA, the Stimulus current subsequent pulse to 0.1 nA, the Pulse Duration to 100 ms, the Inter-stimulus interval to 0, the Number of pulses to 2, and the Total duration to 250 ms. Run the simulation, and observe what happens immediately after the hyperpolarizing pulse ends. Please explain what you observe. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Sag Current, Conductance, and Gates (3 plots).
  • Now re-set the H-current conductance to 0.005 $\mu S$. What do you observe? Please explain this in terms of the sag current and its effect on the Hodgkin Huxley gates. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Sag Current, Conductance, and Gates (3 plots).
  • Now set the Stimulus current first pulse to 0 nA, and again run the simulation. What do you observe? What is the importance of the initial hyperpolarization for the action of the sag current? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Sag Current, Conductance, and Gates (3 plots).

• Question 5: Nerve cells contain a wide variety of voltage-gated channels that allow the ion calcium to enter the nerve cell. Calcium is tightly regulated within nerve cells. The free ionic calcium concentration is usually kept at very low levels inside the cell relative to the extracellular concentration of free calcium using a variety of pumps, buffers, and special organelles that take up and sequester calcium (e.g., the mitochondria and the endoplasmic reticulum). In the next part of this problem set, we will explore three calcium conductances (T, N and P), their effects on excitability and their effects on intracellular calcium levels.
  • To see how the neuron behaves in the absence of any of the calcium conductances, please press the button Simple simulation, and observe the response of the model neuron before and after the current pulse. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots). Now, please press the Calcium currents button, and observe the response of the model neuron before and after the current pulse. Looking at the current plots under the Calcium Currents, Conductances, and Gates, which of the three calcium ion channels generates the largest current? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Calcium Currents, Conductances, and Gates (3 plots).
  • Set that channel's conductance to zero, and again run the simulation. What do you observe? Explain. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Calcium Currents, Conductances, and Gates (3 plots).
  • Which of the two remaining calcium ion channels now generates the largest current? Set its conductance to zero, and again run the simulation. What do you observe? Explain. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Calcium Currents, Conductances, and Gates (3 plots).

• Question 6: We can now examine the effect of the calcium currents on the accumulation of calcium within the nerve cell. To understand this properly, think about a buffer as a kind of "sponge" that soaks up calcium so that it is no longer in the neuron's cytoplasm. If the buffer has a short time constant, it works quickly to clear calcium from the cytoplasm.
  • Please press the Calcium currents button. Now set the T-current conductance, the N-current conductance, and the P-current conductance to zero. Run the simulation. Please measure the amount of calcium at the beginning of the simulation and after 40 ms (i.e. measure values from the Intracellular Calcium Concentration graph). In addition, take a screenshot of the Intracellular Calcium Concentration graph (1 plot).
  • Please press the Calcium currents button again, which will restore the conductances of the calcium currents, and again measure the amount of calcium at the beginning of the simulation and after 40 ms. What do you observe? Explain. Take a screenshot of the Intracellular Calcium Concentration plot (1 plot).
  • How does the buffer affect the long term internal calcium levels? Understanding this will be very important for understanding the next conductance, the calcium-dependent potassium conductance. To explore this, change the value of the Calcium buffering time constant from from 25 ms to 5 ms. Please measure the total amount of calcium that accumulates by the end of 40 ms. What do you observe? Explain. Now change the Calcium buffering time constant to 100 ms. Again, please measure the total amount of calcium that accumulates by the end of 40 ms. What do you observe? Explain. Take screenshots of the Intracellular Calcium Concentration plots for these two scenarios. (1 plot each)

• Question 7: Calcium is not only an ion that permeates channels; it can act as an effector and second messenger within cells. Thus, fluxes of calcium not only contribute to changes in the potential of the membrane, but can also contribute to many intracellular processes. An example of one of its effects is its ability to affect a special potassium conductance. These channels open in response to both voltage and calcium levels. We can now use the simulation to explore the properties of this conductance.
  • Please press the Ca and Ca-dependent K currents button. Please change the Total duration of the simulation to 100 ms, and run the simulation. What do you observe after the current pulse? How does the response of the neuron differ from what you observed in Question 5, when there was no calcium-dependent potassium current, but all three calcium currents were present? Please measure the membrane potential at 50 ms. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots).
  • Now examine the Calcium-Dependent Potassium Current, Conductance, and Gate. Take a screenshot of the Calcium-Dependent Potassium Current, Conductance, and Gate (3 plots). Please use their changes to account for what has happened to the membrane potential. To test your hypothesis, turn off the calcium currents (i.e., set their conductances to zero), and again run the simulation. Measure the membrane potential at 50 ms. How has it changed? Explain. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Calcium-Dependent Potassium Current, Conductance, and Gate (3 plots).
  • How does the time constant of calcium buffering affect the duration of the calcium-dependent potassium current? Please press the Ca and Ca-dependent K currents again. Change the Total duration to 300 ms. Run the simulation and measure the duration of the afterhyperpolarization in the top voltage trace. Take a screenshot of the Membrane Potential (1 plot).
  • Now change the value of the Calcium buffering time constant from 25 ms to 100 ms. Run the simulation. What do you observe about the duration of the afterhyperpolarization? Explain in terms of the calcium buffering and the calcium-dependent potassium current. Take a screenshot of the Membrane Potential (1 plot).
  • Now change the value of the Calcium buffering time constant to 6 ms. Run the simulation. What do you observe about the duration of the afterhyperpolarization? Explain in terms of the calcium buffering and the calcium-dependent potassium current. Take a screenshot of the Membrane Potential (1 plot).

• Question 8: There are multiple kinds of sodium channels. In this simulation, you can study one of them, the persistent sodium current, which (unlike the more usual sodium current that contributes to the action potential) takes a very long time to inactivate.
  • Please press the button Persistent sodium current. Note that no current is now injected into the model neuron. What do you observe? Explain what is happening in terms of the graphs of the Persistent Sodium Current, Conductances, and Gates. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Persistent Sodium Currents, Conductances, and Gates (3 plots).
  • Now, under Current Clamp, please set the Stimulus delay to 1 ms, the Pulse duration to 30 ms, and the Number of pulses to 1, and please set the Persistent sodium conductance to 0 $\mu S$. Run the simulation. How does the neuron respond? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Hodgkin-Huxely Currents, Conductances, and Gates (3 plots).
  • Now restore the Persistent sodium conductance to 0.05 $\mu S$. How does the neuron respond? Please explain both in terms of the persistent sodium conductance and in terms of the Hodgkin Huxley gates. Which of these gates is most likely to account for the observations? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), the Hodgkin-Huxely Currents, Conductances, and Gates (3 plots), and the Persistent Sodium Currents, Conductances, and Gates (3 plots).

• Question 9: There are a wide variety of other potassium conductances, which play a variety of important roles in shaping the behavior of nerve cells. We will explore one of them, known as the fast potassium or $I_{A}$ conductance.
  • Please press the Fast potassium current button. What happens during the current pulse? Under the Current Clamp menu, please increase the Pulse duration to 3 ms. What happens now during the current pulse? Please explain, using the graphs of the Fast Potassium Current, Conductance, and gates. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Fast Potassium Currents, Conductances, and Gates (3 plots).
  • The $I_{A}$ conductance, unlike the delayed rectifier potassium current of the Hodgkin Huxley model, inactivates, and this inactivation can be removed through hyperpolarization. Under Current Clamp, please set the Stimulus current first pulse to 0, the Pulse duration to 100 ms, Inter-stimulus interval to 0, and the Number of pulses to 2. Change the Total duration to 250 ms. Run the simulation. How many spikes do you observe during the depolarizing current pulse? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Fast Potassium Currents, Conductances, and Gates (3 plots).
  • Now, change the Stimulus current first pulse to -0.5 nA, and again run the simulation. How many spikes do you observe during the depolarizing current pulse? Please explain what has happened, again making reference to the graphs of the Fast Potassium Current, Conductance, and gates. Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots), and the Fast Potassium Currents, Conductances, and Gates (3 plots).

• Question 10: What is the functional significance of having so many different conductances? One important feature of nerve cells is their ability to respond with different firing frequencies to a given sensory input. We can explore this using the simulation.
  • Please press the Simple simulation button. Set the Stimulus current first pulse to 0.5 nA, the Pulse duration to 180 ms, and the Total duration to 200 ms. Run the simulation. How many action potentials do you observe? What is the period between the first and the second, and between the last two? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots) You can measure the period from peak to peak, trough to trough, or at the half-width, but you must explain how you did it.
  • Now add in all the currents that we have examined separately by pressing the Full simulation button. Again, set the Stimulus current first pulse to 0.5 nA, the Pulse duration to 180 ms, and the Total duration to 200 ms. Run the simulation. How many action potentials do you observe? What is the period between the first and the second, and between the last two? Take a screenshot of the Membrane Potential and Stimulation Currents (2 plots)
  • Please explain how the different conductances in the full simulation contribute to this result. Which current or currents are most important? Please test your hypotheses by reducing the conductance of a single current, and examining the effects. Then restore the conductance to its original value, before altering a second conductance.
  • Show whichever pictures are necessary to prove your hypothesis. You will know that you understood the unit if you can answer this question.

In the next unit, we will explore these conductances using voltage clamp, and also use this knowledge for the analysis and design of multi-conductance nerve cells.