Sample HHSim Exercises

I. Equilibrium Potential
II. Membrane Potential
III. The Action Potential
IV. The Fast Sodium Channel
V. The Delayed Rectifier
VI. Voltage-Gated Channel Parameters

Part I: Equilibrium Potential

We will first explore the equilibrium potential of a cell with a single channel type. Click on the Channels button in the main window to call up the Channels window. In the Channels window, turn off the all the channels except the first one (passive sodium), by clicking on their respective buttons. The buttons should change color to gray when you turn them off. Notice that the resting potential Vr displayed in the Membrane window is now equal to the reversal potential for sodium, 52.4 mV. A mouse click on the red line will show you the exact value of Vm. Vm may not be exactly equal to Vr, but clicking the yellow Nudge button a few times will nudge things along until the voltage reaches the theoretical asymptote. Or you can click Run, wait a while, and then click Stop.

Hit the Reset button in the Membrane window to restore all parameters to their initial values.

Part II: Membrane Potential

In exercise we will look at the membrane potential with only passive channels (no voltage gated sodium or potassium channels.)

First, reset the simulator. If you are running from source code using Matlab, type "hhsim" again in the Matlab Command Window. (Don't hit the green Run button, and make sure the simulator is stopped before you type "hhsim" in the Command Window). If you're instead running a stand-alone executable version of the simulator, just kill the application and start it again.

Turn off the yellow and green plots in the main window by selecting "-" in their respective pulldown menus. Set the cyan (light blue) plot to "I_leak", the leakage current. I_leak is equal to the flow of current through the passive channels. By looking above and below the cyan pulldown menu, you will see that the scale of the I_leak plot ranges from -0.05 to 0.05 pico-Amp. Display the Membrane and Channels windows as well.

In the Channels window, turn off the two voltage gated channels (pink buttons) so that only the three passive channels (green buttons) are functional. Note that the membrane voltage (red line) rises from its previous level of roughly -62 mv to a new level of around -48 mv. The simulator stops when the membrane voltage is close to asymptote. To get an exact resting value, click the yellow Nudge button a few times until the value displayed for Vm stops changing.

The rise in Vm shows that at the normal resting potential, some voltage-gated channels as well as leak channels were open. When the voltage-gated channels are disabled, the only open channels are the leak channels, and resting membrane voltage rises. Since membrane current must be 0 at Vrest, the leak current (blue line) is zero.

Part III: The Action Potential

Reset the simulator again (see instructions in Part II). Click on the purple Stim1 button in the main window to stimulate the cell and generate a spike. You can see the stimulus plotted as a purplish line, while the spike is visible as the membrane voltage plotted as a red line. Click on the Stim2 button to apply a hyperpolarizing pulse to the cell. Notice that this also results in a spike.

Click on the Stimuli button in the main window to call up the Stimulus Parameters window. Notice that button Stim1 is set up to provide a 1 millisecond stimulus at +10 nano-amps. Using the slider controls, you can add a second pulse to Stim1. Set up a second 10 nA pulse 5 msec after the first. (Note: clicking on the slider changes the value by 5. Clicking on the arrowheads changes the value by 1. You can also type values directly into the box next to each slider. Moving the mouse over a slider or text box dispays some pop-up text explaining its function.)

Notice that two pulses spaced 5 msec apart do not cause two spikes.

A negative pulse occuring soon after a positive pulse can prevent a spike from occurring. Set up the Stim1 parameters to provide a 5 nA pulse folowed 1 msec later by a -5 nA pulse. You'll see that the 1 ms interval is too long to prevent the spike. Shorter intervals can be obtained by typing a fractional value like 0.9 into the text for the middle horizontal slider.

Part IV: The Fast Sodium Channel

Reset the simulator again. Press the Stim1 button to stimulate the cell and generate a spike. The yellow, green, and cyan (light blue) lines in the upper plot display the Hodgkin-Huxley variables m, h, and n, respectively. The variables take on values between 0 and 1. Recall that the fast sodium channel conductance is proportional to m3h, and potassium channel activity is proportional to n4.

In this exercise we will focus on just the sodium channel, which consists of an activation gate and an inactivation gate. Both these gates must be open in order for current to flow. The m3 term describes the state of the activation gate, and the h term describes the state of the inactivation gate. The closer these values are to 1, the more "open" the gate is.

Using the cyan pop-up menu, set the cyan plot to display g_Na, the sodium conductance, which is proportional to m3h. The numbers above and below the yellow, green, and cyan windows represent the y-axis values of the uppermost and lowest dashed lines, respectively, in the upper plot. The range for the cyan plot is now 0 to 30 pico-Siemens. Click on Zoom In to get a better view, and use the scroll bar to position the plot as necessary.

As you can see, the inactivation gate variable h (green line) has a resting value of around 0.5, so it's normally part-way open. But the activation gate variable m (yellow line) normally rests close to 0. This is why the total sodium conductance (cyan line) is close to zero pico-Siemens at rest.

When a stimulus depolarizes the cell slightly, the activation gate opens (yellow line "m" rises), the sodium conductance (cyan line) increases, and the cell depolarizes further (red line Vm rises). However, as the membrane voltage increases, the inactivation gate begins to close (green line "h" decreases to zero), and this causes the overall conductance (cyan line, proportional to m3h) to drop even though the yellow line remains high (activation gate remains open) for quite a while.

What eventually causes the yellow line to drop is the repolarization of membrane potential (indicated by the red line dropping) as a result of opening of voltage dependent potassium channels. Without the potassium channels, the cell would remain depolarized. To demonstrate this, we will do an experiment.

Call up the Channels window and disable the Delayed Rectifier (potassium) channel. There is normally some leakage current through this channel when the cell is at rest; removing it triggers a spike. The cell depolarizes but cannot repolarize, because now there is no active potassium current. The membrane voltage thus remains high, and the yellow line stays high.

The spike peak is around +50 mV. With no active potassium channel conductance, the cell does not fully repolarize, but the membrane voltage does decline some from the peak value. Notice that if Vm is perturbed from its new resting value by a depolarizing or hyperpolarizing stimulus (try the Stim1 and Stim2 buttons) it returns to the new resting value.

The new value of Vm is determined by the conductances of two channel types: the leak channels (which have conductance to Na, K, and Cl), and the (not quite fully closed) fast voltage dependent sodium channels. Although it may appear that in this state h = 0 and hence g_Na = 0, they are actually slightly positive. Because the leakage conductance is so low, the slight steady-state activation of voltage gated Na channels still has a large effect on Vrest.

Part V: The Delayed Rectifier

Reset the simulator again. Press Stim1 to stimulate the cell and generate a "normal" spike, for reference. Then call up the Channels window, and disable the fast sodium channel. The resting potential hyperpolarizes only slightly, indicating that very few voltage dependent Na channels are open under normal resting conditions.

Turn off the yellow plot (select "-") in the pop-up menu and set the green plot to "I_K", the delayed rectifier current.

Call up the Stimulus window and set Stim1 to a 100 nA stimulus lasting 5 msec. Press the Stim1 button in the main simulator window to depolarize the cell, and see what happens.

Part VI: Voltage-Gated Channel Parameters

Hodgkin and Huxley explained the behavior of voltage-gated channels in terms of activation and inactivation gates that opened or closed based on the membrane voltage. These "gates" are pieces of channel structure (i.e., segments of the amino acid chains that make up the channel) which change their shape or position based on the potential across the membrane. The m3 term in the expression m3h for the fast sodium channel indicates that three segments of the channel must change their conformation to open the activation gate, while the h term indicates that only a single piece of structure controls the inactivation gate.

At the molecular level channel components do not just sit in a fixed position at temperatures above absolute zero,. They continually change their conformation. However, some conformations are more likely than others, and therefore occur more frequently. The channel characteristics determine which configurations are more likely, as a function of the current membrane voltage. The parameter m can be viewed as the probability that a particular segment of the activation gate is in the configuration required for the gate to be open. Alternatively, it can be viewed as the fraction of channels in which this segment is in the configuration required for the gate to open. Since all three segments must be in the open position for the activation gate to be open, and the segments are independent, the probability that the gate is open (or equivalently, the fraction of channels with open activation gates) is m3.

Reset the simulator again. Call up the Channels window, and press the Details button for the fast sodium channel. This displays a window showing the parameters responsible for the behavior of the fast sodium channel. Looking at the top of the display, we see that this channel is permeable to the sodium ion, and that its maximum conductance g_max is 120 mico-Siemens.

The left half of the Fast Sodium details window describes the activation gate, while the right half describes the inactivation gate. The behavior of an activation gate segment is governed by two rates: alpha is the rate at which segments move from the closed to the open state (causing m to increase), while beta is the rate at which they move from the open to the closed state (causing m to decrease). Alpha and beta are determined by exponential functions whose parameters are shown in the window. These equations are functions of Vm. In the graph at the bottom, alpha is plotted in red, while beta is plotted in blue.

Looking at this plot, we see that when Vm = -50, alpha is much less than beta. That means segments will move from the open to the closed configuration at a higher rate than they move from closed to open. As a result, the net change in m will be a decrease.

Now press Stim1 to trigger a spike, which we will use for reference.

The state of the inactivation gate is indicated by the variable h, whose parameters are displayed in the right half of the Details window. Suppose we want to lengthen the duration of a spike. We can do that by slowing down the rate at which the inactivation gate closes, so that the channel remains open longer. Beta is the rate at which the inactivation gate will close. Change the beta magnitude parameter (c) from 1.0 to 0.4. Then press the yellow Nudge button in the main simulator window a few times. The cell starts spiking and doesn't stop. Press the red Stop button to stop the simulation. You should notice several things:

  1. The spike width is now greater, as we intended.

  2. The green line (h) decreases more slowly during a spike, and does not get as close to zero as it did previously. This is the direct consequence of our decreasing the h beta magnitude.

  3. The yellow line (m) stays high for longer. This is because the cell is remaining depolarized longer. Remember that it is the closing of the inactivation gate (h) that cuts off the sodium influx and allows the cell to repolarize. If the gate closes more slowly, Vm stays high longer, and hence the activation gate remains open longer.

  4. The peak of the spike is higher than before.

As you can see, the cell is quite sensitive to parameter changes; altering a single value produces a complex set of effects. We may conclude that the channels governing a cell's behavior form an exquisitely tuned system. And here we're only dealing with two channel types; keep in mind that some cells have a dozen!

This work was supported in part by National Science Foundation grant DGE-9987588. Any opinions, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Dave Touretzky
Last modified: Sun Sep 13 22:44:04 EDT 2015