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LAB 4

1. (5 points) Paste screenshots of your front panel for test cases #1 and #2 from Section 3.2.1. Are the membrane voltage responses the same or different? Explain the differences in results, you see and explain why they are different. Figure 1. Test Case #1: Hodgkin-Huxley Model Front Panel

Figure 2. Test Case #2: Hodgkin-Huxley Model Front Panel The objective for this section was to find the minimum applied current for inducing an action potential. It was seen that the sodium and potassium channel conductance behave similarly to the membrane voltage, as the conductance also experiences a sudden increase at threshold. The membrane voltage responses are different as seen in Figures 1 and 2 . This is due to the different t on and t off values altering the period of the waveform. In case 1, there was an on time of 10 seconds and an off time of 0 seconds. Whereas case 2 had an on time of 1 second and an off

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time of 9 seconds. Since the period has decreased, the action potential does not fully propagate. Therefore, case 2 does not decay for a full action potential. 2. (5 points) In your own words, explain how the changing conductance of the ion channels generates the action potential observed? In the lab, these changing values were illustrated with mathscript nodes. A difference in period illustrated a significantly different change in the values of the ion gating variables. This contributed to the observed differences in the voltage waveform. The Hodgkin-Huxley model uses differential equations to represent the ions’ conductance over time, which generate the action potential. The g Na term is for sodium and g K term is for potassium An action potential is generated upon alterations to the conductance of each of the ion channels, potassium and sodium. The sodium, Na+, channels open, commencing the generation of the potential, causing depolarization within the axon. Then, as the potassium, K+, channels open, which causes repolarization. This difference in polarity generates the impulse response down the axon terminal towards the synapse. There are also other cellular ions that cause leakage of these ions, such as chlorine. 3. (5 points) In Section 3.3, you are asked to change the value of the step size and the type of ODE solver. Explain why these changes are necessary. This RK4 works with variable step sizes as the original RK23 ODE does not work well with rapid changing elements. The RK4 solver utilizes a fixed time step input, and allows for modeling the rate of change of the current as a result of the PID controller. The step size had to be decreased to properly model this rate of change, and thus was set to .001. As a result of these changes, the plots are more accurate due to utilizing more data points. The step size is at a faster rate of 1 microsecond from 1 millisecond due to the rapid change of current. 4. (5 points) In your implementation of the PID controller for the voltage, you were asked to include a “Saturation” block. What is the purpose of this block? The saturation block is a point where something being inputted cannot exceed a certain amount depending on the company or in this case would be the user. It is used in order to limit the acceptable range of a PID signal. In this system, a saturation block was used in order to limit the range of the applied current signal the controller allowed to output, from -6000 μA/cm2 to +6000

μA/cm2. This was done by wiring a control block to each of the upper and lower limit terminals of the saturation block, and setting them to the appropriate values. 5. (10 points) Paste screenshots of your front panel for the 4 reference voltage values listed in test case #1 of Section 3.3.1. Also paste the results from test case #2. Discuss the process you used to tune your PID controller in order to be able to generate these outputs. Figure 3. Front Panel for Case #1 (V 0 =25mV)

Figure 4. Front Panel for Case #1 (V 0 =50mV) Figure 5. Front Panel for Case #1 (V 0 =75mV)

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Figure 6. Front Panel for Case #1 (V 0 =100mV) Figure 7. Front Panel for Case #2 Initially, the Ki and Kd were set to 0, and then increased Kp near reference. Tuning the PID controller to produce these waveforms was an iterative process with k p , the proportional gain. This value increased until the steady state value was within 20% of the reference value. A 25mV was the reference voltage, and increased. Then, the k i , or integral gain was set. The purpose of this is to reduce the steady state error. The larger the value, the lesser the error. It was seen that small values were the most ideal, so it was set to 1. Kd was increased, as it was used to control the overshoot. Also, Ki was increased which eliminated oscillations that occurred initially and reduced the error.

6. (5 points) For the reference voltage 25 mV, compare and contrast the results of using a PI, a PD, and a PID controller. You should include screenshots for each type of controller. Explain the differences in the results that you obtain. Figure 8. Voltage Clamp w/ PI Controller

Figure 9. Voltage Clamp w/ PD Controller Figure 10. Voltage Clamp w/ PID Controller A PI controller is used to reduce error of the system, whereas a PD controller is used to reduce oscillations of the system’s and the PID controller is used for both of the reasons above and to further enhance the stability of the system. The PI controller had minimal steady state error compared to the PD; however, it has a large settling time. The PD controller has a significant steady state error, but a small percent overshoot and rise time. However, the PID controller is superior, as it has the smallest steady state error and reduced percent overshoot. 7. (5 points) Insert a copy of the table from Exercise 1 of Section 3.3.2.2 with the data you collected in the voltage clamp mini-experiments. Voltage clamp reference value (mV) Steady-state applied current (μA/cm2 ) Estimate of n_00(v0 ) from 20 172.08 0.620

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40 800 0.805 60 1672.03 0.8919 80 2589.4 0.941 100 3477.2 0.964 Table 1. Steady-State Applied Current Values 8. (5 points) Paste a copy of the plot you created in Exercise 2 of Section 3.3.2.2. How well does the data match the theoretical result? What are the limiting factors in making this “Measurement?” aaaaaaaaaa Figure 11. Measured n ∞ Values versus Predicted n ∞ Values by Reference Voltage Here, the theoretical values and measured values were basically identical. There was only a little difference at 60 and 80 mV, but is negligible. The limiting factor is the value for tau, as it must be determined with high accuracy at the right time. This will allow for the theoretical value to be solved correctly. Furthermore, not having enough voltage reference values can make the actual values inaccurate compared to the theoretical. 9. (5 points) Insert a copy of the table from Exercise 3 of Section 3.3.2.2 with the data you collected in the voltage clamp mini-experiments. 10. (5 points) Why is it important that your PID controller achieves the reference voltage in less than 1 msec?

The steady state should be reached in under 1ms, since the waveform must stabilized. Here, the change in the gating variable with time needs to be zero, which it is at steady state. 11. (5 points) Paste a copy of the plot you created in Exercise 4 of Section 3.3.2.2. How well does the data match the theoretical result? What are the limiting factors in making this “Measurement?” 12. (10 points) Paste a copy of the plot you created in Exercise 5 of Section 3.3.2.2. How well does the data match the theoretical result? What are the limiting factors in making this “Measurement?” 13. (5 points) When studying the potassium ion channel, why is it important to inhibit the sodium channel? In the body, if the sodium, Na+, channels were opened the same time as the potassium, K+, channel then there would be an equal amount of ions coming into and out of the cell simultaneously. Therefore, such an amount would mean equilibrium and no action potential would be generated for the cell. This is proven from the equations given, and the results would give incorrect reading. Inhibiting the sodium channel allows for a better understanding and uninhibited observation of how potassium behaves. 14. (10 points) The experiments of Section 3.3.2.2 were designed to characterize the amount potassium ion channel gating function (i.e., the reaction rate coefficients for n). The most direct approach to this would be to study the current flowing due to potassium ions; however, rather than measuring this current, we actually applied a current (i.e., the Iapp term). Describe in your own words why this approach actually still yields the information we are interested in obtaining. 15. (15 points) Describe in your own words (without using equations) how you use the applied current data to determine the reaction rate coefficients ( and ). In other 𝛼𝑛 𝛽𝑛 words, explain the dependencies between the reaction rate coefficients and the various

intermediate parameters derived from the applied current data

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