Reference no: EM13991324

The lab 1 2 3 in the hw doc is lab 4 5 6 attached. i have the codes of lab 4 5 6 i can send it to you if you need it

?You need to answer Word file only.the labs are just to assist us through it

i have the codes for lab 1 and 2 which are the doc i send named lab 4 5 respectively?

Part -1:

Lab: Neural Modeling

An axon is made up of sequential Nodes of Ranvier ("Ron-ve-a") separated by myelinated "internodes." In this lab, you will learn about the two of the ion channels that exist in a Node of Ranvier. These ion channels are what are responsible for forming the shape of the action potential.

An individual Node of Ranvier is represented by the model to the left, where C_m (µF/cm²) is the membrane capacitance, IC is the capacitive current, INa is the sodium current, IK is the potassium current, IL is the "leakage" (generic or non-specific) ionic current, RNa is the variable resistance to sodium flow, R_K is the variable resistance to potassium flow, RL is the resistance to leakage flow, ENa is the reversal potential for sodium, EK is the reversal potential for potassium, and EL is the reversal potential for leakage current. Note, often you will see Rx expressed as Gx, the conductance of ion x. R=1/G. This is the Hodgkin Huxley model of a Node of Ranvier and it's based on the squid giant axon.

1. Develop a model of the Node of Ravier by using an ODE solver to simultaneously (i.e., in 1 function) solve equations 1-10. However, here you will set dVm/dt (Equation 1) equal to 0. Thus, there is no change in voltage as a function of time. When the voltage of a node is held at a constant, unchanging value, it is said to have been "voltage clamped" to the initial voltage (the initial condition for Vm). Note, because the neuron is voltage clamped, there is no injected current, I.

2. Use a for loop to set V_m(0) and solve the set of differential equations. Sweep the value of V_m(0) across the following values: -15:1:-10, 0:20:100, 112:1:117. Solve the simulation between 0 <= t <= 10.

3. For each iteration of your for loop,

a. Plot the resulting m, h, and n values as a function of time on three subplots of a 3x1 (3 rows, 1 column) figure.

b. Plot the sodium current and potassium current on two subplots of a 2x1 figure.

You'll need to use the hold on command so you can plot the results for each Vm value of the for loop. Be sure the lines are distinguishable (consider using different colors). Be sure your axes are labeled, you use a legend, and you have a title.

Now you're going to slightly alter your Node of Ranvier model. Instead of setting dVm to 0, meaning the membrane voltage is clamped at Vm(0), you will calculate dVm/dt (equation 1). This requires you to calculate the sodium, potassium, and leakage currents as a function of the maximum ionic conductances, m, h, n, the membrane voltage, and the reversal potentials. There are two other adjustments to make:

• Each of these three currents should be multiplied by 1000 to account for a conversion in units.

• I (the injected current) should be set to 0.01 when time is greater than 1s, otherwise it is 0. This can be done with an if statement or a Heaviside function.

4. Set V_m(0) = 0, and sweep g¯Na from 1e-5 to 10e-5 in steps of 0.5e-5. Plot V_m(t) for each value of the maximum sodium conductance on a single plot. Be sure to include labels and a legend.

Questions:

1. Looking at the sodium channel responses (subplots 1 and 2 of figure 1), what can you deduce about the speed of the m-gates? What about the speed of the h-gate? What does a value of 0 mean? What does a value of 1 mean?

2. Looking at the potassium channel response (subplot 3 of figure 1), what can you deduce about the speed of the n-gates relative to the m and h gate speeds?

3. Given your conclusions from the previous two questions, when would you expect sodium current to be maximal? Minimal? When would you expect potassium current to be maximal? Use the response when Vm(0) was +117 mV.

4. Looking at the resulting sodium current (subplot 1, figure 2), what direction do sodium ions flow when the voltage clamp was greater than the sodium reversal potential? At the reversal potential? Less than the reversal potential?

5. Looking at the resulting potassium current (subplot 2, figure 2), what direction do potassium ions flow when the voltage clamp was greater than the potassium reversal potential? At the reversal potential? Less than the reversal potential?

6. Now that you have a sense of what direction the two ionic currents are running (into or out of the cell) and when those currents occur, explain the general shape you see of the action potential in Figure 3.

7. What effect does the maximum sodium conductance have on the sodium current and, therefore, the shape of the action potential that you see in Figure 3?

Part -2:

Constructing an axon

You will now construct an axon that connects multiple Nodes of Ranvier together. The neuron is represented by attached Nodes of Ranvier (NoR), each separated by a section of axon that is myelinated. Ionic current can flow between the Nodes of Ranvier through the axon, which has an internal resistance, Rax. For this assignment, assume that all extracellular voltage is 0 mV.

1. Find the injected current (pulse amplitude) threshold of the axon.

Find the smallest stimulus current that will initiate AND propagate an action potential for the pulse widths (the duration over which the current is injected) listed above. Create a figure of the threshold amplitude versus pulse width.

2. For the prize (optional)

Make sure your main code is a function. The first line after your function definition should be tic. Do not use tic anywhere else within the function. Then, at the very end of your main function, insert the code:

x=toc

y=bench;

disp(['Solve time/ODE benchmark time: ' num2str(x/y(3)) 'x']);

The student who produces the fastest code relative to the ode benchmark time for that computer will receive a large Toblerone bar. Make sure the TA notes your solve time ratio (the solve time divided by the time your computer takes to run the ode benchmark, which is typically the 3rd returned value). For reference, it took me 319.3817 seconds, or 1732.2045x the ode benchmark time, to run the lab on my 7- year-old laptop.

Tips:
• You'll need to pass in 4 initial conditions for each node in your axon model. Therefore, there are 84 values in your initial condition vector.
• Thus, there are 84 columns in your solution matrix.
• Instead of using for loops, which are very slow, you should vectorize all of your calculations.

o Your ionic current, α and β calculations should result in vectors with a value for each node (i.e., they should be 21-element vectors).

o Your injected current, although only occurring at the center node, can be modeled as a vector with zeros everywhere except the where the actual current is injected. Thus, your injected current should also be a 21-element vector.

• You should use odeset() to set the maximum step size such that the ode solver that you choose does not miss the injected current,  which is only on for a very short period of time.

• To determine if an action potential has propagated, make sure that at least 11 nodes had a Vm>20 mV at some point during the simulation.

• If you choose to use a binary search:

o The percent difference between your minimum and maximum pulse amplitude that serves as the criterion for stopping your search should be 5%.

o You should limit your maximum pulse amplitude to 0.1. The units are in μA.

Part -3:

In this lab, you're going to vary two things: electrode polarity (positive or negative) and electrode height (Δy). Read 1)-3) below before starting.

1. Vary the height of the electrode above the axon between 100 μm and 800 μm in steps of 100 μm while ensuring that the electrode is centered over the center Node of Ranvier (the node with an equal number of nodes on each side of it). At each location, find the threshold to activate the axon using a cathodic stimulus (negative polarity) when the pulse width is 0.5 ms and the fiber diameter is 10 μm. Plot the threshold as a function of vertical height of the electrode. Be sure to label your axes and include units. For the purposes of this and the next question, plot the absolute value of the threshold.

2. Now do the same thing as above, but change the polarity of your electrode. That is, make the current positive in sign ("anodic"). Plot the resulting threshold on the same plot you created in question 1. Be sure to include a legend.

3. Plot the ratio of the anodic to cathodic threshold as a function of electrode distance on a second plot. Set the marker style for each point you plot to ‘o' and do not plot a connecting line between points.

Then use lsqcurvefit to estimate a and b in the equation y = a (1 + b), where y in this case is the s ratio of anodic/cathodic threshold and x is the height of the electrode over the axon. Plot this "best fit" line as well. It should capture the trend of your data points.

Questions

1. What conclusion can you draw about the effect of electrode distance on threshold?

2. Does the anodic current follow the same trend as the cathodic current?

3. Which type of electrode (cathodic or anodic) has a lower threshold?

4. What is the ratio of cathodic to anodic thresholds as a function of electrode height?