(20 points) In this set of exercises, you will simulate several versions of the ML equations. I have provided the XPP, Python, and MatLab codes. The MatLab uses Euler's method. In the Python,MatLab code, you will need to change the parameters by editing the code. In XPP, you can just click on File Get-par-set and choose the correct parameters. For completeness, I provide the parameters for the different versions below. The initial conditions are also provided In each case, simulate for 400 msec unless otherwise instructed. It is probably best to draw the voltage vs time for these in order to quantify the period when necessary. - (6 points) Explore the Hopf: Give a 10 msec impulse at t=200 varying the amplitude ( ist) until a pulse is emitted. How much can you delay the spike onset? Next, let I0=90 and set the initial consditions to v(0)=26.59,w(0)=0.129 and give a 10msec pulse with amplitude 40 and report your findings. Repeat with an amplitude of 50 . What happened? Is possible to elicit just one spike with a transient stimulus? Explain your reasoning. Finally set I0=90,v(0)=0,w=0.2, set the transient stimulus, ist =0, and integrate the equations for 1000 msec. How many spikes do you get? This is the frequency in Hz since we solved for 1 second. Try I0=91,92 and more accurately measure the time between spikes to get the period, P and then from this the frequency (1000/P) - (6 points) Explore the SNIC: Give a 10 msec impulse at t=200 varying the amplitude ( ist) until a pulse is emitted. How much can you delay the spike onset? Bonus point to anyone with a delay of more than 40msec after the stimulus was turned off! (I got 75msec, but needed 4 decimal places). Set total time to integrate to 1000 msec. Set I0=41, and the transient stimulus, ist =0, and tell me the frequency. Try I0=42 and I0=43. Note that the frequency is much more sensitive to I0 for the SNIC! - (4 points) For Class III, give a 2 msec pulse stimulus at t=200 and find the minimum amplitude to 1 decimal. By solving the equations with different levels of I0 (say, 100,110,120, 150) discover that there cannot be a limit cycle. - (4 points) For the homoclinic parameters, set I0=37,v(0)=35.7,w(0)= 0.004 and then give a 10msec pulse at t=200 with amplitude ist =20,40,50,55 and report what happens. The explanation of this is tricky, but can be discovered by looking at the phase-plane and the diagrams I drew in class