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Help with nondimensionalization

  1. Feb 23, 2016 #1
    • Member warned about posting without the HW template
    I am working on an assignment for my neuroscience course, and I'm running into a problem with one question which requires me to rewrite an equation into its nondimensionalized form. The equation is given below.

    Screenshot_from_2016_02_23_02_51_56.png

    and I need to convert it to the form
    Screenshot_from_2016_02_23_02_54_11.png

    by rescaling and shifting the given variables in the equation as Screenshot_from_2016_02_23_02_55_40.png

    I have already attempted to follow the nondimensionalization procedure given in the Wikipedia article on it, but however I do it, there are some variables on the R.H.S that simply refuse to go away to give me the form required. I'd be grateful for any help. Thanks!
     
  2. jcsd
  3. Feb 23, 2016 #2

    Samy_A

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    When you replace ##V## by ##\alpha x+ V_0## and ##I_e## by ##\gamma I + I_0## in the equation, what do you get?
    Can't you then choose ##\alpha## in such a way that the term in ##x## vanishes?
    (Is ##\tau_m>0##?)
     
  4. Feb 23, 2016 #3
    Yes, I should have mentioned earlier. ##\tau_m > 0, a_L > 0## and ##V_C > V_L##
     
  5. Feb 23, 2016 #4

    Samy_A

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    In that case, why don't you simply do the suggested computation? You will get a term in ##x²## and a term in ##x##. By choosing appropriate values for ##\alpha## and ##V_0## you can make the coefficient of ##x²## equal to 1, and make the term in ##x## vanish.
     
  6. Feb 23, 2016 #5
    I have tried this. Even if the ##x## term vanishes, how about constant terms like ##V_L##, ##V_C##? Also, no value I choose for ##\alpha## is helping me make the current term ##\frac{R_mI_0}{\alpha\tau_m}## vanish simultaneously.
     
  7. Feb 23, 2016 #6

    Samy_A

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    Can you show your calculation, because I think I managed to do it (of course maybe I have an error in my calculation). Don't forget that after having "fixed" ##x## by choosing appropriate ##\alpha,\ V_0##, you can choose ##\gamma## and ##I_0## to get the desired result.
     
  8. Feb 23, 2016 #7
    I can describe the steps I've tried taking so far.

    1. Substitute for both ##V## and ##I_e## as given on both sides of the equation.
    2. L.H.S then becomes ##\alpha\tau_m\frac{dx}{dt}##, since ##\frac{dV_0}{dt}## term equals 0.
    3. Now divide throughout by ##\alpha\tau_m## to get just ##\frac{dx}{dt}## as needed.
    4. Substitute for ##\alpha## with ##\frac{\tau_m}{a_L}## to get coefficient of ##x^2## term as 1.

    I'm unclear after this. What can I substitute for ##V_0## to make all those terms disappear?
     
  9. Feb 23, 2016 #8

    Samy_A

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    After setting ##\alpha =\frac{\tau_m}{a_L}##, what is the coefficient of ##x## is your RHS? Can you chose ##V_0## so that the term in ##x## disappears?
     
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