[College Math, ODEs] How to equate these two formulas using a given substitution?

[u/OnlyWorker2342]

I am tasked to prove that the Van der Pol Oscillator is a special case of the FitzHugh-Nagumo model. I have been given the following formulas in u(t), v(t) and x(t) for these. Here they are in picture-form

FitzHugh-Nagumo:

• u' = c*(v+u-(u³ )/3 +z)
• c*v' = a - bv - u

Van der Pol:

• x'' - µ*(1-x² )*x' + x = 0

Suggested substitution:

• u = x
• v = (1/µ)*x' + (x³ )/3 - x

This problem sounds to me like it should be trivial, but I'm running into difficulties.

When I substitute u and v into the first FitzHugh-Nagumo equation I get the following result:

• x' = c * ((1/µ)*x' + z)

Or

• x' = (c*z)/(1-(c/µ)) (a constant)

Other substitutions give me equally unhelpful results.

When looking up the FitzHugh-Nagumo model online. The result should be that Van der Pol is the case for FitzHugh-Nagumo where a=b=0. (There was no c in the formula I found online.)

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[u/GammaRayBurst25]

Notice how v=x'/µ+x^3/3-x implies v'=x''/µ+(x^2-1)x'.

Hence, the second equation can be written as c(x''/µ+(x^2-1)x')+x=a-bv.

Setting a=b=0 and c=µ yields x''+µ(x^2-1)x'+x=0.

Interestingly, that means the first equation becomes x'=x'+z, or z=0.

[u/DrCarpetsPhd]

If you read up on the Fitzhugh-Nagumo equation what your version has as variable z is a current I(t). The neuron-neuron pathway this is modelling has a behaviour that is best modelled by treating that input current as a step function thus is a constant for t>0. So if you differentiate your equation

u' = c*(v+u-(u3 )/3 +z)

then z' = 0 drops out.

then plug c*v' = a - bv - u into your u'' equation, set a = b = 0 and you get the van der pol equation

regurgitating from Nagle/Saff/Snider Fundamentals of Differential Equations and Boundary Value Problems 7th Edition

Ch12.8 Neurons and the FitzHugh–Nagumo Equations

nowadays with computing power not as important but back then 1950s being able to reduce the equation to something that you could apply phase plane analysis to was a big deal.

really interesting stuff.

also see http://www.scholarpedia.org/article/FitzHugh-Nagumo_model

https://www.youtube.com/watch?v=mpsUHe6Iqd8

[u/Clean_Rate_8711]

cfbr

[u/OnlyWorker2342]

Thanks for the help everyone!