# FitzHugh - Nagumo caricature of the Hodgkin - Huxley formulas

I've been attempting to address this trouble from Elements of Applied Bifurcation Theory, yet despite having the tips offered I have not figured out just how to continue yet. I would certainly value any kind of more tips or understandings you can offer me in order to make some progression.

It is workout 2.6. (2):

The adhering to system of partial differential formulas is the FitzHugh - Nagumo caricature of the Hodgkin - Huxley formulas modeling the nerve impulse breeding along an axon:

$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}-f_a(u)-v$,

$ \frac{\partial v}{\partial t} = bu $,

where $u(x,t)$ stands for the membrane layer possibility, $v=v(x,t)$ is a. " recuperation" variable, $f_a(u)=u(u-a)(u-1)$, $1 > a > 0, b > 0, - \ infty < x < \ infty$, and $t > 0$.

Taking a trip waves are remedies to these formulas of the kind

$u(x,t)=U(\xi)$, $v(x,t)=V(\xi)$, $\xi = x+ct$, where $c$ is an a. priori unidentified wave breeding rate. The features $U(\xi)$,. $V(\xi)$ are the wave accounts.

(a) Derive a system of 3 ordinary differential equations for the. accounts with "time" $\xi$. (Hint: present an added variable:. $W=\dot{U}$.

(b) Check that for all $c>0$ the system for the accounts (the wave. system) has an one-of-a-kind stability with one favorable eigenvalue and also 2. eigenvalues with adverse actual components. (Hint: First, validate this. thinking that eigenvalues are actual. The, show that the particular. formula can not have origins on the fictional axis, and also ultimately, make use of the. continual dependancy of the eigenvalues on the parameters.

(c) Conclude that the stability can be either a saddle or a saddle - emphasis with a one - dimensional unpredictable and also a 2 - dimensional secure in - alternative manifold, and also locate a problem on the system parameters that de - penalties a border in between these 2 instances. Story numerous borders in the. $(a,c)$ - aircraft for various values of $b$ and also define the area. representing saddle - emphases. (Hint: at the border the. particular polynomial $h(\lambda)$ has a double origin. $\lambda_0:h(\lambda_0)=h'(\lambda_0)=0.$).

Here is something to get you began: $u_t=c U'(\xi)$, $u_{xx}=U''(\xi)$, and also $v_t=c V'(\xi)$. Connecting this right into the system of PDEs offers you a system of 2 ODEs for $U(\xi)$ and also $V(\xi)$. The first of these ODEs is 2nd order in $U$, yet you can transform it right into a first order formula by changing $U'(\xi)$ by a supporting function $W(\xi)$ and also changing $U''(\xi)$ by $W'(\xi)$, and afterwards boosting the system with the added formula $U'=W$. Currently you have a system of 3 first order ODEs (i.e., a dynamical system), and also can continue to component (b).

Related questions