Solving boundary value problems with XPPAUT

We want to solve:

Since XPPAUT can only solve systems of first order equations, we rewrite this as a system of two first order equations:

which translates into the following ODE file

# steady state cable  sscab.ode
dv/dt = vx
dvx/dt = v
# boundary condition at left end:
bndry v-1
# boundary condition at right end:
bndry bl*v'+vx'
#
parameter bl=1
done

Since XPPAUT use t as its independent variable, we have changed the name of ``x'' to ``t''. Boundary conditions are set by using the declaration, bndry. XPPAUT will try to set these to zero. Primes mean to evaluate the variable at the end of the integration while unprimed variables evaluate at the beginning. To generate these picture, I just set the total amount for the integration to either of 0.5, 1, 2, 3, corresponding to different electrotonic distances. Then I used the Bndry (N)o show combination to solve the equations. I used the Graphics Freeze Freeze combination to make a permanent copy of the given curve. I repeated this for each curve and used the Text Text option to draw the text on the curves and the View 2d combination to set the axes and labels.

HOMEWORK

1.

What is the electrotonic length for a cable with $R_M=20000\Omega {cm}^2, R_A=75\Omega cm, d=8 \mu, C_M=1 \mu F/{cm}^2$, $\ell=1mm$

2.

Find the input resistance for the semi-infinite cable with the above parameters

3.

Solve the steady state voltages for B_L= 0,.25,1,10,100 on a cable with electrotonic length 3 using XPPAUT.