## Example

In this section, I offer two ODE files, the Hodgkin-Huxley as usually solved and an exponential Euler version of the same. Here are the HH equations :
```# hhh.ode
init v=0  m=0  h=0  n=0
par vna=50  vk=-77  vl=-54.4  gna=120  gk=36  gl=0.3  c=1  i=10
am(v)=.1*(v+40)/(1-exp(-(v+40)/10))
bm(v)=4*exp(-(v+65)/18)
ah(v)=.07*exp(-(v+65)/20)
bh(v)=1/(1+exp(-(v+35)/10))
an(v)=.01*(v+55)/(1-exp(-(v+55)/10))
bn(v)=.125*exp(-(v+65)/80)
v'=(I - gna*h*(v-vna)*m^3-gk*(v-vk)*n^4-gl*(v-vl))/c
m'=am(v)*(1-m)-bm(v)*m
h'=ah(v)*(1-h)-bh(v)*h
n'=an(v)*(1-n)-bn(v)*n
done
```
and the iterative version using the exponential method:
```# this is the exponential form for HH equations hhexp.dif
init v=0  m=0  h=0  n=0
par vna=50  vk=-77  vl=-54.4  gna=120  gk=36  gl=0.3  c=1  i=10
par delt=.5
am(v)=.1*(v+40)/(1-exp(-(v+40)/10))
bm(v)=4*exp(-(v+65)/18)
ah(v)=.07*exp(-(v+65)/20)
bh(v)=1/(1+exp(-(v+35)/10))
an(v)=.01*(v+55)/(1-exp(-(v+55)/10))
bn(v)=.125*exp(-(v+65)/80)
# x'=-a*x+b ==>   x(t+delt)=exp(-a*delt)*(x(t)-b/a)+b/a
bv=(I+vna*gna*h*m^3+vk*gk*n^4+gl*vl)/c
av=(gna*h*m^3+gk*n^4+gl)/c
v'=exp(-av*delt)*(v-bv/av)+bv/av
amv=am(v)+bm(v)
bmv=am(v)/amv
m'=exp(-amv*delt)*(m-bmv)+bmv
ahv=ah(v)+bh(v)
bhv=ah(v)/ahv
h'=exp(-ahv*delt)*(h-bhv)+bhv
anv=an(v)+bn(v)
bnv=an(v)/anv
n'=exp(-anv*delt)*(n-bnv)+bnv
done
```
In the exponential Euler version, switch the method to Discrete from the numerics menu. Iterate for 500 time steps, which at the step size of 0.5 milliseconds represents (500)(.5)=250. The oscillation occurs at the 85th iterate or about 42 milliseconds. Now repeat this with the true differential equation using a variety of integration methods (Backward Euler, Euler, modified euler, Runge-Kutta, Gear) with different time steps. Compare the results. Try exponential euler by setting the parameter delt=1.

This is a severely abridged version of Sherman's notes. The full version is available as a postscript file by clicking here.

G. Bard Ermentrout
1/9/1998