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 85^th 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.