The `xpp` file has the following format:

# hhh.ode init v=-65 m=0.05 h=0.6 n=0.317 par vna=50 vk=-77 vl=-54.4 gna=120 gk=36 gl=0.3 c=1 phi=1 i0=0 par ip=0 pon=50 poff=150 is(t)=ip*heav(t-pon)*heav(poff-t) am(v)=phi*.1*(v+40)/(1-exp(-(v+40)/10)) bm(v)=phi*4*exp(-(v+65)/18) ah(v)=phi*.07*exp(-(v+65)/20) bh(v)=phi*1/(1+exp(-(v+35)/10)) an(v)=phi*.01*(v+55)/(1-exp(-(v+55)/10)) bn(v)=phi*.125*exp(-(v+65)/80) v'=(I0+is(t) - 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 # track the currents aux ina=gna*(v-vna)*h*m^3 aux ik=gk*(v-vk)*n^4 aux il=gl*(v-vl) # track the stimulus aux stim=is(t) done

*Note that I have set to 1 and thus have assumed the
standard temperature of I have also kept track of the
currents and the stimulus.*

We can use this file to explore the dynamics of the HH model as
various parameters are changed. The most obvious parameter to vary
is the applied current. By setting `I0` we can inject a constant
current and by setting `IP` a pulse of current can be injected at
`PON ` lasting until `POFF`. In class I will play around a
bit but I want you to do the following experiments. Use the
Runge-Kutta integrator with a time-step of 0.05 and set `nOut` to
10 so that output occurs every half a millisecond. Also set the total
amount of time to 200 msec and set the `Bounds` to 10000 so that
the various currents do not exceed them.

- 1.
- With
*I*=0 try to find the threshold by setting all the variables at rest but incrementing_{0}*V*by different amounts. - 2.
- Change the current
*I*until the neuron fires repetitively. What is the critical value of current that you found?_{0} - 3.
- With
*I*=10 integrate the equations. In the_{0}`Data Browser`add a column called`minf`which contains the formula,`am(v)/(am(v)+bm(v)).`Compare this to the value of`m(t)`by plotting the two during a few spikes. The two are almost identical. This tells ypu that it may be possible to approximate the dynamics of*m*by thus making the differential equation one fewer variable. - 4.
- Next, plot a phase-plane of
`n`and`h.`They seem to lie along a line. What is the equation for this line? This says that*n*and*h*may be linearly related. This means that we may be able to reduce this 4 dimensional system to 2 dimensions by eliminating*m*and one of*n*,*h*. - 5.
- Set
*I*=0 and let the membrane start at rest. Set_{0}`Pon=0, poff=50, ip=-5`. This hyperpolarizes the membrane. What happens after the stimulus is removed? You should get a spike. Explain why this happens. (Hint: Look at the variable`h`during the hyperpolarization. ) - 6.
- Repeat the above experiment but use less negative values of
`ip`is there a critical value below which you get no rebound spike? - 7.
- Set
`ip=0, i0=6.5.`Solve the equations for*v*=-61,*m*=0,*h*=0.45,*n*=.4. What happens? Try the same thing with*v*=-45. What happens? What does this tell you about the number of stable states for the membrane?