Math
Biology: Hodgkin Huxley Model
- HH.ode -
xppaut code for the Hodgin-Huxley model
In this version of the Hodgkin-Huxley
model, the transmembrane potential v has been shifted
with respect to the rest potential of the cell so
that v=0 is the rest potential (for convenience).
Units are
- [t]
= msec (milliseconds),
- [V]
= mV (millivolts)
- [I]
= mA (milliamps)
The code:
- Note
the potassium current ik, the sodium current ina,
the leakage current il and the applied current ia
are all defined BEFORE the differential equations.
For larger models this
type of code structure helps to intuitively compartmentalize
expressions.
- After
the differential equations are
declared,
a few auxilliary
or "aux" variables
defined. These are typically functions of the solution one wants to
monitor. Here the capitalized versions
of the currents have been defined: IK, INA, IL, IA.
Experiment
1:
Here
we choose
the applied current ia(t) and "measure" the voltage response
v(t). In this case, the applied current in is the sum of
two "pulse" functions defined using the Heaviside function
"heav". A single "pulse" in the code has the formula:
i(amp,t0,dt)=amp*(heav(t-t0)-heav(t-t0-dt))
which has an amplitude of "amp" only for the times
between t0 and t0+dt. Otherwise, it is zero. Then,
ia=i(ia1,t1,dt1)+i(ia2,t2,dt2)
We get to
choose parameter values (ia1,t1,dt1) for the first "pulse" and
(ia2,t2,dt2) for the second
"pulse".
Run the code experimenting with:
- A
single pulse only varying its amplitude
- excitability
- A
second pulse varying the distance
between first and second pulses - refractoriness
- Plot
v(t) and iax(t) on same graph
- Plot
ia(t) and all "aux" currents
on the same figure
Note that
voltage increases (depolarizes),
then
decreases (hyperpolarizes),
goes below the rest
potential and then has a long recovery period
before going back to rest. This shape is ubiquitous
in neurons and is called an "action potential". Typically synaptic
input (current) from other cells cause action potentials
insitu.
Soon after the
applied current pulse, the sodium current activates. This causes the
cell depolarization. The potassium current is slightly
delayed, i.e. activates "after" the sodium current. The potassium
current is positive and opposite sign to the negative sodium
current. This sign difference reflects the fact that one is an outward current
and one is
an inward
current.
Experiment
2:
Do you want to
add TTX??? That is the dreaded
neurotoxin that shuts down all sodium channels.
To eliminate the sodium channels in the model just set the conductance
gna=0. Repeat Experiment 1
with gna=0 and note qualitative differences. Are there action
potentials? When you are done, set gna=120 again!!!
Experiment
3:
Copy HH.ode
to HH_clamp.ode. Edit HH_clamp.ode to mimic a voltage clamp experiment
in which Na, K and/or leak currents can be phamacologically blocked. In
such experiments v is a (constant) parameter. In particular, v'=0 so
that ia(t) = sum of all the other terms in the original v'.
Use
this to declare ia as an "aux"illiary variable. Once done do the
following two simulations:
- Block
NA and leak currents then plot
several ia(t) currents for many v
- Block
K currents then plot ia(t) currents
for many v.
Experiment 4:
Fitzhugh (1955-61) tried to reduce the order of the HH model while
retaining the same "excitable" dynamics. To do this he made two
observations: a) \tau_m << 1 which leads to m=m_{\infty}(v) or a
QSS approximation and b) numerically n+h = 0.8 throughout an action
potential (approximately). Verify the later and then create HH2.ode
using these assumptions (as in the notes). The resulting planar system
still has action potentials. Using HH2.ode, verify this and then plot
the solution in the (v,n) plane along with its nullclines.
