BIO 4134C Special Topics in Biology III Section C Mathematical Methods in Biology: from Molecules to

1
analysis of FHN-type models phase plane analysis
and singular perturbation theory
2
FitzHugh-Nagumo (FHN) type Models

eltlt1 v is a fast excitation variable and w is
a slow recovery variable.
Example Fast-Slow Dynamics of a reduced HH Model
key ingredient N-shaped v-nullcline, i.e. the
nullcline of the fast variable.
3
FHN-type models
4
response to subthreshold perturbation
v
w
w
time
v
5
response to suprathreshold perturbation
v
w
w
time
v
6
phases of action potential
1. upstroke (fast)
2. plateau (slow)
3. repolarization (fast)
4. recovery (slow)
3
2
1
v
3
4
2
4
w
w
1
time
v
7
GOAL to obtain simplified approximate solutions
for each phase of the action potential and piece
these solutions together to get the full
solution. TECHNIQUE singular perturbation
theory, which exploits the fact that there is are
large separation of the fast-slow time scales,
i.e. eltlt1.
8
defining portions of the v-nullcline
9
upstroke dynamics (fast/short time scale)
1
Because , the approximate
equations are
result a single first order nonlinear ODE that
describes the dynamics of fast subsystem
(parameterized by the slow variable w).

10
upstroke dynamics (fast/short time scale)
1
threshold
rest state
excited state
For v(0) above threshold vM(w0) , the system
quickly goes to the excited state vR(w0).
11
plateau dynamics (slow/long time scale)
2
Rescale time
new initial conditions taken to be values where
the upstroke dynamics left off.
Because , the approximate
equations are
12
2
plateau dynamics (slow/long time scale)
dynamics are restricted to the right branch of
v-nullcline
result a single first order ODE that describes
the dynamics of slow subsystem (flow along
v-nullcline).
13
2
plateau dynamics (slow/long time scale)
(v,w)
This equation is valid until ww.

Duration of the excited state
14
Similarly
repolarization dynamics (fast/short time scale)
3
4
recovery dynamics (slow/long time scale)