Chapter 7
Bard’s AUTO Lab files and
settings
7.1
The Morris Lecar Model
Below is the file ml.ode that was used to illustrate a variety of phase plane and
AUTO bifurcations in lab on May 26.
7.1.1
ml.ode
# Morris-Lecar model Methods Chapter
dv/dt = ( I - gca*minf(V)*(V-Vca)-gk*w*(V-VK)-gl*(V-Vl)+s(t))/c
dw/dt = phi*(winf(V)-w)/tauw(V)
v(0)=-16
w(0)=0.014915minf(v)=.5*(1+tanh((v-va)/vb))
winf(v)=.5*(1+tanh((v-vc)/vd))
tauw(v)=1/cosh((v-vc)/(2*vd))
param vk=-84,vl=-60,vca=120
param i=0,gk=8,gl=2,c=20
param va=-1.2,vb=18
# Uncomment the ones you like!!
par1-3 vc=2,vd=30,phi=.04,gca=4.4
set hopf {vc=2,vd=30,phi=.04,gca=4.4}
set snic {vc=12,vd=17.4,phi=.06666667,gca=4}
set homo {vc=12,vd=17.4,phi=.23,gca=4}
#par4-6 vc=12,vd=17.4,phi=.06666667,gca=4
#par7-8 vc=12,vd=17.4,phi=.23,gca=4
param s1=0,s2=0,t1=50,t2=55,t3=500,t4=550
# double pulse stimulus
s(t)=s1*heav(t-t1)*heav(t2-t)+s2*heav(t-t3)*heav(t4-t)
@ total=150,dt=.25,xlo=-75,xhi=75,ylo=-.25,yhi=.5,xp=v,yp=w
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CHAPTER 7. BARD’S AUTO LAB FILES AND SETTINGS
done
7.1.2
AUTO settings
• Axes HiLo; Xmin -50, Ymin -90, Xmax 120, Ymax 60
• Numerics: Default settings other than the following: ntst 60, Nmax 2000,
NPr 500, EPSL 0.000001. ParMin -80, ParMax 150
• Run
• Grab the HB point
• Run: Periodic (this will produce the bifurc to unstable and stable limit
cycles)
• Zoom in: Axes Zoom (select region with mouse) redraw
• You can select a param setting close to the HB by grab, tab, and then
using the arrow keys to move a bit into the region of interest.
• Return to the XPP simulatoin window and display nullclines (N, N)
• Use Initial cond, mIce to simulate some of the trajectories to see stable
limit cycles, (I L, I L, I, L to get closer to the cycle, then to save it Graphics
Freeze Freeze, (pick a color))
• To see unstable limit cycles: change delta t to negative value: (nUmerics,
Dt, -0.25, enter, escape, Initial cond, Mouse, try a few values, repeat the
ILIL and graphics Freeze to save).
• Now we can quit AUTO and quit XPP and rerun to see some new features
• For ml.ode, there are specific parameter sets close to interesting bifurcation
points that are saved in the ODE file, and we can get XPP/AUto to use
these values as follows:
• From XPP window: File, get param set, SNIC (this loads a set of params
near such a bifurcation).
• Repeat the creation of AUTO as before
• From the Auto Window: Grab the HB; Axes, Frequency, redraw (This
will produce the graph of frequency of ocillation versus the parameter i
(current)).
• Try to change phi to phi=0.25 to see another case.
7.2. HODGKIN HUXLEY EQUATIONS
7.2
3
Hodgkin Huxley Equations
In the Hodgkin-Huxley equations, we can find up to 4 limit cycles, as we saw
using the following file.
7.2.1
hhh.ode
# hhh.ode
init v=-65 m=.05 h=0.6 n=.317
par i0=0
par vna=50 vk=-77 vl=-54.4 gna=120 gk=36 gl=0.3 c=1 phi=1
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)
@ bound=10000
done
7.2.2
AUTO Settings
• Integrate the equations to steady state (Initial conds Go, I L, I L, I L, I
L.)
• Bring up AUTO and select Parm1 to be i0.
• Axes Hi Lo Xmin -10, Ymin -90, Xmax 50, Ymax 55
• Numerics as usual with the following specific choices: Ntst 60, Nmax 2000,
Npr 500, Ds 0.02, EPSL 1e-06, . . . , ParMin -10, ParMax 50
• Run Steady state, grab the HB, Run periodic.
• Select Usr period; 1; type in I0=7.875
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CHAPTER 7. BARD’S AUTO LAB FILES AND SETTINGS
• Rerun AUTO and grab UZ points corresponding to UZ 5, 7, 9, 11
• File; Import orbit
• Go back to the XPP window; Viewaxes; N vs V; window, Fit;
• Freeze the orbit (Graphics Freeze Freeze) and repeat the procedure.