!two.md

Exploring planar systems with xpp

These are just suggestions. Trust your instincts!

---

• select one of the files below and load it in xpp
  • sn.ode
  • snic.ode
  • subHopf.ode
  • supHopf.ode

a) brute force exploration of the dynamics:

• are there stable equilibria?
• are there limit cycles?
• is the system excitable?
• is there a (V) threshold?
• change I and repeat
• are there bifurcations?
• effect of I pulses. Integration?

b) rational exploration of the dynamics

• draw nullclines
• locate equilibria
• determine stability (direct integration / Sing pts menu)
• multistability? thresholds?
• change I and repeat

.xpprc

@ maxstor=100000
@ meth=qualrk,tol=1e-6,atol=1e-6

@ nmesh=150

@ nmax=1000,ntst=50,autoymin=-100,autoymax=100,parmin=-100,parmax=300,autoxmin=-100,autoxmax=300,epsl=1e-7,epsu=1e-7,epss=1e-5, npr=1000 

@ bell=0

@ but=lascon:fq

sn.ode

# saddle-node scenario, Nap - K model

dv/dt = (I - g_Na*minf(V)*(V-V_Na) - g_K*n*(V-V_K) - g_leak*(V-V_leak) + s(t))/C
dn/dt = (ninf(V)-n)/tau

v(0)=-16
n(0)=0.01

minf(v)=boltz(v,vhalf_minf,k_minf)
ninf(v)=boltz(v,vhalf_ninf,k_ninf)
boltz(v,vh,k)=1/(1+exp((vh-v)/k))

param I=0
param g_K=10, g_Na=20, g_leak=8
param V_Na=60, V_K=-90, V_leak=-80
param vhalf_minf=-20, k_minf=15
param vhalf_ninf=-25, tau=0.152
param k_ninf=5
param C=1


# double pulse stimulus 
param s1=0, t1=10, t2=15, 
param s2=0, t3=20, t4=25 
s(t)=s1*heav(t-t1)*heav(t2-t)+s2*heav(t-t3)*heav(t4-t)
@ total=150,dt=.01,xlo=-100,xhi=100,ylo=-.2,yhi=.8,xp=v,yp=n
done

snic.ode

# SNIC scenario, Nap - K model

dv/dt = (I - g_Na*minf(V)*(V-V_Na) - g_K*n*(V-V_K) - g_leak*(V-V_leak) + s(t))/C
dn/dt = (ninf(V)-n)/tau

v(0)=-16
n(0)=0.01

minf(v)=boltz(v,vhalf_minf,k_minf)
ninf(v)=boltz(v,vhalf_ninf,k_ninf)
boltz(v,vh,k)=1/(1+exp((vh-v)/k))

param I=0
param g_K=10, g_Na=20, g_leak=8
param V_Na=60, V_K=-90, V_leak=-80
param vhalf_minf=-20, k_minf=15
param vhalf_ninf=-25, tau=1
param k_ninf=5
param C=1


# double pulse stimulus 
param s1=0, t1=10, t2=15, 
param s2=0, t3=20, t4=25 
s(t)=s1*heav(t-t1)*heav(t2-t)+s2*heav(t-t3)*heav(t4-t)
@ total=150,dt=.01,xlo=-100,xhi=100,ylo=-.2,yhi=.8,xp=v,yp=n
done

subHopf.ode

# subcritical Hopf scenario, Nap - K model

dv/dt = (I - g_Na*minf(V)*(V-V_Na) - g_K*n*(V-V_K) - g_leak*(V-V_leak) + s(t))/C
dn/dt = (ninf(V)-n)/tau

v(0)=-16
n(0)=0.01

minf(v)=boltz(v,vhalf_minf,k_minf)
ninf(v)=boltz(v,vhalf_ninf,k_ninf)
boltz(v,vh,k)=1/(1+exp((vh-v)/k))

param I=0
param g_K=4, g_Na=4, g_leak=1
param V_Na=60, V_K=-90, V_leak=-78
param vhalf_minf=-30, k_minf=7
param vhalf_ninf=-45, tau=1
param k_ninf=5
param C=1

# double pulse stimulus 
param s1=0, t1=10, t2=15, 
param s2=0, t3=20, t4=25 
s(t)=s1*heav(t-t1)*heav(t2-t)+s2*heav(t-t3)*heav(t4-t)
@ total=150,dt=.01,xlo=-100,xhi=50,ylo=-.2,yhi=1.2,xp=v,yp=n
done

supHopf.ode

# supercritical Hopf scenario, Nap - K model

dv/dt = (I - g_Na*minf(V)*(V-V_Na) - g_K*n*(V-V_K) - g_leak*(V-V_leak) + s(t))/C
dn/dt = (ninf(V)-n)/tau

v(0)=-16
n(0)=0.01

minf(v)=boltz(v,vhalf_minf,k_minf)
ninf(v)=boltz(v,vhalf_ninf,k_ninf)
boltz(v,vh,k)=1/(1+exp((vh-v)/k))

param I=0
param g_K=10, g_Na=20, g_leak=8
param V_Na=60, V_K=-90, V_leak=-78
param vhalf_minf=-20, k_minf=15
param vhalf_ninf=-45, tau=1
param k_ninf=5
param C=1

# double pulse stimulus 
param s1=0, t1=10, t2=15, 
param s2=0, t3=20, t4=25 
s(t)=s1*heav(t-t1)*heav(t2-t)+s2*heav(t-t3)*heav(t4-t)
@ total=150,dt=.01,xlo=-100,xhi=100,ylo=-.2,yhi=.8,xp=v,yp=n
done