Homework #3

1. Simulate a pair of integrate and fire oscillators with exponential synapses with both excitatory and inhibitory coupling. Here is a model you can use:

15 v1' = -(v1+65) + I1 - g1 s2 (v1-vrev1)

15 v2' = -(v2+65) + I2 - g2 s1 (v2-vrev2)

with the synapses decaying at rates tsyn1,tsyn2. Use vthr=-55 and vreset=-75. Start them off at slightly different initial conditions with I1=I2=20. First set the coupling to zero (g1=g2=0) and then let the coupling be small (e.g 0.1) and try excitatory (vrev1=vrev2=0) and then inhibitory (vrev1=vrev2=-70) coupling. You probably should simulate for at least a half of a second. Try the same thing with one excitatory and one inhibitory neuron. You may have to speed up the excitatory neuron or slow down the inhibitory neuron. Here is an XPP file

```v1'=(-(v1-vleak)+i1-g1*s2*(v1-vrev1))/tau
v2'=(-(v2-vleak)+i2-g2*s1*(v2-vrev2))/tau
s1'=-s1/tsyn1
s2'=-s2/tsyn2
global 1 v1-vthr {v1=vreset;s1=s1+1}
global 1 v2-vthr {v2=vreset;s2=s2+1}
init v1=-66,v2=-55
par vleak=-66,tau=15,i1=20,i2=20,vthr=-50,vreset=-75
par vrev1=0,vrev2=0,g1=0,g2=0,tsyn1=5,tsyn2=5
@ total=500,njmp=5
done
```

2. Explore a network of 4 sine oscillators with general coupling between them:

x1' = w + a12 sin(x2-x1) + a13 sin(x3-x1) + a14 sin(x4-x1)

x2' = w + a21 sin(x1-x2) + a23 sin(x3-x2) + a24 sin(x4-x2)

x3' = w + a32 sin(x2-x3) + a31 sin(x1-x3) + a34 sin(x4-x3)

x4' = w + a42 sin(x2-x4) + a43 sin(x3-x4) + a41 sin(x1-x4)

Choose either +1,-1, or 0 for the coupling coefficients and try to get a the following types of stable behavior:
• all synchronoous
• two pairs synchronous and 180 degrees out of phase with each other
• a gallop like gait with a nonzero lag between all the oscillators - I managed to get the following 0.54,2.01,3.14,5.15 which is a sort of gallop.
Here again is an XPP file if you want. By the way, a good trick is to fold the phases modulo 2 pi by clicking on phAsespace All and using the default value of 2 pi. This tells XPP to store the values modulo 2 Pi.
```x1' = w + a12 *sin(x2-x1) + a13 *sin(x3-x1) + a14 *sin(x4-x1)
x2' = w + a21 *sin(x1-x2) + a23 *sin(x3-x2) + a24 *sin(x4-x2)
x3' = w + a32 *sin(x2-x3) + a31 *sin(x1-x3) + a34 *sin(x4-x3)
x4' = w + a42 *sin(x2-x4) + a43 *sin(x3-x4) + a41*sin(x1-x4)
par a12=0,a21=0,a23=0,a32=0,a34=0,a43=0,a14=0,a41=0
par a13=0,a31=0,a24=0,a42=0
par w=1
init x1=1,x2=3,x3=4.5,x4=6
done
```
3. Try the following model:

x1' = w1 + sin(x2-x1)

x2' = w2 + sin(x3-x2)+sin(x1-x2)

x3'= w3 + sin(x2-x3)+sin(x4-x3)

x4' = w4 + sin(x3-x4)

Choose w1,w2,w3,w4 so that x1=ct,x2=ct+pi/2,x3=ct+pi,x4=ct+3pi/2. This is a traveling wave going down the chain. Can you generalize this to a chain of length N and such that the phase lag between successive oscillators is exactly 2 Pi/N?
4. A chain of unidirectionally coupled integrate and fire neurons can generate a wave of activity if the coupling is strong enough. Suppose that cell j fires at t=0 and leads to a current of the form

I(t)= g a exp(-a t)

At what point does the integrate and fire neuron fire:

tau dV/dt = -v + I(t)

where the neuron starts at rest, V=0 and has a threshold for firing at 1. (This is actually an impossible problem since you cannot find the roots. However, you can write down the solution to the integrate and fire model.) Here is how to graphically solve this. Plot v(t) and look for intersections. What is the minimal conductance, g, required to guarantee firing. (Hint - what is the maximum value of v(t) and for what values of g is this maximum greater than 1?) Here is some Maple code to solve the differential equation:

```
dsolve({tau*diff(v(t),t)=-v(t)+g*a*exp(-a*t),v(0)=0},v(t));
```