Exercise 2

In this example, we will take the model from Figure 1 and actually add the kinetics. We will then add 4 synapses whose conductances and whose onsets can be individually set (although they will all have the same time courses). The object of this little exercise is to explore thresholds and timing of synapses. The file is called trcomp4.ode and you should integrate it at least for 80 msec. I have used alpha functions for the synapses since there is no post-synaptic cell and these fire only once. The sodium and potassium dynamics are those of Traub's cortical cell model. The parameters you will vary are called gsyn1,gsyn2,gsynb,gsyns which are the conductances for the two apical dendrite compartments, the basal dendrite compartment, and the soma compartment. The other parameters of interest are ts,tb,t1,t2 which are the onset times of the 4 synapses. The voltages are v, va1,va2,vb for the soma, the two apical dendrites, and the basal dendrite. I have chosen the coupling between the compartments for you for the purposes of this exercise. Here are some things to do:

1.

Find the minimum value of the synaptic conductance on each compartment in order to elicit a spike at the soma.

2.

Fix the conductance on the basal dendrite to be below the threshold to elicit a spike. Now increase the conductance on the terminal apical dendrite (labeled gsyn2 ) until the two of them elicit a spike.

3.

With the paired synapses of the above, alter the timing by increasing t2 from 5 to higher values. How close do the two stimuli have to be to elicit a spike. Increase the conductance gsyn2 and repeat this.

4.

Note that the alpha functions are normalized so that their integral in time is 1. Set all the synaptic conductances to 0 except gsyn2. Make a table showing the minimum conductance to elicit a spike at the soma as a function of tau_s the time constant of the synapse. Try fast synapses tau_s=5 and slow tau_s=40 as well as those in between.

5.

Hold gsyn2=4. Now vary tau_s from 1 msec (really fast) to 20 msec. Which stimulus evokes the most action potentials.

Here is the file:

# traub sodium and potassium kinetics and 3 compartments with synapses
# v va1,va2,vb
v'=-(gna*h*m^3*(v-ena)+gk*n^4*(v-ek)+gl*(v-el)+gas*(v-va1)+gbs*(v-vb)+\
     alphas(t)*(v-vsyn))/c
va1'=-(gl*(va1-el)+g21*(va1-va2)+gsa*(va1-v)+alpha1(t)*(va1-vsyn))/c
va2'=-(gl*(va2-el)+g12*(va2-va1)+alpha2(t)*(va2-vsyn))/c
vb'=-(gl*(vb-el)+gsb*(vb-v)+alphab(t)*(vb-vsyn))/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
init v=-67,va1=-67,va2=-67,vb=-67,m=0,n=0,h=1
am(v)=.32*(54+v)/(1-exp(-(v+54)/4))
bm(v)=.28*(v+27)/(exp((v+27)/5)-1)
ah(v)=.128*exp(-(50+v)/18)
bh(v)=4/(1+exp(-(v+27)/5))
an(v)=.032*(v+52)/(1-exp(-(v+52)/5))
bn(v)=.5*exp(-(57+v)/40)
par ek=-100,ena=50,el=-67
par gl=.1,gk=80,gna=100
par c=1,i=0
par gsyns=0,gsynb=0,gsyn1=0,gsyn2=0,vsyn=0
par ts=5,tb=5,t1=5,t2=5
par tau_s=5
par g12=2,g21=1,gsa=.5,gas=2,gbs=2,gsb=.5
alpha(t)=t*exp(-t/tau_s)/(tau_s^2)
alpha1(t)=gsyn1*alpha(max(t-t1,0))
alpha2(t)=gsyn2*alpha(max(t-t2,0))
alphab(t)=gsynb*alpha(max(t-tb,0))
alphas(t)=gsyns*alpha(max(t-ts,0))
@ dt=.25,meth=qualrk,total=80,xhi=80,ylo=-80,yhi=25
done