# Two traub cells coupled together with synapses # cell 1 v1'=-(gna*h1*m1^3*(v1-ena)+gk*n1^4*(v1-ek)+gl*(v1-el) -i1+gsyn2*s2*(v1-vsyn2))/c m1'=am(v1)*(1-m1)-bm(v1)*m1 h1'=ah(v1)*(1-h1)-bh(v1)*h1 n1'=an(v1)*(1-n1)-bn(v1)*n1 s1'=alpha1*tmax*(1-s1)/(1+exp(-(v1-vt)/vs))-beta1*s1 # cell 2 v2'=-(gna*h2*m2^3*(v2-ena)+gk*n2^4*(v2-ek)+gl*(v2-el) -i2+gsyn1*s1*(v2-vsyn1))/c m2'=am(v2)*(1-m2)-bm(v2)*m2 h2'=ah(v2)*(1-h2)-bh(v2)*h2 n2'=an(v2)*(1-n2)-bn(v2)*n2 s2'=alpha2*tmax*(1-s2)/(1+exp(-(v2-vt)/vs))-beta2*s2 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,i1=0,i2=0, par gsyn1=0,gsyn2=0,vsyn1=0,vsyn2=0 par vt=2,vs=5,tmax=3.2 par alpha1=1,beta1=.2,alpha2=1,beta2=.2 init v1=-67,v2=-67,h1=1,h2=1 @ dt=.25,meth=qualrk,total=100,xhi=100,ylo=-85,yhi=25 done

**HOMEWORK**
The model is of two point neurons coupled together by first order
synapses like we have already explored. The neurons are labeled 1 and
2 and neuron 1 synapses onto 2 with synapse `s1` and maximal
conductance `gsyn1` and reversal potential `vsyn1`. Neuron 2
synapses onto neuron 1 with synapse `s2` and maximal
conductance `gsyn2` and reversal potential `vsyn2`. The
synaptic parameters, are `alpha1, alpha2, beta1,
beta2`. Neuron 1 receives an applied current `i1` and neuron 2
receives an applied current `i2.` Other than these parameters, the
neurons are identical. Both neurons are at rest.

- 1.
- Change the initial value of neuron 1
`v1`from -67 to -60. This will cause the neuron to fire a pulse. Use the ``graphics'' ``add curve'' option to add the voltage of cell 2 (`v2`) to the plot. It should be flat. Increase the synaptic conductance`gsyn1=.05`which impinges on cell2 from cell 1. Reintegrate the equations. Note that a spike occurs in both cell1 and cell2 since the firing of cell 1 depolarizes cell 2. Lower the conductance until there is no more spike. Observe a slight depolarization of cell 2 but no firing. What is the minimal conductance needed to elicit a spike? Now set`gsyn1=.1, gsyn2=.1`Integrate the equations. What happens? Set both conductances to 0.15. Integrate the equations. What happens? Can you explain what is going on? What is the phase difference between the cells? Are they synchronized or not? Would this behavior happen if the cells were mutually inhibitory ? How about if one is excitatory and the other inhibitory? - 2.
- Now set
`i1=1,i2=1.05, gsyn1=.05,gsyn2=.05`and integrate the equations with mutual excitatory coupling. What sort of behavior is observed? To see the steady state behavior, you may want to integrate for a longer period of time. - 3.
- Make cell 2 inhibitory by changing
`vsyn2=-80`. Change`i1=.5, i2=0, gsyn1=.1,gsyn2=.2`Integrate for 400 msec. Calculate the period of the oscillation. Change the decay rate of the inhibitory synapse (`beta2=.1`and recalculate the period. Also do the same thing for`beta2=.05`. Explain why the period is getting longer. - 4.
- Here is a last example of the complexities of network
behavior. Set
`vsyn1=0, vsyn2=-80`,`alpha2=.5,beta2=.01, gsyn1=.01``gsyn2=1, i1=3,i2=0.`Set the total integration time to 1000. Integrate the equations. What do you see? Explain the behavior. How would you modify`gsyn1`to get fewer cell 1 spikes per cell 2 spike? How would you get more cell 1 spikes per cell 2 spike?

This exercise should give you an appreciation of how complex even simple networks can get. Synapses provide a powerful computational tool for sculpting complex behaviors out of simple networks. You can now imagine how complex behavior can be when all of the channels that we have previously described are added to the network. The fact that such models are tremendously complex and that solving these ``detailed'' (not really as far as the real biology is concerned) models is computationally expensive makes one ask how can one distinguish between models and how useful can these methods be? For this, reason, in the next set of lectures, we will look at simplified versions of these models and their computational properties.