The basic story is that the action potential travels down the axon and terminates at many presynaptic sites invading regions called synaptic terminals. These terminals contain calcium channels which when depolarized cause (a) release of calcium; (b) calcium activates a calcium binding protein which promotes release by binding to vesicles containing the transmitter; (c) these ``docked'' vesicles release their transmitter into the synaptic cleft. This type of detail is all well and good, but if we are going to model many cells, then we want to simplify things quite a bit. In the chapter by Destexhe, Mainen, and Sejnowski, a detailed description of this model is provided as well as simulations. Click here for a copy.
We adopt here a cruder approach that can be viwed as a mean field or average over many individual synapses. Our simple model is to treat the synapse as just another ionic channel that will be modeled by Ohms law. For every synapse between two neurons, we will add a current of the form:
|Isyn = gsyns(t)(V-Vsyn)||(1)|
# pas_syn.ode init v1=5.08 v2=4.33 v3=3.8 v4=3.46 v5=3.3 par c=1 i=1 gl=0.05 vsyn=0 gsyn=0.1 gc=1 par ton=10 toff=15 v1'=-gl*v1+gc*(v2-v1)+i v2'=-gl*v2+gc*(v1-2*v2+v3) v3'=-gl*v3+gc*(v2-2*v3+v4)-gsyn*heav(t-ton)*heav(toff-t)*(v3-vsyn) v4'=-gl*v4+gc*(v3-2*v4+v5) v5'=-gl*v5+gc*(v4-v5) doneand is called pas_syn5.ode. Plot v1,v3,v5 against time for vsyn=-20mV,0mV,50mV and a variety of different times on/off and maximal conductances. Integrate the equations for 100 msec and define an appropriate window. Note the difference at the ends and at the site of the synapse. The time units are in milliseconds. What are the units of i in this equation? Note also, that the coupling conductances and the leak and the synapses are all in mS/cm2. Observe that even though compartment 1 receives input via a current injection, that the synapse that is downwind from it exerts a significant effect on the membrane potential at that point.
Now that you have explored the effects of these transient conductance changes, I return to modeling the synapse. How do we determine s(t) from the presynaptic activity? As we have noted above, the release of transmitter is quite complex. However, a good fit to the total amount of transmitter release by a single action potential is given by the expression: