More complex synapses

There are many other types of synapses that can have more complex behavior. For example in the bullfrog sympathetic ganglion, synaptic cotransmission occurs in which a cocktail of several neurotransmitters is released which then bind to several different receptor types. With such complex synapses, it is possible to get an initial brief depolarization followed by hyperpolarization and then by a long lasting low amplitude depolarization.

Many cortical neurons have AMPA synapses which depress. That is, when given repeated stimuli, the synapse produces less and less transmitter. We can readily model this by adding a desensitized state to the standard two-state model for the synapse. Instead of the open state going directly back to the closed state, we allow there to be an intermediate state (desensitized) which then returns to the closed state.

$$C \stackrel{\alpha [T]}{\rightharpoonup} O$$

$$O \stackrel{\beta}{\rightharpoonup} X$$

$$X \stackrel{\beta_2}{\rightharpoonup} C$$

Since C+O+X=1 (that is the probability of being any of these three states is 1) we eliminate the equation for the closed states and are left with the following pair of equations:

$$\frac{ds}{dt} = \alpha [T] (1-s-x) -\beta s$$

$$\frac{dx}{dt} = \beta s -\beta_2 x$$

The slower is $\beta_2$, the longer the synapse remains in the desensitized state, x.