Maintaining phase of the crustacean tri-phasic pyloric rhythm

We construct and analyze a model network of the pyloric rhythm of the crustacean stomatogastric ganglion consisting of an oscillator neuron that inhibits two reciprocally inhibitory follower neurons. We derive analytic expressions that determine the phase of firing of the follower neurons with respect to the oscillator. An important aspect of the model is the inclusion of synapses that exhibit short-term synaptic depression. We show that these type of synapses allow there to be a complicated relationship between the intrinsic properties of the neurons and the synapses between them in determining phase relationships. Our analysis reveals the circumstances and ranges of cycle periods under which these properties work in concert with or independently from one another. In particular, we show that phase maintenance over a range of oscillator periods can be enhanced through the interplay of the two follower neurons if the synapses between these neurons are depressing. Since our model represents the core of the oscillatory pyloric network, the results of our analysis can be compared to experimental data and used to make predictions about the biological network.