Short term synaptic plasticity is a phenomenon which is commonly found in the central nervous system. It could contribute to functions of signal processing namely, temporal integration and coincidence detection by modulating the input synaptic strength. This dissertation has two parts. First, we study the effects of short term synaptic plasticity in enhancing coincidence detecting ability of neurons in the avian auditory brainstem. Coincidence detection means a target neuron has a higher firing rate when it receives simultaneous inputs from different neurons as opposed to inputs with large phase delays. This property is used by birds in sound localization. When there is no plasticity from the inputs, the firing rate of the neuron, depends more on input frequencies and less on phase delays between inputs. This leads to ambiguity in localizing the sound source. We derive a mathematical model of a reduced avian brainstem network and show that inputs with synaptic plasticity, to the coincidence detector neuron, play a vital role in enhancing coincidence detecting ability of the bird. We present comparisons to experiments. In the second part of the thesis, we prove the existence and stability of a "n" cluster solution in a "m"-cell network, in the presence of synaptic depression. The model used to represent a single neuron is based on the Hodgkin-Huxley model for the spiking neurons and we use techniques from geometric singular perturbation theory to show that any "n"-cluster solution must satisfy a set of consistency conditions that can be geometrically derived. The results of both problems are validated using numerical simulations. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]