Behavior of a network of neurons is closely linked to the properties of individual neurons and synaptic coupling. Connecting model neurons with inhibitory synapses can cause synchrony or anti-synchrony. A number of studies have attempted to explain what causes these two distinct firing modes, but a consensus has not been reached. Separate model parameters have been shown to change a network's firing pattern, but a complete picture of what causes anti-synchrony is lacking. In this thesis we present a general method for determining if a network of two identical cells can fire synchronously and anti-synchronously. Our results give a complete description of how a number of parameters affect anti-synchronous firing and create a framework for understanding many previously published studies. We develop an augmented spike-time-response (STR) method in which the delay of inhibitory perturbations on the timing of subsequent spikes is used to predict stable firing patterns in interneuron networks. A novel way of relating the shape of the spike time response curve (STRC) to the anti-synchronous firing pattern allows us to find a simple criterion for stable anti-synchrony. The criterion allows us to characterize how individual model parameters and combinations of different parameters affect anti-synchronous firing. Predictions from our augmented spike time response method are verified by using network simulations.

Determining when a two Hodgkin-Huxley cell network can be entrained by oscillating input (with amplitude D and frequency f) has not been studied in detail. We propose a method using a single cell oscillator with instantaneous self-inhibition to determine ( f,D)values that can entrain the network in synchrony (P ^s) and a different single cell oscillator with delayed self-inhibition to designate (f,D)-values that enable anti-synchronous entrainment (P^a). P^s and P^s reveal (f,D)-values where a two cell network receiving oscillatory input displays robust synchrony ((f,D) ∈ P^s\ P^a) and distinct (f,D)-values where a two cell network receiving oscillatory input displays robust anti-synchrony ((f,D) ∈ P^a\ P^s) When (f,D) ∈ P ^a ∩ P_s the two cell network can display both long lasting synchronous firing and long lasting anti-synchronous firing. The network has different output frequency and power for the two distinct firing patterns, illustrating that a single ( f,D)-pair can lead to different network outputs.