In this dissertation, dynamical system analysis and numerical tools are employed in an effort to solve problems in neuroscience on both the behavioral and neurophysiological levels. The motivations range from accurate prediction of periodic patterns in salamander retina to expectation of a future stimulus based on a sequence of stimuli in human subjects. Specific problems include: (1)  Connections among models of decision-making processes. This dissertation begins by exploring the relationships among three successful models widely used for decision-making processes, namely the connectionist model, the ring rate model and the drift diffusion model. The former two models are demonstrated to be equivalent. Then reductions are made from them to the drift diffusion model. Conditions for accurate reduction are discussed. The reduction to the drift diffusion model, for which explicit solutions of reaction time and error rate exist, provides a basis for optimality analysis including the derivation of optimal speed accuracy trade-off and optimal gains believed to be implemented by locus coeruleus. (2) A unified model of sequential effects in serial decision-making tasks. In serial decision making tasks, a subject's performance is influenced not only by the current stimulus but also by stimulus history. The pattern of this sequential effect transits from a benefit-only mode to a cost-benefit mode as response-to-stimulus interval (RSI) increases. In this chapter, the underlying mechanisms are explored under systematic changes of RSI, using a connectionist model to represent the two decision units. In contrast to the traditional view, it is suggested that activity in the anterior cingulate cortex (ACC), rather than residual activities of the decision units, is the neural mechanism underlying the benefit-only pattern. The cost-benefit pattern is proposed to be caused by the activity in the prefrontal cortex (PFC) which develops expectation for the coming stimulus and sends excitatory and inhibitory biases upon expectation confirmation and violation respectively. The dynamics of biases due to ACC and PFC activities during the RSI are also studied. (3) Oscillatory circuit underlying retinal detection of periodic patterns. Expectation of future stimulus occurs not only in the cortex but also in subcortical areas such as the retina. It is observed that isolated salamander and mouse retinas subjected to periodic patterns of dark ashes in the range 6–20 Hz can respond to an omitted flash by emitting ganglion cell spikes and that timing of the firing rate peak is locked to the time of the expected ash. This omitted stimulus response (OSR) is explained by an adaptive resonator model in which an LRC circuit represents the ON bipolar cell terminal. The LRC's resonant frequency is adjusted by calcium concentration to match the frequency of the stimulus. This model captures most of the observations. A resonator bank model is also developed in which different ON bipolar cells can be synchronized by gap junction coupling. (4)  Synchronization of neurons coupled by gap junctions and inhibitory synapses. Motivated by gap-junction-induced synchronization in the retina and in locus coeruleus, a systematic study of synchronization and de-synchronization of coupled neurons is performed. Leaky integrate-and-fire (LIF) neurons with gap junctions and inhibitory synapses are used for the analysis and compared with numerical simulation of a Hodgkin-Huxley type neuron. Bifurcation diagrams are derived based on Poincaré spike-to-spike maps to illustrate the conditions of synchronization and three factors are shown to promote synchronization: external bias current, super-threshold coupling strength (determined by relative size of spikes) and timescale of synaptic coupling.