There are five different types of eye movements: saccades, smooth pursuit, vestibular ocular eye movements, optokinetic eye movements, and vergence eye movements. The purpose of this book is focused primarily on mathematical models of the horizontal saccadic eye movement system and the smooth pursuit system, rather than on how visual information is processed. A saccade is a fast eye movement used to acquire a target by placing the image of the target on the fovea. Smooth pursuit is a slow eye movement used to track a target as it moves by keeping the target on the fovea. The vestibular ocular movement is used to keep the eyes on a target during brief head movements. The optokinetic eye movement is a combination of saccadic and slow eye movements that keeps a full-field image stable on the retina during sustained head rotation. Each of these movements is a conjugate eye movement, that is, movements of both eyes together driven by a common neural source. A vergence movement is a non-conjugate eye movement allowing the eyes to track targets as they come closer or farther away. In this book, early models of saccades and smooth pursuit are presented. The smooth pursuit system allows tracking of a slow moving target to maintain its position on the fovea. Models of the smooth pursuit have been developed using systems control theory, all involving a negative feedback control system that includes a time delay, controller and plant in the forward loop, with unity feedback. The oculomotor plant and saccade generator are the basic elements of the saccadic system. The oculomotor plant consists of three muscle pairs and the eyeball. A number of oculomotor plant models are described here beginning with the Westheimer model published in 1954, and up through our 1995 model involving a 4$^{th}$ order oculomotor plant model. The work presented here is not an exhaustive coverage of the field, but focused on the interests of the author. In Part II, a state-of-art model of the saccade system is presented, including a neural network that controls the system. Table of Contents: Introduction / Smooth Pursuit Models / Early Models of the Horizontal Saccadic Eye Movement System / Velocity and Acceleration Estimation / 1995 Linear Homeomorphic Saccadic Eye Movement Model

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John D. Enderle, Biomedical Engineering Program Director and Professor of Electrical & Computer Engineering at the University of Connecticut, received the B.S., M.E., and Ph.D. degrees in biomedical engineering, and M.E. degree in electrical engineering from Rensselaer Polytechnic Institute, Troy, New York, in 1975, 1977, 1980, and 1978, respectively. After completing his Ph.D. studies, he was a senior staff member at PAR Technology Corporation, Rome, New York, from 1979 to 1981. From 1981-1994, Enderle was a faculty member in the Department of Electrical Engineering and Coordinator for Biomedical Engineering at North Dakota State University (NDSU), Fargo, North Dakota. Dr. Enderle joined the National Science Foundation as Program Director for Biomedical Engineering & Research Aiding Persons with Disabilities Program from January 1994-June 1995. In January 1995, he joined the faculty of the University of Connecticut (UConn) as Professor and Head of the Electrical & SystemsEngineering Department. In June 1997, he became the Director for the Biomedical Engineering Program at UConn. Dr. Enderle is a Fellow of the Institute of Electrical & Electronics Engineers (IEEE), the current Editor-in-Chief of the EMB Magazine, the 2004 EMBS Service Award Recipient, Past-President of the IEEE-Engineering in Medicine and Biology Society (EMBS), EMBS Conference Chair for the 22nd Annual International Conference of the IEEE EMBS and World Congress on Medical Physics and Biomedical Engineering in 2000, a past EMBS Vice-President for Publications & Technical Activities and Vice-President for Member and Student Activities, Fellow of the American Institute for Medical and Biological Engineering (AIMBE), an ABET Program Evaluator for Bioengineering Programs, a member of the Engineering Accreditation Commission, a member of the American Society for Engineering Education and Biomedical Engineering Division Chair for 2005, and a Senior member of the Biomedical Engineering Society. Enderle was elected as a Member of the Connecticut Academy of Science and Engineering in 2003, with membership limited to 200 persons. He is also a Teaching Fellow at the University of Connecticut since 1998.