Digital Commons Network^™

Drawing Place Field Diagrams Of Neural Codes Using Toric Ideals, Nida K. Obatake

Master's Theses

A neural code is a collection of codewords (0-1 vectors) of a given length n; it captures the co-firing patterns of a set of neurons. A neural code is convexly realizable in dimension two if there exist n convex sets in the plane so that each codeword in the code corresponds to a unique intersection carved out by the convex sets. There are some methods to determine whether a neural code is convexly realizable; however, these methods do not describe how to draw a realization, that is, a place field diagram of the code. In this work, we construct toric …

Geometric Control Theory: Nonlinear Dynamics And Applications, Geoffrey A. Zoehfeld

Master's Theses

We survey the basic theory, results, and applications of geometric control theory. A control system is a dynamical system with parameters called controls or inputs. A control trajectory is a trajectory of the control system for a particular choice of the inputs. A control system is called controllable if every two points of the underlying space can be connected by a control trajectory. Two fundamental problems of control theory include:

1) Is the control system controllable?

2) If it is controllable, how can we construct an input to obtain a particular control trajectory? We shall investigate the first problem exclusively …