  While extensive research has been conducted on chaos emerging from a
dynamical system's temporal dynamics, the research documented in this paper
examines extreme sensitivity to initial conditions in discrete-time dynamical
systems from a geometrical perspective. The heart of this paper focuses on two
simple neuron maps developed by Nikolai F. Rulkov in the early 2000s and the
complex geometrical structures that emerge from them. Beginning with a
conversational introduction to the geometry of chaos, this paper integrates
mathematics, physics, neurobiology, computational modeling, and
electrochemistry to present original research that provides a novel perspective
on how types of geometrical sensitivity to initial conditions appear in
discrete-time neuron systems.
  This paper was developed in the Thomas Jefferson High School for Science and
Technology Quantum Lab as part of a senior research project.
