Author Type

Graduate Student

Date of Award

Summer 8-4-2026

Document Type

Dissertation

Publication Status

Version of Record

Submission Date

August 2026

Department

Mathematics and Statistics

College Granting Degree

Charles E. Schmidt College of Science

Department Granting Degree

Mathematics and Statistics

Degree Name

Doctor of Philosophy (PhD)

Thesis/Dissertation Advisor [Chair]

Zvi Rosen

Thesis/Dissertation Co-Chair

Timothy J. Ford

Abstract

Neural data is incredibly rich in combinatorial, topological, and geometrical information, reflecting the intricate shape and connectivity of neural firing patterns. To decipher these structures, neuroscience increasingly relies on advanced mathematical tools to analyze neural activity. Here we study (1) neural codes within the poset PCode of neural codes and (2) the connectivity of neural population activity within the insular cortex when responding to interoceptive information. In (1), we establish combinatorial constructions for all upward covering relations based on what we call “isolated subsets” with supporting theorems and give a slight modification of the existing downward covering relations. We then provide an enumeration algorithm to exhaust all codes covering a given neural code, followed by some computational results. In (2), we infer topological and geometrical differences between the neural connectivity observed around eating events and the activity observed around drinking events of food- and water-deprived mice observed from various analyses via statistical and topological data analysis tools, indicating the use of different neural mechanisms for these two behaviors. We conclude with a discussion of future directions and open problems related to these two areas.

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