Group-like algebraic structures of fuzzy topographic topological mapping for solving neuromagnetic inverse problem

Abstract

Fuzzy Topographic Topological Mapping (FTTM) is a novel mathematical model for solving neuromagnetic inverse problem. It is given as a set of mathematical operations, namely topological transformations with four components and connected by three different algorithms. At this moment, Fuzzy Topographic Topological Mapping 1 (FTTM 1) and Fuzzy Topographic Topological Mapping 2 (FTTM 2), which are used to solve the inverse problem for determining single current source and multiple current sources respectively, have been developed. The purpose of this research is to establish the topological and the algebraic structures of the components of FTTM 1 and FTTM 2. Firstly, the topological structures of the components of FTTM 2 were established and the homeomorphisms between the components of FTTM 2 were shown by using the proving techniques of the topological structures of the components of FTTM 1 and the homeomorphisms between the components of FTTM 1, then followed by the establishment of the algebraic structures of the components of FTTM 1 and FTTM 2. In the process, several definitions and theorems of group theory were adopted and the proving technique by construction was highlighted. In addition, FTTM was then generalized as a set which led to the proving the existence of infinitely many forms of FTTM. Finally, these structures were interpreted physically in order to study the information content of the inverse problem for determining single and multiple current sources.