Neuromagnetic Inverse Problem Research Articles

Tangled Cord of FTTM4

Fuzzy Topological Topographic Mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, denoted as FTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. Later, the FTTMn can also be viewed as a graph. Previously, a group of researchers defined an assembly graph and utilized it to model a DNA recombination process. Some researchers then used this to introduce the concept of tangled cords for assembly graphs. In this paper, the tangled cord for FTTM4 is used to calculate the Eulerian paths. Furthermore, it is utilized to determine the least upper bound of the Hamiltonian paths of its assembly graph. Hence, this study verifies the conjecture made by Burns et al.

Extended Graph of Fuzzy Topographic Topological Mapping Model: G04(FTTMn4)

Fuzzy topological topographic mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. The key to the model is its topological structure that can accommodate electrical or magnetic recorded brain signal. A sequence of FTTM, FTTMn, is an extension of FTTM whereby its form can be arranged in a symmetrical form, i.e., polygon. The special characteristic of FTTM, namely, the homeomorphisms between its components, allows the generation of new FTTM. The generated FTTMs can be represented as pseudo graphs. A pseudo-graph consists of vertices that signify the generated FTTM and edges that connect their incidence components. A graph of pseudo degree zero, G0(FTTMnk ), however, is a special type of graph where each of the FTTM components differs from its adjacent. A researcher posted a conjecture on G03(FTTMn3) in 2014, and it was finally proven in 2021 by researchers who used their novel grid-based method. In this paper, the extended G03(FTTMn3), namely, the conjecture on G04(FTTMn4) that was posed in 2018, is narrated and proven using simple mathematical induction.

Extended Graph of the Fuzzy Topographic Topological Mapping Model

Fuzzy topological topographic mapping (FTTM) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, FTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. The generated FTTMs can be represented as pseudo graphs. A graph of pseudo degree zero is a special type of graph where each of the FTTM components differs from the one adjacent to it. Previous researchers have investigated and conjectured the number of generated FTTM pseudo degree zero with respect to n number of components and k number of versions. In this paper, the conjecture is proven analytically for the first time using a newly developed grid-based method. Some definitions and properties of the novel grid-based method are introduced and developed along the way. The developed definitions and properties of the method are then assembled to prove the conjecture. The grid-based technique is simple yet offers some visualization features of the conjecture.

Monotonicity and topologically group on fuzzy topographic topological mappings

The mathematical model Fuzzy Topographic Topological Mapping (FTTM) was developed to solve the neuromagnetic inverse problem during a seizure for determining the location of epileptic foci. The aim of this paper is to investigate various properties for the components and mappings of FTTM. As a result, various kinds of monotonicity have been assured for its mappings. Furthermore, each component of FTTM was demonstrated as a topological group, unicoherent and indecomposable.

Moduli Space of the Category of Fuzzy Topographic Topological Mappings

The moduli space has concept emerged to classify objects of certain categories by carrying out their parameterization. In fact, the theory of moduli space has been widely appreciated from a distant time in diverse fields not only of mathematics, but also of physics. This paper is devoted to establish the moduli space of the category of the mathematical model fuzzy topographic topological mappings, which is adopted in solving neuromagnetic inverse problem during a seizure. In this study, the moduli functor of fuzzy topographic topological mapping was constructed. The applicability of the theory of moduli space is uniquely revealed with specifying its sort.

Determination of Number of New Elements for Sequence of Fuzzy Topographic Topological Mapping

Problem involving neuro magnetic inverse problem can be solved with Fuzzy Topographic Topological Mapping (FTTM). FTTM is a model consists of four components and connected by three algorithms. FTTM version 1 and version 2 were designed to present 3D view of an unbounded single current and bounded multicurrent sources, respectively. Several definitions related to sequence of FTTM were introduced by Suhana and the feature, namely the cube of FTTM are developed. In this paper, sequence of FTTM namely FTTMn are discussed. Consequently, some theorems are proved to describe the number of the new elements produced from the FTTMn. Besides that, the number of new elements produced from combination of sequence of FTTMn can be written as a combination from cubes of FTTMn.

Sequence of fuzzy topographic topological mapping and k-Fibonacci sequence

Fuzzy Topological Topographic Mapping (FTTM) is a model for solving neuromagnetic inverse problem. FTTM consists of four components and connected by three algorithms. FTTM version 1 and FTTM version 2 were designed to present 3D view of an unbounded single current and bounded multicurrent sources, respectively. In 2008, Suhana proved the conjecture posed by Liau in 2005 such that if there exist n number of FTTM, then n4 - n new elements of FTTM will be generated from it. In this paper, a new proof of sequence of FTTM is presented using fc-Fibonacci sequence.

A new proof on sequence of fuzzy topographic topological mapping

Fuzzy Topological Topographic Mapping (FTTM) is a model for solving neuromagnetic inverse problem. FTTM consists of four components and connected by three algorithms. FTTM version 1 and FTTM version 2 were designed to present 3D view of an unbounded single current and bounded multicurrent source, respectively. In 2008, Suhana proved the conjecture posed by Liau in 2005 such that, if there exists n number of FTTM, then n4-n new elements of FTTM will be generated from it. Suhana also developed some new definitions on geometrical features of FTTM, and discovered some interesting algebraic properties. In this paper, new proof on sequence of FTTM will be presented. In the proof, the sequence of FTTM is transformed into a system of differential equation.

A Subspace Pursuit-based Iterative Greedy Hierarchical solution to the neuromagnetic inverse problem

Magnetoencephalography (MEG) is an important non-invasive method for studying activity within the human brain. Source localization methods can be used to estimate spatiotemporal activity from MEG measurements with high temporal resolution, but the spatial resolution of these estimates is poor due to the ill-posed nature of the MEG inverse problem. Recent developments in source localization methodology have emphasized temporal as well as spatial constraints to improve source localization accuracy, but these methods can be computationally intense. Solutions emphasizing spatial sparsity hold tremendous promise, since the underlying neurophysiological processes generating MEG signals are often sparse in nature, whether in the form of focal sources, or distributed sources representing large-scale functional networks. Recent developments in the theory of compressed sensing (CS) provide a rigorous framework to estimate signals with sparse structure. In particular, a class of CS algorithms referred to as greedy pursuit algorithms can provide both high recovery accuracy and low computational complexity. Greedy pursuit algorithms are difficult to apply directly to the MEG inverse problem because of the high-dimensional structure of the MEG source space and the high spatial correlation in MEG measurements. In this paper, we develop a novel greedy pursuit algorithm for sparse MEG source localization that overcomes these fundamental problems. This algorithm, which we refer to as the Subspace Pursuit-based Iterative Greedy Hierarchical (SPIGH) inverse solution, exhibits very low computational complexity while achieving very high localization accuracy. We evaluate the performance of the proposed algorithm using comprehensive simulations, as well as the analysis of human MEG data during spontaneous brain activity and somatosensory stimuli. These studies reveal substantial performance gains provided by the SPIGH algorithm in terms of computational complexity, localization accuracy, and robustness.

GRAPH OF FINITE SEQUENCE OF FUZZY TOPOGRAPHIC TOPOLOGICAL MAPPING OF ORDER TWO

Fuzzy Topographic Topological Mapping (FTTM) was built to solve the neuromagnetic inverse problem to determine the location of epileptic foci in epilepsy disorder patient. The model which consists of topological and fuzzy structures is composed into three mathematical algorithms. FTTM consists of four topological spaces and connected by three homeomorphisms. FTTM version 1 is also homeomorphic to FTTM version 2. This homeomorphism generates another 14 elements of FTTM. In this study we proved that, if there exist n elements of FTTM, the new elements of order 2 will produce a graph of degree 24n2-16n-8. In this study, the statement is proven by viewing FTTMs as sequence and using its graphical features. In the process, several definitions and theorems were developed.

Semigroup of EEG Signals during Epileptic Seizure

Fuzzy Topographic Topological Mapping (FTTM) is a mathematical model for solving neuromagnetic inverse problem where FTTM is a set consisting of elements with four components and three algorithms which link between the four components. In this study, we show that the first component of FTTM, namely magnetic contour plane which contains electroencephalography signals during epileptic seizure can be viewed as a semigroup of square matrices under matrix multiplication.

Generalized Finite Sequence of Fuzzy Topographic Topological Mapping

Problem statement: Fuzzy Topographic Topological Mapping (FTTM) was developed to solve the neuromagnetic inverse problem. FTTM consisted of four topological spaces and connected by three homeomorphisms. FTTM 1 and FTTM 2 were developed to present 3-D view of an unbounded single current source and bounded multicurrent sources, respectively. FTTM 1 and FTTM 2 were homeomorphic and this homeomorphism will generate another 14 FTTM. We conjectured if there exist n elements of FTTM, then the numbers of new elements are n4-n. Approach: In this study, the conjecture was proven by viewing FTTMs as sequence and using its geometrical features. Results: In the process, several definitions were developed, geometrical and algebraic properties of FTTM were discovered. Conclusion: The conjecture was proven and some features of the sequence appear in Pascal Triangle.

Sequence of Fuzzy Topographic Topological Mapping

Fuzzy Topographic Topological Mapping, shortly FTTM, is a model for solving neuromagnetic inverse problem. FTTM consist of four topological spaces and connected by three homeomorphisms. FTTM 1 and FTTM 2 were developed topresent 3-D view of an unbounded single current source and bounded multicurrent sources, respectively. Liau Li Yun (2006) showed that FTTM 1 and FTTM 2 are homeomorphic and this homeomorphism will generate another 14 FTTM. She then conjectured if there exist n elements of FTTM then the numbers of new elements are n4 − n . The 1purpose of this paper is to study the geometrical features of FTTM. In the process, several definitions were developed which may be used to prove the conjecture. This paper will show the proof of the conjecture and their extension result.

Parallel Minimum p-Norm Solution of the Neuromagnetic Inverse Problem for Realistic Signals Using Exact Hessian-Vector Products

In the neuromagnetic inverse problem, one is interested in determining the current density inside the human brain from measurements of the magnetic field recorded outside the head. From a numerical point of view, the solution of this inverse problem is challenging not only in terms of nonuniqueness and time complexity but also with respect to numerical stability. An efficient and robust computational technique is presented that finds the minimum p-norm solution of the neuromagnetic inverse problem. The approach is based on carefully combining a subspace trust-region algorithm for the solution of an unconstrained nonlinear optimization problem, automatic differentiation for the truncation-error free evaluation of first- and second-order derivatives, and shared-memory parallelization using the OpenMP programming paradigm. Using actual measurements obtained from a head phantom model as well as realistic data sets of middle-latency auditory evoked field data, it is demonstrated that a valid reconstruction of neuromagnetic activity is achieved for values of p less than 2.

Estimation of Solution Accuracy From Leadfield Matrix in Magnetoencephalography

This paper is a report of an investigation of the relationship between the condition number of a leadfield matrix of magnetoencephalography and source reconstruction accuracy. We suggest the use of the condition number as a promising a priori accuracy estimator of a neuromagnetic inverse problem (NIP) since various simulation studies demonstrated that the reconstructed source distribution has the wider region and contains more spurious sources if a leadfield matrix has the higher condition number. The simulation results also verified that the accuracy estimator based upon the condition number can explain well-known phenomena of NIP, which suggests a potential application of the proposed concept

Bayesian analysis of the neuromagnetic inverse problem with ℓp-norm priors

Magnetoencephalography (MEG) allows millisecond-scale non-invasive measurement of magnetic fields generated by neural currents in the brain. However, localization of the underlying current sources is ambiguous due to the so-called inverse problem. The most widely used source localization methods (i.e., minimum-norm and minimum-current estimates (MNE and MCE) and equivalent current dipole (ECD) fitting) require ad hoc determination of the cortical current distribution ( ℓ 2-, ℓ 1-norm priors and point-sized dipolar, respectively). In this article, we perform a Bayesian analysis of the MEG inverse problem with ℓ p -norm priors for the current sources. This way, we circumvent the arbitrary choice between ℓ 1- and ℓ 2-norm prior, which is instead rendered automatically based on the data. By obtaining numerical samples from the joint posterior probability distribution of the source current parameters and model hyperparameters (such as the ℓ p -norm order p) using Markov chain Monte Carlo (MCMC) methods, we calculated the spatial inverse estimates as expectation values of the source current parameters integrated over the hyperparameters. Real MEG data and simulated (known) source currents with realistic MRI-based cortical geometry and 306-channel MEG sensor array were used. While the proposed model is sensitive to source space discretization size and computationally rather heavy, it is mathematically straightforward, thus allowing incorporation of, for instance, a priori functional magnetic resonance imaging (fMRI) information.