Adaptive smoothing of retinotopic maps based on Teichmüller parametrization.

Retinotopic mapping, the mapping between visual inputs on the retina and neural responses on the cortical surface, is one of the fundamental topics in visual neuroscience. In human studies, retinotopic maps are conventionally constructed and processed by decoding blood oxygenation-level dependent (BOLD) functional magnetic resonance imaging (fMRI) responses to designed visual stimuli on the cortical surface. However, these methods frequently generate retinotopic maps that do not preserve topology, contradicting a fundamental property of retinotopic maps observed in neurophysiology. To address this problem, we propose an integrated approach to simultaneously refine the flattening from the 3D cortical surface to the 2D parametric space and adaptively smooth retinotopic perception centers in the visual space to make the retinotopic maps topological. One key element of the approach is the enhanced error tolerant Teichmüller mapping, which refines the parametrization by minimizing angle distortions and maximizing alignment to noisy landmarks. We validated our overall approach with synthetic and real retinotopic mapping datasets and applied it to compute cortical magnification factor (CMF). The results showed that the proposed approach was superior to other conventional retinotopic mapping methods in predicting BOLD fMRI time series and preserving topology.

Protocol for topology-preserving smoothing of BOLD fMRI retinotopic maps of the human visual cortex.

The hierarchical organization of the visual system preserves topology, which is often lost in the "raw" human retinotopic maps derived from BOLD fMRI recordings. Here, we present the protocol for generating topology-preserving and smooth retinotopic maps from human retinotopy fMRI data. We describe data pre-processing, 3D surface flattening, and selection of the region of interest (ROI), followed by smoothing of the retinotopic maps within the ROI. This approach can be applied to visual cortical areas V1, V2, and V3 simultaneously. For complete details on the use and execution of this protocol, please refer to Tu et al. (2021).

Diffeomorphic registration for retinotopic maps of multiple visual regions.

Retinotopic map, the mapping between visual inputs on the retina and neuronal responses on the cortical surface, is one of the central topics in vision science. Typically, human retinotopic maps are constructed by analyzing functional magnetic resonance responses to designed visual stimuli on the cortical surface. Although it is widely used in visual neuroscience, retinotopic maps are limited by the signal-to-noise ratio and spatial resolution of fMRI. One promising approach to improve the quality of retinotopic maps is to register individual subject's retinotopic maps to a retinotopic template. However, none of the existing retinotopic registration methods has explicitly quantified the diffeomorphic condition, that is, retinotopic maps shall be aligned by stretching/compressing without tearing up the cortical surface. Here, we developed Diffeomorphic Registration for Retinotopic Maps (DRRM) to simultaneously align retinotopic maps in multiple visual regions under the diffeomorphic condition. Specifically, we used the Beltrami coefficient to model the diffeomorphic condition and performed surface registration based on retinotopic coordinates. The overall framework preserves the topological condition defined in the template. We further developed a unique evaluation protocol and compared the performance of the new method with several existing registration methods on both synthetic and real datasets. The results showed that DRRM is superior to the existing methods in achieving diffeomorphic  registration in synthetic and empirical data from 3T and 7T MRI systems. DRRM may improve the interpretation of low-quality retinotopic maps and facilitate applications of retinotopic maps in clinical settings.

Quantitative characterization of the human retinotopic map based on quasiconformal mapping.

The retinotopic map depicts the cortical neurons' response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmüller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a "Beltrami coefficient map" (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

Topological Receptive Field Model for Human Retinotopic Mapping.

The mapping between visual inputs on the retina and neuronal activations in the visual cortex, i.e., retinotopic map, is an essential topic in vision science and neuroscience. Human retinotopic maps can be revealed by analyzing the functional magnetic resonance imaging (fMRI) signal responses to designed visual stimuli in vivo. Neurophysiology studies summarized that visual areas are topological (i.e., nearby neurons have receptive fields at nearby locations in the image). However, conventional fMRI-based analyses frequently generate non-topological results because they process fMRI signals on a voxel-wise basis, without considering the neighbor relations on the surface. Here we propose a topological receptive field (tRF) model which imposes the topological condition when decoding retinotopic fMRI signals. More specifically, we parametrized the cortical surface to a unit disk, characterized the topological condition by tRF, and employed an efficient scheme to solve the tRF model. We tested our framework on both synthetic and human fMRI data. Experimental results showed that the tRF model could remove the topological violations, improve model explaining power, and generate biologically plausible retinotopic maps. The proposed framework is general and can be applied to other sensory maps.

Topology-preserving smoothing of retinotopic maps.

Retinotopic mapping, i.e., the mapping between visual inputs on the retina and neuronal activations in cortical visual areas, is one of the central topics in visual neuroscience. For human observers, the mapping is obtained by analyzing functional magnetic resonance imaging (fMRI) signals of cortical responses to slowly moving visual stimuli on the retina. Although it is well known from neurophysiology that the mapping is topological (i.e., the topology of neighborhood connectivity is preserved) within each visual area, retinotopic maps derived from the state-of-the-art methods are often not topological because of the low signal-to-noise ratio and spatial resolution of fMRI. The violation of topological condition is most severe in cortical regions corresponding to the neighborhood of the fovea (e.g., < 1 degree eccentricity in the Human Connectome Project (HCP) dataset), significantly impeding accurate analysis of retinotopic maps. This study aims to directly model the topological condition and generate topology-preserving and smooth retinotopic maps. Specifically, we adopted the Beltrami coefficient, a metric of quasiconformal mapping, to define the topological condition, developed a mathematical model to quantify topological smoothing as a constrained optimization problem, and elaborated an efficient numerical method to solve the problem. The method was then applied to V1, V2, and V3 simultaneously in the HCP dataset. Experiments with both simulated and real retinotopy data demonstrated that the proposed method could generate topological and smooth retinotopic maps.

DIFFEOMORPHIC SMOOTHING FOR RETINOTOPIC MAPPING.

Retinotopic mapping, the mapping of visual input on the retina to cortical neurons, is an important topic in vision science. Typically, cortical neurons are related to visual input on the retina using functional magnetic resonance imaging (fMRI) of cortical responses to slowly moving visual stimuli on the retina. Although it is well known from neurophysiology studies that retinotopic mapping is locally diffeomorphic (i.e., smooth, differentiable, and invertible) within each local area, the retinotopic maps from fMRI are often not diffeomorphic, especially near the fovea, because of the low signal-noise ratio of fMRI. The aim of this study is to develop and solve a mathematical model that produces diffeomorphic retinotopic mapping from fMRI data. Specifically, we adopt a geometry concept, the Beltrami coefficient, as the tool to define diffeomorphism, and model the problem in an optimization framework. Efficient numerical methods are proposed to solve the model. Experimental results with both synthetic and real retinotopy data demonstrate that the proposed method is superior to conventional smoothing methods.

Computing Univariate Neurodegenerative Biomarkers with Volumetric Optimal Transportation: A Pilot Study.

Changes in cognitive performance due to neurodegenerative diseases such as Alzheimer's disease (AD) are closely correlated to the brain structure alteration. A univariate and personalized neurodegenerative biomarker with strong statistical power based on magnetic resonance imaging (MRI) will benefit clinical diagnosis and prognosis of neurodegenerative diseases. However, few biomarkers of this type have been developed, especially those that are robust to image noise and applicable to clinical analyses. In this paper, we introduce a variational framework to compute optimal transportation (OT) on brain structural MRI volumes and develop a univariate neuroimaging index based on OT to quantify neurodegenerative alterations. Specifically, we compute the OT from each image to a template and measure the Wasserstein distance between them. The obtained Wasserstein distance, Wasserstein Index (WI) for short to specify the distance to a template, is concise, informative and robust to random noise. Comparing to the popular linear programming-based OT computation method, our framework makes use of Newton's method, which makes it possible to compute WI in large-scale datasets. Experimental results, on 314 subjects (140 Aß + AD and 174 Aß- normal controls) from the Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline dataset, provide preliminary evidence that the proposed WI is correlated with a clinical cognitive measure (the Mini-Mental State Examination (MMSE) score), and it is able to identify group difference and achieve a good classification accuracy, outperforming two other popular univariate indices including hippocampal volume and entorhinal cortex thickness. The current pilot work suggests the application of WI as a potential univariate neurodegenerative biomarker.

Optimizing Visual Cortex Parameterization with Error-Tolerant Teichmüller Map in Retinotopic Mapping.

The mapping between the visual input on the retina to the cortical surface, i.e., retinotopic mapping, is an important topic in vision science and neuroscience. Human retinotopic mapping can be revealed by analyzing cortex functional magnetic resonance imaging (fMRI) signals when the subject is under specific visual stimuli. Conventional methods process, smooth, and analyze the retinotopic mapping based on the parametrization of the (partial) cortical surface. However, the retinotopic maps generated by this approach frequently contradict neuropsychology results. To address this problem, we propose an integrated approach that parameterizes the cortical surface, such that the parametric coordinates linearly relates the visual coordinate. The proposed method helps the smoothing of noisy retinotopic maps and obtains neurophysiological insights in human vision systems. One key element of the approach is the Error-Tolerant Teichmüller Map, which uniforms the angle distortion and maximizes the alignments to self-contradicting landmarks. We validated our overall approach with synthetic and real retinotopic mapping datasets. The experimental results show the proposed approach is superior in accuracy and compatibility. Although we focus on retinotopic mapping, the proposed framework is general and can be applied to process other human sensory maps.

DIFFEOMORPHIC REGISTRATION FOR RETINOTOPIC MAPPING VIA QUASICONFORMAL MAPPING.

Human visual cortex is organized into several functional regions/areas. Identifying these visual areas of the human brain (i.e., V1, V2, V4, etc) is an important topic in neurophysiology and vision science. Retinotopic mapping via functional magnetic resonance imaging (fMRI) provides a non-invasive way of defining the boundaries of the visual areas. It is well known from neurophysiology studies that retinotopic mapping is diffeomorphic within each local area (i.e. locally smooth, differentiable, and invertible). However, due to the low signal-noise ratio of fMRI, the retinotopic maps from fMRI are often not diffeomorphic, making it difficult to delineate the boundaries of visual areas. The purpose of this work is to generate diffeomorphic retinotopic maps and improve the accuracy of the retinotopic atlas from fMRI measurements through the development of a specifically designed registration procedure. Although there are sophisticated existing cortical surface registration methods, most of them cannot fully utilize the features of retinotopic mapping. By considering unique retinotopic mapping features, we form a quasiconformal geometry-based registration model and solve it with efficient numerical methods. We compare our registration with several popular methods on synthetic data. The results demonstrate that the proposed registration is superior to conventional methods for the registration of retinotopic maps. The application of our method to a real retinotopic mapping dataset also results in much smaller registration errors.

Isometry Invariant Shape Descriptors for Abnormality Detection on Brain Surfaces Affected by Alzheimer's Disease.

Alzheimer's disease (AD), a progressive brain disorder, is the most common neurodegenerative disease in older adults. There is a need for brain structural magnetic resonance imaging (MRI) biomarkers to help assess AD progression and intervention effects. Prior research showed that surface based brain imaging features hold great promise as efficient AD biomarkers. However, the complex geometry of cortical surfaces poses a major challenge to defining such a feature that is sensitive in qualification, robust in analysis, and intuitive in visualization. Here we propose a novel isometry invariant shape descriptor for brain morphometry analysis. First, we calculate a global area-preserving mapping from cortical surface to the unit sphere. Based on the mapping, the Beltrami coefficient shape descriptor is calculated. An analysis of average shape descriptors reveals that our detected features are consistent with some previous AD studies where medial temporal lobe volume was identified as an important AD imaging biomarker. We further apply a novel patch-based spherical sparse coding scheme for feature dimension reduction. Later, a support vector machine (SVM) classifier is applied to discriminate 135 amyloid-beta positive persons with the clinical diagnosis of Mild Cognitive Impairment (MCI) from 248 amyloid-beta-negative normal control subjects. The 5-folder cross-validation accuracy is about 81.82\% on the dataset, outperforming some traditional, Freesurfer based, brain surface features. The results show that our shape descriptor is effective in distinguishing dementia due to AD from age-matched normal aging individuals. Our isometry invariant shape descriptors may provide a unique and intuitive way to inspect cortical surface and its morphometry changes.

MULTI-TASK SPARSE SCREENING FOR PREDICTING FUTURE CLINICAL SCORES USING LONGITUDINAL CORTICAL THICKNESS MEASURES.

Cortical thickness estimation performed in-vivo via magnetic resonance imaging (MRI) is an effective measure of brain atrophy in preclinical individuals at high risk for Alzheimer's disease (AD). However, the high dimensionality of individual cortical thickness data coupled with small population samples make it challenging to perform cortical thickness feature selection for AD diagnosis and prognosis. Thus far, there are very few methods that can accurately predict future clinical scores using longitudinal cortical thickness measures. In this paper, we propose an unsupervised dictionary learning algorithm, termed Multi-task Sparse Screening (MSS) that produces improved results over previous methods within this problem domain. Specifically, we formulate and solve a multi-task problem using extracted top-p significant features from the Alzheimer's Disease Neuroimaging Initiative (ADNI) longitudinal data. Empirical studies on publicly available longitudinal data from ADNI dataset (N = 2797) demonstrate improved correlation coefficients and root mean square errors, when compared to other algorithms.