Structure-seeking multilinear methods for the analysis of fMRI data

A canonical polyadic tensor basis for fast Bayesian estimation of multi-subject brain activation patterns

Task-evoked functional magnetic resonance imaging studies, such as the Human Connectome Project (HCP), are a powerful tool for exploring how brain activity is influenced by cognitive tasks like memory retention, decision-making, and language processing. A fast Bayesian function-on-scalar model is proposed for estimating population-level activation maps linked to the working memory task. The model is based on the canonical polyadic (CP) tensor decomposition of coefficient maps obtained for each subject. This decomposition effectively yields a tensor basis capable of extracting both common features and subject-specific features from the coefficient maps. These subject-specific features, in turn, are modeled as a function of covariates of interest using a Bayesian model that accounts for the correlation of the CP-extracted features. The dimensionality reduction achieved with the tensor basis allows for a fast MCMC estimation of population-level activation maps. This model is applied to one hundred unrelated subjects from the HCP dataset, yielding significant insights into brain signatures associated with working memory.

Exploring the feasibility of tensor decomposition for analysis of fNIRS signals: a comparative study with grand averaging method

The analysis of functional near-infrared spectroscopy (fNIRS) signals has not kept pace with the increased use of fNIRS in the behavioral and brain sciences. The popular grand averaging method collapses the oxygenated hemoglobin data within a predefined time of interest window and across multiple channels within a region of interest, potentially leading to a loss of important temporal and spatial information. On the other hand, the tensor decomposition method can reveal patterns in the data without making prior assumptions of the hemodynamic response and without losing temporal and spatial information. The aim of the current study was to examine whether the tensor decomposition method could identify significant effects and novel patterns compared to the commonly used grand averaging method for fNIRS signal analysis. We used two infant fNIRS datasets and applied tensor decomposition (i.e., canonical polyadic and Tucker decompositions) to analyze the significant differences in the hemodynamic response patterns across conditions. The codes are publicly available on GitHub. Bayesian analyses were performed to understand interaction effects. The results from the tensor decomposition method replicated the findings from the grand averaging method and uncovered additional patterns not detected by the grand averaging method. Our findings demonstrate that tensor decomposition is a feasible alternative method for analyzing fNIRS signals, offering a more comprehensive understanding of the data and its underlying patterns.

High-order tensor flow processing using integrated photonic circuits

Tensor analytics lays the mathematical basis for the prosperous promotion of multiway signal processing. To increase computing throughput, mainstream processors transform tensor convolutions into matrix multiplications to enhance the parallelism of computing. However, such order-reducing transformation produces data duplicates and consumes additional memory. Here, we propose an integrated photonic tensor flow processor (PTFP) without digitally duplicating the input data. It outputs the convolved tensor as the input tensor 'flows' through the processor. The hybrid manipulation of optical wavelengths, space dimensions, and time delay steps, enables the direct representation and processing of high-order tensors in the optical domain. In the proof-of-concept experiment, an integrated processor manipulating wavelengths and delay steps is implemented for demonstrating the key functionalities of PTFP. The multi-channel images and videos are processed at the modulation rate of 20 Gbaud. A convolutional neural network for video action recognition is demonstrated on the processor, which achieves an accuracy of 97.9%.

Constrained CPD of Complex-Valued Multi-Subject fMRI Data via Alternating Rank-R and Rank-1 Least Squares

Complex-valued shift-invariant canonical polyadic decomposition (CPD) under a spatial phase sparsity constraint (pcsCPD) shows excellent separation performance when applied to band-pass filtered complex-valued multi-subject fMRI data. However, some useful information may also be eliminated when using a band-pass filter to suppress unwanted noise. As such, we propose an alternating rank- R and rank-1 least squares optimization to relax the CPD model. Based upon this optimization method, we present a novel constrained CPD algorithm with temporal shift-invariance and spatial sparsity and orthonormality constraints. More specifically, four steps are conducted until convergence for each iteration of the proposed algorithm: 1) use rank- R least-squares fit under spatial phase sparsity constraint to update shared spatial maps after phase de-ambiguity; 2) use orthonormality constraint to minimize the cross-talk between shared spatial maps; 3) update the aggregating mixing matrix using rank- R least-squares fit; 4) utilize shift-invariant rank-1 least-squares on a series of rank-1 matrices reconstructed by each column of the aggregating mixing matrix to update shared time courses, and subject-specific time delays and intensities. The experimental results of simulated and actual complex-valued fMRI data show that the proposed algorithm improves the estimates for task-related sensorimotor and auditory networks, compared to pcsCPD and tensorial spatial ICA. The proposed alternating rank- R and rank-1 least squares optimization is also flexible to improve CPD-related algorithm using alternating least squares.

Discovering hidden brain network responses to naturalistic stimuli via tensor component analysis of multi-subject fMRI data

The study of brain network interactions during naturalistic stimuli facilitates a deeper understanding of human brain function. To estimate large-scale brain networks evoked with naturalistic stimuli, a tensor component analysis (TCA) based framework was used to characterize shared spatio-temporal patterns across subjects in a purely data-driven manner. In this framework, a third-order tensor is constructed from the timeseries extracted from all brain regions from a given parcellation, for all participants, with modes of the tensor corresponding to spatial distribution, time series and participants. TCA then reveals spatially and temporally shared components, i.e., evoked networks with the naturalistic stimuli, their time courses of activity and subject loadings of each component. To enhance the reproducibility of the estimation with the adaptive TCA algorithm, a novel spectral clustering method, tensor spectral clustering, was proposed and applied to evaluate the stability of the TCA algorithm. We demonstrated the effectiveness of the proposed framework via simulations and real fMRI data collected during a motor task with a traditional fMRI study design. We also applied the proposed framework to fMRI data collected during passive movie watching to illustrate how reproducible brain networks are evoked by naturalistic movie viewing.

Multisubject Task-Related fMRI Data Processing via a Two-Stage Generalized Canonical Correlation Analysis

Functional magnetic resonance imaging (fMRI) is one of the most popular methods for studying the human brain. Task-related fMRI data processing aims to determine which brain areas are activated when a specific task is performed and is usually based on the Blood Oxygen Level Dependent (BOLD) signal. The background BOLD signal also reflects systematic fluctuations in regional brain activity which are attributed to the existence of resting-state brain networks. We propose a new fMRI data generating model which takes into consideration the existence of common task-related and resting-state components. We first estimate the common task-related temporal component, via two successive stages of generalized canonical correlation analysis and, then, we estimate the common task-related spatial component, leading to a task-related activation map. The experimental tests of our method with synthetic data reveal that we are able to obtain very accurate temporal and spatial estimates even at very low Signal to Noise Ratio (SNR), which is usually the case in fMRI data processing. The tests with real-world fMRI data show significant advantages over standard procedures based on General Linear Models (GLMs).

Tracing Evolving Networks Using Tensor Factorizations vs. ICA-Based Approaches

Analysis of time-evolving data is crucial to understand the functioning of dynamic systems such as the brain. For instance, analysis of functional magnetic resonance imaging (fMRI) data collected during a task may reveal spatial regions of interest, and how they evolve during the task. However, capturing underlying spatial patterns as well as their change in time is challenging. The traditional approach in fMRI data analysis is to assume that underlying spatial regions of interest are static. In this article, using fractional amplitude of low-frequency fluctuations (fALFF) as an effective way to summarize the variability in fMRI data collected during a task, we arrange time-evolving fMRI data as a subjects by voxels by time windows tensor, and analyze the tensor using a tensor factorization-based approach called a PARAFAC2 model to reveal spatial dynamics. The PARAFAC2 model jointly analyzes data from multiple time windows revealing subject-mode patterns, evolving spatial regions (also referred to as networks) and temporal patterns. We compare the PARAFAC2 model with matrix factorization-based approaches relying on independent components, namely, joint independent component analysis (ICA) and independent vector analysis (IVA), commonly used in neuroimaging data analysis. We assess the performance of the methods in terms of capturing evolving networks through extensive numerical experiments demonstrating their modeling assumptions. In particular, we show that (i) PARAFAC2 provides a compact representation in all modes, i.e., subjects, time, and voxels, revealing temporal patterns as well as evolving spatial networks, (ii) joint ICA is as effective as PARAFAC2 in terms of revealing evolving networks but does not reveal temporal patterns, (iii) IVA's performance depends on sample size, data distribution and covariance structure of underlying networks. When these assumptions are satisfied, IVA is as accurate as the other methods, (iv) when subject-mode patterns differ from one time window to another, IVA is the most accurate. Furthermore, we analyze real fMRI data collected during a sensory motor task, and demonstrate that a component indicating statistically significant group difference between patients with schizophrenia and healthy controls is captured, which includes primary and secondary motor regions, cerebellum, and temporal lobe, revealing a meaningful spatial map and its temporal change.

Low-Rank Tucker-2 Model for Multi-Subject fMRI Data Decomposition With Spatial Sparsity Constraint

Tucker decomposition can provide an intuitive summary to understand brain function by decomposing multi-subject fMRI data into a core tensor and multiple factor matrices, and was mostly used to extract functional connectivity patterns across time/subjects using orthogonality constraints. However, these algorithms are unsuitable for extracting common spatial and temporal patterns across subjects due to distinct characteristics such as high-level noise. Motivated by a successful application of Tucker decomposition to image denoising and the intrinsic sparsity of spatial activations in fMRI, we propose a low-rank Tucker-2 model with spatial sparsity constraint to analyze multi-subject fMRI data. More precisely, we propose to impose a sparsity constraint on spatial maps by using an l_p norm ( ), in addition to adding low-rank constraints on factor matrices via the Frobenius norm. We solve the constrained Tucker-2 model using alternating direction method of multipliers, and propose to update both sparsity and low-rank constrained spatial maps using half quadratic splitting. Moreover, we extract new spatial and temporal features in addition to subject-specific intensities from the core tensor, and use these features to classify multiple subjects. The results from both simulated and experimental fMRI data verify the improvement of the proposed method, compared with four related algorithms including robust Kronecker component analysis, Tucker decomposition with orthogonality constraints, canonical polyadic decomposition, and block term decomposition in extracting common spatial and temporal components across subjects. The spatial and temporal features extracted from the core tensor show promise for characterizing subjects within the same group of patients or healthy controls as well.

Early soft and flexible fusion of electroencephalography and functional magnetic resonance imaging via double coupled matrix tensor factorization for multisubject group analysis

Data fusion refers to the joint analysis of multiple datasets that provide different (e.g., complementary) views of the same task. In general, it can extract more information than separate analyses can. Jointly analyzing electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) measurements has been proved to be highly beneficial to the study of the brain function, mainly because these neuroimaging modalities have complementary spatiotemporal resolution: EEG offers good temporal resolution while fMRI is better in its spatial resolution. The EEG–fMRI fusion methods that have been reported so far ignore the underlying multiway nature of the data in at least one of the modalities and/or rely on very strong assumptions concerning the relation of the respective datasets. For example, in multisubject analysis, it is commonly assumed that the hemodynamic response function is a priori known for all subjects and/or the coupling across corresponding modes is assumed to be exact (hard). In this article, these two limitations are overcome by adopting tensor models for both modalities and by following soft and flexible coupling approaches to implement the multimodal fusion. The obtained results are compared against those of parallel independent component analysis and hard coupling alternatives, with both synthetic and real data (epilepsy and visual oddball paradigm). Our results demonstrate the clear advantage of using soft and flexible coupled tensor decompositions in scenarios that do not conform with the hard coupling assumption.

Taking the 4D Nature of fMRI Data Into Account Promises Significant Gains in Data Completion

Functional magnetic resonance imaging (fMRI) is a powerful, noninvasive tool that has significantly contributed to the understanding of the human brain. FMRI data provide a sequence of whole-brain volumes over time and hence are inherently four dimensional (4D). Missing data in fMRI experiments arise from image acquisition limits, susceptibility and motion artifacts or during confounding noise removal. Hence, significant brain regions may be excluded from the data, which can seriously undermine the quality of subsequent analyses due to the significant number of missing voxels. We take advantage of the four dimensional (4D) nature of fMRI data through a tensor representation and introduce an effective algorithm to estimate missing samples in fMRI data. The proposed Riemannian nonlinear spectral conjugate gradient (RSCG) optimization method uses tensor train (TT) decomposition, which enables compact representations and provides efficient linear algebra operations. Exploiting the Riemannian structure boosts algorithm performance significantly, as evidenced by the comparison of RSCG-TT with state-of-the-art stochastic gradient methods, which are developed in the Euclidean space. We thus provide an effective method for estimating missing brain voxels and, more importantly, clearly show that taking the full 4D structure of fMRI data into account provides important gains when compared with three-dimensional (3D) and the most commonly used two-dimensional (2D) representations of fMRI data.

Robust brain network identification from multi-subject asynchronous fMRI data

We describe a novel method for robust identification of common brain networks and their corresponding temporal dynamics across subjects from asynchronous functional MRI (fMRI) using tensor decomposition. We first temporally align asynchronous fMRI data using the orthogonal BrainSync transform, allowing us to study common brain networks across sessions and subjects. We then map the synchronized fMRI data into a 3D tensor (vertices × time × subject/session). Finally, we apply Nesterov-accelerated adaptive moment estimation (Nadam) within a scalable and robust sequential Canonical Polyadic (CP) decomposition framework to identify a low rank tensor approximation to the data. As a result of CP tensor decomposition, we successfully identified twelve known brain networks with their corresponding temporal dynamics from 40 subjects using the Human Connectome Project's language task fMRI data without any prior information regarding the specific task designs. Seven of these networks show distinct subjects' responses to the language task with differing temporal dynamics; two show sub-components of the default mode network that exhibit deactivation during the tasks; the remaining three components reflect non-task-related activities. We compare results to those found using group independent component analysis (ICA) and canonical ICA. Bootstrap analysis demonstrates increased robustness of networks found using the CP tensor approach relative to ICA-based methods.

Emojis influence autobiographical memory retrieval from reading words: An fMRI-based study

Advances in computer and communications technology have deeply affected the way we communicate. Social media have emerged as a major means of human communication. However, a major limitation in such media is the lack of non-verbal stimuli, which sometimes hinders the understanding of the message, and in particular the associated emotional content. In an effort to compensate for this, people started to use emoticons, which are combinations of keyboard characters that resemble facial expressions, and more recently their evolution: emojis, namely, small colorful images that resemble faces, actions and daily life objects. This paper presents evidence of the effect of emojis on memory retrieval through a functional Magnetic Resonance Imaging (fMRI) study. A total number of fifteen healthy volunteers were recruited for the experiment, during which successive stimuli were presented, containing words with intense emotional content combined with emojis, either with congruent or incongruent emotional content. Volunteers were asked to recall a memory related to the stimulus. The study of the reaction times showed that emotional incongruity among word+emoji combinations led to longer reaction times in memory retrieval compared to congruent combinations. General Linear Model (GLM) and Blind Source Separation (BSS) methods have been tested in assessing the influence of the emojis on the process of memory retrieval. The analysis of the fMRI data showed that emotional incongruity among word+emoji combinations activated the Broca's area (BA44 and BA45) in both hemispheres, the Supplementary Motor Area (SMA) and the inferior prefrontal cortex (BA47), compared to congruent combinations. Furthermore, compared to pseudowords, word+emoji combinations activated the left Broca's area (BA44 and BA45), the amygdala, the right temporal pole (BA48) and several frontal regions including the SMA and the inferior prefrontal cortex.

Shift-Invariant Canonical Polyadic Decomposition of Complex-Valued Multi-Subject fMRI Data With a Phase Sparsity Constraint

Canonical polyadic decomposition (CPD) of multi-subject complex-valued fMRI data can be used to provide spatially and temporally shared components among groups with both magnitude and phase information. However, the CPD model is not well formulated due to the large subject variability in the spatial and temporal modalities, as well as the high noise level in complex-valued fMRI data. Considering that the shift-invariant CPD can model temporal variability across subjects, we propose to further impose a phase sparsity constraint on the shared spatial maps to denoise the complex-valued components and to model the inter-subject spatial variability as well. More precisely, subject-specific time delays are first estimated for the complex-valued shared time courses in the framework of real-valued shift-invariant CPD. Source phase sparsity is then imposed on the complex-valued shared spatial maps. A smoothed l₀ norm is specifically used to reduce voxels with large phase values after phase de-ambiguity based on the small phase characteristic of BOLD-related voxels. The results from both the simulated and experimental fMRI data demonstrate improvements of the proposed method over three complex-valued algorithms, namely, tensor-based spatial ICA, shift-invariant CPD and CPD without spatiotemporal constraints. When comparing with a real-valued algorithm combining shift-invariant CPD and ICA, the proposed method detects 178.7% more contiguous task-related activations.

Exploring individual and group differences in latent brain networks using cross-validated simultaneous component analysis

Component models such as PCA and ICA are often used to reduce neuroimaging data into a smaller number of components, which are thought to reflect latent brain networks. When data from multiple subjects are available, the components are typically estimated simultaneously (i.e., for all subjects combined) using either tensor ICA or group ICA. As we demonstrate in this paper, neither of these approaches is ideal if one hopes to find latent brain networks that cross-validate to new samples of data. Specifically, we note that the tensor ICA model is too rigid to capture real-world heterogeneity in the component time courses, whereas the group ICA approach is too flexible to uniquely identify latent brain networks. For multi-subject component analysis, we recommend comparing a hierarchy of simultaneous component analysis (SCA) models. Our proposed model hierarchy includes a flexible variant of the SCA framework (the Parafac2 model), which is able to both (i) model heterogeneity in the component time courses, and (ii) uniquely identify latent brain networks. Furthermore, we propose cross-validation methods to tune the relevant model parameters, which reduces the potential of over-fitting the observed data. Using simulated and real data examples, we demonstrate the benefits of the proposed approach for finding credible components that reveal interpretable individual and group differences in latent brain networks.

Unraveling Diagnostic Biomarkers of Schizophrenia Through Structure-Revealing Fusion of Multi-Modal Neuroimaging Data

Fusing complementary information from different modalities can lead to the discovery of more accurate diagnostic biomarkers for psychiatric disorders. However, biomarker discovery through data fusion is challenging since it requires extracting interpretable and reproducible patterns from data sets, consisting of shared/unshared patterns and of different orders. For example, multi-channel electroencephalography (EEG) signals from multiple subjects can be represented as a third-order tensor with modes: subject, time, and channel, while functional magnetic resonance imaging (fMRI) data may be in the form of subject by voxel matrices. Traditional data fusion methods rearrange higher-order tensors, such as EEG, as matrices to use matrix factorization-based approaches. In contrast, fusion methods based on coupled matrix and tensor factorizations (CMTF) exploit the potential multi-way structure of higher-order tensors. The CMTF approach has been shown to capture underlying patterns more accurately without imposing strong constraints on the latent neural patterns, i.e., biomarkers. In this paper, EEG, fMRI, and structural MRI (sMRI) data collected during an auditory oddball task (AOD) from a group of subjects consisting of patients with schizophrenia and healthy controls, are arranged as matrices and higher-order tensors coupled along the subject mode, and jointly analyzed using structure-revealing CMTF methods [also known as advanced CMTF (ACMTF)] focusing on unique identification of underlying patterns in the presence of shared/unshared patterns. We demonstrate that joint analysis of the EEG tensor and fMRI matrix using ACMTF reveals significant and biologically meaningful components in terms of differentiating between patients with schizophrenia and healthy controls while also providing spatial patterns with high resolution and improving the clustering performance compared to the analysis of only the EEG tensor. We also show that these patterns are reproducible, and study reproducibility for different model parameters. In comparison to the joint independent component analysis (jICA) data fusion approach, ACMTF provides easier interpretation of EEG data by revealing a single summary map of the topography for each component. Furthermore, fusion of sMRI data with EEG and fMRI through an ACMTF model provides structural patterns; however, we also show that when fusing data sets from multiple modalities, hence of very different nature, preprocessing plays a crucial role.

Blind fMRI source unmixing via higher-order tensor decompositions

BACKGROUND

The growing interest in neuroimaging technologies generates a massive amount of biomedical data of high dimensionality. Tensor-based analysis of brain imaging data has been recognized as an effective analysis that exploits its inherent multi-way nature. In particular, the advantages of tensorial over matrix-based methods have previously been demonstrated in the context of functional magnetic resonance imaging (fMRI) source localization. However, such methods can also become ineffective in realistic challenging scenarios, involving, e.g., strong noise and/or significant overlap among the activated regions. Moreover, they commonly rely on the assumption of an underlying multilinear model generating the data.

NEW METHOD

This paper aims at investigating the possible gains from exploiting the 4-dimensional nature of the brain images, through a higher-order tensorization of the fMRI signal, and the use of less restrictive generative models. In this context, the higher-order block term decomposition (BTD) and the PARAFAC2 tensor models are considered for the first time in fMRI blind source separation. A novel PARAFAC2-like extension of BTD (BTD2) is also proposed, aiming at combining the effectiveness of BTD in handling strong instances of noise and the potential of PARAFAC2 to cope with datasets that do not follow the strict multilinear assumption.

COMPARISON WITH EXISTING METHODS

The methods were tested using both synthetic and real data and compared with state of the art methods.

CONCLUSIONS

The simulation results demonstrate the effectiveness of BTD and BTD2 for challenging scenarios (presence of noise, spatial overlap among activation regions and inter-subject variability in the haemodynamic response function (HRF)).

Scalable and Robust Tensor Decomposition of Spontaneous Stereotactic EEG Data

OBJECTIVE

Identification of networks from resting brain signals is an important step in understanding the dynamics of spontaneous brain activity. We approach this problem using a tensor-based model.

METHODS

We develop a rank-recursive scalable and robust sequential canonical polyadic decomposition (SRSCPD) framework to decompose a tensor into several rank-1 components. Robustness and scalability are achieved using a warm start for each rank based on the results from the previous rank.

RESULTS

In simulations we show that SRSCPD consistently outperforms the multi-start alternating least square (ALS) algorithm over a range of ranks and signal-to-noise ratios (SNRs), with lower computation cost. When applying SRSCPD to resting in-vivo stereotactic EEG (SEEG) data from two subjects with epilepsy, we found components corresponding to default mode and motor networks in both subjects. These components were also highly consistent within subject between two sessions recorded several hours apart. Similar components were not obtained using the conventional ALS algorithm.

CONCLUSION

Consistent brain networks and their dynamic behaviors were identified from resting SEEG data using SRSCPD.

SIGNIFICANCE

SRSCPD is scalable to large datasets and therefore a promising tool for identification of brain networks in long recordings from single subjects.

Unsupervised Discovery of Demixed, Low-Dimensional Neural Dynamics across Multiple Timescales through Tensor Component Analysis

Perceptions, thoughts, and actions unfold over millisecond timescales, while learned behaviors can require many days to mature. While recent experimental advances enable large-scale and long-term neural recordings with high temporal fidelity, it remains a formidable challenge to extract unbiased and interpretable descriptions of how rapid single-trial circuit dynamics change slowly over many trials to mediate learning. We demonstrate a simple tensor component analysis (TCA) can meet this challenge by extracting three interconnected, low-dimensional descriptions of neural data: neuron factors, reflecting cell assemblies; temporal factors, reflecting rapid circuit dynamics mediating perceptions, thoughts, and actions within each trial; and trial factors, describing both long-term learning and trial-to-trial changes in cognitive state. We demonstrate the broad applicability of TCA by revealing insights into diverse datasets derived from artificial neural networks, large-scale calcium imaging of rodent prefrontal cortex during maze navigation, and multielectrode recordings of macaque motor cortex during brain machine interface learning.

A review of tensor-based methods and their application to hospital care data

In many situations, a researcher is interested in the analysis of the scores of a set of observation units on a set of variables. However, in medicine, it is very frequent that the information is replicated at different occasions. The occasions can be time-varying or refer to different conditions. In such cases, the data can be stored in a 3-way array or tensor. The Candecomp/Parafac and Tucker3 methods represent the most common methods for analyzing 3-way tensors. In this work, a review of these methods is provided, and then this class of methods is applied to a 3-way data set concerning hospital care data for a hospital in Rome (Italy) during 15 years distinguished in 3 groups of consecutive years (1892-1896, 1940-1944, 1968-1972). The analysis reveals some peculiar aspects about the use of health services and its evolution along the time.

An improved multi-objective optimization-based CICA method with data-driver temporal reference for group fMRI data analysis

Group independent component analysis (GICA) has been successfully applied to study multi-subject functional magnetic resonance imaging (fMRI) data, and the group independent component (GIC) represents the commonality of all subjects in the group. However, some studies show that the performance of GICA can be improved by incorporating a priori information, which is not always considered when looking for GICs in existing GICA methods. In this paper, we propose an improved multi-objective optimization-based constrained independent component analysis (CICA) method to take advantage of the temporal a priori information extracted from all subjects in the group by incorporating it into the computational process of GICA for group fMRI data analysis. The experimental results of simulated and real data show that the activated regions and the time course detected by the improved CICA method are more accurate in some sense. Moreover, the GIC computed by the improved CICA method has a higher correlation with the corresponding independent component of each subject in the group, which means that the improved CICA method with the temporal a priori information extracted from the group can better reflect the commonality of the subjects. These results demonstrate that the improved CICA method has its own advantages in fMRI data analysis.

Using Matrix and Tensor Factorizations for the Single-Trial Analysis of Population Spike Trains

Advances in neuronal recording techniques are leading to ever larger numbers of simultaneously monitored neurons. This poses the important analytical challenge of how to capture compactly all sensory information that neural population codes carry in their spatial dimension (differences in stimulus tuning across neurons at different locations), in their temporal dimension (temporal neural response variations), or in their combination (temporally coordinated neural population firing). Here we investigate the utility of tensor factorizations of population spike trains along space and time. These factorizations decompose a dataset of single-trial population spike trains into spatial firing patterns (combinations of neurons firing together), temporal firing patterns (temporal activation of these groups of neurons) and trial-dependent activation coefficients (strength of recruitment of such neural patterns on each trial). We validated various factorization methods on simulated data and on populations of ganglion cells simultaneously recorded in the salamander retina. We found that single-trial tensor space-by-time decompositions provided low-dimensional data-robust representations of spike trains that capture efficiently both their spatial and temporal information about sensory stimuli. Tensor decompositions with orthogonality constraints were the most efficient in extracting sensory information, whereas non-negative tensor decompositions worked well even on non-independent and overlapping spike patterns, and retrieved informative firing patterns expressed by the same population in response to novel stimuli. Our method showed that populations of retinal ganglion cells carried information in their spike timing on the ten-milliseconds-scale about spatial details of natural images. This information could not be recovered from the spike counts of these cells. First-spike latencies carried the majority of information provided by the whole spike train about fine-scale image features, and supplied almost as much information about coarse natural image features as firing rates. Together, these results highlight the importance of spike timing, and particularly of first-spike latencies, in retinal coding.

The Functional Segregation and Integration Model: Mixture Model Representations of Consistent and Variable Group-Level Connectivity in fMRI

The brain consists of specialized cortical regions that exchange information between each other, reflecting a combination of segregated (local) and integrated (distributed) processes that define brain function. Functional magnetic resonance imaging (fMRI) is widely used to characterize these functional relationships, although it is an ongoing challenge to develop robust, interpretable models for high-dimensional fMRI data. Gaussian mixture models (GMMs) are a powerful tool for parcellating the brain, based on the similarity of voxel time series. However, conventional GMMs have limited parametric flexibility: they only estimate segregated structure and do not model interregional functional connectivity, nor do they account for network variability across voxels or between subjects. To address these issues, this letter develops the functional segregation and integration model (FSIM). This extension of the GMM framework simultaneously estimates spatial clustering and the most consistent group functional connectivity structure. It also explicitly models network variability, based on voxel- and subject-specific network scaling profiles. We compared the FSIM to standard GMM in a predictive cross-validation framework and examined the importance of different model parameters, using both simulated and experimental resting-state data. The reliability of parcellations is not significantly altered by flexibility of the FSIM, whereas voxel- and subject-specific network scaling profiles significantly improve the ability to predict functional connectivity in independent test data. Moreover, the FSIM provides a set of interpretable parameters to characterize both consistent and variable aspects functional connectivity structure. As an example of its utility, we use subject-specific network profiles to identify brain regions where network expression predicts subject age in the experimental data. Thus, the FSIM is effective at summarizing functional connectivity structure in group-level fMRI, with applications in modeling the relationships between network variability and behavioral/demographic variables.

Ranking brain areas encoding the perceived level of pain from fMRI data

Pain perception is thought to emerge from the integrated activity of a distributed brain system, but the relative contribution of the different network nodes is still incompletely understood. In the present functional magnetic resonance imaging (fMRI) study, we aimed to identify the more relevant brain regions to explain the time profile of the perceived pain intensity in healthy volunteers, during noxious chemical stimulation (ascorbic acid injection) of the left hand. To this end, we performed multi-way partial least squares regression of fMRI data from twenty-two a-priori defined brain regions of interest (ROI) in each hemisphere, to build a model that could efficiently reproduce the psychophysical pain profiles in the same individuals; moreover, we applied a novel three-way extension of the variable importance in projection (VIP) method to summarize each ROI contribution to the model. Brain regions showing the highest VIP scores included the bilateral mid-cingulate, anterior and posterior insular, and parietal operculum cortices, the contralateral paracentral lobule, bilateral putamen and ipsilateral medial thalamus. Most of these regions, with the exception of medial thalamus, were also identified by a statistical analysis on mean ROI beta values estimated using the time course of the psychophysical rating as a regressor at the voxel level. Our results provide the first rank-ordering of brain regions involved in coding the perceived level of pain. These findings in a model of acute prolonged pain confirm and extend previous data, suggesting that a bilateral array of cortical areas and subcortical structures is involved in pain perception.

A comparison of convex and non-convex compressed sensing applied to multidimensional NMR

The resolution of multidimensional NMR spectra can be severely limited when regular sampling based on the Nyquist-Shannon theorem is used. The theorem binds the sampling rate with a bandwidth of a sampled signal and thus implicitly creates a dependence between the line width and the time of experiment, often making the latter one very long. Recently, Candès et al. (2006) [25] formulated a non-linear sampling theorem that determines the required number of sampling points to be dependent mostly on the number of peaks in a spectrum and only slightly on the number of spectral points. The result was pivotal for rapid development and broad use of signal processing method called compressed sensing. In our previous work, we have introduced compressed sensing to multidimensional NMR and have shown examples of reconstruction of two-dimensional spectra. In the present paper we discuss in detail the accuracy and robustness of two compressed sensing algorithms: convex (iterative soft thresholding) and non-convex (iteratively re-weighted least squares with local ℓ(0)-norm) in application to two- and three-dimensional datasets. We show that the latter method is in many terms more effective, which is in line with recent works on the theory of compressed sensing. We also present the comparison of both approaches with multidimensional decomposition which is one of the established methods for processing of non-linearly sampled data.

On the Non-Existence of Optimal Solutions and the Occurrence of "Degeneracy" in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called "degeneracy". That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.

Principal component and linear discriminant analysis of T1 histograms of white and grey matter in multiple sclerosis

Twenty-three relapsing remitting multiple sclerosis (RRMS) patients and 14 controls were imaged to produce normal-appearing white and grey matter T1 histograms. These were used to assess whether histogram measures from principal component analysis (PCA) and linear discriminant analysis (LDA) out-perform traditional histogram metrics in classification of T1 histograms into control and RRMS subject groups and in correlation with the expanded disability status score (EDSS). The histograms were classified into one of two groups using a leave-one-out analysis. In addition, the patients were scanned serially, and the calculated parameters correlated with the EDSS. The classification results showed that the more complex techniques were at least as good at classifying the subjects as histogram mean, peak height and peak location, with PCA/LDA having success rates of 76% for white matter and 68%/65% for grey matter. No significant correlations were found with EDSS for any histogram parameter. These results indicate that there is much information contained within the grey matter as well as the white matter histograms. Although in these histograms PCA and LDA did not add greatly to the discriminatory power of traditional histogram parameters, they provide marginally better performance, while relying only on data-driven feature selection.