Nonlinear estimation of the BOLD signal

State Estimation of Hemodynamic Model for fMRI Under Confounds: SSM Method

Through hemodynamic models, the change of neuronal state can be estimated from functional magnetic resonance imaging (fMRI) signals. Usually, there are confounds in the fMRI signal, which will degrade the performance of the estimation for the neuronal state change. For the reason, this paper introduces a state-space model with confounds, from a conventional hemodynamic model. In this model, a successive state estimation method requires a state value vector, an error covariance, an innovation covariance, and a cross covariance to be re-derived. Thus, a confounds square-root cubature Kalman smoothing (CSCKS) algorithm is proposed in this paper. We use a Balloon-Windkessel model to generate simulation data and add confounds signals to evaluate the performance of the proposed algorithm. The experiment results show that when the signal-to-interference ratio is less than 21 dB, the CSCKS proposed in this paper reduced estimation error to 16%, whereas the traditional algorithm reduced it to only 73%.

An efficient multistage algorithm for full calibration of the hemodynamic model from BOLD signal responses

We propose a computational strategy that falls into the category of prediction/correction iterative-type approaches, for calibrating the hemodynamic model. The proposed method is used to estimate consecutively the values of the two sets of model parameters. Numerical results corresponding to both synthetic and real functional magnetic resonance imaging measurements for a single stimulus as well as for multiple stimuli are reported to highlight the capability of this computational methodology to fully calibrate the considered hemodynamic model.

EEG-fMRI Bayesian framework for neural activity estimation: a simulation study

OBJECTIVE

Due to the complementary nature of electroencephalography (EEG) and functional magnetic resonance imaging (fMRI), and given the possibility of simultaneous acquisition, the joint data analysis can afford a better understanding of the underlying neural activity estimation. In this simulation study we want to show the benefit of the joint EEG-fMRI neural activity estimation in a Bayesian framework.

APPROACH

We built a dynamic Bayesian framework in order to perform joint EEG-fMRI neural activity time course estimation. The neural activity is originated by a given brain area and detected by means of both measurement techniques. We have chosen a resting state neural activity situation to address the worst case in terms of the signal-to-noise ratio. To infer information by EEG and fMRI concurrently we used a tool belonging to the sequential Monte Carlo (SMC) methods: the particle filter (PF).

MAIN RESULTS

First, despite a high computational cost, we showed the feasibility of such an approach. Second, we obtained an improvement in neural activity reconstruction when using both EEG and fMRI measurements.

SIGNIFICANCE

The proposed simulation shows the improvements in neural activity reconstruction with EEG-fMRI simultaneous data. The application of such an approach to real data allows a better comprehension of the neural dynamics.

Joint state and parameter estimation of the hemodynamic model by particle smoother expectation maximization method

OBJECTIVE

In this paper, we aimed for the robust estimation of the parameters and states of the hemodynamic model by using blood oxygen level dependent signal.

APPROACH

In the fMRI literature, there are only a few successful methods that are able to make a joint estimation of the states and parameters of the hemodynamic model. In this paper, we implemented a maximum likelihood based method called the particle smoother expectation maximization (PSEM) algorithm for the joint state and parameter estimation.

MAIN RESULTS

Former sequential Monte Carlo methods were only reliable in the hemodynamic state estimates. They were claimed to outperform the local linearization (LL) filter and the extended Kalman filter (EKF). The PSEM algorithm is compared with the most successful method called square-root cubature Kalman smoother (SCKS) for both state and parameter estimation. SCKS was found to be better than the dynamic expectation maximization (DEM) algorithm, which was shown to be a better estimator than EKF, LL and particle filters.

SIGNIFICANCE

PSEM was more accurate than SCKS for both the state and the parameter estimation. Hence, PSEM seems to be the most accurate method for the system identification and state estimation for the hemodynamic model inversion literature. This paper do not compare its results with Tikhonov-regularized Newton-CKF (TNF-CKF), a recent robust method which works in filtering sense.

A novel approach to calibrate the hemodynamic model using functional Magnetic Resonance Imaging (fMRI) measurements

BACKGROUND

The calibration of the hemodynamic model that describes changes in blood flow and blood oxygenation during brain activation is a crucial step for successfully monitoring and possibly predicting brain activity. This in turn has the potential to provide diagnosis and treatment of brain diseases in early stages.

NEW METHOD

We propose an efficient numerical procedure for calibrating the hemodynamic model using some fMRI measurements. The proposed solution methodology is a regularized iterative method equipped with a Kalman filtering-type procedure. The Newton component of the proposed method addresses the nonlinear aspect of the problem. The regularization feature is used to ensure the stability of the algorithm. The Kalman filter procedure is incorporated here to address the noise in the data.

RESULTS

Numerical results obtained with synthetic data as well as with real fMRI measurements are presented to illustrate the accuracy, robustness to the noise, and the cost-effectiveness of the proposed method.

COMPARISON WITH EXISTING METHOD(S)

We present numerical results that clearly demonstrate that the proposed method outperforms the Cubature Kalman Filter (CKF), one of the most prominent existing numerical methods.

CONCLUSION

We have designed an iterative numerical technique, called the TNM-CKF algorithm, for calibrating the mathematical model that describes the single-event related brain response when fMRI measurements are given. The method appears to be highly accurate and effective in reconstructing the BOLD signal even when the measurements are tainted with high noise level (as high as 30%).

Nonlinear estimation of BOLD signals with the aid of cerebral blood volume imaging

BACKGROUND

The hemodynamic balloon model describes the change in coupling from underlying neural activity to observed blood oxygen level dependent (BOLD) response. It plays an increasing important role in brain research using magnetic resonance imaging (MRI) techniques. However, changes in the BOLD signal are sensitive to the resting blood volume fraction (i.e., [Formula: see text]) associated with the regional vasculature. In previous studies the value was arbitrarily set to a physiologically plausible value to circumvent the ill-posedness of the inverse problem. These approaches fail to explore actual [Formula: see text] value and could yield inaccurate model estimation.

METHODS

The present study represents the first empiric attempt to derive the actual [Formula: see text] from data obtained using cerebral blood volume imaging, with the aim of augmenting the existing estimation schemes. Bimanual finger tapping experiments were performed to determine how [Formula: see text] influences the model estimation of BOLD signals within a single-region and multiple-regions (i.e., dynamic causal modeling). In order to show the significance of applying the true [Formula: see text], we have presented the different results obtained when using the real [Formula: see text] and assumed [Formula: see text] in terms of single-region model estimation and dynamic causal modeling.

RESULTS

The results show that [Formula: see text] significantly influences the estimation results within a single-region and multiple-regions. Using the actual [Formula: see text] might yield more realistic and physiologically meaningful model estimation results.

CONCLUSION

Incorporating regional venous information in the analysis of the hemodynamic model can provide more reliable and accurate parameter estimations and model predictions, and improve the inference about brain connectivity based on fMRI data.

On the distinguishability of HRF models in fMRI

Modeling the Hemodynamic Response Function (HRF) is a critical step in fMRI studies of brain activity, and it is often desirable to estimate HRF parameters with physiological interpretability. A biophysically informed model of the HRF can be described by a non-linear time-invariant dynamic system. However, the identification of this dynamic system may leave much uncertainty on the exact values of the parameters. Moreover, the high noise levels in the data may hinder the model estimation task. In this context, the estimation of the HRF may be seen as a problem of model falsification or invalidation, where we are interested in distinguishing among a set of eligible models of dynamic systems. Here, we propose a systematic tool to determine the distinguishability among a set of physiologically plausible HRF models. The concept of absolutely input-distinguishable systems is introduced and applied to a biophysically informed HRF model, by exploiting the structure of the underlying non-linear dynamic system. A strategy to model uncertainty in the input time-delay and magnitude is developed and its impact on the distinguishability of two physiologically plausible HRF models is assessed, in terms of the maximum noise amplitude above which it is not possible to guarantee the falsification of one model in relation to another. Finally, a methodology is proposed for the choice of the input sequence, or experimental paradigm, that maximizes the distinguishability of the HRF models under investigation. The proposed approach may be used to evaluate the performance of HRF model estimation techniques from fMRI data.

Nonlinear Bayesian estimation of BOLD signal under non-Gaussian noise

Modeling the blood oxygenation level dependent (BOLD) signal has been a subject of study for over a decade in the neuroimaging community. Inspired from fluid dynamics, the hemodynamic model provides a plausible yet convincing interpretation of the BOLD signal by amalgamating effects of dynamic physiological changes in blood oxygenation, cerebral blood flow and volume. The nonautonomous, nonlinear set of differential equations of the hemodynamic model constitutes the process model while the weighted nonlinear sum of the physiological variables forms the measurement model. Plagued by various noise sources, the time series fMRI measurement data is mostly assumed to be affected by additive Gaussian noise. Though more feasible, the assumption may cause the designed filter to perform poorly if made to work under non-Gaussian environment. In this paper, we present a data assimilation scheme that assumes additive non-Gaussian noise, namely, the e-mixture noise, affecting the measurements. The proposed filter MAGSF and the celebrated EKF are put to test by performing joint optimal Bayesian filtering to estimate both the states and parameters governing the hemodynamic model under non-Gaussian environment. Analyses using both the synthetic and real data reveal superior performance of the MAGSF as compared to EKF.

A review of fMRI simulation studies

Simulation studies that validate statistical techniques for fMRI data are challenging due to the complexity of the data. Therefore, it is not surprising that no common data generating process is available (i.e. several models can be found to model BOLD activation and noise). Based on a literature search, a database of simulation studies was compiled. The information in this database was analysed and critically evaluated focusing on the parameters in the simulation design, the adopted model to generate fMRI data, and on how the simulation studies are reported. Our literature analysis demonstrates that many fMRI simulation studies do not report a thorough experimental design and almost consistently ignore crucial knowledge on how fMRI data are acquired. Advice is provided on how the quality of fMRI simulation studies can be improved.

Causal and structural connectivity of pulse-coupled nonlinear networks

We study the reconstruction of structural connectivity for a general class of pulse-coupled nonlinear networks and show that the reconstruction can be successfully achieved through linear Granger causality (GC) analysis. Using spike-triggered correlation of whitened signals, we obtain a quadratic relationship between GC and the network couplings, thus establishing a direct link between the causal connectivity and the structural connectivity within these networks. Our work may provide insight into the applicability of GC in the study of the function of general nonlinear networks.

Identification of nonlinear fMRI models using Auxiliary Particle Filter and kernel smoothing method

Hemodynamic models have a high potential in application to understanding the functional differences of the brain. However, full system identification with respect to model fitting to actual functional magnetic resonance imaging (fMRI) data is practically difficult and is still an active area of research. We present a simulation based Bayesian approach for nonlinear model based analysis of the fMRI data. The idea is to do a joint state and parameter estimation within a general filtering framework. One advantage of using Bayesian methods is that they provide a complete description of the posterior distribution, not just a single point estimate. We use an Auxiliary Particle Filter adjoined with a kernel smoothing approach to address this joint estimation problem.

Reliable and efficient approach of BOLD signal with dual Kalman filtering

By introducing the conflicting effects of dynamic changes in blood flow, volume, and blood oxygenation, Balloon model provides a biomechanical compelling interpretation of the BOLD signal. In order to obtain optimal estimates for both the states and parameters involved in this model, a joint filtering (estimate) method has been widely used. However, it is flawed in several aspects (i) Correlation or interaction between the states and parameters is incorporated despite its nonexistence in biophysical reality. (ii) A joint representation for states and parameters necessarily means the large dimension of state space and will in turn lead to huge numerical cost in implementation. Given this knowledge, a dual filtering approach is proposed and demonstrated in this paper as a highly competent alternative, which can not only provide more reliable estimates, but also in a more efficient way. The two approaches in our discussion will be based on unscented Kalman filter, which has become the algorithm of choice in numerous nonlinear estimation and machine learning applications.

Identifying FMRI model violations with Lagrange multiplier tests

The standard modeling framework in functional magnetic resonance imaging (fMRI) is predicated on assumptions of linearity, time invariance and stationarity. These assumptions are rarely checked because doing so requires specialized software, although failure to do so can lead to bias and mistaken inference. Identifying model violations is an essential but largely neglected step in standard fMRI data analysis. Using Lagrange multiplier testing methods we have developed simple and efficient procedures for detecting model violations such as nonlinearity, nonstationarity and validity of the common double gamma specification for hemodynamic response. These procedures are computationally cheap and can easily be added to a conventional analysis. The test statistic is calculated at each voxel and displayed as a spatial anomaly map which shows regions where a model is violated. The methodology is illustrated with a large number of real data examples.

Quantitative evaluation of activation state in functional brain imaging

Neuronal activity can evoke the hemodynamic change that gives rise to the observed functional magnetic resonance imaging (fMRI) signal. These increases are also regulated by the resting blood volume fraction (V (0)) associated with regional vasculature. The activation locus detected by means of the change in the blood-oxygen-level-dependent (BOLD) signal intensity thereby may deviate from the actual active site due to varied vascular density in the cortex. Furthermore, conventional detection techniques evaluate the statistical significance of the hemodynamic observations. In this sense, the significance level relies not only upon the intensity of the BOLD signal change, but also upon the spatially inhomogeneous fMRI noise distribution that complicates the expression of the results. In this paper, we propose a quantitative strategy for the calibration of activation states to address these challenging problems. The quantitative assessment is based on the estimated neuronal efficacy parameter [Formula: see text] of the hemodynamic model in a voxel-by-voxel way. It is partly immune to the inhomogeneous fMRI noise by virtue of the strength of the optimization strategy. Moreover, it is easy to incorporate regional vascular information into the activation detection procedure. By combining MR angiography images, this approach can remove large vessel contamination in fMRI signals, and provide more accurate functional localization than classical statistical techniques for clinical applications. It is also helpful to investigate the nonlinear nature of the coupling between synaptic activity and the evoked BOLD response. The proposed method might be considered as a potentially useful complement to existing statistical approaches.

Exploiting magnetic resonance angiography imaging improves model estimation of BOLD signal

The change of BOLD signal relies heavily upon the resting blood volume fraction ([Formula: see text]) associated with regional vasculature. However, existing hemodynamic data assimilation studies pretermit such concern. They simply assign the value in a physiologically plausible range to get over ill-conditioning of the assimilation problem and fail to explore actual [Formula: see text]. Such performance might lead to unreliable model estimation. In this work, we present the first exploration of the influence of [Formula: see text] on fMRI data assimilation, where actual [Formula: see text] within a given cortical area was calibrated by an MR angiography experiment and then was augmented into the assimilation scheme. We have investigated the impact of [Formula: see text] on single-region data assimilation and multi-region data assimilation (dynamic cause modeling, DCM) in a classical flashing checkerboard experiment. Results show that the employment of an assumed [Formula: see text] in fMRI data assimilation is only suitable for fMRI signal reconstruction and activation detection grounded on this signal, and not suitable for estimation of unobserved states and effective connectivity study. We thereby argue that introducing physically realistic [Formula: see text] in the assimilation process may provide more reliable estimation of physiological information, which contributes to a better understanding of the underlying hemodynamic processes. Such an effort is valuable and should be well appreciated.

A stochastic linear model for fMRI activation analyses

PURPOSE

The debate regarding how best to model variability of the hemodynamic response function in fMRI data has focussed on the linear vs. nonlinear nature of the optimal signal model, with few studies exploring the deterministic vs. stochastic nature of the dynamics. We propose a stochastic linear model (SLM) of the hemodynamic signal and noise dynamics to more robustly infer fMRI activation estimates.

METHODS

The SLM models the hemodynamic signal by an exogenous input autoregressive model driven by Gaussian state noise. Activation weights are inferred by a joint state-parameter iterative coordinate descent algorithm based on the Kalman smoother.

RESULTS

The SLM produced more accurate parameter estimates than the GLM for event-design simulated data. In application to block-design experimental visuo-motor task fMRI data, the SLM resulted in more punctate and well-defined motor cortex activation maps than the GLM, and was able to track variations in the hemodynamics, as expected from a stochastic model.

CONCLUSIONS

We demonstrate in application to both simulated and experimental fMRI data that in comparison to the GLM, the SLM produces more flexible, consistent and enhanced fMRI activation estimates.

Dynamic modeling of neuronal responses in fMRI using cubature Kalman filtering

This paper presents a new approach to inverting (fitting) models of coupled dynamical systems based on state-of-the-art (cubature) Kalman filtering. Crucially, this inversion furnishes posterior estimates of both the hidden states and parameters of a system, including any unknown exogenous input. Because the underlying generative model is formulated in continuous time (with a discrete observation process) it can be applied to a wide variety of models specified with either ordinary or stochastic differential equations. These are an important class of models that are particularly appropriate for biological time-series, where the underlying system is specified in terms of kinetics or dynamics (i.e., dynamic causal models). We provide comparative evaluations with generalized Bayesian filtering (dynamic expectation maximization) and demonstrate marked improvements in accuracy and computational efficiency. We compare the schemes using a series of difficult (nonlinear) toy examples and conclude with a special focus on hemodynamic models of evoked brain responses in fMRI. Our scheme promises to provide a significant advance in characterizing the functional architectures of distributed neuronal systems, even in the absence of known exogenous (experimental) input; e.g., resting state fMRI studies and spontaneous fluctuations in electrophysiological studies. Importantly, unlike current Bayesian filters (e.g. DEM), our scheme provides estimates of time-varying parameters, which we will exploit in future work on the adaptation and enabling of connections in the brain.

A state space based approach in non-linear hemodynamic response modeling with fMRI data

In this paper we use the modified and integrated version of the balloon model in the analysis of fMRI data. We propose a new state space model realization for this balloon model and represent it with the standard A,B,C and D matrices widely used in system theory. A second order Padé approximation with equal numerator and denominator degree is used for the time delay approximation in the modeling of the cerebral blood flow. The results obtained through numerical solutions showed that the new state space model realization is in close agreement to the actual modified and integrated version of the balloon model. This new system theoretic formulation is likely to open doors to a novel way of analyzing fMRI data with real time robust estimators. With further development and validation, the new model has the potential to devise a generalized measure to make a significant contribution to improve the diagnosis and treatment of clinical scenarios where the brain functioning get altered. Concepts from system theory can readily be used in the analysis of fMRI data and the subsequent synthesis of filters and estimators.

MEG and fMRI Fusion for Non-Linear Estimation of Neural and BOLD Signal Changes

The combined analysis of magnetoencephalography (MEG)/electroencephalography and functional magnetic resonance imaging (fMRI) measurements can lead to improvement in the description of the dynamical and spatial properties of brain activity. In this paper we empirically demonstrate this improvement using simulated and recorded task related MEG and fMRI activity. Neural activity estimates were derived using a dynamic Bayesian network with continuous real valued parameters by means of a sequential Monte Carlo technique. In synthetic data, we show that MEG and fMRI fusion improves estimation of the indirectly observed neural activity and smooths tracking of the blood oxygenation level dependent (BOLD) response. In recordings of task related neural activity the combination of MEG and fMRI produces a result with greater signal-to-noise ratio, that confirms the expectation arising from the nature of the experiment. The highly non-linear model of the BOLD response poses a difficult inference problem for neural activity estimation; computational requirements are also high due to the time and space complexity. We show that joint analysis of the data improves the system's behavior by stabilizing the differential equations system and by requiring fewer computational resources.

Sensitivity analysis for biomedical models

This article discusses the application of sensitivity analysis (SA) in biomedical models. Sensitivity analysis is widely applied in physics, chemistry, economics, social sciences and other areas where models are developed. By assigning a prior probability distribution to each model variable, the SA framework appeals to the posterior probabilities of the model to evaluate the relative importance of these variables on the output distribution based on the principle of general variance decomposition. Within this framework, the SA paradigm serves as an objective platform to quantify the contributions of each model factor relative to their empirical range. We present statistical derivations of variance-based SA in this context and discuss its detailed properties through some practical examples. Our emphasis is on the application of SA in the biomedical field. As we show, it may provide a useful tool for model quality assessment, model reduction and factor prioritization, and improve our understanding of the model structure and underlying mechanisms. When usual approaches for calculating sensitivity index involve the employment of Monte Carlo analysis, which is computationally expensive in the large-sampling paradigm, we develop two effective numerical approximate methods for quick SA evaluations based on the unscented transformation (UT) that utilize a deterministic sampling approach in place of random sampling to calculate posterior statistics. We show that these methods achieve an excellent compromise between computational burden and calculation precision. In addition, a clear guideline is absent to evaluate the importance of variable for model reduction, we also present an objective statistical criterion to quantitatively decide whether or not a descriptive parameter is nominal and may be discarded in ensuing model-based analysis without significant loss of information on model behavior.

On consciousness, resting state fMRI, and neurodynamics

BACKGROUND

During the last years, functional magnetic resonance imaging (fMRI) of the brain has been introduced as a new tool to measure consciousness, both in a clinical setting and in a basic neurocognitive research. Moreover, advanced mathematical methods and theories have arrived the field of fMRI (e.g. computational neuroimaging), and functional and structural brain connectivity can now be assessed non-invasively.

RESULTS

The present work deals with a pluralistic approach to "consciousness'', where we connect theory and tools from three quite different disciplines: (1) philosophy of mind (emergentism and global workspace theory), (2) functional neuroimaging acquisitions, and (3) theory of deterministic and statistical neurodynamics - in particular the Wilson-Cowan model and stochastic resonance.

CONCLUSIONS

Based on recent experimental and theoretical work, we believe that the study of large-scale neuronal processes (activity fluctuations, state transitions) that goes on in the living human brain while examined with functional MRI during "resting state", can deepen our understanding of graded consciousness in a clinical setting, and clarify the concept of "consiousness" in neurocognitive and neurophilosophy research.

Spatially adaptive mixture modeling for analysis of FMRI time series

Within-subject analysis in fMRI essentially addresses two problems, the detection of brain regions eliciting evoked activity and the estimation of the underlying dynamics. In Makni et aL, 2005 and Makni et aL, 2008, a detection-estimation framework has been proposed to tackle these problems jointly, since they are connected to one another. In the Bayesian formalism, detection is achieved by modeling activating and nonactivating voxels through independent mixture models (IMM) within each region while hemodynamic response estimation is performed at a regional scale in a nonparametric way. Instead of IMMs, in this paper we take advantage of spatial mixture models (SMM) for their nonlinear spatial regularizing properties. The proposed method is unsupervised and spatially adaptive in the sense that the amount of spatial correlation is automatically tuned from the data and this setting automatically varies across brain regions. In addition, the level of regularization is specific to each experimental condition since both the signal-to-noise ratio and the activation pattern may vary across stimulus types in a given brain region. These aspects require the precise estimation of multiple partition functions of underlying Ising fields. This is addressed efficiently using first path sampling for a small subset of fields and then using a recently developed fast extrapolation technique for the large remaining set. Simulation results emphasize that detection relying on supervised SMM outperforms its IMM counterpart and that unsupervised spatial mixture models achieve similar results without any hand-tuning of the correlation parameter. On real datasets, the gain is illustrated in a localizer fMRI experiment: brain activations appear more spatially resolved using SMM in comparison with classical general linear model (GLM)-based approaches, while estimating a specific parcel-based HRF shape. Our approach therefore validates the treatment of unsmoothed fMRI data without fixed GLM definition at the subject level and makes also the classical strategy of spatial Gaussian filtering deprecated.

Nonlinear connectivity by Granger causality

The communication among neuronal populations, reflected by transient synchronous activity, is the mechanism underlying the information processing in the brain. Although it is widely assumed that the interactions among those populations (i.e. functional connectivity) are highly nonlinear, the amount of nonlinear information transmission and its functional roles are not clear. The state of the art to understand the communication between brain systems are dynamic causal modeling (DCM) and Granger causality. While DCM models nonlinear couplings, Granger causality, which constitutes a major tool to reveal effective connectivity, and is widely used to analyze EEG/MEG data as well as fMRI signals, is usually applied in its linear version. In order to capture nonlinear interactions between even short and noisy time series, a few approaches have been proposed. We review them and focus on a recently proposed flexible approach has been recently proposed, consisting in the kernel version of Granger causality. We show the application of the proposed approach on EEG signals and fMRI data.

Kernel Granger causality mapping effective connectivity on FMRI data

Although it is accepted that linear Granger causality can reveal effective connectivity in functional magnetic resonance imaging (fMRI), the issue of detecting nonlinear connectivity has hitherto not been considered. In this paper, we address kernel Granger causality (KGC) to describe effective connectivity in simulation studies and real fMRI data of a motor imagery task. Based on the theory of reproducing kernel Hilbert spaces, KGC performs linear Granger causality in the feature space of suitable kernel functions, assuming an arbitrary degree of nonlinearity. Our results demonstrate that KGC captures effective couplings not revealed by the linear case. In addition, effective connectivity networks between the supplementary motor area (SMA) as the seed and other brain areas are obtained from KGC.