Using nonlinear models in fMRI data analysis: model selection and activation detection

There is an increasing interest in using physiologically plausible models in fMRI analysis. These models do raise new mathematical problems in terms of parameter estimation and interpretation of the measured data. In this paper, we show how to use physiological models to map and analyze brain activity from fMRI data. We describe a maximum likelihood parameter estimation algorithm and a statistical test that allow the following two actions: selecting the most statistically significant hemodynamic model for the measured data and deriving activation maps based on such model. Furthermore, as parameter estimation may leave much incertitude on the exact values of parameters, model identifiability characterization is a particular focus of our work. We applied these methods to different variations of the Balloon Model (Buxton, R.B., Wang, E.C., and Frank, L.R. 1998. Dynamics of blood flow and oxygenation changes during brain activation: the balloon model. Magn. Reson. Med. 39: 855-864; Buxton, R.B., Uluda?, K., Dubowitz, D.J., and Liu, T.T. 2004. Modelling the hemodynamic response to brain activation. NeuroImage 23: 220-233; Friston, K. J., Mechelli, A., Turner, R., and Price, C. J. 2000. Nonlinear responses in fMRI: the balloon model, volterra kernels, and other hemodynamics. NeuroImage 12: 466-477) in a visual perception checkerboard experiment. Our model selection proved that hemodynamic models better explain the BOLD response than linear convolution, in particular because they are able to capture some features like poststimulus undershoot or nonlinear effects. On the other hand, nonlinear and linear models are comparable when signals get noisier, which explains that activation maps obtained in both frameworks are comparable. The tools we have developed prove that statistical inference methods used in the framework of the General Linear Model might be generalized to nonlinear models.     

Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics

There is a growing appreciation of the importance of nonlinearities in evoked responses in fMRI, particularly with the advent of event-related fMRI. These nonlinearities are commonly expressed as interactions among stimuli that can lead to the suppression and increased latency of responses to a stimulus that are incurred by a preceding stimulus. We have presented previously a model-free characterization of these effects using generic techniques from nonlinear system identification, namely a Volterra series formulation. At the same time Buxton et al. (1998) described a plausible and compelling dynamical model of hemodynamic signal transduction in fMRI. Subsequent work by Mandeville et al. (1999) provided important theoretical and empirical constraints on the form of the dynamic relationship between blood flow and volume that underpins the evolution of the fMRI signal. In this paper we combine these system identification and model-based approaches and ask whether the Balloon model is sufficient to account for the nonlinear behaviors observed in real time series. We conclude that it can, and furthermore the model parameters that ensue are biologically plausible. This conclusion is based on the observation that the Balloon model can produce Volterra kernels that emulate empirical kernels. To enable this evaluation we had to embed the Balloon model in a hemodynamic input-state-output model that included the dynamics of perfusion changes that are contingent on underlying synaptic activation. This paper presents (i) the full hemodynamic model (ii), how its associated Volterra kernels can be derived, and (iii) addresses the model's validity in relation to empirical nonlinear characterizations of evoked responses in fMRI and other neurophysiological constraints.     

A temporal comparison of BOLD, ASL, and NIRS hemodynamic responses to motor stimuli in adult humans

In this study, we have preformed simultaneous near-infrared spectroscopy (NIRS) along with BOLD (blood oxygen level dependent) and ASL (arterial spin labeling)-based fMRI during an event-related motor activity in human subjects in order to compare the temporal dynamics of the hemodynamic responses recorded in each method. These measurements have allowed us to examine the validity of the biophysical models underlying each modality and, as a result, gain greater insight into the hemodynamic responses to neuronal activation. Although prior studies have examined the relationships between these two methodologies through similar experiments, they have produced conflicting results in the literature for a variety of reasons. Here, by employing a short-duration, event-related motor task, we have been able to emphasize the subtle temporal differences between the hemodynamic parameters with a high contrast-to-noise ratio. As a result of this improved experimental design, we are able to report that the fMRI measured BOLD response is more correlated with the NIRS measure of deoxy-hemoglobin (R = 0.98; P < 10(-20)) than with oxy-hemoglobin (R = 0.71), or total hemoglobin (R = 0.53). This result was predicted from the theoretical grounds of the BOLD response and is in agreement with several previous works [Toronov, V.A.W., Choi, J.H., Wolf, M., Michalos, A., Gratton, E., Hueber, D., 2001. "Investigation of human brain hemodynamics by simultaneous near-infrared spectroscopy and functional magnetic resonance imaging." Med. Phys. 28 (4) 521-527.; MacIntosh, B.J., Klassen, L.M., Menon, R.S., 2003. "Transient hemodynamics during a breath hold challenge in a two part functional imaging study with simultaneous near-infrared spectroscopy in adult humans". NeuroImage 20 1246-1252.; Toronov, V.A.W., Walker, S., Gupta, R., Choi, J.H., Gratton, E., Hueber, D., Webb, A., 2003. "The roles of changes in deoxyhemoglobin concentration and regional cerebral blood volume in the fMRI BOLD signal" Neuroimage 19 (4) 1521-1531]. These data have also allowed us to examine more detailed measurement models of the fMRI signal and comment on the roles of the oxygen saturation and blood volume contributions to the BOLD response. In addition, we found high correlation between the NIRS measured total hemoglobin and ASL measured cerebral blood flow (R = 0.91; P < 10(-10)) and oxy-hemoglobin with flow (R = 0.83; P < 10(-05)) as predicted by the biophysical models. Finally, we note a significant amount of cross-modality, correlated, inter-subject variability in amplitude change and time-to-peak of the hemodynamic response. The observed co-variance in these parameters between subjects is in agreement with hemodynamic models and provides further support that fMRI and NIRS have similar vascular sensitivity.     

Modeling the hemodynamic response to brain activation

Neural activity in the brain is accompanied by changes in cerebral blood flow (CBF) and blood oxygenation that are detectable with functional magnetic resonance imaging (fMRI) techniques. In this paper, recent mathematical models of this hemodynamic response are reviewed and integrated. Models are described for: (1) the blood oxygenation level dependent (BOLD) signal as a function of changes in cerebral oxygen extraction fraction (E) and cerebral blood volume (CBV); (2) the balloon model, proposed to describe the transient dynamics of CBV and deoxy-hemoglobin (Hb) and how they affect the BOLD signal; (3) neurovascular coupling, relating the responses in CBF and cerebral metabolic rate of oxygen (CMRO(2)) to the neural activity response; and (4) a simple model for the temporal nonlinearity of the neural response itself. These models are integrated into a mathematical framework describing the steps linking a stimulus to the measured BOLD and CBF responses. Experimental results examining transient features of the BOLD response (post-stimulus undershoot and initial dip), nonlinearities of the hemodynamic response, and the role of the physiologic baseline state in altering the BOLD signal are discussed in the context of the proposed models. Quantitative modeling of the hemodynamic response, when combined with experimental data measuring both the BOLD and CBF responses, makes possible a more specific and quantitative assessment of brain physiology than is possible with standard BOLD imaging alone. This approach has the potential to enhance numerous studies of brain function in development, health, and disease.     

Nonlinear coupling between evoked rCBF and BOLD signals: a simulation study of hemodynamic responses

The aim of this work was to investigate the dependence of BOLD responses on different patterns of stimulus input/neuronal changes. In an earlier report, we described an input-state-output model that combined (i) the Balloon/Windkessel model of nonlinear coupling between rCBF and BOLD signals, and (ii) a linear model of how regional flow changes with synaptic activity. In the present investigation, the input-state-output model was used to explore the dependence of simulated PET (rCBF) and fMRI (BOLD) signals on various parameters pertaining to experimental design. Biophysical simulations were used to estimate rCBF and BOLD responses as functions of (a) a prior stimulus, (b) epoch length (for a fixed SOA), (c) SOA (for a fixed number of events), and (d) stimulus amplitude. We also addressed the notion that a single neuronal response may differ, in terms of the relative contributions of early and late neural components, and investigated the effect of (e) the relative size of the late or "endogenous" neural component. We were interested in the estimated average rCBF and BOLD responses per stimulus or event, not in the statistical efficiency with which these responses are detected. The BOLD response was underestimated relative to rCBF with a preceding stimulus, increasing epoch length, and increasing SOA. Furthermore, the BOLD response showed some highly nonlinear behaviour when varying stimulus amplitude, suggesting some form of hemodynamic "rectification." Finally, the BOLD response was underestimated in the context of large late neuronal components. The difference between rCBF and BOLD is attributed to the nonlinear transduction of rCBF to BOLD signal. Our simulations support the idea that varying parameters that specify the experimental design may have differential effects in PET and fMRI. Moreover, they show that fMRI can be asymmetric in its ability to detect deactivations relative to activations when an absolute baseline is stipulated. Finally, our simulations suggest that relative insensitivity to BOLD signal in specific regions, such as the temporal lobe, may be partly explained by higher cognitive functions eliciting a relatively large late endogenous neuronal component.     

Delayed hemodynamic responses in schizophrenia

While there are many reports of reduced amplitude of hemodynamic responses in schizophrenia, there are no reports of delayed hemodynamic responses, in spite of event-related brain potential (ERP) evidence of slowed neural responses. Recently, Henson et al. (2002) [Henson, R., Price, C., Rugg, M., Turner, R., Friston, K., 2002. Detecting latency differences in event-related BOLD responses: application to words versus nonwords and initial versus repeated face presentations. NeuroImage, 15:83-97] proposed a new method for testing small latency effects (<2 s) in hemodynamic responses. fMRI data were collected during a visual oddball task with infrequent (12%) targets (K) and frequent (88%) standards (X), presented every 1-3 s pseudorandomly, with 7-24 s between targets. SPM99 yielded parameter estimates for the hemodynamic response and its temporal derivative (TD). Beta images reflecting hemodynamic response magnitude to target stimuli were minimally thresholded (P < 0.05, uncorrected), and latencies were estimated for surviving voxels using TD and hemodynamic response beta values. DSM-IV schizophrenia patients (n = 12) and sex- and age-matched healthy control subjects (n = 12) were recruited from the community. Although groups differed only minimally in activation height, hemodynamic responses were significantly delayed in basal ganglia, thalamus, and anterior cingulate in patients with schizophrenia. Psychomotor slowing reflected in reaction times to targets recorded outside the magnet was related to hemodynamic slowing in basal ganglia, anterior cingulate, thalamus, as well as left cerebellum in controls. Delays in the hemodynamic response have far-reaching implications for understanding the pathophysiology of schizophrenia from slowed neural responses to slowed substrate delivery, and depending of the degree of delay, may raise methodological issues regarding modeling hemodynamic responses in patients and controls.     

Large-scale neural models and dynamic causal modelling

Dynamic causal modelling (DCM) is a method for estimating and making inferences about the coupling among small numbers of brain areas, and the influence of experimental manipulations on that coupling [Friston, K.J., Harrison, L., Penny, W., 2003 Dynamic causal modelling. Neuroimage 19, 1273-1302]. Large-scale neural modelling aims to construct neurobiologically grounded computational models with emergent behaviours that inform our understanding of neuronal systems. One such model has been used to simulate region-specific BOLD time-series [Horwitz, B., Friston, K.J., Taylor, J.G., 2000. Neural modeling and functional brain imaging: an overview. Neural Netw. 13, 829-846]. DCM was used to make inferences about effective connectivity using data generated by a model implementing a visual delayed match-to-sample task [Tagamets, M.A., Horwitz, B., 1998. Integrating electrophysiological and anatomical experimental data to create a large-scale model that simulates a delayed match-to-sample human brain imaging study. Cereb. Cortex 8, 310-320]. The aim was to explore the validity of inferences made using DCM about the connectivity structure and task-dependent modulatory effects, in a system with a known connectivity structure. We also examined the effects of misspecifying regions of interest. Models with hierarchical connectivity and reciprocal connections were examined using DCM and Bayesian Model Comparison [Penny, W.D., Stephan, K.E., Mechelli, A., Friston, K.J., 2004. Comparing dynamic causal models. Neuroimage 22, 1157-1172]. This approach revealed strong evidence for those models with correctly specified anatomical connectivity. Furthermore, Bayesian model comparison favoured those models when bilinear effects corresponded to their implementation in the neural model. These findings generalised to an extended model with two additional areas and reentrant circuits. The conditional uncertainty of coupling parameter estimates increased in proportion to the number of incorrectly specified regions. These results highlight the role of neural models in establishing the validity of estimation and inference schemes. Specifically, Bayesian model comparison confirms the validity of DCM in relation to a well-characterised and comprehensive neuronal model.     

Biophysical model for integrating neuronal activity, EEG, fMRI and metabolism

Our goal is to model the coupling between neuronal activity, cerebral metabolic rates of glucose and oxygen consumption, cerebral blood flow (CBF), electroencephalography (EEG) and blood oxygenation level-dependent (BOLD) responses. In order to accomplish this, two previous models are coupled: a metabolic/hemodynamic model (MHM) for a voxel, linking BOLD signals and neuronal activity, and a neural mass model describing the neuronal dynamics within a voxel and its interactions with voxels of the same area (short-range interactions) and other areas (long-range interactions). For coupling both models, we take as the input to the BOLD model, the number of active synapses within the voxel, that is, the average number of synapses that will receive an action potential within the time unit. This is obtained by considering the action potentials transmitted between neuronal populations within the voxel, as well as those arriving from other voxels. Simulations are carried out for testing the integrated model. Results show that realistic evoked potentials (EP) at electrodes on the scalp surface and the corresponding BOLD signals for each voxel are produced by the model. In another simulation, the alpha rhythm was reproduced and reasonable similarities with experimental data were obtained when calculating correlations between BOLD signals and the alpha power curve. The origin of negative BOLD responses and the characteristics of EEG, PET and BOLD signals in Alzheimer's disease were also studied.     

Influence of heart rate on the BOLD signal: the cardiac response function

It has previously been shown that low-frequency fluctuations in both respiratory volume and cardiac rate can induce changes in the blood-oxygen level dependent (BOLD) signal. Such physiological noise can obscure the detection of neural activation using fMRI, and it is therefore important to model and remove the effects of this noise. While a hemodynamic response function relating respiratory variation (RV) and the BOLD signal has been described [Birn, R.M., Smith, M.A., Jones, T.B., Bandettini, P.A., 2008b. The respiration response function: The temporal dynamics of fMRI signal fluctuations related to changes in respiration. Neuroimage 40, 644-654.], no such mapping for heart rate (HR) has been proposed. In the current study, the effects of RV and HR are simultaneously deconvolved from resting state fMRI. It is demonstrated that a convolution model including RV and HR can explain significantly more variance in gray matter BOLD signal than a model that includes RV alone, and an average HR response function is proposed that well characterizes our subject population. It is observed that the voxel-wise morphology of the deconvolved RV responses is preserved when HR is included in the model, and that its form is adequately modeled by Birn et al.'s previously-described respiration response function. Furthermore, it is shown that modeling out RV and HR can significantly alter functional connectivity maps of the default-mode network.     

Investigating hemodynamic response variability at the group level using basis functions

Introduced is a general framework for performing group-level analyses of fMRI data using any basis set of two functions (i.e., the canonical hemodynamic response function and its first derivative) to model the hemodynamic response to neural activity. The approach allows for flexible implementation of physiologically based restrictions on the results. Information from both basis functions is used at the group level and the limitations avoid physiologically ambiguous or implausible results. This allows for investigation of specific BOLD activity such as hemodynamic responses peaking within a specified temporal range (i.e., 4–5 s). The general nature of the presented approach allows for applications using basis sets specifically designed to investigate various physiologic phenomena, i.e., age-related variability in poststimulus undershoot, hemodynamic responses measured with cerebral blood flow imaging, or subject-specific basis sets. An example using data from a group of healthy young participants demonstrates the methods and the specific steps to study poststimulus variability are discussed. The approach is completely implemented within the general linear model and requires minimal programmatic calculations.     

Investigating the post-stimulus undershoot of the BOLD signal--a simultaneous fMRI and fNIRS study

Measuring the hemodynamic response with functional magnetic resonance imaging (fMRI) together with functional near-infrared spectroscopy (fNIRS) may overcome limitations of single-method approaches. Accordingly, we measured the event-related hemodynamic response with both imaging methods simultaneously in young subjects during visual stimulation. An intertrial interval of 60 s was chosen to include the prolonged post-stimulus undershoot of the blood oxygenation level dependent (BOLD) signal. During visual stimulation, the BOLD signal, oxy-, and total hemoglobin (Hb) increased, whereas deoxy-Hb decreased. The post-stimulus period was characterized by an undershoot of the BOLD signal, oxy-Hb, and an overshoot of deoxy-Hb. Total Hb as measured by fNIRS returned to baseline immediately after the end of stimulation. Results suggest that the post-stimulus events as measured by fNIRS are dominated by a prolonged high-level oxygen consumption in the microvasculature. The contribution of a delayed return of blood volume to the BOLD post-stimulus undershoot in post-capillary veins as suggested by the Balloon and Windkessel models remains ambiguous. Temporal changes in the BOLD signal were highly correlated with deoxy-Hb, with lower correlation values for oxy- and total Hb. Furthermore, data show that fNIRS covers the outer 1 cm of the brain cortex. These results were confirmed by simultaneous fMRI/fNIRS measurements during rest. In conclusion, multimodal imaging approaches may contribute to the understanding of neurovascular coupling.     

Discrepancies between BOLD and flow dynamics in primary and supplementary motor areas: application of the balloon model to the interpretation of BOLD transients

The blood-oxygen-level-dependent (BOLD) signal measured in the brain with functional magnetic resonance imaging (fMRI) during an activation experiment often exhibits pronounced transients at the beginning and end of the stimulus. Such transients could be a reflection of transients in the underlying neural activity, or they could result from transients in cerebral blood flow (CBF), cerebral metabolic rate of oxygen (CMRO2), or cerebral blood volume (CBV). These transients were investigated using an arterial spin labeling (ASL) method that allows simultaneous measurements of BOLD and CBF responses. Responses to a finger-tapping task (40-s stimulus, 80-s rest) were measured in primary motor area (M1) and supplementary motor area (SMA) in five healthy volunteers. In SMA, the average BOLD response was pronounced near the beginning and end of the stimulus, while in M1, the BOLD response was nearly flat. However, CBF responses in the two regions were rather similar, and did not exhibit the same transient features as the BOLD response in SMA. Because this suggests a hemodynamic rather than a neural origin for the transients of the BOLD response in SMA, we used a generalization of the balloon model to test the degree of hemodynamic transients required to produce the measured curves. Both data sets could be approximated with modest differences in the shapes of the CMRO2 and CBV responses. This study illustrates the utility and the limitations of using theoretical models combined with ASL techniques to understand the dynamics of the BOLD response.     

A state-space model of the hemodynamic approach: nonlinear filtering of BOLD signals

In this paper, a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent (BOLD) responses. The proposed method constitutes an alternative to the recently proposed Friston [Neuroimage 16 (2002) 513] method and has some advantages over it. The procedure is based on recent groundbreaking time series analysis techniques that have been, in this case, adopted to characterize hemodynamic responses in functional magnetic resonance imaging (fMRI). This work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons. First, our model includes physiological noise. Previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation. Secondly, by using the innovation method and the local linearization filter, not only the parameters, but also the underlying states of the system generating responses can be estimated. These states can include things like a flow-inducing signal triggered by neuronal activation, de-oxyhemoglobine, cerebral blood flow and volume. Finally, radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach, which could be essential to understanding those brain areas indirectly related to the stimulus. Hence, thirdly, by inferring about the radial basis parameters, we are able to perform a blind deconvolution, which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also, to implicitly reconstruct the underlying synaptic dynamics, induced experimentally, which caused these states variations. From this study, we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps (SPM), nonlinear BOLD phenomena and unspecific input temporal sequences must be included in the fMRI analysis.     

The BOLD response to interictal epileptiform discharges

We studied single-event and average BOLD responses to EEG interictal epileptic discharges (IEDs) in four patients with focal epilepsy, using continuous EEG-fMRI during 80-min sessions. The detection of activated areas was performed by comparing the BOLD signal at each voxel to a model of the expected signal. Since little is known about the BOLD response to IEDs, we modeled it with the response to brief auditory events (G. H., NeuroImage 9, 416-429). For each activated area, we then obtained the time course of the BOLD signal for the complete session and computed the actual average hemodynamic response function (HRF) to IEDs. In two of four patients, we observed clear BOLD responses to single IEDs. The average response was composed of a positive lobe peaking between 6 and 7 s in all patients and a negative undershoot in three patients. There were important variations in amplitude and shape between average HRFs across patients. The average HRF presented a wider positive lobe than the Glover model in three patients and a longer undershoot in two. There was a remarkable similarity in the shape of the HRF across areas for patients presenting multiple activation sites. There was no clear correlation between the amplitude of individual BOLD responses and the amplitude of the corresponding EEG spike. The possibility of a longer HRF could be used to improve statistical detection of activation in simultaneous EEG-fMRI. The variability in average HRFs across patients could reflect in part different pathophysiological mechanisms.     

Dynamical causal modelling for M/EEG: spatial and temporal symmetry constraints

We describe the use of spatial and temporal constraints in dynamic causal modelling (DCM) of magneto- and electroencephalography (M/EEG) data. DCM for M/EEG is based on a spatiotemporal, generative model of electromagnetic brain activity. The temporal dynamics are described by neural-mass models of equivalent current dipole (ECD) sources and their spatial expression is modelled by parameterized lead-field functions. Often, in classical ECD models, symmetry constraints are used to model homologous pairs of dipoles in both hemispheres. These constraints are motivated by assumptions about symmetric activation of bilateral sensory sources. In classical approaches, these constraints are 'hard'; i.e. the parameters of homologous dipoles are shared. Here, in the context of DCM, we illustrate the use of informed Bayesian priors to implement 'soft' symmetry constraints that are expressed in the posterior estimates only when supported by the data. Critically, with DCM one can deploy symmetry constraints in either the temporal or spatial components of the model. This enables one to test for symmetry in temporal (neural-mass) parameters in the presence of non-symmetric spatial expressions of homologous sources (and vice versa). Furthermore, we demonstrate that Bayesian model comparison can be used to identify the best models among a range of symmetric and non-symmetric variants. Our main finding is that the use of 'soft' symmetry priors is recommended for evoked responses to bilateral sensory input. We illustrate the use of symmetry constraints in DCM on synthetic and real EEG data.     

Vascular dynamics and BOLD fMRI: CBF level effects and analysis considerations

Changes in the cerebral blood flow (CBF) baseline produce significant changes to the hemodynamic response. This work shows that increases in the baseline blood flow level produce blood oxygenation-level dependent (BOLD) and blood flow responses that are slower and lower in amplitude, while decreases in the baseline blood flow level produce faster and higher amplitude hemodynamic responses. This effect was characterized using a vascular model of the hemodynamic response that separated arterial blood flow response from the venous blood volume response and linked the input stimulus to the vascular response. The model predicted the baseline blood flow level effects to be dominated by changes in the arterial vasculature. Specifically, it predicted changes in the arterial blood flow time constant and venous blood volume time constant parameters of +294% and -24%, respectively, for a 27% increase in the baseline blood flow. The vascular model performance was compared to an empirical model of the hemodynamic response. The vascular and empirical hemodynamic models captured most of the baseline blood flow level effects observed and can be used to correct for these effects in fMRI data. While the empirical hemodynamic model is easy to implement, it did not incorporate any explicit physiological information.     

Method for functional MRI mapping of nonlinear response

Nonlinear systems analysis combining blood oxygen level dependent (BOLD), functional magnetic resonance imaging (fMRI) and m-sequence stimulation paradigms are proposed as a new method for exploring neuronal responses and interactions. Previous studies of electrical activity in the human visual cortex have observed significant nonlinearities of task-induced activity with temporal dynamics on a timescale of 10-20 ms. Despite the confounding effect of the seconds-long hemodynamic response, it is demonstrated that BOLD fMRI can be used to probe neuronal interactions on a time scale of tens of ms. Visual activation experiments were performed with various stimuli, and amplitude maps of first and second order kernel coefficients were generated using correlation analysis. Second order nonlinearities in BOLD fMRI were observed and attributed to temporal contrast caused by transitions in the stimulus sequence. In addition, the kernel maps showed significant differences between second order nonlinearities of foveal and peripheral vision. By including a reference experiment with a slightly modified stimulus presentation, a distinction could be made between (fast) neuronal nonlinearities and hemodynamic effects on the time scale of the seconds. The results indicate that BOLD fMRI can probe fast neuronal nonlinearities.     

Reducing the unwanted draining vein BOLD contribution in fMRI with statistical post-processing methods

Recent BOLD fMRI data analysis methods show promise in reducing contributions from draining veins. The phase regressor method developed by [Menon, R.S., 2002. Post-acquisition suppression of large-vessel BOLD signals in high-resolution fMRI. Magn. Reson. Med., 47, 1-9] creates phase and magnitude images, regresses magnitude as a function of phase, and subtracts phase-estimated magnitudes from the observed magnitudes. The corrected magnitude images are used to compute cortical activations. The complex constant phase method, developed by [Rowe, D.B., Logan, B.R., 2004. A complex way to compute fMRI activation. NeuroImage, 23, 1078-1092], uses complex-valued reconstructed images and a nonlinear regressor model to compute magnitude cortical activations assuming temporally constant phase. In both methods, the usage of the phase information is claimed to bias against voxels with task-related phase changes caused by some draining veins. The behavior of the statistical methods in data with several task-related magnitude and phase changes is compared. The power of the statistical methods for determining voxels with specific task-related magnitude and phase change combinations are determined in ideal simulated data. The phase regressor and complex constant phase activation determination techniques are examined to characterize the responses of the models to select task-related phase and magnitude change combinations in representative simulated time series. Possible draining veins in human preliminary data are discussed and analyzed with the models and the current challenges which prevent these methods from being reliably implemented are discussed.     

Spatial dependence of the nonlinear BOLD response at short stimulus duration

Most functional magnetic resonance imaging studies use linear models to predict the measured response by convolution of an impulse response with the stimulus profile. Using very short visual presentation times (<2 s), deviation from the linear model in the measured BOLD data from the human brain was found for the response integral, amplitude, and width. In this study, high temporal and spatial resolution were used to quantify nonlinear effects and investigate the spatial dependence. Data at 4 Tesla showed at short stimulus duration a nonlinearity, i.e., deviation from a linear model, with an index up to 400%, whereas data at 7 Tesla exhibited a nonlinearity index up to 40%. The effect was more pronounced for response amplitude than for response area. A reduced width and sharpening of responses at shorter stimulus duration was also found. A voxel-based analysis of 7 Tesla data with 1.2 x 1.2 x 2 mm(3) resolution revealed a correlation between response onset and nonlinearity index. This suggests that the nonlinearity effects are a tissue-specific phenomenon and are likely to be more localized to the site of neuronal activity. The observed magnetic field dependence and the demonstrated nonlinearity in the response width support the hypothesis that the source of the nonlinearity at short stimulus duration has a considerable hemodynamic contribution. The nonlinearity was modeled as a "switch"-type initial hemodynamic response onset. Understanding these nonlinearities in the BOLD response is important for design and the analysis of rapid event-related fMRI experiments with brief stimulus presentations.     

A Bayesian approach to modeling dynamic effective connectivity with fMRI data

A state-space modeling approach for examining dynamic relationship between multiple brain regions was proposed in Ho, Ombao and Shumway (Ho, M.R., Ombao, H., Shumway, R., 2005. A State-Space Approach to Modelling Brain Dynamics to Appear in Statistica Sinica). Their approach assumed that the quantity representing the influence of one neuronal system over another, or effective connectivity, is time-invariant. However, more and more empirical evidence suggests that the connectivity between brain areas may be dynamic which calls for temporal modeling of effective connectivity. A Bayesian approach is proposed to solve this problem in this paper. Our approach first decomposes the observed time series into measurement error and the BOLD (blood oxygenation level-dependent) signals. To capture the complexities of the dynamic processes in the brain, region-specific activations are subsequently modeled, as a linear function of the BOLD signals history at other brain regions. The coefficients in these linear functions represent effective connectivity between the regions under consideration. They are further assumed to follow a random walk process so to characterize the dynamic nature of brain connectivity. We also consider the temporal dependence that may be present in the measurement errors. ML-II method (Berger, J.O., 1985. Statistical Decision Theory and Bayesian Analysis (2nd ed.). Springer, New York) was employed to estimate the hyperparameters in the model and Bayes factor was used to compare among competing models. Statistical inference of the effective connectivity coefficients was based on their posterior distributions and the corresponding Bayesian credible regions (Carlin, B.P., Louis, T.A., 2000. Bayes and Empirical Bayes Methods for Data Analysis (2nd ed.). Chapman and Hall, Boca Raton). The proposed method was applied to a functional magnetic resonance imaging data set and results support the theory of attentional control network and demonstrate that this network is dynamic in nature.