An Overcomplete Approach to Fitting Drift-Diffusion Decision Models to Trial-By-Trial Data.

Drift-diffusion models or DDMs are becoming a standard in the field of computational neuroscience. They extend models from signal detection theory by proposing a simple mechanistic explanation for the observed relationship between decision outcomes and reaction times (RT). In brief, they assume that decisions are triggered once the accumulated evidence in favor of a particular alternative option has reached a predefined threshold. Fitting a DDM to empirical data then allows one to interpret observed group or condition differences in terms of a change in the underlying model parameters. However, current approaches only yield reliable parameter estimates in specific situations (c.f. fixed drift rates vs drift rates varying over trials). In addition, they become computationally unfeasible when more general DDM variants are considered (e.g., with collapsing bounds). In this note, we propose a fast and efficient approach to parameter estimation that relies on fitting a "self-consistency" equation that RT fulfill under the DDM. This effectively bypasses the computational bottleneck of standard DDM parameter estimation approaches, at the cost of estimating the trial-specific neural noise variables that perturb the underlying evidence accumulation process. For the purpose of behavioral data analysis, these act as nuisance variables and render the model "overcomplete," which is finessed using a variational Bayesian system identification scheme. However, for the purpose of neural data analysis, estimates of neural noise perturbation terms are a desirable (and unique) feature of the approach. Using numerical simulations, we show that this "overcomplete" approach matches the performance of current parameter estimation approaches for simple DDM variants, and outperforms them for more complex DDM variants. Finally, we demonstrate the added-value of the approach, when applied to a recent value-based decision making experiment.

Computational neuroimaging strategies for single patient predictions.

Neuroimaging increasingly exploits machine learning techniques in an attempt to achieve clinically relevant single-subject predictions. An alternative to machine learning, which tries to establish predictive links between features of the observed data and clinical variables, is the deployment of computational models for inferring on the (patho)physiological and cognitive mechanisms that generate behavioural and neuroimaging responses. This paper discusses the rationale behind a computational approach to neuroimaging-based single-subject inference, focusing on its potential for characterising disease mechanisms in individual subjects and mapping these characterisations to clinical predictions. Following an overview of two main approaches - Bayesian model selection and generative embedding - which can link computational models to individual predictions, we review how these methods accommodate heterogeneity in psychiatric and neurological spectrum disorders, help avoid erroneous interpretations of neuroimaging data, and establish a link between a mechanistic, model-based approach and the statistical perspectives afforded by machine learning.

Dynamic causal modelling of brain-behaviour relationships.

In this work, we expose a mathematical treatment of brain-behaviour relationships, which we coin behavioural Dynamic Causal Modelling or bDCM. This approach aims at decomposing the brain's transformation of stimuli into behavioural outcomes, in terms of the relative contribution of brain regions and their connections. In brief, bDCM places the brain at the interplay between stimulus and behaviour: behavioural outcomes arise from coordinated activity in (hidden) neural networks, whose dynamics are driven by experimental inputs. Estimating neural parameters that control network connectivity and plasticity effectively performs a neurobiologically-constrained approximation to the brain's input-outcome transform. In other words, neuroimaging data essentially serves to enforce the realism of bDCM's decomposition of input-output relationships. In addition, post-hoc artificial lesions analyses allow us to predict induced behavioural deficits and quantify the importance of network features for funnelling input-output relationships. This is important, because this enables one to bridge the gap with neuropsychological studies of brain-damaged patients. We demonstrate the face validity of the approach using Monte-Carlo simulations, and its predictive validity using empirical fMRI/behavioural data from an inhibitory control task. Lastly, we discuss promising applications of this work, including the assessment of functional degeneracy (in the healthy brain) and the prediction of functional recovery after lesions (in neurological patients).

Bayesian model selection for group studies - revisited.

In this paper, we revisit the problem of Bayesian model selection (BMS) at the group level. We originally addressed this issue in Stephan et al. (2009), where models are treated as random effects that could differ between subjects, with an unknown population distribution. Here, we extend this work, by (i) introducing the Bayesian omnibus risk (BOR) as a measure of the statistical risk incurred when performing group BMS, (ii) highlighting the difference between random effects BMS and classical random effects analyses of parameter estimates, and (iii) addressing the problem of between group or condition model comparisons. We address the first issue by quantifying the chance likelihood of apparent differences in model frequencies. This leads to the notion of protected exceedance probabilities. The second issue arises when people want to ask "whether a model parameter is zero or not" at the group level. Here, we provide guidance as to whether to use a classical second-level analysis of parameter estimates, or random effects BMS. The third issue rests on the evidence for a difference in model labels or frequencies across groups or conditions. Overall, we hope that the material presented in this paper finesses the problems of group-level BMS in the analysis of neuroimaging and behavioural data.

Stochastic dynamic causal modelling of fMRI data: should we care about neural noise?

Dynamic causal modelling (DCM) was introduced to study the effective connectivity among brain regions using neuroimaging data. Until recently, DCM relied on deterministic models of distributed neuronal responses to external perturbation (e.g., sensory stimulation or task demands). However, accounting for stochastic fluctuations in neuronal activity and their interaction with task-specific processes may be of particular importance for studying state-dependent interactions. Furthermore, allowing for random neuronal fluctuations may render DCM more robust to model misspecification and finesse problems with network identification. In this article, we examine stochastic dynamic causal models (sDCM) in relation to their deterministic counterparts (dDCM) and highlight questions that can only be addressed with sDCM. We also compare the network identification performance of deterministic and stochastic DCM, using Monte Carlo simulations and an empirical case study of absence epilepsy. For example, our results demonstrate that stochastic DCM can exploit the modelling of neural noise to discriminate between direct and mediated connections. We conclude with a discussion of the added value and limitations of sDCM, in relation to its deterministic homologue.

Learning and generalization under ambiguity: an fMRI study.

Adaptive behavior often exploits generalizations from past experience by applying them judiciously in new situations. This requires a means of quantifying the relative importance of prior experience and current information, so they can be balanced optimally. In this study, we ask whether the brain generalizes in an optimal way. Specifically, we used Bayesian learning theory and fMRI to test whether neuronal responses reflect context-sensitive changes in ambiguity or uncertainty about experience-dependent beliefs. We found that the hippocampus expresses clear ambiguity-dependent responses that are associated with an augmented rate of learning. These findings suggest candidate neuronal systems that may be involved in aberrations of generalization, such as over-confidence.

An electrophysiological validation of stochastic DCM for fMRI.

In this note, we assess the predictive validity of stochastic dynamic causal modeling (sDCM) of functional magnetic resonance imaging (fMRI) data, in terms of its ability to explain changes in the frequency spectrum of concurrently acquired electroencephalography (EEG) signal. We first revisit the heuristic model proposed in Kilner et al. (2005), which suggests that fMRI activation is associated with a frequency modulation of the EEG signal (rather than an amplitude modulation within frequency bands). We propose a quantitative derivation of the underlying idea, based upon a neural field formulation of cortical activity. In brief, dense lateral connections induce a separation of time scales, whereby fast (and high spatial frequency) modes are enslaved by slow (low spatial frequency) modes. This slaving effect is such that the frequency spectrum of fast modes (which dominate EEG signals) is controlled by the amplitude of slow modes (which dominate fMRI signals). We then use conjoint empirical EEG-fMRI data-acquired in epilepsy patients-to demonstrate the electrophysiological underpinning of neural fluctuations inferred from sDCM for fMRI.

Dynamic causal modelling: a critical review of the biophysical and statistical foundations.

The goal of dynamic causal modelling (DCM) of neuroimaging data is to study experimentally induced changes in functional integration among brain regions. This requires (i) biophysically plausible and physiologically interpretable models of neuronal network dynamics that can predict distributed brain responses to experimental stimuli and (ii) efficient statistical methods for parameter estimation and model comparison. These two key components of DCM have been the focus of more than thirty methodological articles since the seminal work of Friston and colleagues published in 2003. In this paper, we provide a critical review of the current state-of-the-art of DCM. We inspect the properties of DCM in relation to the most common neuroimaging modalities (fMRI and EEG/MEG) and the specificity of inference on neural systems that can be made from these data. We then discuss both the plausibility of the underlying biophysical models and the robustness of the statistical inversion techniques. Finally, we discuss potential extensions of the current DCM framework, such as stochastic DCMs, plastic DCMs and field DCMs.

Ten simple rules for dynamic causal modeling.

Dynamic causal modeling (DCM) is a generic Bayesian framework for inferring hidden neuronal states from measurements of brain activity. It provides posterior estimates of neurobiologically interpretable quantities such as the effective strength of synaptic connections among neuronal populations and their context-dependent modulation. DCM is increasingly used in the analysis of a wide range of neuroimaging and electrophysiological data. Given the relative complexity of DCM, compared to conventional analysis techniques, a good knowledge of its theoretical foundations is needed to avoid pitfalls in its application and interpretation of results. By providing good practice recommendations for DCM, in the form of ten simple rules, we hope that this article serves as a helpful tutorial for the growing community of DCM users.

EEG-fMRI integration: a critical review of biophysical modeling and data analysis approaches.

The diverse nature of cerebral activity, as measured using neuroimaging techniques, has been recognised long ago. It seems obvious that using single modality recordings can be limited when it comes to capturing its complex nature. Thus, it has been argued that moving to a multimodal approach will allow neuroscientists to better understand the dynamics and structure of this activity. This means that integrating information from different techniques, such as electroencephalography (EEG) and the blood oxygenated level dependent (BOLD) signal recorded with functional magnetic resonance imaging (fMRI), represents an important methodological challenge. In this work, we review the work that has been done thus far to derive EEG/fMRI integration approaches. This leads us to inspect the conditions under which such an integration approach could work or fail, and to disclose the types of scientific questions one could (and could not) hope to answer with it.

Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models.

In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.

Diffusion-based spatial priors for functional magnetic resonance images.

We recently outlined a Bayesian scheme for analyzing fMRI data using diffusion-based spatial priors [Harrison, L.M., Penny, W., Ashburner, J., Trujillo-Barreto, N., Friston, K.J., 2007. Diffusion-based spatial priors for imaging. NeuroImage 38, 677-695]. The current paper continues this theme, applying it to a single-subject functional magnetic resonance imaging (fMRI) study of the auditory system. We show that spatial priors on functional activations, based on diffusion, can be formulated in terms of the eigenmodes of a graph Laplacian. This allows one to discard eigenmodes with small eigenvalues, to provide a computationally efficient scheme. Furthermore, this formulation shows that diffusion-based priors are a generalization of conventional Laplacian priors [Penny, W.D., Trujillo-Barreto, N.J., Friston, K.J., 2005. Bayesian fMRI time series analysis with spatial priors. NeuroImage 24, 350-362]. Finally, we show how diffusion-based priors are a special case of Gaussian process models that can be inverted using classical covariance component estimation techniques like restricted maximum likelihood [Patterson, H.D., Thompson, R., 1974. Maximum likelihood estimation of components of variance. Paper presented at: 8th International Biometrics Conference (Constanta, Romania)]. The convention in SPM is to smooth data with a fixed isotropic Gaussian kernel before inverting a mass-univariate statistical model. This entails the strong assumption that data are generated smoothly throughout the brain. However, there is no way to determine if this assumption is supported by the data, because data are smoothed before statistical modeling. In contrast, if a spatial prior is used, smoothness is estimated given non-smoothed data. Explicit spatial priors enable formal model comparison of different prior assumptions, e.g., that data are generated from a stationary (i.e., fixed throughout the brain) or non-stationary spatial process. Indeed, for the auditory data we provide strong evidence for a non-stationary process, which concurs with a qualitative comparison of predicted activations at the boundary of functionally selective regions.

DEM: a variational treatment of dynamic systems.

This paper presents a variational treatment of dynamic models that furnishes time-dependent conditional densities on the path or trajectory of a system's states and the time-independent densities of its parameters. These are obtained by maximising a variational action with respect to conditional densities, under a fixed-form assumption about their form. The action or path-integral of free-energy represents a lower bound on the model's log-evidence or marginal likelihood required for model selection and averaging. This approach rests on formulating the optimisation dynamically, in generalised coordinates of motion. The resulting scheme can be used for online Bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches, such as Kalman and particle filtering. Furthermore, it provides for dual and triple inferences on a system's states, parameters and hyperparameters using exactly the same principles. We refer to this approach as dynamic expectation maximisation (DEM).

Accurate anisotropic fast marching for diffusion-based geodesic tractography.

Using geodesics for inferring white matter fibre tracts from diffusion-weighted MR data is an attractive method for at least two reasons: (i) the method optimises a global criterion, and hence is less sensitive to local perturbations such as noise or partial volume effects, and (ii) the method is fast, allowing to infer on a large number of connexions in a reasonable computational time. Here, we propose an improved fast marching algorithm to infer on geodesic paths. Specifically, this procedure is designed to achieve accurate front propagation in an anisotropic elliptic medium, such as DTI data. We evaluate the numerical performance of this approach on simulated datasets, as well as its robustness to local perturbation induced by fiber crossing. On real data, we demonstrate the feasibility of extracting geodesics to connect an extended set of brain regions.

Recent advances in recording electrophysiological data simultaneously with magnetic resonance imaging.

Simultaneous recording of brain activity by different neurophysiological modalities can yield insights that reach beyond those obtained by each technique individually, even when compared to those from the post-hoc integration of results from each technique recorded sequentially. Success in the endeavour of real-time multimodal experiments requires special hardware and software as well as purpose-tailored experimental design and analysis strategies. Here, we review the key methodological issues in recording electrophysiological data in humans simultaneously with magnetic resonance imaging (MRI), focusing on recent technical and analytical advances in the field. Examples are derived from simultaneous electroencephalography (EEG) and electromyography (EMG) during functional MRI in cognitive and systems neuroscience as well as in clinical neurology, in particular in epilepsy and movement disorders. We conclude with an outlook on current and future efforts to achieve true integration of electrical and haemodynamic measures of neuronal activity using data fusion models.

Concordance between distributed EEG source localization and simultaneous EEG-fMRI studies of epileptic spikes.

In order to analyze where epileptic spikes are generated, we assessed the level of concordance between EEG source localization using distributed source models and simultaneous EEG-fMRI which measures the hemodynamic correlates of EEG activity. Data to be compared were first estimated on the same cortical surface and two comparison strategies were used: (1) MEM-concordance: a comparison between EEG sources localized with the Maximum Entropy on the Mean (MEM) method and fMRI clusters showing a significant hemodynamic response. Minimal geodesic distances between local extrema and overlap measurements between spatial extents of EEG sources and fMRI clusters were used to quantify MEM-concordance. (2) fMRI-relevance: estimation of the fMRI-relevance index alpha quantifying if sources located in an fMRI cluster could explain some scalp EEG data, when this fMRI cluster was used to constrain the EEG inverse problem. Combining MEM-concordance and fMRI-relevance (alpha) indexes, each fMRI cluster showing a significant hemodynamic response (p<0.05 corrected) was classified according to its concordance with EEG data. Nine patients with focal epilepsy who underwent EEG-fMRI examination followed by EEG recording outside the scanner were selected for this study. Among the 62 fMRI clusters analyzed (7 patients), 15 (24%) found in 6 patients were highly concordant with EEG according to both MEM-concordance and fMRI-relevance. EEG concordance was found for 5 clusters (8%) according to alpha only, suggesting sources missed by the MEM. No concordance with EEG was found for 30 clusters (48%) and for 10 clusters (16%) alpha was significantly negative, suggesting EEG-fMRI discordance. We proposed two complementary strategies to assess and classify EEG-fMRI concordance. We showed that for most patients, part of the hemodynamic response to spikes was highly concordant with EEG sources, whereas other fMRI clusters in response to the same spikes were found distant or discordant with EEG sources.

A neural mass model of spectral responses in electrophysiology.

We present a neural mass model of steady-state membrane potentials measured with local field potentials or electroencephalography in the frequency domain. This model is an extended version of previous dynamic causal models for investigating event-related potentials in the time-domain. In this paper, we augment the previous formulation with parameters that mediate spike-rate adaptation and recurrent intrinsic inhibitory connections. We then use linear systems analysis to show how the model's spectral response changes with its neurophysiological parameters. We demonstrate that much of the interesting behaviour depends on the non-linearity which couples mean membrane potential to mean spiking rate. This non-linearity is analogous, at the population level, to the firing rate-input curves often used to characterize single-cell responses. This function depends on the model's gain and adaptation currents which, neurobiologically, are influenced by the activity of modulatory neurotransmitters. The key contribution of this paper is to show how neuromodulatory effects can be modelled by adding adaptation currents to a simple phenomenological model of EEG. Critically, we show that these effects are expressed in a systematic way in the spectral density of EEG recordings. Inversion of the model, given such non-invasive recordings, should allow one to quantify pharmacologically induced changes in adaptation currents. In short, this work establishes a forward or generative model of electrophysiological recordings for psychopharmacological studies.

Evaluation of EEG localization methods using realistic simulations of interictal spikes.

Performing an accurate localization of sources of interictal spikes from EEG scalp measurements is of particular interest during the presurgical investigation of epilepsy. The purpose of this paper is to study the ability of six distributed source localization methods to recover extended sources of activated cortex. Due to the frequent lack of a gold standard to evaluate source localization methods, our evaluation was performed in a controlled environment using realistic simulations of EEG interictal spikes, involving several anatomical locations with several spatial extents. Simulated data were corrupted by physiological EEG noise. Simulations involving pairs of sources with the same amplitude were also studied. In addition to standard validation criteria (e.g., geodesic distance or mean square error), we proposed an original criterion dedicated to assess detection accuracy, based on receiver operating characteristic (ROC) analysis. Six source localization methods were evaluated: the minimum norm, the minimum norm weighted by multivariate source prelocalization (MSP), cortical LORETA with or without additional minimum norm regularization, and two derivations of the maximum entropy on the mean (MEM) approach. Results showed that LORETA-based and MEM-based methods were able to accurately recover sources of different spatial extents, with the exception of sources in temporo-mesial and fronto-mesial regions. Several spurious sources were generated by those methods, however, whereas methods using the MSP always located very accurately the maximum of activity but not its spatial extent. These findings suggest that one should always take into account the results from different localization methods when analyzing real interictal spikes.