Gradient-based MCMC samplers for dynamic causal modelling.

In this technical note, we derive two MCMC (Markov chain Monte Carlo) samplers for dynamic causal models (DCMs). Specifically, we use (a) Hamiltonian MCMC (HMC-E) where sampling is simulated using Hamilton's equation of motion and (b) Langevin Monte Carlo algorithm (LMC-R and LMC-E) that simulates the Langevin diffusion of samples using gradients either on a Euclidean (E) or on a Riemannian (R) manifold. While LMC-R requires minimal tuning, the implementation of HMC-E is heavily dependent on its tuning parameters. These parameters are therefore optimised by learning a Gaussian process model of the time-normalised sample correlation matrix. This allows one to formulate an objective function that balances tuning parameter exploration and exploitation, furnishing an intervention-free inference scheme. Using neural mass models (NMMs)-a class of biophysically motivated DCMs-we find that HMC-E is statistically more efficient than LMC-R (with a Riemannian metric); yet both gradient-based samplers are far superior to the random walk Metropolis algorithm, which proves inadequate to steer away from dynamical instability.

Behavioral modeling of human choices reveals dissociable effects of physical effort and temporal delay on reward devaluation.

There has been considerable interest from the fields of biology, economics, psychology, and ecology about how decision costs decrease the value of rewarding outcomes. For example, formal descriptions of how reward value changes with increasing temporal delays allow for quantifying individual decision preferences, as in animal species populating different habitats, or normal and clinical human populations. Strikingly, it remains largely unclear how humans evaluate rewards when these are tied to energetic costs, despite the surge of interest in the neural basis of effort-guided decision-making and the prevalence of disorders showing a diminished willingness to exert effort (e.g., depression). One common assumption is that effort discounts reward in a similar way to delay. Here we challenge this assumption by formally comparing competing hypotheses about effort and delay discounting. We used a design specifically optimized to compare discounting behavior for both effort and delay over a wide range of decision costs (Experiment 1). We then additionally characterized the profile of effort discounting free of model assumptions (Experiment 2). Contrary to previous reports, in both experiments effort costs devalued reward in a manner opposite to delay, with small devaluations for lower efforts, and progressively larger devaluations for higher effort-levels (concave shape). Bayesian model comparison confirmed that delay-choices were best predicted by a hyperbolic model, with the largest reward devaluations occurring at shorter delays. In contrast, an altogether different relationship was observed for effort-choices, which were best described by a model of inverse sigmoidal shape that is initially concave. Our results provide a novel characterization of human effort discounting behavior and its first dissociation from delay discounting. This enables accurate modelling of cost-benefit decisions, a prerequisite for the investigation of the neural underpinnings of effort-guided choice and for understanding the deficits in clinical disorders characterized by behavioral inactivity.

Gradient-free MCMC methods for dynamic causal modelling.

In this technical note we compare the performance of four gradient-free MCMC samplers (random walk Metropolis sampling, slice-sampling, adaptive MCMC sampling and population-based MCMC sampling with tempering) in terms of the number of independent samples they can produce per unit computational time. For the Bayesian inversion of a single-node neural mass model, both adaptive and population-based samplers are more efficient compared with random walk Metropolis sampler or slice-sampling; yet adaptive MCMC sampling is more promising in terms of compute time. Slice-sampling yields the highest number of independent samples from the target density - albeit at almost 1000% increase in computational time, in comparison to the most efficient algorithm (i.e., the adaptive MCMC sampler).

Transcranial electrical brain stimulation modulates neuronal tuning curves in perception of numerosity and duration.

Transcranial direct current stimulation (tDCS) is a non-invasive brain stimulation method with many putative applications and reported to effectively modulate behaviour. However, its effects have yet to be considered at a computational level. To address this we modelled the tuning curves underlying the behavioural effects of stimulation in a perceptual task. Participants judged which of the two serially presented images contained more items (numerosity judgement task) or was presented longer (duration judgement task). During presentation of the second image their posterior parietal cortices (PPCs) were stimulated bilaterally with opposite polarities for 1.6s. We also examined the impact of three stimulation conditions on behaviour: anodal right-PPC and cathodal left-PPC (rA-lC), reverse order (lA-rC) and no-stimulation condition. Behavioural results showed that participants were more accurate in numerosity and duration judgement tasks when they were stimulated with lA-rC and rA-lC stimulation conditions respectively. Simultaneously, a decrease in performance on numerosity and duration judgement tasks was observed when the stimulation condition favoured the other task. Thus, our results revealed a double-dissociation of laterality and task. Importantly, we were able to model the effects of stimulation on behaviour. Our computational modelling showed that participants' superior performance was attributable to a narrower tuning curve--smaller standard deviation of detection noise. We believe that this approach may prove useful in understanding the impact of brain stimulation on other cognitive domains.

Forward and backward inference in spatial cognition.

This paper shows that the various computations underlying spatial cognition can be implemented using statistical inference in a single probabilistic model. Inference is implemented using a common set of 'lower-level' computations involving forward and backward inference over time. For example, to estimate where you are in a known environment, forward inference is used to optimally combine location estimates from path integration with those from sensory input. To decide which way to turn to reach a goal, forward inference is used to compute the likelihood of reaching that goal under each option. To work out which environment you are in, forward inference is used to compute the likelihood of sensory observations under the different hypotheses. For reaching sensory goals that require a chaining together of decisions, forward inference can be used to compute a state trajectory that will lead to that goal, and backward inference to refine the route and estimate control signals that produce the required trajectory. We propose that these computations are reflected in recent findings of pattern replay in the mammalian brain. Specifically, that theta sequences reflect decision making, theta flickering reflects model selection, and remote replay reflects route and motor planning. We also propose a mapping of the above computational processes onto lateral and medial entorhinal cortex and hippocampus.

Changes in auditory feedback connections determine the severity of speech processing deficits after stroke.

We compared brain structure and function in two subgroups of 21 stroke patients with either moderate or severe chronic speech comprehension impairment. Both groups had damage to the supratemporal plane; however, the severe group suffered greater damage to two unimodal auditory areas: primary auditory cortex and the planum temporale. The effects of this damage were investigated using fMRI while patients listened to speech and speech-like sounds. Pronounced changes in connectivity were found in both groups in undamaged parts of the auditory hierarchy. Compared to controls, moderate patients had significantly stronger feedback connections from planum temporale to primary auditory cortex bilaterally, while in severe patients this connection was significantly weaker in the undamaged right hemisphere. This suggests that predictive feedback mechanisms compensate in moderately affected patients but not in severely affected patients. The key pathomechanism in humans with persistent speech comprehension impairments may be impaired feedback connectivity to unimodal auditory areas.

The problem of low variance voxels in statistical parametric mapping; a new hat avoids a 'haircut'.

Statistical parametric mapping (SPM) locates significant clusters based on a ratio of signal to noise (a 'contrast' of the parameters divided by its standard error) meaning that very low noise regions, for example outside the brain, can attain artefactually high statistical values. Similarly, the commonly applied preprocessing step of Gaussian spatial smoothing can shift the peak statistical significance away from the peak of the contrast and towards regions of lower variance. These problems have previously been identified in positron emission tomography (PET) (Reimold et al., 2006) and voxel-based morphometry (VBM) (Acosta-Cabronero et al., 2008), but can also appear in functional magnetic resonance imaging (fMRI) studies. Additionally, for source-reconstructed magneto- and electro-encephalography (M/EEG), the problems are particularly severe because sparsity-favouring priors constrain meaningfully large signal and variance to a small set of compactly supported regions within the brain. (Acosta-Cabronero et al., 2008) suggested adding noise to background voxels (the 'haircut'), effectively increasing their noise variance, but at the cost of contaminating neighbouring regions with the added noise once smoothed. Following theory and simulations, we propose to modify--directly and solely--the noise variance estimate, and investigate this solution on real imaging data from a range of modalities.

Age-related changes in causal interactions between cortical motor regions during hand grip.

Brain activity during motor performance becomes more widespread and less lateralized with advancing age in response to ongoing degenerative processes. In this study, we were interested in the mechanism by which this change in the pattern of activity supports motor performance with advancing age. We used both transcranial magnetic stimulation (TMS) and functional magnetic resonance imaging (fMRI) to assess age related changes in motor system connectivity during isometric hand grip. Paired pulse TMS was used to measure the change in interhemispheric inhibition (IHI) from contralateral M1 (cM1) to ipsilateral M1 (iM1) during right hand grip. Dynamic Causal Modelling (DCM) of fMRI data was used to investigate the effect of age on causal interactions throughout the cortical motor network during right hand grip. Bayesian model selection was used to identify the causal model that best explained the data for all subjects. Firstly, we confirmed that the TMS and DCM measures both demonstrated a less inhibitory/more facilitatory influence of cM1 on iM1 during hand grip with advancing age. These values correlated with one another providing face validity for our DCM measures of connectivity. We found increasing reciprocal facilitatory influences with advancing age (i) between all ipsilateral cortical motor areas and (ii) between cortical motor areas of both hemispheres and iM1. There were no differences in the performance of our task with ageing suggesting that the ipsilateral cortical motor areas, in particular iM1, play a central role in maintaining performance levels with ageing through increasingly facilitatory cortico-cortical influences.

Bayesian comparison of neurovascular coupling models using EEG-fMRI.

Functional magnetic resonance imaging (fMRI), with blood oxygenation level-dependent (BOLD) contrast, is a widely used technique for studying the human brain. However, it is an indirect measure of underlying neuronal activity and the processes that link this activity to BOLD signals are still a topic of much debate. In order to relate findings from fMRI research to other measures of neuronal activity it is vital to understand the underlying neurovascular coupling mechanism. Currently, there is no consensus on the relative roles of synaptic and spiking activity in the generation of the BOLD response. Here we designed a modelling framework to investigate different neurovascular coupling mechanisms. We use Electroencephalographic (EEG) and fMRI data from a visual stimulation task together with biophysically informed mathematical models describing how neuronal activity generates the BOLD signals. These models allow us to non-invasively infer the degree of local synaptic and spiking activity in the healthy human brain. In addition, we use Bayesian model comparison to decide between neurovascular coupling mechanisms. We show that the BOLD signal is dependent upon both the synaptic and spiking activity but that the relative contributions of these two inputs are dependent upon the underlying neuronal firing rate. When the underlying neuronal firing is low then the BOLD response is best explained by synaptic activity. However, when the neuronal firing rate is high then both synaptic and spiking activity are required to explain the BOLD signal.

An introduction to thermodynamic integration and application to dynamic causal models.

In generative modeling of neuroimaging data, such as dynamic causal modeling (DCM), one typically considers several alternative models, either to determine the most plausible explanation for observed data (Bayesian model selection) or to account for model uncertainty (Bayesian model averaging). Both procedures rest on estimates of the model evidence, a principled trade-off between model accuracy and complexity. In the context of DCM, the log evidence is usually approximated using variational Bayes. Although this approach is highly efficient, it makes distributional assumptions and is vulnerable to local extrema. This paper introduces the use of thermodynamic integration (TI) for Bayesian model selection and averaging in the context of DCM. TI is based on Markov chain Monte Carlo sampling which is asymptotically exact but orders of magnitude slower than variational Bayes. In this paper, we explain the theoretical foundations of TI, covering key concepts such as the free energy and its origins in statistical physics. Our aim is to convey an in-depth understanding of the method starting from its historical origin in statistical physics. In addition, we demonstrate the practical application of TI via a series of examples which serve to guide the user in applying this method. Furthermore, these examples demonstrate that, given an efficient implementation and hardware capable of parallel processing, the challenge of high computational demand can be overcome successfully. The TI implementation presented in this paper is freely available as part of the open source software TAPAS. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11571-021-09696-9.

Comparing families of dynamic causal models.

Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This "best model" approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data.

Retrospective Inference as a Form of Bounded Rationality, and Its Beneficial Influence on Learning.

Probabilistic models of cognition typically assume that agents make inferences about current states by combining new sensory information with fixed beliefs about the past, an approach known as Bayesian filtering. This is computationally parsimonious, but, in general, leads to suboptimal beliefs about past states, since it ignores the fact that new observations typically contain information about the past as well as the present. This is disadvantageous both because knowledge of past states may be intrinsically valuable, and because it impairs learning about fixed or slowly changing parameters of the environment. For these reasons, in offline data analysis it is usual to infer on every set of states using the entire time series of observations, an approach known as (fixed-interval) Bayesian smoothing. Unfortunately, however, this is impractical for real agents, since it requires the maintenance and updating of beliefs about an ever-growing set of states. We propose an intermediate approach, finite retrospective inference (FRI), in which agents perform update beliefs about a limited number of past states (Formally, this represents online fixed-lag smoothing with a sliding window). This can be seen as a form of bounded rationality in which agents seek to optimize the accuracy of their beliefs subject to computational and other resource costs. We show through simulation that this approach has the capacity to significantly increase the accuracy of both inference and learning, using a simple variational scheme applied to both randomly generated Hidden Markov models (HMMs), and a specific application of the HMM, in the form of the widely used probabilistic reversal task. Our proposal thus constitutes a theoretical contribution to normative accounts of bounded rationality, which makes testable empirical predictions that can be explored in future work.

Bayesian model selection for group studies.

Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. BMS has recently found widespread application in neuroimaging, particularly in the context of dynamic causal modelling (DCM). However, so far, combining BMS results from several subjects has relied on simple (fixed effects) metrics, e.g. the group Bayes factor (GBF), that do not account for group heterogeneity or outliers. In this paper, we compare the GBF with two random effects methods for BMS at the between-subject or group level. These methods provide inference on model-space using a classical and Bayesian perspective respectively. First, a classical (frequentist) approach uses the log model evidence as a subject-specific summary statistic. This enables one to use analysis of variance to test for differences in log-evidences over models, relative to inter-subject differences. We then consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new variational Bayes method rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution which describes the probabilities for all models considered. These probabilities then define a multinomial distribution over model space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject as well as the exceedance probability of one model being more likely than any other model. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects, is more informative and appropriate than both the GBF and frequentist tests of the log-evidences. In particular, we found that the hierarchical Bayesian approach is considerably more robust than either of the other approaches in the presence of outliers. We expect that this new random effects method will prove useful for a wide range of group studies, not only in the context of DCM, but also for other modelling endeavours, e.g. comparing different source reconstruction methods for EEG/MEG or selecting among competing computational models of learning and decision-making.

Interhemispheric integration of visual processing during task-driven lateralization.

The mechanisms underlying interhemispheric integration (IHI) remain poorly understood, particularly for lateralized cognitive processes. To test competing theories of IHI, we constructed and fitted dynamic causal models to functional magnetic resonance data from two visual tasks that operated on identical stimuli but showed opposite hemispheric dominance. Using a systematic Bayesian model selection procedure, we found that, in the ventral visual stream, which was activated by letter judgments, interhemispheric connections mediated asymmetric information transfer from the nonspecialized right to the specialized left hemisphere when the latter did not have direct access to stimulus information. Notably, this form of IHI did not engage all areas activated by the task but was specific for areas in the lingual and fusiform gyri. In the dorsal stream, activated by spatial judgments, it did not matter which hemisphere received the stimulus: interhemispheric coupling increased bidirectionally, reflecting recruitment of the nonspecialized left hemisphere. Again, not all areas activated by the task were involved in this form of IHI; instead, it was restricted to interactions between areas in the superior parietal gyrus. Overall, our results provide direct neurophysiological evidence, in terms of effective connectivity, for the existence of context-dependent mechanisms of IHI that are implemented by specific visual areas during task-driven lateralization.

Integrated Bayesian models of learning and decision making for saccadic eye movements.

The neurophysiology of eye movements has been studied extensively, and several computational models have been proposed for decision-making processes that underlie the generation of eye movements towards a visual stimulus in a situation of uncertainty. One class of models, known as linear rise-to-threshold models, provides an economical, yet broadly applicable, explanation for the observed variability in the latency between the onset of a peripheral visual target and the saccade towards it. So far, however, these models do not account for the dynamics of learning across a sequence of stimuli, and they do not apply to situations in which subjects are exposed to events with conditional probabilities. In this methodological paper, we extend the class of linear rise-to-threshold models to address these limitations. Specifically, we reformulate previous models in terms of a generative, hierarchical model, by combining two separate sub-models that account for the interplay between learning of target locations across trials and the decision-making process within trials. We derive a maximum-likelihood scheme for parameter estimation as well as model comparison on the basis of log likelihood ratios. The utility of the integrated model is demonstrated by applying it to empirical saccade data acquired from three healthy subjects. Model comparison is used (i) to show that eye movements do not only reflect marginal but also conditional probabilities of target locations, and (ii) to reveal subject-specific learning profiles over trials. These individual learning profiles are sufficiently distinct that test samples can be successfully mapped onto the correct subject by a naïve Bayes classifier. Altogether, our approach extends the class of linear rise-to-threshold models of saccadic decision making, overcomes some of their previous limitations, and enables statistical inference both about learning of target locations across trials and the decision-making process within trials.

Dynamic causal models of neural system dynamics:current state and future extensions.

Complex processes resulting from interaction of multiple elements can rarely be understood by analytical scientific approaches alone; additional, mathematical models of system dynamics are required. This insight, which disciplines like physics have embraced for a long time already, is gradually gaining importance in the study of cognitive processes by functional neuroimaging. In this field, causal mechanisms in neural systems are described in terms of effective connectivity. Recently, dynamic causal modelling (DCM) was introduced as a generic method to estimate effective connectivity from neuroimaging data in a Bayesian fashion. One of the key advantages of DCM over previous methods is that it distinguishes between neural state equations and modality-specific forward models that translate neural activity into a measured signal. Another strength is its natural relation to Bayesian model selection (BMS) procedures. In this article, we review the conceptual and mathematical basis of DCM and its implementation for functional magnetic resonance imaging data and event-related potentials. After introducing the application of BMS in the context of DCM, we conclude with an outlook to future extensions of DCM. These extensions are guided by the long-term goal of using dynamic system models for pharmacological and clinical applications, particularly with regard to synaptic plasticity.

Event-related brain dynamics.

Event-related potentials (ERPs) provide evidence of a direct link between cognitive events and brain electrical activity in a wide range of cognitive paradigms. It has generally been held that an ERP is the result of a set of discrete stimulus-evoked brain events. A recent study, however, provides new evidence to suggest that some ERP components might be generated by stimulus-induced changes in ongoing brain dynamics. This is consistent with views emerging from several neuroscientific fields, suggesting that phase synchronization of ongoing rhythms across different spatio-temporal scales mediates the functional integration necessary to perform higher cognitive tasks.

Biophysical models of fMRI responses.

Functional magnetic resonance imaging (fMRI) is used to investigate where the neural implementation of specific cognitive processes occurs. The standard approach uses linear convolution models that relate experimentally designed inputs, through a haemodynamic response function, to observed blood oxygen level dependent (BOLD) signals. Such models are, however, blind to the causal mechanisms that underlie observed BOLD responses. Recent developments have focused on how BOLD responses are generated and include biophysical input-state-output models with neural and haemodynamic state equations and models of functional integration that explain local dynamics through interactions with remote areas. Forward models with parameters at the neural level, such as dynamic causal modelling, combine both approaches, modelling the whole causal chain from external stimuli, via induced neural dynamics, to observed BOLD responses.

mpdcm: A toolbox for massively parallel dynamic causal modeling.

BACKGROUND: Dynamic causal modeling (DCM) for fMRI is an established method for Bayesian system identification and inference on effective brain connectivity. DCM relies on a biophysical model that links hidden neuronal activity to measurable BOLD signals. Currently, biophysical simulations from DCM constitute a serious computational hindrance. Here, we present Massively Parallel Dynamic Causal Modeling (mpdcm), a toolbox designed to address this bottleneck. NEW METHOD: mpdcm delegates the generation of simulations from DCM's biophysical model to graphical processing units (GPUs). Simulations are generated in parallel by implementing a low storage explicit Runge-Kutta's scheme on a GPU architecture. mpdcm is publicly available under the GPLv3 license. RESULTS: We found that mpdcm efficiently generates large number of simulations without compromising their accuracy. As applications of mpdcm, we suggest two computationally expensive sampling algorithms: thermodynamic integration and parallel tempering. COMPARISON WITH EXISTING METHOD(S): mpdcm is up to two orders of magnitude more efficient than the standard implementation in the software package SPM. Parallel tempering increases the mixing properties of the traditional Metropolis-Hastings algorithm at low computational cost given efficient, parallel simulations of a model. CONCLUSIONS: Future applications of DCM will likely require increasingly large computational resources, for example, when the likelihood landscape of a model is multimodal, or when implementing sampling methods for multi-subject analysis. Due to the wide availability of GPUs, algorithmic advances can be readily available in the absence of access to large computer grids, or when there is a lack of expertise to implement algorithms in such grids.

Investigating the functional role of callosal connections with dynamic causal models.

The anatomy of the corpus callosum has been described in considerable detail. Tracing studies in animals and human postmortem experiments are currently complemented by diffusion-weighted imaging, which enables noninvasive investigations of callosal connectivity to be conducted. In contrast to the wealth of anatomical data, little is known about the principles by which interhemispheric integration is mediated by callosal connections. Most importantly, we lack insights into the mechanisms that determine the functional role of callosal connections in a context-dependent fashion. These mechanisms can now be disclosed by models of effective connectivity that explain neuroimaging data from paradigms that manipulate interhemispheric interactions. In this article, we demonstrate that dynamic causal modeling (DCM), in conjunction with Bayesian model selection (BMS), is a powerful approach to disentangling the various factors that determine the functional role of callosal connections. We first review the theoretical foundations of DCM and BMS before demonstrating the application of these techniques to empirical data from a single subject.