Lee M. Harrison

Dynamic causal modelling.

In this paper we present an approach to the identification of nonlinear input-state-output systems. By using a bilinear approximation to the dynamics of interactions among states, the parameters of the implicit causal model reduce to three sets. These comprise (1) parameters that mediate the influence of extrinsic inputs on the states, (2) parameters that mediate intrinsic coupling among the states, and (3) [bilinear] parameters that allow the inputs to modulate that coupling. Identification proceeds in a Bayesian framework given known, deterministic inputs and the observed responses of the system. We developed this approach for the analysis of effective connectivity using experimentally designed inputs and fMRI responses. In this context, the coupling parameters correspond to effective connectivity and the bilinear parameters reflect the changes in connectivity induced by inputs. The ensuing framework allows one to characterise fMRI experiments, conceptually, as an experimental manipulation of integration among brain regions (by contextual or trial-free inputs, like time or attentional set) that is revealed using evoked responses (to perturbations or trial-bound inputs, like stimuli). As with previous analyses of effective connectivity, the focus is on experimentally induced changes in coupling (cf., psychophysiologic interactions). However, unlike previous approaches in neuroimaging, the causal model ascribes responses to designed deterministic inputs, as opposed to treating inputs as unknown and stochastic.

Dynamic causal modeling of evoked responses in EEG and MEG.

Neuronally plausible, generative or forward models are essential for understanding how event-related fields (ERFs) and potentials (ERPs) are generated. In this paper, we present a new approach to modeling event-related responses measured with EEG or MEG. This approach uses a biologically informed model to make inferences about the underlying neuronal networks generating responses. The approach can be regarded as a neurobiologically constrained source reconstruction scheme, in which the parameters of the reconstruction have an explicit neuronal interpretation. Specifically, these parameters encode, among other things, the coupling among sources and how that coupling depends upon stimulus attributes or experimental context. The basic idea is to supplement conventional electromagnetic forward models, of how sources are expressed in measurement space, with a model of how source activity is generated by neuronal dynamics. A single inversion of this extended forward model enables inference about both the spatial deployment of sources and the underlying neuronal architecture generating them. Critically, this inference covers long-range connections among well-defined neuronal subpopulations. In a previous paper, we simulated ERPs using a hierarchical neural-mass model that embodied bottom-up, top-down and lateral connections among remote regions. In this paper, we describe a Bayesian procedure to estimate the parameters of this model using empirical data. We demonstrate this procedure by characterizing the role of changes in cortico-cortical coupling, in the genesis of ERPs. In the first experiment, ERPs recorded during the perception of faces and houses were modeled as distinct cortical sources in the ventral visual pathway. Category-selectivity, as indexed by the face-selective N170, could be explained by category-specific differences in forward connections from sensory to higher areas in the ventral stream. We were able to quantify and make inferences about these effects using conditional estimates of connectivity. This allowed us to identify where, in the processing stream, category-selectivity emerged. In the second experiment, we used an auditory oddball paradigm to show that the mismatch negativity can be explained by changes in connectivity. Specifically, using Bayesian model selection, we assessed changes in backward connections, above and beyond changes in forward connections. In accord with theoretical predictions, there was strong evidence for learning-related changes in both forward and backward coupling. These examples show that category- or context-specific coupling among cortical regions can be assessed explicitly, within a mechanistic, biologically motivated inference framework.

A free energy principle for the brain.

By formulating Helmholtzs ideas about perception, in terms of modern-day theories, one arrives at a model of perceptual inference and learning that can explain a remarkable range of neurobiological facts: using constructs from statistical physics, the problems of inferring the causes of sensory input and learning the causal structure of their generation can be resolved using exactly the same principles. Furthermore, inference and learning can proceed in a biologically plausible fashion. The ensuing scheme rests on Empirical Bayes and hierarchical models of how sensory input is caused. The use of hierarchical models enables the brain to construct prior expectations in a dynamic and context-sensitive fashion. This scheme provides a principled way to understand many aspects of cortical organisation and responses. In this paper, we show these perceptual processes are just one aspect of emergent behaviours of systems that conform to a free energy principle. The free energy considered here measures the difference between the probability distribution of environmental quantities that act on the system and an arbitrary distribution encoded by its configuration. The system can minimise free energy by changing its configuration to affect the way it samples the environment or change the distribution it encodes. These changes correspond to action and perception respectively and lead to an adaptive exchange with the environment that is characteristic of biological systems. This treatment assumes that the systems state and structure encode an implicit and probabilistic model of the environment. We will look at the models entailed by the brain and how minimisation of its free energy can explain its dynamics and structure.

Multiple sparse priors for the M/EEG inverse problem

This paper describes an application of hierarchical or empirical Bayes to the distributed source reconstruction problem in electro- and magnetoencephalography (EEG and MEG). The key contribution is the automatic selection of multiple cortical sources with compact spatial support that are specified in terms of empirical priors. This obviates the need to use priors with a specific form (e.g., smoothness or minimum norm) or with spatial structure (e.g., priors based on depth constraints or functional magnetic resonance imaging results). Furthermore, the inversion scheme allows for a sparse solution for distributed sources, of the sort enforced by equivalent current dipole (ECD) models. This means the approach automatically selects either a sparse or a distributed model, depending on the data. The scheme is compared with conventional applications of Bayesian solutions to quantify the improvement in performance.

Multivariate autoregressive modeling of fMRI time series

We propose the use of multivariate autoregressive (MAR) models of functional magnetic resonance imaging time series to make inferences about functional integration within the human brain. The method is demonstrated with synthetic and real data showing how such models are able to characterize interregional dependence. We extend linear MAR models to accommodate nonlinear interactions to model top-down modulatory processes with bilinear terms. MAR models are time series models and thereby model temporal order within measured brain activity. A further benefit of the MAR approach is that connectivity maps may contain loops, yet exact inference can proceed within a linear framework. Model order selection and parameter estimation are implemented by using Bayesian methods.

Nonlinear dynamic causal models for fMRI

Models of effective connectivity characterize the influence that neuronal populations exert over each other. Additionally, some approaches, for example Dynamic Causal Modelling (DCM) and variants of Structural Equation Modelling, describe how effective connectivity is modulated by experimental manipulations. Mathematically, both are based on bilinear equations, where the bilinear term models the effect of experimental manipulations on neuronal interactions. The bilinear framework, however, precludes an important aspect of neuronal interactions that has been established with invasive electrophysiological recording studies; i.e., how the connection between two neuronal units is enabled or gated by activity in other units. These gating processes are critical for controlling the gain of neuronal populations and are mediated through interactions between synaptic inputs (e.g. by means of voltage-sensitive ion channels). They represent a key mechanism for various neurobiological processes, including top-down (e.g. attentional) modulation, learning and neuromodulation. This paper presents a nonlinear extension of DCM that models such processes (to second order) at the neuronal population level. In this way, the modulation of network interactions can be assigned to an explicit neuronal population. We present simulations and empirical results that demonstrate the validity and usefulness of this model. Analyses of synthetic data showed that nonlinear and bilinear mechanisms can be distinguished by our extended DCM. When applying the model to empirical fMRI data from a blocked attention to motion paradigm, we found that attention-induced increases in V5 responses could be best explained as a gating of the V1-->V5 connection by activity in posterior parietal cortex. Furthermore, we analysed fMRI data from an event-related binocular rivalry paradigm and found that interactions amongst percept-selective visual areas were modulated by activity in the middle frontal gyrus. In both practical examples, Bayesian model selection favoured the nonlinear models over corresponding bilinear ones.

Modelling event-related responses in the brain

The aim of this work was to investigate the mechanisms that shape evoked electroencephalographic (EEG) and magneto-encephalographic (MEG) responses. We used a neuronally plausible model to characterise the dependency of response components on the models parameters. This generative model was a neural mass model of hierarchically arranged areas using three kinds of inter-area connections (forward, backward and lateral). We investigated how responses, at each level of a cortical hierarchy, depended on the strength of connections or coupling. Our strategy was to systematically add connections and examine the responses of each successive architecture. We did this in the context of deterministic responses and then with stochastic spontaneous activity. Our aim was to show, in a simple way, how event-related dynamics depend on extrinsic connectivity. To emphasise the importance of nonlinear interactions, we tried to disambiguate the components of event-related potentials (ERPs) or event-related fields (ERFs) that can be explained by a linear superposition of trial-specific responses and those engendered nonlinearly (e.g., by phase-resetting). Our key conclusions were; (i) when forward connections, mediating bottom-up or extrinsic inputs, are sufficiently strong, nonlinear mechanisms cause a saturation of excitatory interneuron responses. This endows the system with an inherent stability that precludes nondissipative population dynamics. (ii) The duration of evoked transients increases with the hierarchical depth or level of processing. (iii) When backward connections are added, evoked transients become more protracted, exhibiting damped oscillations. These are formally identical to late or endogenous components seen empirically. This suggests that late components are mediated by reentrant dynamics within cortical hierarchies. (iv) Bilateral connections produce similar effects to backward connections but can also mediate zero-lag phase-locking among areas. (v) Finally, with spontaneous activity, ERPs/ERFs can arise from two distinct mechanisms: For low levels of (stimulus related and ongoing) activity, the systems response conforms to a quasi-linear superposition of separable responses to the fixed and stochastic inputs. This is consistent with classical assumptions that motivate trial averaging to suppress spontaneous activity and disclose the ERP/ERF. However, when activity is sufficiently high, there are nonlinear interactions between the fixed and stochastic inputs. This interaction is expressed as a phase-resetting and represents a qualitatively different explanation for the ERP/ERF.

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.

Trial-by-Trial Fluctuations in the Event-Related Electroencephalogram Reflect Dynamic Changes in the Degree of Surprise

The P300 component of the human event-related brain potential has often been linked to the processing of rare, surprising events. However, the formal computational processes underlying the generation of the P300 are not well known. Here, we formulate a simple model of trial-by-trial learning of stimulus probabilities based on Information Theory. Specifically, we modeled the surprise associated with the occurrence of a visual stimulus to provide a formal quantification of the “subjective probability” associated with an event. Subjects performed a choice reaction time task, while we recorded their brain responses using electroencephalography (EEG). In each of 12 blocks, the probabilities of stimulus occurrence were changed, thereby creating sequences of trials with low, medium, and high predictability. Trial-by-trial variations in the P300 component were best explained by a model of stimulus-bound surprise. This model accounted for the data better than a categorical model that parametrically encoded the stimulus identity, or an alternative model of surprise based on the Kullback–Leibler divergence. The present data demonstrate that trial-by-trial changes in P300 can be explained by predictions made by an ideal observer keeping track of the probabilities of possible events. This provides evidence for theories proposing a direct link between the P300 component and the processing of surprising events. Furthermore, this study demonstrates how model-based analyses can be used to explain significant proportions of the trial-by-trial changes in human event-related EEG responses.

Influence of uncertainty and surprise on human corticospinal excitability during preparation for action.

Summary Actions are guided by prior sensory information [1–10], which is inherently uncertain. However, how the motor system is sculpted by trial-by-trial content of current sensory information remains largely unexplored. Previous work suggests that conditional probabilities, learned under a particular context, can be used preemptively to influence the output of the motor system [11–14]. To test this we used transcranial magnetic stimulation (TMS) to read out corticospinal excitability (CSE) during preparation for action in an instructed delay task [15, 16]. We systematically varied the uncertainty about an impending action by changing the validity of the instructive visual cue. We used two information-theoretic quantities to predict changes in CSE, prior to action, on a trial-by-trial basis: entropy (average uncertainty) and surprise (the stimulus-bound information conveyed by a visual cue) [17–19]. Our data show that during preparation for action, human CSE varies according to the entropy and surprise conveyed by visual events guiding action. CSE increases on trials with low entropy about the impending action and low surprise conveyed by an event. Commensurate effects were observed in reaction times. We suggest that motor output is biased according to contextual probabilities that are represented dynamically in the brain.