Structure Learning in Coupled Dynamical Systems and Dynamic Causal Modelling

Identifying a coupled dynamical system out of many plausible candidates, each of which could serve as the underlying generator of some observed measurements, is a profoundly ill posed problem that commonly arises when modelling real world phenomena. In this review, we detail a set of statistical procedures for inferring the structure of nonlinear coupled dynamical systems (structure learning), which has proved useful in neuroscience research. A key focus here is the comparison of competing models of (ie, hypotheses about) network architectures and implicit coupling functions in terms of their Bayesian model evidence. These methods are collectively referred to as dynamical casual modelling (DCM). We focus on a relatively new approach that is proving remarkably useful; namely, Bayesian model reduction (BMR), which enables rapid evaluation and comparison of models that differ in their network architecture. We illustrate the usefulness of these techniques through modelling neurovascular coupling (cellular pathways linking neuronal and vascular systems), whose function is an active focus of research in neurobiology and the imaging of coupled neuronal systems

MCMC for Bayesian uncertainty quantification from time-series data

In computational neuroscience, Neural Population Models (NPMs) are mechanistic models that describe brain physiology in a range of different states. Within computational neuroscience there is growing interest in the inverse problem of inferring NPM parameters from recordings such as the EEG (Electroencephalogram). Uncertainty quantification is essential in this application area in order to infer the mechanistic effect of interventions such as anaesthesia. This paper presents Open image in new window software for Bayesian uncertainty quantification in the parameters of NPMs from approximately stationary data using Markov Chain Monte Carlo (MCMC). Modern MCMC methods require first order (and in some cases higher order) derivatives of the posterior density. The software presented offers two distinct methods of evaluating derivatives: finite differences and exact derivatives obtained through Algorithmic Differentiation (AD). For AD, two different implementations are used: the open source Stan Math Library and the commercially licenced Open image in new window tool distributed by NAG (Numerical Algorithms Group). The use of derivative information in MCMC sampling is demonstrated through a simple example, the noise-driven harmonic oscillator. And different methods for computing derivatives are compared. The software is written in a modular object-oriented way such that it can be extended to derivative based MCMC for other scientific domains

Dynamic causal modeling of spontaneous fluctuations in skin conductance

Spontaneous fluctuations (SF) in skin conductance are often used to index sympathetic arousal and emotional states. SF are caused by sudomotor nerve activity (SNA), which is a direct indicator of sympathetic arousal. Here, we describe a dynamic causal model (DCM) of how SNA causes SF, and apply variational Bayesian model inversion to infer SNA, given empirically observed SF. The estimated SNA bears a relationship to the number of SF as derived from conventional (semi-visual) analysis. Crucially, we show that, during public speaking induced anxiety, the estimated number of SNA bursts is a better predictor of the (known) psychological state than the number of SF. We suggest dynamic causal modeling of SF potentially allows a more precise and informed inference about arousal than purely descriptive methods

Oscillatory, Computational, and Behavioral Evidence for Impaired GABAergic Inhibition in Schizophrenia

The dysconnection hypothesis of schizophrenia (SZ) proposes that psychosis is best understood in terms of aberrant connectivity. Specifically, it suggests that dysconnectivity arises through aberrant synaptic modulation associated with deficits in GABAergic inhibition, excitation-inhibition balance and disturbances of high-frequency oscillations. Using a computational model combined with a graded-difficulty visual orientation discrimination paradigm, we demonstrate that, in SZ, perceptual performance is determined by the balance of excitation-inhibition in superficial cortical layers. Twenty-eight individuals with a DSM-IV diagnosis of SZ, and 30 age- and gender-matched healthy controls participated in a psychophysics orientation discrimination task, a visual grating magnetoencephalography (MEG) recording, and a magnetic resonance spectroscopy (MRS) scan for GABA. Using a neurophysiologically informed model, we quantified group differences in GABA, gamma measures, and the predictive validity of model parameters for orientation discrimination in the SZ group. MEG visual gamma frequency was reduced in SZ, with lower peak frequency associated with more severe negative symptoms. Orientation discrimination performance was impaired in SZ. Dynamic causal modeling of the MEG data showed that local synaptic connections were reduced in SZ and local inhibition correlated negatively with the severity of negative symptoms. The effective connectivity between inhibitory interneurons and superficial pyramidal cells predicted orientation discrimination performance within the SZ group; consistent with graded, behaviorally relevant, disease-related changes in local GABAergic connections. Occipital GABA levels were significantly reduced in SZ but did not predict behavioral performance or oscillatory measures. These findings endorse the importance, and behavioral relevance, of GABAergic synaptic disconnection in schizophrenia that underwrites excitation-inhibition balance

Estimating directed connectivity from cortical recordings and reconstructed sources

In cognitive neuroscience, electrical brain activity is most commonly recorded at the scalp. In order to infer the contributions and connectivity of underlying neuronal sources within the brain, it is necessary to reconstruct sensor data at the source level. Several approaches to this reconstruction have been developed, thereby solving the so-called implicit inverse problem Michel et al. (Clin Neurophysiol 115:2195-2222, 2004). However, a unifying premise against which to validate these source reconstructions is seldom available. The dataset provided in this work, in which brain activity is simultaneously recorded on the scalp (non-invasively) by electroencephalography (EEG) and on the cortex (invasively) by electrocorticography (ECoG), can be of a great help in this direction. These multimodal recordings were obtained from a macaque monkey under wakefulness and sedation. Our primary goal was to establish the connectivity architecture between two sources of interest (frontal and parietal), and to assess how their coupling changes over the conditions. We chose these sources because previous studies have shown that the connections between them are modified by anaesthesia Boly et al. (J Neurosci 32:7082-7090, 2012). Our secondary goal was to evaluate the consistency of the connectivity results when analyzing sources recorded from invasive data (128 implanted ECoG sources) and source activity reconstructed from scalp recordings (19 EEG sensors) at the same locations as the ECoG sources. We conclude that the directed connectivity in the frequency domain between cortical sources reconstructed from scalp EEG is qualitatively similar to the connectivity inferred directly from cortical recordings, using both data-driven (directed transfer function) and biologically grounded (dynamic causal modelling) methods. Furthermore, the connectivity changes identified were consistent with previous findings Boly et al. (J Neurosci 32:7082-7090, 2012). Our findings suggest that inferences about directed connectivity based upon non-invasive electrophysiological data have construct validity in relation to invasive recordings

Computational modelling of movement-related beta-oscillatory dynamics in human motor cortex

Oscillatory activity in the beta range, in human primary motor cortex (M1), shows interesting dynamics that are tied to behaviour and change systematically in disease. To investigate the pathophysiology underlying these changes, we must first understand how changes in beta activity are caused in healthy subjects. We therefore adapted a canonical (repeatable) microcircuit model used in dynamic causal modelling (DCM) previously used to model induced responses in visual cortex. We adapted this model to accommodate cytoarchitectural differences between visual and motor cortex. Using biologically plausible connections, we used Bayesian model selection to identify the best model of measured MEG data from 11 young healthy participants, performing a simple handgrip task. We found that the canonical M1 model had substantially more model evidence than the generic canonical microcircuit model when explaining measured MEG data. The canonical M1 model reproduced measured dynamics in humans at rest, in a manner consistent with equivalent studies performed in mice. Furthermore, the changes in excitability (self-inhibition) necessary to explain beta suppression during handgrip were consistent with the attenuation of sensory precision implied by predictive coding. These results establish the face validity of a model that can be used to explore the laminar interactions that underlie beta-oscillatory dynamics in humans in vivo. Our canonical M1 model may be useful for characterising the synaptic mechanisms that mediate pathophysiological beta dynamics associated with movement disorders, such as stroke or Parkinson's disease

Characterising seizures in anti-NMDA-receptor encephalitis with dynamic causal modelling

We characterised the pathophysiology of seizure onset in terms of slow fluctuations in synaptic efficacy using EEG in patients with anti-N-methyl-d-aspartate receptor (NMDA-R) encephalitis. EEG recordings were obtained from two female patients with anti-NMDA-R encephalitis with recurrent partial seizures (ages 19 and 31). Focal electrographic seizure activity was localised using an empirical Bayes beamformer. The spectral density of reconstructed source activity was then characterised with dynamic causal modelling (DCM). Eight models were compared for each patient, to evaluate the relative contribution of changes in intrinsic (excitatory and inhibitory) connectivity and endogenous afferent input. Bayesian model comparison established a role for changes in both excitatory and inhibitory connectivity during seizure activity (in addition to changes in the exogenous input). Seizures in both patients were associated with a sequence of changes in inhibitory and excitatory connectivity; a transient increase in inhibitory connectivity followed by a transient increase in excitatory connectivity and a final peak of excitatory-inhibitory balance at seizure offset. These systematic fluctuations in excitatory and inhibitory gain may be characteristic of (anti NMDA-R encephalitis) seizures. We present these results as a case study and replication to motivate analyses of larger patient cohorts, to see whether our findings generalise and further characterise the mechanisms of seizure activity in anti-NMDA-R encephalitis