arXiv subjects

Karl J. Friston

Publications and source records attributed to Karl J. Friston.

43 records · Page 3Linked to original sources

arXiv2019

A tutorial on group effective connectivity analysis, part 2: second level analysis with PEB

This tutorial provides a worked example of using Dynamic Causal Modelling (DCM) and Parametric Empirical Bayes (PEB) to characterise inter-subject variability in neural circuitry (effective connectivity). This involves specifying a hierarchical model with two or more levels. At the first level, state space models (DCMs) are used to infer the effective connectivity that best explains a subject's neuroimaging timeseries (e.g. fMRI, MEG, EEG). Subject-specific connectivity parameters are then taken to the group level, where they are modelled using a General Linear Model (GLM) that partitions between-subject variability into designed effects and additive random effects. The ensuing (Bayesian) hierarchical model conveys both the estimated connection strengths and their uncertainty (i.e., posterior covariance) from the subject to the group level; enabling hypotheses to be tested about the commonalities and differences across subjects. This approach can also finesse parameter estimation at the subject level, by using the group-level parameters as empirical priors. We walk through this approach in detail, using data from a published fMRI experiment that characterised individual differences in hemispheric lateralization in a semantic processing task. The preliminary subject specific DCM analysis is covered in detail in a companion paper. This tutorial is accompanied by the example dataset and step-by-step instructions to reproduce the analyses.

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arXiv2018

Breaking the bonds of weak coupling: the dynamic causal modelling of oscillator amplitudes

Models of coupled oscillators are useful in describing a wide variety of phenomena in physics, biology and economics. These models typically rest on the premise that the oscillators are weakly coupled, meaning that amplitudes can be assumed to be constant and dynamics can therefore be described purely in terms of phase differences. Whilst mathematically convenient, the restrictive nature of the weak coupling assumption can limit the explanatory power of these phase-coupled oscillator models. We therefore propose an extension to the weakly-coupled oscillator model that incorporates both amplitude and phase as dependent variables. We use the bilinear neuronal state equations of dynamic causal modelling as a foundation in deriving coupled differential equations that describe the activity of both weakly and strongly coupled oscillators. We show that weakly-coupled oscillator models are inadequate in describing the processes underlying the temporally variable signals observed in a variety of systems. We demonstrate that phase-coupled models perform well on simulations of weakly coupled systems but fail when connectivity is no longer weak. On the other hand, using Bayesian model selection, we show that our phase-amplitude coupling model can describe non-weakly coupled systems more effectively despite the added complexity associated with using amplitude as an extra dependent variable. We demonstrate the advantage of our phase-amplitude model in the context of model-generated data, as well as of a simulation of inter-connected pendula, neural local field potential recordings in rodents under anaesthesia and international economic gross domestic product data.

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arXiv2018

Optimising data for modelling neuronal responses

In this technical note, we address an unresolved challenge in neuroimaging statistics: how to determine which of several datasets is the best for inferring neuronal responses. Comparisons of this kind are important for experimenters when choosing an imaging protocol - and for developers of new acquisition methods. However, the hypothesis that one dataset is better than another cannot be tested using conventional statistics (based on likelihood ratios), as these require the data to be the same under each hypothesis. Here we present Bayesian data comparison, a principled framework for evaluating the quality of functional imaging data, in terms of the precision with which neuronal connectivity parameters can be estimated and competing models can be disambiguated. For each of several candidate datasets, neuronal responses are inferred using Dynamic Casual Modelling (DCM) - a commonly used Bayesian procedure for modelling neuroimaging data. Next, the parameters from subject-specific models are summarised at the group level using a Bayesian General Linear Model (GLM). A series of measures, which we introduce here, are then used to evaluate each dataset in terms of the precision of (group-level) parameter estimates and the ability of the data to distinguish similar models. To exemplify the approach, we compared four datasets that were acquired in a study evaluating multiband fMRI acquisition schemes. To enable people to reproduce these analyses using their own data and experimental paradigms, we provide general-purpose Matlab code via the SPM software.

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arXiv2018

How Robust are Deep Neural Networks?

Convolutional and Recurrent, deep neural networks have been successful in machine learning systems for computer vision, reinforcement learning, and other allied fields. However, the robustness of such neural networks is seldom apprised, especially after high classification accuracy has been attained. In this paper, we evaluate the robustness of three recurrent neural networks to tiny perturbations, on three widely used datasets, to argue that high accuracy does not always mean a stable and a robust (to bounded perturbations, adversarial attacks, etc.) system. Especially, normalizing the spectrum of the discrete recurrent network to bound the spectrum (using power method, Rayleigh quotient, etc.) on a unit disk produces stable, albeit highly non-robust neural networks. Furthermore, using the $ε$-pseudo-spectrum, we show that training of recurrent networks, say using gradient-based methods, often result in non-normal matrices that may or may not be diagonalizable. Therefore, the open problem lies in constructing methods that optimize not only for accuracy but also for the stability and the robustness of the underlying neural network, a criterion that is distinct from the other.

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arXiv2017

A computational hierarchy in human cortex

Hierarchies feature prominently in anatomical accounts of cortical organisation. An open question is which computational (algorithmic) processes are implemented by these hierarchies. One renowned hypothesis is that cortical hierarchies implement a model of the world's causal structure and serve to infer environmental states from sensory inputs. This view, which casts perception as hierarchical Bayesian inference, has become a highly influential concept in both basic and clinical neuroscience. So far, however, a direct correspondence between the predicted order of hierarchical Bayesian computations and the sequence of evoked neuronal activity has not been demonstrated. Here, we present evidence for this correspondence from neuroimaging and electrophysiological data in healthy volunteers. Trial-wise sequences of hierarchical computations were inferred from participants' behaviour during a social learning task that required multi-level inference about intentions. We found that the temporal sequence of neuronal activity matched the order of computations as predicted by the theory. These findings provide strong evidence for the operation of hierarchical Bayesian inference in human cortex. Furthermore, our approach offers a novel strategy for the combined computational-physiological phenotyping of patients with disorders of perception, such as schizophrenia or autism.

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arXiv2016

Analysis of family-wise error rates in statistical parametric mapping using random field theory

This technical report revisits the analysis of family-wise error rates in statistical parametric mapping - using random field theory - reported in (Eklund et al., 2015). Contrary to the understandable spin that these sorts of analyses attract, a review of their results suggests that they endorse the use of parametric assumptions - and random field theory - in the analysis of functional neuroimaging data. We briefly rehearse the advantages parametric analyses offer over nonparametric alternatives and then unpack the implications of (Eklund et al., 2015) for parametric procedures.

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arXiv2010

Topological inference for EEG and MEG

Neuroimaging produces data that are continuous in one or more dimensions. This calls for an inference framework that can handle data that approximate functions of space, for example, anatomical images, time--frequency maps and distributed source reconstructions of electromagnetic recordings over time. Statistical parametric mapping (SPM) is the standard framework for whole-brain inference in neuroimaging: SPM uses random field theory to furnish $p$-values that are adjusted to control family-wise error or false discovery rates, when making topological inferences over large volumes of space. Random field theory regards data as realizations of a continuous process in one or more dimensions. This contrasts with classical approaches like the Bonferroni correction, which consider images as collections of discrete samples with no continuity properties (i.e., the probabilistic behavior at one point in the image does not depend on other points). Here, we illustrate how random field theory can be applied to data that vary as a function of time, space or frequency. We emphasize how topological inference of this sort is invariant to the geometry of the manifolds on which data are sampled. This is particularly useful in electromagnetic studies that often deal with very smooth data on scalp or cortical meshes. This application illustrates the versatility and simplicity of random field theory and the seminal contributions of Keith Worsley (1951--2009), a key architect of topological inference.

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