Kay Henning Brodersen

The Balanced Accuracy and Its Posterior Distribution

Evaluating the performance of a classification algorithm critically requires a measure of the degree to which unseen examples have been identified with their correct class labels. In practice, generalizability is frequently estimated by averaging the accuracies obtained on individual cross-validation folds. This procedure, however, is problematic in two ways. First, it does not allow for the derivation of meaningful confidence intervals. Second, it leads to an optimistic estimate when a biased classifier is tested on an imbalanced dataset. We show that both problems can be overcome by replacing the conventional point estimate of accuracy by an estimate of the posterior distribution of the balanced accuracy.

Hierarchical Prediction Errors in Midbrain and Basal Forebrain during Sensory Learning

In Bayesian brain theories, hierarchically related prediction errors (PEs) play a central role for predicting sensory inputs and inferring their underlying causes, e.g., the probabilistic structure of the environment and its volatility. Notably, PEs at different hierarchical levels may be encoded by different neuromodulatory transmitters. Here, we tested this possibility in computational fMRI studies of audio-visual learning. Using a hierarchical Bayesian model, we found that low-level PEs about visual stimulus outcome were reflected by widespread activity in visual and supramodal areas but also in the midbrain. In contrast, high-level PEs about stimulus probabilities were encoded by the basal forebrain. These findings were replicated in two groups of healthy volunteers. While our fMRI measures do not reveal the exact neuron types activated in midbrain and basal forebrain, they suggest a dichotomy between neuromodulatory systems, linking dopamine to low-level PEs about stimulus outcome and acetylcholine to more abstract PEs about stimulus probabilities.

Decoding the perception of pain from fMRI using multivariate pattern analysis

Pain is known to comprise sensory, cognitive, and affective aspects. Despite numerous previous fMRI studies, however, it remains open which spatial distribution of activity is sufficient to encode whether a stimulus is perceived as painful or not. In this study, we analyzed fMRI data from a perceptual decision-making task in which participants were exposed to near-threshold laser pulses. Using multivariate analyses on different spatial scales, we investigated the predictive capacity of fMRI data for decoding whether a stimulus had been perceived as painful. Our analysis yielded a rank order of brain regions: during pain anticipation, activity in the periaqueductal gray (PAG) and orbitofrontal cortex (OFC) afforded the most accurate trial-by-trial discrimination between painful and non-painful experiences; whereas during the actual stimulation, primary and secondary somatosensory cortex, anterior insula, dorsolateral and ventrolateral prefrontal cortex, and OFC were most discriminative. The most accurate prediction of pain perception from the stimulation period, however, was enabled by the combined activity in pain regions commonly referred to as the ‘pain matrix’. Our results demonstrate that the neural representation of (near-threshold) pain is spatially distributed and can be best described at an intermediate spatial scale. In addition to its utility in establishing structure-function mappings, our approach affords trial-by-trial predictions and thus represents a step towards the goal of establishing an objective neuronal marker of pain perception.

Generative Embedding for Model-Based Classification of fMRI Data

Decoding models, such as those underlying multivariate classification algorithms, have been increasingly used to infer cognitive or clinical brain states from measures of brain activity obtained by functional magnetic resonance imaging (fMRI). The practicality of current classifiers, however, is restricted by two major challenges. First, due to the high data dimensionality and low sample size, algorithms struggle to separate informative from uninformative features, resulting in poor generalization performance. Second, popular discriminative methods such as support vector machines (SVMs) rarely afford mechanistic interpretability. In this paper, we address these issues by proposing a novel generative-embedding approach that incorporates neurobiologically interpretable generative models into discriminative classifiers. Our approach extends previous work on trial-by-trial classification for electrophysiological recordings to subject-by-subject classification for fMRI and offers two key advantages over conventional methods: it may provide more accurate predictions by exploiting discriminative information encoded in ‘hidden’ physiological quantities such as synaptic connection strengths; and it affords mechanistic interpretability of clinical classifications. Here, we introduce generative embedding for fMRI using a combination of dynamic causal models (DCMs) and SVMs. We propose a general procedure of DCM-based generative embedding for subject-wise classification, provide a concrete implementation, and suggest good-practice guidelines for unbiased application of generative embedding in the context of fMRI. We illustrate the utility of our approach by a clinical example in which we classify moderately aphasic patients and healthy controls using a DCM of thalamo-temporal regions during speech processing. Generative embedding achieves a near-perfect balanced classification accuracy of 98% and significantly outperforms conventional activation-based and correlation-based methods. This example demonstrates how disease states can be detected with very high accuracy and, at the same time, be interpreted mechanistically in terms of abnormalities in connectivity. We envisage that future applications of generative embedding may provide crucial advances in dissecting spectrum disorders into physiologically more well-defined subgroups.

Segregated encoding of reward-identity and stimulus-reward associations in human orbitofrontal cortex.

A dominant focus in studies of learning and decision-making is the neural coding of scalar reward value. This emphasis ignores the fact that choices are strongly shaped by a rich representation of potential rewards. Here, using fMRI adaptation, we demonstrate that responses in the human orbitofrontal cortex (OFC) encode a representation of the specific type of food reward predicted by a visual cue. By controlling for value across rewards and by linking each reward with two distinct stimuli, we could test for representations of reward–identity that were independent of associative information. Our results show reward–identity representations in a medial-caudal region of OFC, independent of the associated predictive stimulus. This contrasts with a more rostro-lateral OFC region encoding reward–identity representations tied to the predicate stimulus. This demonstration of adaptation in OFC to reward specific representations opens an avenue for investigation of more complex decision mechanisms that are not immediately accessible in standard analyses, which focus on correlates of average activity.

Uncertainty in perception and the Hierarchical Gaussian Filter

In its full sense, perception rests on an agents model of how its sensory input comes about and the inferences it draws based on this model. These inferences are necessarily uncertain. Here, we illustrate how the Hierarchical Gaussian Filter (HGF) offers a principled and generic way to deal with the several forms that uncertainty in perception takes. The HGF is a recent derivation of one-step update equations from Bayesian principles that rests on a hierarchical generative model of the environment and its (in)stability. It is computationally highly efficient, allows for online estimates of hidden states, and has found numerous applications to experimental data from human subjects. In this paper, we generalize previous descriptions of the HGF and its account of perceptual uncertainty. First, we explicitly formulate the extension of the HGFs hierarchy to any number of levels; second, we discuss how various forms of uncertainty are accommodated by the minimization of variational free energy as encoded in the update equations; third, we combine the HGF with decision models and demonstrate the inversion of this combination; finally, we report a simulation study that compared four optimization methods for inverting the HGF/decision model combination at different noise levels. These four methods (Nelder–Mead simplex algorithm, Gaussian process-based global optimization, variational Bayes and Markov chain Monte Carlo sampling) all performed well even under considerable noise, with variational Bayes offering the best combination of efficiency and informativeness of inference. Our results demonstrate that the HGF provides a principled, flexible, and efficient—but at the same time intuitive—framework for the resolution of perceptual uncertainty in behaving agents.

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.

Laminar activity in the hippocampus and entorhinal cortex related to novelty and episodic encoding

The ability to form long-term memories for novel events depends on information processing within the hippocampus (HC) and entorhinal cortex (EC). The HC–EC circuitry shows a quantitative segregation of anatomical directionality into different neuronal layers. Whereas superficial EC layers mainly project to dentate gyrus (DG), CA3 and apical CA1 layers, HC output is primarily sent from pyramidal CA1 layers and subiculum to deep EC layers. Here we utilize this directionality information by measuring encoding activity within HC/EC subregions with 7 T high resolution functional magnetic resonance imaging (fMRI). Multivariate Bayes decoding within HC/EC subregions shows that processing of novel information most strongly engages the input structures (superficial EC and DG/CA2–3), whereas subsequent memory is more dependent on activation of output regions (deep EC and pyramidal CA1). This suggests that while novelty processing is strongly related to HC–EC input pathways, the memory fate of a novel stimulus depends more on HC–EC output.

The Binormal Assumption on Precision-Recall Curves

The precision-recall curve (PRC) has become a widespread conceptual basis for assessing classification performance. The curve relates the positive predictive value of a classifier to its true positive rate and often provides a useful alternative to the well-known receiver operating characteristic (ROC). The empirical PRC, however, turns out to be a highly imprecise estimate of the true curve, especially in the case of a small sample size and class imbalance in favour of negative examples. Ironically, this situation tends to occur precisely in those applications where the curve would be most useful, e.g., in anomaly detection or information retrieval. Here, we propose to estimate the PRC on the basis of a simple distributional assumption about the decision values that generalizes the established binormal model for estimating smooth ROC curves. Using simulations, we show that our approach outperforms empirical estimates, and that an account of the class imbalance is crucial for obtaining unbiased PRC estimates.

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.