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One of the main challenges in computational neuroscience is to map the ways in which information flows through the networks of the brain. We present an approach to this challenge by tracing inter-regional entropy transmission within the murine visual cortex. In particular, we show that the entropy associated with the Fokker-Planck equation can be decomposed into two components, where one accounts for inter-regional flow and the other describes a diffusive spread. Following a proof of principle simulation using synthetic data, we apply this entropy decomposition to calcium imaging data collected in the murine visual cortex, revealing distinct patterns of information flow across cortical regions. We show that there is a consistent lateral redistribution of information from central to peripheral areas and a posterior-medial flow of processed information from frontal regions to specialized sensory areas. These findings offer insight into the structured propagation of neural information, contributing to our understanding of the functional architecture of the murine visual cortex.
In Friston et al. ((2002) Neuroimage 16: 465–483) we introduced empirical Bayes as a potentially useful way to estimate and make inferences about effects in hierarchical models. In this paper we present a series of models that exemplify the diversity of problems that can be addressed within this framework. In hierarchical linear observation models, both classical and empirical Bayesian approaches can be framed in terms of covariance component estimation (e.g., variance partitioning). To illustrate the use of the expectation–maximization (EM) algorithm in covariance component estimation we focus first on two important problems in fMRI: nonsphericity induced by (i) serial or temporal correlations among errors and (ii) variance components caused by the hierarchical nature of multisubject studies. In hierarchical observation models, variance components at higher levels can be used as constraints on the parameter estimates of lower levels. This enables the use of parametric empirical Bayesian (PEB) estimators, as distinct from classical maximum likelihood (ML) estimates. We develop this distinction to address: (i) The difference between response estimates based on ML and the conditional means from a Bayesian approach and the implications for estimates of intersubject variability. (ii) The relationship between fixed- and random-effect analyses. (iii) The specificity and sensitivity of Bayesian inference and, finally, (iv) the relative importance of the number of scans and subjects. The forgoing is concerned with within- and between-subject variability in multisubject hierarchical fMRI studies. In the second half of this paper we turn to Bayesian inference at the first (within-voxel) level, using PET data to show how priors can be derived from the (between-voxel) distribution of activations over the brain. This application uses exactly the same ideas and formalism but, in this instance, the second level is provided by observations over voxels as opposed to subjects. The ensuing posterior probability maps (PPMs) have enhanced anatomical precision and greater face validity, in relation to underlying anatomy. Furthermore, in comparison to conventional SPMs they are not confounded by the multiple comparison problem that, in a classical context, dictates high thresholds and low sensitivity. We conclude with some general comments on Bayesian approaches to image analysis and on some unresolved issues.