Revisiting Multi-Subject Random Effects in fMRI: Advocating Prevalence Estimation

Random Effects analysis has been introduced into fMRI research in order to generalize findings from the study group to the whole population. Generalizing findings is obviously harder than detecting activation in the study group since in order to be significant, an activation has to be larger than the inter-subject variability. Indeed, detected regions are smaller when using random effect analysis versus fixed effects. The statistical assumptions behind the classic random effects model are that the effect in each location is normally distributed over subjects, and "activation" refers to a non-null mean effect. We argue this model is unrealistic compared to the true population variability, where, due to functional plasticity and registration anomalies, at each brain location some of the subjects are active and some are not. We propose a finite-Gaussian--mixture--random-effect. A model that amortizes between-subject spatial disagreement and quantifies it using the "prevalence" of activation at each location. This measure has several desirable properties: (a) It is more informative than the typical active/inactive paradigm. (b) In contrast to the hypothesis testing approach (thus t-maps) which are trivially rejected for large sample sizes, the larger the sample size, the more informative the prevalence statistic becomes. In this work we present a formal definition and an estimation procedure of this prevalence. The end result of the proposed analysis is a map of the prevalence at locations with significant activation, highlighting activations regions that are common over many brains

From brain to earth and climate systems: Small-world interaction networks or not?

We consider recent reports on small-world topologies of interaction networks derived from the dynamics of spatially extended systems that are investigated in diverse scientific fields such as neurosciences, geophysics, or meteorology. With numerical simulations that mimic typical experimental situations we have identified an important constraint when characterizing such networks: indications of a small-world topology can be expected solely due to the spatial sampling of the system along with commonly used time series analysis based approaches to network characterization

Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.Comment: 23 pages. Together with arXiv:1305.6260, this version replaces the old. The main results have been strengthened and an earlier error in the statement corrected. To appear in J. Theoret. Proba