Cryo-electron microscopy provides 2D projection images of the 3D electron scattering intensity of many instances of the particle under study (e.g., a virus). Both symmetry (rotational point groups) and heterogeneity are important aspects of biological particles and both aspects can be combined by describing the electron scattering intensity of the particle as a stochastic process with a symmetric probability law and, therefore, symmetric moments. A maximum likelihood estimator implemented by an expectation-maximization algorithm is described, which estimates the unknown statistics of the electron scattering intensity stochastic process from the images of instances of the particle. The algorithm is demonstrated on the bacteriophage HK97 and the virus [Formula: see text]. The results are contrasted with the existing algorithms, which assume that each instance of the particle has the symmetry rather than the less restrictive assumption that the probability law has the symmetry.

Cryo electron microscopy leads to 3-D image reconstruction problems. The data is one projection of each of many different instances of the object. When the object has symmetry and is also heterogeneous, it is natural to describe the object as stochastic with symmetrical statistics. This paper describes methods for incorporating the symmetry into expectation-maximization algorithms for the maximum likelihood solution of the reconstruction problem.

Resting-state functional MRI (rs fMRI) is widely used to non-invasively study human brain networks. Network functional connectivity is estimated by calculating the standard correlation between blood-oxygen-level dependent (BOLD) signals in specific regions of interests (ROIs). However, standard correlation fails to characterize the causality and the direction of information flow between regions, which are important factors in characterizing a network. Here, we use causal linear time-invariant models, with the impulse response duration estimated by Information Criteria, to describe the effective connectivity between ROIs. To do so, we replace the standard correlation between BOLD signals with a correlation between a BOLD signal and a prediction via the model of that BOLD signal. Prediction correlation is then used in a network analysis similar to that used with standard correlation. Our results include the causality information, the direction of information flow, and the possibility of delays in information flow. This approach replicates the local and distributed network architecture of the human brain previously observed with standard correlations, as well as providing novel insight into the directed interactivity of regions comprising these networks.