**MOTIVATION: **Modern population genetics studies typically involve genome-wide genotyping of individuals from a diverse network of ancestries. An important problem is how to formulate and estimate probabilistic models of observed genotypes that account for complex population structure. The most prominent work on this problem has focused on estimating a model of admixture proportions of ancestral populations for each individual. Here, we instead focus on modeling variation of the genotypes without requiring a higher-level admixture interpretation.

**RESULTS: **We formulate two general probabilistic models, and we propose computationally efficient algorithms to estimate them. First, we show how principal component analysis can be utilized to estimate a general model that includes the well-known Pritchard-Stephens-Donnelly admixture model as a special case. Noting some drawbacks of this approach, we introduce a new {\textquoteright}logistic factor analysis{\textquoteright} framework that seeks to directly model the logit transformation of probabilities underlying observed genotypes in terms of latent variables that capture population structure. We demonstrate these advances on data from the Human Genome Diversity Panel and 1000 Genomes Project, where we are able to identify SNPs that are highly differentiated with respect to structure while making minimal modeling assumptions.

**AVAILABILITY AND IMPLEMENTATION: **A Bioconductor R package called lfa is available at http://www.bioconductor.org/packages/release/bioc/html/lfa.html

**CONTACT: **jstorey@princeton.edu

**SUPPLEMENTARY INFORMATION: **Supplementary data are available at Bioinformatics online.

The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance.

}, keywords = {Algorithms, Animals, Brain, Entropy, Hot Temperature, Models, Neurological, Models, Statistical, Monte Carlo Method, Nerve Net, Neurons, Probability, Reproducibility of Results, Retina, Thermodynamics, Urodela}, issn = {1091-6490}, doi = {10.1073/pnas.1514188112}, author = {Tka{\v c}ik, Ga{\v s}per and Mora, Thierry and Marre, Olivier and Amodei, Dario and Palmer, Stephanie E and Berry, Michael J and Bialek, William} } @article {2951, title = {Searching for collective behavior in a large network of sensory neurons.}, journal = {PLoS Comput Biol}, volume = {10}, year = {2014}, month = {2014 Jan}, pages = {e1003408}, abstract = {Maximum entropy models are the least structured probability distributions that exactly reproduce a chosen set of statistics measured in an interacting network. Here we use this principle to construct probabilistic models which describe the correlated spiking activity of populations of up to 120 neurons in the salamander retina as it responds to natural movies. Already in groups as small as 10 neurons, interactions between spikes can no longer be regarded as small perturbations in an otherwise independent system; for 40 or more neurons pairwise interactions need to be supplemented by a global interaction that controls the distribution of synchrony in the population. Here we show that such "K-pairwise" models--being systematic extensions of the previously used pairwise Ising models--provide an excellent account of the data. We explore the properties of the neural vocabulary by: 1) estimating its entropy, which constrains the population{\textquoteright}s capacity to represent visual information; 2) classifying activity patterns into a small set of metastable collective modes; 3) showing that the neural codeword ensembles are extremely inhomogenous; 4) demonstrating that the state of individual neurons is highly predictable from the rest of the population, allowing the capacity for error correction.

}, keywords = {Action Potentials, Animals, Computational Biology, Entropy, Fishes, Models, Neurological, Movement, Nerve Net, Probability, Retina, Sensory Receptor Cells, Urodela}, issn = {1553-7358}, doi = {10.1371/journal.pcbi.1003408}, author = {Tka{\v c}ik, Ga{\v s}per and Marre, Olivier and Amodei, Dario and Schneidman, Elad and Bialek, William and Berry, Michael J} }