Clustering is a powerful and commonly used technique that organizes and elucidates the structure of biological data. Clustering data from gene expression, metabolomics and proteomics experiments has proven to be useful at deriving a variety of insights, such as the shared regulation or function of biochemical components within networks. However, experimental measurements of biological processes are subject to substantial noise-stemming from both technical and biological variability-and most clustering algorithms are sensitive to this noise. In this article, we explore several methods of accounting for noise when analyzing biological data sets through clustering. Using a toy data set and two different case studies-gene expression and protein phosphorylation-we demonstrate the sensitivity of clustering algorithms to noise. Several methods of accounting for this noise can be used to establish when clustering results can be trusted. These methods span a range of assumptions about the statistical properties of the noise and can therefore be applied to virtually any biological data source.

Integrate-and-fire models of biological neurons combine differential equations with discrete spike events. In the simplest case, the reset of the neuronal voltage to its resting value is the only spike event. The response of such a model to constant input injection is limited to tonic spiking. We here study a generalized model in which two simple spike-induced currents are added. We show that this neuron exhibits not only tonic spiking at various frequencies but also the commonly observed neuronal bursting. Using analytical and numerical approaches, we show that this model can be reduced to a one-dimensional map of the adaptation variable and that this map is locally contractive over a broad set of parameter values. We derive a sufficient analytical condition on the parameters for the map to be globally contractive, in which case all orbits tend to a tonic spiking state determined by the fixed point of the return map. We then show that bursting is caused by a discontinuity in the return map, in which case the map is piecewise contractive. We perform a detailed analysis of a class of piecewise contractive maps that we call bursting maps and show that they robustly generate stable bursting behavior. To the best of our knowledge, this work is the first to point out the intimate connection between bursting dynamics and piecewise contractive maps. Finally, we discuss bifurcations in this return map, which cause transitions between spiking patterns.

A novel metric between linear dynamical systems, the alignment distance, was recently introduced, with promising results in many computer vision tasks. The computation of the alignment distance requires a minimization over the orthogonal group. In this paper, we present a fast and accurate Jacobi-type algorithm that solves this problem. Each step of the algorithm is equivalent to finding the roots of a quartic polynomial. We show that this rooting may be done efficiently and accurately using a careful implementation of Ferrari's classical closedform solution for quartic polynomials. For linear systems with orders that commonly arise in computer vision scenarios, our algorithm is roughly twenty times faster than a fast Riemannian gradient descent algorithm implementation and has comparable accuracy.