Correlation functions quantify super-resolution images and estimate apparent clustering due to over-counting

We present an analytical method to quantify clustering in super-resolution localization images of static surfaces in two dimensions. The method also describes how over-counting of labeled molecules contributes to apparent self-clustering and how the effective lateral resolution of an image can be determined. This treatment applies to clustering of proteins and lipids in membranes, where there is significant interest in using super-resolution localization techniques to probe membrane heterogeneity. When images are quantified using pair correlation functions, the magnitude of apparent clustering due to over-counting will vary inversely with the surface density of labeled molecules and does not depend on the number of times an average molecule is counted. Over-counting does not yield apparent co-clustering in double label experiments when pair cross-correlation functions are measured. We apply our analytical method to quantify the distribution of the IgE receptor (Fc{\epsilon}RI) on the plasma membranes of chemically fixed RBL-2H3 mast cells from images acquired using stochastic optical reconstruction microscopy (STORM) and scanning electron microscopy (SEM). We find that apparent clustering of labeled IgE bound to Fc{\epsilon}RI detected with both methods arises from over-counting of individual complexes. Thus our results indicate that these receptors are randomly distributed within the resolution and sensitivity limits of these experiments.Comment: 22 pages, 5 figure

Zipf's Law in Short-Time Timbral Codings of Speech, Music, and Environmental Sound Signals

Timbre is a key perceptual feature that allows discrimination between different sounds. Timbral sensations are highly dependent on the temporal evolution of the power spectrum of an audio signal. In order to quantitatively characterize such sensations, the shape of the power spectrum has to be encoded in a way that preserves certain physical and perceptual properties. Therefore, it is common practice to encode short-time power spectra using psychoacoustical frequency scales. In this paper, we study and characterize the statistical properties of such encodings, here called timbral code-words. In particular, we report on rank-frequency distributions of timbral code-words extracted from 740 hours of audio coming from disparate sources such as speech, music, and environmental sounds. Analogously to text corpora, we find a heavy-tailed Zipfian distribution with exponent close to one. Importantly, this distribution is found independently of different encoding decisions and regardless of the audio source. Further analysis on the intrinsic characteristics of most and least frequent code-words reveals that the most frequent code-words tend to have a more homogeneous structure. We also find that speech and music databases have specific, distinctive code-words while, in the case of the environmental sounds, this database-specific code-words are not present. Finally, we find that a Yule-Simon process with memory provides a reasonable quantitative approximation for our data, suggesting the existence of a common simple generative mechanism for all considered sound sources