Invariant measure selection by noise. An example

We consider a deterministic system with two conserved quantities and infinity many invariant measures. However the systems possess a unique invariant measure when enough stochastic forcing and balancing dissipation are added. We then show that as the forcing and dissipation are removed a unique limit of the deterministic system is selected. The exact structure of the limiting measure depends on the specifics of the stochastic forcing.

Regularity of invariant densities for 1D systems with random switching

© 2015 IOP Publishing Ltd & London Mathematical Society.This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove the smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points.

Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

Morphologically cryptic biological species within the liverwort Frullania asagrayana.

UNLABELLED: PREMISE OF THE STUDY: The Frullania tamarisci complex includes eight Holarctic liverwort species. One of these, F. asagrayana, is distributed broadly throughout eastern North America from Canada to the Gulf Coast. Preliminary genetic data suggested that the species includes two groups of populations. This study was designed to test whether the two groups are reproductively isolated biological species. • METHODS: Eighty-eight samples from across the range of F. asagrayana, plus 73 samples from one population, were genotyped for 13 microsatellite loci. Sequences for two plastid loci and nrITS were obtained from 13 accessions. Genetic data were analyzed using coalescent models and Bayesian inference. • KEY RESULTS: Frullania asagrayana is sequence-invariant at the two plastid loci and ITS2, but two clear groups were resolved by microsatellites. The two groups are largely reproductively isolated, but there is a low level of gene flow from the southern to the northern group. No gene flow was detected in the other direction. A local population was heterogeneous but displayed strong genetic structure. • CONCLUSIONS: The genetic structure of F. asagrayana in eastern North America reflects morphologically cryptic differentiation between reproductively isolated groups of populations, near-panmixis within groups, and clonal propagation at local scales. Reproductive isolation between groups that are invariant at the level of nucleotide sequences shows that caution must be exercised in making taxonomic and evolutionary inferences from reciprocal monophyly (or lack thereof) between putative species.

Spike avalanches exhibit universal dynamics across the sleep-wake cycle.

BACKGROUND: Scale-invariant neuronal avalanches have been observed in cell cultures and slices as well as anesthetized and awake brains, suggesting that the brain operates near criticality, i.e. within a narrow margin between avalanche propagation and extinction. In theory, criticality provides many desirable features for the behaving brain, optimizing computational capabilities, information transmission, sensitivity to sensory stimuli and size of memory repertoires. However, a thorough characterization of neuronal avalanches in freely-behaving (FB) animals is still missing, thus raising doubts about their relevance for brain function. METHODOLOGY/PRINCIPAL FINDINGS: To address this issue, we employed chronically implanted multielectrode arrays (MEA) to record avalanches of action potentials (spikes) from the cerebral cortex and hippocampus of 14 rats, as they spontaneously traversed the wake-sleep cycle, explored novel objects or were subjected to anesthesia (AN). We then modeled spike avalanches to evaluate the impact of sparse MEA sampling on their statistics. We found that the size distribution of spike avalanches are well fit by lognormal distributions in FB animals, and by truncated power laws in the AN group. FB data surrogation markedly decreases the tail of the distribution, i.e. spike shuffling destroys the largest avalanches. The FB data are also characterized by multiple key features compatible with criticality in the temporal domain, such as 1/f spectra and long-term correlations as measured by detrended fluctuation analysis. These signatures are very stable across waking, slow-wave sleep and rapid-eye-movement sleep, but collapse during anesthesia. Likewise, waiting time distributions obey a single scaling function during all natural behavioral states, but not during anesthesia. Results are equivalent for neuronal ensembles recorded from visual and tactile areas of the cerebral cortex, as well as the hippocampus. CONCLUSIONS/SIGNIFICANCE: Altogether, the data provide a comprehensive link between behavior and brain criticality, revealing a unique scale-invariant regime of spike avalanches across all major behaviors.

Emotion-attention network interactions during a visual oddball task.

Emotional and attentional functions are known to be distributed along ventral and dorsal networks in the brain, respectively. However, the interactions between these systems remain to be specified. The present study used event-related functional magnetic resonance imaging (fMRI) to investigate how attentional focus can modulate the neural activity elicited by scenes that vary in emotional content. In a visual oddball task, aversive and neutral scenes were presented intermittently among circles and squares. The squares were frequent standard events, whereas the other novel stimulus categories occurred rarely. One experimental group [N=10] was instructed to count the circles, whereas another group [N=12] counted the emotional scenes. A main effect of emotion was found in the amygdala (AMG) and ventral frontotemporal cortices. In these regions, activation was significantly greater for emotional than neutral stimuli but was invariant to attentional focus. A main effect of attentional focus was found in dorsal frontoparietal cortices, whose activity signaled task-relevant target events irrespective of emotional content. The only brain region that was sensitive to both emotion and attentional focus was the anterior cingulate gyrus (ACG). When circles were task-relevant, the ACG responded equally to circle targets and distracting emotional scenes. The ACG response to emotional scenes increased when they were task-relevant, and the response to circles concomitantly decreased. These findings support and extend prominent network theories of emotion-attention interactions that highlight the integrative role played by the anterior cingulate.

The abelianization of the Johnson kernel

We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym H 1(Tg,C)-module when g ≥ 6. We do this by proving that, for a finitely generated group G satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of G. In this setup, we also obtain a precise nilpotence test. © European Mathematical Society 2014.

Automatic verification of linear controller software

© 2015 IEEE.We consider the problem of verification of software implementations of linear time-invariant controllers. Commonly, different implementations use different representations of the controller's state, for example due to optimizations in a third-party code generator. To accommodate this variation, we exploit input-output controller specification captured by the controller's transfer function and show how to automatically verify correctness of C code controller implementations using a Frama-C/Why3/Z3 toolchain. Scalability of the approach is evaluated using randomly generated controller specifications of realistic size.

WormSizer: high-throughput analysis of nematode size and shape.

The fundamental phenotypes of growth rate, size and morphology are the result of complex interactions between genotype and environment. We developed a high-throughput software application, WormSizer, which computes size and shape of nematodes from brightfield images. Existing methods for estimating volume either coarsely model the nematode as a cylinder or assume the worm shape or opacity is invariant. Our estimate is more robust to changes in morphology or optical density as it only assumes radial symmetry. This open source software is written as a plugin for the well-known image-processing framework Fiji/ImageJ. It may therefore be extended easily. We evaluated the technical performance of this framework, and we used it to analyze growth and shape of several canonical Caenorhabditis elegans mutants in a developmental time series. We confirm quantitatively that a Dumpy (Dpy) mutant is short and fat and that a Long (Lon) mutant is long and thin. We show that daf-2 insulin-like receptor mutants are larger than wild-type upon hatching but grow slow, and WormSizer can distinguish dauer larvae from normal larvae. We also show that a Small (Sma) mutant is actually smaller than wild-type at all stages of larval development. WormSizer works with Uncoordinated (Unc) and Roller (Rol) mutants as well, indicating that it can be used with mutants despite behavioral phenotypes. We used our complete data set to perform a power analysis, giving users a sense of how many images are needed to detect different effect sizes. Our analysis confirms and extends on existing phenotypic characterization of well-characterized mutants, demonstrating the utility and robustness of WormSizer.