41 research outputs found
Mod-two cohomology of symmetric groups as a Hopf ring
We compute the mod-2 cohomology of the collection of all symmetric groups as
a Hopf ring, where the second product is the transfer product of Strickland and
Turner. We first give examples of related Hopf rings from invariant theory and
representation theory. In addition to a Hopf ring presentation, we give
geometric cocycle representatives and explicitly determine the structure as an
algebra over the Steenrod algebra. All calculations are explicit, with an
additive basis which has a clean graphical representation. We also briefly
develop related Hopf ring structures on rings of symmetric invariants and end
with a generating set consisting of Stiefel-Whitney classes of regular
representations v2. Added new results on varieties which represent the
cocycles, a graphical representation of the additive basis, and on the Steenrod
algebra action. v3. Included a full treatment of invariant theoretic Hopf
rings, refined the definition of representing varieties, and corrected and
clarified references.Comment: 31 pages, 6 figure
Clique topology reveals intrinsic geometric structure in neural correlations
Detecting meaningful structure in neural activity and connectivity data is
challenging in the presence of hidden nonlinearities, where traditional
eigenvalue-based methods may be misleading. We introduce a novel approach to
matrix analysis, called clique topology, that extracts features of the data
invariant under nonlinear monotone transformations. These features can be used
to detect both random and geometric structure, and depend only on the relative
ordering of matrix entries. We then analyzed the activity of pyramidal neurons
in rat hippocampus, recorded while the animal was exploring a two-dimensional
environment, and confirmed that our method is able to detect geometric
organization using only the intrinsic pattern of neural correlations.
Remarkably, we found similar results during non-spatial behaviors such as wheel
running and REM sleep. This suggests that the geometric structure of
correlations is shaped by the underlying hippocampal circuits, and is not
merely a consequence of position coding. We propose that clique topology is a
powerful new tool for matrix analysis in biological settings, where the
relationship of observed quantities to more meaningful variables is often
nonlinear and unknown.Comment: 29 pages, 4 figures, 13 supplementary figures (last two authors
contributed equally
Choosing Wavelet Methods, Filters, and Lengths for Functional Brain Network Construction
Wavelet methods are widely used to decompose fMRI, EEG, or MEG signals into
time series representing neurophysiological activity in fixed frequency bands.
Using these time series, one can estimate frequency-band specific functional
connectivity between sensors or regions of interest, and thereby construct
functional brain networks that can be examined from a graph theoretic
perspective. Despite their common use, however, practical guidelines for the
choice of wavelet method, filter, and length have remained largely
undelineated. Here, we explicitly explore the effects of wavelet method (MODWT
vs. DWT), wavelet filter (Daubechies Extremal Phase, Daubechies Least
Asymmetric, and Coiflet families), and wavelet length (2 to 24) - each
essential parameters in wavelet-based methods - on the estimated values of
network diagnostics and in their sensitivity to alterations in psychiatric
disease. We observe that the MODWT method produces less variable estimates than
the DWT method. We also observe that the length of the wavelet filter chosen
has a greater impact on the estimated values of network diagnostics than the
type of wavelet chosen. Furthermore, wavelet length impacts the sensitivity of
the method to detect differences between health and disease and tunes
classification accuracy. Collectively, our results suggest that the choice of
wavelet method and length significantly alters the reliability and sensitivity
of these methods in estimating values of network diagnostics drawn from graph
theory. They furthermore demonstrate the importance of reporting the choices
utilized in neuroimaging studies and support the utility of exploring wavelet
parameters to maximize classification accuracy in the development of biomarkers
of psychiatric disease and neurological disorders.Comment: working pape