1,501 research outputs found
Sampling Properties of the Spectrum and Coherency of Sequences of Action Potentials
The spectrum and coherency are useful quantities for characterizing the
temporal correlations and functional relations within and between point
processes. This paper begins with a review of these quantities, their
interpretation and how they may be estimated. A discussion of how to assess the
statistical significance of features in these measures is included. In
addition, new work is presented which builds on the framework established in
the review section. This work investigates how the estimates and their error
bars are modified by finite sample sizes. Finite sample corrections are derived
based on a doubly stochastic inhomogeneous Poisson process model in which the
rate functions are drawn from a low variance Gaussian process. It is found
that, in contrast to continuous processes, the variance of the estimators
cannot be reduced by smoothing beyond a scale which is set by the number of
point events in the interval. Alternatively, the degrees of freedom of the
estimators can be thought of as bounded from above by the expected number of
point events in the interval. Further new work describing and illustrating a
method for detecting the presence of a line in a point process spectrum is also
presented, corresponding to the detection of a periodic modulation of the
underlying rate. This work demonstrates that a known statistical test,
applicable to continuous processes, applies, with little modification, to point
process spectra, and is of utility in studying a point process driven by a
continuous stimulus. While the material discussed is of general applicability
to point processes attention will be confined to sequences of neuronal action
potentials (spike trains) which were the motivation for this work.Comment: 33 pages, 9 figure
Spectral estimation on a sphere in geophysics and cosmology
We address the problem of estimating the spherical-harmonic power spectrum of
a statistically isotropic scalar signal from noise-contaminated data on a
region of the unit sphere. Three different methods of spectral estimation are
considered: (i) the spherical analogue of the one-dimensional (1-D)
periodogram, (ii) the maximum likelihood method, and (iii) a spherical analogue
of the 1-D multitaper method. The periodogram exhibits strong spectral leakage,
especially for small regions of area , and is generally unsuitable
for spherical spectral analysis applications, just as it is in 1-D. The maximum
likelihood method is particularly useful in the case of nearly-whole-sphere
coverage, , and has been widely used in cosmology to estimate
the spectrum of the cosmic microwave background radiation from spacecraft
observations. The spherical multitaper method affords easy control over the
fundamental trade-off between spectral resolution and variance, and is easily
implemented regardless of the region size, requiring neither non-linear
iteration nor large-scale matrix inversion. As a result, the method is ideally
suited for most applications in geophysics, geodesy or planetary science, where
the objective is to obtain a spatially localized estimate of the spectrum of a
signal from noisy data within a pre-selected and typically small region.Comment: Submitted to the Geophysical Journal Internationa
Analysis of Dynamic Brain Imaging Data
Modern imaging techniques for probing brain function, including functional
Magnetic Resonance Imaging, intrinsic and extrinsic contrast optical imaging,
and magnetoencephalography, generate large data sets with complex content. In
this paper we develop appropriate techniques of analysis and visualization of
such imaging data, in order to separate the signal from the noise, as well as
to characterize the signal. The techniques developed fall into the general
category of multivariate time series analysis, and in particular we extensively
use the multitaper framework of spectral analysis. We develop specific
protocols for the analysis of fMRI, optical imaging and MEG data, and
illustrate the techniques by applications to real data sets generated by these
imaging modalities. In general, the analysis protocols involve two distinct
stages: `noise' characterization and suppression, and `signal' characterization
and visualization. An important general conclusion of our study is the utility
of a frequency-based representation, with short, moving analysis windows to
account for non-stationarity in the data. Of particular note are (a) the
development of a decomposition technique (`space-frequency singular value
decomposition') that is shown to be a useful means of characterizing the image
data, and (b) the development of an algorithm, based on multitaper methods, for
the removal of approximately periodic physiological artifacts arising from
cardiac and respiratory sources.Comment: 40 pages; 26 figures with subparts including 3 figures as .gif files.
Originally submitted to the neuro-sys archive which was never publicly
announced (was 9804003
Temporal structure in neuronal activity during working memory in Macaque parietal cortex
A number of cortical structures are reported to have elevated single unit
firing rates sustained throughout the memory period of a working memory task.
How the nervous system forms and maintains these memories is unknown but
reverberating neuronal network activity is thought to be important. We studied
the temporal structure of single unit (SU) activity and simultaneously recorded
local field potential (LFP) activity from area LIP in the inferior parietal
lobe of two awake macaques during a memory-saccade task. Using multitaper
techniques for spectral analysis, which play an important role in obtaining the
present results, we find elevations in spectral power in a 50--90 Hz (gamma)
frequency band during the memory period in both SU and LFP activity. The
activity is tuned to the direction of the saccade providing evidence for
temporal structure that codes for movement plans during working memory. We also
find SU and LFP activity are coherent during the memory period in the 50--90 Hz
gamma band and no consistent relation is present during simple fixation.
Finally, we find organized LFP activity in a 15--25 Hz frequency band that may
be related to movement execution and preparatory aspects of the task. Neuronal
activity could be used to control a neural prosthesis but SU activity can be
hard to isolate with cortical implants. As the LFP is easier to acquire than SU
activity, our finding of rich temporal structure in LFP activity related to
movement planning and execution may accelerate the development of this medical
application.Comment: Originally submitted to the neuro-sys archive which was never
publicly announced (was 0005002
Dynamic Decomposition of Spatiotemporal Neural Signals
Neural signals are characterized by rich temporal and spatiotemporal dynamics
that reflect the organization of cortical networks. Theoretical research has
shown how neural networks can operate at different dynamic ranges that
correspond to specific types of information processing. Here we present a data
analysis framework that uses a linearized model of these dynamic states in
order to decompose the measured neural signal into a series of components that
capture both rhythmic and non-rhythmic neural activity. The method is based on
stochastic differential equations and Gaussian process regression. Through
computer simulations and analysis of magnetoencephalographic data, we
demonstrate the efficacy of the method in identifying meaningful modulations of
oscillatory signals corrupted by structured temporal and spatiotemporal noise.
These results suggest that the method is particularly suitable for the analysis
and interpretation of complex temporal and spatiotemporal neural signals
Spectral analysis of stationary random bivariate signals
A novel approach towards the spectral analysis of stationary random bivariate
signals is proposed. Using the Quaternion Fourier Transform, we introduce a
quaternion-valued spectral representation of random bivariate signals seen as
complex-valued sequences. This makes possible the definition of a scalar
quaternion-valued spectral density for bivariate signals. This spectral density
can be meaningfully interpreted in terms of frequency-dependent polarization
attributes. A natural decomposition of any random bivariate signal in terms of
unpolarized and polarized components is introduced. Nonparametric spectral
density estimation is investigated, and we introduce the polarization
periodogram of a random bivariate signal. Numerical experiments support our
theoretical analysis, illustrating the relevance of the approach on synthetic
data.Comment: 11 pages, 3 figure
Slepian functions and their use in signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden,
Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla
Scalar and vector Slepian functions, spherical signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and, particularly for applications in the
geosciences, for scalar and vectorial signals defined on the surface of a unit
sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics,
edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be
published by Springer Verlag. This is a slightly modified but expanded
version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the
Handbook, when it was called: Slepian functions and their use in signal
estimation and spectral analysi
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