Extracting Information from the Power Spectrum of Synaptic Noise

In cortical neurons, synaptic "noise" is caused by the nearly random release of thousands of synapses. Few methods are presently available to analyze synaptic noise and deduce properties of the underlying synaptic inputs. We focus here on the power spectral density (PSD) of several models of synaptic noise. We examine different classes of analytically solvable kinetic models for synaptic currents, such as the "delta kinetic models," which use Dirac delta functions to represent the activation of the ion channel. We first show that, for this class of kinetic models, one can obtain an analytic expression for the PSD of the total synaptic conductance and derive equivalent stochastic models with only a few variables. This yields a method for constraining models of synaptic currents by analyzing voltage-clamp recordings of synaptic noise. Second, we show that a similar approach can be followed for the PSD of the the membrane potential (Vm) through an effective-leak approximation. Third, we show that this approach is also valid for inputs distributed in dendrites. In this case, the frequency scaling of the Vm PSD is preserved, suggesting that this approach may be applied to intracellular recordings of real neurons. In conclusion, using simple mathematical tools, we show that Vm recordings can be used to constrain kinetic models of synaptic currents, as well as to estimate equivalent stochastic models. This approach, therefore, provides a direct link between intracellular recordings in vivo and the design of models consistent with the dynamics and spectral structure of synaptic noise.     

How does environmental variation translate into biological processes?

Birth and death rates, as so many other biological processes, are usually not linearly related to environmental variation. Common examples of non-linear response forms include unimodal "optimum-type" responses and various saturating responses. These responses filter the signal coming from the environment to a corresponding biological process. We explored how different types of environmental signal may be transformed to a biological process. We were interested in the effect of the filter on modulation of (1) the variance of the signal, on (2) the variance-covariance structure between the signal and the filtered signal, and on (3) the match between the power spectra of the signal and the filtered signal. We found that the filters will change the frequency distribution (mean, variance, modality) of the signal. Especially symmetric filters that have a single peak of optimum will change signal structure so that there either exists or does not exist a correlation between the signal and the filtered signal. When the correlation exists it may be either positive or negative depending on the signal's mode relative to the filter structure. Also, the power spectrum properties of the signal may be dramatically transformed after passing the filter, e.g., blue noise can turn into red noise. Our results strongly suggest that studies on the influence of external signals on biological processes, such as population dynamics, should explicitly consider how the signal is transferred to biological processes.     

Stationary gene frequency distribution in the environment fluctuating between two distinct states

Summary A general method is given to obtain a stationary distribution in a stochastic one-dimensional dynamical system in which an environmental parameter specifying the dynamical system is a stationary Markov process with only two states. By applying this method, the exact stationary gene frequency distribution is obtained for a genic selection model in the environment fluctuating between two distinct states. Several limiting stationary distributions are obtained therefrom, and one of them is shown to coincide with a stationary solution of the diffusion equation heuristically derived by us for more general cases. Discussion is given on the relationship between the diffusion equations obtained by various authors starting from discrete, non-overlapping generation models.     

A Fourier characteristic of coding sequences: origins and a non-Fourier approximation.

The 3-base periodicity, identified as a pronounced peak at the frequency N/3 (N is the length of the DNA sequence) of the Fourier power spectrum of protein coding regions, is used as a marker in gene-finding algorithms to distinguish protein coding regions (exons) and noncoding regions (introns) of genomes. In this paper, we reveal the explanation of this phenomenon which results from a nonuniform distribution of nucleotides in the three coding positions. There is a linear correlation between the nucleotide distributions in the three codon positions and the power spectrum at the frequency N/3. Furthermore, this study indicates the relationship between the length of a DNA sequence and the variance of nucleotide distributions and the average Fourier power spectrum, which is the noise signal in gene-finding methods. The results presented in this paper provide an efficient way to compute the Fourier power spectrum at N/3 and the noise signal in gene-finding methods by calculating the nucleotide distributions in the three codon positions.     

Filtering the surface EMG signal: Movement artifact and baseline noise contamination

The surface electromyographic (sEMG) signal that originates in the muscle is inevitably contaminated by various noise signals or artifacts that originate at the skin-electrode interface, in the electronics that amplifies the signals, and in external sources. Modern technology is substantially immune to some of these noises, but not to the baseline noise and the movement artifact noise. These noise sources have frequency spectra that contaminate the low-frequency part of the sEMG frequency spectrum. There are many factors which must be taken into consideration when determining the appropriate filter specifications to remove these artifacts; they include the muscle tested and type of contraction, the sensor configuration, and specific noise source. The band-pass determination is always a compromise between (a) reducing noise and artifact contamination, and (b) preserving the desired information from the sEMG signal. This study was designed to investigate the effects of mechanical perturbations and noise that are typically encountered during sEMG recordings in clinical and related applications. The analysis established the relationship between the attenuation rates of the movement artifact and the sEMG signal as a function of the filter band pass. When this relationship is combined with other considerations related to the informational content of the signal, the signal distortion of filters, and the kinds of artifacts evaluated in this study, a Butterworth filter with a corner frequency of 20 Hz and a slope of 12 dB/oct is recommended for general use. The results of this study are relevant to biomechanical and clinical applications where the measurements of body dynamics and kinematics may include artifact sources.     

Applying hidden Markov models to the analysis of single ion channel activity

Hidden Markov models have recently been used to model single ion channel currents as recorded with the patch clamp technique from cell membranes. The estimation of hidden Markov models parameters using the forward-backward and Baum-Welch algorithms can be performed at signal to noise ratios that are too low for conventional single channel kinetic analysis; however, the application of these algorithms relies on the assumptions that the background noise be white and that the underlying state transitions occur at discrete times. To address these issues, we present an "H-noise" algorithm that accounts for correlated background noise and the randomness of sampling relative to transitions. We also discuss three issues that arise in the practical application of the algorithm in analyzing single channel data. First, we describe a digital inverse filter that removes the effects of the analog antialiasing filter and yields a sharp frequency roll-off. This enhances the performance while reducing the computational intensity of the algorithm. Second, the data may be contaminated with baseline drifts or deterministic interferences such as 60-Hz pickup. We propose an extension of previous results to consider baseline drift. Finally, we describe the extension of the algorithm to multiple data sets.     

On the Relationship of Steady States of Continuous and Discrete Models Arising from Biology

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.     

Optimal noise filtering in the chemotactic response of Escherichia coli

Information-carrying signals in the real world are often obscured by noise. A challenge for any system is to filter the signal from the corrupting noise. This task is particularly acute for the signal transduction network that mediates bacterial chemotaxis, because the signals are subtle, the noise arising from stochastic fluctuations is substantial, and the system is effectively acting as a differentiator which amplifies noise. Here, we investigated the filtering properties of this biological system. Through simulation, we first show that the cutoff frequency has a dramatic effect on the chemotactic efficiency of the cell. Then, using a mathematical model to describe the signal, noise, and system, we formulated and solved an optimal filtering problem to determine the cutoff frequency that bests separates the low-frequency signal from the high-frequency noise. There was good agreement between the theory, simulations, and published experimental data. Finally, we propose that an elegant implementation of the optimal filter in combination with a differentiator can be achieved via an integral control system. This paper furnishes a simple quantitative framework for interpreting many of the key notions about bacterial chemotaxis, and, more generally, it highlights the constraints on biological systems imposed by noise.     

Protein dynamics and 1/f noise.

It has long been recognized that protein dynamical processes occur over a wide temporal range. However, the functionality of this spectrum of events remains unclear. In this work, a generalized noise function analysis is applied to a collection of diverse protein dynamical systems. It is shown that a power law model with an oscillatory component can adequately describe the time course of a variety of processes. These results suggest that under the appropriate conditions, proteins are in a metastable state. A microscopic, chemical kinetic model based on a Poisson distribution of activation energies is presented. From this model specific functional forms for the parameters of the generalized noise model can be derived. Additionally, a model is presented to described kinetic hole burning effects observed at low temperatures. Scaling laws are derived for these models that provide a connection with the generalized noise analysis.     

Decreasing motion artifacts in calcium-dependent fluorescence transients from the perfused mouse heart using frequency filtering

A strategy has been developed for the removal of motion artifact and noise in calcium-dependent fluorescence transients from the perfused mouse heart using frequency filtering. An analytical model indicates that the spectral removal of motion artifacts is independent of the phase shift of the motion waveform in the frequency domain, and thus to the time shift (or delay) of motion in the time domain. This is based on the "shift theorem" of Fourier analysis, which avoids erroneous correction of motion artifact when using the motion signal obtained using reflectance from the heart. Several major steps are adopted to implement this model for elimination of motion as well as detection noise from the fluorescence transient signals from the calcium-sensitive probe Rhod-2. These include (1) extracting the fluorescence calcium transient signal from the raw data by using power spectrum density (PSD) in the frequency domain by subtracting the motion recorded using the reflectance of excitation light, (2) digitally filtering out the random noise using multiple bandpass filters centralized at harmonic frequencies of the transients, and (3) extracting high frequency noise with a Gaussian Kernel filter method. The processed signal of transients acquired with excessive motion artifact is comparable to transients acquired with minimal motion obtained by immobilizing the heart against the detection window, demonstrating the usefulness of this technique.     

表面肌电信号采集与降噪处理

目的:本文以设计的表面肌电(sEMG)信号采集系统为基础,探讨sEMG信号中的降噪处理问题。方法:结合sEMG信号的噪声影响情况,首先利用带通滤波器消除肌电信号频带外噪声,再通过频谱插值法来抑制工频干扰分量,最后使用小波分析方法来削弱肌电信号频带内噪声。结果:通过对检测sEMG信号的降噪处理,信号噪声得到明显抑制。结论:所设计采集系统能够获得满意的sEMG信号检测效果,所采用降噪方法能够有效提高sEMG信号的质量。     

Electroencephalogram structure

The representation of an electroencephalogram as a convolution of harmonic functions having negative decrements with discrete (evenly discontinuous) white Gaussian noise was considered. A substantiation of this representation is given. It was numerically shown that the convolution of the decrement damping alpha-rhythm of one frequency with discrete white noise is a narrow-band chaos, an irregular spindle-shaped activity, which results in the appearance of close spectral components in the spectrum of power. The estimation of the autocorrelation function represented a damping cosinusoid shifted in phase depending of the factor of fading. The absolute value of convolution described an envelope of the output signal. The average duration of spindles decreased as the module of the decrement increased.     

Deep analysis of perception through dynamic structures that emerge in cortical activity from self-regulated noise

The statistical properties of the spontaneous background electrocorticogram (ECoG) were modeled, starting with random numbers, constraining the distributions, and identifying characteristic deviations from randomness in ECoG from subjects at rest and during intentional behaviors. The ECoG had been recorded through 8 × 8 arrays of 64 electrodes, from the surfaces of auditory, visual, or somatic cortices of 9 rabbits, and from the inferotemporal cortex of a human subject. Power spectral densities (PSD) in coordinates of log₁₀ power versus log₁₀ frequency of ECoG from subjects at rest usually conformed to noise in power-law distributions in a continuum. PSD of ECoG from active subjects usually deviated from noise in having peaks in log₁₀ power above the power-law line in various frequency bands. The analytic signals from the Hilbert transform after band pass filtering in the beta and gamma ranges revealed beats from interference among distributed frequencies in band pass filtered noise called Rayleigh noise. The beats were displayed as repetitive down spikes in log₁₀ analytic power. Repetition rates were proportional to filter bandwidths for all center frequencies. Resting ECoG often gave histograms of the magnitudes and intervals of down spikes that conformed to noise. Histograms from active ECoG often deviated from noise in Rayleigh distributions of down spike intervals by giving what are called Rice (Mathematical analysis of random noise—and appendixes—technical publications monograph B-1589. Bell Telephone Labs Inc., New York, 1950) distributions. Adding power to noise as signals at single frequencies simulated those deviations. The beats in dynamic theory are deemed essential for perception, by gating beta and gamma bursts at theta rates through enhancement of the cortical signal-to-noise ratio in exceptionally deep down spikes called null spikes.     

Sensory transduction in an insect mechanoreceptor: Linear and nonlinear properties

Mechanotransduction in the femoral tactile spine of the cockroach, Periplaneta americana, was examined as a function of displacement of the spine axially in its socket. Linear behaviour was analyzed by measurement of the frequency response function between displacement and action potential output using sinusoidal stimulation and random noise stimulation. The measured frequency response functions can be well fitted by a relationship which is a fractional power of complex frequency. This power was close to 0.5 for all experiments. To distinguish between the effects of nonlinearity and of inherent variability, the averaged responses of the preparation to repeated sequences of pseudorandom noise were compared to those from experiments in which continuous pseudorandom noise were used. The lack of sensitivity of the coherence function to these two methods of measurement suggests that mechanical stimuli are encoded into action potentials with a large signal-to-noise ratio. The low value of the coherence function which is characteristics of insect mechanoreceptors is therefore due to the strong non-linearity of their responses. To investigate the nonlinear properties of transduction, the second-order frequency response function of the tactile spine was measured for random noise stimulation experiments. Two models of the transduction process were considered in which a linear element with memory was cascaded with a nonlinear element without memory in the two possible configurations. Comparison of the experimental second-order frequency response functions with predictions based upon these two models and the measured first-order frequency response suggests that the transduction mechanism can be modelled by a linear element, which may be associated with the viscoelastic properties of the dendritic tubular body, and a zeromemory nonlinearity, which is most likely to be rectification by the dendritic membrane.     

Poor environmental tracking can make extinction risk insensitive to the colour of environmental noise

The relative importance of environmental colour for extinction risk compared with other aspects of environmental noise (mean and interannual variability) is poorly understood. Such knowledge is currently relevant, as climate change can cause the mean, variability and temporal autocorrelation of environmental variables to change. Here, we predict that the extinction risk of a shorebird population increases with the colour of a key environmental variable: winter temperature. However, the effect is weak compared with the impact of changes in the mean and interannual variability of temperature. Extinction risk was largely insensitive to noise colour, because demographic rates are poor in tracking the colour of the environment. We show that three mechanisms-which probably act in many species-can cause poor environmental tracking: (i) demographic rates that depend nonlinearly on environmental variables filter the noise colour, (ii) demographic rates typically depend on several environmental signals that do not change colour synchronously, and (iii) demographic stochasticity whitens the colour of demographic rates at low population size. We argue that the common practice of assuming perfect environmental tracking may result in overemphasizing the importance of noise colour for extinction risk. Consequently, ignoring environmental autocorrelation in population viability analysis could be less problematic than generally thought.     

Sturdy cycles in the chaotic Tribolium castaneum data series

Because of the inherent discreteness of individuals, population dynamical models must be discrete variable systems. In case of strong nonlinearity, such systems interacting with noise can generate a great variety of patterns from nearly periodic behavior through complex combination of nearly periodic and chaotic patterns to noisy chaotic time series. The interaction of a population consisting of discrete individuals and demographic noise has been analyzed in laboratory population data Henson et al. (Science 294 (2001) 602; Proc. Roy. Soc. Ser. B 270 (2003) 1549). In this paper we point out that some of the cycles are fragile, i.e. they are sensitive to the discretization algorithm and to small variation of the model parameters, while others remain "sturdy" against the perturbations. We introduce a statistical algorithm to detect disjoint, nearly-periodic patterns in data series. We show that only the sturdy cycles of the discrete variable models appear in the data series significantly. Our analysis identified the quasiperiodic 11-cycle (emerging in the continuous model) to be present significantly only in one of the three experimental data series. Numerical simulations confirm that cycles can be detected only if noise is smaller than a certain critical level and population dynamics display the largest variety of nearly-periodic patterns if they are on the border of "grey" and "noisy" regions, defined in Domokos and Scheuring (J. Theor. Biol. 227 (2004) 535).     

Sweeps in time: leveraging the joint distribution of branch lengths

Current methods of identifying positively selected regions in the genome are limited in two key ways: the underlying models cannot account for the timing of adaptive events and the comparison between models of selective sweeps and sequence data is generally made via simple summaries of genetic diversity. Here, we develop a tractable method of describing the effect of positive selection on the genealogical histories in the surrounding genome, explicitly modeling both the timing and context of an adaptive event. In addition, our framework allows us to go beyond analyzing polymorphism data via the site frequency spectrum or summaries thereof and instead leverage information contained in patterns of linked variants. Tests on both simulations and a human data example, as well as a comparison to SweepFinder2, show that even with very small sample sizes, our analytic framework has higher power to identify old selective sweeps and to correctly infer both the time and strength of selection. Finally, we derived the marginal distribution of genealogical branch lengths at a locus affected by selection acting at a linked site. This provides a much-needed link between our analytic understanding of the effects of sweeps on sequence variation and recent advances in simulation and heuristic inference procedures that allow researchers to examine the sequence of genealogical histories along the genome.     

表面肌电信号采集与降噪处理

目的：本文以设计的表面~g（sEMG）信号采集系统为基础,探讨sEMG信号中的降噪处理问题。方法：结合sEMG信号的噪声影响情况,首先利用带通滤波器消除肌电信号频带外噪声,再通过频谱插值法来抑制工频干扰分量,最后使用小波分析方法来削弱肌电信号频带内噪声。结果：通过对检测sEMG信号的降噪处理,信号噪声得到明显抑制。结论：所设计采集系统能够获得满意的sEMG信号检测效果,所采用降噪方法能够有效提高sEMG信号的质量。     

Signal/noise ratio at the visual recognition threshold of incomplete figures

We measured recognition thresholds of incomplete figure perception (the Gollin test). This test we regarded as a visual masking problem. Digital image processing permits us to measure the spatial properties and spatial frequency spectrum of the absent part of the image as the mask. Using a noise paradigm, we have measured the signal/noise ratio for Incomplete Figure. Recognition was worse with better spectral "similarity" between the figure and the "invisible" mask. At threshold, the spectrum of the fragmented image was equally similar to that of the "invisible" mask and complete image. We think the recognition thresholds for Gollin stimuli reflect the signal/noise ratio.