Noisy Threshold Research Articles

Analysis of Theoretical Bounds in Noisy Threshold Group Testing

The objective of this study was to describe a noisy threshold group testing model where positive and negative cases could occur depending on virus concentration in coronavirus disease 2019 (COVID-19) diagnosis with output results flipped due to measurement noise. We investigated lower bounds for successful reconstruction of a small set of defective samples in the noisy threshold group testing framework. To this end, using Fano’s inequality, we derived the minimum number of tests required to find unknown signals with defective samples. Our results showed that the minimum number of tests on probability of error for reconstruction of unknown signals was a function of the defective rate and noise probability. We obtained lower bounds for performance of the noisy threshold group testing framework with respect to noise intervals. In addition, the relationship between defective rate of signals and sparsity of group matrices to design optimal noisy threshold group testing systems is presented.

Theoretical Bounds on the Number of Tests in Noisy Threshold Group Testing Frameworks

We consider a variant of group testing (GT) models called noisy threshold group testing (NTGT), in which when there is more than one defective sample in a pool, its test result is positive. We deal with a variant model of GT where, as in the diagnosis of COVID-19 infection, if the virus concentration does not reach a threshold, not only do false positives and false negatives occur, but also unexpected measurement noise can reverse a correct result over the threshold to become incorrect. We aim to determine how many tests are needed to reconstruct a small set of defective samples in this kind of NTGT problem. To this end, we find the necessary and sufficient conditions for the number of tests required in order to reconstruct all defective samples. First, Fano’s inequality was used to derive a lower bound on the number of tests needed to meet the necessary condition. Second, an upper bound was found using a MAP decoding method that leads to giving the sufficient condition for reconstructing defective samples in the NTGT problem. As a result, we show that the necessary and sufficient conditions for the successful reconstruction of defective samples in NTGT coincide with each other. In addition, we show a trade-off between the defective rate of the samples and the density of the group matrix which is then used to construct an optimal NTGT framework.

Pair approximation for the noisy threshold q-voter model.

In the standard q-voter model, a given agent can change its opinion only if there is a full consensus of the opposite opinion within a group of influence of size q. A more realistic extension is the threshold q voter, where a minimal agreement (at least 0<q_{0}≤q opposite opinions) is sufficient to flip the central agent's opinion, including also the possibility of independent (nonconformist) choices. Variants of this model including nonconformist behavior have been previously studied in fully connected networks (mean-field limit). Here we investigate its dynamics in random networks. Particularly, while in the mean-field case it is irrelevant whether repetitions in the influence group are allowed, we show that this is not the case in networks, and we study the impact of both cases, with or without repetition. Furthermore, the results of computer simulations are compared with the predictions of the pair approximation derived for uncorrelated networks of arbitrary degree distributions.

Gaussian Process Classification Using Posterior Linearization

This paper proposes a new algorithm for Gaussian process classification based on posterior linearisation (PL). In PL, a Gaussian approximation to the posterior density is obtained iteratively using the best possible linearisation of the conditional mean of the labels and accounting for the linearisation error. PL has some theoretical advantages over expectation propagation (EP): all calculated covariance matrices are positive definite and there is a local convergence theorem. In experimental data, PL has better performance than EP with the noisy threshold likelihood and the parallel implementation of the algorithms.

How does transient signaling input affect the spike timing of postsynaptic neuron near the threshold regime: an analytical study

The noisy threshold regime, where even a small set of presynaptic neurons can significantly affect postsynaptic spike-timing, is suggested as a key requisite for computation in neurons with high variability. It also has been proposed that signals under the noisy conditions are successfully transferred by a few strong synapses and/or by an assembly of nearly synchronous synaptic activities. We analytically investigate the impact of a transient signaling input on a leaky integrate-and-fire postsynaptic neuron that receives background noise near the threshold regime. The signaling input models a single strong synapse or a set of synchronous synapses, while the background noise represents a lot of weak synapses. We find an analytic solution that explains how the first-passage time (ISI) density is changed by transient signaling input. The analysis allows us to connect properties of the signaling input like spike timing and amplitude with postsynaptic first-passage time density in a noisy environment. Based on the analytic solution, we calculate the Fisher information with respect to the signaling input’s amplitude. For a wide range of amplitudes, we observe a non-monotonic behavior for the Fisher information as a function of background noise. Moreover, Fisher information non-trivially depends on the signaling input’s amplitude; changing the amplitude, we observe one maximum in the high level of the background noise. The single maximum splits into two maximums in the low noise regime. This finding demonstrates the benefit of the analytic solution in investigating signal transfer by neurons.

Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model.

Providing an analytical treatment to the stochastic feature of neurons' dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrate-and-fire model and its associated Fokker-Planck equation are probably the most popular way to deal with neural variability. Another well-known formalism is the escape-rate model: a model giving the probability that a neuron fires at a certain time knowing the time elapsed since its last action potential. This model leads to a so-called age-structured system, a partial differential equation with non-local boundary condition famous in the field of population dynamics, where the age of a neuron is the amount of time passed by since its previous spike. In this theoretical paper, we investigate the mathematical connection between the two formalisms. We shall derive an integral transform of the solution to the age-structured model into the solution of the Fokker-Planck equation. This integral transform highlights the link between the two stochastic processes. As far as we know, an explicit mathematical correspondence between the two solutions has not been introduced until now.

Fluctuation spectra of weakly driven nonlinear systems

We show that in periodically driven systems, along with the delta-peak at the driving frequency, the spectral density of fluctuations displays extra features. These can be peaks or dips with height quadratic in the driving amplitude, for weak driving. For systems where inertial effects can be disregarded, the peaks/dips are generally located at zero frequency and at the driving frequency. The shape and intensity of the spectra very sensitively depend on the parameters of the system dynamics. To illustrate this sensitivity and the generality of the effect, we study three types of systems: an overdamped Brownian particle (e.g., an optically trapped particle), a two-state system that switches between the states at random, and a noisy threshold detector. The analytical results are in excellent agreement with numerical simulations.

Neuronal spike-train responses in the presence of threshold noise

The variability of neuronal firing has been an intense topic of study for many years. From a modelling perspective it has often been studied in conductance based spiking models with the use of additive or multiplicative noise terms to represent channel fluctuations or the stochastic nature of neurotransmitter release. Here we propose an alternative approach using a simple leaky integrate-and-fire model with a noisy threshold. Initially, we develop a mathematical treatment of the neuronal response to periodic forcing using tools from linear response theory and use this to highlight how a noisy threshold can enhance downstream signal reconstruction. We further develop a more general framework for understanding the responses to large amplitude forcing based on a calculation of first passage times. This is ideally suited to understanding stochastic mode-locking, for which we numerically determine the Arnol'd tongue structure. An examination of data from regularly firing stellate neurons within the ventral cochlear nucleus, responding to sinusoidally amplitude modulated pure tones, shows tongue structures consistent with these predictions and highlights that stochastic, as opposed to deterministic, mode-locking is utilised at the level of the single stellate cell to faithfully encode periodic stimuli.

The ghost of stochastic resonance: an introductory review

Nonlinear systems driven by noise and periodic forces with more than one frequency exhibit the phenomenon of Ghost Stochastic Resonance (GSR) found in a wide and disparate variety of fields ranging from biology to geophysics. The common novel feature is the emergence of a ‘ghost’ frequency in the system's output which is absent in the input. As reviewed here, the uncovering of this phenomenon helped to understand a range of problems, from the perception of pitch in complex sounds or visual stimuli, to the explanation of climate cycles. Recent theoretical efforts show that a simple mechanism with two ingredients are at work in all these observations. The first one is the linear interference between the periodic inputs and the second a nonlinear detection of the largest constructive interferences, involving a noisy threshold. These notes are dedicated to review the main aspects of this phenomenon, as well as its different manifestations described on a bewildering variety of systems ranging from neurons, semiconductor lasers, electronic circuits to models of glacial climate cycles.

Design and Optimization of Degree Distributions of Irregular LDPC

摘要: 一对好的度分布可以有效降低LDPC的错误平层和编译码复杂度，在AWGN信道下，通过高斯近似分析方法可近似计算给定度分布的LDPC译码门限，利用差分进化算法可优化度分布以获得具有最大门限的度分布，仿真结果表明获得的度分布的译码门限比线性算法优化结果要好0.15dB左右。 关键词: LDPC / 差分进化 / 高斯近似 / 门限值 / 度分布

Gene regulation in the intraerythrocytic cycle of Plasmodium falciparum

To date, there is little knowledge about one of the processes fundamental to the biology of Plasmodium falciparum, gene regulation including transcriptional control. We use noisy threshold models to identify regulatory sequence elements explaining membership to a gene expression cluster where each cluster consists of genes active during the part of the developmental cycle inside a red blood cell. Our approach is both able to capture the combinatorial nature of gene regulation and to incorporate uncertainty about the functionality of putative regulatory sequence elements. We find a characteristic pattern where the most common motifs tend to be absent upstream of genes active in the first half of the cycle and present upstream of genes active in the second half. We find no evidence that motif's score, orientation, location and multiplicity improves prediction of gene expression. Through comparative genome analysis, we find a list of potential transcription factors and their associated motifs. Supplementary data are available at Bioinformatics online.

Excitation of coherent oscillations in a noisy medium

We numerically study the influence of neuronal threshold modulation on the properties of cortical traveling waves. For that reason we simplify a Wilson-Cowan-type integrodifferential equation model of propagating neocortical activity to a spatially discrete version. Further we introduce a noisy threshold. Depending on the noise level we find different states of the network activity, ranging from asynchronous oscillations, traveling waves, to synchronous oscillations. Finally, we induce the transition between these different states by an external modulation.

Predicting spike timing of neocortical pyramidal neurons by simple threshold models

Neurons generate spikes reliably with millisecond precision if driven by a fluctuating current--is it then possible to predict the spike timing knowing the input? We determined parameters of an adapting threshold model using data recorded in vitro from 24 layer 5 pyramidal neurons from rat somatosensory cortex, stimulated intracellularly by a fluctuating current simulating synaptic bombardment in vivo. The model generates output spikes whenever the membrane voltage (a filtered version of the input current) reaches a dynamic threshold. We find that for input currents with large fluctuation amplitude, up to 75% of the spike times can be predicted with a precision of +/-2 ms. Some of the intrinsic neuronal unreliability can be accounted for by a noisy threshold mechanism. Our results suggest that, under random current injection into the soma, (i) neuronal behavior in the subthreshold regime can be well approximated by a simple linear filter; and (ii) most of the nonlinearities are captured by a simple threshold process.

Noise-Assisted Signal Reception in Threshold Neuron

In certain cases, noises can improve signal transmission or signal processing. This phenomenon is the so-called stochastic resonance. In this paper, we firstly present two theorems to prove that the noisy threshold neuron shows stochastic resonance in terms of the probability of correct reception. Secondly, we analytically discuss stochastic resonance effects and give the probability-optimal noise levels for four representative noises. Finally, we discuss the stochastic gradient ascent learning law, which can be used to find the probability-optimal noise levels. We also present our simulation results for the four representative noises. These results indicate that stochastic resonance is favorable both in biological neurons and in signal processing.

Optimal information transmission in nonlinear arrays through suprathreshold stochastic resonance

We examine the optimal threshold distribution in populations of noisy threshold devices. When the noise on each threshold is independent, and sufficiently large, the optimal thresholds are realized by the suprathreshold stochastic resonance effect, in which case all threshold devices are identical. This result has relevance for neural population coding, as such noisy threshold devices model the key dynamics of nerve fibres. It is also relevant to quantization and lossy source coding theory, since the model provides a form of stochastic signal quantization. Furthermore, it is shown that a bifurcation pattern appears in the optimal threshold distribution as the noise intensity increases. Fisher information is used to demonstrate that the optimal threshold distribution remains in the suprathreshold stochastic resonance configuration as the population size approaches infinity.

Directed percolation in a two-dimensional stochastic fire-diffuse-fire model

We establish, from extensive numerical experiments, that the two dimensional stochastic fire-diffuse-fire model belongs to the directed percolation universality class. This model is an idealized model of intracellular calcium release that retains both the discrete nature of calcium stores and the stochastic nature of release. It is formed from an array of noisy threshold elements that are coupled only by a diffusing signal. The model supports spontaneous release events that can merge to form spreading circular and spiral waves of activity. The critical level of noise required for the system to exhibit a nonequilibrium phase transition between propagating and nonpropagating waves is obtained by an examination of the local slope delta (t) of the survival probability Pi(t) proportional exp [-delta(t)] for a wave to propagate for a time t .

Adaptive Stochastic Resonance in Noisy Neurons Based on Mutual Information

Noise can improve how memoryless neurons process signals and maximize their throughput information. Such favorable use of noise is the so-called "stochastic resonance" or SR effect at the level of threshold neurons and continuous neurons. This paper presents theoretical and simulation evidence that 1) lone noisy threshold and continuous neurons exhibit the SR effect in terms of the mutual information between random input and output sequences, 2) a new statistically robust learning law can find this entropy-optimal noise level, and 3) the adaptive SR effect is robust against highly impulsive noise with infinite variance. Histograms estimate the relevant probability density functions at each learning iteration. A theorem shows that almost all noise probability density functions produce some SR effect in threshold neurons even if the noise is impulsive and has infinite variance. The optimal noise level in threshold neurons also behaves nonlinearly as the input signal amplitude increases. Simulations further show that the SR effect persists for several sigmoidal neurons and for Gaussian radial-basis-function neurons.

Stochastic resonance in a sinusoidally forced LIF model with noisy threshold

In this report, the LIF neural model driven by underthreshold sinusoidal signals but with a gaussian-distributed noise on the threshold, is approximated by suitably defining an instantaneous firing (or escape) rate, which depends only on the momentary value of the voltage variable. This allows us to obtain, by analytically solving the relevant equations, the main statistical functions describing the “firing activity”; namely, the probability density function of firing phases and that of interspike intervals. From these functions two quantities can be derived, whose dependence on the noise intensity allows the Stochastic Resonance (SR) to be demonstrated. Besides the “regular” SR, the analysed system was found to produce, either for low frequencies and large amplitudes of modulation or for high modulation frequencies, resonance curves displaying two peaks. This bimodal feature of the resonance curves is accounted for on the basis of phase locked firing patterns.

Stochastic resonance in noisy threshold neurons

Stochastic resonance occurs when noise improves how a nonlinear system performs. This paper presents two general stochastic-resonance theorems for threshold neurons that process noisy Bernoulli input sequences. The performance measure is Shannon mutual information. The theorems show that small amounts of independent additive noise can increase the mutual information of threshold neurons if the neurons detect subthreshold signals. The first theorem shows that this stochastic-resonance effect holds for all finite-variance noise probability density functions that obey a simple mean constraint that the user can control. A corollary shows that this stochastic-resonance effect occurs for the important family of (right-sided) gamma noise. The second theorem shows that this effect holds for all infinite-variance noise types in the broad family of stable distributions. Stable bell curves can model extremely impulsive noise environments. So the second theorem shows that this stochastic-resonance effect is robust against violent fluctuations in the additive noise process.

Frequency and phase locking of noise-sustained oscillations in coupled excitable systems: array-enhanced resonances.

We study the interplay among noise, weak driving signal and coupling in excitable FitzHugh-Nagumo neurons. Due to coupling, noise-sustained oscillations become locked to the signal as functions of both signal frequency and noise intensity. Higher order m:n locking tongues and various array-enhanced resonance features are demonstrated. This resonance and locking behavior due to a time scale matching between noise-sustained oscillations and the signal is fundamentally different from stochastic resonance in usual noisy threshold elements.