Superpolynomial growth in the number of attractors in Kauffman networks

The Kauffman model describes a particularly simple class of random Boolean networks. Despite the simplicity of the model, it exhibits complex behavior and has been suggested as a model for real world network problems. We introduce a novel approach to analyzing attractors in random Boolean networks, and applying it to Kauffman networks we prove that the average number of attractors grows faster than any power law with system size.     

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.     

Percolation and spreading of damage in a simplified Kauffman model

A simplified version of the Kauffman cellular automaton is introduced. As in the usual Kauffman model, there is a transition between a frozen phase and a chaotic phase where damage may spread. We associate the onset of chaos in this model with a percolation transition of certain rules occurring in the model. It seems to be in a different universality class from the usual Kauffman cellular automaton.     

Shipping noise in whale habitat: characteristics, sources, budget, and impact on belugas in Saguenay-St. Lawrence Marine Park hub

A continuous car ferry line crossing the Saguenay Fjord mouth and traffic from the local whale-watching fleet introduce high levels of shipping noise in the heart of the Saguenay-St. Lawrence Marine Park. To characterize this noise and examine its potential impact on belugas, a 4-hydrophone array was deployed in the area and continuously recorded for five weeks in May-June 2009. The source levels of the different vessel types showed little dependence on vessel size or speed increase. Their spectral range covered 33 dB. Lowest noise levels occurred at night, when ferry crossing pace was reduced, and daytime noise peaked during whale-watching tour departures and arrivals. Natural ambient noise prevailed 9.4% of the time. Ferry traffic added 30-35 dB to ambient levels above 1 kHz during crossings, which contributed 8 to 14 dB to hourly averages. The whale-watching fleet added up to 5.6 dB during peak hours. Assuming no behavioral or auditory compensation, half of the time, beluga potential communication range was reduced to less than ~30% of its expected value under natural noise conditions, and to less than ~15% for one quarter of the time, with little dependence on call frequency. The echolocation band for this population of belugas was also affected by the shipping noise.     

String Theory and the Kauffman Polynomial

We propose a new, precise integrality conjecture for the colored Kauffman polynomial of knots and links inspired by large N dualities and the structure of topological string theory on orientifolds. According to this conjecture, the natural knot invariant in an unoriented theory involves both the colored Kauffman polynomial and the colored HOMFLY polynomial for composite representations, i.e. it involves the full HOMFLY skein of the annulus. The conjecture sheds new light on the relationship between the Kauffman and the HOMFLY polynomials, and it implies for example Rudolph’s theorem. We provide various non-trivial tests of the conjecture and we sketch the string theory arguments that lead to it.     

Spatial correlations in permeability distributions due to extreme dynamics restructuring of unconsolidated sandstone

We propose a model for porous sandstone formation from unconsolidated sand based on a series of restructuring events where the local pressure difference due to flow in the sand is the largest. We investigate the local and global permeability distributions after steady state has been reached. Whereas we find no spatial correlations in the local permeability distribution, the distribution of inverse permeability shows spatial correlations consistent with a fractional Brownian noise characterized by a Hurst exponent of 0.88(9). The global permeability of the system shows time fluctuations as restructuring proceeds consistent with self-affinity characterized by a Hurst exponent of 0.25(3), crossing over to white noise at larger time scales.     

Mechanisms of stochastic phase locking

Periodically driven nonlinear oscillators can exhibit a form of phase locking in which a well-defined feature of the motion occurs near a preferred phase of the stimulus, but a random number of stimulus cycles are skipped between its occurrences. This feature may be an action potential, or another crossing by a state variable of some specific value. This behavior can also occur when no apparent external periodic forcing is present. The phase preference is then measured with respect to a time scale internal to the system. Models of these behaviors are briefly reviewed, and new mechanisms are presented that involve the coupling of noise to the equations of motion. Our study investigates such stochastic phase locking near bifurcations commonly present in models of biological oscillators: (1) a supercritical and (2) a subcritical Hopf bifurcation, and, under autonomous conditions, near (3) a saddle-node bifurcation, and (4) chaotic behavior. Our results complement previous studies of aperiodic phase locking in which noise perturbs deterministic phase-locked motion. In our study however, we emphasize how noise can induce a stochastic phase-locked motion that does not have a similar deterministic counterpart. Although our study focuses on models of excitable and bursting neurons, our results are applicable to other oscillators, such as those discussed in the respiratory and cardiac literatures. (c) 1995 American Institute of Physics.     

Stochastic Resonance and First Arrival Time for Excitable Systems

We study the noise induced thermally activated barrier crossing of Brownian particles that hop in a piecewise linear potential. Using the exact analytic solutions and via numerical simulations not only we explore the dependence for the first passage time of a single particle but also we calculate the first arrival time for one particle out of N particles. The first arrival time decreases as the number of particles increases as expected. We then explore the thermally activated barrier crossing rate of the system in the presence of time varying signal. The dependence of signal to noise ratio SNR as well as the power amplification (\(\eta \)) on model parameters is explored. \(\eta \) and SNR depict a pronounced peak at particular noise strength. In the presence of N particles, \(\eta \) is considerably amplified as N steps up showing the weak periodic signal plays a vital role in controlling the noise induced dynamics of the system. Moreover, for the sake of generality, the viscous friction \(\gamma \) is considered to decrease exponentially when the temperature T of the medium increases (\(\gamma =Be^{-A T}\)) as proposed originally by Reynolds (Philos Trans R Soc Lond 177:157, 1886).     

Noise reduction by recycling dynamically coupled time series

We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. We show that much of the information lost due to measurement noise in a target time series can be recovered with a noise reduction algorithm by crossing the time series with another time series with which it is dynamically coupled. The method is particularly useful for reduction of measurement noise in short length time series with high uncertainties.     

Stochastic Calculus: Application to Dynamic Bifurcations and Threshold Crossings

For the dynamic pitchfork bifurcation in the presence of white noise, the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter . The threshold crossing problem used, for example, to model the firing of a single cortical neuron is considered, concentrating on quantities that may be experimentally measurable but have so far received little attention. Expressions for the statistics of pre-threshold excursions, occupation density, and last crossing time of zero are compared with results from numerical generation of paths.     

A time series analysis and a non-homogeneous Poisson model with multiple change-points applied to acoustic data

High levels of the so-called community noise may produce hazardous effect on the health of a population exposed to them for large periods of time. Hence, the study of the behaviour of those noise measurements is very important. In this work we analyse that in terms of the probability of exceeding a given threshold level a certain number of times in a time interval of interest. Since the datasets considered contain missing measurements, we use a time series model to estimate the missing values and complete the datasets. Once the data is complete, we use a non-homogeneous Poisson model with multiple change-points to estimate the probability of interest. Estimation of the parameters of the models are made using the usual time series methodology as well as the Bayesian point of view via Markov chain Monte Carlo algorithms. The models are applied to data obtained from two measuring sites in Messina, Italy.     

Unstable periodic orbits and noise in chaos computing

Different methods to utilize the rich library of patterns and behaviors of a chaotic system have been proposed for doing computation or communication. Since a chaotic system is intrinsically unstable and its nearby orbits diverge exponentially from each other, special attention needs to be paid to the robustness against noise of chaos-based approaches to computation. In this paper unstable periodic orbits, which form the skeleton of any chaotic system, are employed to build a model for the chaotic system to measure the sensitivity of each orbit to noise, and to select the orbits whose symbolic representations are relatively robust against the existence of noise. Furthermore, since unstable periodic orbits are extractable from time series, periodic orbit-based models can be extracted from time series too. Chaos computing can be and has been implemented on different platforms, including biological systems. In biology noise is always present; as a result having a clear model for the effects of noise on any given biological implementation has profound importance. Also, since in biology it is hard to obtain exact dynamical equations of the system under study, the time series techniques we introduce here are of critical importance.     

Prediction of Time Series Gene Expression and Structural Analysis of Gene Regulatory Networks Using Recurrent Neural Networks

Methods for time series prediction and classification of gene regulatory networks (GRNs) from gene expression data have been treated separately so far. The recent emergence of attention-based recurrent neural network (RNN) models boosted the interpretability of RNN parameters, making them appealing for the understanding of gene interactions. In this work, we generated synthetic time series gene expression data from a range of archetypal GRNs and we relied on a dual attention RNN to predict the gene temporal dynamics. We show that the prediction is extremely accurate for GRNs with different architectures. Next, we focused on the attention mechanism of the RNN and, using tools from graph theory, we found that its graph properties allow one to hierarchically distinguish different architectures of the GRN. We show that the GRN responded differently to the addition of noise in the prediction by the RNN and we related the noise response to the analysis of the attention mechanism. In conclusion, this work provides a way to understand and exploit the attention mechanism of RNNs and it paves the way to RNN-based methods for time series prediction and inference of GRNs from gene expression data.     

What distinguishes individual stocks from the index?

Stochastic volatility models decompose the time series of financial returns into the product of a volatility factor and an iid noise factor. Assuming a slow dynamic for the volatility factor, we show via nonparametric tests that both the index as well as its individual stocks share a common volatility factor. While the noise component is Gaussian for the index, individual stock returns turn out to require a leptokurtic noise. Thus we propose a two-component model for stocks, given by the sum of Gaussian noise, which reflects market-wide fluctuations, and Laplacian noise, which incorporates firm-specific factors such as firm profitability or growth performance, both of which are known to be Laplacian distributed. In the case of purely Gaussian noise, the chi-squared probability for the density of individual stock returns is typically on the order of 10^-20, while it increases to values of O(1) by adding the Laplace component.     

表面催化反应模型中关联噪声诱导非平衡相变

建立含有关联噪声的双分子-单分子(DM)表面催化反应延迟反馈模型,该模型能同时显示一级和二级非平衡动力学相变,即在一级和二级非平衡动力学相变之间的反应窗口展现.讨论双分子在DM延迟反馈模型中两种吸附机制,即局域和随机吸附模型.研究结果表明:1)外部噪声及两噪声关联性致使反应窗口的宽度收缩;2)内部噪声对非平衡动力学相变行为的影响依赖两噪声关联性,即当两噪声负关联,内部噪声致使反应窗口的宽度变宽;而当两噪声正关联时,内部噪声致使反应窗口的宽度收缩;3)关联噪声致使反应窗口变化对DM模型中一级和二级非平衡动力学相变研究具有重要的科学意义.     

Fitting of two-tensor models without ad hoc assumptions to detect crossing fibers using clinical DWI data

Analysis of crossing fibers is a challenging topic in recent diffusion-weighted imaging (DWI). Resolving crossing fibers is expected to bring major changes to present tractography results based on the standard tensor model. Model free approaches, like Q-ball or diffusion spectrum imaging, as well as multi-tensor models are used to unfold the different diffusion directions mixed in a voxel of DWI data. Due to its seeming simplicity, the two-tensor model (TTM) is applied frequently to provide two positive-definite tensors and the relative population fraction modeling two crossing fiber branches. However, problems with uniqueness and noise instability are apparent. To stabilize the fit, several of the 13 physical parameters are fixed ad hoc, before fitting the model to the data. Our analysis of the TTM aims at fitting procedures where ad hoc parameters are avoided. Revealing sources of instability, we show that the model's inherent ambiguity can be reduced to one scalar parameter which only influences the fraction and the eigenvalues of the TTM, whereas the diffusion directions are not affected. Based on this, two fitting strategies are proposed: the parsimonious strategy detects the main diffusion directions without extra parameter fixation, to determine the eigenvalues and the population fraction an empirically motivated condition must be added. The expensive strategy determines all 13 physical parameters of the TTM by a fit to DWIs alone; no additional assumption is necessary. Ill-posedness of the model in case of noisy data is cured by denoising of the data and by L-curve regularization combined with global minimization performing a least-squares fit of the full model. By model simulations and real data applications, we demonstrate the feasibility of our fitting strategies and achieve convincing results. Using clinically affordable diffusion acquisition paradigms (encoding numbers: 21, 2*15, 2*21) and b values (b = 500–1500 s/mm²), this methodology can place the TTM parameters involved in crossing fibers on a more empirical basis than fitting procedures with technical assumptions.     

No Statistical-Computational Gap in Spiked Matrix Models with Generative Network Priors

We provide a non-asymptotic analysis of the spiked Wishart and Wigner matrix models with a generative neural network prior. Spiked random matrices have the form of a rank-one signal plus noise and have been used as models for high dimensional Principal Component Analysis (PCA), community detection and synchronization over groups. Depending on the prior imposed on the spike, these models can display a statistical-computational gap between the information theoretically optimal reconstruction error that can be achieved with unbounded computational resources and the sub-optimal performances of currently known polynomial time algorithms. These gaps are believed to be fundamental, as in the emblematic case of Sparse PCA. In stark contrast to such cases, we show that there is no statistical-computational gap under a generative network prior, in which the spike lies on the range of a generative neural network. Specifically, we analyze a gradient descent method for minimizing a nonlinear least squares objective over the range of an expansive-Gaussian neural network and show that it can recover in polynomial time an estimate of the underlying spike with a rate-optimal sample complexity and dependence on the noise level.