A decision-making Fokker–Planck model in computational neuroscience

In computational neuroscience, decision-making may be explained analyzing models based on the evolution of the average firing rates of two interacting neuron populations, e.g., in bistable visual perception problems. These models typically lead to a multi-stable scenario for the concerned dynamical systems. Nevertheless, noise is an important feature of the model taking into account both the finite-size effects and the decision’s robustness. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker–Planck partial differential equation. In particular, in the Fokker–Planck setting, we analytically discuss the asymptotic behavior for large times towards a unique probability distribution, and we propose a numerical scheme capturing this convergence. These simulations are used to validate deterministic moment methods recently applied to the stochastic differential system. Further, proving the existence, positivity and uniqueness of the probability density solution for the stationary equation, as well as for the time evolving problem, we show that this stabilization does happen. Finally, we discuss the convergence of the solution for large times to the stationary state. Our approach leads to a more detailed analytical and numerical study of decision-making models applied in computational neuroscience.     

Stochastic signal processing and transduction in chemotactic response of eukaryotic cells

Single-molecule imaging analysis of chemotactic response in eukaryotic cells has revealed a stochastic nature in the input signals and the signal transduction processes. This leads to a fundamental question about the signaling processes: how does the signaling system operate under stochastic fluctuations or noise? Here, we report a stochastic model of chemotactic signaling in which noise and signal propagation along the transmembrane signaling pathway by chemoattractant receptors can be analyzed quantitatively. The results obtained from this analysis reveal that the second-messenger-production reactions by the receptors generate noisy signals that contain intrinsic noise inherently generated at this reaction and extrinsic noise propagated from the ligand-receptor binding. Such intrinsic and extrinsic noise limits the directional sensing ability of chemotactic cells, which may explain the dependence of chemotactic accuracy on chemical gradients that has been observed experimentally. Our analysis also reveals regulatory mechanisms for signal improvement in the stochastically operating signaling system by analyzing how the SNR of chemotactic signals can be improved on or deteriorated by the stochastic properties of receptors and second-messenger molecules. Theoretical consideration of noisy signal transduction by chemotactic signaling systems can further be applied to signaling systems in general.     

Selection to minimise noise in living systems and its implications for the evolution of gene expression

Gene expression, like many biological processes, is subject to noise. This noise has been measured on a global scale, but its general importance to the fitness of an organism is unclear. Here, I show that noise in gene expression in yeast has evolved to prevent harmful stochastic variation in the levels of genes that reduce fitness when their expression levels change. Therefore, there has probably been widespread selection to minimise noise in gene expression. Selection to minimise noise, because it results in gene expression that is stable to stochastic variation in cellular components, may also constrain the ability of gene expression to respond to non‐stochastic variation. I present evidence that this has indeed been the case in yeast. I therefore conclude that gene expression noise is an important biological trait, and one that probably limits the evolvability of complex living systems.     

The Interplay of Intrinsic and Extrinsic Bounded Noises in Biomolecular Networks

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a biomolecular network. The influence of intrinsic and extrinsic noises on biomolecular networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, a model of enzymatic futile cycle and a genetic toggle switch. In and we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possible functional role of bounded noises.     

Irregular dynamics in up and down cortical states

Complex coherent dynamics is present in a wide variety of neural systems. A typical example is the voltage transitions between up and down states observed in cortical areas in the brain. In this work, we study this phenomenon via a biologically motivated stochastic model of up and down transitions. The model is constituted by a simple bistable rate dynamics, where the synaptic current is modulated by short-term synaptic processes which introduce stochasticity and temporal correlations. A complete analysis of our model, both with mean-field approaches and numerical simulations, shows the appearance of complex transitions between high (up) and low (down) neural activity states, driven by the synaptic noise, with permanence times in the up state distributed according to a power-law. We show that the experimentally observed large fluctuation in up and down permanence times can be explained as the result of sufficiently noisy dynamical synapses with sufficiently large recovery times. Static synapses cannot account for this behavior, nor can dynamical synapses in the absence of noise.     

模拟计算平行样对流域生物信息流估算的影响

流域生物信息流是流域生态学研究中的重要内容,是流域生态系统中的物质输移和能量输移过程的信息标记,是用eDNA技术调查评估河流水体中物种组成空间特征的基础。估算流域生物信息流是流域生态系统过程研究和eDNA技术调查评估河流水体中物种组成空间特征的关键。在有限的调查采样中,平行样的数量如何影响流域生物信息流的估算,尚待解答。基于随机抽样调查的基本原理,提出假设--采样数量不影响流域生物信息流估算结果的准确度,但会影响其精密度,然后通过问题简化转化和模拟计算,对该假设进行了检验。模拟计算结果显示,随着样点生物信息检出度(平行样数量)的增大,流域生物信息流估算结果会从偏小逐渐靠近流域生物信息流实际值,同时其99.9%置信区间也逐渐集中于流域生物信息流实际值。即样点生物信息检出度(平行样数量)对流域生物信息流估算的准确度和精密度均有影响。在实际调查研究过程中,建议先在所研究区域对平行样数量和样点生物信息检出度的关系进行预评估,然后基于流域生物信息流估算可信度目标在正式实施方案中经济有效地设置平行样,基于多平行样调查结果估算流域生物信息流,再根据各样点生物信息检出状况对流域生物信息流估算结果进行后验评估。     

An Information-Based Approach to Change-Point Analysis with Applications to Biophysics and Cell Biology

This article describes the application of a change-point algorithm to the analysis of stochastic signals in biological systems whose underlying state dynamics consist of transitions between discrete states. Applications of this analysis include molecular-motor stepping, fluorophore bleaching, electrophysiology, particle and cell tracking, detection of copy number variation by sequencing, tethered-particle motion, etc. We present a unified approach to the analysis of processes whose noise can be modeled by Gaussian, Wiener, or Ornstein-Uhlenbeck processes. To fit the model, we exploit explicit, closed-form algebraic expressions for maximum-likelihood estimators of model parameters and estimated information loss of the generalized noise model, which can be computed extremely efficiently. We implement change-point detection using the frequentist information criterion (which, to our knowledge, is a new information criterion). The frequentist information criterion specifies a single, information-based statistical test that is free from ad hoc parameters and requires no prior probability distribution. We demonstrate this information-based approach in the analysis of simulated and experimental tethered-particle-motion data.     

Size-Independent Differences between the Mean of Discrete Stochastic Systems and the Corresponding Continuous Deterministic Systems

In this paper, it is shown that for a class of reaction networks, the discrete stochastic nature of the reacting species and reactions results in qualitative and quantitative differences between the mean of exact stochastic simulations and the prediction of the corresponding deterministic system. The differences are independent of the number of molecules of each species in the system under consideration. These reaction networks are open systems of chemical reactions with no zero-order reaction rates. They are characterized by at least two stationary points, one of which is a nonzero stable point, and one unstable trivial solution (stability based on a linear stability analysis of the deterministic system). Starting from a nonzero initial condition, the deterministic system never reaches the zero stationary point due to its unstable nature. In contrast, the result presented here proves that this zero-state is a stable stationary state for the discrete stochastic system, and other finite states have zero probability of existence at large times. This result generalizes previous theoretical studies and simulations of specific systems and provides a theoretical basis for analyzing a class of systems that exhibit such inconsistent behavior. This result has implications in the simulation of infection, apoptosis, and population kinetics, as it can be shown that for certain models the stochastic simulations will always yield different predictions for the mean behavior than the deterministic simulations.     

Juxtacrine Signaling Is Inherently Noisy

Juxtacrine signaling is an important class of signaling systems that plays a crucial role in various developmental processes ranging from coordination of differentiation between neighboring cells to guiding axon growth during neurogenesis. Such signaling systems rely on the interaction between receptors on one cell and trans-membrane ligands on the membrane of a neighboring cell. Like other signaling systems, the ability of signal-receiving cells to accurately determine the concentration of ligands, is affected by stochastic diffusion processes. However, it is not clear how restriction of ligand movement to the two-dimensional (2D) cell membrane in juxtacrine signaling affects the accuracy of ligand sensing. In this study, we use a statistical mechanics approach, to show that long integration times, from around one second to several hours, are required to reach high-sensing accuracy (better than 10%). Surprisingly, the accuracy of sensing cannot be significantly improved, neither by increasing the number of receptors above three to five receptors per contact area, nor by increasing the contact area between cells. We show that these results impose stringent constraints on the dynamics of processes relying on juxtacrine signaling systems, such as axon guidance mediated by Ephrins and developmental patterns mediated by the Notch pathway.     

Extracellular noise-induced stochastic synchronization in heterogeneous quorum sensing network

Quorum sensing is a bacterial mechanism used to synchronize the coordinated response of a microbial population. Because quorum sensing in Gram-negative bacteria depends on release and detection of a diffusible signaling molecule (autoinducer) among a multicellular group, it is considered a simple form of cell-cell communication for the purposes of mathematical analysis. Stochastic equation systems have provided a common approach to model biochemical or biophysical processes. Recently, the effect of noise to synchronize a specific homogeneous quorum sensing network was successfully modeled using a stochastic equation system with fixed parameters. The question remains of how to model quorum sensing networks in a general setting. To address this question, we first set a stochastic equation system as a general model for a heterogeneous quorum sensing network. Then, using two relevant biophysical characteristics of Gram-negative bacteria (the permeability of the cell membrane to the autoinducer and the symmetry of autoinducer diffusion) we construct the solution of the stochastic equation system at an abstract level. The solution indicates that stable synchronization of a quorum sensing network is robustly induced by an environment with a heterogenous distribution of extracellular and intracellular noise. The synchronization is independent of the initial state of the system and is solely the result of the connectivity of the cell network established through the effects of extracellular noise.     

Alternative (un)stable states in a stochastic predator–prey model

Stochastic models sometimes behave qualitatively differently from their deterministic analogues. We explore the implications of this in ecosystems that shift suddenly from one state to another. This phenomenon is usually studied through deterministic models with multiple stable equilibria under a single set of conditions, with stability defined through linear stability analysis. However, in stochastic systems, some unstable states can trap stochastic dynamics for long intervals, essentially masquerading as additional stable states. Using a predator–prey model, we demonstrate that this effect is sufficient to make a stochastic system with one stable state exhibit the same characteristics as an analogous system with alternative stable states. Although this result is surprising with respect to how stability is defined by standard analyses, we show that it is well-anticipated by an alternative approach based on the system's “quasi-potential.” Broadly, understanding the risk of sudden state shifts will require a more holistic understanding of stability in stochastic systems.     

Effect of syncytium structure of receptor systems on stochastic resonance induced by chaotic potential fluctuation.

To study a role of syncytium structure of sensory receptor systems in the detection of weak signals through stochastic resonance, we present a model of a receptor system with syncytium structure in which receptor cells are interconnected by gap junctions. The apical membrane of each cell includes two kinds of ion channels whose gating processes are described by the deterministic model. The membrane potential of each cell fluctuates chaotically or periodically, depending on the dynamical state of collective channel gating. The chaotic fluctuation of membrane potential acts as internal noise for the stochastic resonance. The detection ability of the system increases as the electric conductance between adjacent cells generated by the gap junction increases. This effect of gap junctions arises mainly from the fact that the synchronization of chaotic fluctuation of membrane potential between the receptor cells is strengthened as the density of gap junctions is increased.     

Stochastic S-system modeling of gene regulatory network

Microarray gene expression data can provide insights into biological processes at a system-wide level and is commonly used for reverse engineering gene regulatory networks (GRN). Due to the amalgamation of noise from different sources, microarray expression profiles become inherently noisy leading to significant impact on the GRN reconstruction process. Microarray replicates (both biological and technical), generated to increase the reliability of data obtained under noisy conditions, have limited influence in enhancing the accuracy of reconstruction . Therefore, instead of the conventional GRN modeling approaches which are deterministic, stochastic techniques are becoming increasingly necessary for inferring GRN from noisy microarray data. In this paper, we propose a new stochastic GRN model by investigating incorporation of various standard noise measurements in the deterministic S-system model. Experimental evaluations performed for varying sizes of synthetic network, representing different stochastic processes, demonstrate the effect of noise on the accuracy of genetic network modeling and the significance of stochastic modeling for GRN reconstruction . The proposed stochastic model is subsequently applied to infer the regulations among genes in two real life networks: (1) the well-studied IRMA network, a real-life in-vivo synthetic network constructed within the Saccharomycescerevisiae yeast, and (2) the SOS DNA repair network in Escherichiacoli.     

The human amygdala and orbital prefrontal cortex in behavioural regulation

Survival in complex environments depends on an ability to optimize future behaviour based on past experience. Learning from experience enables an organism to generate predictive expectancies regarding probable future states of the world, enabling deployment of flexible behavioural strategies. However, behavioural flexibility cannot rely on predictive expectancies alone and options for action need to be deployed in a manner that is responsive to a changing environment. Important moderators on learning-based predictions include those provided by context and inputs regarding an organism's current state, including its physiological state. In this paper, I consider human experimental approaches using functional magnetic resonance imaging that have addressed the role of the amygdala and prefrontal cortex (PFC), in particular the orbital PFC, in acquiring predictive information regarding the probable value of future events, updating this information, and shaping behaviour and decision processes on the basis of these value representations.     

Precritical State Transition Dynamics in the Attractor Landscape of a Molecular Interaction Network Underlying Colorectal Tumorigenesis

From the perspective of systems science, tumorigenesis can be hypothesized as a critical transition (an abrupt shift from one state to another) between proliferative and apoptotic attractors on the state space of a molecular interaction network, for which an attractor is defined as a stable state to which all initial states ultimately converge, and the region of convergence is called the basin of attraction. Before the critical transition, a cellular state might transit between the basin of attraction for an apoptotic attractor and that for a proliferative attractor due to the noise induced by the inherent stochasticity in molecular interactions. Such a flickering state transition (state transition between the basins of attraction for alternative attractors from the impact of noise) would become more frequent as the cellular state approaches near the boundary of the basin of attraction, which can increase the variation in the estimate of the respective basin size. To investigate this for colorectal tumorigenesis, we have constructed a stochastic Boolean network model of the molecular interaction network that contains an important set of proteins known to be involved in cancer. In particular, we considered 100 representative sequences of 20 gene mutations that drive colorectal tumorigenesis. We investigated the appearance of cancerous cells by examining the basin size of apoptotic, quiescent, and proliferative attractors along with the sequential accumulation of gene mutations during colorectal tumorigenesis. We introduced a measure to detect the flickering state transition as the variation in the estimate of the basin sizes for three-phenotype attractors from the impact of noise. Interestingly, we found that this measure abruptly increases before a cell becomes cancerous during colorectal tumorigenesis in most of the gene mutation sequences under a certain level of stochastic noise. This suggests that a frequent flickering state transition can be a precritical phenomenon of colorectal tumorigenesis.     

How noisy adaptation of neurons shapes interspike interval histograms and correlations

Channel noise is the dominant intrinsic noise source of neurons causing variability in the timing of action potentials and interspike intervals (ISI). Slow adaptation currents are observed in many cells and strongly shape response properties of neurons. These currents are mediated by finite populations of ionic channels and may thus carry a substantial noise component. Here we study the effect of such adaptation noise on the ISI statistics of an integrate-and-fire model neuron by means of analytical techniques and extensive numerical simulations. We contrast this stochastic adaptation with the commonly studied case of a fast fluctuating current noise and a deterministic adaptation current (corresponding to an infinite population of adaptation channels). We derive analytical approximations for the ISI density and ISI serial correlation coefficient for both cases. For fast fluctuations and deterministic adaptation, the ISI density is well approximated by an inverse Gaussian (IG) and the ISI correlations are negative. In marked contrast, for stochastic adaptation, the density is more peaked and has a heavier tail than an IG density and the serial correlations are positive. A numerical study of the mixed case where both fast fluctuations and adaptation channel noise are present reveals a smooth transition between the analytically tractable limiting cases. Our conclusions are furthermore supported by numerical simulations of a biophysically more realistic Hodgkin-Huxley type model. Our results could be used to infer the dominant source of noise in neurons from their ISI statistics.     

Probability in biology: Overview of a comprehensive theory of probability in living systems

Probability is closely related to biological organization and adaptation to the environment. Living systems need to maintain their organizational order by producing specific internal events non-randomly, and must cope with the uncertain environments. These processes involve increases in the probability of favorable events for these systems by reducing the degree of uncertainty of events. Systems with this ability will survive and reproduce more than those that have less of this ability. Probabilistic phenomena have been deeply explored using the mathematical theory of probability since Kolmogorov's axiomatization provided mathematical consistency for the theory. However, the interpretation of the concept of probability remains both unresolved and controversial, which creates problems when the mathematical theory is applied to problems in real systems. In this article, recent advances in the study of the foundations of probability from a biological viewpoint are reviewed, and a new perspective is discussed toward a comprehensive theory of probability for understanding the organization and adaptation of living systems.     

Approximate reduction of linear population models governed by stochastic differential equations: application to multiregional models

In this work we develop approximate aggregation techniques in the context of slow-fast linear population models governed by stochastic differential equations and apply the results to the treatment of populations with spatial heterogeneity. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of ‘global’ variables, in such a way that the dynamics of the former can be approximated by that of the latter. In our model we contemplate a linear fast deterministic process together with a linear slow process in which the parameters are affected by additive noise, and give conditions for the solutions corresponding to positive initial conditions to remain positive for all times. By letting the fast process reach equilibrium we build a reduced system with a lesser number of variables, and provide results relating the asymptotic behaviour of the first- and second-order moments of the population vector for the original and the reduced system. The general technique is illustrated by analysing a multiregional stochastic system in which dispersal is deterministic and the rate growth of the populations in each patch is affected by additive noise.