arXiv subjects

Abhyudai Singh

Publications and source records attributed to Abhyudai Singh.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of inter-spike interval statistics in neuronal networks with depolarizing and hyperpolarizing threshold potentials

Neuronal communication is mediated in part by changes in neuronal firing rates. The time interval between successive neuronal firings is referred to as the inter-spike interval (ISI), and quantifying its statistics is important for understanding neuronal communication. This paper studies the ISI statistics of a postsynaptic neuron receiving independent excitatory and inhibitory presynaptic action potentials (EI circuit). This circuit is modeled as a classical integrate-and-fire neuron, and the ISI statistics are investigated for both fixed and adaptive threshold potentials. First, a depolarizing adaptive threshold model is studied, where the threshold potential increases with the postsynaptic membrane potential. Our analysis shows that the ISI noise, quantified as the coefficient of variation, is larger in the adaptive threshold model compared to the fixed threshold model for the same mean ISI. Additionally, simulations reveal that the ISI noise can be either hypo- or hyper-exponential (defined as ISI noise smaller or larger than one, respectively) depending on the frequencies of excitatory and inhibitory inputs. Next, a hyperpolarizing adaptive threshold potential is studied, where the threshold decreases as the membrane potential hyperpolarizes. Interestingly, this model shows that the postsynaptic neuron can generate action potentials (APs) when driven solely by inhibitory inputs. Furthermore, mean and noise signatures are characterized across model parameters for both excitatory and inhibitory inputs. In summary, this work provides a systematic stochastic analysis of adaptive threshold models for AP generation to understand their role in interneuronal information processing.

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arXiv2026

Analyzing Post-transcriptional Regulation in Stochastic Gene Expression Models Using Partitioned Poisson Arrivals

Gene expression is a stochastic process that allows for fluctuations in protein levels that can give rise to phenotypic heterogeneity within a population of genetically identical cells. Thus, there is great interest in quantifying how natural variation (noise) in gene expression is impacted by cellular control mechanisms, such as the various mechanisms pertaining to post-transcriptional regulation. Although previous research has developed a general analytical framework to compute the exact moments of mRNA distributions for any promoter-based regulatory motif, and the exact mRNA distribution itself in some cases, a similar framework for protein fluctuations is currently lacking. Here, we invoke the partitioning property of Poisson arrivals to map a general class of stochastic models of post-transcriptional regulation onto models that resemble promoter-based regulation. This approach leads to exact analytical results for the moments of protein distributions, and in certain cases the full distribution itself, using known exact results for mRNA distributions undergoing arbitrary promoter-based regulation. We further extend the framework to incorporate transcriptional bursting, leading to a versatile, unifying analytical framework for analyzing post-transcriptional regulation in stochastic gene expression.

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arXiv2025

Stochastic multi-step cell size homeostasis model for cycling human cells

Measurements of cell size dynamics have established the adder principle as a robust mechanism of cell size homeostasis. In this framework, cells add a nearly constant amount of size during each cell cycle, independent of their size at birth. Theoretical studies have shown that the adder principle can be achieved when cell-cycle progression is coupled to cell size. Here, we extend this framework by considering a general growth law modeled as a Hill-type function of cell size. This assumption introduces growth saturation to the model, such that very large cells grow approximately linearly rather than exponentially. Additionally, to capture the sequential nature of division, we implement a stochastic multi-step adder model in which cells progress through internal regulatory stages before dividing. From this model, we derive exact analytical expressions for the moments of cell size distributions. Our results show that stronger growth saturation increases the mean cell size in steady state, while slightly reducing fluctuations compared to exponential growth. Importantly, despite these changes, the adder property is preserved. This emphasizes that the reduction in size variability is a consequence of~the growth law rather than simple scaling with mean size. Finally, we analyze stochastic clonal proliferation and find that growth saturation influences both single-cell size statistics and variability across populations. Our results provide a generalized framework for connecting multi-step adder mechanisms with proliferation dynamics, extending size control theory beyond exponential growth.

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arXiv2025

Intrinsic limits of timekeeping precision in gene regulatory cascades

Multiple cellular processes are triggered when the concentration of a regulatory protein reaches a critical threshold. Previous analyses have characterized timing statistics for single-gene systems. However, many biological timers are based on cascades of genes that activate each other sequentially. Here, we develop an analytical framework to describe the timing precision of such cascades using a burst-dilution hybrid stochastic model. We first revisit the single-gene case and recover the known result of an optimal activation threshold that minimizes first-passage-time (FPT) variability. Extending this concept to two-gene cascades, we identify three distinct optimization regimes determined by the ratio of intrinsic noise levels and the protein dilution rate, defining when coupling improves or worsens timing precision compared to a single-gene strategy. Generalizing to cascades of arbitrary gene length, we obtain a simple mathematical condition that determines when a new gene in the cascade can decrease the timing noise based on its intrinsic noise and protein dilution rate. In the specific case of a cascade of identical genes, our analytical results predict suppression of FPT noise with increasing cascade length and the existence of a mean time that decreases relative timing fluctuations. Together, these results define the intrinsic limits of timekeeping precision in gene regulatory cascades and provide a minimal analytical framework to explore timing control in biological systems.

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arXiv2025

Dynamical Inference of Cell Size Regulation Parameters

Cells achieve size homeostasis by regulating their division timing based on their size, added size, and cell cycle time. Previous research under steady-state conditions demonstrated the robustness of these mechanisms. However, their dynamic responses in fluctuating environments, such as nutrient depletion due to population growth, remain challenging to fully characterize. Currently, advances in single-cell microscopy have revealed various cellular division strategies whose underlying molecular mechanisms are complex and not always available. This study introduces a novel approach to model cell size dynamics using a piecewise deterministic Markov chain framework, where cell division events are modeled as stochastic jumps determined by a division propensity dependent on both current cell size and added size since birth. We propose a three-parameter characterization for the division process: scale (target added size at division), shape (division stochasticity), and division strategy (relevance of cell size, added size, or cell cycle duration). We derive analytical formulas for the probability of division, and with this probability, we develop a maximum likelihood estimation (MLE) framework. We implement a systematic investigation of the accuracy of inference as a function of sample size. The model's performance is studied across various scenarios, including those exhibiting dynamical changes in one or more parameters, suggesting its broad applicability for analyzing new experimental data on cell size regulation in dynamic environments.

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arXiv2025

Information and fitness in two-state systems: self-replicating individuals in a fluctuating environment

A population of individuals with the same genes can present heterogeneous traits (phenotypes). The prevalence of this heterogeneity can be explained as a bet-hedging strategy that improves the population proliferation rate (fitness) in fluctuating environments. The phenotype distribution is influenced by factors such as competition between phenotypes, the duration of environmental states, and the rate of phenotype-switching. We illustrate these effects in a system where both the environment and the phenotype can adopt two states. This system includes scenarios such as symmetric bet-hedging and dormant-proliferating phenotypes. We examine how environmental and phenotypic states share mutual information, measured in bits, and explore the relationship between this information and population fitness. We propose that when fitness is measured relative to the case where phenotype and environment are independent, information and fitness can be treated as equivalent measures. We investigate strategies that individuals can use to improve this information, such as adjusting the rates of proliferation and phenotype-switching relative to the environmental fluctuation rate. Through these strategies, with fixed marginal distributions, an increase in information implies an increase in population fitness. We also identify limits to the maximum achievable fitness and information and discuss the value of the information in terms of this new normalized fitness. Our framework offers new insights into how organisms adapt to fluctuating environmental conditions.

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arXiv2024

Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction

We introduce a biologically detailed, stochastic model of gene expression describing the multiple rate-limiting steps of transcription, nuclear pre-mRNA processing, nuclear mRNA export, cytoplasmic mRNA degradation and translation of mRNA into protein. The processes in sub-cellular compartments are described by an arbitrary number of processing stages, thus accounting for a significantly finer molecular description of gene expression than conventional models such as the telegraph, two-stage and three-stage models of gene expression. We use two distinct tools, queueing theory and model reduction using the slow-scale linear-noise approximation, to derive exact or approximate analytic expressions for the moments or distributions of nuclear mRNA, cytoplasmic mRNA and protein fluctuations, as well as lower bounds for their Fano factors in steady-state conditions. We use these to study the phase diagram of the stochastic model; in particular we derive parametric conditions determining three types of transitions in the properties of mRNA fluctuations: from sub-Poissonian to super-Poissonian noise, from high noise in the nucleus to high noise in the cytoplasm, and from a monotonic increase to a monotonic decrease of the Fano factor with the number of processing stages. In contrast, protein fluctuations are always super-Poissonian and show weak dependence on the number of mRNA processing stages. Our results delineate the region of parameter space where conventional models give qualitatively incorrect results and provide insight into how the number of processing stages, e.g. the number of rate-limiting steps in initiation, splicing and mRNA degradation, shape stochastic gene expression by modulation of molecular memory.

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arXiv2019

Global redistribution and local migration in semi-discrete host-parasitoid population dynamic models

Host-parasitoid population dynamics is often probed using a semi-discrete/hybrid modeling framework. Here, the update functions in the discrete-time model connecting year-to-year changes in the population densities are obtained by solving ordinary differential equations that mechanistically describe interactions when hosts become vulnerable to parasitoid attacks. We use this semi-discrete formalism to study two key spatial effects: local movement (migration) of parasitoids between patches during the vulnerable period; and yearly redistribution of populations across patches outside the vulnerable period. Our results show that in the absence of any redistribution, constant density-independent migration and parasitoid attack rates are unable to stabilize an otherwise unstable host-parasitoid population dynamics. Interestingly, inclusion of host redistribution (but not parasitoid redistribution) before the start of the vulnerable period can lead to stable coexistence of both species. Next, we consider a Type-III functional response (parasitoid attack rate increases with host density), where the absence of any spatial effects leads to a neutrally stable host-parasitoid equilibrium. As before, density-independent parasitoid migration by itself is again insufficient to stabilize the population dynamics and host redistribution provides a stabilizing influence. Finally, we show that a Type-III functional response combined with density-dependent parasitoid migration leads to stable coexistence, even in the absence of population redistributions. In summary, we have systematically characterized parameter regimes leading to stable/unstable population dynamics with different forms of spatial heterogeneity coupled to the parasitoid's functional response using mechanistically formulated semi-discrete models.

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arXiv2019

Network Inference in Systems Biology: Recent Developments, Challenges, and Applications

One of the most interesting, difficult, and potentially useful topics in computational biology is the inference of gene regulatory networks (GRNs) from expression data. Although researchers have been working on this topic for more than a decade and much progress has been made, it remains an unsolved problem and even the most sophisticated inference algorithms are far from perfect. In this paper, we review the latest developments in network inference, including state-of-the-art algorithms like PIDC, Phixer, and more. We also discuss unsolved computational challenges, including the optimal combination of algorithms, integration of multiple data sources, and pseudo-temporal ordering of static expression data. Lastly, we discuss some exciting applications of network inference in cancer research, and provide a list of useful software tools for researchers hoping to conduct their own network inference analyses.

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arXiv2019

Evaluating Pruning Methods in Gene Network Inference

One challenge in gene network inference is distinguishing between direct and indirect regulation. Some algorithms, including ARACNE and Phixer, approach this problem by using pruning methods to eliminate redundant edges in an attempt to explain the observed data with the simplest possible network structure. However, we hypothesize that there may be a cost in accuracy to simplifying the predicted networks in this way, especially due to the prevalence of redundant connections, such as feed forward loops, in gene networks. In this paper, we evaluate the pruning methods of ARACNE and Phixer, and score their accuracy using receiver operating characteristic curves and precision-recall curves. Our results suggest that while pruning can be useful in some situations, it may have a negative effect on overall accuracy that has not been previously studied. Researchers should be aware of both the advantages and disadvantages of pruning when inferring networks, in order to choose the best inference strategy for their experimental context.

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arXiv2018

Moment analysis of linear time-varying dynamical systems with renewal transitions

Stochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear time-varying dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for the inclusion of state-dependent and independent noise terms. Our key contribution is derivation of necessary and sufficient conditions for the stability of statistical moments, along with exact analytical expressions for the steady-state moments. These results are illustrated on an example from cell biology, where deterministic synthesis and decay of a gene product (RNA or protein) is coupled to random timing of cell-division events. As experimentally observed, cell-division events occur based on an internal timer that measures the time elapsed since the start of cell cycle (i.e., last event). Upon division, the gene product level is halved, together with a state-dependent noise term that arises due to randomness in the partitioning of molecules between two daughter cells. We show that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and derive unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms. Systematic analysis of the formulas reveal counterintuitive insights, such as, if the partitioning noise is large then making the timing of cell division more random reduces noise in gene product levels.

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arXiv2018

Moment Analysis of Stochastic Hybrid Systems Using Semidefinite Programming

This paper proposes a semidefinite programming based method for estimating moments of a stochastic hybrid system (SHS). For polynomial SHSs -- which consist of polynomial continuous vector fields, reset maps, and transition intensities -- the dynamics of moments evolve according to a system of linear ordinary differential equations. However, it is generally not possible to solve the system exactly since time evolution of a specific moment may depend upon moments of order higher than it. One way to overcome this problem is to employ so-called moment closure methods that give point approximations to moments, but these are limited in that accuracy of the estimations is unknown. We find lower and upper bounds on a moment of interest via a semidefinite program that includes linear constraints obtained from moment dynamics, along with semidefinite constraints that arise from the non-negativity of moment matrices. These bounds are further shown to improve as the size of semidefinite program is increased. The key insight in the method is a reduction from stochastic hybrid systems with multiple discrete modes to a single-mode hybrid system with algebraic constraints. We further extend the scope of the proposed method to a class of non-polynomial SHSs which can be recast to polynomial SHSs via augmentation of additional states. Finally, we illustrate the applicability of results via examples of SHSs drawn from different disciplines.

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arXiv2017

The Linear-Noise Approximation and moment-closure approximations for stochastic chemical kinetics

This is a short review of two common approximations in stochastic chemical and biochemical kinetics. It will appear as Chapter 6 in the book "Quantitative Biology: Theory, Computational Methods and Examples of Models" edited by Brian Munsky, Lev Tsimring and Bill Hlavacek (to be published in late 2017/2018 by MIT Press). All chapter references in this article refer to chapters in the aforementioned book.

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arXiv2017

Estimating stationary characteristic functions of stochastic systems via semidefinite programming

This paper proposes a methodology to estimate characteristic functions of stochastic differential equations that are defined over polynomials and driven by Lévy noise. For such systems, the time evolution of the characteristic function is governed by a partial differential equation; consequently, the stationary characteristic function can be obtained by solving an ordinary differential equation (ODE). However, except for a few special cases such as linear systems, the solution to the ODE consists of unknown coefficients. These coefficients are closely related with the stationary moments of the process, and bounds on these can be obtained by utilizing the fact that the characteristic function is positive definite. These bounds can be further used to find bounds on other higher order stationary moments and also estimate the stationary characteristic function itself. The method is finally illustrated via examples.

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arXiv2017

Linear piecewise-deterministic Markov processes with families of random discrete events

We consider a class of piecewise-deterministic Markov processes where the state evolves according to a linear dynamical system. This continuous time evolution is interspersed by discrete events that occur at random times and change (reset) the state based on a linear affine map. In particular, we consider two families of discrete events, with the first family of resets occurring at exponentially-distributed times. The second family of resets is generally-distributed, in the sense that, the time intervals between events are independent and identically distributed random variables that follow an arbitrary continuous positively-valued probability density function. For this class of stochastic systems, we provide explicit conditions that lead to finite stationary moments, and the corresponding exact closed-form moment formulas. These results are illustrated on an example drawn from systems biology, where a protein is expressed in bursts at exponentially-distributed time intervals, decays within the cell-cycle, and is randomly divided among daughter cells when generally-distributed cell-division events occur. Our analysis leads to novel results for the mean and noise levels in protein copy numbers, and we decompose the noise levels into components arising from stochastic expression, random cell-cycle times, and partitioning. Interestingly, these individual noise contributions behave differently as cell division times become more random. In summary, the paper expands the class of stochastic hybrid systems for which statistical moments can be derived exactly without any approximations, and these results have applications for studying random phenomena in diverse areas.

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arXiv2017

Prediction and Control of Projectile Impact Point using Approximate Statistical Moments

In this paper, trajectory prediction and control design for a desired hit point of a projectile is studied. Projectiles are subject to environment noise such as wind effect and measurement noise. In addition, mathematical models of projectiles contain a large number of important states that should be taken into account for having a realistic prediction. Furthermore, dynamics of projectiles contain nonlinear functions such as monomials and sine functions. To address all these issues we formulate a stochastic model for the projectile. We showed that with a set of transformations projectile dynamics only contains nonlinearities of the form of monomials. In the next step we derived approximate moment dynamics of this system using mean-field approximation. Our method still suffers from size of the system. To address this problem we selected a subset of first- and second-order statistical moments and we showed that they give reliable approximations of the mean and standard deviation of the impact point for a real projectile. Finally we used these selected moments to derive a control law that reduces error to hit a desired point.

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arXiv2017

Control design and analysis of a stochastic network control system

A Network Control System (NCS) consists of control components that interact with the plant over a shared network. The system dynamics of a NCS could be subject to noise arising from randomness in the times at which the data is transmitted over the network, corruption of the transmitted data by the communication network, and external disturbances that might affect the plant. A question of interest is to understand how the statistics of the data transmission times affects the system dynamics, and under what conditions the system is stable. Another related issue is designing a controller that meets desired performance specifications (e.g., a specific mean and variance of the system state). Here, we consider a minimal NCS that consists of a plant and a controller, and it is subject to random transmission times, channel corruption and external disturbances. We derive exact dynamics of the first two moments of the system, and use them to derive the stability conditions of the system. We further design a control law that steers the system to a desired mean and variance. Finally, we demonstrate our results using different examples, and show that under some specific conditions, randomness in the data transmission times can even reduce the variability contributed from disturbance.

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arXiv2017

Approximate moment dynamics for polynomial and trigonometric stochastic systems

Stochastic dynamical systems often contain nonlinearities which make it hard to compute probability density functions or statistical moments of these systems. For the moment computations, nonlinearities in the dynamics lead to unclosed moment dynamics; in particular, the time evolution of a moment of a specific order may depend both on moments of order higher than it and on some nonlinear function of other moments. The moment closure techniques are used to find an approximate, close system of equations the moment dynamics. In this work, we extend a moment closure technique based on derivative matching that was originally proposed for polynomial stochastic systems with discrete states to continuous state stochastic systems to continuous state stochastic differential equations, with both polynomial and trigonometric nonlinearities. We validate the technique using two examples of nonlinear stochastic systems.

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