arXiv subjects

Shaunak Sen

Publications and source records attributed to Shaunak Sen.

13 recordsLinked to original sources

arXiv2026

Comparing Point and Interval Methods for Equilibrium Computation under Parametric Uncertainty

Equilibrium points define operating conditions for nonlinear dynamical and control systems. Their existence, multiplicity, and stability under parametric uncertainty determine feasible operating regimes and the validity of robustness claims. With parameters constrained to a bounded set, one can (i) compute equilibria at sampled parameter values, (ii) trace equilibria along a prescribed path in parameter space, or (iii) identify states in a given operating domain that are equilibria for at least one admissible parameter realization. We compare standard pointwise workflows-direct simulation, numerical continuation, residual minimization, and a multistart Newton-Raphson method-with validated interval-analysis-based workflows. The latter (a) provide formal certificates of exclusion, existence, and uniqueness of equilibria for fixed parameters and, under parametric inclusion conditions, uniformly over entire parameter boxes, and (b) construct rigorous outer enclosures in state space that provably contain all equilibria associated with the full admissible parameter set. Biomolecular circuit models governed by nonlinear ODEs serve as a representative application domain. We benchmark three canonical architectures across four levels of parameter uncertainty, including a genetic toggle switch near a symmetry-breaking bifurcation. Sampling- and slice-based approaches can miss or underrepresent multistability, whereas interval-based outer enclosures yield mathematically rigorous bounds on the equilibrium set induced by parametric uncertainty.

eess.SY↗

arXiv2026

Certified Detection of Bifurcation Candidates in Uncertain Nonlinear Systems using Interval Analysis

Qualitative transitions in nonlinear dynamical systems (e.g., loss of stability, onset of oscillations, emergence of multistability) delimit operating regimes and can arise as implicit constraints in robust analysis and design under parametric uncertainty. When parameters are inferred from data, admissible values are naturally represented as uncertainty sets, motivating certified tests for the presence or absence of regime-transition candidates. We propose a validated interval workflow that encodes saddle-node and Hopf candidate conditions as square augmented algebraic systems and applies the Krawczyk operator to certify, over a prescribed state-parameter box, either (i) existence and local uniqueness of a candidate solution or (ii) certified absence. Numerical experiments on uncertain synthetic gene-network ODE models yield locally certified saddle-node candidate enclosures on a two-parameter slice for a bistable circuit and certified Hopf candidate enclosures for a three-state oscillator using a Routh-Hurwitz specialization. The resulting certificates are intended to support regime-aware analysis and design under bounded uncertainty by complementing non-validated, pointwise baselines (e.g., Newton method solves at discrete parameter values) and sampling-based workflows with rigorous presence/absence guarantees on user-specified parameter slices.

eess.SY↗

arXiv2026

Block Diagram Analysis of a Design Principle for Amplitude-Frequency Profiles in Biological Oscillations

An important design principle for biological oscillators divides the oscillators into two classes: fixed frequency, variable amplitude and fixed amplitude, variable frequency. Because of the interplay of nonlinearity and feedback, both positive and negative, analytical investigations of this design principle are primarily based on numerical simulations of ordinary differential equations. To enhance the qualitative and quantitative characterization, we adapted and developed a block diagram modeling framework. We showed how the observed amplitude-frequency characteristics could be obtained from the block diagram models. We obtained constraints on the positive feedback and negative feedback strengths for the oscillations to exist. These results should contribute to a systems and control perspective on oscillations in biology and related contexts.

eess.SY↗

arXiv2026

Computational Complexity Analysis of Interval Methods in Solving Uncertain Nonlinear Systems

This paper analyzes the computational complexity of validated interval methods for uncertain nonlinear systems and steady-state enclosure. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its adoption is often limited by computational cost in high dimensions. We develop an algorithm-level worst-case framework that makes explicit the dependence on the problem dimension $n$, the initial search region size $\mathrm{Vol}(X_0)$, the target tolerance $\varepsilon$, and the costs of validated primitives (inclusion-function evaluation, Jacobian evaluation, and interval linear algebra). Within this framework, we derive worst-case time and space bounds for interval bisection, subdivision$+$filter, interval constraint propagation, interval Newton, and interval Krawczyk, and identify dominant cost drivers. We also show that the computation of the determinant and inverse of interval matrices via naive Laplace expansion exhibits factorial growth with increasing matrix dimension, motivating specialized interval linear algebra. We complement the worst-case bounds with computational results on two application-motivated biochemical steady-state models (a Hill-type regulatory network and an enzyme-saturation-based winner-take-all circuit) in dimensions $n\in\{2,5,10\}$, including instances that process millions of boxes. The resulting analysis and experiments support the practical design of validated solvers for uncertainty-aware steady-state screening tasks such as robust operating-point certification and multistability assessment.

cs.DS↗

arXiv2025

A Kalman Filter Algorithm with Process Noise Covariance Update

Stochastic models in biomolecular contexts can have a state-dependent process noise covariance. The choice of the process noise covariance is an important parameter in the design of a Kalman Filter for state estimation and the theoretical guarantees of updating the process noise covariance as the state estimate changes are unclear. Here we investigated this issue using the Minimum Mean Square Error estimator framework and an interpretation of the Kalman Filter as minimizing a weighted least squares cost using Newton's method. We found that a Kalman Filter-like algorithm with a process noise covariance update is the best linear unbiased estimator for a class of systems with linear process dynamics and a square root-dependence of the process noise covariance on the state. We proved the result for discrete-time system dynamics and then extended it to continuous-time dynamics using a limiting procedure. For nonlinear dynamics with a general dependence of process noise covariance on the state, we showed that this algorithm minimizes a quadratic approximation to a least squares cost weighted by the noise covariance. The algorithm is illustrated with an example.

eess.SY↗

arXiv2024

A Necessary and Sufficient Condition for Local Synchronization in Nonlinear Oscillator Networks

Determining conditions on the coupling strength for the synchronization in networks of interconnected oscillators is a challenging problem in nonlinear dynamics. While sophisticated mathematical methods have been used to derive conditions, these conditions are usually only sufficient and/ or based on numerical methods. We addressed the gap between the sufficient coupling strength and numerically observations using the Lyapunov-Floquet Theory and the Master Stability Function framework. We showed that a positive coupling strength is a necessary and sufficient condition for local synchronization in a network of identical oscillators coupled linearly and in full state fashion. For partial state coupling, we showed that a positive coupling constant results in an asymptotic contraction of the trajectories in the state space, which results in synchronisation for two-dimensional oscillators. We extended the results to networks with non-identical coupling over directed graphs and showed that positive coupling constants is a sufficient condition for synchronisation. These theoretical results are validated using numerical simulations and experimental implementations. Our results contribute to bridging the gap between the theoretically derived sufficient coupling strengths and the numerically observed ones.

eess.SY↗

arXiv2022

Comparative Analysis and Calibration of Low Cost Resistive and Capacitive Soil Moisture Sensor

Soil moisture is an essential parameter in agriculture. It determines several environmental and agricultural activities such as climate change, drought prediction, irrigation, etc. Smart irrigation management requires continuous soil moisture monitoring to reduce unnecessary water usage. In recent times, the use of low-cost sensors is becoming popular among farmers for soil moisture monitoring. In this paper, a comparison of low-cost resistive and capacitive soil moisture sensors is demonstrated in two ways. One way is the calibration of the sensors in gravimetric and volumetric water content, and the other is the sensors' response analysis when different quantities of water are added to the same amount of soil. The analysis shown in this work is essential before choosing cost-effective sensors for any soil moisture monitoring platform.

cs.NI↗

arXiv2020

On the length scale dependence of DNA conformational change under local perturbation

Conformational change of a DNA molecule is frequently observed in multiple biological processes and has been modelled using a chain of strongly coupled oscillators with a nonlinear bistable potential. While the mechanism and properties of conformational change in the model have been investigated and several reduced order models developed, the conformational dynamics as a function of the length of the oscillator chain is relatively less clear. To address this, we used a modified Lindstedt-Poincare method and numerical computations. We calculate a perturbation expansion of the frequency of the model's nonzero modes, finding that approximating these modes with their unperturbed dynamics, as in a previous reduced order model, may not hold when the length of the DNA model increases. We investigate the conformational change to local perturbation in models of varying lengths, finding that for chosen input and parameters, there are two regions of DNA length in the model, first where the minimum energy required to undergo the conformational change increases with DNA length; and second, where it is almost independent of the length of the DNA model. We analyze the conformational change in these models by adding randomness to the local perturbation, finding that the tendency of the system to remain in a stable conformation against random perturbation decreases with an increase in the DNA length. These results should help to understand the role of the length of a DNA molecule in influencing its conformational dynamics.

physics.bio-ph↗

arXiv2020

Robustness of a Biomolecular Oscillator to Pulse Perturbations

Biomolecular oscillators can function robustly in the presence of environmental perturbations, which can either be static or dynamic. While the effect of different circuit parameters and mechanisms on the robustness to steady perturbations has been investigated, the scenario for dynamic perturbations is relatively unclear. To address this we use a benchmark three protein oscillator design - the repressilator - and investigate its robustness to pulse perturbations, computationally as well as using analytical tools of Floquet theory. We find that the metric provided by direct computations of the time it takes for the oscillator to settle after a pulse perturbation is applied, correlates well with the metric provided by Floquet theory. We investigate the parametric dependence of the Floquet metric, finding that the parameters that increase the effective delay enhance robustness to pulse perturbation. We find that the structural changes such as increasing the number of proteins in a ring oscillator as well as adding positive feedback, both of which increase effective delay, facilitates such robustness. These results highlight such design principles, especially the role of delay, for designing an oscillator that is robust to pulse perturbation.

q-bio.MN↗

arXiv2018

A Kalman Filter Approach for Biomolecular Systems with Noise Covariance Updating

An important part of system modeling is determining parameter values, particularly for biomolecular systems, where direct measurements of individual parameters are typically hard. While Extended Kalman Filters have been used for this purpose, the choice of the process noise covariance is generally unclear. In this chapter, we address this issue for biomolecular systems using a combination of Monte Carlo simulations and experimental data, exploiting the dependence of the process noise covariance on the states and parameters, as given in the Langevin framework. We adapt a Hybrid Extended Kalman Filtering technique by updating the process noise covariance at each time step based on estimates. We compare the performance of this framework with different fixed values of process noise covariance in biomolecular system models, including an oscillator model, as well as in experimentally measured data for a negative transcriptional feedback circuit. We find that the Extended Kalman Filter with such process noise covariance update is closer to the optimality condition in the sense that the innovation sequence becomes white and in achieving a balance between the mean square estimation error and parameter convergence time. The results of this chapter may help in the use of Extended Kalman Filters for systems where process noise covariance depends on states and/or parameters.

q-bio.QM↗

arXiv2018

Period-Amplitude Co-variation in Biomolecular Oscillators

The period and amplitude of biomolecular oscillators are functionally important properties in multiple contexts. For a biomolecular oscillator, the overall constraints in how tuning of amplitude affects period, and vice versa, are generally unclear. Here we investigate this co-variation of the period and amplitude in mathematical models of biomolecular oscillators using both simulations and analytical approximations. We computed the amplitude-period co-variation of eleven benchmark biomolecular oscillators as their parameters were individually varied around a nominal value, classifying the various co-variation patterns such as a simultaneous increase/ decrease in period and amplitude. Next, we repeated the classification using a power norm-based amplitude metric, to account for the amplitudes of the many biomolecular species that may be part of the oscillations, finding largely similar trends. Finally, we calculate "scaling laws" of period-amplitude co-variation for a subset of these benchmark oscillators finding that as the approximated period increases, the upper bound of the amplitude increases, or reaches a constant value. Based on these results, we discuss the effect of different parameters on the type of period-amplitude co-variation as well as the difficulty in achieving an oscillation with large amplitude and small period.

q-bio.MN↗

arXiv2017

Describing Function-based Approximations of Biomolecular Systems

Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of analytical and computational approaches, we adapt and develop the method of describing functions to represent the input-output responses of biomolecular signalling systems. We approximate representative systems exhibiting various saturating and hysteretic dynamics in a way that is better than the standard linearization. Further, we develop analytical upper bounds for the computational error estimates. Finally, we use these error estimates to augment the limit cycle analysis with a simple and quick way to bound the predicted oscillation amplitude. These results provide system approximations that can add more insight into the local behaviour of these systems than standard linearization, compute responses to other periodic inputs, and to analyze limit cycles.

q-bio.MN↗

arXiv2017

Non-normality Can Facilitate Pulsing in Biomolecular Circuits

Non-normality can underlie pulse dynamics in many engineering contexts. However, its role in pulses generated in biomolecular contexts is generally unclear. Here, we address this issue using the mathematical tools of linear algebra and systems theory on simple computational models of biomolecular circuits. We find that non-normality is present in standard models of feedforward loops. We used a generalized framework and pseudospectrum analysis to identify non-normality in larger biomolecular circuit models, finding that it correlates well with pulsing dynamics. Finally, we illustrate how these methods can be used to provide analytical support to numerical screens for pulsing dynamics as well as provide guidelines for design.

q-bio.MN↗