arXiv subjects

G Manjunath

Publications and source records attributed to G Manjunath.

9 recordsLinked to original sources

arXiv2026

Ghost Dynamics in Receptor Signalling Networks: A Fast--Slow Adaptive Extension of Competitive Cancer Inhibition Models

Receptor occupancy models quantify inhibition of cancer signalling, often treating target occupancy as a proxy for downstream activity. Although this simplification yields low-dimensional models, it cannot represent delayed pathway shutdown, transient resistance, or non-monotone viability responses generated by intracellular networks. We formulate a fast--slow framework that distinguishes drug--target occupancy from downstream signalling activity. The fast variable $X$ represents occupancy, the activity variable $A$ represents pro-survival signalling, and the slow variable $B$ represents adaptive feedback including phosphatase induction, stress adaptation, or signalling rewiring. Rapid occupancy relaxation permits a quasi-steady reduction, while weak feedback guarantees global convergence to a unique equilibrium. Stronger feedback may bring the frozen activity subsystem near a saddle-node fold, producing a ghost regime in which trajectories linger near the remnant of a vanished high-activity equilibrium. We determine when slow adaptive passage preserves the inverse-square-root ghost law or, under transverse crossing, produces the dynamic delay scale $O(\varepsilon^{-1/3})$. Coupling activity to viability translates these delays into shoulders in dose--response curves, while adaptive lag may generate early-time overshoot without ad hoc forcing. A proof-of-concept fit to time-resolved viability data reproduces patterns across measured concentrations and observation times and reveals practical identifiability limitations. The framework also predicts exposure-time-dependent shifts in apparent potency, including $IC_{50}(t)$, because viability integrates signalling activity over time rather than receptor occupancy alone.

math.DS↗

arXiv2026

Data-Specific Hyper-Parameter Design: A Paradigm Shift in Reservoir Computing

Reservoir computing typically relies on large, randomly generated reservoirs, enabling simple, often linear readouts. Over the past two decades, most constructions have exploited the freedom to select the reservoir, constrained primarily by stability conditions based on state contraction or memory capacity. However, these designs are largely independent of the input data and learning objective, resulting in a trial-and-error methodology driven by randomness. In high dimensions, the reservoir acts as a random embedding of the input history, implicitly relying on Johnson--Lindenstrauss--type concentration phenomena to preserve information. In contrast, we develop reservoir design principles from a geometric perspective for inputs generated by deterministic dynamical systems. Rather than relying on random embeddings, we require reservoir state increments to align within a cone around an input-determined vector subspace, and prove that such a cone concentration reduces ridge-regression training error. When the cone angle is small, the variance of reservoir states concentrates in the input-determined subspace, improving conditioning of the empirical second-moment matrix and strengthening alignment between dominant covariance directions and the state-target cross-covariance. For echo state networks, we provide a constructive approach to reservoir design. The reservoir matrix is chosen so that associated Krylov-chain directions remain nearly closed within an input-determined subspace while permitting controlled mixing in its orthogonal complement. We also provide a spectral diagnostic for ridge regression training that identifies when reservoir geometry concentrates predictive information into a few dominant covariance modes and when ``spectral pollution'' inhibits forecasting. Numerical experiments demonstrate consistent performance gains over arbitrary reservoir constructions.

math.DS↗

arXiv2024

Data-driven cold starting of good reservoirs

Using short histories of observations from a dynamical system, a workflow for the post-training initialization of reservoir computing systems is described. This strategy is called cold-starting, and it is based on a map called the starting map, which is determined by an appropriately short history of observations that maps to a unique initial condition in the reservoir space. The time series generated by the reservoir system using that initial state can be used to run the system in autonomous mode, to produce accurate forecasts of the time series under consideration immediately. By utilizing this map, the lengthy "washouts" that are necessary to initialize reservoir systems can be eliminated, enabling the generation of forecasts using any selection of appropriately short histories of the observations.

math.DS↗

arXiv2022

Transport in reservoir computing

Reservoir computing systems are constructed using a driven dynamical system in which external inputs can alter the evolving states of a system. These paradigms are used in information processing, machine learning, and computation. A fundamental question that needs to be addressed in this framework is the statistical relationship between the input and the system states. This paper provides conditions that guarantee the existence and uniqueness of asymptotically invariant measures for driven systems and shows that their dependence on the input process is continuous when the set of input and output processes are endowed with the Wasserstein distance. The main tool in these developments is the characterization of those invariant measures as fixed points of naturally defined Foias operators that appear in this context and which have been profusely studied in the paper. Those fixed points are obtained by imposing a newly introduced stochastic state contractivity on the driven system that is readily verifiable in examples. Stochastic state contractivity can be satisfied by systems that are not state-contractive, which is a need typically evoked to guarantee the echo state property in reservoir computing. As a result, it may actually be satisfied even if the echo state property is not present.

cs.NE↗

arXiv2022

Quantifying assays: A Modeling tale of variability in cancer therapeutics assessed on cancer cells

Inhibiting a signalling pathway concerns controlling the cellular processes of a cancer cell's viability, cell division, and death. Assay protocols created to see if the molecular structures of the drugs being tested have the desired inhibition qualities often show great variability across experiments, and it is imperative to diminish the effects of such variability while inferences are drawn. In this paper we propose the study of experimental data through the lenses of a mathematical model depicting the inhibition mechanism and the activation-inhibition dynamics. The method is exemplified through assay data obtained from the study of inhibition of the CXCL12/CXCR4 activation axis for the melanoma cells. To mitigate the effects of the variability of the data on the cell viability measurement, the cell viability is theoretically constructed as a function of time depending on several parameters. The values of these parameters are estimated by using the experimental data. Deriving approximation for the cell viability in a theoretically pre-determined form has the advantages of (i) being less sensitive to data variability (ii) the estimated values of the parameters are interpreted directly in the biological processes, (iii) the amount of variability explained via the approximation validates the quality of the model, (iv) with the data integrated into the model one can derive a more complete view over the whole process. These advantages are demonstrated in the step-by-step implementation of the outlined approach.

q-bio.QM↗

arXiv2021

Universal set of Observables for Forecasting Physical Systems through Causal Embedding

We demonstrate when and how an entire left-infinite orbit of an underlying dynamical system or observations from such left-infinite orbits can be uniquely represented by a pair of elements in a different space, a phenomenon which we call \textit{causal embedding}. The collection of such pairs is derived from a driven dynamical system and is used to learn a function which together with the driven system would: (i). determine a system that is topologically conjugate to the underlying system (ii). enable forecasting the underlying system's dynamics since the conjugacy is computable and universal, i.e., it does not depend on the underlying system (iii). guarantee an attractor containing the image of the causally embedded object even if there is an error made in learning the function. By accomplishing these we herald a new forecasting scheme that beats the existing reservoir computing schemes that often lead to poor long-term consistency as there is no guarantee of the existence of a learnable function, and overcomes the challenges of stability in Takens delay embedding. We illustrate accurate modeling of underlying systems where previously known techniques have failed.

math.DS↗

arXiv2021

Embedding Information onto a Dynamical System

The celebrated Takens' embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalization of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.

math.DS↗

arXiv2020

Stability and Memory-loss go Hand-in-Hand: Three Results in Dynamics & Computation

The search for universal laws that help establish a relationship between dynamics and computation is driven by recent expansionist initiatives in biologically inspired computing. A general setting to understand both such dynamics and computation is a driven dynamical system that responds to a temporal input. Surprisingly, we find memory-loss a feature of driven systems to forget their internal states helps provide unambiguous answers to the following fundamental stability questions that have been unanswered for decades: what is necessary and sufficient so that slightly different inputs still lead to mostly similar responses? How does changing the driven system's parameters affect stability? What is the mathematical definition of the edge-of-criticality? We anticipate our results to be timely in understanding and designing biologically inspired computers that are entering an era of dedicated hardware implementations for neuromorphic computing and state-of-the-art reservoir computing applications.

cs.LG↗

arXiv2016

Evolving Network Model that Almost Regenerates Epileptic Data

In many realistic networks, the edges representing the interactions between the nodes are time-varying. There is growing evidence that the complex network that models the dynamics of the human brain has time-varying interconnections, i.e., the network is evolving. Based on this evidence, we construct a patient and data specific evolving network model (comprising discrete-time dynamical systems) in which epileptic seizures or their terminations in the brain are also determined by the nature of the time-varying interconnections between the nodes. A novel and unique feature of our methodology is that the evolving network model remembers the data from which it was conceived from, in the sense that it evolves to almost regenerate the patient data even upon presenting an arbitrary initial condition to it. We illustrate a potential utility of our methodology by constructing an evolving network from clinical data that aids in identifying an approximate seizure focus -- nodes in such a theoretically determined seizure focus are outgoing hubs that apparently act as spreaders of seizures. We also point out the efficacy of removal of such spreaders in limiting seizures.

q-bio.NC↗