About

Eduardo Sontag is a university distinguished professor of electrical and computer engineering and bioengineering at Northeastern University, with additional affiliations as a professor of mathematics and chemical engineering. Recognized for his distinguished and continuing contributions to original research in applied mathematics, he was elected a Member of the National Academy of Sciences, an honor that signifies a high level of achievement in science. Sontag's work has profoundly shaped modern control theory, particularly through his concept of input-to-state stability for nonlinear systems, which has become foundational in the field. He has authored more than 500 research papers, book chapters, and monographs, and his work has been cited nearly 65,000 times and referenced in hundreds of patents. His research bridges rigorous mathematics with systems biology, neural networks, and nonlinear dynamics, illustrating how engineering ideas can be transferred to biological and neuroscientific fields. Sontag's approach involves thinking about big ideas in basic concepts with a rigorous, abstract, and logical methodology. Growing up in Buenos Aires, Argentina, he was drawn to science and mathematics early on for their intellectual challenge and their power to address problems in health, engineering, and society. He maintains a profound curiosity about many fields, especially biology and neuroscience, and often explores analogies between engineering and biological systems.

Research topics

• Computer Science
• Medicine
• Pathology
• Physics
• Cancer research
• Internal medicine
• Bioinformatics
• Biology
• Environmental health
• Psychology

Selected publications

• Delaying cancer progression by integrating toxicity constraints in a model of adaptive therapy

  npj Systems Biology and Applications · 2026-01-07

  articleOpen access

  Cancer therapies often fail when intolerable toxicity or drug-resistant cancer cells undermine otherwise effective treatment strategies. Over the past decade, adaptive therapy has emerged as a promising approach to postpone emergence of resistance by altering dose timing based on tumor burden thresholds. Despite encouraging results, these protocols often overlook the crucial role of toxicity-induced treatment breaks, which may permit tumor regrowth. Herein, we explore the following question: would incorporating toxicity feedback improve or hinder the efficacy of adaptive therapy? To address this question, we propose a mathematical framework for incorporating toxic feedback into treatment design. We find that the degree of competition between sensitive and resistant populations, along with the growth rate of resistant cells, critically modulates the impact of toxicity feedback on time to progression. Further, our conceptual model identifies circumstances where strategic treatment breaks, which may be based on either tumor size or toxicity, can mitigate overtreatment and extend time to progression, both at the baseline parameterization and across a heterogeneous virtual population. Taken together, these findings highlight the importance of integrating toxicity considerations into the design of adaptive therapy.

  PublisherOA PDFDOI

• Delaying Cancer Progression by Integrating Toxicity Constraints in a Model of Adaptive Therapy

  bioRxiv (Cold Spring Harbor Laboratory) · 2025-04-27 · 2 citations

  preprintOpen access

  Abstract Cancer therapies often fail when intolerable toxicity or drug-resistant cancer cells undermine otherwise effective treatment strategies. Over the past decade, adaptive therapy has emerged as a promising approach to postpone emergence of resistance by altering dose timing based on tumor burden thresholds. Despite encouraging results, these protocols often overlook the crucial role of toxicity-induced treatment breaks, which may permit tumor regrowth. Herein, we explore the following question: would incorporating toxicity feedback improve or hinder the efficacy of adaptive therapy? To address this question, we propose a mathematical framework for incorporating toxic feedback into treatment design. We find that the degree of competition between sensitive and resistant populations, along with the growth rate of resistant cells, critically modulates the impact of toxicity feedback on time to progression. Further, our model identifies circumstances where strategic treatment breaks, which may be based on either tumor size or toxicity, can mitigate overtreatment and extend time to progression, both at the baseline parameterization and across a heterogeneous virtual population. Taken together, these findings highlight the importance of integrating toxicity considerations into the design of adaptive therapy.

  PublisherOA PDFDOI

• Optimization of heuristic logic synthesis by iteratively reducing circuit substructures using a database of optimal implementations

  bioRxiv (Cold Spring Harbor Laboratory) · 2025-11-29

  preprintOpen accessSenior authorCorresponding

  Minimal synthesis of Boolean functions is an NP-hard problem, and heuristic approaches typically give suboptimal circuits. However, in the emergent field of synthetic biology, genetic logic designs that use even a single additional Boolean gate can render a circuit unimplementable in a cell. This has led to a renewed interest in the field of optimal multilevel Boolean synthesis. For small numbers (1-4) of inputs, an exhaustive search is possible, but this is impractical for large circuits. In this work, we demonstrate that even though it is challenging to build a database of optimal implementations for anything larger than 4-input Boolean functions, a database of 4-input optimal implementations can be used to greatly reduce the number of logical gates required in larger heuristic logic synthesis implementations. The proposed algorithm combines the heuristic results with an optimal implementation database and yields average improvements in fractional gate-count reduction relative to ABC of 5.16% for 5-input circuits and 4.54% for 6-input circuits on outputs provided by the logic synthesis tool ABC . In addition to the gains in the efficiency of the implemented circuits, this work also attests to the importance and practicality of the field of optimal synthesis, even if it cannot directly provide results for larger circuits. The focus of this work is on circuits made exclusively of 2-input NOR gates but the presented results are readily applicable to 2-input NAND circuits as well as (2-input) AND/NOT circuits. The framework proposed here is likely to be adaptable to other types of circuits. Moreover, a small computational pipeline is provided for integration with synthetic biology tools such as Cello . An implementation of the described algorithm, HLM (Hybrid Logic Minimizer), is available at https://github.com/sontaglab/HLM/ .

  PublisherOA PDFDOI

• Cumulative dose responses for adapting biological systems

  Journal of The Royal Society Interface · 2025-08-01

  articleOpen accessSenior author

  Physiological adaptation is a fundamental property of biological systems across all levels of organization, ensuring survival and proper function. Adaptation is typically formulated as an asymptotic property of the dose response (DR ), defined as the level of a response variable with respect to an input parameter. In pharmacology, the input could be a drug concentration; in immunology, it might correspond to an antigen level. In contrast to the DR, this paper develops the concept of a transient, finite-time, cumulative dose response (cDR) , which is obtained by integrating the response variable over a fixed time interval and viewing that integral—area under the curve—as a function of the input parameter. This study is motivated by experimental observations of cytokine accumulation under T-cell stimulation, which exhibit a non-monotonic cDR. It is known from the systems biology literature that only two types of network motifs, incoherent feedforward loops and negative integral feedback (IFB) mechanisms, can generate adaptation. Three paradigmatic such motifs—two types of incoherent loops and one integral feedback—have been the focus of much study. Surprisingly, it is shown here that these two incoherent feedforward loop motifs—despite their capacity for non-monotonic DR—always yield a monotonic cDR, and are therefore inconsistent with these experimental data. On the other hand, this work reveals that the IFB motif is indeed capable of producing a non-monotonic cDR, and is thus consistent with these data.

  PublisherDOI

• Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems

  ArXiv.org · 2025-03-31

  preprintOpen accessSenior author

  This work explores generalizations of the Polyak-Lojasiewicz inequality (PLI) and their implications for the convergence behavior of gradient flows in optimization problems. Motivated by the continuous-time linear quadratic regulator (CT-LQR) policy optimization problem -- where only a weaker version of the PLI is characterized in the literature -- this work shows that while weaker conditions are sufficient for global convergence to, and optimality of the set of critical points of the cost function, the "profile" of the gradient flow solution can change significantly depending on which "flavor" of inequality the cost satisfies. After a general theoretical analysis, we focus on fitting the CT-LQR policy optimization problem to the proposed framework, showing that, in fact, it can never satisfy a PLI in its strongest form. We follow up our analysis with a brief discussion on the difference between continuous- and discrete-time LQR policy optimization, and end the paper with some intuition on the extension of this framework to optimization problems with L1 regularization and solved through proximal gradient flows.

  PublisherOA PDFDOI

• Understanding therapeutic tolerance through a mathematical model of drug-induced resistance

  npj Systems Biology and Applications · 2025-04-09 · 5 citations

  articleOpen accessSenior author

  There is growing recognition that phenotypic plasticity enables cancer cells to adapt to various environmental conditions. An example of this adaptability is the ability of an initially sensitive population of cancer cells to acquire resistance and persist in the presence of therapeutic agents. Understanding the implications of this drug-induced resistance is essential for predicting transient and long-term tumor dynamics subject to treatment. This paper introduces a mathematical model of drug-induced resistance which provides excellent fits to time-resolved in vitro experimental data. From observational data of total numbers of cells, the model unravels the relative proportions of sensitive and resistance subpopulations and quantifies their dynamics as a function of drug dose. The predictions are then validated using data on drug doses that were not used when fitting parameters. Optimal control techniques are then utilized to discover dosing strategies that could lead to better outcomes as quantified by lower total cell volume.

  PublisherOA PDFDOI

• Quantitative Pharmacology Methods for Bispecific T Cell Engagers

  bioRxiv (Cold Spring Harbor Laboratory) · 2025-04-26

  preprintOpen accessSenior authorCorresponding

  Abstract T Cell Engager (TCE)s are an exciting therapeutic modality in immuno-oncology that acts to bypass antigen presentation and forms a direct link between cancer and immune cells in the Tumor Microenvironment (TME). TCEs are efficacious only when the drug is bound to both immune and cancer cell targets. Therefore, approaches that maximize the formation of the drug-target trimer in the TME are expected to increase the drug’s efficacy. In this study, we quantitatively investigate how the concentration of ternary complex and its biodistribution depend on both the targets’ specific properties and the design characteristics of the TCE, and specifically on the binding kinetics of the drug to its targets. A simplified mathematical model of drug-target interactions is considered here, with insights from the “three-body” problem applied to the model. Parameter identifiability analysis performed on the model demonstrates that steady state data, which is often available at the early pre-clinical stages, is sufficient to estimate the binding affinity of the TCE molecule to both targets. We used the model to analyze several existing antibodies, both clinically approved and under development, to explore their common kinetic features. The manuscript concludes with an assessment of a full quantitative pharmacology model that accounts for drug disposition into the peripheral compartment.

  PublisherOA PDFDOI

• Convergence analysis of gradient flow for overparameterized LQR formulations

  Automatica · 2025-08-25

  articleOpen accessSenior author

  This paper analyzes the intersection between results from gradient methods for the model-free linear quadratic regulator (LQR) problem, and linear feedforward neural networks (LFFNNs). More specifically, it looks into the case where one wants to find an LFFNN feedback that minimizes an LQR cost. It starts by deriving a key conservation law of the system, which is then leveraged to generalize existing results on boundedness and global convergence of solutions, and invariance of the set of stabilizing LFFNNs under the training dynamics (gradient flow). For the single hidden layer LFFNN, the paper proves that the solution converges to the optimal feedback control law for all but a set of Lebesgue measure zero of the initializations. These results are followed by an analysis of a simple version of the problem – the “vector case” – proving the theoretical properties of accelerated convergence and a type of input-to-state stability (ISS) result for this simpler example. Finally, the paper presents numerical evidence of faster convergence of the gradient flow of general LFFNNs when compared to non-overparameterized formulations, showing that the acceleration of the solution is observable even when the gradient is not explicitly computed, but estimated from evaluations of the cost function.

  PublisherDOI

• On the (almost) Global Exponential Convergence of the Overparameterized Policy Optimization for the LQR Problem

  ArXiv.org · 2025-10-02

  preprintOpen accessSenior author

  In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations.

  PublisherOA PDFDOI

• Small-Covariance Noise-to-State Stability of Stochastic Systems and Its Applications to Stochastic Gradient Dynamics

  ArXiv.org · 2025-09-29

  preprintOpen accessSenior author

  This paper studies gradient dynamics subject to additive random noise, which may arise from sources such as stochastic gradient estimation, measurement noise, or stochastic sampling errors. To analyze the robustness of such stochastic gradient systems, the concept of small-covariance noise-to-state stability (NSS) is introduced, along with a Lyapunov-based characterization. Furthermore, the classical Polyak-Lojasiewicz (PL) condition on the objective function is generalized to the $\mathcal{K}$-PL condition via comparison functions, thereby extending its applicability to a broader class of optimization problems. It is shown that the stochastic gradient dynamics exhibit small-covariance NSS if the objective function satisfies the $\mathcal{K}$-PL condition and possesses a globally Lipschitz continuous gradient. This result implies that the trajectories of stochastic gradient dynamics converge to a neighborhood of the optimum with high probability, with the size of the neighborhood determined by the noise covariance. Moreover, if the $\mathcal{K}$-PL condition is strengthened to a $\mathcal{K}_\infty$-PL condition, the dynamics are NSS; whereas if it is weakened to a general positive-definite-PL condition, the dynamics exhibit integral NSS. The results further extend to objectives without globally Lipschitz gradients through appropriate step-size tuning. The proposed framework is further applied to the robustness analysis of policy optimization for the linear quadratic regulator (LQR) and logistic regression.

  PublisherOA PDFDOI

Recent grants

• NIH Grant R01HL106788

  NIH · $3.1M · 2015

• Transient behaviors of adapting biological systems

  NIH · $1.6M · 2011–2017

• Monotone Input/Output Systems in Mathematical Biology

  NSF · $180k · 2006–2011

• New techniques for analyzing the long-term behavior of intracellular networks

  NSF · $305k · 2021–2025

• Collaborative Research: Nonlinear Control Analysis and Design Based on Input to State Stability

  NSF · $151k · 2005–2009

Frequent coauthors

• David Angeli

  96 shared

• M. Ali Al-Radhawi

  84 shared

• Rajeev Alur

  51 shared

• Thomas A. Henzinger

  51 shared

• Gerhard Goos

  University of Nottingham

  49 shared

• Jan Van Leeuwen

  Utrecht University

  49 shared

• James M. Greene

  46 shared

• Michael Margaliot

  37 shared

Education

• Ph.D., Mathematics

  University of Florida

  1977

Awards & honors

• Member of the National Academy of Sciences