Internal reports
  1. E.D. Sontag. A remark on incoherent feedforward circuits as change detectors and feedback controllers. Technical report, arXiv:1602.00162, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology.
    This note analyzes incoherent feedforward loops in signal processing and control. It studies the response properties of IFFL's to exponentially growing inputs, both for a standard version of the IFFL and for a variation in which the output variable has a positive self-feedback term. It also considers a negative feedback configuration, using such a device as a controller. It uncovers a somewhat surprising phenomenon in which stabilization is only possible in disconnected regions of parameter space, as the controlled system's growth rate is varied.

  2. E.D. Sontag. Examples of computation of exact moment dynamics for chemical reaction networks. Technical report, arXiv:1612.02393, 2016. [PDF] Keyword(s): systems biology, biochemical networks, stochastic systems, Chemical Master Equation, chemical reaction networks, moments, molecular networks, complex-balanced networks.
    We review in a unified way results for two types of stochastic chemical reaction systems for which moments can be effectively computed: feedforward networks and complex-balanced networks.

  3. E.D. Sontag. Two-zone tumor tolerance can arise from a simple immunological feedforward motif that estimates tumor growth rates. Technical report, bioRxiv, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, cancer.
    Preprint version of "A dynamical model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination", to appear in Cell Systems 2017. However, the journal version does not include Section 9 on degradation-based IFFL's from this preprint.

  1. E.D. Sontag. Incoherent feedforward motifs as immune change detectors. Technical report, bioRxiv, December 2015. [PDF] Keyword(s): scale invariance, fcd, fold change detection, T cells, incoherent feedforward loops, immunology.
    We speculate that incoherent feedforward loops may be phenomenologically involved in self/nonself discrimination in immune-infection and immune-tumor interactions, acting as "change detectors". In turn, this may result in logarithmic sensing (Weber phenomenon) and even scale invariance (fold-change detection).

  1. J. Barton and E.D. Sontag. Remarks on the energy costs of insulators in enzymatic cascades. Technical report,, December 2014. [PDF] Keyword(s): retroactivity, systems biology, biochemical networks, futile cycles, singular perturbations, modularity.
    The connection between optimal biological function and energy use, measured for example by the rate of metabolite consumption, is a current topic of interest in the systems biology literature which has been explored in several different contexts. In [J. P. Barton and E. D. Sontag, Biophys. J. 104, 6 (2013)], we related the metabolic cost of enzymatic futile cycles with their capacity to act as insulators which facilitate modular interconnections in biochemical networks. There we analyzed a simple model system in which a signal molecule regulates the transcription of one or more target proteins by interacting with their promoters. In this note, we consider the case of a protein with an active and an inactive form, and whose activation is controlled by the signal molecule. As in the original case, higher rates of energy consumption are required for better insulator performance.

  1. Z. Aminzare and E. D. Sontag. Remarks on a population-level model of chemotaxis: advection-diffusion approximation and simulations. Technical report, arxiv:1302.2605, 2013. [PDF]
    This note works out an advection-diffusion approximation to the density of a population of E. coli bacteria undergoing chemotaxis in a one-dimensional space. Simulations show the high quality of predictions under a shallow-gradient regime.

  2. E.D. Sontag. A remark about polynomials with specified local minima and no other critical points. Technical report, arxiv 1302.0759, 2013. [PDF]
    The following observation must surely be "well-known", but it seems worth giving a simple and quite explicit proof. Take any finite subset X of Rn, n>1. Then, there is a polynomial function P:Rn -> R which has local minima on the set X, and has no other critical points. Applied to the negative gradient flow of P, this implies that there is a polynomial vector field with asymptotically stable equilibria on X and no other equilibria. Some trajectories of this vector field are not pre-compact; a complementary observation says that, again for arbitrary X, one can find a vector field with asymptotically stable equilibria on X, no other equilibria except saddles, and all omega-limit sets consisting of singletons.

  1. J. Barton and E.D. Sontag. The energy costs of biological insulators. Technical report,, October 2012. Keyword(s): retroactivity, systems biology, biochemical networks, futile cycles, singular perturbations, modularity.
    Biochemical signaling pathways can be insulated from impedance and competition effects through enzymatic "futile cycles" which consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption, and provide evidence, through the computational analysis of a simplified physical model, to support this hypothesis.

  2. M. Marcondes de Freitas and E.D. Sontag. Remarks on random dynamical systems with inputs and outputs and a small-gain theorem for monotone RDS. Technical report,, July 2012. Keyword(s): random dynamical systems, monotone systems.

  1. E.D. Sontag. Remarks on invariance of population distributions for systems with equivariant internal dynamics. Technical report, arxiv.1108.3245, August 2011. [PDF] Keyword(s): scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, jump Markov processes.

  1. E.D. Sontag. An observation regarding systems which converge to steady states for all constant inputs, yet become chaotic with periodic inputs. Technical report, arxiv 0906.2166, 2009. [PDF]

  1. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Technical report 381, Institute for Dynamical Systems, University of Bremen, 1996.

  1. F. Albertini and E.D. Sontag. Some connections between chaotic dynamical systems and control systems. Technical report SYCON-90-13, Rutgers Center for Systems and Control, 1990.

  1. E.D. Sontag. Sigmoids distinguish more efficiently than Heavisides. Technical report SYCON-89-12, Rutgers Center for Systems and Control, 1989. Keyword(s): neural networks.

  1. E.D. Sontag. Integrability of certain distributions associated to actions on manifolds and an introduction to Lie-algebraic control. Technical report SYCON-88-04, Rutgers Center for Systems and Control, 1988.

  2. E.D. Sontag. Some remarks on the backpropagation algorithm for neural net learning. Technical report SYCON-88-02, Rutgers Center for Systems and Control, 1988. [PDF] Keyword(s): neural networks, neural networks.
    This is a very old informal report that discusses the study of local minima of quadratic loss functions for fitting errors in sigmoidal neural net learning. It also includes several remarks concerning the growth of weights during gradient descent. There is nothing very interesting here - far better knowledge is now available - but the report was placed here by request.

  1. E.D. Sontag and H.J. Sussmann. Optimization algorithms for image restoration and segmentation. Technical report 34, Rutgers Center for Computer Aids for Industrial Productivity, 1987.

  1. E.D. Sontag and D.E. Stevenson. Remarks on multi-server, multi-priority queuing models related to MVS job scheduling. Technical report TM-81-45281-1, Bell Telephone Labs., 1981.



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