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Publications about 'identifiability'
Articles in journal or book chapters
  1. J. M. Greene, C. Sanchez-Tapia, and E.D. Sontag. Mathematical details on a cancer resistance model. Frontiers in Bioengineering and Biotechnology, 8:501: 1-27, 2020. [PDF] [doi:10.3389/fbioe.2020.00501] Keyword(s): resistance, chemotherapy, phenotype, optimal control, singular controls, cancer, oncology.
    Abstract:
    One of the most important factors limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy. In this work, we expound on the details relating to an optimal control problem outlined in our previous paper (Greene et al., 2018). The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebraic techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.


  2. D.K. Agrawal, R. Marshall, V. Noireaux, and E.D. Sontag. In vitro implementation of robust gene regulation in a synthetic biomolecular integral controller. Nature Communications, 10:1-12, 2019. [PDF] Keyword(s): tracking, synthetic biology, integral feedback, TX/TL, systems biology, dynamical systems, adaptation, internal model principle, identifiability.
    Abstract:
    Cells respond to biochemical and physical internal as well as external signals. These signals can be broadly classified into two categories: (a) ``actionable'' or ``reference'' inputs that should elicit appropriate biological or physical responses such as gene expression or motility, and (b) ``disturbances'' or ``perturbations'' that should be ignored or actively filtered-out. These disturbances might be exogenous, such as binding of nonspecific ligands, or endogenous, such as variations in enzyme concentrations or gene copy numbers. In this context, the term robustness describes the capability to produce appropriate responses to reference inputs while at the same time being insensitive to disturbances. These two objectives often conflict with each other and require delicate design trade-offs. Indeed, natural biological systems use complicated and still poorly understood control strategies in order to finely balance the goals of responsiveness and robustness. A better understanding of such natural strategies remains an important scientific goal in itself and will play a role in the construction of synthetic circuits for therapeutic and biosensing applications. A prototype problem in robustly responding to inputs is that of ``robust tracking'', defined by the requirement that some designated internal quantity (for example, the level of expression of a reporter protein) should faithfully follow an input signal while being insensitive to an appropriate class of perturbations. Control theory predicts that a certain type of motif, called integral feedback, will help achieve this goal, and this motif is, in fact, a necessary feature of any system that exhibits robust tracking. Indeed, integral feedback has always been a key component of electrical and mechanical control systems, at least since the 18th century when James Watt employed the centrifugal governor to regulate steam engines. Motivated by this knowledge, biological engineers have proposed various designs for biomolecular integral feedback control mechanisms. However, practical and quantitatively predictable implementations have proved challenging, in part due to the difficulty in obtaining accurate models of transcription, translation, and resource competition in living cells, and the stochasticity inherent in cellular reactions. These challenges prevent first-principles rational design and parameter optimization. In this work, we exploit the versatility of an Escherichia coli cell-free transcription-translation (TXTL) to accurately design, model and then build, a synthetic biomolecular integral controller that precisely controls the expression of a target gene. To our knowledge, this is the first design of a functioning gene network that achieves the goal of making gene expression track an externally imposed reference level, achieves this goal even in the presence of disturbances, and whose performance quantitatively agrees with mathematical predictions.


  3. S. Barish, M.F. Ochs, E.D. Sontag, and J.L. Gevertz. Evaluating optimal therapy robustness by virtual expansion of a sample population, with a case study in cancer immunotherapy. Proc Natl Acad Sci USA, 114:E6277-E6286, 2017. [WWW] [PDF] [doi:10.1073/pnas.1703355114] Keyword(s): cancer, oncolytic therapy, immunotherapy, optimal therapy, identifiability.
    Abstract:
    This paper proposes a technique that combines experimental data, mathematical modeling, and statistical analyses for identifying optimal treatment protocols that are robust with respect to individual variability. Experimental data from a small sample population is amplified using bootstrapping to obtain a large number of virtual populations that statistically match the expected heterogeneity. Alternative therapies chosen from among a set of clinically-realizable protocols are then compared and scored according to coverage. As proof of concept, the method is used to evaluate a treatment with oncolytic viruses and dendritic cell vaccines in a mouse model of melanoma. The analysis shows that while every scheduling variant of an experimentally-utilized treatment protocol is fragile (non-robust), there is an alternative region of dosing space (lower oncolytic virus dose, higher dendritic cell dose) for which a robust optimal protocol exists.


  4. E.D. Sontag. Dynamic compensation, parameter identifiability, and equivariances. PLoS Computational Biology, 13:e1005447, 2017. Note: Preprint was in bioRxiv https://doi.org/0.1101/095828, 2016.[WWW] [PDF] Keyword(s): fcd, fold-change detection, scale invariance, dynamic compensation, identifiability, observability.
    Abstract:
    A recent paper by Karin et al. introduced a mathematical notion called dynamical compensation (DC) of biological circuits. DC was shown to play an important role in glucose homeostasis as well as other key physiological regulatory mechanisms. Karin et al.\ went on to provide a sufficient condition to test whether a given system has the DC property. Here, we show how DC is a reformulation of a well-known concept in systems biology, statistics, and control theory -- that of parameter structural non-identifiability. Viewing DC as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for DC. We obtain as a special case the sufficient criterion discussed by Karin et al. We also draw connections to system equivalence and to the fold-change detection property.


  5. M. Chaves, A. M. Sengupta, and E.D. Sontag. Geometry and topology of parameter space: investigating measures of robustness in regulatory networks. J. of Mathematical Biology, 59:315-358, 2009. [PDF] Keyword(s): identifiability, robust, robustness, geometry.
    Abstract:
    The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed.


  6. A. Dayarian, M. Chaves, E.D. Sontag, and A. M. Sengupta. Shape, Size and Robustness: Feasible Regions in the Parameter Space of Biochemical Networks. PLoS Computational Biology, 5:e10000256, 2009. [PDF] Keyword(s): identifiability, robust, robustness, geometry.
    Abstract:
    The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In recent work, we emphasized that topology and geometry matter, as well as volume. In this paper, and using the segment polarity gene network to illustrate our approach, we show that random walks in parameter space and how they exit the feasible region provide a rich perspective on the different modes of failure of a model. In particular, for the segment polarity network, we found that, between two alternative ways of activating Wingless, one is more robust. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.


  7. E.D. Sontag, Y. Wang, and A. Megretski. Input classes for identification of bilinear systems. IEEE Transactions Autom. Control, 54:195-207, 2009. Note: Also arXiv math.OC/0610633, 20 Oct 2006, and short version in ACC'07.[PDF] Keyword(s): realization theory, observability, identifiability, bilinear systems.
    Abstract:
    This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification.


  8. M. Chaves, E.D. Sontag, and R. Albert. Methods of robustness analysis for Boolean models of gene control networks. IET Systems Biology, 153:154-167, 2006. [PDF] Keyword(s): systems biology, biochemical networks, boolean systems, identifiability, robust, robustness, geometry, Boolean, segment polarity network, gene and protein networks, hybrid systems.
    Abstract:
    As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes.


  9. E.D. Sontag. For differential equations with r parameters, 2r+1 experiments are enough for identification. J. Nonlinear Sci., 12(6):553-583, 2002. [PDF] Keyword(s): identifiability, observability, systems biology, biochemical networks, parameter identification.
    Abstract:
    Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.


  10. E.D. Sontag. Recurrent neural networks: Some systems-theoretic aspects. In M. Karny, K. Warwick, and V. Kurkova, editors, Dealing with Complexity: a Neural Network Approach, pages 1-12. Springer-Verlag, London, 1997. [PDF] Keyword(s): neural networks, recurrent neural networks.
    Abstract:
    This paper provides an exposition of some recent results regarding system-theoretic aspects of continuous-time recurrent (dynamic) neural networks with sigmoidal activation functions. The class of systems is introduced and discussed, and a result is cited regarding their universal approximation properties. Known characterizations of controllability, observability, and parameter identifiability are reviewed, as well as a result on minimality. Facts regarding the computational power of recurrent nets are also mentioned.


  11. Y. Wang and E.D. Sontag. Orders of input/output differential equations and state-space dimensions. SIAM J. Control Optim., 33(4):1102-1126, 1995. [PDF] [doi:http://dx.doi.org/10.1137/S0363012993246828] Keyword(s): identifiability, observability, realization theory.
    Abstract:
    This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems.


  12. F. Albertini and E.D. Sontag. State observability in recurrent neural networks. Systems Control Lett., 22(4):235-244, 1994. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(94)90054-X] Keyword(s): neural networks, recurrent neural networks, observability, identifiability.
    Abstract:
    This paper concerns recurrent networks x'=s(Ax+Bu), y=Cx, where s is a sigmoid, in both discrete time and continuous time. Our main result is that observability can be characterized, if one assumes certain conditions on the nonlinearity and on the system, in a manner very analogous to that of the linear case. Recall that for the latter, observability is equivalent to the requirement that there not be any nontrivial A-invariant subspace included in the kernel of C. We show that the result generalizes in a natural manner, except that one now needs to restrict attention to certain special "coordinate" subspaces.


  13. F. Albertini and E.D. Sontag. For neural networks, function determines form. Neural Networks, 6(7):975-990, 1993. [PDF] Keyword(s): neural networks, identifiability, recurrent neural networks, realization theory, observability, neural networks.
    Abstract:
    This paper shows that the weights of continuous-time feedback neural networks x'=s(Ax+Bu), y=Cx (where s is a sigmoid) are uniquely identifiable from input/output measurements. Under very weak genericity assumptions, the following is true: Assume given two nets, whose neurons all have the same nonlinear activation function s; if the two nets have equal behaviors as "black boxes" then necessarily they must have the same number of neurons and -except at most for sign reversals at each node- the same weights. Moreover, even if the activations are not a priori known to coincide, they are shown to be also essentially determined from the external measurements.


  14. Y. Wang and E.D. Sontag. Algebraic differential equations and rational control systems. SIAM J. Control Optim., 30(5):1126-1149, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
    Abstract:
    It is shown that realizability of an input/output operators by a finite-dimensional continuous-time rational control system is equivalent to the existence of a high-order algebraic differential equation satisfied by the corresponding input/output pairs ("behavior"). This generalizes, to nonlinear systems, the classical equivalence between autoregressive representations and finite dimensional linear realizability.


  15. Y. Wang and E.D. Sontag. Generating series and nonlinear systems: analytic aspects, local realizability, and i/o representations. Forum Math., 4(3):299-322, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
    Abstract:
    This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations.


  16. E.D. Sontag and Y. Wang. Input/output equations and realizability. In Realization and modelling in system theory (Amsterdam, 1989), volume 3 of Progr. Systems Control Theory, pages 125-132. Birkhäuser Boston, Boston, MA, 1990. [PDF] Keyword(s): identifiability, observability, realization theory.


  17. Y. Wang and E.D. Sontag. On two definitions of observation spaces. Systems Control Lett., 13(4):279-289, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90116-3] Keyword(s): observability, identifiability, observables, observation space, realization theory.
    Abstract:
    This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls.


  18. E.D. Sontag. Bilinear realizability is equivalent to existence of a singular affine differential I/O equation. Systems Control Lett., 11(3):181-187, 1988. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(88)90057-6] Keyword(s): identification, identifiability, observability, observation space.
    Abstract:
    For continuous time analytic input/output maps, the existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability. A similar result holds for the problem of immersion into bilinear systems. The proof is very analogous to that of the corresponding, and previously known, result for discrete time.


  19. E.D. Sontag. On the observability of polynomial systems. I. Finite-time problems. SIAM J. Control Optim., 17(1):139-151, 1979. [PDF] Keyword(s): observability, observability, polynomial systems.
    Abstract:
    Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.


  20. E.D. Sontag and Y. Rouchaleau. On discrete-time polynomial systems. Nonlinear Anal., 1(1):55-64, 1976. [PDF] Keyword(s): identifiability, observability, polynomial systems, realization theory, discrete-time.
    Abstract:
    Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and introduces the subject of realization theory for this class of systems.


Conference articles
  1. E.D. Sontag, Y. Wang, and A. Megretski. Remarks on Input Classes for Identification of Bilinear Systems. In Proceedings American Control Conf., New York, July 2007, pages 4345-4350, 2007. Keyword(s): realization theory, observability, identifiability, bilinear systems.


  2. E.D. Sontag and Y. Wang. Orders of I/O equations and uniformly universal inputs. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 1994, IEEE Publications, 1994, pages 1270-1275, 1994. Keyword(s): identifiability, observability, realization theory.


  3. F. Albertini and E.D. Sontag. Identifiability of discrete-time neural networks. In Proc. European Control Conf., Groningen, June 1993, pages 460-465, 1993. Keyword(s): neural networks, recurrent neural networks.


  4. F. Albertini and E.D. Sontag. Uniqueness of weights for recurrent nets. In Systems and Networks: Mathematical Theory and Applications, Proc. MTNS '93, Vol. 2, Akad. Verlag, Regensburg, pages 599-602, 1993. Note: Full version, never submitted for publication, is here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/93mtns-nn-extended.pdf. [PDF] Keyword(s): neural networks, identifiability, recurrent neural networks.
    Abstract:
    This paper concerns recurrent networks x'=s(Ax+Bu), y=Cx, where s is a sigmoid, in both discrete time and continuous time. The paper establishes parameter identifiability under stronger assumptions on the activation than in "For neural networks, function determines form", but on the other hand deals with arbitrary (nonzero) initial states.


  5. E.D. Sontag and Y. Wang. I/O equations in discrete and continuous time. In Proc. IEEE Conf. Decision and Control, Tucson, Dec. 1992, IEEE Publications, 1992, pages 3661-3662, 1992. Keyword(s): identifiability, observability, realization theory.


  6. E.D. Sontag and Y. Wang. I/O equations for nonlinear systems and observation spaces. In Proc. IEEE Conf. Decision and Control, Brighton, UK, Dec. 1991, IEEE Publications, 1991, pages 720-725, 1991. [PDF] Keyword(s): identifiability, observability, realization theory.
    Abstract:
    This paper studies various types of input/output representations for nonlinear continuous time systems. The algebraic and analytic i/o equations studied in previous papers by the authors are generalized to integral and integro-differential equations, and an abstract notion is also considered. New results are given on generic observability, and these results are then applied to give conditions under which that the minimal order of an equation equals the minimal possible dimension of a realization, just as with linear systems but in contrast to the discrete time nonlinear theory.


  7. Y. Wang and E.D. Sontag. Realization of families of generating series: differential algebraic and state space equations. In Proc. 11th IFAC World Congress, Tallinn, former USSR, 1990, pages 62-66, 1990. Keyword(s): identifiability, observability, realization theory.


  8. Y. Wang and E.D. Sontag. A new result on the relation between differential-algebraic realizability and state space realizations. In Proc. Conf. Info. Sciences and Systems, Johns Hopkins University Press, 1989, pages 143-147, 1989. Keyword(s): observables, observation space, identifiability, observability, realization theory.


  9. Y. Wang and E.D. Sontag. Realization and input/output relations: the analytic case. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 1975-1980, 1989. IEEE. Keyword(s): identifiability, observability, realization theory.


Internal reports
  1. J.M. Greene, C. Sanchez-Tapia, and E.D. Sontag. Mathematical details on a cancer resistance model. Technical report, bioRxiv 2018/475533, 2018. [PDF] Keyword(s): identifiability, drug resistance, chemotherapy, optimal control theory, singular controls, oncology, cancer.
    Abstract:
    The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy . In this work, the control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie techniques. A structural identfiability analysis is also presented, demonstrating that patient-specfic parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy.



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