C. Letellier, I. Sendi˝a-Nadal, E. Bianco-Martinez & M. S. Baptista
A symbolic network-based nonlinear theory for dynamical systems observability
Scientific Reports, 8, 3785, 2018.
When the state of the whole reaction network can be inferred by just measuring the dynamics of a limited set of nodes the system is said to be fully observable. However, as the number of all possible combinations of measured variables and time derivatives spanning the reconstructed state of the system exponentially increases with its dimension, the observability becomes a computationally prohibitive task. Our approach consists in computing the observability coefficients from a symbolic Jacobian matrix whose elements encode the linear, nonlinear polynomial or rational nature of the interaction among the variables. The novelty we introduce in this paper, required for treating large-dimensional systems, is to identify from the symbolic Jacobian matrix the minimal set of variables (together with their time derivatives) candidate to be measured for completing the state space reconstruction. Then symbolic observability coefficients are computed from the symbolic observability matrix. Our results are in agreement with the analytical computations, evidencing the correctness of our approach. Its application to efficiently exploring the dynamics of real world complex systems such as power grids, socioeconomic networks or biological networks is quite promising.
C. Letellier, S. Mangiarotti, I. Sendi˝a-Nadal & O. E. R÷ssler
Topological characterization versus synchronization for assessing (or not) dynamical equivalence
Chaos, 28, 045107, 2018. Online
Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson’s group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson’s early career in interaction with Otto E. R÷ssler.
C. Letellier, I. Sendina-Nadal & L. A. Aguirre
A nonlinear graph-based theory for dynamical network observability
Physical Review E, 98, 020303(R), 2018. Online
A faithful description of the state of a complex dynamical network would require, in principle, the measurement of all its d variables, an infeasible task for high dimensional systems due to practical limitations. However the network dynamics might be observable from a reduced set of measured variables but how to reliably identify the minimum set of variables providing full observability still remains an unsolved problem. In order to tackle this issue from the Jacobian matrix of the governing equations, we construct a pruned fluence graph in which the nodes are the state variables and the links represent only the linear dynamical interdependences after having ignored the nonlinear ones. From this graph, we identify the largest connected subgraphs with no outgoing links in which every node can be reached from any other node in the subgraph. In each one of them, at least one node must be measured to correctly monitor the state of the system in a d-dimensional reconstructed space. Our procedure is here tested by investigating large-dimensional reaction networks. Our results are validated by comparing them with the determinant of the observability matrix which provides a rigorous assessment of the system’s observability.
L. A. Aguirre, L. L. Portes & C. Letellier
Structural, Dynamical and Symbolic Observability : From Dynamical Systems to Networks
eprint arXiv:1806.08909 Online
Abstract The concept of observability of linear systems initiated with Kalman in the mid 1950s. Roughly a decade later, the observability of nonlinear systems appeared. By such definitions a system is either observable or not. Continuous measures of observability for linear systems were proposed in the 1970s and two decades ago were adapted to deal with nonlinear dynamical systems. Related topics developed either independently or as a consequence of these. Observability has been recognized as an important feature to study complex networks, but as for dynamical systems in the beginning the focus has been on determining conditions for a network to be observable. In this relatively new field previous and new results on observability merge either producing new terminology or using terms, with well established meaning in other fields, to refer to new concepts. Motivated by the fact that twenty years have passed since some of these concepts were introduced in the field of nonlinear dynamics, in this paper (i) various aspects of observability will be reviewed, and (ii) it will be discussed in which ways networks could be ranked in terms of observability. The aim is to make a clear distinction between concepts and to understand what does each one contribute to the analysis and monitoring of networks. Some of the main ideas are illustrated with simulations.
C.Draghi, F. Denis, A. TolÚdano & C. Letellier
Parameter identification of a model for prostate cancer treated by intermittent therapy,
Journal of Theoretical Biology, Available online 5 October 2018
Adenocarcinoma is the most frequent cancer affecting the prostate walnut-size gland in the male reproductive system. Such cancer may have a very slow progression or may be associated with a “dark prognosis” when tumor cells are spreading very quickly. Prostate cancers have the particular properties to be marked by the level of prostate specific antigen (PSA) in blood which allows to follow its evolution. At least in its first phase, prostate adenocarcinoma is most often hormone-dependent and, consequently, hormone therapy is a possible treatment. Since few years, hormone therapy started to be provided intermittently for improving patient’s quality of life. Today, durations of on- and off-treatment periods are still chosen empirically, most likely explaining why there is no clear benefit from the survival point of view. We therefore developed a model for describing the interaction between the tumor environment, the PSA produced by hormone-dependent and hormone-independent tumor cells, respectively, and the level of androgens. Model parameters were identified using a genetic algorithm applied to the PSA time series measured in a few patients who initially received prostatectomy and were then treated by intermittent hormone therapy (LHRH analogs and anti-androgen). The measured PSA time series is quite correctly reproduced by free runs over the whole follow-up. Model parameter values allow for distinguishing different types of patient (age and Gleason score) meaning that the model can be individualized. We thus showed that the long-term evolution of the cancer can be affected by durations of on- and off-treatment periods.