Identification of Piecewise Linear Models of Complex Dynamical Systems

The paper addresses the realization and identification problem or a subclass of piecewise-affine hybrid systems. The paper provides necessary and sufficient conditions for existence of a realization, a characterization of minimality, and an identification algorithm for this subclass of hybrid systems. The considered system class and the identification problem are motivated by applications in systems biology

Analysis of parametric biological models with non-linear dynamics

In this paper we present recent results on parametric analysis of biological models. The underlying method is based on the algorithms for computing trajectory sets of hybrid systems with polynomial dynamics. The method is then applied to two case studies of biological systems: one is a cardiac cell model for studying the conditions for cardiac abnormalities, and the second is a model of insect nest-site choice.Comment: In Proceedings HSB 2012, arXiv:1208.315

Hybrid system modeling and identification of cell biology systems: perspectives and challenges

Abstract: Some perspectives and challenges in research on the crossing of system biology, hybrid system formulations and system identification are outlined. Emphasis is given on the hybrid, gray box modeling of interactions between the different abstract levels of organization typically recognized in (micro-)organisms, its associated identification problem and optimal experimental (input) design

Alternative rammeverk for analyse av genregulatoriske nettverk med tidsforsinkelse

When trying to understand the role and functioning of a gene regulatory network (GRN), the first step is to assemble components of the network and interactions between them. It is important that models are kept simple but nevertheless capture the key processes of the real system. There is a large body of theoretical and experimental results showing that underlying processes of gene regulation, such as transcription and translation, do not occur instantaneously. Therefore the delay effects are everywhere in GRNs, but they are not always well-represented in mathematical models. The scope of the present work is to incorporate delays into a well-established differential equation model for GRNs and to apply alternative mathematical frameworks for analysis of the obtained delayed system. Due to a huge amount of equations and parameters involved, it is widely accepted that no analysis is possible without a considerable simplification of the underlying model. The non-linear, switch-like character of many of interactions in gene expression has motivated the most common simplification, so-called Boolean-like formalism. To simplify the model one uses the step functions and the corresponding limit system. It leads to the subdivision of the phase space into regions at the boundary of which discontinuities may occur. Using this simplification for analysis of delayed GRNs we face two main mathematical challenges: to analyze the stability properties of steady states and to reconstruct the limit trajectories in switching domains. Papers I and II of my thesis are addressed to answer these two questions. There is one more effect which is indisputably important for any reasonable model of GRNs, namely an effect of stochasticity, which may be caused by uncertainty in data, random fluctuations in the system, or simply due to a large number of interacting genes. In Paper III we propose an analytic stochastic modeling approach, which incorporates intrinsic noise effects directly into a well established deterministic models of GRNs with and without delay, and study the dynamics of the resulting systems. In Paper IV we suggest a method which covers very general Boolean genetic networks with delay and thus opens for a more complete qualitative analysis of such networks. The method extends the Filippov theory of differential inclusions to the case of multivalued Volterra operators. We believe that the proposed frameworks can provide good insights into deeper understanding of the complicated biological and chemical processes associated with genetic regulation.For å forstå rollen og funksjonene av et genregulatorisk nettverk (GRN) er det først og fremst nødvendig å sette sammen komponentene av nettverket og å analysere samhandlinger mellom dem. Det er viktig at modeller beholdes enkle, men samtidig gir et realistisk bilde av nøkkelprosessene i det reelle systemet. Det finnes flere teoretiske og eksperimentelle resultater som viser til at de genregulatoriske prosessene som transkripsjon og translasjon ikke skjer simultant. Tidsforsinkelser er normalt i GRN, men de er ikke representert i de fleste matematiske modeller. Hensikten med denne avhandlingen er å inkorporere tidsforsinkelser inn i veletablerte differensialligning-modeller av GRN og å benytte alternative rammeverk for analyse av de nyutviklede modellene med tidsforsinkelse. Grunnet mange ligninger og parametre involvert i systemet er det vanlig å forenkle den underliggende modellen. Den ikke-lineære, sprangvise oppførselen av mange variabler i gen uttrykk har motivert den mest utbredte forenklingen, såkalt Boolsk formalisme. For å forenkle modellen bruker man i så fall trinnfunksjoner og det tilhørende grensesystemet. Det fører til en oppdeling av faserommet i regulære områder, og ved grensene mellom disse områdene kan diskontinuitet forekomme. Bruk av denne forenklingen for å analysere tidsforsinket GRN medfører to matematiske utfordringer: å undersøke stabilitet til likevektspunkter og å rekonstruere løsningskurver i singulære domener. Artikler I og II av min avhandling har til hensikt å svare på disse to spørsmålene. Spesielt viktig for en god GRN modell er stokastiske effekter. Disse stokastiske effektene kan forekomme på grunn av usikkerhet i dataene, tilfeldige endringer i systemet eller av den grunn at antall av gen interaksjoner er stort. I artikkel III setter vi opp en analytisk stokastisk modell ved å inkorporere indre støy inn i veletablerte modeller av GRN med og uten tidsforsinkelse samt å undersøke dynamikk til de resulterende systemene. I artikkel IV foreslår vi en metode som dekker generelle Boolske genetiske nettverk med tidsforsinkelse. Dette åpner for en mer komplett kvalitativ analyse av slike nettverk. Metoden utvider Filippovs teori av differensialinklusjoner til multivaluerte Volterra operatorer. Vi mener at de foreslåtte rammeverkene vil kunne gi innsikt i en grundigere forståelse av de kompliserte biologiske og kjemiske prosessene som beskriver gen regulering.Center for Integrative Genetic

Change point detection with application to the identification of a switching process

International audienceThis paper deals with the change point detection problem with application to filtering and on-line identification of simple hybrid systems. The proposed method allows to obtain fast estimators for the unknown switch times and parameters. Numerical simulations with noisy data are provided

Qualitative modeling in computational systems biology

The human body is composed of a large collection of cells,\the building blocks of life". In each cell, complex networks of biochemical processes contribute in maintaining a healthy organism. Alterations in these biochemical processes can result in diseases. It is therefore of vital importance to know how these biochemical networks function. Simple reasoning is not su±cient to comprehend life's complexity. Mathematical models have to be used to integrate information from various sources for solving numerous biomedical research questions, the so-called systems biology approach, in which quantitative data are scarce and qualitative information is abundant. Traditional mathematical models require quantitative information. The lack in ac- curate and su±cient quantitative data has driven systems biologists towards alternative ways to describe and analyze biochemical networks. Their focus is primarily on the anal- ysis of a few very speci¯c biochemical networks for which accurate experimental data are available. However, quantitative information is not a strict requirement. The mutual interaction and relative contribution of the components determine the global system dy- namics; qualitative information is su±cient to analyze and predict the potential system behavior. In addition, mathematical models of biochemical networks contain nonlinear functions that describe the various physiological processes. System analysis and parame- ter estimation of nonlinear models is di±cult in practice, especially if little quantitative information is available. The main contribution of this thesis is to apply qualitative information to model and analyze nonlinear biochemical networks. Nonlinear functions are approximated with two or three linear functions, i.e., piecewise-a±ne (PWA) functions, which enables qualitative analysis of the system. This work shows that qualitative information is su±cient for the analysis of complex nonlinear biochemical networks. Moreover, this extra information can be used to put relative bounds on the parameter values which signi¯cantly improves the parameter estimation compared to standard nonlinear estimation algorithms. Also a PWA parameter estimation procedure is presented, which results in more accurate parameter estimates than conventional parameter estimation procedures. Besides qualitative analysis with PWA functions, graphical analysis of a speci¯c class of systems is improved for a certain less general class of systems to yield constraints on the parameters. As the applicability of graphical analysis is limited to a small class of systems, graphical analysis is less suitable for general use, as opposed to the qualitative analysis of PWA systems. The technological contribution of this thesis is tested on several biochemical networks that are involved in vascular aging. Vascular aging is the accumulation of changes respon- sible for the sequential alterations that accompany advancing age of the vascular system and the associated increase in the chance of vascular diseases. Three biochemical networks are selected from experimental data, i.e., remodeling of the extracellular matrix (ECM), the signal transduction pathway of Transforming Growth Factor-¯1 (TGF-¯1) and the unfolded protein response (UPR). The TGF-¯1 model is constructed by means of an extensive literature search and con- sists of many state equations. Model reduction (the quasi-steady-state approximation) reduces the model to a version with only two states, such that the procedure can be visual- ized. The nonlinearities in this reduced model are approximated with PWA functions and subsequently analyzed. Typical results show that oscillatory behavior can occur in the TGF-¯1 model for speci¯c sets of parameter values. These results meet the expectations of preliminary experimental results. Finally, a model of the UPR has been formulated and analyzed similarly. The qualitative analysis yields constraints on the parameter values. Model simulations with these parameter constraints agree with experimental results

Data Driven Techniques for Modeling Coupled Dynamics in Transient Processes

We study the problem of modeling coupled dynamics in transient processes that happen in a network. The problem is considered at two levels. At the node level, the coupling between underlying sub-processes of a node in a network is considered. At the network level, the direct influence among the nodes is considered. After the model is constructed, we develop a network-based approach for change detection in high dimension transient processes. The overall contribution of our work is a more accurate model to describe the underlying transient dynamics either for each individual node or for the whole network and a new statistic for change detection in multi-dimensional time series. Specifically, at the node level, we developed a model to represent the coupled dynamics between the two processes. We provide closed form formulas on the conditions for the existence of periodic trajectory and the stability of solutions. Numerical studies suggest that our model can capture the nonlinear characteristics of empirical data while reducing computation time by about 25% on average, compared to a benchmark modeling approach. In the last two problems, we provide a closed form formula for the bound in the sparse regression formulation, which helps to reduce the effort of trial and error to find an appropriate bound. Compared to other benchmark methods in inferring network structure from time series, our method reduces inference error by up to 5 orders of magnitudes and maintain better sparsity. We also develop a new method to infer dynamic network structure from a single time series. This method is the basis for introducing a new spectral graph statistic for change detection. This statistic can detect changes in simulation scenario with modified area under curve (mAUC) of 0.96. When applying to the problem of detecting seizure from EEG signal, our statistic can capture the physiology of the process while maintaining a detection rate of 40% by itself. Therefore, it can serve as an effective feature to detect change and can be added to the current set of features for detecting seizures from EEG signal