Tackling the Inverse Problem for Non-Autonomous Systems: Application to the Life Sciences

145 Pages·2014·2.48 MB·English

by  Tomislav Stankovski (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Tomislav Stankovski
Tackling the Inverse
Problem for
Non-Autonomous
Systems
Application to the Life Sciences
Springer Theses
Recognizing Outstanding Ph.D. Research
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Tomislav Stankovski
Tackling the Inverse
Problem for
Non-Autonomous Systems
Application to the Life Sciences
Doctoral Thesis accepted by
Lancaster University, UK
123
Author Supervisor
Dr. TomislavStankovski Prof.Aneta Stefanovska
Department of Physics Department of Physics
Lancaster University Lancaster University
Lancaster Lancaster
UK UK
ISSN 2190-5053 ISSN 2190-5061 (electronic)
ISBN 978-3-319-00752-6 ISBN 978-3-319-00753-3 (eBook)
DOI 10.1007/978-3-319-00753-3
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Supervisor’s Foreword
In simple terms science might be defined as the systematic observation of nature
andofhownaturalprocessesevolveintimeandspace.Suchprocessescaninclude
thebeatingoftheheartofalivinghumanorothermammal,themovementsofthe
planets, or simply how human society works. In all these cases, physics tries to
generatea model based on data collectedovertime—a time interval thatdepends
on how fast the processes occur, which may in some cases happen over centuries
and in others over seconds or microseconds. Science attempts to develop the
models that can most comprehensively link the causes and consequences of the
processes in question. One of the most frequently-used approaches, and arguably
the most useful one, is the Bayesian approach. It is based on Bayes’ theorem,
whichiscentraltotheinverseproblemapproachanddynamicalinference,seeking
toanswerthequestion:givenaseriesofdataresultingfromobservations,whatcan
we deduce about the nature of the system or the process that generated that data?
Thomas Bayes was fortunate that his friend Richard Price significantly edited
and updated his work and read it to the Royal Society in 1763, a year after his
death. It was published in the Philosophical Transactions of the Royal Society of
London the following year. The ideas gained limited exposure until they were
independentlyrediscoveredandfurtherdevelopedbyLaplace,whofirstpublished
the modern formulation in his 1812 Théorie analytique des probabilités.
The classical approach to statistics defines the probability of an event as ‘‘The
numberoftimestheeventoccursoverthetotalnumberoftrials,inthelimitofan
infinite series of equiprobable repetitions’’, which is quite limited. Many of these
limitations can be avoided and paradoxes resolved by taking a Bayesian stance
about probabilities. Bayes defines probability as:
The probability of any event is the ratio between the value at which an
expectation depending on the happening of the event ought to be computed, and
the value of the thing expected upon its happening.
HoweverevenBayeshimselfmightnothaveembracedthebroadinterpretation
now called Bayesian. It is difficult to assess Bayes’ philosophical views on
probability, since his work does not go into questions of interpretation.
TodayBayesianprobabilityisthenamegiventoseveralrelatedinterpretations
of probability. To evaluate the probability of a hypothesis, the Bayesian
v
vi Supervisor’sForeword
probability specifies some prior probability, which is then updated in the light of
new, relevant data. ‘‘Bayesian’’hasbeen used in this sense since its rebirth in the
1950s. Advances in computer technology have allowed scientists from many
disciplines to extend and use this approach in diverse fields. Sir Harold Jeffreys ,
whowrotethebookTheoryofProbability,whichfirstappearedin1939,playedan
important role in the revival of the Bayesian view of probability. He wrote that
Bayes’ theorem ‘‘is to the theory of probability what Pythagoras’s theorem is to
geometry’’.
Recently,wehavefacedhugedevelopmentsinboththetheoryandapplications
of Bayes theorem. The applications span practically every aspect of science,
including particle physics, astrophysics, cosmology, geophysics, and medical and
biological physics. The work by Tomislav Stankovski deals with one particular
aspect of Bayesian inference: it focuses on inference of the dynamical properties
of oscillatory systems.
The work is based on the development of Feynman’s path integral whose
central idea is that, for the motion of a particle between two points in space, all
possible connecting trajectories should be considered and a probability amplitude
assignedtoeachoneofthem.Thispathintegralgiveslikelihoodofobservationof
thedynamicaltrajectoryforagivensetofdistributionsofmodelparameters.Once
theactualdynamicaltrajectoryismeasuredintheexperimentthedistributionsfor
the set of model parameters can be improved using Bayes theorem.
The other important ingredient is based on recent developments in the under-
standing of the nonlinear dynamics of oscillatory processes, and the theory of
synchronization in particular. The main objective of Tomislav’s thesis, now pre-
sentedasabook,wastostudy,detectandunderstandingreaterdetailtheeffectof
external dynamical influences on interacting selfsustained oscillators.
The work was motivated by problems associated with biological systems and
thecardiovascularsysteminparticular.Notsurprisingly,lifecannotbeunderstood
properly without a self-consistent theory of non-autonomous systems and associ-
ated methods to infer such dynamical characteristics from measured data. How-
ever,non-autonomicityischaracteristicofmanysystems.Inreality,practicallyall
systemsarenon-autonomous,thoughsomecansafelybesimplifiedandstudiedas
closed or isolated. Hence, I can envisage very wide applicability of the method
proposed in this book.
Despite a long-standing general awareness of non-autonomous systems, rela-
tively little has been done in this field. Mathematics has included mainly the
process and the skew product flow formalism, while the physics approach to date
has mainly been to reduce the dynamics to being autonomous by adding an extra
dimension for the time-dependence.
Supervisor’sForeword vii
Tomislav’s work is one of the first systematic approaches to the treatment of
non-autonomoussystemsfromaphysicsperspectiveand,inparticular,todoingso
inaninverseapproach. Asamultitudeofdataarewidelyavailable today,wecan
expect highly significant advances in many fields to be facilitated by the results
presented in this book.
Lancaster, May 2013 Prof. Aneta Stefanovska
Acknowledgments
First and foremost I offer my sincerest gratitude to my supervisor, Prof. Aneta
Stefanovska, who introduced me to the world of nonlinear biomedical physics.
I am very grateful for her unreserved support throughout my thesis, guiding me
with her knowledge, experience and enthusiasm.
IwishtosincerelythankProf.PeterV.E.McClintockforhissupport,guidance
and willingness to share his great experience and wisdom.
IespeciallywishtothankDr.AndreaDuggento,withwhomIhavecollaborated
on much of the work presented. He was great tutor, colleague and friend, and it
was a great pleasure to work with him.
IwishtothankProf.DwainEckbergforthecollaborationandforprovidingthe
data for the physiological study. I appreciate greatly the valuable discussion with
Dr. Martin Rasmussen, Prof. Peter Kloeden and Dr. Martin Wechselberger
regardingthestudyofnon-autonomoussystems.IwishalsotothankProf.Metodija
Kamilovskiforhisadvicesonanaloguesimulation.
A special gratitude for both scientific assistance and kind friendship goes to
manyofmycolleaguesDr.AlanBernjak,Dr.AlirezaBahraminasab,Dr.Lawrence
Sheppard, Dr. David Kenwright, Dr. Rodrigue Tindjong, Dr. Martin Horvat,
Dr. Uchechukwu Vincent, Adam Bradbury, Dmytro Iatsenko. For valuable
discussionandimprovingthewrittenthesis,Ialsowishtothankmycolleaguesand
friendsPhilClemson,GemmaLancaster,PhilipStephens,YvannStephens,Rachel
Sparks,Spase Petkoski andDr.YevhenSuprunenko.
Iamespecially grateful tomyfriends andfamily, for their encouragementand
support. I thank my father for inspiring me, and my mother for giving me the
mental strength to cope with the challenges throughout my studies.
Last,butbynomeansleast,Iwishtoexpressmygratitudetomybelovedwife.
Her unreserved and unconditional support, understanding and encouragement
helpedmetogetthroughoverthelast4years.Iwillneverbegratefulenoughtoher.
ix
Abstract
The common assumption that a dynamical system found in nature can be
considered as isolated and autonomous is frequently a poor approximation. In
reality, there are always external influences, and these are often too strong to
ignore. In the case of an interacting oscillatory systems, they may, e.g. modify
theirnaturalfrequenciesorcouplingamplitudes.Themainobjectiveofthisthesis
istostudy,detectandunderstandingreaterdetailtheeffectofexternaldynamical
influences on interacting self-sustained oscillators.
Theoretical framework for the analysis of synchronization between non-
autonomousoscillatingsystemsisdiscussed.Multiple-scaleanalysisisappliedon
a phase oscillators model with slowly varying frequency. This analysis revealed
the analytic form of the synchronization state with respect to slow and fast time-
variations. Limit-cycle oscillators are used to study amplitude dynamics and to
investigate synchronization transitions, which occur in the bifurcation points
where the equilibrium solution for the phase difference and amplitudes changes
their stability. Bifurcation diagrams as functions of coupling parameters are also
constructed. In a case of non-autonomous interacting oscillators, the phase
difference varies dynamically, the external influences can be the cause for
synchronization transitions between different synchronization orders, and lag
synchronization is hardly achievable. It is also demonstrated that the time-
variations of the form of the coupling function alone can be the cause for
synchronization transitions.
A method is introduced for analysis of interactions between time-dependent
coupled oscillators, based on the signals they generate. It distinguishes
unsynchronized dynamics from noise-induced phase slips, and enables the
evolution of the coupling functions and other parameters to be followed. The
technique is based on Bayesian inference of the time-evolving parameters,
achieved by shaping the prior densities to incorporate knowledge of previous
samples.Thedynamicscanbeinferredfromphasevariables,inwhichcaseafinite
number of Fourier base functions are used, or from state variables exploiting the
model state base functions. The latter is used for detection of generalized
synchronization. The method is tested numerically and applied to reveal and
quantify the time-varying nature of synchronization, directionality and coupling
functionsfromcardiorespiratoryandanalogue signals.Itisfound that,incontrast
xi