Subject: Subject: ARTICLES - Part (1/5) of UK Nonlinear News

Date: Wed, 01 Sep 2004 15:27:45 +0100

From: UK Nonlinear News <uk-nonl@...>

Organization: UCL Centre for Nonlinear Dynamics

Newsgroups: sci.nonlinear

Subject: ARTICLES - Part (1/5) of UK Nonlinear News

UK Nonlinear News, August 2004

------------------------------------------------------------------------

Articles and Reviews

* Book Review: Simulating, Analyzing, and Animating Dynamical

Systems. A

Guide to XPPAUT for Researchers and Students .

Reviewed by Harvinder Sidhu.

* Book Review: Critical Phenomena in Natural Sciences. Chaos,

Fractals,

Selforganization and Disorder: Concepts and Tools .

Reviewed by Henrik Jensen.

* Boor Review: Weighing the Odds: A Course in Probability and

Statistics

Reviewed by Jaroslav Stark

* Book Review: Nonlinear Dynamics in Physiology and Medicine

Reviewed by Alona Ben-Tal

* Report: Noisy Oscillators - Workshop on the Dynamics of Coupled

Oscillatory & Complex Systems

By Peter V.E. McClintock

* Report: Modelling of Neuronal Dendritic Trees

By Yulia Timofeeva

* A listing of reviews of nonlinear books can be found at

http://www.amsta.leeds.ac.uk/Applied/news.dir/issue7.dir/art/books.html

(this article is periodically updated).

* An index of UK Nonlinear News can be found at

http://www.amsta.leeds.ac.uk/Applied/news.dir/uknonl-index.html.

------------------------------------------------------------------------

Simulating, Analyzing, and Animating Dynamical Systems. A Guide to

XPPAUT

for Researchers and Students

Bard Ermentrout

Reviewed by Harvinder Sidhu

SIAM. Softback, 290 pages.

ISBN: 0-89871-506-7.

Researchers in diverse areas such as Biology, Physics, Engineering etc

are

constantly investigating nonlinear phenomena in order to understand the

behaviour of their specific physical system. Furthermore, nonlinear

dynamics

is now an integral part of many undergraduate Mathematics and Physics

curricula in universities all over the world. Summer schools in areas

such

as ecology, physiology, medicine etc have also devoted much time to

exploring nonlinear phenomena. Regardless of the application, the most

important issue in the understanding of nonlinear behaviour is the

combination of both analytical and computational techniques. However, a

significant number of researchers and students from these diverse fields

may

have insufficient computational background to program in MATLAB, MAPLE,

MATHEMATICA etc to solve the governing equations for the system that

they

are modelling. This is where XPPAUT is extremely useful. Bard

Ermentrout,

the author of this book and the developer of the software, states that

XPPAUT offers several advantages over the above-mentioned softwares.

These

include:

* a simple syntax for setting up various systems of equations such as

ODEs, maps and some PDEs,

* a convenient interface with AUTO - a powerful continuation and

bifurcation package,

* free downloading of the source code.

I must concur with Bard regarding the advantages of XPPAUT and see these

as

compelling reasons for anyone interested in studying nonlinear dynamics

to

use this software. Personally I feel that XPPAUT is an excellent

software

for both researchers and students. Prior to reviewing this book, I have

always used MATLAB and AUTO in my research. However, I am now a

``convert''

and have begun to use XPPAUT extensively in both my research and

teaching of

dynamical systems. I must now stop talking about the software and

proceed to

review this book, although I find it very difficult to divorce one from

the

other.

The book consists of nine chapters and seven appendices. The author is

to be

commended for his excellent organization of the subject matter and his

clarity in writing. I found the background of the physical models used

to

illustrate particular aspects of XPPAUT to be extremely refreshing.

Although

contrived examples are often used in books, this is not the case here.

The author begins with a useful explanation of how to install the

package on

computers with various operating systems. Chapter 2 explains the basics

of

creating an ODE file and solving the system and plotting the solutions.

Due

to the simplicity of the package, chapter 2 provides the reader with

sufficient information to analyse their own system using XPPAUT. Chapter

3

provides a more detailed explanation of the syntax of the ODE files.

Here

the author explains how to set up user-defined functions as well as the

handling of discontinuous differential equations in XPPAUT. Chapter 4

which

is aptly entitled ``XPPAUT in the Classroom'' is one of the most useful

chapters in the book. The chapter begins by showing how XPPAUT can be

used

to plot functions, before focussing on one-dimensional maps. Here the

author

shows how to obtain cobweb diagrams, bifurcation diagrams, liapunov

exponents, and the devil's staircase for such maps. The final part of

this

chapter is dedicated to nonlinear differential equations. The author

explains very clearly how three- and higher-dimensional dynamical

systems

can be analysed using this package. Chapter 5 explains how to define ODE

files for ``more complicated model systems'' such as systems with delay,

integral equations, stochastic equations and differential algebraic

equations. Chapter 6 focuses on the use of XPPAUT to solve boundary

value

problems and special cases of spatially distributed systems. I was very

impressed with the ease with which the governing equations can be set up

for

such systems using XPPAUT. Chapter 7 is one of the major chapters as it

clearly explains how to use XPPAUT's simple interface to the

continuation

package AUTO. Although I have previously used AUTO, I found the XPPAUT's

interface much easier to use. Animating systems is the focus of chapter

8.

Even I was able to produce good quality animations for some of my

systems by

following the author's clear step-by-step instructions and examples.

Finally, chapter 9 is aptly entitled ``Tricks and Advanced Methods'' as

it

provides users with very useful tips including exporting data, linking

with

external C routines, and numerous other ways in which ``tricky'' systems

or

situations can be handled.

Overall I believe that this is an excellent book for researchers and

students who are interested in learning how to analyse dynamical systems

using XPPAUT -- a powerful, simple-to-use and free software package. By

using this book readers will be able to discover the software's full

range

of capabilities, and like me, they will certainly be impressed.

UK Nonlinear News would like to thank SIAM for providing a review copy

of

this book.

------------------------------------------------------------------------

Critical Phenomena in Natural Sciences. Chaos, Fractals,

Selforganization

and Disorder: Concepts and Tools

By Didier Sornette.

Reviewed by Henrik Jensen

Springer, 2nd ed. 2004, 102 figs, 528pp., EUR 79.95, GBP 61.50, US $

99.00

Hardcover, ISBN 3-540-40754-5

This is an extremely impressive and comprehensive review of large areas

of

research into the collective behaviour of systems with many degrees of

freedom. The author, Didier Sornette, is an exceptionally productive

researcher who has contributed to most of the topics covered in the

book.

This doesn't by any means imply that the presentation is limited to

Sornette's own work. On the contrary vast numbers of models and

approaches

are discussed in remarkable detail. The reference list contains a

staggering

1067 items. One gets the impression that Sornette knows these references

in

and out and presents the reader with a digested explanation of the

essential

content of this huge list of papers and books.

The book does what the title promises, namely equip the reader with an

arsenal of concepts and tools which will be a great help when reading

the

research literature or attacking research problems in the field of, what

one

might call, applied statistical mechanics. The focus is - again as the

title

implies - on the correlated behaviour of systems with many components.

The

term criticality is meant to imply that essential correlations exist

between

the different parts of the system and, hence, its behaviour cannot be

deduced by a simple summation of the properties of the individual

components.

The tools and concepts supplied include a basic and worthwhile

introduction

to statistics. The first chapter introduces the very foundation of

probability theory with a refreshing discussion of the frequency

interpretation contrary to the Bayesian school. The material is

organised in

an unusual but very effective manner with an emphasis on characteristic

functions, moments and cumulants. I like very much the discussion of

Extreme

Value Statistics and Large Deviations in Chap. 1 and 3 sandwiching the

obligatory discussion of the Central Limit Theorem in Chap. 2. The

peculiarities of power law distributions (Levy Laws) are detailed in

Chap.

4, Fractals and Multifractals are explained with great lucidity in Chap.

5.

Chap. 6 rounds off the mathematical tool box with a discussion of

Rank-Ordering Statistics and Heavy Tails, topics which are central to

the

current activities in complex systems research.

The remaining 11 Chapters are concerned with physics concepts and

models. A

broad range of very relevant topics are covered: the relevance of the

temperature concept to out-of-equilibrium systems, the role of

long-range

correlations and phase transitions. The latter is the prototype example

of

macroscopic coherent behaviour in physics; here the concept and its

mathematical description are explained at a level that should be

accessible

to readers with no physics background. Two more methods chapters follow:

one

from dynamical systems theory on bifurcations and one from statistical

mechanics on The Renormalisation Group.

Chaps. 12 and 13 present two important models: The percolation model

which

is a paradigmatic model from statistical mechanics and the Rupture

Model,

which is an interesting model with a special appeal to materials

scientists

and geo-physicists. Chap. 14 and 15 contains probably the material with

widest current appeal and inspiration. Back in 1987 Bak, Tang and

Wiesenfeld

published a paper in which they introduced the concept of Self-Organized

Criticality which they suggested to be the generic explanation of the

power

laws characterising many natural phenomena. This inspired a huge

research

effort which is still very active. Sornette devotes Chap. 14 to a

detailed

discussion of a large number of mechanisms that may lead to power laws

and

focus next on specific Self-Organized Criticality models in Chap. 15. I

find

these two chapters to be very useful nearly up-to-date summaries of the

present research situation.

The book's two last chapters contains for completeness discussions of

random

systems. Again well introduced and well explain material.

This book is a treasure horn. Without assuming much mathematical

background

Sornette manages to supply the reader with the most essential

mathematical

tools and scientific concepts to be able to participate in the current

research effort on developing mathematical modelling and understanding

of

extended system in which the behaviour is a result of cooperative

interaction between the components. A very good book to have in the bag!

This is the second edition and it is even by Springer, it therefore

surprises me that there are small mistakes here and there. None of these

are

essential; a few might cause momentarily confusion for the reader

unfamiliar

with the material - I wonder if the copy editing has been sufficiently

careful? Last point, do we use British or American spelling? Or both

perhaps? I found on p. 96 line 9 and 11 a certain amount of oscillation

between the two conventions. This is of course not important, but nor is

it

elegant.

------------------------------------------------------------------------

Weighing the Odds: A Course in Probability and Statistics

D. Williams

Reviewed by Jaroslav Stark

Cambridge University Press 2001, 0-521-00618, 566 pages.

£26 (paperback); £75 (hardback)

ISBN: 0-521-80356

As those who have read some of my other book reviews in UK Nonlinear

News

will know, one of my key concerns about the state of the mathematical

sciences today is the gulf that exists between applied mathematics and

statistics. I find it astonishing that someone can get a 1st class

degree in

Applied Mathematics from almost any university in the UK (and most other

countries) but when confronted by real data will have practically no

idea

how to compare it to a mathematical model. Conversely, modern statistics

concentrates primarily on developing ever more sophisticated techniques

for

fitting and comparing models which from an applied mathematics

perspective

(and particularly from a nonlinear dynamics point of view) are rather

simplistic. As a consequence, ask either an applied mathematician or a

statistician how to fit data to a set of coupled nonlinear differential

equations and, with a few notable exceptions, you are likely to be met

by a

puzzled shrug of the shoulders.

I was thus intrigued to find a close parallel in the mischievous first

sentence of the volume under review:

Probability and Statistics used to be married; then they

separated; then they got divorced; now they hardly ever see each

other.

The book?s stated aim is to provide support for a much-needed

reconciliation. As a, perhaps unintended, by-product the book?s unusual

approach and entertaining style results in an introduction to statistics

which is unusually accessible to an applied mathematics audience. As an

added bonus, the final chapter on quantum probability and computing will

be

of interest to anyone with a physics background. I doubt that there is

another book anywhere that manages to combine topics as diverse as ANOVA

and

quantum entanglement in such an effortless way. Essentially no prior

knowledge is assumed in either probability or statistics, resulting in a

work that is suitable for a very wide range of backgrounds.

Given the author?s background, the underlying emphasis is on concepts in

probability, ranging from an intuitive first introduction to topics as

advanced as martingales. This gives the volume a strong mathematical

flavour, with analysis and linear algebra playing a key role.

Statistical

ideas are introduced gradually, and the author maintains a balance

between

the traditional frequentist approach and the Bayesian point of view. I

suspect that he himself is more inclined philosophically towards the

latter,

but he is not above criticizing a pure Bayesian methodology where

appropriate. One interesting aspect is his strong dislike of hypothesis

testing, and preference for reporting confidence intervals of estimated

parameters instead. This is a point of view that I increasingly sense in

modern statistics, and it is useful to have it argued here so

persuasively.

A variety of other thought provoking and probably controversial ideas

are

sprinkled throughout the book, always presented in a lively and

challenging

manner. This makes for a very readable book, from which I learned a

great

deal and into which I have been continuously dipping since I finished

it. I

would recommend it to anyone, from final year undergraduate, to an

established researcher in nonlinear science. They might be surprised how

interesting statistics can be, and perhaps respond more positively next

time

they are asked to fit data to a model.

UK Nonlinear News would like to thank Cambridge University Press for

providing a copy of this volume for review.

------------------------------------------------------------------------

Nonlinear Dynamics in Physiology and Medicine

Ann Beuter, Leon Glass, Michael C. Mackey and Michèle S. Titcombe

(editors),

Reviewed by Alona Ben-Tal

Springer-Verlag, Interdisciplinary Applied Mathematics, Volume 25, 2003,

434

pages, 162 illustrations, Hardcover.

ISBN: 0-387-00449-1

I was delighted when the book "Nonlinear Dynamics in Physiology and

Medicine" arrived in the mail. This pleasant looking book is the most

recent

volume of the Interdisciplinary Applied Mathematics series by

Springer-Verlag and since I know some other books in this series (for

example, [1], [2], [3]) I looked forward to reading this new addition.

The book has evolved out of notes written in three summer schools

organized

by The Centre for Nonlinear Dynamics in Physiology and Medicine at

McGill

University. The summer schools were held in 1996, 1997 and 2000. The

result

is an edited book with 12 contributors (mentioned later) that contains

cutting edge subjects of research and (perhaps not surprisingly)

reflects

mainly the work done by the McGill group.

Chapter 1, by M. C. Mackey and A. Beuter, gives "A wee bit of history to

motivate things". The chapter contains some interesting examples showing

the

role mathematics has played in the life sciences and vice versa (but see

also: highlights of a keynote address by Dr. Joel Cohen "Mathematics Is

Biology's Next Microscope, Only Better; Biology Is Mathematics' Next

Physics, Only Better." http://www.bisti.nih.gov/mathregistration/ ).

Chapter 2, by J. Bélair and L. Glass, "Introduction to Dynamics in

Nonlinear

Difference and Differential Equations", has nice examples of nonlinear

phenomena seen in physiological systems, among them, the co-existence of

two

stable states in the human heart. Overall it is a reasonable

introduction to

the field but I was disappointed to find the following statement (given

in

p. 17 when the authors discuss the period three window seen in the

logistic

map):

..."So why did Li and Yorke (1975) claim "period three implies chaos"?

(Li

and York 1975). The answer lies in the definition of chaos. For Li and

Yorke, chaos meant an infinite number of cycles ....That definition does

not

involve the stability of the cycles."

In the context of the book it is not clear that "claim" actually means

"proved" [4]. I also thought that for the purposes of the book it would

have

been sufficient to give an intuitive definition of chaos, as the authors

did

on p. 16, skip the exact definitions of chaos altogether and refer the

reader to a technical definition of chaos, given for example in [5, 6].

But

isn't this an example where mathematics is a better microscope? We KNOW

that

an unstable chaotic solution exists even though we cannot observe it. I

felt

that a student reading this statement (and others on the definition of

chaos) may get the wrong impression that only the observable solutions

are

important (and see for example, [7]).

Chapter 3, by M. R. Guevara, is another introductory chapter to

non-linear

dynamics and contains some very nice examples of non-linear phenomena in

physiology. Chapter 4, also by M. R. Guevara gives a nice introduction

to

the Hodgkin-Huxley equations and serves as a specific example to

illustrate

the subjects covered in Chapter 3.

Chapter 5, by L. Glass looks at periodic solutions and the phenomena

seen

when an oscillating autonomous system is forced by a single stimulus or

by

periodic train of stimuli. A related subject, "Reentry in Excitable

Media",

is presented in Chapter 7 by M. Courtemanche and A. Vinet. Chapter 7

describes the dynamics of excitable cardiac cell by three families of

models: Cellular Automata, delay equations and partial differential

equations.

The important subject "Effects of Noise on Nonlinear Dynamics" is

covered in

Chapter 6 by A. Longtin. Different kinds of noise are discussed.

Postponement of a Hopf bifurcation is demonstrated in a first order

delayed

equation (model for a pupil light reflex) and the phenomenon of

stochastic

phase locking (usually known as "skipping") is described. The phenomenon

of

bursting as a result of noise is also presented but, to my

disappointment,

there is no discussion (or even mention) of bursting without noise.

It seemed natural to me to read next Chapter 9 by J. Milton. This is a

nice

chapter that describes the pupil light reflex. I thought that there is a

good balance here between physiology and mathematics but there is a

relatively large number of typos.

I next read Chapter 10 by A. Beuter, R. Edwards and M. S. Titcombe

describing the interesting phenomenon of tremor (an approximately

rhythmical

movement of a body part). I thought it was interesting that the Van der

Pol

equation was proposed as a model for Parkinsonian tremor but there is

hardly

any discussion at all about this model. A six-unit Hopfield-type neural

network model is discussed here in more detail.

Finally, I made my way through the jargon of Chapter 8 (granulopoiesis,

erythroid precursors, erythropoietin, neutrophil, am I reading English?,

megakaryocytic, eosinophil, reticulocyte, idiopathic, promyelocytes, the

list goes on, see http://cancerweb.ncl.ac.uk/omd/ for an On-Line Medical

Dictionary). Chapter 8 by M. C. Mackey, C. Haurie and J. Bélair

describes

the control of blood cell production. Here you can find more examples of

periodic behaviour on the scale of days that arise from Hopf

bifurcations in

delay equations.

Throughout the book experimental results are presented and the authors

share

with the readers the joys and sorrows of dealing with real data

including

some practical aspects of estimating parameters (for example, a useful

tip

on obtaining data from published graphs by using Ghostview is given in

Chapter 8 p. 248). All the chapters contain computer exercises with

source

codes and data files that are available on line at

http://www.cnd.mcgill.ca/books_nonlinear.html. The book also has three

appendixes (an introduction to XPP, an introduction to Matlab and time

series analysis) which I found handy. But the book lacks the coherence

and

general perspective that a single authored book has (despite the noted

effort by the authors). Still, this is a stimulating book. I don't

recommend

it as a text book but certainly as a research and teaching resource.

UK Nonlinear News would like to thank Springer-Verlag for providing a

copy

of this volume for review.

References:

1. J. Keener and J. Sneyd, "Mathematical Physiology", Springer-Verlag,

1998.

2. J. D. Murray, "Mathematical Biology I. An Introduction",

Springer-Verlag, 3rd edition, 2002.

3. R. Seydel, "Practical Bifurcation and Stability Analysis. From

Equilibrium to Chaos", Springer-Verlag, 2nd edition, 1994.

4. T-Y Li and J. A. Yorke, "Period Three Implies Chaos", The American

Mathematical Monthly, Vol. 82 (10), 1975, pp. 985-992.

5. S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems

and

Chaos", Texts in Applied Mathematics, Springer-Verlag, 2nd edition,

2003.

6. M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations,

Dynamical Systems & An Introduction to Chaos", Elsevier, 2nd

edition,

2004.

7. C. Robert, K. T. Alligood, E. Ott and J. A. Yorke, "Explosions of

Chaotic Sets", Physica D, Vol. 144, pp. 44-61, 2000.

------------------------------------------------------------------------

Noisy Oscillators: Workshop on the Dynamics of Coupled Oscillatory &

Complex

Systems

Ljubljana, 10-13 December 2003

By Peter V.E. McClintock

The Workshop on the Dynamics of Coupled Oscillatory and Complex Systems

was

held at the University of Ljubljana, Slovenia, 10-13 December 2003. The

venue was the Zborni?na dvorana in the historic central building of the

University in the middle of the city, a stone's-throw from the

Philharmonie

where Mahler worked for many years (and Beethoven's job application was

rejected!).

This was a small select gathering, mostly by invitation, and highly

interdisciplinary. The 37 participants, drawn from 17 countries,

included

all ages and stages from selected PhD students up to senior scientists.

The

meeting was opened by Professor Katja Breskvar, Vice-Rector of the

University of Ljubljana. Professor Toma? Slivnik, Dean of its Faculty of

Electrical Engineering, contributed some further welcoming remarks and

the

first scientific session then followed immediately.

The meeting reflected the fact that, in the modern theory of complex

dynamical systems in physics (fluctuations in lasers), biology

(transport of

proteins), physiology (autonomous nervous control of the cardiovascular

dynamics), and econometrics the effects of both chaotic and stochastic

dynamics, and their mutual interactions, must be taken fully into

account.

It was also motivated by a growing awareness that reaching an

understanding

of the general features of chaos-noise interactions in complex dynamical

systems may introduce entirely new ideas of how to control the system

dynamics, and of how to exploit its complexity usefully in a variety of

applications on all scales from molecules to the global economy.

Examples

discussed were numerous, including e.g. applications of stochastic

resonance

and Brownian ratchets in biology, chaotic communications, and chaos

control.

All these systems operate in regimes far away from thermal equilibrium,

and

the problems of modelling are highly challenging.

The function of the meeting was thus to bring together fluctuational

dynamicists working on nonequilibrium and complex systems to engage in

direct discussions with experimental scientists working in areas that

can

potentially be described by the theories that are under development,

especially in biology. It was felt to have succeeded admirably in these

aims. The sessions on cardiovascular dynamics were especially valuable,

with

data obtained in Lancaster, Ljubljana, Oslo and Pisa being exposed to

general scrutiny and open discussion with theorists expert in e.g.

stochastic dynamics, synchronisation, information theory, relaxation

oscillations, excitable systems and physiology.

The local Director, Professor Aneta Stefanovska did a superlative job in

organising all aspects of the meeting { both scientific and social { the

latter including a tour of central Ljubljana, two excellent concerts, a

magnificent conference dinner and, after the conference, an excursion

taking

in Cerknica (the "disappearing lake"), the Postojna caves, a

wine-tasting,

and another memorable dinner. The meeting was supported by INTAS and by

the

European Science Foundation under its STOCHDYN programme.

Details can be found on the conference web page:

http://osc.fe.uni-lj.si/ljubljana/index.htm

------------------------------------------------------------------------

Report: Modelling of Neuronal Dendritic Trees

ICMS, 14 India St, Edinburgh, 18 June 2004

By Dr Yulia Timofeeva

A one-day meeting on Modelling of Neuronal Dendritic Trees was held on

Friday 18th June at 14 India St. Edinburgh, the birth place of James

Clerk

Maxwell and home to the International Centre for Mathematical Sciences.

Organised by Dr Stephen Coombes (Nottingham) and Dr Gabriel Lord

(Heriot-Watt), the workshop focused on the application of mathematical

tools

from modelling and numerical analysis to information processing and

learning

within a single neuron with branched dendritic structure. The meeting

brought together around 40 researchers in computational neuroscience,

applied mathematics and the life sciences.

Professor John Rinzel (Center! for Neural Science, NYU), started the

meeting

by giving a thorough background about the compartmental method and the

seminal contributions of Wilfrid Rall. He emphasised a minimalistic

approach

to investigating dendritic cable properties using only a few

spatial/electrical compartments. Using a comprehensive compartmental

simulation study, Dr Arnd Roth (UCL) then explained the link between

dendritic morphology and the experimentally observed patterns of forward

and

backward propagating action potentials seen in real neurons. The

usefulness

of mathematical analysis was highlighted by the talk of Professor Steve

Cox

(Rice University, Texas), who described a rigorous approach for

estimating

the location and time course of synaptic input from multi-site potential

recordings. Dr Mickey London (UCL) ill ustrated the success of

information

theory in uncovering the role of dendritic structure in neuronal

computation

and Dr Bruce Graham (Stirling University) spoke about dynamics of

synaptic

integration in a hippocampal pyramidal neuron model. Some new

mathematical

ideas in compartmental modelling were introduced by Dr Ken Lindsay

(Glasgow)

and the meeting concluded with a talk by Dr David Barber (IDIAP,

Switzerland) on a mathematical framework for learning with dendritic

spines.

Overall the workshop was very successful, with a large attendance,

excellent

presentations and fruitful discussions. The meeting was financially

supported by the ICIAM99 fund.

=============================================================================

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