Introduction to computational neurobiology and clustering

242 Pages·2007·2.054 MB·English

by  Bianchi, Daniela, Ferraro, Enrico, Tirozzi, Brunello

INTRODUCTION TO
COMPUTATIONAL
NEUROBIOLOGY AND
CLUSTERING
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Series on Advances in Mathematics for Applied Sciences - Vol. 73
INTRODUCTION TO
COMPUTATIONAL
NEUROBIOLOGY AND
CLUSTERING
Brunello Tirozzi
Daniela Bianchi
Department of Physics
University of Rome “La Sapienza”, Italy
Enrico Ferraro
Department of Biology
University of Rome “Tor Vergata”, Italy
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INTRODUCTION TO COMPUTATIONAL NEUROBIOLOGY AND CLUSTERING
Series on Advances in Mathematics for Applied Sciences — Vol. 73
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-270-539-6
ISBN-10 981-270-539-2
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Preface
The book is divided into two parts. The first concerns the simulations of
neurobiologicalsystems,theseconddescribesthecurrentlyusedalgorithms
of genes and protein classification. The first part of this book corresponds
tothelecturesthatoneofusisgivingforthecourseofFisica Applicata alle
Biotecnologie (AppliedPhysicstoBiotechnology). Biotechnologyisaquite
recent discipline taught in Italian Universities, as well as in other parts of
theworld,andmanystudentsmovedfromBiologycoursestoit. Theaimof
the book, as well as the lectures, is to teach simulations of neurobiological
systems to students with a little background of physics and mathematics,
a usual situation for the students of biology. In the lectures I succeeded to
teach only the first part but the most curious students and readers should
not stop and should go also through the second one. These two themes
are very important and there is a huge amount of work on them in the
currentresearchliterature. Themathematicalmodelsofneurobiologyhave
beenatthe centerofattentionofthe researchinneurobiologyfor 20years.
Actually the story is much older since we can establish a starting point
for this discipline with the model of Hodgkin and Huxley of the fifties in
which the neural activity of the axon of the squid was fully described. So
there is an old tradition in this field which has been renewed many times.
In the last decade the models describing the neurons have become more
sophisticated by the introduction of the stochastic inputs described by the
Poisson and the Wiener processes. There was a transition from ordinary
differential equations to stochastic differential equations and the mathe-
matics become more complicated,stochastic contributionswere insertedin
the non-linearsystem ofequations of the HodgkinHuxley model (or of the
Fitzhugh Nagumo model) as well as in the simpler Integrate & Fire model
describedinthefirstchapterofthebook. Actuallythereisalsoahugevari-
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viii Introduction to Computational Neurobiology & Clustering
etyofnon-linearmodelsdescribingtheneuronsindifferentsituationsofthe
nervesystem(brain,spinalchord,etc.). Eachsystemofneuronshasavery
complicated structure and interactions and the stochastic terms are intro-
ducedinordertodescribetheinputthateachneuronreceivesfromtherest
ofthesystem. Anequallysophisticatedbackgroundofprobabilitytheoryis
thebasisoftheclusteringtechniquesintroducedforgenesandproteinclas-
sification. The necessity to use more complex algorithms comes from the
huge amount of data obtained by microarraytechniques. In fact this tech-
nology is able to extract thousands of genes from the DNA of a cell at the
sametimeandsotheproblemofcollectingthegenesingroups(orclusters)
withthe samefunctionhasbecomecrucial. Butbeforearrivingatthe level
ofunderstanding these problemsthe students, as many students inbiology
andbiotechnology,havetobecomemoreacquaintedwiththeusualcalculus
and probability. Even if they already had an introduction to physics and
to calculus, they need other training in order to understand the matter of
thisbook. Thuseachchapterisaccompaniedbyappendixescontainingthe
details of some necessary calculus, a brief description of the mathematics
orstatistics orphysicsnecessaryforunderstandingthe meaningofthe text
containedinthe chapters. Inotherwordsthe chapterscontainthe descrip-
tion of the model used and the main results without giving all the details
ofcalculationswhichcanbefoundintheappendixes. Therearealsosolved
problems in order to give examples of all the techniques and models in-
troduced. Since mathematical or physical concepts become less abstract if
realizedwithcomputerprograms,animportantpartofthebookisdevoted
to illustrate the models andtheir results by means ofprogramswith useful
graphic outputs. So the appendixes also contain elementary matlab pro-
gramswhichrealizemanymodelsandtechniquesdescribedinthechapters.
PartoftheexercisesaresolvedbymeansofMatlabprograms. Ifonewants
to comparethis book with the usualbooks describingbiologicalmodels we
cansay that this book brings the beginner,the student of the firstyearsof
physics, mathematics and biotechnology to the level of understanding the
basic of the models used in neurobiology and cluster analysis, giving him
alsoallthenecessaryinstrumentsandconceptsforsolvingsomeinteresting
realisticmodelordoingclusteranalysis. Apersonorstudentwiththisbook
canbuild new models or understandresearcharticles onthe simulations of
neuronal systems. The rest of the literature is more advanced so it is not
for the beginners to whom this book is addressed.
Brunello Tirozzi, Daniela Bianchi, Enrico Ferraro
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Contents
Preface vii
Neurobiological models 1
1. RC circuit, spiking times and interspike interval 3
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Electric properties of a neuron . . . . . . . . . . . . . . . 3
1.3 Lapicque or I& F model . . . . . . . . . . . . . . . . . . . 6
2. Calculation of interspike intervals for deterministic inputs 13
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Case of constant input current . . . . . . . . . . . . . . . 13
2.3 Constant input current for a finite time . . . . . . . . . . 17
2.4 Constant input current with a periodic pattern . . . . . . 20
2.5 Periodic instantaneous inputs . . . . . . . . . . . . . . . . 23
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3. The Fitzhugh-Nagumo and Hodgkin-Huxley models 31
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 TheFitzhugh-Nagumomodelandthegeneralpropertiesof
differential equations . . . . . . . . . . . . . . . . . . . . . 32
3.3 The generation of spikes and the Hopf bifurcation . . . . 37
3.4 A more realistic model: the Hodgkin-Huxley model
(HH model) . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. Definition and simulation of the main random variables 55
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