Neurons: A Mathematical Ignition

231 Pages·2014·2.49 MB·English

by  Masayoshi Hata

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Series on Number Theory and Its Applications ISSN 1793-3161
Series Editor: Shigeru Kanemitsu (Kinki University, Japan)
Editorial Board Members:
V. N. Chubarikov (Moscow State University, Russian Federation)
Christopher Deninger (Universität Münster, Germany)
Chaohua Jia (Chinese Academy of Sciences, PR China)
Jianya Liu (Shangdong University, PR China)
H. Niederreiter (National University of Singapore, Singapore)

Advisory Board:
A. Schinzel (Polish Academy of Sciences, Poland)
M. Waldschmidt (Université Pierre et Marie Curie, France)

Published
Vol. 1 Arithmetic Geometry and Number Theory
edited by Lin Weng & Iku Nakamura
Vol. 2 Number Theory: Sailing on the Sea of Number Theory
edited by S. Kanemitsu & J.-Y. Liu
Vol. 4 Problems and Solutions in Real Analysis
by Masayoshi Hata
Vol. 5 Algebraic Geometry and Its Applications
edited by J. Chaumine, J. Hirschfeld & R. Rolland
Vol. 6 Number Theory: Dreaming in Dreams
edited by T. Aoki, S. Kanemitsu & J.-Y. Liu
Vol. 7 Geometry and Analysis of Automorphic Forms of Several Variables
Proceedings of the International Symposium in Honor of Takayuki Oda
on the Occasion of His 60th Birthday
edited by Yoshinori Hamahata, Takashi Ichikawa, Atsushi Murase &
Takashi Sugano
Vol. 8 Number Theory: Arithmetic in Shangri-La
Proceedings of the 6th China–Japan Seminar
edited by S. Kanemitsu, H.-Z. Li & J.-Y. Liu
Vol. 9 Neurons: A Mathematical Ignition
by Masayoshi Hata

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World Scientific
9263_9789814618618_TP.indd 2 18/8/14 10:21 am
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Number Theory and Its Applications — Vol. 9
NEURONS
A Mathematical Ignition
Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
ISBN 978-981-4618-61-8
Printed in Singapore
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August12,2014 16:54 9263-Neurons:AMathematicalIgnition 9789814618618 pagev
Preface
Thepresentbookisaconsiderablyrevisedversionofmaterialsgivenbytheauthor
in some intensive courses at Hokkaido University, the University of Tokyo and
OsakaCityUniversityin1996,andatYamaguchiUniversityin2010.
Itissaidthatourbrainconsistsofmorethanahundredbillionneurons,which
are connectedmutuallybysynapsesandbehavetremendouslyin its complexity,
and that, if one connectsall the axonsand dendritesin a brain end to end, then
itattainsalengthinexcessofamillionkilometers. In1961E.R.Caianiellopub-
lishedamonumentalpaperentitled“Outlineofatheoryofthought-processesand
thinkingmachines”inJournalof TheoreticalBiology,inwhichhehadstrength-
ened the convictionthat the human brain yet obeysdynamicallaws that are not
necessarilycomplicated,ifonelooksattheoperationofindividualneurons.
This book deals with two mathematicalsides of Caianiello’s neuronic equa-
tions; one is from the point of view of dynamical systems and the other from
thenumber-theoreticpointofview.Inparticular,thelattershowsthatCaianiello’s
equationsarecloselyrelatedtosometopicsinelementarynumbertheoryandeven
intranscendentalnumbertheory.
TheauthorhascomefullcirclebacktoCaianiello’sneuronicequations,which
were the first research theme in his career. The author expresses deep gratitude
to the late Professor Masaya Yamaguti, his supervisorat KyotoUniversity, who
broughthimtothissubjectofresearchandhandedhisfirstpapertothelatePro-
fessorCaianiellodirectlyattheUniversityofSalernointhelate1980s.
EachchapterexceptforChapter2endswithsomeinstructiveandmostlyorig-
inal exercises at various levels. Some contain examples, additional results and
related topics, whichwouldprovidea sense ofperspective,andseveralare used
in thetext. Foralmostallexercisesthe detailedsolutionsaregiveninHintsand
Solutionsattheendofthebook.Thetableoftheassociatedpolynomialswillhelp
forreaderstoverifyandinvestigatetheirrelations.
v
August12,2014 16:54 9263-Neurons:AMathematicalIgnition 9789814618618 pagevi
vi Neurons:AMathematicalIgnition
Allthefiguresinthebookweredrawnbyusing“Grapher2.1”bundledwith
MacOSX.
WerefertothewebsiteofI.I.A.S.S.(http://www.iiassvietri.it)fortheprofile
andhistoricalnotesaboutProfessorCaianiello.
TheauthorcordiallyappreciatesProfessorShigeruKanemitsuwhoprovided
uninterruptedencouragementand the exquisite title of this book with a twofold
significance.TheauthorisalsodeeplyindebtedtoProfessorMichelWaldschmidt
for his careful reading of the manuscript and for giving detailed comments and
suggestions. Also thanksare due to the staff of World Scientific Publishing Co.
fortheexcellenthelpandcooperation.
This work is supported by the Japan Society for the Promotion of Science
(JSPS) through the “Funding Program for World-Leading Innovative R&D on
ScienceandTechnology(FIRSTProgram)”.
Kyoto, JAPAN M.Hata
August12,2014 16:54 9263-Neurons:AMathematicalIgnition 9789814618618 pagevii
Contents
Preface v
SymbolsandNotations xi
1. BasicsofDiscreteDynamicalSystems 1
1.1 OrbitsandPeriodicPoints . . . . . . . . . . . . . . . . . . . . 1
1.2 OrbitsintheSenseofKuratowski . . . . . . . . . . . . . . . . 3
1.3 PiecewiseLinearMaps . . . . . . . . . . . . . . . . . . . . . . 4
1.4 InvariantPropertiesunderIterations . . . . . . . . . . . . . . . 6
1.5 TopologicalConjugacy . . . . . . . . . . . . . . . . . . . . . . 7
1.6 SymbolicDynamics. . . . . . . . . . . . . . . . . . . . . . . . 8
ExercisesinChapter1 . . . . . . . . . . . . . . . . . . . . . . . 9
2. Caianiello’sEquations 11
2.1 BriefHistory . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Caianiello’sNeuronicEquations . . . . . . . . . . . . . . . . . 12
2.3 Nagumo-Sato’sEquation . . . . . . . . . . . . . . . . . . . . . 14
2.4 Higher-DimensionalAnalogy . . . . . . . . . . . . . . . . . . 15
3. RotationNumbers 19
3.1 SetB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 DiscontinuityPointsof fn . . . . . . . . . . . . . . . . . . . . 20
3.3 Itineraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 RotationNumbers. . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 InvariantCoordinates . . . . . . . . . . . . . . . . . . . . . . . 28
ExercisesinChapter3 . . . . . . . . . . . . . . . . . . . . . . . 30
vii
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viii Neurons:AMathematicalIgnition
4. Classificationof B 33
4.1 NumberofSubintervals. . . . . . . . . . . . . . . . . . . . . . 33
4.2 FundamentalProperties . . . . . . . . . . . . . . . . . . . . . . 34
4.3 PeriodicBehaviorfor f ∈B . . . . . . . . . . . . . . . . . . . 36
q
4.4 ClassificationofB . . . . . . . . . . . . . . . . . . . . . . . . 38
q
4.5 Classificationof E(p/q) . . . . . . . . . . . . . . . . . . . . . 39
4.6 BasicPropertiesof f ∈B∞ . . . . . . . . . . . . . . . . . . . . 42
4.7 CharacterizationofEφ andE0 . . . . . . . . . . . . . . . . . . 42
ExercisesinChapter4 . . . . . . . . . . . . . . . . . . . . . . . 44
5. FareySeries 45
5.1 FareySeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 FareyIntervals . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Predecessors . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4 FloorandCeilingFunctions . . . . . . . . . . . . . . . . . . . 49
5.5 TailInversionSequences . . . . . . . . . . . . . . . . . . . . . 51
5.6 ApproximationSequences . . . . . . . . . . . . . . . . . . . . 52
5.7 ContinuedFractionExpansions. . . . . . . . . . . . . . . . . . 57
ExercisesinChapter5 . . . . . . . . . . . . . . . . . . . . . . . 60
6. FurtherInvestigationof f ∈B∞ 63
6.1 Cases (a)and (b) . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 BehaviorinCase (a) . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 DiscontinuationofCase (b) . . . . . . . . . . . . . . . . . . . 67
6.4 Cases (A)and (B) . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5 InvariantCoordinatesfor f ∈B∞ . . . . . . . . . . . . . . . . 69
ExercisesinChapter6 . . . . . . . . . . . . . . . . . . . . . . . 73
7. LimitSets Ω and ω (x) 75
f f
7.1 Definitionof Ω . . . . . . . . . . . . . . . . . . . . . . . . . 75
f
7.2 AperiodicBehavior . . . . . . . . . . . . . . . . . . . . . . . . 77
7.3 ω-LimitSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.4 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.5 ComplexityofItineraries . . . . . . . . . . . . . . . . . . . . . 82
ExercisesinChapter7 . . . . . . . . . . . . . . . . . . . . . . . 84
8. PiecewiseLinearMaps 85
8.1 BasicFormulaeoftheFirstKind . . . . . . . . . . . . . . . . . 85
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Contents ix
8.2 LimitSets Ωψ . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.3 BasicFormulaeoftheSecondKind . . . . . . . . . . . . . . . 92
8.4 HausdorffDimensionof Ωψ . . . . . . . . . . . . . . . . . . . 95
ExercisesinChapter8 . . . . . . . . . . . . . . . . . . . . . . . 99
9. OrbitalandItineraryFunctions 103
9.1 OrbitalFunctionsu andv . . . . . . . . . . . . . . . . . . . . 103
n n
9.2 u-andv-DiscontinuityPoints . . . . . . . . . . . . . . . . . . 105
9.3 ItineraryFunctionsU andV . . . . . . . . . . . . . . . . . . . 107
n n
ExercisesinChapter9 . . . . . . . . . . . . . . . . . . . . . . . 111
10. FareyStructure 113
10.1 Constructionwhenra,b =1 . . . . . . . . . . . . . . . . . . . . 113
10.2 Constructionwhenra,b <1 . . . . . . . . . . . . . . . . . . . . 118
10.3 ResidualSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.4 LengthofΔ(p/q) . . . . . . . . . . . . . . . . . . . . . . . . . 124
10.5 DistancebetweenTwoΔ’s . . . . . . . . . . . . . . . . . . . . 125
10.6 SomePropertieswhenra,b <1 . . . . . . . . . . . . . . . . . . 128
10.7 Proofof |Γa,b|=0whena<1 . . . . . . . . . . . . . . . . . . 130
10.8 Proofof |Γa,b|=0whena≥1 . . . . . . . . . . . . . . . . . . 133
10.9 SummationFormulae . . . . . . . . . . . . . . . . . . . . . . . 134
10.10 HausdorffDimensionofΓa,b . . . . . . . . . . . . . . . . . . . 137
10.11 RotationNumberFunctions . . . . . . . . . . . . . . . . . . . 139
ExercisesinChapter10 . . . . . . . . . . . . . . . . . . . . . . 142
11. α-and β-Leaves 143
11.1 DefinitionofLeaves . . . . . . . . . . . . . . . . . . . . . . . 143
11.2 Limitsofα-Leaves . . . . . . . . . . . . . . . . . . . . . . . . 144
11.3 Limitsof β-Leaves . . . . . . . . . . . . . . . . . . . . . . . . 152
ExercisesinChapter11 . . . . . . . . . . . . . . . . . . . . . . 160
12. ApproximationstoHecke-MahlerSeries 161
12.1 MinimalIndices. . . . . . . . . . . . . . . . . . . . . . . . . . 161
12.2 ApproximationstoMμandHμ . . . . . . . . . . . . . . . . . . 162
12.3 IrrationalityExponents . . . . . . . . . . . . . . . . . . . . . . 166
12.4 LeavesoutsideFirstQuadrant . . . . . . . . . . . . . . . . . . 171
12.5 InfiniteOscillationofH1/ϕ . . . . . . . . . . . . . . . . . . . . 178
12.6 SumsinvolvingFibonacciNumbers . . . . . . . . . . . . . . . 181