Table Of ContentAnalysis of a Model for
Epilepsy
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Analysis of a Model for Epilepsy
Application of a Max-Type Difference Equation to Mesial Temporal Lobe Epilepsy
Candace M. Kent, David M. Chan
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Analysis of a Model for
Epilepsy
Application of a Max-Type Difference
Equation to Mesial Temporal Lobe
Epilepsy
Candace M. Kent
Virginia Commonwealth University
David M. Chan
Virginia Commonwealth University
First edition published 2022
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© 2022 Candace M. Kent, David M. Chan
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Library of Congress Cataloging-in-Publication Data
Names: Kent, Candace M., author. | Chan, David M., author.
Title: Analysis of a model for epilepsy : application of a max-type
difference equation to mesial temporal lobe epilepsy / authored by
Candace M. Kent, David M. Chan.
Description: First edition. | Boca Raton : C&H/CRC Press, 2022. | Series:
Chapman & Hall/CRC monographs and research notes in mathematics |
Includes bibliographical references and index.
Identifiers: LCCN 2021059471 (print) | LCCN 2021059472 (ebook) | ISBN
9781032255385 (hardback) | ISBN 9781032258683 (paperback) | ISBN
9781003285380 (ebook)
Subjects: LCSH: Temporal lobe epilepsy--Mathematical models. |
Convulsions--Mathematical models. | Difference equations. | Predictive
analytics.
Classification: LCC RC373 .K46 2022 (print) | LCC RC373 (ebook) | DDC
616.85/3--dc23/eng/20220303
LC record available at https://lccn.loc.gov/2021059471
LC ebook record available at https://lccn.loc.gov/2021059472
ISBN: 978-1-032-25538-5 (hbk)
ISBN: 978-1-032-25868-3 (pbk)
ISBN: 978-1-003-28538-0 (ebk)
DOI: 10.1201/9781003285380
Typeset in Latin Modern font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface vii
1 Introduction: Epilepsy 1
1.1 Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mesial Temporal Lobe Epilepsy and Other Examples . . . . 3
2 The Model 13
2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Connection to a Simpler Model . . . . . . . . . . . . . . . . . 16
2.3 Connection between the Model and Epileptic Seizures . . . . 20
2.3.1 Fundamentals of Seizures . . . . . . . . . . . . . . . . 21
2.3.2 Seizure Characteristics . . . . . . . . . . . . . . . . . . 26
2.4 Open Problem: Seizure Threshold as a Function of Time . . 31
3 Eventual Periodicity of the Model 37
3.1 Bounded and Persistent Solutions . . . . . . . . . . . . . . . 38
3.2 Eventually Periodic Solutions with Periods Multiples of Six . 55
3.3 Eventually Periodic Solutions with Period 4 . . . . . . . . . 65
3.4 Partially and Completely Seizure-Free States . . . . . . . . . 70
4 Rippled Almost Periodic Solutions 79
4.1 Rippled Behavior . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Rippled Almost Periodic Solutions . . . . . . . . . . . . . . . 92
4.3 Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 The State of Status Epilepticus . . . . . . . . . . . . . . . . . 100
4.5 On Termination of Repetition . . . . . . . . . . . . . . . . . 109
5 Numerical Results and Biological Conclusions 117
5.1 Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . 117
5.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 117
5.1.2 Bifurcation Values b and b . . . . . . . . . . . . . . 118
0 1
5.2 Variability in Seizure Characteristics . . . . . . . . . . . . . . 124
5.3 A Case of Variability in Region 1 . . . . . . . . . . . . . . . 130
5.4 The Hyperexcitable State . . . . . . . . . . . . . . . . . . . . 133
5.5 Impact of Individual Historical Differences . . . . . . . . . . 138
6 Epilogue 147
v
vi Contents
Bibliography 151
Index 157
Preface
In the 1960s and 1970s, mathematical biologists Sir Robert M. May1, E.C.
Pielou2, and others popularized the use of difference equations as models of
real-life situations. At the time, their models were of population dynamics,
involving ecology and epidemiology. Since then, with or without applications,
the mathematics of difference equations has evolved into a field unto itself.
Differenceequationswiththemaximum(ortheminimumorthe“rank-type”)
functionwererigorouslyinvestigatedfromthemid-1990sintothe2000s,with-
outanyapplicationsinmind.Theseequationsofteninvolvedargumentsvary-
ing from reciprocal terms with parameters in the numerators to other spe-
cial functions. Recently, Kent and colleagues3 in 2018 investigated the first
known application of a “max-type” difference equation. Their equation is a
phenomenologicalmodelofepilepticseizures.Hereweexpandonthatresearch
and present a more comprehensive development of mathematical, numerical,
and biological results.
Weemployanonautonomous(i.e.,havingperiodiccoefficientparameters),
second-order,reciprocal,max-typedifferenceequationincorporatingathresh-
old function, specifically the Heaviside function, to represent the focal (i.e.,
local) seizures of a particularly common form of epilepsy called mesial, or
medial, temporal lobe epilepsy. The Heaviside function, with its threshold ϵ,
prevents solutions from becoming unbounded so that they can realistically
represent the initiation and termination of focal seizures. We refer to ϵ as
the “seizure threshold,” which is central to our mathematical, biological, and
numerical analyses. Our model is phenomenological, and thus does not con-
cern itself with seizures at the chemical or subcellular level. Instead, it seeks
to describe the empirical connections between the seizure threshold and the
elicitationoreliminationofseizures.Itsinsightsarenotgivenintermsofmea-
suredvaluesofvariablesbutinsteadaredescriptionsofobservedrelationships
betweenphenomenathataregeneralenoughtopavethewayfordialogueand
1R M May, ed. Theoretical Ecology: Principles and Applications. 2nd ed. Sunderland,
Massachussetts:SinauerAssociates,1981.
2ECPielou.PopulationandCommunityEcology.NewYork:GordonandBreachScience
Publishers,1974.
3D M Chan, C M Kent, V L Koci´c, and S Stevi´c. “A proposal for an application of a
max-type difference equation to epilepsy”. In: Differential and Difference Equations with
Applications.Ed.bySPinelas,TCaraballo,PKloeden,andJRGraef.NewYork:Springer,
2018,pp.193–210.
vii
viii Preface
further questions. Also, our model is recursive, and so nicely reflects what we
maintaintobetherecursive natureofoscillatoryneuronalsignalingprocesses.
The numerical studies consist of bifurcation diagrams with respect to the
parameterϵ.Weobserveessentiallythreekindsofsolutionsoverthespectrum
ofϵvaluesdisclosedbythebifurcationdiagrams,andaremostlysubstantiated
by mathematical analysis using in unique and intricate ways the tools at the
disposalof difference equationssuchas semicycle analysis. Overthe “middle”
section of the observed spectrum, solutions are eventually periodic with rela-
tivelysmallperiodsrepresentingtheinvariantrecurrenceofseizures.Overthe
rightendoftheobservedspectrum,solutionsarealsoeventuallyperiodic,usu-
ally with period 4, but represent a partially or completely seizure-free state.
Over the left end of the spectrum, there exist anomalous solutions that may
or may not be eventually periodic. Such anomalies are addressed as akin to
but not exactly the same as almost periodic solutions of other types of differ-
enceequations.Thesesolutionsattheleftendoftheobservedϵspectrumalso
represent the presence of recurrent seizures whose characteristics are highly
variable.
Chapter 1 begins with a brief overview of the epilepsies (the word
“epilepsy” is a misnomer and actually refers to a whole collection of disease
entities), with an emphasis on types of seizures. Five examples of the epilep-
sies,whicharerelatedbythefactthattheirseizuresoccurmainlyregionallyin
thebrainwithinitiationlinkedtoawell-definedlesion(e.g.,tumor)andwhich
include mesial temporal lobe epilepsy, are described especially with regard to
their semiology, or the clinical manifestations of their seizures. In particular,
the neurochemical groundwork is laid for a controversially proposed etiology
intheliteraturefortheseizuresofmesialtemporallobeepilepsy,whichislater
addressed more conclusively, based on suggestive findings from the model, in
the Epilogue, Chapter 6.
Chapter 2 describes the model itself and its connection to earlier nonau-
tonomous second-order reciprocal max-type difference equations and also its
connection to the biology of seizures. The model’s construction by incorpo-
ration of the Heaviside function, with threshold parameter ϵ (doubling as
a seizure threshold), into previously established nonautonomous second-order
reciprocalmax-typedifferenceequationswithbehaviorthatisunboundedand
non-persistentand/oreventuallyperiodicisdescribedindetail.Howsolutions
of the model represent recurrent unprovoked seizures is precisely defined and
examples are given. Biological meaning is attached to the model in defining
the“ictalphase”(i.e.,periodduringaseizure)and“interictalphase”(i.e.,pe-
riod between seizures) and in defining seizure characteristics, which we limit
to duration, intensity, and frequency. Finally, an open problem concerning a
time dependent seizure threshold function, ϵ(n), is given.
Chapter 3 covers the mathematical analysis of the model and first shows
that solutions are bounded (and persistent), at least for ϵ≤1, as they should
be in order to represent seizures, which of course are bounded in their spread
Preface ix
throughouttheconfinesofthehumanbrain.Next,itisshownthatthereexists
eventually periodic solutions with periods that are multiples of six again at
least for ϵ ≤ 1. Exhibited by such solutions is the intermittent initiation and
termination of alternating tendencies toward unbounded growth (representa-
tive of an increase in densities of activated neurons) and non-persistent decay
(representative of an increase in densities of neurons in their deactivation
phase). The intermittent occurrences of growth and decay represent ongoing
seizure activity. The overall presence of eventual periodicity perpetuates the
intermittent occurrences so that they represent the indefinite or lifetime re-
currenceofseizuresthataremoreoverinvariable andsopredictable.Itisthen
shown that for any given pair of initial values and any given set of coefficient
parameter values, there exists a number c > 0 such that at every ϵ > c, the
solution is eventually periodic with period 4 and is such that intermittent oc-
currences of growth and decay are completely absent, thereby representing a
seizure-free state (and the normal oscillatory functioning of neural networks)
at higher seizure thresholds. The concluding section of this chapter concerns
itself with the development of a technical definition of “partially and com-
pletely seizure-free states,” based on the result above on eventually period-4
solutions.Anisolatedexampleisgivenwherebythereexistseventuallyperiod-
12 solutions which represent the absence of seizures.
In Chapter 4, we prove that if what are observed to be atypical solutions
atthelowendofthespectrumofvaluesofϵsatisfyacertainsetofproperties
basedonextrapolationsofempiricalobservations,thentheyarenot eventually
periodic. We refer to the idealized solutions that satisfy the properties of
extrapolations as “rippled almost periodic,” and the actual atypical solutions
themselvesas“possessingrippledalmostperiodicbehavior.”Withanidealized
rippled almost periodic solution, we have that for every δ > 0, there exists
an integer p ∈ {1,2,...} such that infinitely often a string of p terms in
the solution “approximately” repeats with a degree of accuracy given by δ
for a finite number of repetitions. These finite sets of repetitions that occur
infinitelyoftenrepresentindefinitelyoccurringseizuresthatarehighlyvariable
andsounpredictable.Solutionswithrippledalmostperiodicbehavior,ifthey
in reality satisfy the properties of extrapolations, comprise a novel category
of solutions of difference equations. Such solutions are associated with unique
andunusualtimeseriesplotsthatarecomposedof“undulating”or“rippling”
layers of horizontal bands of plotted points, which may or may not extend
without limit. Chaos is ruled out using a crude estimate of the Lyapunov
exponent in Section 4.3. However, as Section 4.4 in this chapter illustrates, at
the smallest values of ϵ, these solutions with rippled almost periodic behavior
representthedangerousstateofstatus epilepticus,inwhicheitherthereisone
prolonged seizure or there are many seizures with little time for recovery in
between. The last section, Section 4.5, contains a result that elucidates what
we consider to be the underlying “mechanics” of a property associated with
the definition of a rippled almost periodic solution.
