While alternans in a single cardiac cell appears through a simple
period-doubling bifurcation, in extended tissue the exact nature
of the bifurcation is unclear. In particular, the phase of
alternans can exhibit wave-like spatial dependence, either
stationary or traveling, which is known as
alternans. We study these phenomena in simple cardiac models
through a modulation equation proposed by Echebarria-Karma. In
this dissertation, we perform bifurcation analysis for their
modulation equation.
Suppose we have a cardiac fiber of length l, which is
stimulated periodically at its x=0 end. When the pacing period
(basic cycle length) B is below some critical value Bc,
alternans emerges along the cable. Let a(x,n) be the amplitude
of the alternans along the fiber corresponding to the n-th
stimulus. Echebarria and Karma suppose that a(x,n) varies
slowly in time and it can be regarded as a time-continuous
function a(x,t). They derive a weakly nonlinear modulation
equation for the evolution of a(x,t) under some approximation,
which after nondimensionization is as follows:
&partialt a = σ a +
where the linear operator
In the equation, σ is dimensionless and proportional to
Bc - B, i.e. σ indicates how rapid the pacing is,
Λ
propagation and the nonlinear term -ga
onset of linear instability. No flux boundary conditions are
imposed on both ends.
The zero solution of their equation may lose stability, as the
pacing rate is increased. To study the bifurcation, we calculate
the spectrum of operator
bifurcation may be Hopf or steady-state. Which bifurcation occurs
first depends on parameters in the equation, and for one critical
case both modes bifurcate together at a degenerate (codimension 2)
bifurcation.
For parameters close to the degenerate case, we investigate the
competition between modes, both numerically and analytically. We
find that at sufficiently rapid pacing (but assuming a 1:1
response is maintained), steady patterns always emerge as the only
stable solution. However, in the parameter range where Hopf
bifurcation occurs first, the evolution from periodic solution
(just after the bifurcation) to the eventual standing wave
solution occurs through an interesting series of secondary
bifurcations.
We also find that for some extreme range of parameters, the
modulation equation also includes chaotic solutions. Chaotic waves
in recent years has been regarded to be closely related with
dreadful cardiac arrhythmia. Proceeding work illustrated some
chaotic phenomena in two- or three-dimensional space, for instance
spiral and scroll waves. We show the existence of chaotic waves in
one dimension by the Echebarria-Karma modulation equation for
cardiac alternans. This new discovery may provide a different
mechanism accounting for the instabilities in cardiac dynamics.
","abstract_html":"<p>While alternans in a single cardiac cell appears through a simple</p><p>period-doubling bifurcation, in extended tissue the exact nature</p><p>of the bifurcation is unclear. In particular, the phase of</p><p>alternans can exhibit wave-like spatial dependence, either</p><p>stationary or traveling, which is known as <italic>discordant</italic></p><p>alternans. We study these phenomena in simple cardiac models</p><p>through a modulation equation proposed by Echebarria-Karma. In</p><p>this dissertation, we perform bifurcation analysis for their</p><p>modulation equation.</p><p>Suppose we have a cardiac fiber of length l, which is</p><p>stimulated periodically at its x=0 end. When the pacing period</p><p>(basic cycle length) B is below some critical value B<sub>c</sub>,</p><p>alternans emerges along the cable. Let a(x,n) be the amplitude</p><p>of the alternans along the fiber corresponding to the n-th</p><p>stimulus. Echebarria and Karma suppose that a(x,n) varies</p><p>slowly in time and it can be regarded as a time-continuous</p><p>function a(x,t). They derive a weakly nonlinear modulation</p><p>equation for the evolution of a(x,t) under some approximation,</p><p>which after nondimensionization is as follows: </p><p> &partial<sub>t</sub> a = σ a + <bold>L</bold> a - g a <super>3</super>,</p><p>where the linear operator</p><p> <bold>L</bold> a = &partial<sub>xx</sub>a - &partial<sub>x</sub> a -Λ<super>-1</super> ∫ <super>0</super> <sub>x</sub> a(x',t)dx'.</p><p>In the equation, σ is dimensionless and proportional to</p><p>B<sub>c</sub> - B, i.e. σ indicates how rapid the pacing is,</p><p>Λ<super>-1</super> is related to the conduction velocity (CV) of the</p><p>propagation and the nonlinear term -ga<super>3</super> limits growth after the</p><p>onset of linear instability. No flux boundary conditions are</p><p>imposed on both ends.</p><p>The zero solution of their equation may lose stability, as the</p><p>pacing rate is increased. To study the bifurcation, we calculate</p><p>the spectrum of operator <bold>L</bold>. We find that the</p><p>bifurcation may be Hopf or steady-state. Which bifurcation occurs</p><p>first depends on parameters in the equation, and for one critical</p><p>case both modes bifurcate together at a degenerate (codimension 2)</p><p>bifurcation.</p><p>For parameters close to the degenerate case, we investigate the</p><p>competition between modes, both numerically and analytically. We</p><p>find that at sufficiently rapid pacing (but assuming a 1:1</p><p>response is maintained), steady patterns always emerge as the only</p><p>stable solution. However, in the parameter range where Hopf</p><p>bifurcation occurs first, the evolution from periodic solution</p><p>(just after the bifurcation) to the eventual standing wave</p><p>solution occurs through an interesting series of secondary</p><p>bifurcations.</p><p>We also find that for some extreme range of parameters, the</p><p>modulation equation also includes chaotic solutions. Chaotic waves</p><p>in recent years has been regarded to be closely related with</p><p>dreadful cardiac arrhythmia. Proceeding work illustrated some</p><p>chaotic phenomena in two- or three-dimensional space, for instance</p><p>spiral and scroll waves. We show the existence of chaotic waves in</p><p>one dimension by the Echebarria-Karma modulation equation for</p><p>cardiac alternans. This new discovery may provide a different</p><p>mechanism accounting for the instabilities in cardiac dynamics.</p>","abstract_has_math":false,"creators":["Dai, Shu"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Schaeffer, David G."],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009","date_published":"2009","updated_at":"2026-07-24T02:07:07Z","subjects":["Mathematics","Biology, Physiology","bifurcation","cardiac alternans","modulation equation"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/1320","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Schaeffer, David G."]},{"key":"dc:creator","label":"Author","values":["Dai, Shu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2009-08-27T18:33:54Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2009-08-27T18:33:54Z"]},{"key":"dc:date.issued","label":"Date","values":["2009"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Biology, Physiology","bifurcation","cardiac alternans","modulation equation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/1320"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["While alternans in a single cardiac cell appears through a simple
period-doubling bifurcation, in extended tissue the exact nature
of the bifurcation is unclear. In particular, the phase of
alternans can exhibit wave-like spatial dependence, either
stationary or traveling, which is known as
alternans. We study these phenomena in simple cardiac models
through a modulation equation proposed by Echebarria-Karma. In
this dissertation, we perform bifurcation analysis for their
modulation equation.
Suppose we have a cardiac fiber of length l, which is
stimulated periodically at its x=0 end. When the pacing period
(basic cycle length) B is below some critical value Bc,
alternans emerges along the cable. Let a(x,n) be the amplitude
of the alternans along the fiber corresponding to the n-th
stimulus. Echebarria and Karma suppose that a(x,n) varies
slowly in time and it can be regarded as a time-continuous
function a(x,t). They derive a weakly nonlinear modulation
equation for the evolution of a(x,t) under some approximation,
which after nondimensionization is as follows:
&partialt a = σ a +
where the linear operator
In the equation, σ is dimensionless and proportional to
Bc - B, i.e. σ indicates how rapid the pacing is,
Λ
propagation and the nonlinear term -ga
onset of linear instability. No flux boundary conditions are
imposed on both ends.
The zero solution of their equation may lose stability, as the
pacing rate is increased. To study the bifurcation, we calculate
the spectrum of operator
bifurcation may be Hopf or steady-state. Which bifurcation occurs
first depends on parameters in the equation, and for one critical
case both modes bifurcate together at a degenerate (codimension 2)
bifurcation.
For parameters close to the degenerate case, we investigate the
competition between modes, both numerically and analytically. We
find that at sufficiently rapid pacing (but assuming a 1:1
response is maintained), steady patterns always emerge as the only
stable solution. However, in the parameter range where Hopf
bifurcation occurs first, the evolution from periodic solution
(just after the bifurcation) to the eventual standing wave
solution occurs through an interesting series of secondary
bifurcations.
We also find that for some extreme range of parameters, the
modulation equation also includes chaotic solutions. Chaotic waves
in recent years has been regarded to be closely related with
dreadful cardiac arrhythmia. Proceeding work illustrated some
chaotic phenomena in two- or three-dimensional space, for instance
spiral and scroll waves. We show the existence of chaotic waves in
one dimension by the Echebarria-Karma modulation equation for
cardiac alternans. This new discovery may provide a different
mechanism accounting for the instabilities in cardiac dynamics.
"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Bifurcations in the Echebarria-Karma Modulation Equation for Cardiac Alternans in One Dimension"]}]}],"canonical_facts":{"dc:contributor.advisor":["Schaeffer, David G."],"dc:creator":["Dai, Shu"],"dc:date.accessioned":["2009-08-27T18:33:54Z"],"dc:date.available":["2009-08-27T18:33:54Z"],"dc:date.issued":["2009"],"dc:description.abstract":["While alternans in a single cardiac cell appears through a simple
period-doubling bifurcation, in extended tissue the exact nature
of the bifurcation is unclear. In particular, the phase of
alternans can exhibit wave-like spatial dependence, either
stationary or traveling, which is known as
alternans. We study these phenomena in simple cardiac models
through a modulation equation proposed by Echebarria-Karma. In
this dissertation, we perform bifurcation analysis for their
modulation equation.
Suppose we have a cardiac fiber of length l, which is
stimulated periodically at its x=0 end. When the pacing period
(basic cycle length) B is below some critical value Bc,
alternans emerges along the cable. Let a(x,n) be the amplitude
of the alternans along the fiber corresponding to the n-th
stimulus. Echebarria and Karma suppose that a(x,n) varies
slowly in time and it can be regarded as a time-continuous
function a(x,t). They derive a weakly nonlinear modulation
equation for the evolution of a(x,t) under some approximation,
which after nondimensionization is as follows:
&partialt a = σ a +
where the linear operator
In the equation, σ is dimensionless and proportional to
Bc - B, i.e. σ indicates how rapid the pacing is,
Λ
propagation and the nonlinear term -ga
onset of linear instability. No flux boundary conditions are
imposed on both ends.
The zero solution of their equation may lose stability, as the
pacing rate is increased. To study the bifurcation, we calculate
the spectrum of operator
bifurcation may be Hopf or steady-state. Which bifurcation occurs
first depends on parameters in the equation, and for one critical
case both modes bifurcate together at a degenerate (codimension 2)
bifurcation.
For parameters close to the degenerate case, we investigate the
competition between modes, both numerically and analytically. We
find that at sufficiently rapid pacing (but assuming a 1:1
response is maintained), steady patterns always emerge as the only
stable solution. However, in the parameter range where Hopf
bifurcation occurs first, the evolution from periodic solution
(just after the bifurcation) to the eventual standing wave
solution occurs through an interesting series of secondary
bifurcations.
We also find that for some extreme range of parameters, the
modulation equation also includes chaotic solutions. Chaotic waves
in recent years has been regarded to be closely related with
dreadful cardiac arrhythmia. Proceeding work illustrated some
chaotic phenomena in two- or three-dimensional space, for instance
spiral and scroll waves. We show the existence of chaotic waves in
one dimension by the Echebarria-Karma modulation equation for
cardiac alternans. This new discovery may provide a different
mechanism accounting for the instabilities in cardiac dynamics.
"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/10161/1320"],"dc:language.iso":["en_US"],"dc:subject":["Mathematics","Biology, Physiology","bifurcation","cardiac alternans","modulation equation"],"dc:title":["Bifurcations in the Echebarria-Karma Modulation Equation for Cardiac Alternans in One Dimension"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:07:07Z"}