# Spectrum of a linearized amplitude equation for alternans in a cardiac fiber

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2008-12-01

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Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suffer reversals of phase at various points along the fiber, a phenomenon called (spatially) discordant alternans. Either stationary or moving patterns are possible. Using a weak approximation, Echebarria and Karma proposed an equation to describe the spatiotemporal dynamics of small-amplitude alternans in a class of simple cardiac models, and they showed that an instability in this equation predicts the spontaneous formation of discordant alternans. To study the bifurcation, they computed the spectrum of the relevant linearized operator numerically, supplemented with partial analytical results. In the present paper we calculate this spectrum with purely analytical methods in two cases where a small parameter may be exploited: (i) small dispersion or (ii) a long fiber. From this analysis we estimate the parameter ranges in which the phase reversals of discordant alternans are stationary or moving. © 2008 Society for Industrial and Applied Mathematics.

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Dai, S, and DG Schaeffer (2008). Spectrum of a linearized amplitude equation for alternans in a cardiac fiber. *SIAM Journal on Applied Mathematics*, 69(3). pp. 704–719. 10.1137/070711384 Retrieved from https://hdl.handle.net/10161/6957.

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#### David G. Schaeffer

**Granular flow**

Although I worked in granular flow for 15 years, I largely stopped working in this area around 5 years ago. Part of my fascination with this field derived from the fact that typically constitutive equations derived from engineering approximations lead to ill-posed PDE. However, I came to believe that the lack of well-posed governing equations was the major obstacle to progress in the field, and I believe that finding appropriate constitutive relations is a task better suited for physicists than mathematicians, so I reluctantly moved on.

One exception: a project analyzing periodic motion in a model for landslides as a Hopf bifurcation. This work is joint with Dick Iverson of the Cascades Volcanic Laboratory in Vancouver Washington. This paper [1] was a fun paper for an old guy because we were able to solve the problem with techniques I learned early in my career--separation of variables and one complex variable.

**Fluid mechanics**

In my distant bifurcation-theory past I studied finite-length effects in Taylor vortices. Questions of this sort were first raised by Brooke Benjamin. My paper [2] shed some light on these issues, but some puzzles remained. Over the past few years I have conducted a leisurely collaboration with Tom Mullin trying to tie up the loose ends of this problem. With the recent addition of Tom Witelski to the project, it seems likely that we will soon complete it.

**Mathematical problems in electrocardiology**

About 10 years ago I began to study models for generation of cardiac rhythms. (Below I describe how I got interested in this area.) This work has been in collaboration with Wanda Krassowska (BME), Dan Gauthier (Physics) and Salim Idress (Med School). Postdocs Lena Tolkacheva and Xiaopeng Zhou contributed greatly to the projects, as well as grad students John Cain and Shu Dai. The first paper [3], with Colleen Mitchell was a simple cardiac model, similar in spirit and complexity to the FitzHugh-Nagumo model, but based on the heart rather than nerve fibers. Other references [4--9] are given below.

A general theme of our group's work has been trying to understand the origin of *alternans*. This term refers to a response of the heart at rapid periodic pacing in which action potentials alternate between short and long durations. This bifurcation is especially interesting in extended tissue because during propagation the short-long alternation can suffer phase reversals at different locations, which is called *discordant* alternans. Alternans is considered a precursor to more serious arrythmias.

Let me describe one current project [9]. My student, Shu Dai, is analyzing a weakly nonlinear modulation equation modeling discordant alternans that was proposed by Echebarria and Karma. First we show that, for certain parameter values, the system exhibits a degenerate (codimension 2) bifurcation in which Hopf and steady-state bifurcations occur simultaneously. Then we show, as expected on grounds of genericity (see Guckenheimer and Holmes, Ch. 7) that chaotic solutions can appear. The appearance of chaos in this model is noteworthy because it contains only one space dimension; by contrast the usual route to chaos in cardiac systems is believed to be through the breakup of spiral or scroll waves, which of course requires two or more dimensions.

**Other biological problems**

Showing less caution than appropriate for a person my age, I have recently begun to supervise a student, Kevin Gonzales, on a project modeling gene networks. Working with Paul Magwene (Biology), we seek to understand the network through which yeast cells, if starved for nitrogen, choose between sporulation and pseudohyphal growth. (Whew!) This work is an outgrowth of my participation in the recently funded Center for Systems Biology at Duke.

I have gotten addicted to applying bifurcation theory to differential equations describing biological systems. For example, my colleagues Harold and Anita Layton are tempting my with some fascinating bifurcations exhibited by the kidney. Here is a whimsical catch phrase that describes my addiction: "Have bifurcation theory but won't travel". (Are you old enough--and sufficiently tuned in to American popular culture--to understand the reference?)

**Research growing out of teaching**

Starting in 1996 I have sometimes taught a course that led to an expansion of my research. The process starts by my sending a memo to the science and engineering faculty at Duke, asking if they would like the assistance of a group of math graduate students working on mathematical problems arising in their (the faculty member's) research. I choose one area from the responses, and I teach a case-study course for math grad students focused on problems in that area. In broad terms, during the first half of the course I lecture on scientific and mathematical background for the area; and during the second half student teams do independent research, with my collaboration, on the problems isolated earlier in the semester. I also give supplementary lectures during the second half, and at the end of the semester each team lectures to the rest of the class on what it has discovered. This course was written up in the SIAM Review [11].

Topics and their proposers have been: LithotripsyL. Howle, P. Zhong (ME)Population models in ecologyW. Wilson (Zoology)Electrophysiology of the heart IC. Henriquez (BME)Electrophysiology of the heart IID. Gauthier (Physics). Lithotripsy is an alternative to surgery for treating kidney stones--focused ultrasound pulses are used to break the stones into smaller pieces that can be passed naturally.

Multiple research publications, including a PhD. thesis, have come out of these courses, especially my work in electrophysiology.

I hope to offer this course in the future. *Duke faculty:* Do you have a problem area to propose?

- [1] D.G. Schaeffer and R. Iverson, Steady and intermittent slipping in a model of landslide motion regulated by pore-pressure feedback, SIAM Applied Math 2008 (to appear)
- [2] Schaeffer, David G., Qualitative analysis of a model for boundary effects in the Taylor problem, Math. Proc. Cambridge Philos. Soc., vol. 87, no. 2, pp. 307--337, 1980 [MR81c:35007]
- [3] Colleen C. Mitchell, David G. Schaeffer, A two-current model for the dynamics of cardiac membrane, Bulletin Math Bio, vol. 65 (2003), pp. 767--793
- [4] D.G. Schaeffer, J. Cain, E. Tolkacheva, D. Gauthier, Rate-dependent waveback velocity of cardiac action potentials in a done-dimensional cable, Phys Rev E, vol. 70 (2004), 061906
- [5] D.G. Schaeffer, J. Cain, D. Gauthier,S. Kalb, W. Krassowska, R. Oliver, E. Tolkacheva, W. Ying, An ionically based mapping model with memory for cardiac restitution, Bull Math Bio, vol. 69 (2007), pp. 459--482
- [6] D.G. Schaeffer, C. Berger, D. Gauthier, X. Zhao, Small-signal amplification of period-doubling bifurcations in smooth iterated mappings, Nonlinear Dynamics, vol. 48 (2007), pp. 381--389
- [7] D.G. Schaeffer, X. Zhao, Alternate pacing of border-collision period-doubling bifurcations, Nonlinear Dynamics, vol. 50 (2007), pp. 733--742
- [8] D.G. Schaeffer, M. Beck, C. Jones, and M. Wechselberger, Electrical waves in a one-dimensional model of cardiac tissue, SIAM Applied Dynamical Systems (Submitted, 2007)
- [9] D.G. Schaeffer and Shu Dai, Spectrum of a linearized amplitude equation for alternans in a cardiac fiber, SIAM Analysis 2008 (to appear)
- [10] D.G. Schaeffer, A. Catlla, T. Witelski, E. Monson, A. Lin, Annular patterns in reaction-diffusion systems and their implications for neural-glial interactions (Preprint, 2008)
- [11] L. Howle, D. Schaeffer, M. Shearer, and P. Zhong, Lithotripsy, The treatment of kidney stones with shock waves, SIAM Review vol. 40 (1998), pp356--371

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