# David G. Schaeffer

- James B. Duke Professor Emeritus of Mathematics
- Professor Emeritus of Mathematics

**External address:**231 Physics Bldg, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2814

**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 physicsts 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 biologogical 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:

Lithotripsy | L. Howle, P. Zhong (ME) |

Population models in ecology | W. Wilson (Zoology) |

Electrophysiology of the heart I | C. Henriquez (BME) |

Electrophysiology of the heart II | D. 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

### Selected Grants

EMSW21-RTG: awarded by National Science Foundation (Co-Principal Investigator). 2010 to 2017

Border-Collision Bifurcations in Cardiac Muscle awarded by National Science Foundation (Co-Principal Investigator). 2006 to 2009

FRG-Collaborative Research: Physical, Mathematical, and Engineering Problems in Slow Granular Flow awarded by National Science Foundation (Principal Investigator). 2003 to 2007

Supplement:Collaborative Proposal:Physical,Mathematical,and Engineering Problems in Slow Granular Flow awarded by National Science Foundation (Principal Investigator). 2002 to 2004

Collaborative Proposal: Physical, Mathematical, and Engineering Problems in Slow Granular Flow awarded by National Science Foundation (Principal Investigator). 2002 to 2003

Fundamental and Applied Problems in Granular Flow awarded by National Science Foundation (Principal Investigator). 1998 to 2002

(96-1035) Multidimensional Problems in Dynamic Plasticity awarded by National Science Foundation (Principal Investigator). 1995 to 1999

(97-0828) Multidimensional Problems in Dynamic Plasticity awarded by National Science Foundation (Principal Investigator). 1995 to 1999

(95-0285) Multidimensional Problems in Dynamic Plasticity awarded by National Science Foundation (Principal Investigator). 1995 to 1996

(94-0972) Multidimensional Problems In Dynamic Plasticity awarded by National Science Foundation (Principal Investigator). 1992 to 1996

## Pages

Gonzales, K, Kayikci, O, Schaeffer, DG, and Magwene, P. "Modeling mutant phenotypes and oscillatory dynamics in the \emph{Saccharomyces cerevisiae} cAMP-PKA pathway (Submitted)." *PLoS Computational Biology* (December 2010). (Academic Article)

Dai, S, and Schaeffer, DG. "Bifurcations in a modulation equation for alternans in a cardiac fiber." *Esaim: Mathematical Modelling and Numerical Analysis* 44.6 (November 2010): 1225-1238.
Full Text Open Access Copy

Farjoun, Y, and Schaeffer, DG. "The hanging thin rod: a singularly perturbed eigenvalue problem (Submitted)." *SIAM Sppl. Math.* (July 2010). (Academic Article)

Dai, S, and Schaeffer, DG. "Chaos in a one-dimensional model for cardiac dynamics." *Chaos* 20.2 (June 2010). (Academic Article)

Dai, S, and Schaeffer, DG. "Spectrum of a linearized amplitude equation for alternans in a cardiac fiber." *Siam Journal on Applied Mathematics* 69.3 (December 1, 2008): 704-719.
Full Text Open Access Copy

Schaeffer, DG, and Iverson, RM. "Steady and intermittent slipping in a model of landslide motion regulated by pore-pressure feedback." *Siam Journal on Applied Mathematics* 69.3 (December 1, 2008): 769-786.
Full Text

Beck, M, Jones, CKRT, Schaeffer, D, and Wechselberger, M. "Electrical waves in a one-dimensional model of cardiac tissue." *Siam Journal on Applied Dynamical Systems* 7.4 (December 1, 2008): 1558-1581.
Full Text

Schaeffer, DG, Ying, W, and Zhao, X. "Asymptotic approximation of an ionic model for cardiac restitution." *Nonlinear Dynamics* 51.1-2 (January 2008): 189-198.
Full Text

Schaeffer, DG, and Cain, J. "Shortening of action potential duraction near an insulating boundary." *Math Medicine and Biology* 25.21--36 (2008). (Academic Article)

Schaeffer, DG, Catlla, A, Witelski, T, Monson, E, and Lin, A. "On spiking models of synaptic activity and impulsive differential equations." *SIAM Review* 50.553--569 (2008). (Academic Article)