UK Nonlinear News, February 1998

The heartbeat: An Interdisciplinary Problem

By Caroline Chopra

Caroline Chopra Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT England

The pioneering experimental work of Alexandre Fabiato in 1975 saw the birth of new opportunities for thousands of scientists in the area of intracellular calcium dynamics. In examining skinned cardiac muscle cell preparations Fabiato showed that the elevated intracellular calcium concentration preceding muscle contraction was largely due to the release of calcium from intracellular stores and modulated by intracellular calcium itself, hence introducing the important role of calcium as an intracellular signaler. (For calcium reviews see[2-3,7,11,13].)

This finding opposed the then current hypothesis that intracellular calcium concentration increased mainly due to a flow of ions through electrically activated channels in the cell membrane. These revelations offered a new lease of life to mathematicians and computational biologists striving to better understand the heart beat through modelling.

The 1980s saw the development of a whole host of dynamical systems ranging in scope from simple qualitative models to very complicated biophysical models, taking the form of highly nonlinear coupled ordinary differential equations, which, more or less, accurately described Fabiato's results. Linear stability analysis and bifurcation theory have helped further understanding of these systems, together with the use of other general results relating to the existence and stability of periodic orbits (see [1,4-5] for useful texts in this area). In the wake of technological advances, such as super computers, it has become possible to perform extensive numerical experiments on these models incorporating both time and space variations of the calcium transient. In particular models have been spatially extended to form reaction-diffusion systems, the analytical and numerical work performed on generic excitable systems (extensively reviewed in [12]) has been adapted to these systems providing useful qualitative results [8-9]. Such simulations have provided illuminating results which in turn has set a precedent for further experimental work.

The advent of experimental techniques such as intracellular fluorescent dyes and voltage clamping in the late 1980s meant not only that the behaviour of the calcium transient, in the absence of electrical activity, could be studied whilst retaining the membrane but also that a direct measurement of intracellular calcium concentration could be obtained based on the fluorescence of the inserted dye. These advances have provided mathematicians with further evidence on which to base new and more accurate models.

The 1990s saw the discovery of intracellular calcium spiral waves in the presence of an obstacle, such as the nucleus[6]. Mathematicians working in this area are currently trying to obtain more information on these types of waves by relating the speed normal to a curved travelling wave front to its curvature, known as the eikonal equation, in an attempt to characterise this behaviour[12]. It is of interest to know whether calcium dynamics follow the same speed-curvature result as other dynamical systems and to postulate as to why differences may occur. Also of current interest are the conditions required for the anchoring of wave fronts to obstacle's. Although little is known about these conditions the current belief is that they are intrinsically linked to the excitability of the system[10] and anchoring can therefore be achieved, or eliminated, by tweaking the model parameters.

From plane waves to spiral waves, the heartbeat is an interdisciplinary mystery evoking many questions with the answers lying in further collaborative research between experts from many scientific backgrounds. Further details of current research in and relating to this area are to be found at http://www.cbiol.leeds.ac.uk/.

References

  1. P. Drazin. Nonlinear Systems. Cambridge, 1992.
  2. A. Fabiato and F. Fabiato. Contractions induced by a calcium-triggered release of calcium from the sarcoplasmic reticulum of single skinned cardiac cells. J. Physiol., 249:469--495, 1975.
  3. A. Golbeter. Cell to Cell Signaling: From Experiments to Theoretical Models. London: Academic, 1989.
  4. J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 1983.
  5. D. Kaplan and L. Glass. Understanding Nonlinear Dynamics. Springer-Verlag, 1995.
  6. P. Lipp and E. Niggli. Microscopic spiral waves reveal positive feedback in subcellular calcium signaling. Biophysical Journal, 65(6):2272--2276, 1993.
  7. C.H. Orchard, M.R. Mustafa, and E. White. Oscillations and waves of intracellular calcium in cardiac muscle cells. Chaos, Solitons and Fractals, 5(3/4):447--458, 1995.
  8. J. Sneyd and A. Atri. Curvature dependence of a model for calcium wave propagation. Physica D, 65(4):365--372, 1993.
  9. J. Sneyd, S. Girard, and D. Clapham. Calcium wave propagation by calcium induced calcium release: an unusual excitable system. Bulletin of Mathematical Biology, 55(2):315--344, 1993.
  10. J.M. Starobin and C.F. Starmer. Boundary-layer analysis of waves propagating in an excitable medium: Medium conditions for wave-front-obstacle separation. Physical Review E, 54(6):430--437, 1996.
  11. M.D. Stern. Theory of ec coupling in cardiac muscle. Biophysical Journal, 71(63):497--517, 1992.
  12. J.J. Tyson and J.P. Keener. Singular perturbation theory of travelling waves in excitable media. Physica D, 32(3):327--361, 1988.
  13. W.G. Wier. Cytoplasmic calcium in mammalian ventricle: dynamical control by cellular processes. Annual Rev. Physiol., 52:467--485, 1990.


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