A numerical method for the fractional Fitzhugh–Nagumo monodomain model


  • Fawang Liu Queensland University of Technology
  • Ian Turner Queensland University of Technology
  • Vo Anh Queensland University of Technology
  • Qianqian Yang Queensland University of Technology
  • Kevin Burrage Queensland University of Technology




fractional FitzHugh--Nagumo Monodomain Model, fractional Riesz space nonlinear reaction-diffusion model, stability and convergence


A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach. References
  • B. Baeumer, M. Kovaly, and M. M. Meerschaert, Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bulletin of Mathematical Biology 69:2281–2297, 2007. doi:10.1007/s11538-007-9220-2
  • B. Baeumer, M. Kovaly, and M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Computers and Mathematics with Applications 55:2212–2226, 2008. doi:10.1016/j.camwa.2007.11.012
  • N. Badie and N. Bursac, Novel micropatterned cardiac cell cultures with realistic ventricular microstructure, Biophys J 96:3873–3885, 2009. doi:10.1016/j.bpj.2009.02.019
  • A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, Technical report, University of Oxford, 2013.
  • A. Bueno-Orovioy, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional dffusion models of electrical propagation in cardiac tissue: electrotonic effects and the modulated dispersion of repolarization, Technical report, University of Oxford, 2013.
  • K. F. Decker, J. Heijman, J. R. Silva, T. J. Hund and Y. Rudy, Properties and ionic mechanisms of action potential adaptation, restitution, and accommodation in canine epicardium, Am. J. Physiol Heart Circ. Physiol. 296:H1017–H1026, 2009. doi:10.1152/ajpheart.01216.2008
  • J. S. Frank and G. A. Langer, The myocardial interstitium: its structure and its role in ionic exchange, J Cell Biol 60:586–601, 1974. doi:10.1083/jcb.60.3.586
  • A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond), 117:500–544, 1952. http://jp.physoc.org/content/117/4/500.html
  • R. FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J., 1:445–466, 1961. doi:10.1016/S0006-3495(61)86902-6
  • D. Kay, I. W. Turner, N. Cusimano and K. Burrage, Reflections from a boundary: reflecting boundary conditions for space-fractional partial differential equations on bounded domains, Technical report, University of Oxford, 2013. .
  • F. Liu, V. Anh and I. Turner, Numerical solution of space fractional Fokker-Planck equation. J. Comp. and Appl. Math., 166:209–219, 2004. doi:10.1016/j.cam.2003.09.028
  • F. Liu, P. Zhuang, V. Anh and I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comp., 191:12–20, 2007. doi:10.1016/j.amc.2006.08.162
  • R. Magin, O. Abdullah, D. Baleanu and X. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation, Journal of Magnetic Resonance 190:255–270, 2008. doi:10.1016/j.jmr.2007.11.007
  • M. M. Meerschaert, J. Mortensenb and S. W. Wheatcraft, Fractional vector calculus for fractional advection-dispersion, Physica A, 367:181–190, 2006. doi:10.1016/j.physa.2005.11.015
  • L. C. McSpadden, R. D. Kirkton and N. Bursac, Electrotonic loading of anisotropic cardiac monolayers by unexcitable cells depends on connexin type and expression level, Am. J. Physiol. Cell Physiol. 297:C339–C351, 2009. doi:10.1152/ajpcell.00024.2009
  • J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50:2061–2070, 1962. doi:10.1109/JRPROC.1962.288235
  • S. F. Roberts, J. G. Stinstra and C. S. Henriquez, Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model, Biophys J 95:3724–3737, 2008. doi:10.1529/biophysj.108.137349
  • J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K. A. Mardal and A. Tveitio, Computing the electrical activity in the heart, Springer-Verlag, 2006.
  • G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985.
  • F. J. Valdes-Parada, J. A. Ochoa-Tapia and J. Alvarez-Ramirez, Effective medium equations for fractional Fick law in porous media, Physica A, 373:339–353, 2007. doi:10.1016/j.physa.2006.06.007
  • Q. Yang, F. Liu and I. Turner, Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term, International Journal of Differential Equations, 2010:464321, 2010, doi:10.1155/2010/464321
  • W. Ying, A multilevel adaptive approach for computational cardiology, PhD thesis, Duke University, 2005.
  • Q. Yu, F. Liu, I. Turner and K. Burrage, A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D, Appl. Math. Comp., 219:4082–4095, 2012. doi:10.1016/j.amc.2012.10.056
  • Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation, the special issue of Fractional Calculus and Its Applications in-Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371:20120150, 2013. doi:10.1098/rsta.2012.0150
  • Q. Yu, F. Liu, I. Turner and K. Burrage, Numerical simulation of the fractional Bloch equations, J. Comp. Appl. Math., 255:635–651, 2014. doi:10.1016/j.cam.2013.06.027
  • P. Zhuang, F. Liu, V. Anh and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Num. Anal., 47:1760–1781, 2009. doi:10.1137/080730597





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