Ionic Models

General framework

Ionic models describe how the transmembrane voltage and ionic species interact across cellular membranes. These exchanges are generally described by ODE systems, with gate variables representing the permeability of ion channels (transmembrane proteines) to different species, and additional variables describing the state of the system, that are called miscallenous variables in CEPS implementation. Together with the transmembrane voltage, this forms the ODE system:

\[ \displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix} W \\ C \\ v \end{pmatrix} = \begin{pmatrix} H(X,t) \\ G(X,t) \\ \frac{1}{C_\mathrm{m}}\left(\sum_{i}I_i(X,t)+I_{\mathrm{app}}(t)\right)\end{pmatrix}, \]


  • $ W = (w_k)_{k=1,N_w}^t $ are the gate variables,
  • $ C = (c_k)_{k=1,N_c}^t $ are the misc variables,
  • $ v $ is the scalar transmembrane voltage,
  • $ X $ designates the whole vector $(W,C,v)^t$, FIXME or space var ? unclear from previous doc.
  • $ H = (h_k)_{k=1,N_w}^t,\quad h(X,t)_k = \displaystyle\frac{w_{k,\infty}(X,t)-w_k(X,t)}{\tau_k} $ is the standard evolution function for gate variables (sliding-door models), with limit values $w_{k,\infty}$ and dynamics $\tau_k$,
  • $ I_i $ are ionic currents considered by the model to act on $ v $,
  • $ I_{\mathrm{app}} $ is an external applied current,
  • $ C_\mathrm{m} $ is the cell membrane capacitance.

Each value is defined everywhere in regions of the computational domain that are defined by the main input file and mesh attributes. Several ionic models can then coexist, and some regions can have no ionic model at all. Usually, there is a model defined at each degree of freedom of the PDE problem, meaning that there exists a large number of arrays that have the size of the discretization mesh.

For more information on how the ODE systems are solved, please refer to the numerics page.

Implemented models

  • Beeler-Reuter (BR77) : Reconstruction of the action potential of ventricular myocardial fibres, Beeler, G.W. and Reuter, H. 1977 Journal of Physiology , 268, 177-210. PubMed ID: 874889
  • Courtemanche-Ramirez-Nattel (CRN98) : Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. M. Courtemanche, R.J. Ramirez, and S. Nattel.
    Am. J. Physiol., 275 (Heart Circ. Physiol.44):H301–H321, 1998. PubMed ID: 9688927
  • Mitchell-Schaeffer (MS03) : A two-current model for the dynamics of cardiac membrane, Colleen C. Mitchell, David G. Schaeffer, 2003, Bulletin of Mathematical Biology , 65, (5), 767-793. PubMed ID: 12909250
  • Ten-Tusscher Panvilov 2006 (TTP06), with its three variants for endo-, epi- and midmyo-cardium : Alternans and spiral breakup in a human ventricular tissue model, K.H.W.J. ten Tusscher, A.V. Panfilov, Sep 2006, American Journal of Physiology, Heart and Circulatory Physiology, 291 3, H1088-1100. PubMed ID: 16565318