Fenton & Karma 1998

Key: fenton1998vortex

A simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potential (Beeler-Reuter and others)

The transmembrane potential can be found using the relation: Vm = V_0 + u * (V_fi - V_0).

References

  1. https://doi.org/10.1063/1.166311

Figure

Figure illustrating the Fenton & Karma 1998 model.

Variables

  1. u = 0

  2. v = 1

  3. w = 1

Parameters

  • diffusivity_u = 1

  • k = 10

  • tau_0 = 12.5

  • tau_d = 0.25

  • tau_r = 33.33

  • tau_si = 29

  • tau_vm1 = 1250

  • tau_vm2 = 19.6

  • tau_vp = 3.33

  • tau_wm = 41

  • tau_wp = 870

  • u_c = 0.13

  • u_csi = 0.85

  • u_v = 0.04

  • C_si = 1

Source code

OpenCL kernel
const Real tc = u >= u_c;
const Real tv = u >= u_v;
const Real tc_ = 1.0 - tc;
const Real tv_ = 1.0 - tv;
const Real tau_vm = tv * tau_vm1 + tv_ * tau_vm2;
const Real Jfi = -v * tc * (1.0 - u) * (u - u_c) / tau_d;
const Real Jsi = C_si * ( -w * (1.0 + tanh(k * (u - u_csi))) / (2.0 * tau_si));
const Real Jso = u * tc_ / tau_0 + tc / tau_r;
*_new_u = u + dt*(_diffuse_u - (Jfi + Jso + Jsi));
*_new_v = v + dt*(tc_ * (1.0 - v) / tau_vm - tc * v / tau_vp);
*_new_w = w + dt*(tc_ * (1.0 - w) / tau_wm - tc * w / tau_wp);

Additional metadata

keywords:
- excitable media
- electrophysiology
- heart
- phenomenological
extra parameters:
  V_0: -85
  V_fi: 15
  g_fi_max: C_m/tau_d
  C_m: 1
metadata for parameter sets:
  BR:
    name: Beeler Reuter
    dois:
    - https://doi.org/10.1113/jphysiol.1977.sp011853
    - https://doi.org/10.1063/1.166311
  MBR:
    name: Modified Beeler Reuter
    dois:
    - https://doi.org/10.1113/jphysiol.1977.sp011853
    - https://doi.org/10.1063/1.166311
  MLR-I:
    name: Luo Rudy
    dois:
    - https://doi.org/10.1161/01.RES.68.6.1501
    - https://doi.org/10.1063/1.166311
  GP:
    name: Girouard guinea pig
    dois:
    - https://doi.org/10.1161/01.CIR.93.3.603
    - https://doi.org/10.1063/1.166311
  others:
    name: More parameters sets from Fenton et al. 2022.
    note: In this paper, `tau_vm1` and `tau_vm2` have swapped names.
    dois:
    - https://doi.org/10.1063/1.1504242
parameter sets:
  BR:
    k: 10
    tau_0: 12.5
    tau_d: 0.25
    tau_r: 33
    tau_si: 30
    tau_vm1: 1250
    tau_vm2: 19.6
    tau_vp: 3.33
    tau_wm: 41
    tau_wp: 870
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.04
    C_si: 1
  MBR:
    k: 10
    tau_0: 8.3
    tau_d: 0.25
    tau_r: 50
    tau_si: 45
    tau_vm1: 1000
    tau_vm2: 19.2
    tau_vp: 3.33
    tau_wm: 11
    tau_wp: 667
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.055
    C_si: 1
  MLR-I:
    k: 10
    tau_0: 12.5
    tau_d: 0.172
    tau_r: 130
    tau_si: 127
    tau_vm1: 18.2
    tau_vm2: 18.2
    tau_vp: 10
    tau_wm: 80
    tau_wp: 1020
    u_c: 0.13
    u_csi: 0.85
    u_v: 0
    C_si: 1
  GP:
    k: 10
    tau_0: 12.5
    tau_d: 0.115
    tau_r: 25
    tau_si: 22
    tau_vm1: 333
    tau_vm2: 40
    tau_vp: 10
    tau_wm: 65
    tau_wp: 1000
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.025
    C_si: 1
  Set 1:
    k: 10
    tau_0: 8.3
    tau_d: 0.42
    tau_r: 50
    tau_si: 45
    tau_vm1: 1000
    tau_vm2: 19.6
    tau_vp: 3.33
    tau_wm: 11
    tau_wp: 667
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.055
    C_si: 1
  Set 2:
    k: 10
    tau_0: 10
    tau_d: 0.25
    tau_r: 190
    tau_si: 45
    tau_vm1: 10
    tau_vm2: 10
    tau_vp: 10
    tau_wm: 11
    tau_wp: 667
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.055
    C_si: 0
  Set 3:
    k: 10
    tau_0: 12.5
    tau_d: 0.25
    tau_r: 33.33
    tau_si: 29
    tau_vm1: 1250
    tau_vm2: 19.6
    tau_vp: 3.33
    tau_wm: 41
    tau_wp: 870
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.04
    C_si: 1
  Set 4:
    k: 15
    tau_0: 9
    tau_d: 0.407
    tau_r: 34
    tau_si: 26.5
    tau_vm1: 5
    tau_vm2: 15.6
    tau_vp: 3.33
    tau_wm: 80
    tau_wp: 350
    u_c: 0.15
    u_csi: 0.45
    u_v: 0.04
    C_si: 1
  Set 5:
    k: 15
    tau_0: 5
    tau_d: 0.362
    tau_r: 33.33
    tau_si: 29
    tau_vm1: 2
    tau_vm2: 12
    tau_vp: 3.33
    tau_wm: 100
    tau_wp: 1000
    u_c: 0.13
    u_csi: 0.7
    u_v: 0.04
    C_si: 1
  Set 6:
    k: 15
    tau_0: 9
    tau_d: 0.395
    tau_r: 33.33
    tau_si: 29
    tau_vm1: 8
    tau_vm2: 9
    tau_vp: 3.33
    tau_wm: 60
    tau_wp: 250
    u_c: 0.13
    u_csi: 0.5
    u_v: 0.04
    C_si: 1
  Set 7:
    k: 15
    tau_0: 12
    tau_d: 0.25
    tau_r: 100
    tau_si: 29
    tau_vm1: 7
    tau_vm2: 7
    tau_vp: 10
    tau_wm: 60
    tau_wp: 250
    u_c: 0.13
    u_csi: 0.5
    u_v: 0.04
    C_si: 0
  Set 8:
    k: 10
    tau_0: 12.5
    tau_d: 0.45
    tau_r: 33.25
    tau_si: 29
    tau_vm1: 1250
    tau_vm2: 19.6
    tau_vp: 13.03
    tau_wm: 40
    tau_wp: 800
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.04
    C_si: 1
  Set 9:
    k: 10
    tau_0: 12.5
    tau_d: 0.25
    tau_r: 28
    tau_si: 29
    tau_vm1: 2
    tau_vm2: 15
    tau_vp: 3.33
    tau_wm: 61
    tau_wp: 670
    u_c: 0.13
    u_csi: 0.45
    u_v: 0.05
    C_si: 1
  Set 10:
    k: 10
    tau_0: 12.5
    tau_d: 0.115
    tau_r: 25
    tau_si: 22.22
    tau_vm1: 333
    tau_vm2: 40
    tau_vp: 10
    tau_wm: 65
    tau_wp: 1000
    u_c: 0.13
    u_csi: 0.85
    u_v: 0.025
    C_si: 1
example discretisation:
  dt: 0.21
  dx: 1.3
  stimulus: 0.2
units:
  t: ms
  x: mm
  u: 1