Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

Figure 1.  Nullclines of the local FHN system (1.1) in the monostable, excitable and bistable regime. (a) The monostable case, with $ \varepsilon = 1, a = -0.05 $, $ \beta = 0.1 $ and $ \gamma = 1.1 $. (b) The excitable case, with $ \varepsilon = 1, a = -0.3 $, $ \beta = 0.1 $ and $ \gamma = 1.02 $. (c) The bistable case, with $ \varepsilon = 1, a = 0 $, $ \beta = 0.5 $ and $ \gamma = 0.7 $. The labels $ m $ and $ M $ indicate the minimum and the maximum of the $ u $-nullcline (2.3). The matrices give the signs of the derivatives $ f_u, f_v, g_u, g_v $ evaluated at the equilibrium point

Figure 2.  Geometrical representation of the conditions for the diffusive instability. (a)-(b) For two different choices of $ d_v $, the dark gray regions in the $ (d_u, d) $-plane delimited by the two straight lines $ d = d_u\, d_v $ (dashed line) and $ d = \bar{d} $ (dotted line) correspond to the fulfillment of conditions (2.10)-(2.16a). The other parameters are chosen as $ \beta = 0.1, a = 0.0001, \gamma = 1.02 , \varepsilon = 2 $, so that $ E^* = (0.0051, 0.0051) $. (a) $ d_v = 0.1 $, which gives $ \delta_u^{(2)} = 2.2051 , \delta_u^{(1)} = 2.0045, I_c = 2.2030 $. (b) $ d_v = 1 $, which gives $ \delta_u^{(2)} = 4.0421 , \delta_u^{(1)} = 2.0201, I_c = 4.0382 $. (c)-(d) For two different values of $ d_v $, the gray shaded areas represent the diffusive instability regions in the $ (d_u, d) $-plane, corresponding to the fulfillment of both (2.10)-(2.16a) and (2.16b). The boundaries of the Turing region are $ d = d_c $, or $ P(d) = 0 $, (solid line) and $ d = d_ud_v $ (dashed line). (c) The parameters are chosen as in (a). (d) The parameters are chosen as in (b)

Figure 3.  The kinetics parameters are chosen as in Fig. 2(a). (a) The dark-gray and light-gray region indicate the presence of Turing and cross-Turing patterns, respectively, separated by the dash-dot curve expressed by (2.34). The dashed line marks the boundary above which $ d>d_ud_v $. (b) Turing pattern obtained by the numerical simulation of the system (1.1), with $ d_v = 0.1 $, $ d_u = 2.0194 $ and $ d = 0.32> d_c^{+} = 0.3191 $ (so that $ k_c^{+} = 0.75 $), corresponding to the point marked by an asterisk in (a). The profile of the activator (inhibitor) is represented by a solid (dotted) line. (c) Cross-Turing pattern obtained by the numerical simulation of the system (1.1) with $ d_v = 0.1 $, $ d_u = 2.0194 $ and $ d = 0.20423<d_c^{-} = 0.20432 $ (so that $ k_c^{-} = 2 $), corresponding to the point marked by a plus sign in (a). The profile of the activator $ u $ (inhibitor $ v $) is represented by a solid (dotted) line

Figure 4.  (a)-(b)-(c) Turing patterns observed along the upper branch of the bifurcation parabola. (d)-(e)-(f) Cross-Turing patterns observed along the lower branch of the bifurcation parabola. The parameters are chosen as in Fig. 2(a). The profile of the activator $ u $ (inhibitor $ v $) is represented by a solid (dotted) line. (a) $ d_u = 2.012 $ and $ d = 0.3398>d_c^+ = 0.3364 $, so that $ k_c^+ = 0.7 $. (b) $ d_u = 2.0198 $ and $ d = 0.3213>d_c^+ = 0.3181 $, so that $ k_c^+ = 0.7568 $. (c) $ d_u = 2.0385 $ and $ d = 0.2527>d_c^+ = 0.2553 $, so that $ k_c^+ = 0.9 $. (d) $ d_u = 2.0198 $ and $ d = 0.201689<d_c^- = 0.2017 $, so that $ k_c^- = 3 $. (e) $ d_u = 2.0194 $ and $ d = 0.2040<d_c^- = 0.2044 $, so that $ k_c^- = 1.9881 $. (f) $ d_u = 2.0385 $ and $ d = 0.2319<d_c^- = 0.2324 $, so that $ k_c^- = 1.07 $

Figure 5.  (a) Bifurcation diagram of the system (1.1) in the $ (d_u, d) $-plane as given by the WNL theory. The black, gray and dark gray regions correspond to cross-Turing subcritical bifurcation, cross-Turing supercritical bifurcation and Turing supercritical bifurcation, respectively. The dashed line marks the boundary above which $ d>d_ud_v $. The dash-dot curve, expressed by (2.34), that separates Turing from cross-Turing patterns is also reported. The parameters are chosen as in Fig. 2. (b) Turing instability for $ d\ge d_c^+ $: graph of the coefficient $ L $ of the amplitude equation (3.7) versus $ d_u $ for different choices of $ d_v $. The kinetic parameter are chosen as in (a). (c) Cross-Turing instability for $ d\le d_c^- $: graph of the coefficient $ L $ of the amplitude equation (3.7) versus $ d_u $ for different choices of $ d_v $. The kinetic parameter are chosen as in (a)

Figure 6.  Cross-Turing stationary pattern supported by the system (1.1) at a regular bifurcation. The parameters are chosen as $ \beta = 0.1 $, $ \gamma = 1.02 $, $ \varepsilon = 2 $, $ a = 0.0001 $, so that $ E^* = (0.0051, 0.0051) $ and $ d_u = 2.035 $, $ d_v = 0.1 $ and $ d = 0.2176<d_c^-\approx 0.2198 $. In the square domain $ [0, {4\sqrt{2}\pi}/{k_c}]\times[0, {4\sqrt{2}\pi}/{k_c}] $ the Turing bifurcation is regular: the critical wavenumber $ k_c\approx 1.2356 $ corresponds to the unique couple of modes $ (m, n) = (4, 4) $ satisfying the condition (3.11). (a)-(b) Initial condition assigned as a small random perturbation of the homogeneous equilibrium. (b)-(c) Numerical solution of the system (1.1) computed via spectral methods. (d)-(e) Spectrum of the solution

Figure 7.  Cross-Turing stationary subcritical pattern supported by the system (1.1) at a regular bifurcation. The parameters of the reaction term are chosen as in Fig. 6 and $ d_v = 0.1 $, $ d_u = 2.0151 $, $ d = 0.2024< d_c^-\approx0.2026 $. In the square domain $ [0, \pi]\times[0, \sqrt{2}\pi] $ the most unstable mode $ k_c\approx 2.4495 $ corresponds to the unique couple of integers $ (m, n) = (2, 2) $ satisfying the condition (3.11). The simulation reveals that several modes, other than the critical one, are excited. (a)-(c) Numerical solution of the system (1.1) computed using spectral methods and assigning as initial condition a small random perturbation of the homogeneous equilibrium. (b)-(d) Spectrum of the solution

Figure 8.  Cross-Turing stationary mixed-mode pattern supported by the system (1.1) when the monostable equilibrium loses stability via a degerate bifurcation and non-resonance conditions (3.13) hold. The parameters of the reaction term are chosen as in Fig. 6 and $ d_u = 2.035 $ and $ d_v = 0.1 $, $ d = 0.219<d_c^-\approx 0.2198 $. In the rectangular domain $ \left[0, {8\pi}/{k_c}\right]\times\left[0, {8\pi}/{k_c}\right] $ the most unstable mode $ k_c\approx 1.2356 $ corresponds to the two couples of integers $ (m_1, n_1) = (0, 8) $ and $ (m_2, n_2) = (8, 0) $ satisfying the condition (3.11). (a)-(c) Numerical solution of the system (1.1) computed using spectral methods and assigning as initial condition a small random perturbation of the homogeneous equilibrium. (b)-(d) Spectrum of the solution

Figure 9.  Resonant degenerate cross-Turing bifurcation: bifurcation diagrams of the amplitude system (3.18). The parameters of the reaction term in (1.1) are chosen as in Fig. 6 and $ d_u = 2.035 $ and $ d_v = 0.1 $, $ d_c^-\approx 0.2198 $. A solid red (dashed black) line represents stable (unstable) branches of stationary equlibria. The branches labeled by $ H^\pm_s $ represent stable hexagons of the FHN system (1.1) that bifurcate subcritically from the homogeneous equilibrium at the point $ B_p $. A pair of rolls of (1.1) bifurcates at $ B_p $ in a saddle-node bifurcation so leading, on the left of $ B_p $, to coexistence of stable rolls and hexagons. (a) Bifurcation diagram of $ A_1 $. (b) Bifurcation diagram of $ A_2 $

Figure 10.  Resonant degenerate cross-Turing bifurcation: phase space diagrams of the amplitude system (3.18). (a) The parameters are chosen as in Fig. 9 with $ d = d_c^- +0.0002 = 0.22 $. The coordinates of equilibrium points are $ H^\pm_s = (\pm 0.0801882, -0.061473) $ (stable), $ H^\pm_u = (\pm 0.00185716, -0.00094344) $ (unstable). (b) The parameters are chosen as in Fig. 9 with $ d = d_c^- -0.0008 = 0.219 $. The coordinates of equilibrium points are $ R^+_s = (0, 0.02246) $ (stable), $ R^-_u = (0, -0.02246) $ (unstable), $ H^\pm_s = (\pm 0.088722, -0.06956) $ (stable), $ H^\pm_u = (\pm 0.003852, 0.008295) $ (unstable)

Figure 11.  Resonant degenerate cross-Turing bifurcation: stationary hexagonal pattern supported by the system (1.1) when resonance conditions (3.17) hold. The parameters are chosen as in Fig. 9 with $ d = d_c^- -0.0008 = 0.219 $. In the rectangular domain $ [0, {8\pi}/{k_c}]\times[0, 8\sqrt{3}\pi/{k_c}] $ the most unstable mode $ k_c\approx 0.8625 $ corresponds to the two couples of integers $ (m_1, n_1) = (4, 12) $ and $ (m_2, n_2) = (8, 0) $ satisfying the condition (3.11). (a)-(c) Numerical solution of the system (1.1) computed using spectral methods. The initial condition is chosen of the form (3.19), where $ (A_{1\infty}, A_{2\infty}) $ is in the basin of attraction of the stable equilibrium point $ H^+_s $. (b)-(d) Spectrum of the solution

Figure 12.  Resonant degenerate cross-Turing bifurcation: stationary roll pattern supported by the system (1.1) when resonance conditions (3.17) hold. The parameters and the domain are chosen as in Fig. 11. The initial condition is chosen of the form (3.19), where $ (A_{1\infty}, A_{2\infty}) $ is in the basin of attraction of the stable equilibrium point $ R^+_s = (0, 0.02246) $ and $ (m, n) = (8, 0) $. (a)-(c) Numerical solution of the system (1.1) computed via spectral methods. (b)-(d) Spectrum of the solution