Chimera patterns in two-dimensional networks of coupled neurons

We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.

Figure 1

Schematic depiction of the coupling topology used for the FHN model of Eqs. (2) in panel (a) and for the LIF model of Eq. (9) in panel (b) according to Eqs. (3) and (10), respectively.

Figure 2

FHN system: Chimera state of (a) incoherent-spot and (b) coherent-spot type. The left panels depict a snapshot of the activator variable $x_{ij}$ at $t=2000$. The center and right panels show the mean phase velocity ${\omega }_{ij}$ and a horizontal cut of ${\omega }_{ij}$. Coupling parameters: (a) $r=33,\phi =\pi /2-0.2$ and (b) $r=42,\phi =\pi /2$. Other parameters are $\sigma =0.1,a=0.5$, and $\epsilon =0.05$.

Figure 3

FHN system: Radius $\rho$ of the spotlike chimera states, (a) incoherent spot and (b) coherent spot, in dependence on the coupling radius $r$ for different coupling phases $\phi$ given as a color code. The error bars indicate an uncertainty of (a) $\mathrm{\Delta }\rho =0.3$ and (b) $\mathrm{\Delta }\rho =0.5$ in the measurements of the radii. Parameters as in Fig. 2.

Figure 4

FHN system: Ring chimera, found for $r=33$ and $\phi =\pi /2-0.24$. A snapshot at $t=1900$ of the (color-coded) activator variable $x_{ij}$ is shown on the left side and the mean phase velocity and a section through the ring in the center and right panels, respectively. Other parameters as in Fig. 2.

Figure 5

FHN system: Inner (red triangles) and outer (black squares) diameter $D$ of the incoherent ring in dependence on the coupling radius $r$. The error bars indicate an uncertainty of $\mathrm{\Delta }D=0.9$ in the measurements of the diameters. The lines are linear fits as guides to the eye. All ring patterns found for $\phi \in [\pi /2-0.24,\pi /2-0.2]$ are shown. The inset shows the data with rescaled axes: Fraction of incoherent elements $N_{\text{inc}}/N^2$ vs number of coupled neighbors $N_r/N^2$. The brightness of the dots corresponds to the number of observed ring patterns for every $\left(D,r\right)$ pair in the considered $\phi$ range. Other parameters as in Fig. 2.

Figure 6

LIF system: Chimera state of incoherent-spot type. Snapshot of $u_{ij}$ (left) and the corresponding mean phase velocity (middle and right) for small values of the coupling constant $\sigma =0.1$ and for a coupling range $R=10$. Other parameters are $\mu =1.0,p_r=0$, and $N=100$.

Figure 7

LIF system. Ring chimera: Snapshot of $u_{ij}$ (left) and the corresponding mean phase velocity (middle and right) for a coupling range of $R=10$ and $p_r=0.22T_s$. Other parameters as in Fig. 6.

Figure 8

LIF system: Inner and outer diameter $D$ of the incoherent ring patterns as a function of the coupling range $R$. Inset: Relative number of incoherent elements $N_{\text{inc}}/N^2$ as a function of the relative number of coupled neighbors $N_R/N^2$. Parameters as in Fig. 7.

Figure 9

FHN system: Stripe chimera for $r=39,\phi =\pi /2-0.04$, and $t=1900$. The (color-coded) activator $x_{ij}$ (left panel) shows a synchronous stripe region, while the other panels show the mean phase velocity distribution ${\omega }_{ij}$ (middle panel) and the section of ${\omega }_{ij}$ (right panel). Other parameters as in Fig. 2.

Figure 10

FHN system: Width $W$ of the coherent region in the fully synchronized stripes in dependence on the coupling range $r$ for different coupling phases $\phi$ as a color code. The size of the error bars refers to an uncertainty of $\mathrm{\Delta }W=0.9$ in the measurements. Parameters as in Fig. 2.

Figure 11

LIF system: Snapshot of $u_{ij}$ (left), the corresponding mean phase velocity (middle), and a section of it (right). Parameters are $\sigma =0.6,\mu =1.0,p_r=0.6T_s,N=100$, and $R=20$.

Figure 12

FHN system: (a) Grid state with $6\times 6$ spots for $r=30$ and $\phi =\pi /2+0.82$. The $k\times k$ structure is visible in the (color-coded) activator variable $x_{ij}$ (left) at $t=2000$ and the (color-coded) mean phase velocity (center). The right column shows a section of ${\omega }_{ij}$. (b) Lines of incoherent spots for $r=22,\phi =\pi /2+0.96$, and $t=1900$. Other parameters as in Fig. 2.

Figure 13

LIF system: (a) Grid chimera and (b) double-spotted line. The Patterns (a) and (b) are formed starting from two different random initial conditions. Left panels: snapshots of $u_{ij}$; middle panels: corresponding mean phase velocities; right panels: section of mean phase velocities. Parameters: $R=22,\sigma =0.7,\mu =1.0,p_r=0.22T_s$, and $N=100$.

Figure 14

LIF system: The snapshots of $u_{ij}$ (left panels), corresponding mean phase velocities (middle panels), and sections of mean phase velocities (right panels) in the limit of strong coupling for two different coupling ranges: (a) $R=23$ and (b) $R=45$. Parameters: $\sigma =0.7,\mu =1.0,p_r=0.22T_s$, and $N=100$.

Figure 15

LIF system: The ratio of incoherent elements in all observed $6\times 6$ grids as a function of the coupling strength $\sigma$, using a fixed coupling range at $R=25$ and other parameters as in Fig. 7.

Figure 16

FHN system: full parameter space $(\phi ,r)$. The colors indicate groups of chimera states (blue), frequency-locked isochronous chimera states (green), cluster states and patterns with a gradual phase shift (orange), and solitary states (red). Other parameters as in Fig. 2.

Figure 17

FHN system: overview of some chimeras for the cross-coupled region with $1.1\le \phi \le 2.15$. The regions with bold borders are discussed in the main text in Secs. 3a and 4a. The colors indicate different types of chimeras, while the arrows and numbers point out the number of coherent heads (for the leopard chimeras), the number of spirals (for the spiral states), and the alignment (horizontal, diagonal) of the ninepin chimera. Other parameters as in Fig. 2.

Figure 18

FHN system: Enlargement of the parameter space where grid chimera states are found (cf. Sec. 5a). The color indicates the number of incoherent heads in a single line. Every grid chimera pattern is partially surrounded by a corresponding grid pattern, which does not include the incoherent cores (black). Other parameters as in Fig. 2.

Figure 19

FHN system. Examples of chimeras and other states: The left panels show the activator variable $x_{ij}$, while the other panels show the mean phase velocity distribution ${\omega }_{ij}$ (left panel) and a horizontal section of ${\omega }_{ij}$ (right panel). (a) Solitary state, $r=24,\phi =\pi /2-1.04$ at $t=1900$; (b) double spiral chimera, $r=17,\phi =\pi /2-0.54$ at $t=1900$; (c) cross chimera, $r=19,\phi =\pi /2-0.14$ at $t=6000$; (d) coherent spot with a spiral, $r=49,\phi =\pi /2-0.02$ at $t=2000$; (e) leopard chimera $\left(4\times 3\right)$, $r=37,\phi =\pi /2+0.16$ at $t=2000$; (f) ninepin chimera $(2\times 2$ diagonal), $r=48,\phi =\pi /2+0.34$ at $t=2000$; (g) mosaic chimera, $r=31,\phi =\pi /2+0.36$ at $t=6000$; (h) wave chimera with spots (five waves), $r=17,\phi =\pi /2+0.48$ at $t=2000$; (i) twisted 5,3-chimera, $r=25,\phi =\pi /2+0.42$ at $t=2000$; (j) organic chimera, $r=2,\phi =5\pi /2-2.8$ at $t=6000$; (k) diagonal antiphase stripe and spots, $r=5,\phi =5\pi /2-2.7$ at $t=2000$; (l) spiral in antiphase spot, $r=15,\phi =5\pi /2-2.7$ at $t=1900$. Other parameters are $\sigma =0.1,a=0.5$, and $\epsilon =0.05$.

Figure 20

FHN system. Grid states: Dependence of the number of asynchronous heads $k$ in a single line on the coupling radius $r$. The red line is an exponential fit. The brightness of the dots indicates the number of occurrences of $k\times k$ grid chimeras for $\phi \in (1.71,4.65)$. Parameters as in Fig. 2.