Chimera states in networks of type-I Morris-Lecar neurons

Chimeras are complex spatiotemporal patterns that emerge as coexistence of both coherent and incoherent groups of coupled dynamical systems. Here, we investigate the emergence of chimera states in nonlocal networks of type-I Morris-Lecar neurons coupled via chemical synapses. This constitutes a more realistic neuronal modeling framework than previous studies of chimera states, since the Morris-Lecar model provides biophysically more relevant control parameters to describe the activity in actual neural systems. We explore systematically the transitions of dynamic behavior and find that different types of synchrony appear depending on the excitability level and nonlocal network features. Furthermore, we map the transitions between incoherent states, traveling waves, chimeras, coherent states, and global amplitude death in the parameter space of interest. This work contributes to a better understanding of biological conditions giving rise to the emergence of chimera states in neural medium.

Figure 1

Panel (a) shows the bifurcation diagram of the Morris-Lecar neuron with type-I excitability as a function of the external bias current $I_0$. (thick solid lines: stable equilibria; dotted line: unstable ones; full and empty circles: maxima and minima of stable and unstable limit cycles, respectively). SNIC, HB, and SN mark the saddle-node on invariant circle, Hopf, and saddle-node of limit cycle bifurcations, respectively. Panel (b) represents the corresponding firing frequency vs. current curve ($I_0$, $f$). Evidently, the neuron starts to fire at an arbitrary low firing frequency, which is the distinctive characteristic of type-I excitability. Parameters as in Table 1.

Figure 2

Influence of the excitability level $I_0$ on the emergence of chimera states: Increasing the level of excitability gives rise to different system behaviors in nonlocally coupled identical neuron population, i.e., (a) incoherent state, (b) traveling wave, (c) chimera state, (d) coherent state, and (e) global amplitude death for $I_0=8\mu {\mathrm{A}/\mathrm{cm}}^2$, $I_0=10\mu {\mathrm{A}/\mathrm{cm}}^2$, $I_0=11\mu {\mathrm{A}/\mathrm{cm}}^2$, $I_0=15\mu {\mathrm{A}/\mathrm{cm}}^2$, and $I_0=22\mu {\mathrm{A}/\mathrm{cm}}^2$, respectively. Other system parameters are set as $r=0.1$ and $g=0.1{\mathrm{mS}/\mathrm{cm}}^2$. Note that each column pictures the space-time plots (top panels) and typical snapshots of membrane potentials (middle panels) as well as corresponding average firing frequency profiles (bottom panels) for a given population that is subjected to a fixed $I_0$.

Figure 3

Strength of incoherence $S$ as a function of the excitability level $I_0$. The color shading indicates five different levels of synchrony: incoherence (yellow), traveling wave (green), chimera (magenta), coherence (red), global amplitude death (blue). System parameters are as described in the caption of Fig. 2.

Figure 4

Effect of coupling radius $r$ on the emergence of chimera states. Figure shows dynamical behavior transitions with strength of incoherence ($S$) as a function of $r$ for a fixed coupling strength $g=0.1{\mathrm{mS}/\mathrm{cm}}^2$ and different excitability levels which are set as: (a) $I_0=8.5\mu {\mathrm{A}/\mathrm{cm}}^2$; (b) $I_0=10\mu {\mathrm{A}/\mathrm{cm}}^2$; (c) $I_0=11\mu {\mathrm{A}/\mathrm{cm}}^2$; (d) $I_0=13\mu {\mathrm{A}/\mathrm{cm}}^2$. The color shading indicates different dynamical regimes: incoherence (yellow), traveling wave (green), chimera (magenta), coherence (red).

Figure 5

Effect of excitability level $I_0$ and coupling radius $r$ on the emergence of chimera states: incoherent state (yellow), traveling wave (green), chimera state (magenta), coherent state (red), and global amplitude death (blue). The coupling strength is fixed at $g=0.1{\mathrm{mS}/\mathrm{cm}}^2$.

Figure 6

Behavioral map on ($r$, $g$) plane for a population which has a fixed excitability level at $I_0=10\mu {\mathrm{A}/\mathrm{cm}}^2$. Yellow, green, magenta and red color coded regions represent incoherent state, traveling wave, chimera state and coherent state, respectively. The vertical dashed line marked at $r=0.1$ is used as a reference for explanation (see text).

Figure 7

Emergence of multichimera states with the variation of $r$ and $g$. Top panels show the snapshots of the chimeric population activity splitting into distinct incoherent domains featuring chimera heads. Bottom panels represent average frequency profiles of multichimeras with different numbers of chimera heads. System parameters are fixed as $I_0=10\mu {\mathrm{A}/\mathrm{cm}}^2$, $r=0.1$ in columns (a–c) and $g=0.15{\mathrm{mS}/\mathrm{cm}}^2$ in columns (d-f). Coupling strength is set as (a) $g=0.16{\mathrm{mS}/\mathrm{cm}}^2$, (b) $g=0.185{\mathrm{mS}/\mathrm{cm}}^2$, and (c) $g=0.19{\mathrm{mS}/\mathrm{cm}}^2$, while the coupling radius is chosen as (d) $r=0.133$, (e) $r=0.138$, and (f) $r=0.143$.

Figure 8

Effect of $r$ and $g$ on the emergence of multichimera states. The figure illustrates multiheaded chimeric behavior of type-I Morris-Lecar neuron population emerging with increasing intensity of network interaction. A single label, for instance 6H, refers to 6-headed chimera states. The excitability level is fixed at $I_0=10\mu {\mathrm{A}/\mathrm{cm}}^2$, $g=0.15{\mathrm{mS}/\mathrm{cm}}^2$ in panel (a) and $r=0.1$ in panel (b).