Chimera-like States in Modular Neural Networks

Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider a neural network inspired by the connectome of the C. elegans soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter ρ, we also employ other measures of coherence, such as the chimera-like χ and metastability λ indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the C. elegans neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks.

Supplementary Movie S2. It shows the time evolution of the Hindmarsh-Rose model, as it is described by Eqs.
(1). The strength of electrical and chemical coupling is g el = 0.7 and g ch = 0.18, respectively that correspond to point B, marked in the parameter space in Figure 2. The time growth is illustrated by a gray horizontal line which traverses the plots of spacetime and time-series and moves vertically towards increasing time. The dynamical behavior shown in this movie is desynchronous (see also Figure 3(b)).
Supplementary Movie S3. It shows the time evolution of the Hindmarsh-Rose model, as it is described by Eqs.
(1). The strength of electrical and chemical coupling is g el = 0.5 and g ch = 0.015, respectively that correspond to point C, marked in the parameter space in Figure 2. The time growth is illustrated by a gray horizontal line which traverses the plots of spacetime and time-series and moves vertically towards increasing time. The dynamical behavior shown in this movie is chimera-like, the communities 1 (brown), 3 (red) and 5 (purple) are desynchronized, whereas the rest are synchronized (see also Figure 3(c)).

II. Analysis of the modular network with communities detected with the Louvain method
Here, we proceed to the same analysis presented in the main text by using a modular network with 277 neurons grouped into six communities. The communities are detected by employing the Louvain method [1]; they are shown in Figure S1. As is discussed in the text, neurons of the same community are assumed to be connected with electrical coupling, while accross communities with chemical coupling.

Community 1
Community 2 Community 3 Community 6 Community 5 Community 4 Figure

III. Analysis for modular networks with small-world communities
Here, we proceed to the same analysis presented in the main text by using a modular network with 277 neurons grouped into six communities. Each community is a subnetwork created with the configuration model by randomly assigning links to match the degrees sequences (i.e. the number of electrical synapses of the i-th neuron) as they were detected in the C.elegans neural network using the walktrap algorithm. We eliminate parallel links and self loops. Then, we connect neurons across communities with a random network, whose links represent the chemical synapses. The number of these synapses is the same as in the network used in the main text.

Community 1
Community 2 Community 3 Community 6 Community 5 Community 4 Figure S4: Modular network with small-word communities. A modular network with six interconnected communities is shown. We assign a number and color to each community (1: brown, 2: blue, 3: red, 4: yellow, 5: purple, 6: green). Black links indicate electrical couplings between neurons within the same community, whereas gray links represent chemical couplings between neurons across different communities. A chimera-like state is shown for g el = 0.5 and g ch = 0.04, values which correspond to point C in Fig. S5.

IV. Analysis for modular networks with random communities
Here, we proceed to the same analysis presented in the main text by using a modular network with 277 neurons grouped into six Erdős-Rényi communities. Each of the six communities has the same number of neurons with the communities detected in the C.elegans neural network using the walktrap algorithm. Moreover, we use the same number of electrical synapses, and we randomly assign them to a pair of neurons with equal probability. By doing this, each community is an Erdős-Rényi subnetwork. Chemical synapses used here are the same as in the modular networks with small-world communities, discussed previously in the supplementary material.

Community 1
Community 2 Community 3 Community 6 Community 5 Community 4 Figure S7: Modular network with random communities. A modular network with six interconnected Erdős-Rényi communities is shown. We assign a number and color to each community (1: brown, 2: blue, 3: red, 4: yellow, 5: purple, 6: green). Black links indicate electrical coupling between neurons within the same community and gray represent chemical couplings between neurons across different communities.