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Chimera-like states in modular neural networks
Johanne Hizanidis, Nikos E. Kouvaris, Gorka Zamora-López, Albert Díaz-Guilera, Chris G. Antonopoulos
TL;DR
The paper addresses the limited study of chimera states in modular neural networks. It simulates chaotic bursting neurons in the six-community C. elegans network with electrical and chemical coupling, finding chimera-like states driven by larger, more influential communities and qualitatively similar behavior in related modular networks.
Problem
Chimera states have been studied in complex networks but not extensively in modular networks, whose neural dynamics depend on community structure and coupling.
Method
The study simulates Hindmarsh–Rose neurons in a six-community C. elegans network with electrical within-community and chemical across-community coupling, using synchronization indices to analyze collective behavior.
Results
Chimera-like states form spontaneously, with larger communities driving synchronization while smaller communities remain less synchronized; related modular networks show qualitatively similar results.
Takeaways & Limitations
Under the modeled assumptions, chimera-like states are prominent phenomena in modular networks and may provide insight into more complex modular-network behavior.
Takeaways & Limitations
The model assumes excitatory chemical synapses and, for simplicity, bidirectional chemical coupling.
Abstract
from arXiv · showhide
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 the neural network of the \textit{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 \textit{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.
Introduction
The paper situates chimera-like synchronization within modular neural networks and examines it in a six-community C. elegans network using electrical and chemical coupling. It focuses on how these dynamics relate to network topology and community structure.
- Motivation: Modular brain networks combine locally differentiated processors with global connections supporting coherent functions.
- Motivation: C. elegans exhibits diverse learning behaviors and sensory adaptation, motivating its use as a neural-network model.
- Synchronization: Synchronization in neural networks depends on intrinsic node dynamics, network topology, and connection functions.
- Chimera states: Chimera states are hybrid synchronization patterns in which coherent and incoherent parts coexist, and they occur in neural systems as well as other oscillators.
- Study design: The study models the six-community C. elegans network with electrical synapses within communities and chemical synapses across them.
- Study design: Its primary focus is the conditions supporting chimera-like states and their relationship to the network’s topological characteristics.
Results
The simulations map synchronization across electrical and chemical coupling and identify parameter regions with coherent, metastable, or chimera-like behavior. Chimera-like states are associated with larger communities synchronizing while smaller communities remain less synchronized, and similar patterns occur in other modular networks.
- Synchronization measures: The Kuramoto order parameter ρ measures synchronization within each community and across the entire network as coupling strengths vary.
- Synchronization patterns: High synchronization occurs for low chemical and high electrical coupling, while communities 2 and 4 synchronize across a broader electrical-coupling range.
- Synchronization patterns: Additional synchronization islands occur for communities 3 and 6, even when the remaining communities and the network globally are incoherent.
- Collective regimes: The metastability index λ approaches zero when all communities synchronize and increases when the system switches between synchronous and asynchronous states.
- Collective regimes: The chimera-like index χ is highest in synchronization islands and along the boundary separating coherent from incoherent behavior.
- Representative regimes: At point A, low λ and χ coincide with broad intercommunity synchrony and a high global order parameter under high electrical and low chemical coupling.
- Representative regimes: At point B, low χ accompanies temporal alternation between synchronous and incoherent bursting behavior.
- Representative regimes: The chimera-like state at point C has low metastability, with larger communities more synchronized and smaller communities more difficult to synchronize because of their inputs.
Discussion
The study quantifies synchronization, metastability, and chimera-like behavior in a six-community C. elegans-based neural network. It finds that chimera-like states are driven by the largest, most influential communities and also occur in related modular networks.
- The analysis quantifies synchronization, metastability, and chimera-like behavior using corresponding dynamical indices.The measures characterize coherence, metastability, and chimera-like behavior across the system and its communities.
- Structural analysis links chimera-like behavior to community size, chemical coupling, participation, and node influence.The study examines chemical-synapse ratios, community structure, participation, and global hubness.
- Hubs have the largest participation and extend connections across most communities, while low-participation nodes are not network hubs.Intermediate-participation neurons also share connections across several communities.
- Chimera-like states form spontaneously when the chimera-like index exceeds the metastability index and are driven by the largest communities.They appear in synchronization islands and near the boundary separating coherent from incoherent behavior.
- Related modular networks with Erdős-Rényi and small-world communities also exhibit chimera-like states, suggesting prominence under the studied assumptions.The comparison supports relevance beyond the C. elegans-based network, within the modeled modular-network settings.
Methods
The study models six C. elegans neural communities with Hindmarsh–Rose chaotic bursting neurons and electrical and chemical synapses. It combines community detection, network-role measures, and coherence indices to analyze how topology and coupling shape collective dynamics.
- Network construction: Six communities are identified in the C. elegans neural network using the walktrap community-detection method.The method uses short random walks to find densely connected subgraphs.
- Synaptic coupling: Neurons within communities are connected by electrical synapses, while chemical synapses connect neurons across communities.Electrical coupling is modeled as linear diffusive coupling, whereas chemical coupling is nonlinear and sigmoidal.
- Neural dynamics: Hindmarsh–Rose neurons are parameterized in a chaotic spike-bursting regime, with electrical and chemical coupling strengths controlling network interactions.The model uses a=1, b=3, c=1, d=5, s=4, p0=−1.6, Iext=3.25, and r=0.005.
- Simulation design: The simulations analyze six small-world communities and relate emergent dynamics to the coaction of electrical and chemical synapses.Community sizes and intercommunity chemical connectivity are used to represent the modular topology.
- Network measures: Node roles are characterized using hubness and participation measures that compare degree or link distribution across communities.Hubness compares a node’s degree with an equivalent random graph, while participation measures how links are distributed among communities.
- Dynamical measures: Synchronization is measured with the time-averaged Kuramoto order parameter ρ, while metastability λ quantifies temporal fluctuations in community synchrony.ρ=1 denotes complete synchrony and ρ=0 complete desynchronization; λ averages the variance of community order parameters over time.
Author contributions statement
The study was conceived, designed, simulated, analyzed, and written by the listed authors through divided responsibilities.
- J. H., N. E. K., and C. G. A. conceived and designed the study and performed the numerical simulations.
- G. Z.-L. and A. D.-G. analyzed node roles, while all listed authors contributed to analysis and writing.
Additional information
The authors report no competing financial interests and provide correspondence contacts for the study.
- The authors declare no competing financial interest.
- Correspondence and material requests should be directed to the listed authors.
Figures Legends
The figures establish the network’s six-community modular organization and map parameter regimes to synchronous, metastable, and chimera-like dynamics. Together, they show how community structure and coupling conditions correspond to distinct collective behaviors.
- Figure 1: Figure 1 depicts six C. elegans neural communities, with electrical links within communities and chemical links across communities.Community colors and sizes encode membership and neuron counts, while link styles distinguish synapse types.
- Figure 2: Figure 2 maps community and whole-network order parameters alongside metastability λ and chimera-like χ across chemical and electrical coupling strengths.Points A, B, and C mark synchronous, desynchronous-metastable, and chimera-like regimes, respectively.
- Figures 2–3: At point C, communities 2 and 4 are synchronized while communities 1, 3, and 5 are desynchronized, with low λ and high χ.The corresponding state is identified as chimera-like rather than metastable.