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Chimera states in a multilayer network of coupled and uncoupled neurons
Soumen Majhi, Matjaz Perc, Dibakar Ghosh
TL;DR
The paper asks how chimera states arise when coupled and uncoupled neuronal layers interact through different synapse types. It models Hindmarsh–Rose neurons in a multilayer network with electrical within-layer and chemical inter-layer coupling, including transmission delay. The model produces between-layer synchronous chimera and cluster states, while increasing inter-layer delay expands chimera regions and removes cluster and coherent regions in the examined parameter planes.
Problem
The paper addresses how to model chimera dynamics in neuronal networks containing coupled and uncoupled neurons, multiple synapse types, and multilayer structure.
Method
The authors use two Hindmarsh–Rose neuron layers, with electrical coupling within the coupled layer and chemical synapses linking replica neurons across layers, while varying inter-layer delay.
Results
The network exhibits between-layer synchronous chimera and cluster states, including chimera states for local and global coupled-layer interactions; increasing delay expands chimera regions and suppresses cluster and coherent states.
Takeaways & Limitations
Hybrid synapses and multilayer organization can support synchronous chimera and cluster patterns involving coupled and uncoupled neuronal layers.
Abstract
from arXiv · showhide
We study the emergence of chimera states in a multilayer neuronal network, where one layer is composed of coupled and the other layer of uncoupled neurons. Through the multilayer structure, the layer with coupled neurons acts as the medium by means of which neurons in the uncoupled layer share information in spite of the absent physical connections among them. Neurons in the coupled layer are connected with electrical synapses, while across the two layers neurons are connected through chemical synapses. In both layers the dynamics of each neuron is described by the Hindmarsh-Rose square wave bursting dynamics. We show that the presence of two different types of connecting synapses within and between the two layers, together with the multilayer network structure, plays a key role in the emergence of between-layer synchronous chimera states and patterns of synchronous clusters. In particular, we find that these chimera states can emerge in the coupled layer regardless of the range of electrical synapses. Even in all-to-all and nearest-neighbor coupling within the coupled layer, we observe qualitatively identical between-layer chimera states. Moreover, we show that the role of information transmission delay between the two layers must not be neglected, and we obtain precise parameter bounds at which chimera states can be observed. The expansion of the chimera region and annihilation of cluster and fully coherent states in the parameter plane for increasing values of inter-layer chemical synaptic time delay are illustrated using effective range measurement. These results are discussed in the light of neuronal evolution, where the coexistence of coherent and incoherent dynamics during the developmental stage is particularly likely.
I. INTRODUCTION
The introduction motivates studying chimera states in multilayer neuronal networks because brain activity involves coupled and uncoupled neurons, multiple synapse types, and coherent–incoherent dynamics with biological relevance.
- Chimera states combine spatially separated coherent and incoherent domains, whereas cluster states split coherence into multiple mutually synchronized domains.
- Multilayer neural structures are relevant because brain networks may contain multiple structural and functional connection types.
- Neuronal chimera patterns are connected with brain diseases and unihemispheric sleep, where synchronized and desynchronized hemispheres coexist.
- Most prior neural-network studies considered either electrical or chemical synapses, although structural neuronal networks contain both types.
- The paper studies a multilayer network with coupled and uncoupled neuron layers, nonlocal electrical interaction in one layer, and chemical inter-layer connections.
- The study targets between-layer synchronous chimera and cluster states, in which replica nodes across layers share dynamical behavior while individual layers may differ.
II. MATHEMATICAL FORM OF THE NETWORK
The model contains two equal-sized Hindmarsh–Rose neuron layers: an isolated upper layer and a generally nonlocally coupled lower layer linked through chemical synapses.
- The network has two layers with N identical neurons each; layer I neurons are isolated, while layer II neurons are generally nonlocally coupled.
- Each layer-I neuron connects to its replica in layer II, which acts as the medium transmitting interactions to otherwise isolated neurons.
- All neurons follow Hindmarsh–Rose dynamics; electrical synapses couple layer-II neurons, while chemical synapses connect the layers.
- Figure 1 depicts nonlocal lower-layer connections and direct vertical links between each upper-layer node and its corresponding lower-layer node.
- Layer-II neurons connect to P neighbors on both sides of a ring, with coupling range R determined by P and N.
- The inter-layer information-transfer delay is τ, and the chemical synapses use an excitatory sigmoidal response with threshold Θs = −0.25 and slope λ = 10.
III. RESULTS
The study examines instantaneous and delayed inter-layer chemical coupling, beginning from isolated bursting and an incoherent coupled layer before introducing inter-layer interaction.
- The results compare instantaneous inter-layer chemical coupling with coupling that includes transmission delay.
- With Kel = 0 and Kch = 0, an isolated layer-II neuron exhibits regular square-wave bursting dynamics.
- With Kel = 0.005, Kch = 0, P = 30, and N = 100, layer II displays incoherent membrane-potential behavior.
A. Instantaneous inter-layer interaction
With instantaneous inter-layer interaction, increasing chemical coupling drives both layers from incoherence through synchronous chimera and cluster states to coherence. These patterns persist across electrical coupling ranges, including nearest-neighbor and global coupling.
- Dynamical-state transitions: At Kel = 0.005, Kch = 0.5 produces incoherence, Kch = 1.1 chimera, Kch = 2.0 clusters, and Kch = 3.0 coherence in both layers.The corresponding angular-frequency profiles distinguish scattered incoherent domains from identical coherent domains.
- Dynamical-state transitions: For 1.0 ≤ Kch ≤ 1.75, both layers exhibit chimera states with 0 < SIk, Sk < 1; cluster states occur for 1.75 ≤ Kch ≤ 2.9.For Kch > 2.9, the layers become coherent as SIk reaches zero.
- Between-layer synchrony: The zero inter-layer difference ΔSI and matching snapshots indicate synchronous chimera and cluster patterns across the two layers.The layers also maintain a bounded difference between membrane-potential amplitudes.
- Parameter dependence: Across nearly all Kel values in [0.0, 0.015], the system transitions from disordered dynamics through chimera and cluster states to coherence.The result indicates that these patterns are not restricted to specific intra-layer electrical coupling strengths.
- Coupling-range robustness: Synchronous chimera and cluster states arise across wide Kch intervals for every coupling range R, including nearest-neighbor and globally coupled electrical networks.The inter-layer chemical coupling and multilayer structure support chimera states at both limiting coupling topologies.
B. Delayed inter-layer interaction
Inter-layer chemical synaptic delay reshapes the network’s dynamical regimes by expanding synchronous-chimera regions while suppressing cluster and coherent states. Across parameter-plane analyses, increasing delay progressively favors chimera patterns and establishes thresholds for the disappearance of other states.
- Delay-induced state transitions: At τ = 2.4, cluster states disappear and chimera states occupy parameter regions that previously showed clustering or coherence.The transformation is illustrated for Kch = 2.3, where delay changes a cluster pattern into a chimera state.
- Delay-induced state transitions: For τ ∈ [0, 5], cluster states occur only for τ ∈ [0, 1.1], while the Kch range supporting chimera states expands as τ increases.The τ−Kch plane fixes P = 30 and Kel = 0.005.
- Effective-range analysis: The effective range of chimera states increases with τ, whereas cluster states have ER = 0 after τ = 1.10 and coherent states diminish for τ ≥ 3.5.ER is the fraction of sampled points in the prescribed Kel−Kch parameter plane reaching each state.
- Coupling-range dependence: At τ = 0.5, delay enlarges the chimera region and narrows the cluster region in the R−Kch plane; at τ = 2.4, clusters no longer occur.At τ = 4.0, coherence also disappears, leaving incoherent and chimera states as Kch increases for any coupling range R.
- Overall effect: The authors characterize chemical-synaptic transmission delay as enlarging synchronous-chimera domains while slowing the development of cluster states.This conclusion applies to the modeled network of coupled and uncoupled neuronal layers.
IV. DISCUSSION AND CONCLUSION
The study demonstrates that hybrid electrical–chemical synapses in a multilayer network of coupled and uncoupled Hindmarsh–Rose neurons generate synchronous chimera and cluster patterns. Inter-layer delay further broadens chimera regions and can eliminate cluster and coherent regimes.
- IV. DISCUSSION AND CONCLUSION: The multilayer architecture produces synchronous chimera and synchronous cluster states between coupled and uncoupled neuronal layers.The model combines electrical synapses within the connected layer and chemical synapses between layers.
- IV. DISCUSSION AND CONCLUSION: Hybrid synapses generate between-layer synchronous patterns even when the coupled layer uses local or global interactions.The authors contrast this with the absence of chimera emergence under electrical synapses alone.
- IV. DISCUSSION AND CONCLUSION: Inter-layer delay can induce chimera states where instantaneous interaction cannot, while sufficiently large delay eliminates cluster and coherent profiles.The authors report that delay broadens the chimera range in parameter space.