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Chimera states in networks of nonlocally coupled Hindmarsh-Rose neuron models
Johanne Hizanidis, Vasilis Kanas, Anastasios Bezerianos, Tassos Bountis
TL;DR
The paper addresses whether chimera states occur in realistic neuronal models and investigates them in nonlocally coupled two- and three-dimensional Hindmarsh–Rose networks using several coupling schemes. It identifies chimera states across these models, verifies them in coupled bistable elements, and reports a mixed oscillatory state alongside evidence of relevance to neuroscience applications.
Problem
The paper examines whether chimera states occur in realistic neuronal networks and whether they extend beyond oscillator models with a single attracting limit cycle.
Method
The study analyzes nonlocally coupled two- and three-dimensional Hindmarsh–Rose oscillator networks under multiple coupling schemes and parameter settings.
Results
The authors identify chimera states in two- and three-dimensional Hindmarsh–Rose networks, including cases with membrane-potential-only coupling, and observe mixed oscillatory states.
Takeaways & Limitations
The findings provide evidence that chimeras may be relevant to neuroscience, occur in coupled bistable elements, and can coexist with stationary or synchronously oscillating neurons in mixed states.
Abstract
from arXiv · showhide
We have identified the occurrence of chimera states for various coupling schemes in networks of two-dimensional and three-dimensional Hindmarsh-Rose oscillators, which represent realistic models of neuronal ensembles. This result, together with recent studies on multiple chimera states in nonlocally coupled FitzHugh-Nagumo oscillators, provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications. Moreover, our work verifies the existence of chimera states in coupled bistable elements, whereas to date chimeras were known to arise in models possessing a single stable limit cycle. Finally, we have identified an interesting class of mixed oscillatory states, in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion.
1. Introduction
Chimera states are a counterintuitive coexistence of synchronized and incoherent oscillators that has been studied across many oscillator models. This paper extends that investigation to physiologically more realistic Hindmarsh–Rose neuron networks, including two- and three-dimensional versions.
- Background: Chimera states describe the coexistence of coherent and incoherent oscillators under nonlocal symmetric coupling.The phenomenon was first observed in populations of identical phase oscillators and later named after the Greek mythological creature Chimera.
- Related work: Research on chimera states has expanded from phase oscillators to maps and Stuart–Landau oscillators, with experimental evidence in chemical and optical oscillator systems.
- Paper aim: The paper identifies single and multi-chimera states in networks of nonlocally coupled Hindmarsh–Rose oscillators.
- Model motivation: Hindmarsh–Rose models are used because they reproduce neuronal behaviors such as rapid firing and regular or chaotic bursting that two-dimensional FitzHugh–Nagumo models do not.The study considers both two-dimensional and three-dimensional Hindmarsh–Rose versions, with the latter including a slowly varying current associated with bursting.
2. Two–dimensional HR models
The two-dimensional Hindmarsh–Rose network exhibits mixed oscillatory, chimera, and multi-domain patterns under nonlocal coupling. These states depend on coupling architecture, phase, coupling range, and network size.
- Model and coupling: The model couples N Hindmarsh–Rose oscillators to R nearest neighbors on each side through a ring topology with direct and cross-coupling.The study examines equal coupling in both variables and coupling restricted to the membrane-potential variable.
- Mixed oscillatory states: At φ = −π, a mixed oscillatory state places stationary and oscillating neurons interspersed across the network.This behavior is linked to bistability: uncoupled oscillators have three fixed points and a stable limit cycle.
- Phase-dependent patterns: At φ = 0, diagonal coupling produces a classical chimera with two incoherent domains.For φ = −π/4, the neurons shift into spiking dynamics and form a wave-like spatial pattern.
- Network-size and range effects: Reducing R while increasing N produces multiple coherent and incoherent domains at φ = 0.Under φ = −π/4, the corresponding spatial patterns have larger wave numbers.
- Membrane-potential coupling: When coupling is restricted to x, chimera states occur at intermediate coupling strength between desynchronization and complete synchronization.The spatial snapshots and corresponding (x_k, y_k)-plane show this progression as σ_x increases.
3. Three–dimensional HR models
The 3D Hindmarsh-Rose model adds a slowly varying current that enables bursting, and its network dynamics depend strongly on which variables are coupled. Coupling both fast variables yields mixed or fully synchronized states, whereas coupling only the membrane-potential variable produces chimera states across intermediate strengths.
- Model: The 3D Hindmarsh-Rose model adds a slowly varying current that changes the applied current, enables firing-frequency adaptation, and produces bursting modes unavailable to the 2D model.The parameter z represents the slowly varying current, while s governs adaptation.
- Coupling both variables: With symmetric coupling in x and y, low coupling produces mixed oscillatory states with synchronized regular spiking alongside irregularly spiking neurons.The network is prepared in the spiking regime with b = 3 and J = 5.
- Coupling both variables: At higher equal coupling strengths in x and y, the system becomes fully synchronized.Figure 5 compares σx = σy = 0.14 and σx = σy = 0.29 for N = 1000 and R = 350.
- Coupling only x: With coupling only in x, low coupling preserves regular spiking while desynchronization alternates with complete synchronization as σx increases.The reported patterns are shown for σy = 0 while increasing σx.
- Coupling only x: At intermediate x-only coupling, one incoherent domain forms while individual neurons produce irregular bursts; higher σx increases spikes per burst and eventually restores full synchronization.The study reports that chimera states disappear and reappear as σx varies, likely because of multistability and sensitive dependence on initial conditions.
4. Conclusions
The paper identifies chimera states across coupling schemes in 2D and 3D Hindmarsh-Rose networks, supporting their relevance to neuronal-ensemble models. It also extends the known setting of chimeras to coupled bistable elements and reports a novel mixed oscillatory state.
- Main conclusions: Chimera states occur for various coupling schemes in networks of two-dimensional and three-dimensional Hindmarsh-Rose models.The paper presents these models as realistic representations of neuronal ensembles.
- Relevance: Nonlocal connectivity in brain-like complex systems and the relationship between chimeras and synchronization motivate studying these states in neuron-dynamics models.The paper connects nonlocal coupling with the occurrence of chimera states and notes its compatibility with human-brain connectivity.
- Theoretical significance: The study verifies chimera states in coupled bistable elements, whereas earlier known examples involved oscillator models with a single attracting limit-cycle state.This is presented as a theoretical significance of the work.
- Mixed oscillatory states: The paper identifies a mixed oscillatory state in which desynchronized neurons are interspersed among neurons that are stationary or oscillate synchronously.The authors describe this as a novel type of mixed oscillatory state.