Chimera states, characterized by the coexistence of synchronized and desynchronized phases in networks of coupled oscillators, have garnered significant attention. Initially believed to require stringent conditions like non-local interactions (as in the Kuramoto model), recent research, particularly with spiking neural networks (SNNs), has shown that chimera states can emerge under broader conditions, including local or global interactions, and even with adaptive interactions. While theoretical SNN studies are prevalent, experimental research, especially using SNN-like systems, remains limited. This paper presents experimental observations of various chimera states in a photonic SNN, using coupled DOPOs as neuromorphic neurons. Each DOPO neuron exhibits two spiking modes (Class-I and Class-II, as classified by Hodgkin), mirroring real biological neurons. Synchronization in the network causes a spontaneous spiking-mode shift accompanied by a substantial frequency change, a unique property that stabilizes and influences the movement of chimera states. This study aims to provide an experimental platform for researching chimera states in SNNs, offering insights into real-world chimera-like phenomena and potentially advancing understanding of brain diseases.

The phenomenon of chimera states, where synchronized and desynchronized oscillators coexist, has been extensively studied in various systems. Kuramoto and Battogtokh's work highlighted the counter-intuitive possibility of such inhomogeneous states in identical oscillators with homogeneous interactions. Abrams and Strogatz further formalized the concept of chimera states. Real-world observations of chimera-like behavior have been reported in diverse areas, including power grids and condensed matter systems. Neuromorphic and biological systems also exhibit chimera-like phenomena, including social systems, neuronal bump states, heart cell contractions, unihemispheric sleep, and brain-related diseases. Theoretical studies on neural networks have increasingly focused on SNN models, utilizing various neuron models like Fitzhugh-Nagumo, Hindmarsh-Rose, Morris-Lecar, and Wilson-Cowan. Early belief that chimera states demanded strict conditions, including non-local interactions, has been challenged. Research on SNNs demonstrated the possibility of chimera states under broader conditions, encompassing local or global interactions, and also adaptive interactions that dynamically change. The discovery of amplitude chimeras broadened the definition of chimera states, and a categorization scheme improved the classification of chimera states. While theoretical studies on chimera states in SNN systems have expanded, experimental research remains limited, particularly in systems mimicking SNNs using electronic circuits or other physical implementations. This gap underscores the need for experimental investigation to validate theoretical findings and explore real-world implications.

The experimental setup utilizes a photonic SNN composed of 256 DOPO neurons. Each neuron consists of a pair of DOPOs, nonlinear oscillators based on parametric down-conversion in χ(2) nonlinear material. The DOPOs exhibit two spiking modes (Class-I and Class-II) depending on operating parameters (pump amplitude and basic frequency). Optical coherence between bistable states (0 and π phases) represents the membrane potential. Energy transfer between paired DOPOs (v- and w-DOPOs) through antisymmetric coupling implements spiking dynamics. Intra- and inter-neuron interactions are controlled using a measurement feedback (MFB) method based on an optical and electronic hybrid system. This allows for arbitrary network structures and weights (signed 8-bit integers). Parallel and cross-type inter-neuron interactions with weights α and β are implemented. Without inter-neuron coupling (α=β=0), changing *P*/ω₀ (renormalized pump amplitude and basic frequency) allows switching between Class-I and Class-II spiking modes. The photonic SNN is built using a 1-km fiber ring cavity, PSA module, and MFB system. A DOPO pulse is generated through parametric amplification in periodically poled lithium-niobate (PPLN) waveguides. A pump pulse at 768 nm is generated from a 1536 nm pulse via second harmonic generation (PPLN1) and used for amplification in PPLN2. An optical band-pass filter selects the 1536 nm DOPO pulse. Time-domain multiplexing generates a large number of DOPO pulses. The MFB system enables arbitrary network structures by injecting optical feedback pulses. In-phase components are measured (balanced homodyne detector), processed (FPGA module), and converted back to optical feedback pulses (optical modulator). The system permits 512 DOPO pulses, thus 256 neurons (pairs of DOPOs), with 56 neurons acting as references. Two network structures were investigated: a ring lattice with constant long-range interaction and an exponentially decaying interaction. Initial states are prepared by initially increasing pump amplitudes under a randomized bias field without couplings, leading to random phase distributions (0, 0), (0, π), (π, 0), and (π, π). After initialization, the bias field is removed, and interactions are introduced. The local curvature (D) is calculated to distinguish synchronized (D ≈ 0) and desynchronized (D ≠ 0) regions. Indices g₀(t) (proportion of synchronized components) and h₀ (proportion of temporarily stable components) categorize chimera states. Spiking mode changes are analyzed in the v-w plane and the phase dynamics θᵢ.

Two types of chimera states were experimentally observed: 1. **Moving-stationary chimera state (constant long-range interaction):** A ring lattice with constant long-range interaction (dmax = 35) exhibited a double-headed, meandering chimera state. The local curvature measure clearly distinguished synchronized and desynchronized regions. Analysis using g₀(t) and h₀ indices categorized this as a moving-stationary chimera, characterized by the meandering movement of the chimera. The meandering originates from spontaneous spiking mode changes (Class-I to Class-II) at the boundary between synchronized and desynchronized regions. Class-I neurons (desynchronized) show stepwise phase changes, while Class-II neurons (synchronized) exhibit linear phase changes, with occasional class changes at the boundary. This spontaneous mode shift causes a time-scale difference that stabilizes the chimera. 2. **Moving-turbulent chimera state (exponentially decaying interaction):** A ring lattice with exponentially decaying interaction (dmax = 6) showed a moving-turbulent chimera state. Desynchronized regions were short-lived, with bubble-like patterns of synchronization and desynchronization. Analysis of g₀(t) and h₀ revealed this to be a moving-turbulent chimera, where g₀(t) fluctuates irregularly, indicating unstable desynchronized regions. A strong pump amplitude (P/ω₀ = 2.6) generated a chimera state with synchronized firing and desynchronized non-firing regions, analogous to "chimera death". This behavior involved a spontaneous mode shift, with the non-firing region linked to stable fixed points. Antiphase correlation was also observed in some desynchronized regions. The study demonstrated that the spontaneous spiking-mode shift, rather than multiple time scales inherent to standard SNNs, plays a key role in stabilizing the chimera states. This self-organized time-scale change generates high-frequency synchronized and low-frequency desynchronized regions, contrasting with many previous studies showing low-frequency synchronized regions. The experimental system offers high controllability over spiking modes, time scales, and interaction geometry, providing a valuable platform for investigating chimera states and their implications.

The experimental observation of chimera states in a photonic SNN based on coupled DOPOs addresses the need for experimental validation of theoretical models and expands our understanding of chimera state dynamics. The results demonstrate the emergence of chimera states under relatively simple conditions, with static interactions rather than adaptive or non-local interactions often proposed in theoretical models. The spontaneous spiking-mode shift, induced by synchronization, is a novel mechanism for stabilizing and creating complex chimera dynamics, particularly the observed meandering behavior. This finding challenges the assumptions that fast-slow dynamics are prerequisites for chimera states. The observation of both moving-stationary and moving-turbulent chimera states highlights the richness of dynamical behaviors possible in even simple SNN architectures. The system's high controllability allows for systematic investigations of chimera states, including the creation of a chimera phase diagram, and opens avenues to explore the influence of chimera states on computation.

This study experimentally demonstrated diverse chimera states in a photonic SNN based on coupled DOPOs, revealing a novel self-organized mechanism for their generation and stability. The spontaneous shift in spiking modes between Class-I and Class-II behavior, driven by synchronization, creates high-frequency synchronized and low-frequency desynchronized regions, contrasting with many previous studies. The high degree of control in this experimental platform opens up new opportunities for research into chimera states and their potential impact on computation. Future work could involve a more systematic exploration of parameter space to construct a chimera phase diagram and investigate the effects of chimera states on computation within this controlled SNN.

While this study offers significant advancements in the experimental observation of chimera states, some limitations should be noted. The precise control of the pump parameter in the experiments was challenging, limiting the ability to perform highly detailed parameter sweeps. The interaction weights were limited to signed 8-bit integers due to experimental constraints. Further research is needed to explore a wider range of parameter values and network architectures to comprehensively understand the dynamics of chimera states in these systems.