Dynamical properties of neuromorphic Josephson junctions.

Neuromorphic computing exploits the dynamical analogy between many physical systems and neuron biophysics. Superconductor systems, in particular, are excellent candidates for neuromorphic devices due to their capacity to operate at great speeds and with low energy dissipation compared to their silicon counterparts. In this paper, we revisit a prior work on Josephson Junction-based neurons to identify the exact dynamical mechanisms underlying the system's neuronlike properties and reveal complex behaviors which are relevant for neurocomputation and the design of superconducting neuromorphic devices. Our paper lies at the intersection of superconducting physics and theoretical neuroscience, both viewed under a common framework-that of nonlinear dynamics theory.

Synchronization transitions in a hyperchaotic SQUID trimer.

The phenomena of intermittent and complete synchronization between two out of three identical, magnetically coupled Superconducting QUantum Interference Devices (SQUIDs) are investigated numerically. SQUIDs are highly nonlinear superconducting oscillators/devices that exhibit strong resonant and tunable response to applied magnetic field(s). Single SQUIDs and SQUID arrays are technologically important solid-state devices, and they also serve as a testbed for exploring numerous complex dynamical phenomena. In SQUID oligomers, the dynamic complexity increases considerably with the number of SQUIDs. The SQUID trimer, considered here in a linear geometrical configuration using a realistic model with experimentally accessible control parameters, exhibits chaotic and hyperchaotic behavior in wide parameter regions. Complete chaos synchronization as well as intermittent chaos synchronization between two SQUIDs of the trimer is identified and characterized using the complete Lyapunov spectrum of the system and appropriate measures. The passage from complete to intermittent synchronization seems to be related to chaos-hyperchaos transitions as has been conjectured in the early days of chaos synchronization.

Multi-branched resonances, chaos through quasiperiodicity, and asymmetric states in a superconducting dimer.

A system of two identical superconducting quantum interference devices (SQUIDs) symmetrically coupled through their mutual inductance and driven by a sinusoidal field is investigated numerically with respect to dynamical properties such as its multibranched resonance curve, its bifurcation structure and transition to chaos as well as its synchronization behavior. The SQUID dimer is found to exhibit a hysteretic resonance curve with a bubble connected to it through Neimark-Sacker (torus) bifurcations, along with coexisting chaotic branches in their vicinity. Interestingly, the transition of the SQUID dimer to chaos occurs through a torus-doubling cascade of a two-dimensional torus (quasiperiodicity-to-chaos transition). Periodic, quasiperiodic, and chaotic states are identified through the calculated Lyapunov spectrum and illustrated using Lyapunov charts on the parameter plane of the coupling strength and the frequency of the driving field. The basins of attraction for chaotic and non-chaotic states are determined. Bifurcation diagrams are constructed on the parameter plane of the coupling strength and the frequency of the driving field, and they are superposed to maps of the three largest Lyapunov exponents on the same plane. Furthermore, the route of the system to chaos through torus-doubling bifurcations and the emergence of Hénon-like chaotic attractors are demonstrated in stroboscopic diagrams obtained with varying driving frequency. Moreover, asymmetric states that resemble localized synchronization have been detected using the correlation function between the fluxes threading the loop of the SQUIDs.

Pattern formation and chimera states in 2D SQUID metamaterials.

The Superconducting QUantum Interference Device (SQUID) is a highly nonlinear oscillator with rich dynamical behavior, including chaos. When driven by a time-periodic magnetic flux, the SQUID exhibits extreme multistability at frequencies around the geometric resonance, which is manifested by a "snakelike" form of the resonance curve. Repeating motifs of SQUIDs form metamaterials, i.e., artificially structured media of weakly coupled discrete elements that exhibit extraordinary properties, e.g., negative diamagnetic permeability. We report on the emergent collective dynamics in two-dimensional lattices of coupled SQUID oscillators, which involves a rich menagerie of spatiotemporal dynamics, including Turing-like patterns and chimera states. Using Fourier analysis, we characterize these patterns and identify characteristic spatial and temporal periods. In the low coupling limit, the Turing-like patterns occur near the synchronization-desynchronization transition, which can be related to the bifurcation scenarios of the single SQUID. Chimeras emerge due to the multistability near the geometric resonance, and by varying the dc component of the external force, we can make them appear and reappear and, also, control their location. A detailed analysis of the parameter space reveals the coexistence of Turing-like patterns and chimera states in our model, as well as the ability to transform between these states by varying the system parameters.

Flux bias-controlled chaos and extreme multistability in SQUID oscillators.

The radio frequency (rf) Superconducting QUantum Interference Device (SQUID) is a highly nonlinear oscillator exhibiting the rich dynamical behavior. It has been studied for many years and it has found numerous applications in magnetic field sensors, in biomagnetism, in non-destructive evaluation, and gradiometers, among others. Despite its theoretical and practical importance, there is relatively very little work on its multistability, chaotic properties, and bifurcation structure. In the present work, the dynamical properties of the SQUID in the strongly nonlinear regime are demonstrated using a well-established model whose parameters lie in the experimentally accessible range of values. When driven by a time-periodic (ac) flux either with or without a constant (dc) bias, the SQUID exhibits extreme multistability at frequencies around the (geometric) resonance. This effect is manifested by a "snake-like" form of the resonance curve. In the presence of both ac and dc flux, multiple bifurcation sequences and secondary resonance branches appear at frequencies above and below the geometric resonance. In the latter case, the SQUID exhibits chaotic behavior in large regions of the parameter space; it is also found that the state of the SQUID can be switched from chaotic to periodic or vice versa by a slight variation of the dc flux.

Three-dimensional chimera patterns in networks of spiking neuron oscillators.

We study the stable spatiotemporal patterns that arise in a three-dimensional (3D) network of neuron oscillators, whose dynamics is described by the leaky integrate-and-fire (LIF) model. More specifically, we investigate the form of the chimera states induced by a 3D coupling matrix with nonlocal topology. The observed patterns are in many cases direct generalizations of the corresponding two-dimensional (2D) patterns, e.g., spheres, layers, and cylinder grids. We also find cylindrical and "cross-layered" chimeras that do not have an equivalent in 2D systems. Quantitative measures are calculated, such as the ratio of synchronized and unsynchronized neurons as a function of the coupling range, the mean phase velocities, and the distribution of neurons in mean phase velocities. Based on these measures, the chimeras are categorized in two families. The first family of patterns is observed for weaker coupling and exhibits higher mean phase velocities for the unsynchronized areas of the network. The opposite holds for the second family, where the unsynchronized areas have lower mean phase velocities. The various measures demonstrate discontinuities, indicating criticality as the parameters cross from the first family of patterns to the second.

Turbulent chimeras in large semiconductor laser arrays.

Semiconductor laser arrays have been investigated experimentally and theoretically from the viewpoint of temporal and spatial coherence for the past forty years. In this work, we are focusing on a rather novel complex collective behavior, namely chimera states, where synchronized clusters of emitters coexist with unsynchronized ones. For the first time, we find such states exist in large diode arrays based on quantum well gain media with nearest-neighbor interactions. The crucial parameters are the evanescent coupling strength and the relative optical frequency detuning between the emitters of the array. By employing a recently proposed figure of merit for classifying chimera states, we provide quantitative and qualitative evidence for the observed dynamics. The corresponding chimeras are identified as turbulent according to the irregular temporal behavior of the classification measure.

Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions.

The dynamics of a large array of coupled semiconductor lasers is studied numerically for a nonlocal coupling scheme. Our focus is on chimera states, a self-organized spatiotemporal pattern of coexisting coherence and incoherence. In laser systems, such states have been previously found for global and nearest-neighbor coupling, mainly in small networks. The technological advantage of large arrays has motivated us to study a system of 200 nonlocally coupled lasers with respect to the emerging collective dynamics. Moreover, the nonlocal nature of the coupling allows us to obtain robust chimera states with multiple (in)coherent domains. The crucial parameters are the coupling strength, the coupling phase and the range of the nonlocal interaction. We find that multiclustered chimera states exist in a wide region of the parameter space and we provide quantitative characterization for the obtained spatiotemporal patterns. By proposing two different experimental setups for the realization of the nonlocal coupling scheme, we are confident that our results can be confirmed in the laboratory.

Robust chimera states in SQUID metamaterials with local interactions.

We report on the emergence of robust multiclustered chimera states in a dissipative-driven system of symmetrically and locally coupled identical superconducting quantum interference device (SQUID) oscillators. The "snakelike" resonance curve of the single SQUID is the key to the formation of the chimera states and is responsible for the extreme multistability exhibited by the coupled system that leads to attractor crowding at the geometrical resonance (inductive-capacitive) frequency. Until now, chimera states were mostly believed to exist for nonlocal coupling. Our findings provide theoretical evidence that nearest-neighbor interactions are indeed capable of supporting such states in a wide parameter range. SQUID metamaterials are the subject of intense experimental investigations, and we are highly confident that the complex dynamics demonstrated in this paper can be confirmed in the laboratory.

Noise-induced front motion: signature of a global bifurcation.

We show that front motion can be induced by noise in a spatially extended excitable system with a global constraint. Our model system is a semiconductor superlattice exhibiting complex dynamics of electron accumulation and depletion fronts. The presence of noise induces a global change in the dynamics of the system forcing stationary fronts to move through the entire device. We demonstrate the effect of coherence resonance in our model; i.e., there is an optimal level of noise at which the regularity of front motion is enhanced. Physical insight is provided by relating the space-time dynamics of the fronts with a phase-space analysis.