Elsevier

Chaos, Solitons & Fractals

Volume 162, September 2022, 112415
Chaos, Solitons & Fractals

Synchronization phenomena in dual-transistor spiking oscillators realized experimentally towards physical reservoirs

https://doi.org/10.1016/j.chaos.2022.112415Get rights and content

Highlights

  • The spiking dynamics of a ring of transistor-based oscillators are considered.

  • Experiments show multiple pattern-formation phenomena via synchronization.

  • Noise injection, additional links and node patterning confer additional complexity.

  • The system has several features desirable for use in physical reservoir computing.

Abstract

Transistor-based chaotic oscillators are known to realize highly diverse dynamics despite having elementary circuit topologies. This work investigates, numerically and experimentally using a ring network, a recently-introduced dual-transistor circuit that generates neural-like spike trains. A multitude of non-trivial effects are observed as a function of the supply voltage and coupling strength, including pattern formation under incomplete synchronization and sensitivity to additional long-distance links. Globally-applied noise exerts a synchronizing effect that interacts with the other control parameters. When the network is partitioned in halves at different levels of granularity, their interplay gives rise to adversarial route-to-synchronization phenomena. These results highlight the generative ability of this circuit and motivate its consideration towards the future realization of physical reservoirs.

Introduction

Synchronization is a widespread phenomenon that occurs in natural and artificial systems wherein the units (e.g., fireflies, neurons, or nodes of a telecommunication or power grid) interact with each other, that is, exchange energy to coordinate their dynamical behavior [1], [2]. Given its universality and fundamental importance, countless studies have advanced our theoretical understanding of the underlying mechanisms promoting or hindering its emergence, as well as of its possible applications in engineered systems and networks [3], [4], [5], [6]. One aspect that is of particular interest is the formation of spatial patterns, whereby the symmetry of structural connections is broken and a more spatially-diversified arrangement appears. While many studies have provided a theoretical analysis alongside an experimental characterization of synchronization for two coupled units only (see, for instance, Refs. [7], [8], [9], [10], [11], [12], [13], [14], [15]), the number of experimental investigations considering the general setting of more than two interacting units is considerably more limited. Because, in the presence of multiple units, the mismatches and non-idealities that permeate natural and physical entities can have a profound effect on global properties such as the emergence of synchronization patterns, this represents an important shortcoming [16].

Common approaches to the experimental analysis of synchronization make use of mechanical systems such as metronomes or pendulums, [17], [18], [19], lasers [20], [21], and analog electronic circuits [22], [23], [24], [25], [26], [27], [28], [29]. Experimental setups based on electronic circuits are particularly suited for the purpose, as they often offer a higher degree of reconfigurability of the connectivity and tunability of the control parameters compared to their mechanical or optical counterparts. However, also in electronic systems, the cost and practicality of realizing the interconnections among the oscillators are limiting factors that cap the number of coupled units and the links considered in the experiments. To implement the flexible interconnections among the electronic oscillators, which are needed for characterizing the role of the topology of interactions on synchronization, different solutions have been adopted, including plugins for manual insertion of the coupling elements (e.g., resistors, or capacitors) [30], sets of switchable resistors [22], and digital potentiometers [29]. Similarly, also in large-scale integrated arrays of neuromorphic circuits, the realization of reconfigurable synapses represents a bottleneck, and it can also be noted that the preponderant fraction of the human brain is made up by white matter, that is, most volume is invested in realizing the connectivity between neurons [31].

Notably, the use of electronic apparatuses allows querying the influences of parameter mismatches and other non-idealities in the units that, in simulation, may be difficult to model or lead to numerical issues, and are consequently often neglected. The need for an experimental validation is established in the case of global synchronization (i.e., when all the units of the system are synchronized with each other) and becomes even more prominent when non-trivial patterns of synchronization or control techniques for inducing synchronization are investigated under the regime of incomplete synchronization (i.e., when entrainment is detectable but the orbits in phase space remain mostly non-overlapping) [24], [25]. In this regard, recent works have provided experimental evidence of the robustness, in networks of non-identical electronic circuits, of global synchronization [26], [27], pinning control techniques [22], explosive transitions [23], synchronization patterns in multiplex networks [28], [32], and chimera states [29], [33]. Finally, synchronization has been experimentally investigated also in presence of large mismatches between the oscillators [34]. This framework yields a network of heterogeneous oscillators, as the units have either largely different parameters or fundamentally diverse dynamics, and makes it possible to study regimes such as remote synchronization [35], [36].

In the authors' view, experimental investigation in this area should be intended beyond merely attempting to seek confirmation of numerical findings, but as a generative process in itself, particularly in light of the countless discoveries that can be made by serendipity while manipulating a physical system, later leading to the formulation of simplified models, of which remote synchronization can be taken as an example [16], [25], [37]. It can be said that this, after all, has represented a key mechanism of advancement in physics since the times of Galilei [38].

The effect of noise on coupled nonlinear circuits is another aspect that is particularly relevant to investigate experimentally, as noise can either promote or hinder synchronization. When coherent, it can represent an arbitrary external input and provide a globally attractive force. When incoherent, as is normally the case in physical apparatuses, it usually provides a repulsive force that hinders synchronization, potentially enlarging a region of incomplete synchronization, but may also have the non-trivial effect of inducing synchronization at a critical intensity, via effects such as stochastic resonance [39], [40], [41], [42], [43]. In general, the study of noise in nonlinear dynamical systems is not trivial because it poses important practical issues, turning ordinary differential equations into stochastic differential equations and therefore requiring the usage of specific solvers, such as the Euler-Maruyama method, which are less desirable than consolidated iterative solvers such as the family of Runge-Kutta methods [44]. Therefore, the experimental investigation appears even more worthwhile in the presence of noise. In recent works, this issue has been dealt with by considering noise acting on a single unit of the system [45], [46], a setting that finds applications in power grids [47] and in the theoretical analysis of consensus and synchronization [48].

Owing to the relatively limited number of experimental studies venturing beyond oscillator pairs, many configurations remain to be explored. In particular, concerning single-transistor oscillators, which are minimalist entities potentially well-suited for constructing large networks, to the authors' knowledge only one experimental study of a large ring configuration is available [49]. The present paper provides an extensive experimental characterization of synchronization in oscillators having dynamics previously considered only in isolation. Namely, we consider an experimental setup based on coupled atypical transistor-based chaotic oscillators. These are a family of recently discovered nonlinear circuits, wherein the nonlinearity consists of one or two bipolar-junction transistors, the dynamics are controllable by the supply voltage together with a series-connected resistor, and a handful of reactive components complete the circuit, leading to particularly simple configurations. Such circuits are discovered via “numerical serendipity” through large-scale simulations and can replicate diverse known attractor shapes, such as funnel-like, double-scroll, and spiking [24], [50], [51]. In particular, we concentrate on one configuration that can readily generate spiking behaviors, which render it relevant as a possible building block for constructing ensembles having neuron-like dynamics. This configuration represents one of the smallest known types of spike-generating oscillators [50].

Throughout this paper, the focus is on presenting the properties of a small network of these circuits and, in the process, discussing its potential suitability as a substrate for the realization of physical reservoir computing. In general terms, reservoir computing is a powerful computational framework wherein an ensemble of coupled dynamical elements is used as means of mapping inputs onto a suitable high-dimensional space wherein a given classification or prediction can be solved efficiently [52]. The architecture comprises an input layer and a non-linear reservoir where all weights are fixed, cascaded by a trained output layer [53], [54]. This has shown a computational ability similar to recurrent neural networks in forecasting chaotic dynamics with a significant reduction in training time and memory [55], [56]. More specifically, physical reservoir computing is an emerging field wherein one or more advantageous properties of a physical (as opposed to mathematical) system are leveraged to perform the computation: these may include specific dynamics, together with the possibility of low-power operation and very high-density realization, as offered by photonic networks and nanomechanical devices [57]. Physical reservoirs, in fact, can be implemented using various and often elementary apparatuses, allowing them to be realized via small equivalent mechanical systems or electronic circuits [57]. Compelling examples are given by the wave dynamics in a water bucket [58], and the dynamics of bacterial growth [59]. While reservoir computing based on analog circuits is well-established, and the suitability of experimentally-realized chaotic circuits has been demonstrated, to date, the application of networks of transistor-based chaotic oscillators remains unexplored [60].

Below, we realize a complete experimental setup that we deem to have the required versatility enabling future reservoir computing experiments using the elementary oscillator circuit under consideration. We focus on exemplifying the phenomena that can emerge in the network and relating them to the potential suitability for constructing a physical reservoir. In particular, we investigate the role of two primary control parameters, the influences of long-distance links and noise, and the behavior in the presence of heterogeneous dynamics.

Section snippets

Circuit topology

The topology of the oscillator circuit, shown in Fig. 1a for a hypothetical coupled pair, is atypical in that it cannot be reduced to any known canonical arrangement. It consists of two cascaded NPN bipolar transistors, with the collector of the second (Q2) connected to the base of the first (Q1). A string of three inductors, akin to a double-tapped inductor, connects the supply node to the base of the second transistor (via L1), then reaches the emitter of the same (via L2) and finally its

Chaotic network board design

The oscillator used for physical realization had the same component values and circuit topology as in the previous study wherein it was discovered, that is, the transistors were flipped compared to the numerical simulations (Fig. 3a) [50]. To minimize the deviations from an ideal scenario, shielded inductors suitable for high-frequency operation were used, together with correspondingly high-performance transistors. Namely, the inductor values and models were selected as L1=15 μH (type

Temporal irregularity and synchronization

Having illustrated the combined effects of the supply voltage VS and coupling voltage VC, it is opportune to gain deeper insight into the node- and network-level dynamics. Histograms obtained pooling inter-spike times from the entire network are useful for this purpose. First, sweeping VS at three levels of coupling (Fig. 6a), revealed a situation wherein for low values, the dynamics were essentially period-1 (Δt ≈ 0.9 μs, Δt not to be confused with the integration step), hallmarking the

From coupled oscillators to reservoirs

The present work introduces a small network of coupled chaotic oscillators realized experimentally, and discusses its in-principle suitability for realizing physical reservoir computing. It appears interesting to consider a historical perspective briefly. Computation with oscillators originates from the 1950s, when LC-based oscillators were used as phase-dependent logic gates. One early example is the von Neumann oscillatory computer that was further developed into fully-functional Boolean

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partially supported by JSPS KAKENHI Grant Number 19H02191 and JST-SPRING Grant Number JPMJSP2106. L.M. conceived, designed and fabricated with self-funding the hardware when based in Trento, Italy, later conducting measurements and preparing the manuscript while in Tokyo, Japan. The authors are grateful to S. Aldrigo and C. Roncolato of Tecno77 Srl (Vicenza, Italy), respectively, for excellent assistance in board layout design and prototype assembly.

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    L. Minati and J. Bartels contributed in equal measure to this study.

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