Experimental synchronization of chaos in a large ring of mutually coupled single-transistor oscillators: Phase, amplitude, and clustering effects

Abstract

In this paper, experimental evidence of multiple synchronization phenomena in a large (n = 30) ring of chaotic oscillators is presented. Each node consists of an elementary circuit, generating spikes of irregular amplitude and comprising one bipolar junction transistor, one capacitor, two inductors, and one biasing resistor. The nodes are mutually coupled to their neighbours via additional variable resistors. As coupling resistance is decreased, phase synchronization followed by complete synchronization is observed, and onset of synchronization is associated with partial synchronization, i.e., emergence of communities (clusters). While component tolerances affect community structure, the general synchronization properties are maintained across three prototypes and in numerical simulations. The clusters are destroyed by adding long distance connections with distant notes, but are otherwise relatively stable with respect to structural connectivity changes. The study provides evidence that several fundamental synchronization phenomena can be reliably observed in a network of elementary single-transistor oscillators, demonstrating their generative potential and opening way to potential applications of this undemanding setup in experimental modelling of the relationship between network structure, synchronization, and dynamical properties.

Synchronization, non-linear dynamics and low-frequency fluctuations: Analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

“…It generates chaotic signals having approximately stable periodicity but highly variable cycle amplitude and is easily synchronized via resistive coupling at the collector node. 32 Due to the presence of multiple LC networks instanced not only by the discrete components but also by stray capacitances, junction capacitances, and other effects, diverse resonance frequencies are available; while R1 is varied different oscillation modes are visited, and chaos can ensue through quasiperiodicity. 26,27,31 By tuning R1, it is possible to obtain oscillation that is chaotic irrespective of coupling with other oscillators or periodic but close to phase transition to chaos.…”

“…At the same time, because in the absence of long-distance connections neighbouring oscillators tend to form extended synchronized communities, direct coupling with a distant node generating a highly uncorrelated signal is likely to have greater impact on dynamics than coupling with a closer node with respect to which activity is already partially synchronized. 32 D. Relevance of experimental physical models and connectivity-Non-linear dynamics correspondence in other networks Experimental physical modelling in analog electronic circuits has potential to play a complementary role with respect to numerical simulations of brain dynamics. Through the presence of complex non-idealities, device tolerances and dynamical noise experimental systems such as transistor oscillators readily capture the reality that networks of identical oscillators are never encountered in living organisms, while being inherently free from temporal and magnitude discretization.…”

“…32 Here, to allow cascading the three boards, the initial chain consisting of an OPA633KP video buffer (Texas Instruments, Inc., Dallas, TX, USA) and DG506B multiplexer (Vishay Electronic GmbH, Selb, Germany) was extended with LT1227 addressable buffers (Linear Technology, Inc., Milipitas, CA, USA); with respect to Ref.…”

“…32, waveforms were smoothed through a running average window over 6 ns (3 samples) and local maxima were extracted over a 60 ns (30 samples) window. Corresponding values were interpolated via cubic spline and synchronization was quantified by means of normalized mutual information, defined according to…”

“…31 An experimental investigation of a ring of 30 diffusively coupled such oscillators, each consisting of a bipolar junction transistor, three reactive components and a resistor, has furthermore demonstrated the spontaneous formation of multi-scale community structure as a function of coupling strength, with elements of similarity to the modular organization observed in brain networks. 32 While that study was conducted for an elementary structural connectivity scenario (a ring), it is possible to construct arbitrarily complex networks. Coupled singletransistor oscillators therefore represent an undemanding experimental platform with which one may attempt to recapture associations between brain connectivity and non-linear dynamics observed through BOLD time-series either empirically or in numerical simulation.…”