Using Noise to Augment Synchronization among Oscillators

Jaykumar Vaidya; Mohammad Khairul Bashar; Nikhil Shukla*

Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA

Corresponding author e-mail: ns6pf@virginia.edu

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Abstract

Noise is expected to play an important role in the dynamics of analog systems such as coupled oscillators which have recently been explored as a hardware platform for application in computing. In this work, we experimentally investigate the effect of noise on the synchronization of relaxation oscillators and their computational properties. Specifically, in contrast to its typically expected adverse effect, we first demonstrate that a common white noise input induces frequency locking among uncoupled oscillators. Experiments show that the minimum noise voltage required to induce frequency locking increases linearly with the amplitude of the oscillator output whereas it decreases with increasing number of oscillators. Further, our work reveals that in a coupled system of oscillators – relevant to solving computational problems such as graph coloring, the injection of white noise helps reduce the minimum required capacitive coupling strength. With the injection of noise, the coupled system demonstrates frequency locking along with the desired phase-based computational properties at 5× lower coupling strength than that required when no external noise is introduced. Consequently, this can reduce the footprint of the coupling element and the corresponding area-intensive coupling architecture. Our work shows that noise can be utilized as an effective knob to optimize the implementation of coupled oscillator-based computing platforms.2

Introduction

Coupled oscillators have experienced renewed interest in computation owing to their rich spatial-temporal properties 1,2. Besides their potential application in realizing associative memory 3,4 and oscillator neural networks (ONNs) 5–8 for tasks such as image processing 9,10, these systems have recently been explored for solving hard combinatorial optimization problems which are still considered intractable to solve using conventional digital computers. Examples of such problems include graph coloring 11 (representative problem considered here), computing the maximum independent set 12 and maximum cut of a graph 13–16 among others. While coupled oscillators can provide an alternate, and potentially more efficient, pathway to solving such problems, one of the important factors in the design and implementation of such analog systems is noise. Normally, the injection of external noise should have adverse effects on the performance of electronic circuits with analog circuits such as oscillators being particularly susceptible. In fact, this was an important consideration in the adoption of digital circuits over analog ones in the 1950s 17.However, in contrast to its typically undesirable effects, noise can play a constructive role in promoting the highly non-linear process of synchronization among oscillators. Prior work has studied the effects of different types of noise on the synchronization of oscillators, both, theoretically (example, [18–27]) and experimentally (example, [28–30]); identical and non-identical oscillators have been shown to exhibit synchronization in response to both white and colored noise. Noise induced synchronization has been explored in various neural networks [31–33]. In fact, the effect of noise on synchronization has even been explored in biological systems such as spike generation in neurons of neocortical slices of rats 34, firing patterns of two uncoupled neurons in paddlefish 35, as3... as well as in other physical systems such as lasers [36,37]. However, the effect of noise injection on the synchronization of oscillators in the context of their computational properties, particularly for solving combinatorial optimization problems (here, graph coloring), remains largely unexplored. Therefore, in this work, we investigate using experiments andsimulations, the role of noise in the coupling dynamics of oscillators and on their resulting computational properties. Specifically, we demonstrate that the injection of noise lowers the minimum coupling strength required to induce frequency locking and the subsequent phase properties required for computation.

Figure 1. (a) Schematic of the Schmitt-trigger oscillator along with the values of the components used in the experimental realization. Time domain output and frequency spectrum of the two oscillators for various white noise inputs: (b) No noise; (c) VNOISE = 100 mV RMS; (d) VNOISE = 270 mV RMS. No frequency locking is observed until VNOISE ≥ 270 mV RMS.

(a) Schematic of the Schmitt-trigger oscillator with labeled elements: VNOISE (white-noise source), CNOISE (injection capacitor), VOUT (oscillator output), RF (positive feedback resistor), CL (load capacitor), R2 and R1 (threshold-setting divider), ground reference, and supply VDD. The Schmitt trigger uses positive feedback via RF and the R1–R2 divider to set hysteresis, with CL determining oscillation dynamics; VNOISE is coupled through CNOISE to study noise-driven synchronization effects.

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Page number: 818

Results

Synchronization of two uncoupled oscillators.We first investigate the effect of noise on the synchronization of two uncoupled oscillators with a common (white) noise input. Fig. 1a shows the schematic of a Schmitt-trigger based relaxation oscillator along with the component values used in the discrete (breadboard-based) experimental realization; the Schmitt-trigger is designed using an OPAMP (LM741) and the oscillators are stabilized using a negative RC feedback. The frequency of oscillations can be tuned using the resistor (R_F) and the capacitor C_L in the negative feedback loop; we intentionally introduce a small change in R_F of the two oscillators to ensure they have slightly different frequencies and are not synchronized trivially. White noise generated using a function generator (Keysight 81160A) is injected at the output of the oscillator through a capacitor (C_NOISE = 1 pF); the value C_NOISE is chosen such that the oscillators, in the absence of noise, do not exhibit frequency locking using only C_NOISE. The oscillator outputs are measured using a digital oscilloscope (Keithley DSO104A).Figure 1a (in-text reference): Schematic of a Schmitt-trigger based relaxation oscillator and component values used in a discrete breadboard realization. The design uses an OPAMP (LM741) with negative RC feedback; oscillation frequency is tunable via R_F and C_L. Two oscillators are intentionally detuned by a small change in R_F. Common white noise from a Keysight 81160A is injected at the oscillator output through C_NOISE = 1 pF (chosen so that, without noise, C_NOISE alone does not induce frequency locking). Outputs are measured with a Keithley DSO104A oscilloscope.Figure 1b-d shows the time domain waveforms and the corresponding frequency spectrum of the oscillators when subjected to different levels of external white noise.Figure 1b–d (in-text reference): Time-domain waveforms and corresponding frequency spectra of the two oscillators for increasing levels of externally injected white noise. Observations include that, without injected noise (Fig. 1b), the oscillators do not frequency lock; at an intermediate condition (Fig. 1c), the oscillators still fail to frequency lock.In the absence of externally injected noise (Fig. 1b), no frequency locking is observed among the oscillators. Furthermore, the oscillators fail to frequency lock (Fig. 1c)5Figure with four subplots labeled (a)–(d). The figure contains the following plotted content and on-plot annotations: (a) Legend indicates two bias conditions V_DD = 3V and V_DD = 4V. Coupling state is indicated with labels C: Coupled and NC: Not coupled. Axes: x-axis is V_NOISE (mV_RMS) with ticks at 0, 80, 160, 240, 320; y-axis is Coupling State with labels C at the top and NC at the bottom. (b) Axes: x-axis is V_OUT (V); y-axis is V_T,NOISE (mV_RMS) with y-ticks at 350, 700, 1050, 1400. (c) On-plot text indicates V_DD = 3V. Axes: x-axis is V_NOISE (mV_RMS) with ticks at 270, 300, 330, 360; y-axis is |Ø| (degrees). (d) On-plot text indicates V_DD = 3V. Axes: x-axis is # of oscillators with ticks at 2, 3, 4; y-axis is V_T,NOISE (mV_RMS) with y-ticks at 270, 240, 210, 180. Caption: Figure 2. (a) Effect of noise voltage on the synchronization of the two oscillators for two different biases (V_DD=3V, 4V). (b) Variation of noise threshold (for frequency locking) with amplitude of the output. (c) Evolution of the relative phase difference (Ø) between the synchronized oscillators as function of the noise voltage. (d) Minimum noise threshold (V_T,NOISE) required for synchronization as a function of the number of oscillators; the threshold decreases with increasing number of oscillators.until the noise voltage reaches the threshold value of 270 mV_RMS (Fig. 1d) even though the frequency mismatch among the oscillators decreases as the intensity of the injected noise is increased. However, when the noise voltage equals or exceeds this threshold input, the oscillators exhibit (noise induced) frequency locking as evident by the fact that the resonant peaks coalesce to the same frequency in the spectrum. Moreover, the frequency locking is also accompanied by a notable reduction of the full width at half maximum (FWHM) which reduces from 20 Hz (oscillator 1) and 32 Hz (oscillator 2) to 10 Hz in the frequency locked system. This indicates the improved immunity of the…synchronized system to internal phase noise, and agrees with the phase noise reduction shown in several other coupled and self-injection locked oscillator systems [38–41].[38–41] Prior work on coupled and self-injection-locked oscillators reporting phase-noise reduction.Figure 2a shows the evolution of the synchronization state (coupled vs. uncoupled) of the two oscillators as a function of the injected white-noise amplitude for two different supply voltages (V_DD = 3 V and 4 V). In both cases, a minimum threshold noise voltage (V_T,NOISE) is required to induce frequency locking, and this critical noise increases with the oscillator output amplitude (increased by raising V_DD).Figure 2a (referenced): Synchronization state (coupled vs. uncoupled) of two oscillators versus injected white-noise amplitude, shown for V_DD = 3 V and V_DD = 4 V. Displays a threshold V_T,NOISE required for frequency locking, which increases with oscillator output amplitude.Figure 2b further quantifies the minimum V_T,NOISE required as a function of the oscillator signal amplitude, showing V_T,NOISE increases linearly with the oscillator amplitude. This indicates that a minimum signal-to-input-noise ratio of approximately 9:1 is needed to induce frequency locking.Figure 2b (referenced): Minimum threshold noise voltage V_T,NOISE versus oscillator signal amplitude, exhibiting a linear dependence; corresponds to a required SNR of approximately 9:1 for locking.Figure 2c examines how the relative phase difference between the two noise-locked oscillators evolves with noise amplitude. Here, phase is defined using the relative time difference between the voltage troughs of the waveforms; each oscillation corresponds to a phase change of 2π radians (≡ 360°), and the relative phase difference is defined as Δφ based on the adjacent troughs’ timing. As the noise increases, Oscillator 2 (orange in Fig. 1b), which has a higher stand-alone resonant frequency, initially leads Oscillator 1 (green) until the pair exhibits nearly anti-phase locking; beyond this point, Oscillator 2 lags Oscillator 1.Figure 2c (referenced): Relative phase difference between the two oscillators versus injected noise amplitude, with phase defined by trough-to-trough timing; demonstrates transition from lead to near anti-phase and then lag as noise increases.Oscillator 2 (orange in Fig. 1b), with higher stand-alone resonant frequency, initially leads Oscillator 1 (green) until they exhibit nearly anti-phase locking; beyond this point, Oscillator 2 lags Oscillator 1.Figure 1b (referenced): Prior depiction of the two oscillators and their color coding (Oscillator 1: green; Oscillator 2: orange) and relative stand-alone resonant frequencies.Synchronization of a larger oscillator system.Further, we experimentally evaluate the synchronization of a larger system of up to four oscillators using white-noise injection (Fig. 2d). The noise injection not only enables frequency locking among the oscillators, but the critical noise voltage (V_T,NOISE) reduces …Figure 2d (referenced): Demonstration of noise-enabled synchronization in a larger network of up to four oscillators, showing frequency locking and a reduced critical noise voltage V_T,NOISE compared to smaller systems.…with increasing number of oscillators. This can be attributed to the reduced internal phase noise, which exhibits an inverse dependence on the number of oscillators in the system [42, 43].Inline citation to references [42, 43] supporting the inverse dependence of internal phase noise on oscillator count.Figure 3. (a) Schematic of the oscillator network. Graph Network with nodes labeled 1, 2, 3, 4 (oscillators); coupling capacitor denoted as Cc; cross-coupled connections among nodes. (b) Frequency spectrum for Cc = 1 pF with no external noise injected. Legend indicates Osc1, Osc2, Osc3, Osc4. Axes: Magnitude (dB) versus Frequency (kHz). Note: No Frequency Synchronization. (c) Frequency spectrum for Cc = 5 pF with no external noise. Axes: Magnitude (dB) versus Frequency (kHz). Note: Frequency Synchronized. (d) Frequency spectrum for Cc = 1 pF with external noise injected. Axes: Magnitude (dB) versus Frequency (kHz). Note: Frequency Synchronized. (e) Time-domain waveforms corresponding to (c). Axes: Vout (V) versus Time (ms). A polar phase plot at right shows angular markers at 90, 180, and −90 degrees with node labels 1–4. Text annotation: No. of Colors detected: 3. (f) Time-domain waveforms corresponding to (d). Axes: Vout (V) versus Time (ms). A polar phase plot at right shows angular markers at 90, 180, and −90 degrees with node labels 1–4. Text annotation: No. of Colors detected: 3. Caption: Schematic and measurement/simulation results for a four-oscillator network. (b) Cc = 1 pF; no frequency locking without injection of noise. (c) Cc = 5 pF; oscillators exhibit frequency locking due to larger coupling capacitance. (d) Cc = 1 pF; noise-induced frequency locking is observed. (e) and (f) show time-domain waveforms corresponding to (c) and (d), respectively, along with phase plots that demonstrate correct phase ordering leading to the optimal graph coloring solution.spectrum of the oscillators for the illustrative graph in Fig. 3a for Cc = 1pF and 5pF, respectively. In the absence of external noise, a minimum Cc = 5pF is required to induce synchronization as shown in Fig. 3c. The corresponding time domain waveforms of the frequency locked oscillators and the phase plots, shown in Fig. 3e, demonstrate a cyclic phase ordering (1, 2, 4, 3, 1,...) where independent nodes (i.e. without an edge; 2,4 here) appear consecutively. Using simple post processing, the nodes can be separated into different clusters (= 3, in this problem) of independent sets ({1}, {2,4}, {3}) each of which can be assigned an independent color. Thus, the solution to the graph in Fig. 3a is equal to 3 (colors).However, the oscillators fail to exhibit frequency locking when Cc < 5pF. The minimum coupling strength requirement puts a constraint on the minimum size and area of theFigure 4. Experimentally observed noise induced synchronization, resulting phase plot and observed graph coloring solutions for various coupled oscillator networks (Cc = 1pF). - Column headers (left to right): Graph | Output — Noise induced synchronization (Cc = 1pF) | Phase plot | Phase Ordering | Graph Coloring solution | Optimal solution - Legend (within figure): — Edge Present · · · Edge Absent - Rows (textual entries visible in figure): 1) Phase Ordering: 1 → 3 → 2; Graph Coloring solution: 3; Optimal solution: 3 2) Phase Ordering: 1 → 3 → 2 → ⋯ → 4; Graph Coloring solution: 3; Optimal solution: 3 3) Phase Ordering: 1 → 4 → 2 → 3; Graph Coloring solution: 4; Optimal solution: 4 4) Phase Ordering: 3 → ~1 → 2 → ⋯ → 4; Graph Coloring solution: 2; Optimal solution: 2However, when white noise of appropriate amplitude (180 mV_RMS) is injected (Fig. 3d), it can be observed that the coupled oscillators not only exhibit frequency locking but also show the optimal phase ordering ({1}, {3}, {2,4}), giving rise to the optimal coloring solution at a significantly lower C c (=1 pF) (Fig. 3d, f). It is to be noted that even though the relative ordering observed in Fig. 3f is different from Fig. 3e, the ordering is still optimal.Figure 3d, Figure 3e, and Figure 3f are referenced in connection with noise-induced frequency locking and optimal phase ordering; Fig. 3f shows a different yet still optimal ordering compared to Fig. 3e.Figure 5. Simulations showing coloring of K=4 nearest-neighbor graphs of various sizes (N) using coupled oscillators, with and without the injection of noise (180 mV_RMS). Noise injection reduces the required coupling capacitance to observe frequency locking and the associated computation-relevant phase properties. C c = 50 pF (without noise) and C c = 0.5 pF (with noise) were chosen since stable frequency locking occurred at these values for all analyzed graphs. With C c = 0.5 pF, the system does not exhibit synchronization before the injection of noise. Panel annotations: K=4 Nearest Neighbor Graphs; diagram with nodes labeled 1, 2, 3, 4; annotations “N” and “N-1”.

Noise induced synchronization in coupled oscillators.

While the uncoupled oscillator system (synchronized by noise) illustrates how noise promotes synchronization, its effect on the dynamics of a coupled system of oscillators is particularly relevant to computational applications. Consequently, we explore the effect of noise on the frequency locking dynamics of coupled oscillators and their resulting computational properties relevant to solving the graph coloring problem.The graph coloring problem entails computing the minimum number of colors (labels) required to be assigned to the nodes of a graph such that no two nodes having a common edge are assigned the same color. The problem is NP-hard and is still considered intractable to solve using digital computers, motivating the exploration of alternate approaches. This problem can be elegantly mapped to the oscillator hardware by creating a topologically equivalent coupled oscillator network (graph node ≡ oscillator, and edge ≡ coupling capacitor Cc). Subsequently, when the coupling strength is strong enough to frequency lock the oscillators to a common frequency, the resulting phase dynamics exhibit a unique phase ordering such that clusters of nodes without an edge (independent set) and can be assigned the same color, appear consecutively in the circular ordering. The nodes of the same color can then be separated using a simple polynomial time sorting algorithm. Details of this approach have been discussed and demonstrated in our prior work^11. It is important to emphasize that a critical coupling strength indicated by the magnitude of the coupling capacitance is required to induce frequency locking among the oscillators and observe the desired phase dynamics.^11 (prior work demonstrating oscillator-based graph coloring and methodology)To understand the effect of external noise injection, we first evaluate the minimum coupling strength required to induce synchronization. Figures 3b, c shows the frequencyFigures 3b, c (referenced): shows the frequency (context truncated).9

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…coupling element. Since the number of elements in the coupling network of a reconfigurable coupled-oscillator–based computational platform exhibits a square-law dependence (= P(N,2)) on the number of oscillators (N), a large footprint for an individual element can dramatically limit the platform’s scalability and reconfigurability.

This further supports the experimental observation that the injection of external noise into the coupled system lowers the minimum coupling capacitance threshold while facilitating the phase dynamics relevant to computation.In summary, we elucidate the critical role of injected noise in the synchronization dynamics of uncoupled and coupled (but not frequency locked) oscillators. Moreover, our work demonstrates empirically that noise reduces the minimum coupling strength required to realize the computational properties of the oscillators for solving combinatorial optimization problems such as graph coloring, thus, enabling an additional ‘knob’ to optimize the implementation and scalability of the area-hungry coupling network in an oscillator platform. Finally, these results also motivate the exploration of the role of noise on the computational performance of other non-Boolean dynamical systems such as spiking neural networks.

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Acknowledgements

This research was supported in part by the National Science Foundation (Grant No. 1914730).

Author contributions

J. V. performed the experiments, simulations, and analyzed the data. M. K. B. helped with the performance of the experiments. N. S. supervised the study. J. V. and N. S. wrote the manuscript. All authors discussed the results and commented on the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Correspondence and requests for materials should be addressed to N. S.Reprints and permissions information is available at www.nature.com/reprints.