Sensitive Signal Detection Using a Feed-Forward Oscillator Network

We present the results of an experimental investigation of a network of nonlinear coupled oscillators which are coupled in feed-forward mode. By exploiting the nonlinear response of each oscillator near its intrinsic Hopf bifurcation point, we have found remarkable amplification of small signals over a narrow bandwidth with a large dynamic range. The effect is exploited to extract a small amplitude periodic signal from an input time series which is dominated by noise. Specifically, we have used this relatively simple experimental system to measure responses with a bandwidth of $\sim 1\%$ of the central frequency, amplifications of $\sim 60\mathrm{dB}$, and a dynamic range of $\sim 80\mathrm{dB}$ and can extract signals from a time series with a signal to noise ratio of $\sim -50\mathrm{dB}$.

Figure 1

A schematic of the 3-cell feed-forward network with periodic forcing. Three identical oscillators (or cells) are coupled unidirectionally as indicated by the arrows. A small forcing signal with amplitude $\epsilon$ and frequency ${\omega }_F$ is put into cell 1, triggering oscillation in the system.

Figure 2

Responses of the 3-cell feed-forward network, on variation of the drive frequency $\omega$. The responses of cells 2 and 3 are plotted separately to show the enhancement in amplitude obtained by adding a third cell. A sharp peak can be seen when the forcing frequency ${\omega }_F$ is around ${\omega }_F={\omega }_H$, the natural frequency of the system at the Hopf bifurcation.

Figure 3

The amplitude found in cell 3 relative to the noise floor is shown on a log scale, plotted as a function of the signal-noise ratio. The line has a slope of one, which indicates that the growth in amplitude is approximately linear for small $\epsilon$. The decrease in gradient for large values of $\epsilon$ demonstrates the dynamic compression which arises from the nonlinear response of the system.

Figure 4

Plots of typical power spectra of the forcing signal and output of cell 3, respectively. In (a), a linear scale is used, and a log scale is used in (b), demonstrating that the output signal is almost purely harmonic. The time series were sampled at 1 kHz for ten seconds to enable good averages for the spectra to be obtained. The insets show example portions of the time sequences of the respective signals.

Figure 5

The amplification found in cells 2 and 3 (${\Gamma }_2$ and ${\Gamma }_3$, respectively) in the range of ${\omega }_F$ between 129.5 and 130 Hz are shown, in decibels. The line indicates a gradient of $1/3$ over this range of ${\omega }_F$.