Fig. 3: A weakly nonlinear model with adaptation.

a–c Single-cell response. a A noisy two-component model with negative feedback. b Frequency-resolved phase shift \({\phi }_{a}=-\arg ({\tilde{R}}_{a})\). A sign change takes place at \(\omega ={\omega }^{* }\simeq {({\tau }_{a}{\tau }_{y})}^{-1/2}\), with \(a\) leading \(s\) on the low frequency side. c Real (\({\tilde{R}}{}_{a}^{{\prime} }\)) and imaginary (\({\tilde{R}}{}_{a}^{{\prime\prime} }\)) components of the response spectrum. \({\tilde{R}}{}_{a}^{{\prime} }\) is of order \(\epsilon\) in the zero frequency limit, while \({\tilde{R}}{}_{a}^{{\prime\prime} }\) changes sign at \(\omega ={\omega }^{* }\). Also shown is the correlation spectrum \({\tilde{C}}_{a}(\omega )\) multiplied by \(\omega /(2T)\), where \(T\) is the noise strength. The fluctuation-dissipation theorem \({\tilde{R}}{}_{a}^{{\prime\prime} }=\omega {\tilde{C}}_{a}(\omega )/(2T)\) for thermal equilibrium systems is satisfied on the high frequency side, but violated at low frequencies. d–f Simulations of coupled adaptive circuits. d Time traces of the signal (red) and of the activity (blue) and memory (cyan) from one of the participating cells at various values of the coupling strength \(\overline{N}={\alpha }_{1}{\alpha }_{2}N\). e The oscillation amplitude \(A\) (of activity \(a\)) and frequency \(\omega\) against \(\overline{N}\). The amplitude \(A\) grows as \({(\overline{N}-{\overline{N}}_{{\rm{o}}})}^{1/2}\) here, a signature of Hopf bifurcation. f Determination of oscillation frequency from the renormalised phase matching condition at finite oscillation amplitudes: \({\phi }_{a}^{+}(\omega ,A)=-{\phi }_{s}^{+}(\omega ,A)\). The linear model for \(s\) yields \({\phi }_{s}^{+}(\omega ,A)=-{\phi }_{s}(\omega )\). Parameters: \({\tau }_{a}={\tau }_{y}=\gamma =K={c}_{3}=1\), \({\alpha }_{1}={\alpha }_{2}=0.5\), and \(\epsilon =0.1\). The strength of noise terms is set at \(T=0.01\).