Contributors:Ole Jensen, Marcel A.J. van Gerven, Haiteng Jiang, Ali Bahramisharif
Cross-frequency coupling... Steps applied to compute both CFC and CFD. (A) High frequency power at frequency v is estimated from the original signal by applying a sliding Hanning tapered time window followed by a Fourier transformation (red line). After that, both the original signal and the power envelope of the high frequency signal are divided into segments. Within each segment, the original signal and the power envelope of the high frequency signal are Fourier-transformed and cross-spectra between them are computed. (B) CFC and CFD quantification. CFC is quantified by coherence and CFD is calculated from the PSI between the phase of slow oscillation fi and power of fast oscillation vj. The red segment indicates the frequency range over which the PSI is calculated. The PSI is calculated for the bandwidth β.
... Statistical assessment of the CFC and CFD when controlling for multiple comparisons over frequencies. (A) Observed CFC/D and clustering threshold. All observed CFC/D values were pooled together (i.e. all frequency by frequency bins) and the threshold is set at the 99.5th percentile of the resulting distribution (right panel). Contiguous CFC/D values exceeding the threshold formed a cluster (left panel). The summed CFC/D values from a given cluster were considered the cluster score. (B) Circular shifted CFC/D and the cluster reference distribution. Random number of the Fourier-transferred phase segment sequences was circular shifted with respect to the amplitude envelope segments and the CFC/D values were recomputed 1000 times. For each randomization, the CFC/D contiguous values exceeding the threshold were used to form reference clusters (e.g., cluster1, cluster2, and so on in the left panel) and the respective cluster scores were calculated. The resulting 1000 maximum cluster scores formed the cluster-level reference distribution. For the observed cluster score, the p value was determined by considering the fraction of cluster scores in the reference distribution exceeding the observed cluster score (right panel).
... Neuronal oscillations... Phase spectra between low frequency signal and high frequency envelope. The red curves represent the envelope of high frequency signals. Fig. 1 is adapted from Schoffelen et al. (2005). Left panel: The low frequency signal is leading the high frequency envelope by 10ms. This constant lead translates into a phase-lead that linearly increases with frequency (e.g., 0.25rad for 4Hz, 0.50rad for 8Hz and 0.75rad for 12Hz). Right panel: The low frequency signal is lagging the high frequency envelope by 10ms. This constant lag translates into a phase-lag that linearly decreases with frequency (e.g., −0.25rad for 4Hz, −0.50rad for 8Hz and −0.75rad for 12Hz).
... Cross-frequency directionality
The period of oscillation T0 in a QR as a function of the transverse and longitudinal effective confinement lengths of the QR lp and lv.
... The period of oscillation T0 in a QR as a function of the electron–phonon coupling strength α and the Coulomb bound potential β.
... Qubit... The period of oscillation T0 in a QR as a function of the ellipsoid aspect ratio e′ and the electron–phonon coupling strength α.
Contributors:Jonas Buchli, Ludovic Righetti, Auke Jan Ijspeert
Adaptive frequencyoscillator... The left plot of this figure represents the evolution of ω(t) when the adaptive Hopf oscillator is coupled to the z variable of the Lorenz attractor. The right plot represents the z variable of the Lorenz attractor. We clearly see that the adaptive Hopf oscillators can correctly learn the pseudo-frequency of the Lorenz attractor. See the text for more details.
... Plots of the frequency of the oscillations of the Van der Pol oscillator according to ω. Here α=50. There are two plots, for the dotted line the oscillator is not coupled and for the plain line the oscillator is coupled to F=sin30t. The strength of coupling is ϵ=2. We clearly see basins of phase-locking, the main one for frequency of oscillations 30. The other major basins appear each 30n (dotted horizontal lines). We also notice small entrainment basins for some frequencies of the form 30pq. For a more detailed discussion of these results refer to the text.
... We show the adaptation of the Van der Pol oscillator to the frequencies of various input signals: (a) a simple sinusoidal input (F=sin(40t)), (b) a sinusoidal input with uniformly distributed noise (F=sin(40t)+uniform noise in [−0.5,0.5]), (c) a square input (F=square(40t)) and (d) a sawtooth input (F=sawtooth(40t)). For each experiment, we set ϵ=0.7 and α=100 and we show three plots. The right one shows the evolution of ω(t). The upper left graph is a plot of the oscillations, x, of the system, at the beginning of the learning. The lower graph shows the oscillations at the end of learning. In both graphs, we also plotted the input signal (dashed). In each experiment, ω converges to ω≃49.4, which corresponds to oscillations with a frequency of 40 rad s−1 like the input and thus the oscillator correctly adapts its frequency to the frequency of the input.
... Frequency spectra of the Van der Pol oscillator, both plotted with ω=10. The left figure is an oscillator with α=10 and on the right the nonlinearity is higher, α=50. On the y-axis we plotted the square root of the power intensity, in order to be able to see smaller frequency components.
... This figure shows the convergence of ω for several initial frequencies. The Van der Pol oscillator is perturbed by F=sin(30t), with coupling ϵ=0.7, α=50. We clearly see that the convergence directly depends on the initial conditions and as expected the different kinds of convergence correspond to the several entrainment basins of Fig. 7.
Contributors:C.R. Locke, D. Cros, A.C. Fowler, M.E. Tobar, E.N. Ivanov, J.D. Anstie, J.G. Hartnett
Traveling wave resonator with two ports incorporated into interferometric frequency discriminator for oscillator stabilisation.
... Modified Galani oscillator stabilisation technique utilising travelling wave resonator with standing wave ratio.
... Frequency Standards and Metrology Group, School of Physics, University of Western Australia, 35 Stirling Hwy, Crawley 6009, Australia
Contributors:F.A. van Goor, A.G. Khachatryan, K.-J. Boller
The phase of the oscillator after action of a linearly-chirped pulsed force as a function of the chirp strength. In this case Ω0=4, σ=5.
... The amplitude of a harmonic oscillator after the action of a pulsed force with a Gaussian envelope and a linear chirp in dependence on the chirp strength, ΔΩ. In this case Ω0=5 and σ=5, 10, and 20; A=1 in all figures.
... The amplitude of the oscillator after the action of a force with an asymmetrical Gaussian envelope, σ1=5, Ω0=5, σ2=10 and 20.
... The amplitude of the oscillator vs. ΔΩ in the case of a periodical chirp in the force. The parameters of the force are: Ω0=5, σ=20, b=4.
... Classical oscillator... Frequency chirp
Dominant frequencies of 10mm nozzle.
... Condensation regime map by Cho et al.  (C–chugging, TC—transitional region from chugging to CO, CO—condensation oscillation, SC—stable condensation, BCO—bubble condensation oscillation, IOC—interfacial oscillation condensation).
... Condensation oscillation... Frequencies at different test conditions—250kgm−2s−1.
... Frequency... Prediction accuracy of simultaneous equations for oscillationfrequency.
... Frequencies at different test conditions—300kgm−2s−1.
Frequency modulation... Frequency modulated Ca2+ oscillations. (A) A computer generated (in silico) oscillating wave with the parameters: period (T), frequency (f), full duration half maximum (FDHM), and duty cycle is depicted. (B) Oscillating wave frequency modulated by agonist concentration. (C) Oscillating wave frequency modulated by the different agonists X, Y, and Z. Three single cell Ca2+ recordings of a Fluo-4/AM-loaded neuroblastoma cell (D), HeLa cell (E), and cardiac cell (F) with the parameters T, f, FDHM, and duty cycle stated. Scale bars are 100s.
... Frequency decoders and host cells. Illustration showing the frequencies and periods that modulate the different frequency decoders and host cells.
... Frequency decoding
Contributors:K.-L. Kompa, R. de Vivie-Riedle, C. Gollub, B.M.R. Schneider
PES of the qubit system (a) and total dipole surface (b). For both surfaces: −52.8 pm⩽rA1⩽+52.8pm and −37.4pm⩽rE⩽+37.4pm.
... Normal modes included in the quantum dynamical calculation. (a) Coordinates of the qubit modes, (b) coordinates of the non-qubit modes.
... Spectral analysis of the NOT (top) and CNOT (bottom) gate. The solid lines correspond to the spectra of the optimized pulses, the dashed lines to the spectra of the sub pulses. The vertical lines indicate the relevant qubit basis transition frequencies for the quantum gates.
... spectroscopical data of the qubit vibrational modes E and A1 and the non-qubit modes, the δ-deformation mode (E) and the dissociative mode (A1)
We consider the interaction between a single cavity mode and N≫1 identical qubits, assuming that any system parameter can be rapidly modulated in situ by external bias. It is shown that, for the qubits initially in the ground states, three photons can be coherently annihilated in the dispersive regime for harmonic modulation with frequency 3ω0−Ω0, where ω0 (Ω0) is the bare cavity (qubit) frequency. This phenomenon can be called “Anti-dynamical Casimir effect”, since a pair of excitations is destroyed without dissipation due to the external modulation. For the initial vacuum cavity state, three qubit excitations can also be annihilated for the modulation frequency 3Ω0−ω0.