### 25677 results for qubit oscillator frequency

Contributors: B.M.R. Schneider, C. Gollub, K.-L. Kompa, R. de Vivie-Riedle

Date: 2007-09-25

Quantum gates are optimized for the IR active high **frequency** modes of MnBr(CO)5-complexes. We investigate whether the selectivity for quantum gates is conserved for energetically close lying **qubits** of different symmetry that are nevertheless simultaneously addressable with the same laser pulse. The **qubits** are encoded in different vibrational normal modes, which are separated only by 50cm−1. Furthermore the influence of additional non-**qubit** modes on the efficiency of the quantum gates optimized for the pure **qubit** system is studied. Potential perturbers are low **frequency** vibrational modes highly populated at room temperature or vibrational modes that seems predestined to interfere during the gate operation. To prevent translational motion, the possibility of spatial localization is explored by incorporation of the carbonyl complex in the unit cell of a MFI zeolite....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)
...Robustness of quantum gates operating on the high **frequency** modes of MnBr(CO)5...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.
...Normal modes included in the quantum dynamical calculation. (a) Coordinates of the qubit modes, (b) coordinates of the non-qubit modes.
...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)
...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.
... Quantum gates are optimized for the IR active high **frequency** modes of MnBr(CO)5-complexes. We investigate whether the selectivity for quantum gates is conserved for energetically close lying **qubits** of different symmetry that are nevertheless simultaneously addressable with the same laser pulse. The **qubits** are encoded in different vibrational normal modes, which are separated only by 50cm−1. Furthermore the influence of additional non-**qubit** modes on the efficiency of the quantum gates optimized for the pure **qubit** system is studied. Potential perturbers are low **frequency** vibrational modes highly populated at room temperature or vibrational modes that seems predestined to interfere during the gate operation. To prevent translational motion, the possibility of spatial localization is explored by incorporation of the carbonyl complex in the unit cell of a MFI zeolite.

Files:

Contributors: Wei Xiao, Jing-Lin Xiao

Date: 2012-10-01

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.
...We study the eigenenergies and the eigenfunctions of the ground and the first excited states of an electron, which is strongly coupled to LO-phonon in a quantum rod with a hydrogen-like impurity at the center by using the variational method of Pekar type. This quantum rod system may be used as a two-level quantum **qubit**. When the electron is in the superposition state of the ground and the first-excited states, the probability density of the electron **oscillates** in the quantum rod. It is found that the probability density and the **oscillation** period are individually increased and decreased due to the presence of the Coulomb interaction between the electron and the hydrogen-like impurity. The **oscillation** period is an increasing function of the ellipsoid aspect ratio and the effective confinement lengths of the quantum rod, whereas it is a decreasing one of the electron–phonon coupling strength....Coulomb bound potential quantum rod **qubit**...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.
...We study the eigenenergies and the eigenfunctions of the ground and the first excited states of an electron, which is strongly coupled to LO-phonon in a quantum rod with a hydrogen-like impurity at the center by using the variational method of Pekar type. This quantum rod system may be used as a two-level quantum **qubit**. When the electron is in the superposition state of the ground and the first-excited states, the probability density of the electron oscillates in the quantum rod. It is found that the probability density and the oscillation period are individually increased and decreased due to the presence of the Coulomb interaction between the electron and the hydrogen-like impurity. The oscillation period is an increasing function of the ellipsoid aspect ratio and the effective confinement lengths of the quantum rod, whereas it is a decreasing one of the electron–phonon coupling strength. ... We study the eigenenergies and the eigenfunctions of the ground and the first excited states of an electron, which is strongly coupled to LO-phonon in a quantum rod with a hydrogen-like impurity at the center by using the variational method of Pekar type. This quantum rod system may be used as a two-level quantum **qubit**. When the electron is in the superposition state of the ground and the first-excited states, the probability density of the electron **oscillates** in the quantum rod. It is found that the probability density and the **oscillation** period are individually increased and decreased due to the presence of the Coulomb interaction between the electron and the hydrogen-like impurity. The **oscillation** period is an increasing function of the ellipsoid aspect ratio and the effective confinement lengths of the quantum rod, whereas it is a decreasing one of the electron–phonon coupling strength.

Files:

Contributors: Ch. Wunderlich, Ch. Balzer

Date: 2003-01-01

Illustration of a linear ion trap including an axial magnetic field gradient. The static field makes individual ions distinguishable in **frequency** space by Zeeman-shifting their internal energy levels (solid horizontal lines represent **qubit** states). In addition, it mediates the coupling between internal and external degrees of freedom when a driving field is applied (dashed horizontal lines stand for vibrational energy levels of the ion string, see text).
...Rabi **oscillations** on the optical E2 transition S1/2-D5/2 in Ba + . A fit of the data (solid line) yields a Rabi **frequency** of 71.4 × 2πkHz and a transversal relaxation time of 100 μs (determined by the coherence time of the ir light used to drive the E2 resonance).
...Illustration of a linear ion trap including an axial magnetic field gradient. The static field makes individual ions distinguishable in frequency space by Zeeman-shifting their internal energy levels (solid horizontal lines represent qubit states). In addition, it mediates the coupling between internal and external degrees of freedom when a driving field is applied (dashed horizontal lines stand for vibrational energy levels of the ion string, see text).
...Illustration of the coupled system ‘qubit ⊗ harmonic oscillator’ in a trap with magnetic field gradient. Internal qubit transitions lead to a displacement dz of the ion from its initial equilibrium position and consequently to the excitation of vibrational motion. In the formal description the usual Lamb–Dicke parameter is replaced by a new effective one (see text).
...Illustration of the coupled system ‘**qubit** ⊗ harmonic **oscillator**’ in a trap with magnetic field gradient. Internal **qubit** transitions lead to a displacement dz of the ion from its initial equilibrium position and consequently to the excitation of vibrational motion. In the formal description the usual Lamb–Dicke parameter is replaced by a new effective one (see text).
...This chapter discusses quantum measurements and new concepts for experiments with trapped ions. Quantum mechanics is a tremendously successful theory playing a central role in natural sciences even beyond physics, and has been verified in countless experiments, some of which were carried out with very high precision. Quantum theory predicts correlations between two or more quantum systems once an entangled state of these systems has been generated. The chapter introduces experiments with 171Yb+ ions demonstrating the precise manipulation of hyperfine states of single ions essentially free of longitudinal and transverse relaxation. A new concept for ion traps is described that allows for experiments requiring individual addressing of ions and conditional dynamics with several ions even with radiation in the radio **frequency** (rf) or microwave (mw) regime. It is shown how an additional magnetic field gradient applied to an electrodynamic trap individually shifts ionic **qubit** resonances making them distinguishable in **frequency** space. Thus, individual addressing for the purpose of single **qubit** operations becomes possible using long-wavelength radiation. At the same time, a coupling term between internal and motional states arises even when rf or mw radiation is applied to drive **qubit** transitions. Thus, conditional quantum dynamics can be carried out in this modified electrodynamic trap and in such a new type of trap all schemes originally devised for optical QIP in ion traps can be applied in the rf or mw regime, too....Schematic drawing of the resonances of qubits j and j + 1 with some accompanying sideband resonances. The angular frequency vN corresponds to** the Nth **axial vibrational mode, and the frequency separation between carrier resonances is denoted by δω.
...(a) Relevant energy levels and transitions in 138Ba + . (b) Schematic drawing of major experimental elements. OPO: Optical parametric oscillator; YAG: Nd:YAG laser; LD: laser diode; DSP: Digital signal processing system allows for real time control of experimental parameters; AOM: Acousto-optic modulators used as optical switches and for tuning of laser light; PM: Photo multiplier tube, serves for detection of resonance fluorescence. All lasers are frequency and intensity stabilized (not shown).
...(a) Relevant energy levels and transitions in 138Ba + . (b) Schematic drawing of major experimental elements. OPO: Optical parametric **oscillator**; YAG: Nd:YAG laser; LD: laser diode; DSP: Digital signal processing system allows for real time control of experimental parameters; AOM: Acousto-optic modulators used as optical switches and for tuning of laser light; PM: Photo multiplier tube, serves for detection of resonance fluorescence. All lasers are **frequency** and intensity stabilized (not shown).
...Schematic drawing of the resonances of **qubits** j and j + 1 with some accompanying sideband resonances. The angular **frequency** vN corresponds to the Nth axial vibrational mode, and the **frequency** separation between carrier resonances is denoted by δω.
...Spin–spin coupling constants Jij/ 2π in units ofHz for 10 171Yb+ ions in a linear trap characterized by the angular frequency of the COM vibrational mode v1 = 100 × 2πkHz using a static field gradient of 25 T/m.
... This chapter discusses quantum measurements and new concepts for experiments with trapped ions. Quantum mechanics is a tremendously successful theory playing a central role in natural sciences even beyond physics, and has been verified in countless experiments, some of which were carried out with very high precision. Quantum theory predicts correlations between two or more quantum systems once an entangled state of these systems has been generated. The chapter introduces experiments with 171Yb+ ions demonstrating the precise manipulation of hyperfine states of single ions essentially free of longitudinal and transverse relaxation. A new concept for ion traps is described that allows for experiments requiring individual addressing of ions and conditional dynamics with several ions even with radiation in the radio **frequency** (rf) or microwave (mw) regime. It is shown how an additional magnetic field gradient applied to an electrodynamic trap individually shifts ionic **qubit** resonances making them distinguishable in **frequency** space. Thus, individual addressing for the purpose of single **qubit** operations becomes possible using long-wavelength radiation. At the same time, a coupling term between internal and motional states arises even when rf or mw radiation is applied to drive **qubit** transitions. Thus, conditional quantum dynamics can be carried out in this modified electrodynamic trap and in such a new type of trap all schemes originally devised for optical QIP in ion traps can be applied in the rf or mw regime, too.

Files:

Contributors: Hongjing Ma, Weiqing Liu, Ye Wu, Meng Zhan, Jinghua Xiao

Date: 2014-08-01

Ragged **oscillation** death...The critical curves of OD domain from analysis in N coupled Landau–Stuart **oscillators** for different N’s: (a) N=2, (b) N=3, and (c)–(e) N=4 for G={1,2,3,4},G={1,2,4,3}, and G={1,3,2,4}, respectively. The ragged OD domain is clear in (d). The numerical results with points within the domains perfectly verify the analytical results.
...The critical curves of OD domain from analysis in N coupled Landau–Stuart oscillators for different N’s: (a) N=2, (b) N=3, and (c)–(e) N=4 for G={1,2,3,4},G={1,2,4,3}, and G={1,3,2,4}, respectively. The ragged OD domain is clear in (d). The numerical results with points within the domains perfectly verify the analytical results.
...Ragged oscillation death in coupled nonidentical **oscillators**...In this paper, the effect of spatial **frequencies** distributions on the oscillation death in a ring of coupled nonidentical **oscillators** is studied. We find that the rearrangement of the spatial **frequencies** may deform the domain of oscillation death and give rise to a ragged oscillation death in some parameter spaces. The usual critical curves with shape V in the parameter space of **frequency**-mismatch vs coupling-strength may become the shape W (or even shape WV). This phenomenon has been not only numerically observed in coupled nonidentical nonlinear systems, but also well supported by our theoretical analysis....The OD regions in the parameter space of ε-δω for a ring of coupled Rossler systems with different frequency distributions: (a) G={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}, (b) G={26,16,25,18,5,14,10,4,6,7,21,12,23,8,1,15,9,29,28,11,2,20,27,30,3,13,17,22,24,19}, and (c) G={19,22,18,13,10,28,7,15,17,8,30,12,26,11,20,9,27,21,25,6,29,1,23,5,3,24,16,14,4,2}. N=30. In all three insets, the values of ωj are plotted for given ω0=0 and δω=N.
...Spatial **frequencies** distributions...The phase synchronization domains (areas enclosed by the red lines) and the OD regions (black areas) in the parameter space of ε-δω for a ring of coupled Rossler systems with different **frequency** distributions: (a) G={1,2,3,4,5,6,7,8}, (b) G={1,4,3,6,2,8,5,7}, and (c) G={1,2,3,6,8,4,7,5}. N=8. The ragged OD sates are clear in (b) and (c) within a certain interval of δω indicated by two vertical dashed lines. In all three insets, the values of ωj are plotted for given ω0=0 and δω=N. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
...The bifurcation diagram and the largest Lyapunov exponent λ of the coupled Rossler **oscillators** versus the coupling strength ε with the same spatial arrangement of natural **frequencies** as in Fig. 1(a)–(c), respectively for δω=0.58. The bifurcation diagram is realized by the soft of XPPAUT [33] where the black dots are fixed points and the red dots are the maximum and minimum values of x1 for the stable periodic solution while the blue dots means the max/min values of x1 for the unstable periodical states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
...The OD regions in the parameter space of ε-δω for a ring of coupled Rossler systems with different **frequency** distributions: (a) G={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}, (b) G={26,16,25,18,5,14,10,4,6,7,21,12,23,8,1,15,9,29,28,11,2,20,27,30,3,13,17,22,24,19}, and (c) G={19,22,18,13,10,28,7,15,17,8,30,12,26,11,20,9,27,21,25,6,29,1,23,5,3,24,16,14,4,2}. N=30. In all three insets, the values of ωj are plotted for given ω0=0 and δω=N.
...Coupled nonidentical **oscillators**...The bifurcation diagram and the largest Lyapunov exponent λ of the coupled Rossler oscillators versus the coupling strength ε with the same spatial arrangement of natural frequencies as in Fig. 1(a)–(c), respectively for δω=0.58. The bifurcation diagram is realized by the soft of XPPAUT [33] where the black dots are fixed points and the red dots are the maximum and minimum values of x1 for the stable periodic solution while the blue dots means the max/min values of x1 for the unstable periodical states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
...In this paper, the effect of spatial **frequencies** distributions on the **oscillation** death in a ring of coupled nonidentical **oscillators** is studied. We find that the rearrangement of the spatial **frequencies** may deform the domain of **oscillation** death and give rise to a ragged **oscillation** death in some parameter spaces. The usual critical curves with shape V in the parameter space of **frequency**-mismatch vs coupling-strength may become the shape W (or even shape WV). This phenomenon has been not only numerically observed in coupled nonidentical nonlinear systems, but also well supported by our theoretical analysis....The phase synchronization domains (areas enclosed by the red lines) and the OD regions (black areas) in the parameter space of ε-δω for a ring of coupled Rossler systems with different frequency distributions: (a) G={1,2,3,4,5,6,7,8}, (b) G={1,4,3,6,2,8,5,7}, and (c) G={1,2,3,6,8,4,7,5}. N=8. The ragged OD sates are clear in (b) and (c) within a certain interval of δω indicated by two vertical dashed lines. In all three insets, the values of ωj are plotted for given ω0=0 and δω=N. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
... In this paper, the effect of spatial **frequencies** distributions on the **oscillation** death in a ring of coupled nonidentical **oscillators** is studied. We find that the rearrangement of the spatial **frequencies** may deform the domain of **oscillation** death and give rise to a ragged **oscillation** death in some parameter spaces. The usual critical curves with shape V in the parameter space of **frequency**-mismatch vs coupling-strength may become the shape W (or even shape WV). This phenomenon has been not only numerically observed in coupled nonidentical nonlinear systems, but also well supported by our theoretical analysis.

Files:

Contributors: Tadashi Watanabe

Date: 2008-01-21

Variation of **frequency** shift due to the amplitude and the rotation rate.
...Conditions for zero **frequency** shift and zero pressure difference. Broken lines indicate linear fitting lines through the origin.
...Free-decay **oscillations** and rotations of a levitated liquid droplet are simulated numerically, and the **frequency** shift of drop-shape **oscillations** is studied. It is shown for an **oscillating**-rotating liquid droplet that the **oscillation** **frequency** decreases as the amplitude of drop-shape **oscillations** increases, while it increases as the rotation rate increases. The pressure difference between the equator and the pole of the droplet is found to correspond to the **frequency** shift. It is also found that the relation between the amplitude and the rotation rate is linear both for zero **frequency** shift and for zero pressure difference....Zero **frequency** shift of an oscillating-rotating liquid droplet...**Oscillation**...**Frequency** shift...Free-decay oscillations and rotations of a levitated liquid droplet are simulated numerically, and the **frequency** shift of drop-shape oscillations is studied. It is shown for an oscillating-rotating liquid droplet that the oscillation **frequency** decreases as the amplitude of drop-shape oscillations increases, while it increases as the rotation rate increases. The pressure difference between the equator and the pole of the droplet is found to correspond to the **frequency** shift. It is also found that the relation between the amplitude and the rotation rate is linear both for zero **frequency** shift and for zero pressure difference. ... Free-decay **oscillations** and rotations of a levitated liquid droplet are simulated numerically, and the **frequency** shift of drop-shape **oscillations** is studied. It is shown for an **oscillating**-rotating liquid droplet that the **oscillation** **frequency** decreases as the amplitude of drop-shape **oscillations** increases, while it increases as the rotation rate increases. The pressure difference between the equator and the pole of the droplet is found to correspond to the **frequency** shift. It is also found that the relation between the amplitude and the rotation rate is linear both for zero **frequency** shift and for zero pressure difference.

Files:

Contributors: E. Il’ichev

Date: 2007-10-01

Dependence of the phase shift α on the two parameters ng and Φe. The **qubit** is irradiated by microwaves with a **frequency** of 8.0GHz. The periodic circular structure is due to the variation of the total interferometer-tank impedance caused by transitions from the lower to the upper energy band. The “crater ridges” (solid-line ellipse) correspond to all combinations of the parameters ng and Φe that give the same energy gap (8.0GHz) between the respective states [14].
...We present here recently obtained results with the theoretical and experimental investigations of charge-flux **qubits**. A charge-flux **qubit** consists of a single-Cooper-pair transistor closed by a superconducting loop. In this arrangement a **qubit** is effectively a tuneable two-level system. The **qubit** inductance was probed by a superconducting high-quality tank circuit. Under resonance irradiation, with a **frequency** of the order of the **qubit** energy level separation, change of the **qubit** inductance was observed. We have demonstrated that this effect is caused by the redistribution of the **qubit** level population. We have extracted from the measured data the energy gap of a **qubit** as a function of the quasicharge in the transistor island as well as the total Josephson phase difference across both junctions. The excitation of the **qubit** by one-, two-, and three-photon processes was detected. Quantitative agreement between theoretical predictions and experimental data was found. Relaxation as well as dephasing rates were reconstructed from the fitting procedure....Tank phase shift α dependence on gate parameter ng for different magnetic flux applied to the **qubit** loop . The data correspond to the flux Φ/Φ0=0.5, 0.53, 0.54, 0.56, 0.57. 0.61, 0.62, 0.65 (from bottom to top). For clarity, the upper curves are shifted.
...Experimentally observed dependence of the tank voltage phase shift α on the phase difference δ. The curves correspond to the fixed **frequency** Ω/2π=7.05GHz with the different power of the excitation (from bottom to top: -80, -60, -57(dB)). For clarity, the upper curves are shifted.
...Superconducting **qubits**...Experimental investigation of an interferometer type charge-flux **qubit**...Integrated design: Al **qubit** fabricated in the middle of the Nb coil (left-hand side), and single-Cooper-pair transistor (right-hand side).
...Left-hand side: tank phase shift α dependence on gate parameter ng without microwave power (lowest curve) and with microwave power at different excitation **frequencies**. The data correspond to the **frequency** of the microwave ΩMW/2π=8.9, 7.5, 6.0GHz (from top to bottom) [14]. Here the applied external magnetic flux was fixed Φdc=Φ0/2. For clarity, the upper curves are shifted. Right-hand side: energy gap Δ between the ground and upper states of the **qubit** determined from the experimental data for the case δ=π (Φdc=Φ0/2) [14]. The dots represent the experimental data, the solid line corresponds to the fit (cf. text).
...Calculated dependence of the tank voltage phase shift α on the phase difference δ. The curves correspond to the fixed **frequency** Ω/2π=7.05GHz with the different amplitude of the excitation (from bottom to top n˜g is: 0.1, 0.2, 0.4) [11]. For clarity, the upper curves are shifted.
... We present here recently obtained results with the theoretical and experimental investigations of charge-flux **qubits**. A charge-flux **qubit** consists of a single-Cooper-pair transistor closed by a superconducting loop. In this arrangement a **qubit** is effectively a tuneable two-level system. The **qubit** inductance was probed by a superconducting high-quality tank circuit. Under resonance irradiation, with a **frequency** of the order of the **qubit** energy level separation, change of the **qubit** inductance was observed. We have demonstrated that this effect is caused by the redistribution of the **qubit** level population. We have extracted from the measured data the energy gap of a **qubit** as a function of the quasicharge in the transistor island as well as the total Josephson phase difference across both junctions. The excitation of the **qubit** by one-, two-, and three-photon processes was detected. Quantitative agreement between theoretical predictions and experimental data was found. Relaxation as well as dephasing rates were reconstructed from the fitting procedure.

Files:

Contributors: Xu-Chu Cai, Jun-Fang Liu

Date: 2011-04-01

Application of the modified **frequency** formulation to a nonlinear **oscillator**...Nonlinear **oscillator**...He’s **frequency** formulation is used to obtain the relationship between the **frequency** and amplitude of a nonlinear **oscillator**. The general approach is to choose two linear **oscillators**; in this paper, however, one linear **oscillator** and the Duffing **oscillator** are chosen as trial equations. The solution procedure is of utter simplicity, while the result is of high accuracy....He’s **frequency** formulation ... He’s **frequency** formulation is used to obtain the relationship between the **frequency** and amplitude of a nonlinear **oscillator**. The general approach is to choose two linear **oscillators**; in this paper, however, one linear **oscillator** and the Duffing **oscillator** are chosen as trial equations. The solution procedure is of utter simplicity, while the result is of high accuracy.

Files:

Contributors: Markku Penttonen, György Buzsáki

Date: 2003-04-01

**Oscillation** bands form an arithmetic progression on the logarithmic scale. For each band the **frequency** (Hz) or period ranges are shown together with their commonly used names.
...Oscillation bands form an arithmetic progression on the logarithmic scale. For each band the frequency (Hz) or period ranges are shown together with their commonly used names.
...Brain **oscillators**...Alpha, gamma and theta **oscillations**...Behaviorally relevant brain **oscillations** relate to each other in a specific manner to allow neuronal networks of different sizes with wide variety of connections to cooperate in a coordinated manner. For example, thalamo-cortical and hippocampal **oscillations** form numerous **frequency** bands, which follow a general rule. Specifically, the center **frequencies** and **frequency** ranges of **oscillation** bands with successively faster **frequencies**, from ultra-slow to ultra-fast **frequency** **oscillations**, form an arithmetic progression on the natural logarithmic scale. Due to mathematical properties of natural logarithm, the cycle lengths (periods) of **oscillations**, as an inverse of **frequency**, also form an arithmetic progression after natural logarithmic transformation. As a general rule, the neuronal excitability is larger during a certain phase of the **oscillation** period. Because the intervals between these activation phases and the temporal window of activation vary in proportion to the length of the **oscillation** period, lower **frequency** **oscillations** allow for an integration of neuronal effects with longer delays and larger variability in delays and larger areas of involvement. Neural representations based on these **oscillations** could therefore be complex. In contrast, high **frequency** **oscillation** bands allow for a more precise and spatially limited representation of information by incorporating synaptic events from closely located regions with short synaptic delays and limited variability. The large family of **oscillation** **frequency** bands with a constant relation may serve to overcome the information processing limitations imposed by the synaptic delays....Natural logarithmic relationship between brain **oscillators**...Behaviorally relevant brain oscillations relate to each other in a specific manner to allow neuronal networks of different sizes with wide variety of connections to cooperate in a coordinated manner. For example, thalamo-cortical and hippocampal oscillations form numerous **frequency** bands, which follow a general rule. Specifically, the center **frequencies** and **frequency** ranges of oscillation bands with successively faster **frequencies**, from ultra-slow to ultra-fast **frequency** oscillations, form an arithmetic progression on the natural logarithmic scale. Due to mathematical properties of natural logarithm, the cycle lengths (periods) of oscillations, as an inverse of **frequency**, also form an arithmetic progression after natural logarithmic transformation. As a general rule, the neuronal excitability is larger during a certain phase of the oscillation period. Because the intervals between these activation phases and the temporal window of activation vary in proportion to the length of the oscillation period, lower **frequency** oscillations allow for an integration of neuronal effects with longer delays and larger variability in delays and larger areas of involvement. Neural representations based on these oscillations could therefore be complex. In contrast, high **frequency** oscillation bands allow for a more precise and spatially limited representation of information by incorporating synaptic events from closely located regions with short synaptic delays and limited variability. The large family of oscillation **frequency** bands with a constant relation may serve to overcome the information processing limitations imposed by the synaptic delays. ... Behaviorally relevant brain **oscillations** relate to each other in a specific manner to allow neuronal networks of different sizes with wide variety of connections to cooperate in a coordinated manner. For example, thalamo-cortical and hippocampal **oscillations** form numerous **frequency** bands, which follow a general rule. Specifically, the center **frequencies** and **frequency** ranges of **oscillation** bands with successively faster **frequencies**, from ultra-slow to ultra-fast **frequency** **oscillations**, form an arithmetic progression on the natural logarithmic scale. Due to mathematical properties of natural logarithm, the cycle lengths (periods) of **oscillations**, as an inverse of **frequency**, also form an arithmetic progression after natural logarithmic transformation. As a general rule, the neuronal excitability is larger during a certain phase of the **oscillation** period. Because the intervals between these activation phases and the temporal window of activation vary in proportion to the length of the **oscillation** period, lower **frequency** **oscillations** allow for an integration of neuronal effects with longer delays and larger variability in delays and larger areas of involvement. Neural representations based on these **oscillations** could therefore be complex. In contrast, high **frequency** **oscillation** bands allow for a more precise and spatially limited representation of information by incorporating synaptic events from closely located regions with short synaptic delays and limited variability. The large family of **oscillation** **frequency** bands with a constant relation may serve to overcome the information processing limitations imposed by the synaptic delays.

Files:

Contributors: Rina Zelmann, Maeike Zijlmans, Julia Jacobs, Claude-E. Châtillon, Jean Gotman

Date: 2009-08-01

High **Frequency** **Oscillations**...Improving the identification of High **Frequency** Oscillations...High **Frequency** **Oscillations** (HFOs), including Ripples (80–250Hz) and Fast Ripples (250–500Hz), can be recorded from intracranial macroelectrodes in patients with intractable epilepsy. We implemented a procedure to establish the duration for which a stable measurement of rate of HFOs is achieved....High **Frequency** Oscillations...High **Frequency** Oscillations (HFOs), including Ripples (80–250Hz) and Fast Ripples (250–500Hz), can be recorded from intracranial macroelectrodes in patients with intractable epilepsy. We implemented a procedure to establish the duration for which a stable measurement of rate of HFOs is achieved. ... High **Frequency** **Oscillations** (HFOs), including Ripples (80–250Hz) and Fast Ripples (250–500Hz), can be recorded from intracranial macroelectrodes in patients with intractable epilepsy. We implemented a procedure to establish the duration for which a stable measurement of rate of HFOs is achieved.

Files:

Contributors: Catherine Oikonomou, Frederick R Cross

Date: 2010-12-01

A phase-locking model for entrainment of peripheral **oscillators** to the cyclin–CDK **oscillator**. (a) Molecular mechanism of the Cdc14 release **oscillator**. The mitotic phosphatase Cdc14 is activated upon release from sequestration in the nucleolus. This release is controlled by a negative feedback loop in which Cdc14 release, promoted by the polo kinase Cdc5, activates APC-Cdh1, which then promotes Cdc5 degradation, allowing Cdc14 resequestration. This negative-feedback **oscillator** is entrained to the cyclin–CDK cycle at multiple points: by cyclin–CDK promotion of CDC5 transcription and Cdc5 kinase activation, and by cyclin–CDK inhibition of Cdh1 activity. (b) Schematic of multiple peripheral **oscillators** coupled to the CDK **oscillator** in budding yeast. As described above, coupling entrains such peripheral **oscillators** to cell cycle progression; peripheral **oscillators** also feed back on the cyclin–CDK **oscillator** itself. For example, major genes in the periodic transcription program include most cyclins, CDC20, and CDC5; Cdc14 directly promotes establishment of the low-cyclin–CDK positive feedback loop by activating Cdh1 and Sic1 as well as more indirectly antagonizing cyclin–CDK activity by dephosphorylating cyclin–CDK targets; the centrosome and budding cycles could communicate with the cyclin–CDK cycle via the spindle integrity and morphogenesis checkpoints. (c) **Oscillator** coupling ensures once-per-cell-cycle occurrence of events. Three hypothetical **oscillators** are shown: a master cycle in black, a faster peripheral cycle in blue, and a slower peripheral cycle in red. In the absence of phase-locking (top), the **oscillators** trigger events (colored circles) without a coherent phase relationship. In the presence of **oscillator** coupling (bottom), the peripheral **oscillators** are slowed or accelerated within their critical periods to produce a locked phase relationship, with events occurring once and only once within each master cycle.
...Positive and negative feedback loops in the cyclin–CDK **oscillator**. (a) Inset: a negative feedback loop which can give rise to oscillations. Such an **oscillator** is thought to form the core of eukaryotic cell cycles, with cyclin–Cyclin Dependent Kinase (cyclin–CDK) acting as activator, Anaphase Promoting Complex-Cdc20 (APC-Cdc20) acting as repressor, and non-linearity in APC-Cdc20 activation preventing the system from settling into a steady state. Below is shown the cyclin–CDK machinery in eukaryotic cell cycles. CDKs, present throughout the cell cycle, require the binding of a cyclin subunit for activity. These cyclin partners can also determine the localization of the complex and its specificity for targets. At the beginning of the cell cycle, cyclin–CDK activity is low, and ramps up over most of the cycle. Early cyclins trigger production of later cyclins and these later cyclins then turn off the earlier cyclins, so that control is passed from one set of cyclin–CDKs to the next. The last set of cyclins to be activated, the G2/M-phase cyclins, initiate mitosis, and also initiate their own destruction by activating the APC-Cdc20 negative feedback loop. APC-Cdc20 targets the G2/M-phase cyclins for destruction, resetting the cell to a low-CDK activity state, ready for the next cycle. (b) Positive feedback is added to the **oscillator** in multiple ways. Left: a highly conserved but non-essential mechanism consists of ‘handoff’ of cyclin proteolysis from APC-Cdc20 to APC-Cdh1. Cdh1 is a relative of Cdc20 which activates the APC late in mitosis and into the ensuing G1. Cdh1 is inhibited by cyclin–CDK activity, resulting in mutual inhibition (which is logically equivalent to positive feedback). Middle: antagonism between cyclin–CDK and stoichiometric CDK inhibitors (CKIs) results in positive feedback. These loops stabilize high- and low-CDK activity states. Right: a double positive feedback loop comprising CDK-mediated inhibition of the Wee1 kinase (which inhibits CDK) and activation of the Cdc25 phosphatase (which activates CDK by removing the phosphorylation added by Wee1) is proposed to stabilize intermediate CDK activity found in mid-cycle, and an alternative stable state of high mitotic CDK activity.
...The cell cycle **oscillator**, based on a core negative feedback loop and modified extensively by positive feedback, cycles with a **frequency** that is regulated by environmental and developmental programs to encompass a wide range of cell cycle times. We discuss how positive feedback allows **frequency** tuning, how size and morphogenetic checkpoints regulate **oscillator** **frequency**, and how extrinsic **oscillators** such as the circadian clock gate cell cycle **frequency**. The master cell cycle regulatory **oscillator** in turn controls the **frequency** of peripheral **oscillators** controlling essential events. A recently proposed phase-locking model accounts for this coupling....Positive and negative feedback loops in the cyclin–CDK **oscillator**. (a) Inset: a negative feedback loop which can give rise to **oscillations**. Such an **oscillator** is thought to form the core of eukaryotic cell cycles, with cyclin–Cyclin Dependent Kinase (cyclin–CDK) acting as activator, Anaphase Promoting Complex-Cdc20 (APC-Cdc20) acting as repressor, and non-linearity in APC-Cdc20 activation preventing the system from settling into a steady state. Below is shown the cyclin–CDK machinery in eukaryotic cell cycles. CDKs, present throughout the cell cycle, require the binding of a cyclin subunit for activity. These cyclin partners can also determine the localization of the complex and its specificity for targets. At the beginning of the cell cycle, cyclin–CDK activity is low, and ramps up over most of the cycle. Early cyclins trigger production of later cyclins and these later cyclins then turn off the earlier cyclins, so that control is passed from one set of cyclin–CDKs to the next. The last set of cyclins to be activated, the G2/M-phase cyclins, initiate mitosis, and also initiate their own destruction by activating the APC-Cdc20 negative feedback loop. APC-Cdc20 targets the G2/M-phase cyclins for destruction, resetting the cell to a low-CDK activity state, ready for the next cycle. (b) Positive feedback is added to the **oscillator** in multiple ways. Left: a highly conserved but non-essential mechanism consists of ‘handoff’ of cyclin proteolysis from APC-Cdc20 to APC-Cdh1. Cdh1 is a relative of Cdc20 which activates the APC late in mitosis and into the ensuing G1. Cdh1 is inhibited by cyclin–CDK activity, resulting in mutual inhibition (which is logically equivalent to positive feedback). Middle: antagonism between cyclin–CDK and stoichiometric CDK inhibitors (CKIs) results in positive feedback. These loops stabilize high- and low-CDK activity states. Right: a double positive feedback loop comprising CDK-mediated inhibition of the Wee1 kinase (which inhibits CDK) and activation of the Cdc25 phosphatase (which activates CDK by removing the phosphorylation added by Wee1) is proposed to stabilize intermediate CDK activity found in mid-cycle, and an alternative stable state of high mitotic CDK activity.
...A phase-locking model for entrainment of peripheral oscillators to the cyclin–CDK **oscillator**. (a) Molecular mechanism of the Cdc14 release **oscillator**. The mitotic phosphatase Cdc14 is activated upon release from sequestration in the nucleolus. This release is controlled by a negative feedback loop in which Cdc14 release, promoted by the polo kinase Cdc5, activates APC-Cdh1, which then promotes Cdc5 degradation, allowing Cdc14 resequestration. This negative-feedback **oscillator** is entrained to the cyclin–CDK cycle at multiple points: by cyclin–CDK promotion of CDC5 transcription and Cdc5 kinase activation, and by cyclin–CDK inhibition of Cdh1 activity. (b) Schematic of multiple peripheral oscillators coupled to the CDK **oscillator** in budding yeast. As described above, coupling entrains such peripheral oscillators to cell cycle progression; peripheral oscillators also feed back on the cyclin–CDK **oscillator** itself. For example, major genes in the periodic transcription program include most cyclins, CDC20, and CDC5; Cdc14 directly promotes establishment of the low-cyclin–CDK positive feedback loop by activating Cdh1 and Sic1 as well as more indirectly antagonizing cyclin–CDK activity by dephosphorylating cyclin–CDK targets; the centrosome and budding cycles could communicate with the cyclin–CDK cycle via the spindle integrity and morphogenesis checkpoints. (c) Oscillator coupling ensures once-per-cell-cycle occurrence of events. Three hypothetical oscillators are shown: a master cycle in black, a faster peripheral cycle in blue, and a slower peripheral cycle in red. In the absence of phase-locking (top), the oscillators trigger events (colored circles) without a coherent phase relationship. In the presence of **oscillator** coupling (bottom), the peripheral oscillators are slowed or accelerated within their critical periods to produce a locked phase relationship, with events occurring once and only once within each master cycle.
...**Frequency** control of cell cycle **oscillators** ... The cell cycle **oscillator**, based on a core negative feedback loop and modified extensively by positive feedback, cycles with a **frequency** that is regulated by environmental and developmental programs to encompass a wide range of cell cycle times. We discuss how positive feedback allows **frequency** tuning, how size and morphogenetic checkpoints regulate **oscillator** **frequency**, and how extrinsic **oscillators** such as the circadian clock gate cell cycle **frequency**. The master cell cycle regulatory **oscillator** in turn controls the **frequency** of peripheral **oscillators** controlling essential events. A recently proposed phase-locking model accounts for this coupling.

Files: