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Circadian clocks have long been known to be essential for the maintenance of physiological and behavioral processes in a variety of organisms ranging from plants to humans. Dysfunctions that subvert gene expression of oscillatory circadian-clock components may result in severe pathologies, including tumors and metabolic disorders. While the underlying molecular mechanisms and dynamics of complex gene behavior are not fully understood, synthetic approaches have provided substantial insight into the operation of complex control circuits, including that of oscillatory networks. Using iterative cycles of mathematical model-guided design and experimental analyses, we have developed a novel low-**frequency** mammalian **oscillator**. It incorporates intronically encoded siRNA-based silencing of the tetracycline-dependent transactivator to enable the autonomous and robust expression of a fluorescent transgene with periods of 26 h, a circadian clock-like oscillatory behavior. Using fluorescence-based time-lapse microscopy of engineered CHO-K1 cells, we profiled expression dynamics of a destabilized yellow fluorescent protein variant in single cells and real time. The novel **oscillator** design may enable further insights into the system dynamics of natural periodic processes as well as into siRNA-mediated transcription silencing. It may foster advances in design, analysis and application of complex synthetic systems in future gene therapy initiatives.,Nucleic acids research, 38 (8),

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Quantum states can contain correlations which are stronger than is possible in classical systems. Quantum information technologies use these correlations, which are known as entanglement, as a resource for implementing novel protocols in a diverse range of fields such as cryptography, teleportation and computing. However, current methods for generating the required entangled states are not necessarily robust against perturbations in the proposed systems. In this thesis, techniques will be developed for robustly generating the entangled states needed for these exciting new technologies.
The thesis starts by presenting some basic concepts in quantum information proccessing. In Ch. 2, the numerical methods which will be used to generate solutions for the dynamic systems in this thesis are presented. It is argued that using a GPU-accelerated staggered leapfrog technique provides a very efficient method for propagating the wave function.
In Ch. 3, a new method for generating maximally entangled two-**qubit** states using a pair of interacting particles in a one-dimensional harmonic **oscillator** is proposed. The robustness of this technique is demonstrated both analytically and numerically for a variety of interaction potentials. When the two **qubits** are initially in the same state, no entanglement is generated as there is no direct **qubit**-**qubit**
interaction. Therefore, for an arbitrary initial state, this process implements a root-of-swap entangling quantum gate. Some possible physical implementations of this proposal for low-dimensional semiconductor
systems are suggested.
One of the most commonly used **qubits** is the spin of an electron. However, in semiconductors, the spin-orbit interaction can couple this **qubit** to the electron's momentum. In order to incorporate this e ffect
into our numerical simulations, a new discretisation of this interaction is presented in Ch. 4 which is signi ficantly more accurate than traditional methods. This technique is shown to be similar to the standard discretisation for magnetic fields.
In Ch. 5, a simple spin-precession model is presented to predict the eff ect of the spin-orbit interaction on the entangling scheme of Ch. 3. It is shown that the root-of-swap quantum gate can be restored by introducing an additional constraint on the system. The robustness of the gate to perturbations in this constraint is demonstrated by presenting numerical solutions using the methods of Ch. 4.

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Brain regions showing significant group and **frequency** (slow-4 and slow-5) interaction effects on ALFF.
... The group and **frequency** (slow-4 and slow-5) interaction effects on ALFF. The regions showing significant group and **frequency** interaction effects on ALFF (hot colors): the left ventromedial prefrontal cortex (a), the left inferior frontal gyrus/precentral gyrus (b), and the bilateral posterior cingulate cortex/precuneus (c). The bar maps show the mean ALFF values for these regions.
... Low-**frequency** **oscillation**... **Frequency** dependence... Main effects of group and **frequency** on ALFF. (a) The group main effects on ALFF. Hot colors represent increased ALFF in the MDD group compared with HC, while the blue colors represent the opposite. (b) **Frequency** main effects on ALFF. Hot colors represent increased ALFF in the slow-5 as compared to slow-4 band, while the blue colors represent the opposite. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Odor Concentration Determines **Oscillation** Coherence, Not **Frequency**
(A) EAG traces revealed total ORN output increased with odor concentration. Example from one antenna; horizontal bar: 4 s.
(B) Summary. EAG amplitude (first 1 s, see bracket in A) evoked by a range of odor concentrations. Mean ± SE; n = 8; two-way ANOVA: fodor_concentration = 16.84, p **oscillations**. Initial portions of the odor response are shown. Scale bar: 50 ms.
(D) The **frequency** of fast **oscillation** changed not at all or only slightly across a broad range of odor concentrations. All results are shown (dots); bar graph shows means, n = 9. Leftmost bars: basal oscillatory power in absence of odorant. Hexanol: two-way ANOVA: fhexanol_concentration = 6.16, p 0.25, ns.
... Odors Evoked LFP **Oscillations** in the Moth MB and AL
(A) Recording site for LFP: center of the calyx in the MB. MB, mushroom body; mnsc, medial neurosecretory cells; OL, optic lobe; AL, antennal lobe.
(B) LFP **oscillations** (black traces) with simultaneously recorded electroantennogram (EAG, green traces) evoked by different pulse durations of 1% benzyl alcohol, a plant volatile. Black horizontal bars: odor pulses. Color bars: time windows (500 ms) used to calculate the power spectra in (D).
(C) Brief odor pulses evoked fast **oscillations**; lengthy pulses evoked first fast, then slow **oscillations**. Normalized, average spectrograms from 18 trials obtained from six animals with three trials each (see Experimental Procedures). Black horizontal bars above each spectrogram: odor pulses.
(D) Power spectra of oscillatory LFP responses averaged from 22 moths and eight odors, total of 820 trials. Color brackets: 14 Hz-wide bands used to calculate the total oscillatory powers of fast (red, 30–44 Hz) and slow (blue, 10–24 Hz) **oscillations** in (E).
(E) Total oscillatory power of fast and slow LFP shifted significantly over lengthy odor pulses. Twenty trials tested for each odor were averaged before pooling, mean ± SE, n = 41; two-way ANOVA: fwindow(2) = 26.62, p **oscillations**); fwindow(2) = 9.09, p **oscillations**). Asterisks: significant differences (Tukey-Kramer multiple comparisons).
(F) LFP **oscillations** in the AL and MB were highly coherent. (Left) Example of odor-evoked LFP **oscillations** recorded simultaneously in the AL and MB; odorant: 1% cyclohexanone (4 s). Areas a and b are expanded in insets. Horizontal red (0.25–1 s) and blue (1–4 s) bars: times used for coherence analysis at right. (Right) Magnitude squared coherence between the AL and MB. Thin black line: coherence of the response shown. Thick black and dotted lines: average coherence and its one standard deviation range (five AL-MB combinations in four preparations, 20 trials each of two odorants), respectively.
... PN and LN Responses Were Strongly Phase Locked to the LFP
(A) Example simultaneous intracellular recordings from PN and LN, with LFP recorded in the MB. First 2 s after the odor onset shown; brackets: portions expanded beneath. Odorant: 1% benzyl alcohol.
(B) Subthreshold **oscillations**: five-trial average sliding window cross-correlograms show reliable LFP and subthreshold membrane potential **oscillations** for the PN (top) and LN (bottom) in (A). Spikes were clipped. Vertical bars: odor pulses.
(C) Spike-LFP phase relationships: Polar histograms show phase position, relative to LFP, of spikes recorded in PNs (n = 14) and LNs (n = 30) for fast and slow **oscillations**. Concentric circles: firing probability. Black arrows: mean direction.
(D) All recorded neurons were filled with dye and later morphologically identified. Example of PN and LN morphology. An Alexa Fluor-633 (red) filled PN and an Alexa Fluor-568 (yellow) filled LN are shown. Scale bar: 50 μm. AN: antennal nerve.
... Spiking in KCs Is Sparse, Odor Specific, and Tightly Phase Locked to the LFP
(A) KCs showed odor-elicited subthreshold membrane potential fluctuations that were tightly correlated with LFP **oscillations**. Example: top, gray: LFP; bottom, black: simultaneous intracellular record of a KC. Bottom: details of fast and slow periods during oscillatory response. Odor: 4 s, 1% benzyl alcohol. Gray broken line: resting potential.
(B) Cross-correlations between LFP **oscillations** and KC subthreshold activity. Cross-correlation was calculated for times bracketed in (A). Black lines: correlation for the trial shown in (A); gray lines: 21 other trials from this cell. All eight KCs showing subthreshold **oscillations** revealed similarly shaped correlation functions, three with coefficients >0.3.
(C) Polar histograms show strong phase locking between spikes in KCs and the LFP **oscillations**. Histograms show spikes recorded from 20 KCs during fast and slow **oscillations**. Arrows: mean phase position.
(D) Example of KC morphology; posterior view of MB; KC filled with Alexa Fluor-633. Scale bar: 50 μm. Arrow: soma; CaM: medial calyx; CaL: lateral calyx.
... Odor-Evoked **Oscillations** in Model of Moth AL
(A) Full-scale, map-based model included randomly connected populations of 820 PNs and 360 LNs. Odor pulse input was simulated by external currents delivered to a subset of neurons.
(B) Amplitude of the input was set to resemble the EAG (bottom). LFP (top) and neuronal (middle) responses resembled those recorded in vivo. The input to the model was tuned to match results of our physiological recordings and corresponded to points “1” and “2” in the parameter space shown in (E).
(C) Raster plots show spikes in all PNs (top) and all LNs (bottom) evoked by one odor pulse (applied from 500 to 2500 ms).
(D) Interspike interval (ISI) distributions during fast and slow phases of LFP **oscillations**. Many PNs fired two spikes in a single oscillatory cycle (ISI **frequency** was typically limited to the LFP **frequency**.
(E) **Frequency** of LFP, PN, and LN **oscillations** as a function of input from ORNs to PNs and LNs. Sweeping the points between “1” and “2” in parameter space mimicked the changes in the ISI distribution (compare D and Figure S7) and the abrupt change in oscillatory **frequency** (compare B and the Figure 1C) we observed in vivo.

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In the analysis of the activation period wPLFs, 36 reliable PAC patterns were identified: two subjects had four reliable PAC patterns, eight had two, 12 had one, and four had none. In the analysis of the baseline period wPLFs, 17 reliable PAC patterns were identified: four subjects had two reliable PAC patterns, nine had one, and 13 had none. Reliability was defined in terms of the split-half correlation between two independent estimates of the spatial and the **frequency** spectra produced by the tensor decomposition. All identified PAC patterns had reliabilities much larger than what can be expected under the hypothesis of a random PAC (see Materials and methods). A representative selection of reliable PAC patterns is shown in Supplemental Fig. 3.... Our conclusion about the number of sources involved in PAC depends on our definition of a source. We defined a source in terms of two patterns: (1) a spatial map that specifies how strongly source activity affects the measurements at the sensor level, and (2) its **frequency** spectrum. We showed that, with this source definition, the array of PAC-measures (wPLFs) can be written as a tensor product of two spatial maps (one complex- and one real-valued) and two **frequency** spectra (both real-valued), which is exactly the structure that is extracted by our tensor decomposition (see Supplemental Methods). However, we cannot exclude PAC-generating source configurations that cannot be characterized in this way. In fact, we have argued that PAC-patterns may also be generated by a source configuration that would be considered a single source when viewed from the perspective of the mechanism that generates the physiological signal. This confronts us with the problem that sources can be defined both in terms of their formal characteristics (i.e., in terms of a spatial map and a **frequency** spectrum) and in terms of the neuronal network that generates the physiological signal. The difference between the two definitions is most clear if the physiological mechanism consist of multiple components, such as networks of inhibitory neurons that are connected to one or multiple classes of principal neurons, each with its own network topology. This whole multi-component network may be considered as a single source, but also as multiple sources, each one corresponding to one component. Importantly, if these components differ with respect to their spatial maps and **frequency** spectra, then they can be extracted by means of tensor decomposition. This shows there may be a need for a linking of the set of extracted source configurations on the basis of the neuronal interactions that may have produced them.... Package of figures showing the spatial maps and the **frequency** spectra of a representative set of reliable cross-**frequency** patterns extracted from the activation period wPLFs. The figures belonging to one cross-**frequency** patterns are each in one folder, of which the name refers to the patient. The files that have freqspectra_and_compass as a part of their name contain a figure of which the left panel show the **frequency** spectra of a coupling **oscillation** and the associated phase-coupled bursts, and the right panel shows the preferred phases of the PAC in a compass plots. The files that have _IA_ (instantaneous amplitude) as a part of their name contain figures of the magnitudes of the spatial maps for the phase-coupled bursts. Different figures show different subsets of the electrodes, each from on optimal viewpoint. Electrode subsets on the medial side of the brain are shown from two viewpoints, lateral and occipital. The files that have _AWIP_ (amplitude-weighted instantaneous phase) as a part of their name contain figures of the spatial maps of the coupling **oscillation**. Again, different figures show different subsets of the electrodes, each from on optimal viewpoint.
... Spatial maps of the phase-coupled bursts are smaller than the spatial maps of the associated coupling **oscillation**. Both panels show scatter plots of the extent of the spatial maps of the phase-coupled bursts (horizontal axis) against the extent of the corresponding spatial maps of the coupling **oscillation** (vertical axis). Panels a and b show the scatter plots for, respectively, the baseline and the activation period. Panels a and b show the scatter plots for, respectively, the baseline and the activation period. In both periods, for most of the PAC patterns, the extent of the spatial map of the phase-coupled bursts is substantially smaller than the extent of the spatial map of the associated coupling **oscillation**.
... Illustration of a tensor decomposition of the four-dimensional array of weighted phase-locking factors (wPLFs). The two spatial maps and the two **frequency** spectra (see text) are each denoted by a different color (red, yellow, green, and blue) and a different index (i, i', f, f'). The same colors are used both for the boundaries of the panels and the symbols in the formula for the wPLF. In panels a and b, with red and index i, we show the complex-valued spatial map of the high-**frequency** bursts that are coupled to a common low-**frequency** **oscillation** (the coupling **oscillation**). In panel a, we show the magnitude (absolute value) of this complex-valued spatial map (one colored circle per channel), which expresses the strength of the coupling. In panel b we show the phases of the coupling **oscillation** to which the high-**frequency** bursts are locked (one arrow per channel). In panel c, with yellow and index i', we show the spatial map of the coupling **oscillation** (one colored circle per channel). The more a coefficient bi’ deviates from zero, the more this channel is affected by the coupling **oscillation**. In panel d, with green and index f, we show the **frequency** spectrum of the phase-coupled bursts (shown on a logarithmic scale). In panel e, with blue and index f', we show the **frequency** spectrum of the coupling **oscillation**. The spectra shown in panels a, b, c, d and e are all in arbitrary units (a.u.). This is because the spectra are produced by a tensor decomposition which involves an arbitrary multiplicative scaling (see Supplemental Material). In panels f, h, and j, we show the magnitudes of the complex wPLFs for three selected channel pairs (see text). By means of arrows, we connect the channels in panels a and c for which these wPLFs were calculated, with the corresponding x-axes (showing the **frequency** of the coupling **oscillation**), respectively, y-axes (showing the **frequency** of the phase-coupled bursts), in panels f, h, and j. In panels g and i, we show the phases of the complex wPLFs that correspond to panels f and h, respectively.
... Simulated field potentials with bursts of gamma **oscillations** (60Hz) that are phase-coupled to the rising phase of a theta **oscillation** (5Hz).
... Spectral signature of the PAC patterns. Panels a and b show the results for, respectively, the baseline and the activation period. Both panels show a scatter plot of the central **frequencies** of the phase-coupled bursts (horizontal axis) against the central **frequencies** of the coupling **oscillations** (vertical axis). The central **frequency** of the phase-coupled bursts is always smaller than the one for the associated coupling **oscillation**. For all PAC patterns above the thick black line, the central **frequency** for the phase-coupled bursts is larger than the central **frequency** for the associated coupling **oscillation**. There is not a dominant **frequency**, neither for the phase-coupled bursts, nor for the associated coupling **frequencies**.

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Excitation Balanced by Proportional Inhibition during Gamma **Oscillations** In Vivo
(A) Whole-cell recording of EPSCs in CA3 cell (red) and simultaneously recorded LFP (black, positivity is up) during gamma **oscillations** in anesthetized rat. IPSCs (cyan) and inverted LFP recorded from the same cell. Note correlated fluctuations in the amplitude of LFP and synaptic currents.
(B) Coherence between LFP and IPSCs (cyan) or EPSC (red); jack-knifed 95% confidence interval (thin lines); arrows mark peak coherence **frequencies**. Summary of peak coherence **frequency** (bottom) and peak coherence (right). Average shown as a vertical or horizontal bar (n = 7 cells).
(C) **Oscillation** triggered average (OTA) of EPSC (red), IPSC (cyan) and LFP. LFP was recorded simultaneously with EPSCs, IPSC (black and dotted traces, respectively). EPSC is inverted for illustration purposes. Overlaid POTH (green, data from Figure 1D, aligned to the LFP also in green) illustrates spike timing during **oscillation** cycle. Note that maximal spiking precedes peak of inhibition. (Bottom) Summary of EPSC-IPSC lag during an **oscillation** cycle; vertical bar is average.
(D) OTA of EPSCs (red), IPSCs (cyan) computed for four different bins of LFP **oscillation** amplitude (black; dotted and solid traces were recorded simultaneously with IPSC and EPSCs, respectively, same cell as A–C).
(E) Summary of correlation between average inhibitory (gI) and excitatory (gE) conductance in vivo; individual cells are each represented by a different color linear regression. Note, although excitation and inhibition are proportional, the inhibitory conductance is approximately five times larger (dotted line is at unity).
(F) OTA of IPSC (middle) computed for four different bins of LFP **oscillation** interevent interval (top). Vertical arrows illustrate IPSC amplitude and horizontal arrows the correlated changes in the time to the next **oscillation** event (IEI). (Bottom) IPSC amplitude during an **oscillation** event correlated with the time to the next **oscillation** in the LFP (IEI); blue dots correspond to the four OTA shown above.
... Gamma **Oscillation** Amplitude Predicts Latency to Next **Oscillation** Cycle
(A) (Top) Broadband (gray) and gamma-band filtered local field potential (LFP, 5–100 Hz) recorded in the stratum radiatum of area CA3 of an anesthetized rat. Raster plot marks the peak of each **oscillation** cycle. (Bottom, left) Autocorrelation of LFP and power spectral density of gamma-band LFP. (Bottom, right) Histograms of **oscillation** amplitude and interevent interval (IEI). (Inset) LFP recording illustrating the measurement of peak-to-peak amplitude and IEI (expansion of the recording marked by a horizontal bracket in the top panel). Positivity is up.
(B) (Top) IEI correlated against amplitude of the previous cycle illustrated in histogram. Note the correlation between **oscillation** amplitude and IEI. (Bottom) Summary of correlations, n = 6 rats. Vertical bar is average.
(C) Broadband extracellular recording (top), gamma-band LFP (middle, 5–100 Hz band-pass), multiunit spiking (green, 0.2–2 kHz) from stratum pyramidale of area CA3. Negativity is up.
(D) **Oscillation** triggered average of LFP, peri-**oscillation** spike-time histogram (POTH), and local linear fit to POTH (green).
(E) (Left) Average LFP and POTH fit calculated separately for large (mean amplitude = 313 μV) and small (99 μV, dotted) **oscillation** cycles. Arrows illustrate the increased latency between spiking events after large-amplitude cycles. (Inset) Small POTH scaled to the peak of the large POTH. (Right) Summary of full-width at half-maximum (FWHM) of POTH for large (solid) and small (open) **oscillation** cycles (n = 6 rats). Averages are illustrated with horizontal bars. Note that spiking occurs in a narrow time window during each **oscillation** cycle independent of **oscillation** amplitude.
... Larger, Longer Hyperpolarization of Pyramidal Cells following Large-Amplitude **Oscillation** Cycles
(A) LFP and simultaneously recorded membrane potential (Vm; whole-cell current-clamp configuration: IC) during in vitro gamma **oscillations** (dotted line is mean Vm). Positivity is up.
(B) **Oscillation** cycles were binned according to LFP amplitude and the **oscillation** triggered average (OTA) of Vm computed for each bin (different colors): average time course of LFP in four bins of increasing amplitude (top) and corresponding (color coded) Vm averages (middle). Note that Vm undergoes larger and longer hyperpolarization during large-amplitude **oscillation** cycles. (Bottom, left) Cycle-by-cycle correlation between the peak hyperpolarization and LFP amplitude. Bins in upper panels are illustrated with solid dots of respective colors. (Bottom, right) Summary of correlation (n = 11 cells).
(C) (Top) **Oscillation** cycles were binned according to LFP interevent interval and the OTA of membrane potential computed for each bin (different cell than A and B). Arrows illustrate “recovery time,” i.e., time from onset of **oscillation** cycle till membrane potential recovers to mean Vm (horizontal dotted line). (Bottom) LFP interevent interval plotted as a function of Vm recovery time. Colored dots and black line correspond to the above cell, other cells shown in gray. Note, mean slope, m = 1.16; SD = 0.3, suggesting that changes in the time for recovery from hyperpolarization in individual cells can account for the entire range of **oscillation** intervals observed in the LFP.
... Excitation Instantaneously Balanced by Proportional Inhibition during Each Gamma **Oscillation** Cycle
(A) (Top) Broadband (gray) and gamma-band filtered (black) LFP recorded in the stratum radiatum of area CA3 in acute hippocampal slice. Raster plot marks the peak of each **oscillation** cycle. (Bottom, left) Autocorrelation of LFP and power spectral density of gamma-band LFP. (Bottom, right) Histograms of **oscillation** amplitude and interevent interval (IEI). (Inset) LFP recording illustrating the measurement of peak-to-peak amplitude and IEI (expansion of the recording marked by a horizontal bracket in the top panel). Positivity is up.
(B) (Top) IEI correlated against amplitude of the previous cycle. (Bottom) Summary of correlations, n = 6 slices. Vertical bar is the average. Note the correlation between **oscillation** amplitude and IEI.
(C) Dual patch-clamp recording from two neighboring CA3 pyramidal cells. **Oscillations** are monitored with an LFP electrode (black, positivity is up). EPSCs (red) and IPSCs (cyan) simultaneously recorded by holding two cells at the reversal potential for inhibition (−3 mV) and excitation (−87 mV), respectively. Note the correlated fluctuations in the amplitude of excitation and inhibition.
(D) (Left) Average time course of EPSC and IPSC (same cell as C) during an **oscillation** cycle recorded in the LFP, i.e., **oscillation** triggered average. EPSC is inverted for illustration purposes. LFPs recorded simultaneously with EPSCs and IPSCs are shown as black and gray traces, respectively. (Right) Summary of EPSC-IPSC lag during an **oscillation** cycle. Horizontal bar is the average.
(E) (Top) Cycle-by-cycle correlation between excitatory and inhibitory conductances recorded in the pair shown in (C). Summary of correlation between excitation and inhibition (bottom) and ratio of mean excitatory and inhibitory conductances (right) (n = 8 pairs). Vertical and horizontal bars illustrate respective averages.
... Correlated Amplitude and **Frequency** in Simple Model of CA3 Circuit
(A) Average excitatory (gE, red) and inhibitory (gI, cyan) synaptic conductance received by model pyramidal cells. LFP (black) is approximated as the sum of the two conductances.
(B) (Top) Autocorrelation and power spectrum of simulated LFP. (Bottom) Interevent interval correlated against amplitude of the previous cycle.
(C) The membrane potential (Vm) of an individual pyramidal cell in modeled circuit (spike truncated), gE (red) and gI (cyan); dotted line illustrates the average Vm.
(D) **Oscillation** cycles were binned according to gI amplitude and the **oscillation** triggered amplitude of Vm computed for each bin (different colors): average time course of gI in four bins of increasing amplitude (middle) and corresponding (color coded) Vm averages (top). The arrows illustrate that it takes longer for Vm to recover to the average potential (horizontal dotted line) after large-amplitude cycles. (Bottom) Cycle-by-cycle correlation between Vm hyperpolarization and the gI. Bins in upper panels are illustrated with solid dots of respective colors.

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Portico **Frequency** of free **oscillations** Resonance Stiffness matrix method.

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