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High-**frequency** **oscillations**... An example of the implantation schedule (patient #1) demonstrating areas with conventional **frequency** ictal patterns, ictal high-**frequency** **oscillations**, hyperexcitability, and radiological lesions.
... An example of the implantation schedule (patient #7) demonstrating areas with conventional **frequency** ictal patterns, ictal high-**frequency** **oscillations**, hyperexcitability, and radiological lesions.
... Summary table for statistical analysis. HFO=high **frequency** **oscillations**, CFIP=conventional **frequency** ictal patterns.

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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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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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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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Quantification of amplitude-to-phase cycle relationship. (A) Mean amplitudes of high **frequency** **oscillations**, sorted by concurrent low-**frequency** phase into 60 bins of 0.105rad, for an example time window during a seizure; (B) an ideal cosine; and (C) a sine is modeled. (D) Phasor demonstrating the amplitude–phase cycle relationship. (E) The argument (angle) of the example phasor is a single contribution to be incremented onto a cumulative polar histogram spanning multiple subjects for one of ten given time periods in the seizure.
... Cross-**frequency** coupling... Modulation of high **frequency** amplitude by low-**frequency** phase. In the seizure-onset zone, significant modulation of high-**frequency** amplitude (40–300Hz) is observed, mainly by the phase of theta and alpha **oscillations** during the ictal period. In the interictal period, no specific CFC with slower **oscillations** is observed. There is also less cross-**frequency** coupling in the early propagation zone during seizures and no significant coupling is noted in the non-epileptogenic cortex. The z-axis demonstrates the modulation of amplitudes of different narrow-band **frequencies** (x-axis) by the phases of other narrow-band **frequencies** (y-axis). Lower and upper planes represent uncorrected and corrected statistical thresholds at p<0.05, respectively.
... High **frequency** **oscillations**... Simulated data demonstrating expected polar histogram distribution. The high **frequency** amplitude is represented by the blue solid line, whereas the low-**frequency** phase is represented by the black dashed line. When the high **frequency** amplitude is maximal at the peak and trough of the low **frequency** phase, the polar histograms will indicate pi and 0, respectively.
... Individual seizure short-time Fourier transform spectrograms. The seizure onset zone contained predominantly low-**frequency** power, which was fairly heterogenous across the population. High **frequency** activity was also evident in all seizures as bursts of high power oscillatory activity.
... Topographic mapping of cross-**frequency** interactions in a representative subject. (A) Intraoperative image of grid demonstrating seizure onset and early propagation zones. (B) Fast-ripple amplitudes sorted by alpha phase for all grid electrodes. Cosine wave represents alpha phase from −π to π. Increased pHFO-to-low-**frequency** coupling occurs in the resected cortex (black borders). Values normalized by 95% confidence interval such values above 0 are significant at pfrequency modulation index for all electrodes, where the X-axis represents low **frequency** phase (1 to 40Hz; left to right), and the Y-axis denotes envelope amplitude (1 to 300Hz; top to bottom). Values exceeding Bonferroni correction threshold shown. Significant modulation of pHFO amplitudes by low-**frequency** phase is observed in the epileptogenic cortex.
... To characterize the ictal dynamics of relations between pHFO amplitude and low **frequency** phase, we measured the preferred slow oscillatory phase at which high amplitude pHFOs occurred at various times throughout the seizure. When the preferred phases from all bins from all subjects were plotted cumulatively on polar histograms, it was observed that pathological fast-ripple amplitudes preferentially occurred during the trough of alpha **oscillations**, whereas pathological ripple amplitudes preferentially occurred between 0rad and π/2rad of alpha and theta oscillatory cycles (Fig. 5; poscillations (p=0.14 and p=0.68, respectively). At seizure termination (i.e. the last bin), pHFO amplitudes occurred at the trough of the alpha oscillatory cycle (pathological ripple amplitude: p<0.01; pathological fast-ripple amplitude: p=0.03). Ripple amplitude maxima were also found at the peak of delta phase irrespective of the seizure progression (Supplementary Fig. S9). To ensure that differences in bin length did explain the measures of CFC, a reanalysis of the data with fixed length segments comprised of the first and last 2000ms of seizures, revealed the same pattern.

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Variations of resonance **frequency** according to drop viscosity. Dash denotes absence of resonance **frequency**.
... Temporal evolution of the base radius (colored symbol) of **oscillating** drops with different viscosities for different **frequencies** of (a) 32Hz and (b) 98Hz at η=0.25; (c) 28 and (d) 94Hz at η=0.39. The dashed lines denote subharmonic **oscillation**.
... Maximum amplitudes of **oscillating** drops with different viscosities along with AC **frequency** at different η; (a) and (b): η=0.25, (c) and (d) η=0.39.
... **Oscillation** patterns of 5μL drops with different viscosities at η=0.25 for different **frequencies** of (a) 32Hz and (b) 98Hz. The patterns are obtained by superposing more than 20 images. The solid lines in the right column depict instantaneous drop deformation for theoretical P2 and P4 mode shape **oscillation**; short dashed lines represent initial quiescent shapes; arrows denote nodal points.
... Resonance **frequency**... **Frequency** response at harmonic and subharmonic **frequencies**.
... Drop **oscillation**

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Amplitude and Intertrial Phase Coherence of Theta and Gamma **Oscillations**
... theta **oscillations**... The amplitude of stimulus-driven gamma **oscillations** is modulated by the phase of ongoing theta **oscillations**. This cross-**frequency** coupling indicates a hierarchical organization of cortical oscillatory dynamics in both healthy control subjects (black line) and schizophrenia patients (red line). The x axis indicates theta phase. The y axis indicates gamma amplitude.
... Heuristic model of phase-amplitude cross-**frequency** coupling. Gamma **oscillations** (red and blue lines) are largest in the excitatory versus inhibitory phase of ongoing theta **oscillations** (black line). Note that excitatory and inhibitory phase may vary according to tasks and neural sources.
... cross-**frequency** coupling... gamma **oscillations**... Schizophrenia patients have normal theta-phase/gamma-amplitude cross-**frequency** coupling. The modulation index demonstrates the relative strength of cross-**frequency** coupling via comparison of observed (O) versus resampled or surrogate (S) electroencephalography data in healthy control subjects (black circle) and schizophrenia patients (red squares). The y axis indicates log transform of modulation index.
... neural **oscillations**... Schizophrenia patients (SZ) have increased theta amplitude and decreased gamma synchrony. The left column shows time-**frequency** maps from healthy control subjects (HC) and the middle column shows time-**frequency** maps from schizophrenia patients. The x axis indicates time in milliseconds and the y axis indicates **frequency**. Color indicates amplitude in the top row and intertrial phase coherence (ITC) in the bottom row. The right column shows difference between schizophrenia patients and healthy control subjects. Difference maps show only time-**frequency** points at p < .01.

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IP3 turnover time controls feedbacks. (A) Positive-feedback model. Dynamics of [Ca2+]c, [IP3], and the fraction of open IP3Rs (solid, dashed, and dot-dashed lines, respectively) during an **oscillation** period; the fraction of open IP3Rs is given by [rcp/(c+Ka)/(p+Kp)]3 (see Eq. 6). Fast IP3 turnover yields a pronounced spike (left panel, τp=0.1 s), whereas slow IP3 turnover supports only a small-amplitude response (right panel, τp=15 s). (B) The negative-feedback model shows the opposite behavior, with a small-amplitude response for fast IP3 turnover (left panel, τp=0.1s) and a sharp spike for slow IP3 turnover (left panel, τp=15s). (C) Positive-feedback model. Bifurcation diagram showing the maxima and minima of the [Ca2+]c **oscillations** as a function of the stimulus for different values of the IP3 turnover. The bifurcation diagrams for different values of τp are compared by plotting them against the product V¯PLC=VPLCτp;; in this way, the steady-state concentrations of Ca2+ and IP3 are identical for a given V¯PLC (solid and dashed lines indicate stable and unstable states, respectively; the stability of the steady state is shown for τp=15s). Both amplitude and range of stimuli leading to **oscillations** increase with faster IP3 turnover. (D) The corresponding bifurcation diagrams for the negative-feedback model show the opposite behavior. The amplitude and range of stimuli leading to **oscillations** increase with slower IP3 turnover. (E, F) Dependence of **frequency** encoding on IP3 turnover in the positive and negative feedback models, respectively. Shown are the differences ΔT between the largest (for low stimulation) and smallest (for high stimulation) **oscillation** period.
... Agonist-induced IP3 and Ca2+ **oscillations** in the positive and negative feedback models. (A) Positive feedback model with Ca2+ activation of PLC. Changes in [Ca2+]c, [IP3], and in the fraction of active receptors r (top, middle, and bottom panels) after stepwise increases in the agonist concentration (arrowheads), modeled by an increase in the maximal rate of PLC (VPLC=0.3μM/s for toscillations (thick lines) and the [Ca2+]c steady states (thin lines) as a function of the stimulus (VPLC). Solid and dashed lines indicate stable and unstable states, respectively. HB, Hopf bifurcation; HC, homoclinic bifurcation; SN, saddle-node bifurcation; FB, saddle node of periodics. (D) Bifurcation diagram for negative feedback model. PD, period doubling; TR, torus bifurcation. Between PD and HB1 and TR and FB there exist complex **oscillations** (omitted for clarity). The parameter values used are listed in Table 1.
... Complex responses to an IP3 buffer in the positive-feedback model. (A) Bifurcation diagram showing the region of **oscillations** as function of stimulus (VPLC) and IP3 buffer concentration (gray-shaded area; the solid lines indicate the locus where the steady state becomes unstable via a Hopf bifurcation). In region I, regular **oscillations** have a decreased rate of [Ca2+]c rise with increased [IP3 buffer] as shown in Fig. 6 A. In region II, the IP3 buffer abolishes the Ca2+ **oscillations** completely, as shown in Fig. 6 C. In region III, bursting [Ca2+]c **oscillations** are observed (the lower boundary of this region is determined by a period doubling bifurcation, dotted line). We have indicated an additional region IV, which is characterized by **oscillations** persisting even at high [IP3 buffer]. The parameters are as in Fig. 6, with ε=2 When the strength of the Ca2+ plasma-mem... Control coefficients for the **oscillation** period. (A, B) Positive and negative feedback models, respectively, in the absence of Ca2+ fluxes across the plasma membrane (ε=0); control coefficients of Ca2+ exchange across the ER membrane (Cer, solid line), IP3 metabolism (Cp, dashed line), and IP3R dynamics (Cr, dotted line) as function of the period of the **oscillations**. A positive period control coefficient signifies that a slowing of the corresponding process increases the **oscillation** period. (C, D) Period control coefficients in the positive and negative feedback models, respectively, in the presence of plasma-membrane fluxes of Ca2+(ε=1). The dash-dotted line indicates the control exerted by Ca2+ exchange across the plasma membrane (Cpm).
... **Frequency** encoding of agonist stimulus. (A) Positive feedback: **oscillation** periods observed at different stimulation strengths (varying VPLC). Increasing the half-saturation constant of PLC for Ca2+, KPLC, from 0 (no functional positive feedback) to 0.2μM (functional feedback) greatly enhances **frequency** encoding. (B) Negative feedback. Increasing the amount of IP3K relative to IP3P (η) enhances **frequency** encoding. (C, D) The feedback effects shown in panels A and B are preserved when plasma-membrane fluxes of Ca2+ are included in the models (ε=1). (E) Range of **oscillation** periods, ΔT=Tmax−Tmin, in the presence (+) and absence (w/o) of positive feedback for two different strengths of the plasma-membrane Ca2+ fluxes (ε=1,4). (F) Range of **oscillation** periods in the presence (−) and absence (w/o) of negative feedback for two different strengths of the plasma-membrane Ca2+ fluxes (ε=1,4). We have found that the IP3K has an impact on the **oscillation** period only when the Ca2+ fluxes between ER and cytoplasm are comparatively slow and the IP3R is less sensitive to Ca2+ activation. To expose the period effect of the negative feedback, we have chosen different parameter values than in the positive feedback model (see Table 1).

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Phase–Amplitude Coupling (PAC) and Amplitude–Amplitude Coupling (AAC). (A) Dense-spiking continuous fast **oscillation** (FO). Plot shows FO amplitude as a function of slow **oscillation** (SO) phase for distinct FO **frequencies** (from 10Hz to 100Hz) for the dense-spiking continuous FO model of Figure 1E. Red, high FO power; blue, low FO power. Peak FO **frequency** varies throughout SO cycle (PFC) but the overall FO power remains constant. (B) Sparse-spiking continuous FO. PAC is present in a model identical to that of panel A with simply an increased level of noise, producing sparse-spiking FO **oscillations**. (Left) Raster plot showing that the number of FO spikes at each cycle varies as a function of SO phase. (Right) FO amplitude as a function of SO phase, showing strong PAC with maximal FO amplitude close to SO peak, on top of PFC. (C) Sparse-spiking continuous FO with adaptation. The model in (B) was modified by introducing an adaptation current, slowing down the dynamics of the FO **oscillation** in response to SO modulation. PAC plot shows that maximal FO power occurs at a later SO cycle compared with the no-adaptation model of (B). (D) Dual ING/PING alternation. (Leftmost and middle-left) The FO network alternates within a SO cycle between a period of pyramidal-interneuron-generated **oscillations** (PING mechanism) and another period of faster interneuron-generated **oscillations** (ING) (this is obtained by a SO modulation applied to the inhibitory instead of the excitatory population). (Middle-right) FO amplitude as a function of SO phase. PAC occurs for two distinct FO **frequencies**, with maximal FO power for these two **frequencies** in antiphase. Such a pattern with distinct **frequencies** (pointing to distinct **oscillation**-generation mechanisms) should not be confounded with the case of a **frequency** modulation within a single FO **frequency** band, as shown in (A). (Rightmost) Data from rat hippocampus recordings showing a similar alternation between slow gamma and fast gamma within a theta cycle (adapted from [41]). (D) Positive AAC for positive asymmetric SO. (Left panel) The cross-**frequency** coupling (CFC) network with sparse-spiking FO **oscillations** was simulated with a SO sinusoid which amplitude was modulated at a 1Hz rate, whereby an increase in SO amplitude is accompanied by a general increase in SO level (green curve, left panel). This leads to general increase of FO amplitude on top of an increase of PAC strength. (Middle panel) FO amplitude as a function of SO phase, separately for high SO amplitude cycles (red curve) and low SO amplitude cycles (black curve). PAC was present as for the non-modulated SO sinusoid (Figure 3B), but here PAC modulation increased with higher SO amplitude. (Right panel) FO amplitude as a function of SO amplitude, showing strong positive AAC. Figure S1 in the supplemental information online shows the absence of AAC for a symmetric SO. (F) Negative AAC for negative asymmetric SO. Same model as in (E) but here the increase in SO amplitude is accompanied by a general decrease in SO level. This in turn leads to general decrease of FO amplitude on top of an increase of PAC strength, hence a negative correlation between SO and FO amplitudes.... cross-**frequency** coupling... neural **oscillations**... Phase–**Frequency** Coupling (PFC) and Phase–Phase Coupling (PPC). (A) Continuous fast **oscillation** (FO). (Top) PPC in the continuous FO model of Figure 1E. The top panel shows the histogram of FO phases as a function of slow **oscillation** (SO) phase for a simulation of 30 s. Darker areas show higher phase concentration. The overall largely homogeneous pattern indicates that very limited PPC emerges, which is confirmed by computing the FO phase-locking factor (PLF) as a function of SO phase [69]: low PLF indicates that SO phase had virtually no influence on FO phase. The green line shows the outline of the SO modulating signal. (Bottom) PFC in the same model. The black curve shows the **frequency** of FO bursts as a function of SO phase. (B) Intermittent oscillatory/quiescent FO. (Top) PPC in the intermittent oscillatory/quiescent FO model of Figure 1F. The top panel shows very strong FO phase concentration throughout the SO cycle, with nearly non-existent phase change during the quiescent period (i.e., **frequency** vanishes) and rapid phase dynamics in the oscillatory period. FO-PLF remains high during the whole SO cycle, with highest values during the quiescent period. (Bottom) PFC in the same model. FO **frequency** is modulated during the oscillatory period and vanishes during the quiescent period. (C) Intermittent oscillatory/asynchronous FO displaying PFC without PPC. (Top) PPC in the intermittent oscillatory/asynchronous FO model of Figure 1G. PPC hardly emerges in this network because no phase-resetting occurs during the asynchronous phase. (Bottom) PFC in the same model. FO **frequency** is modulated during the oscillatory period and vanishes during the asynchronous period. (D) Type II FO displaying PPC without PFC. (Top left) Raster plot of a network of a single class of interconnected excitatory neurons (adaptive quadratic integrate-and-fire) receiving rhythmic SO impulse. (Bottom left) Nearly symmetrical phase response curve (PRC) of FO neurons showing phase lead for later phases and phase lag for earlier phase, corresponding to a type II **oscillator** (Box 1). (Top right) Strong PPC in the network because type II **oscillators** phase-lock easily to external excitatory input. (Lower right) PFC is absent in the same network. Because of the symmetrical PRC, on average there is no acceleration or slowing down of the FO because phase leads and delays cancel out, and FO remains at a constant **frequency** throughout the SO cycle.
... (A,B) Spiking **frequency** of type I and type II neurons as a function of driving-current magnitude. (A) Cells with type I membrane excitability can spike at any arbitrary **frequency** once passed beyond the bifurcation point. (B) When a type II neuron goes through the bifurcation point it starts firing from a finite **frequency**. (C) Phase response curve of a type I neuronal **oscillator**: any small excitatory input induces a phase advance within the spiking cycle of the **oscillator**. (D) Phase response curve of a type II neuron: an excitatory pulse either advances or delays the next spike depending on the particular phase at which the input arrives within the spiking cycle. (E) When solicited with an external sinusoidal input (SO), individual neurons from Aplysia display Arnold tongue maps (i.e., different forms of m:n coupling depending on the input **frequency** and amplitude). Experimental results are in fair agreement with model simulations (adapted from [29]). (F,G) Phase–amplitude coupling modeled through a set of Wilson–Cowan equations to represent the activity of a specific population of interneurons in rat hippocampus (adapted from [96]). Experimenters recorded the response of excitatory and inhibitory neurons to sinusoidal inputs induced using optogenetics tools. (F) Schematic illustration of the excitatory–inhibitory hippocampal network and spontaneous FO visible in E and I population activity in absence of external noise. (G) Phase portrait of the FO model showing the stable limit cycle of the system (blue), the nullclines (i.e., the points where either the excitatory or the inhibitory activity would be constant; black: dashed, excitatory; solid, inhibitory), and the isochrones (the set of initial points from which the dynamics would evolve towards **oscillations** having the same phase; red). Phase and amplitude modulations of FO can be inferred from the phase portrait, given a specific set of initial conditions.
... Functions of Cross-**Frequency** Coupling (CFC). (A) Multi-item/sequence representation. (a) Multi-item representations through CFC is envisaged for working memory (Top, in hippocampus and prefrontal cortex) [54,116], spatial memory for previously and subsequently visited items (in hippocampus) [60,61], and visual attention to a series of items in visual field (Bottom, in visual cortex) [10]. (b) Putative CFC architecture for multi-item representation. Intertwined CFC network with a common population of pyramidal neurons connecting to two populations of distinct types of inhibitory neurons, each responsible for the generation of one of the two **oscillations** (slow, SO; and fast, FO). Some models also include direct connections between the two inhibitory populations [6,20]. (c) Multi-item representation relies on intermittent/quiescent CFC regime: the nested FO **oscillation** only occurs during a defined phase range, with one item being represented at each FO burst. AAC (through asymmetrical SO) allows modulating the phase range of FO and thus the number of represented items. (B) Long-distance communication. (a) Putative ... Architectures for Cross-**Frequency** Neural Coupling. (A–D) Cross-**frequency** coupling (CFC) architectures. (A) Intertwined **oscillators**. Neural **oscillations** in distinct **frequency** bands are generated by partially overlapping neural populations. In the depicted example, excitatory neurons (represented by the triangle) participates in the generation of both a slow **oscillation** (SO) and a fast **oscillation** (FO), whereas separate inhibitory populations (represented by green and orange circles) are involved in the generation of respectively SO and FO. CFC arises though the dynamics of the neural population common to both **oscillations**. Intermingled CFC has been proposed to explain the emergence theta/gamma coupled **oscillations** in hippocampus [6,30,100], where a common excitatory population is coupled to fast-spiking (FS) cells (generating gamma **oscillations**) and oriens-lacunosum moleculare (O-LM) cells (generating theta **oscillations**). (B) Bidirectional coupling. Segregated populations are implicated in the generation of the SO and FO (in the depicted example, a population of excitatory and inhibitory neurons for both **oscillations**), and coupling is mediated by reciprocal coupling between SO neurons and FO neurons. This architecture was used in the first computational model of coupled theta–gamma **oscillations** in hippocampus, which featured two coupled inhibitory subpopulations with distinct GABA decay time [101]. Moreover, precise spiking and local field potential (LFP) dynamics of in vitro cortical slices was explained by a sophisticated model whereby a beta1 rhythm concatenates two bidirectionally coupled **oscillations**, a gamma rhythm generated in superficial layers and a beta2 rhythm generated in deep layers [102,103]. (C) Unidirectional coupling. Distinct populations are implicated in the generation of SO and FO, and coupling arises though one population projecting onto the other population (here SO to FO). In a recent model of speech perception in auditory cortex, coupled theta–gamma **oscillations** were modeled by two separate excitatory–inhibitory subpopulations responsible for the generation of both rhythms [39], with the theta module projecting onto the gamma module. (D) Sensory entrainment. Sensory entrainment of neural **oscillators** is a special case of unidirectional coupling where a neural **oscillator** is modulated by slow modulations in sensory stimulus (e.g., visual rhythmic movements or amplitude-modulated sounds). This model has been tested in the context of visual processing: a PING module generating broad gamma rhythm responds to visual activity experimentally recorded in monkey thalamus that carries strong slow modulations [28]. (E–G) Temporal dynamics of FO. (E) Continuous FO. (Left) Schematic phase diagram of FO dynamical state: continuous FO occurs when SO modulation shapes a trajectory within the region of existence of FO. (Right) The continuous model of CFC from Fontolan et al. [16] (see their Figure 5) was transformed into a mathematically equivalent model of quadratic-and-fire neurons and simulated (with membrane white noise of variance 0.01). SO is modeled as a simple modulatory signal while FO is composed of a population of pyramidal and inhibitory cells generating gamma **oscillations** through the PING mechanism. (Top right) SO modulating signal. (Bottom right) raster plot of FO spikes: pyramidal neurons (n=50, light orange); inhibitory neurons (n=50, dark orange). FO spiking occurs throughout the SO cycle. (F) **Oscillation**/quiescent-state intermittent FO. (Left) This type of intermittent FO occurs when SO modulation shapes a trajectory between the oscillatory and quiescent regions of FO. (Right) The intermittent model of CFC from [16] (see their Figure 7) was transformed into network of quadratic-and-fire neurons and simulated. FO alternates between a period of gamma **oscillations** when excitation from SO is large enough, and a quiescent state when SO signal is lower. (G) **Oscillation**/asynchronous-state intermittent FO. (Left) This type of intermittent FO occurs when SO modulation shapes a trajectory between the oscillatory and asynchronous regimes of FO. (Right) We modified the parameters of the gamma network of [16] (constant current, synaptic conductance, and membrane noise) such that the network alternates within SO cycle between a period of strong **oscillation** and a period of asynchronous spiking activity.

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