Analysis and modeling of time-variant amplitude–frequency couplings of and between oscillations of EEG bursts

Low-frequency (0.5-2.5 Hz) and individually defined high-frequency (7-11 or 8-12 Hz; 11-15 or 14-18 Hz) oscillatory components of the electroencephalogram (EEG) burst activity derived from thiopental-induced burst-suppression patterns (BSP) were investigated in seven sedated patients (17-26 years old) with severe head injury. The predominant high-frequency burst oscillations (>7 Hz) were detected for each patient by means of time-variant amplitude spectrum analysis. Thereafter, the instantaneous envelope (IE) and the instantaneous frequency (IF) were computed for these low- and high-frequency bands to quantify amplitude-frequency dependencies (envelope-envelope, envelope-frequency, and frequency-frequency correlations). Time-variant phase-locking, phase synchronization, and quadratic phase couplings are associated with the observed amplitude-frequency characteristics. Additionally, these time-variant analyses were carried out for modeled burst patterns. Coupled Duffing oscillators were adapted to each EEG burst and by means of these models data-based burst simulations were generated. Results are: (1) strong envelope-envelope correlations (IE courses) can be demonstrated; (2) it can be shown that a rise of the IE is associated with an increase of the IF (only for the frequency bands 0.5-2.5 and 7-11 or 8-12 Hz); (3) the rise characteristics of all individually averaged envelope-frequency courses (IE-IF) are strongly correlated; (4) for the 7-11 or 8-12 Hz oscillation these associations are weaker and the variation between the time courses of the patients is higher; (5) for both frequency ranges a quantitative amplitude-frequency dependency can be shown because higher IE peak maxima are accompanied by stronger IF changes; (6) the time range of significant phase-locking within the 7-11 or 8-12 Hz frequency bands and of the strongest quadratic phase couplings (between 0.5-2.5 and 7-11 or 8-12 Hz) is between 0 and 1,000 ms; (7) all phase coupling characteristics of the modeled bursts accord well with the corresponding characteristics of the measured EEG burst data. All amplitude-frequency dependencies and phase locking/coupling properties described here are known from and can be discussed using coupled Duffing oscillators which are characterized by autoresonance properties.     

Canonical bicoherence analysis of dynamic EEG data

Bicoherence has been used in quantifying quadratic phase coupling (QPC) in electroencephalography (EEG) signals. However, for high-dimensional EEG signals, the calculations of traditional auto– and cross–bicoherences of signals from multiple electrodes are computationally very expensive. This has been compounded by the recognition of the non-stationary character of EEG signals. This paper introduces a new approach, the time-varying canonical bicoherence (CBC) by short-time weighted Fourier transforms, for analyzing QPC nonlinearities of dynamic EEG signals. This new method shows both computational efficiency and simple interpretation of estimated canonical bicoherences. The canonical bicoherence analysis of EEG records, during a human visual stimulus-driven cognitive process, put into evidence of quadratic phase couplings of Beta waves and Delta waves in the frontal regions.     

Activation-Inhibition dynamics of the oscillatory bursts of the human EEG during resting state. The macroscopic temporal range of few seconds

The ubiquitous brain oscillations occur in bursts of oscillatory activity. The present report tries to define the statistical characteristics of electroencephalographical (EEG) bursts of oscillatory activity during resting state in humans to define (i) the statistical properties of amplitude and duration of oscillatory bursts, (ii) its possible correlation, (iii) its frequency content, and (iv) the presence or not of a fixed threshold to trigger an oscillatory burst. The open eyes EEG recordings of five subjects with no artifacts were selected from a sample of 40 subjects. The recordings were filtered in frequency ranges of 2 Hz wide from 1 to 99 Hz. The analytic Hilbert transform was computed to obtain the amplitude envelopes of oscillatory bursts. The criteria of thresholding and a minimum of three cycles to define an oscillatory burst were imposed. Amplitude and duration parameters were extracted and they showed durations between hundreds of milliseconds and a few seconds, and peak amplitudes showed a unimodal distribution. Both parameters were positively correlated and the oscillatory burst durations were explained by a linear model with the terms peak amplitude and peak amplitude of amplitude envelope time derivative. The frequency content of the amplitude envelope was contained in the 0–2 Hz range. The results suggest the presence of amplitude modulated continuous oscillations in the human EEG during the resting conditions in a broad frequency range, with durations in the range of few seconds and modulated positively by amplitude and negatively by the time derivative of the amplitude envelope suggesting activation-inhibition dynamics. This macroscopic oscillatory network behavior is less pronounced in the low-frequency range (1–3 Hz).     

Fronto-Central Theta Oscillations Are Related to Oscillations in Saccadic Response Times (SRT): An EEG and Behavioral Data Analysis

The phase reset hypothesis states that the phase of an ongoing neural oscillation, reflecting periodic fluctuations in neural activity between states of high and low excitability, can be shifted by the occurrence of a sensory stimulus so that the phase value become highly constant across trials (Schroeder et al., 2008). From EEG/MEG studies it has been hypothesized that coupled oscillatory activity in primary sensory cortices regulates multi sensory processing (Senkowski et al. 2008). We follow up on a study in which evidence of phase reset was found using a purely behavioral paradigm by including also EEG measures. In this paradigm, presentation of an auditory accessory stimulus was followed by a visual target with a stimulus-onset asynchrony (SOA) across a range from 0 to 404 ms in steps of 4 ms. This fine-grained stimulus presentation allowed us to do a spectral analysis on the mean SRT as a function of the SOA, which revealed distinct peak spectral components within a frequency range of 6 to 11 Hz with a modus of 7 Hz. The EEG analysis showed that the auditory stimulus caused a phase reset in 7-Hz brain oscillations in a widespread set of channels. Moreover, there was a significant difference in the average phase at which the visual target stimulus appeared between slow and fast SRT trials. This effect was evident in three different analyses, and occurred primarily in frontal and central electrodes.     

Technique for the quantification of transient quadratic phase couplings between heart rate components.

A technique for the time-variant analysis of quadratic phase coupling (QPC) in heart rate data is introduced and tested in 6 human neonates during quiet sleep. The set up of the approach is based up on the assumption that QPCs in the heart rate variability (HRV) are related to amplitude modulation effects. The application of the biamplitude deals with the detection of the coupling pattern and the bicoherence is used for the statistical quantification of coupling. By means of the results of bispectral analysis the time-variant processing has been adapted. The frequency-selective complex demodulation of the HRV leads to the envelope of the respiratory sinus arrhythmia (RSA), this has been used as one input for a time-variant coherence analysis. The other input is the low-pass filtered 10-second-rhythm of the HRV. A time-continuous quantification of the QPC, caused by amplitude modulation (10-second-rhythm modulates the RSA), is possible using this approach. According to our observed results in neonatal HRV both a phase co-ordination between the 10-second-rhythm and RSA as well as a non-linear coupling (amplitude modulation) between these HRV components can be seen.     

冲突适应的神经振荡机制

靶刺激和干扰刺激的不一致性会造成人类的行为冲突.特别地,连续的冲突情境能诱发大脑对冲突的适应.然而,目前的研究结果还不能清楚地阐释冲突适应的认知和神经机制.为了考察冲突适应的神经振荡过程,采用冲突观察实验范式,记录了15个健康成人被试在完成字母Flanker任务时的行为和脑电数据.对EEG数据进行时频分析,结果揭示了冲突适应的神经振荡机制.在观察任务中,对于不一致条件,α频带（9~13 Hz,480~980 ms）在左前额叶和中前区展现了显著的事件相关异步性（ERD）;对于一致条件,α频带在这些区域展现了显著的事件相关同步性（ERS）.在反应任务中,θ频带（6~8 Hz,50~1000 ms）的ERS在左前额叶和中前区表现为iI〈cI,反映了冲突适应.这些结果表明,冲突能诱发特定频带范围内大脑能量的调整,从而使大脑对冲突进行有效的控制.     

Coupling between Intrinsic Prefrontal HbO2 and Central EEG Beta Power Oscillations in the Resting Brain

There is increasing interest in the intrinsic activity in the resting brain, especially that of ultraslow and slow oscillations. Using near-infrared spectroscopy (NIRS), electroencephalography (EEG), blood pressure (BP), respiration and heart rate recordings during 5 minutes of rest, combined with cross spectral and sliding cross correlation calculations, we identified a short-lasting coupling (duration [Formula: see text] s) between prefrontal oxyhemoglobin (HbO2) in the frequency band between 0.07 and 0.13 Hz and central EEG alpha and/or beta power oscillations in 8 of the 9 subjects investigated. The HbO2 peaks preceded the EEG band power peaks by 3.7 s in 6 subjects, with moderate or no coupling between BP and HbO2 oscillations. HbO2 and EEG band power oscillations were approximately in phase with BP oscillations in the 2 subjects with an extremely high coupling (squared coherence [Formula: see text]) between BP and HbO2 oscillation. No coupling was identified in one subject. These results indicate that slow precentral (de)oxyhemoglobin concentration oscillations during awake rest can be temporarily coupled with EEG fluctuations in sensorimotor areas and modulate the excitability level in the brains' motor areas, respectively. Therefore, this provides support for the idea that resting state networks fluctuate with frequencies of between 0.01 and 0.1 Hz (Mantini et.al. PNAS 2007).     

A Brain-Machine Interface for Control of Medically-Induced Coma

Medically-induced coma is a drug-induced state of profound brain inactivation and unconsciousness used to treat refractory intracranial hypertension and to manage treatment-resistant epilepsy. The state of coma is achieved by continually monitoring the patient''s brain activity with an electroencephalogram (EEG) and manually titrating the anesthetic infusion rate to maintain a specified level of burst suppression, an EEG marker of profound brain inactivation in which bursts of electrical activity alternate with periods of quiescence or suppression. The medical coma is often required for several days. A more rational approach would be to implement a brain-machine interface (BMI) that monitors the EEG and adjusts the anesthetic infusion rate in real time to maintain the specified target level of burst suppression. We used a stochastic control framework to develop a BMI to control medically-induced coma in a rodent model. The BMI controlled an EEG-guided closed-loop infusion of the anesthetic propofol to maintain precisely specified dynamic target levels of burst suppression. We used as the control signal the burst suppression probability (BSP), the brain''s instantaneous probability of being in the suppressed state. We characterized the EEG response to propofol using a two-dimensional linear compartment model and estimated the model parameters specific to each animal prior to initiating control. We derived a recursive Bayesian binary filter algorithm to compute the BSP from the EEG and controllers using a linear-quadratic-regulator and a model-predictive control strategy. Both controllers used the estimated BSP as feedback. The BMI accurately controlled burst suppression in individual rodents across dynamic target trajectories, and enabled prompt transitions between target levels while avoiding both undershoot and overshoot. The median performance error for the BMI was 3.6%, the median bias was -1.4% and the overall posterior probability of reliable control was 1 (95% Bayesian credibility interval of [0.87, 1.0]). A BMI can maintain reliable and accurate real-time control of medically-induced coma in a rodent model suggesting this strategy could be applied in patient care.     

Principal component analysis of electroencephalographic activity during sleep and wakefulness in the spider monkey (Ateles geoffroyi)

The study of electroencephalographic (EEG) activity during sleep in the spider monkey has provided new insights into primitive arboreal sleep physiology and behavior in anthropoids. Nevertheless, studies conducted to date have maintained the frequency ranges of the EEG bands commonly used with humans. The aim of the present work was to determine the EEG broad bands that characterize sleep and wakefulness in the spider monkey using principal component analysis (PCA). The EEG activity was recorded from the occipital, central, and frontal EEG derivations of six young-adult male spider monkeys housed in a laboratory setting. To determine which frequencies covaried and which were orthogonally independent during sleep and wakefulness, the power EEG spectra and interhemispheric and intrahemispheric EEG correlations from 1 to 30 Hz were subjected to PCA. Findings show that the EEG bands detection differed from those reported previously in both spider monkeys and humans, and that the 1–3 and 2–13 Hz frequency ranges concur with the oscillatory activity elucidated by cellular recordings of subcortical regions. Results show that applying PCA to the EEG spectrum during sleep and wakefulness in the spider monkey led to the identification of frequencies that covaried with, and were orthogonally independent of, other frequencies in each behavioral vigilance state. The new EEG bands differ from those used previously with both spider monkeys and humans. The 1–3 and 2–13 Hz frequency ranges are in accordance with the oscillatory activity elucidated by cellular recordings of subcortical regions in other mammals.     

Investigation of the modulation between EEG alpha waves and slow/fast delta waves in children in different depths of Desflurane anesthesia

ObjectivesInvestigation of the amplitude modulation of alpha-band EEG oscillations (i.e., grouping of alpha-band activities) by delta-band EEG activities in various depths of anesthesia (DOA).MethodsThis modulation, which is a sort of phase dependent amplitude modulation, is studied in 10 children in various depths of Desflurane anesthesia. Two parameters are defined to quantify the modulation: strength of modulation (SOM) and phase of modulation (POM). SOM indicates to what extent delta and alpha activities are related to each other, and POM is the delta phase in which the alpha amplitude is maximal. These parameters are analyzed in different DOA for various formations of delta sub-bands.ResultsThe ability of POM and SOM were explored to characterize mechanisms contributing to delta activities and their correlation with the level of anesthesia. These parameters are influenced by DOA and frequency intervals of delta sub-bands. SOM takes higher values around certain frequency ranges of delta band. According to this, delta band comprises three main sub-bands in various unconsciousness levels. Although boundaries of these sub-bands change with DOA, they are almost in [0.1–0.5] Hz (very slow delta), [0.7–1.7] Hz (slow delta) and [2–4] Hz (fast delta) intervals. POMs relating to slow and fast delta bands increase with consciousness level. This is an indication that delta waves differently modulate alpha EEG activities (in terms of phase lag) in different DOA. In deep anesthesia, POM relating to fast delta correlates with DOA better than POM relating to slow delta does. In light anesthesia this correlation is inversed. Investigation regarding to different formations of delta sub-bands shows that POM relating to [1.8–4] Hz is a proper choice for distinguishing deep, moderate and light anesthesia.ConclusionSOM allows separating mechanisms underlying delta band activities, and POM can be seen as a complementary neurophysiologic-based parameter for quantifying DOA.     

Dynamic characteristics of the human resonance EEG responses to rhythmic photostimulation

The development of the resonance EEG responses of the left and right occipital areas was studied in right-handed men during prolonged (12 or 120 s) rhythmic, photostimulation with the intensity of 0.7 J and frequencies of 6, 10, and 16 Hz. Analysis of the EEG fine spectral structure was applied to compare the accumulated baseline EEG spectra and EEG spectra during photostimulation, to observe the dynamics of the short-term spectra and to detect power changes in the EEG narrow spectral band sharply coincident with the stimulation frequency. The more pronounced EEG responses to photostimulation were observed in subjects with the initially low EEG baseline, α-rhythm. Two-minute flash trains produced a substantial increase in the EEG power within the stimulation frequency with superposed oscillatory processes with different periods. These fluctuations are considered a reflection of intricate interaction between the adaptive and resonance EEG responses to the presented intermittent stimulation. Under 12-s stimulation the resonance EEG responses are steadily recorded within the first 3 s of stimulation and immediately after the flash cessation EEG power at the stimulation frequency returns to the initial level. The resonance EEG responses were more pronounced in the right hemisphere than in the left one, especially, at the stimulation frequencies of 6 and 16 Hz. With increasing the stimulation frequency, the maximum of resonance EEG responses was reached earlier. Under the stimulation frequency of 6 Hz, the maximal response was recorded 9–12 s after the beginning of flashes, at the frequencies of 10 and 16 Hz, it was recorded within 3–6 and 3 s, respectively.     

Test of statistical stability of the electroencephalogram

The stability of the EEG during eyes closed and eyes open were tested by the statistical method. The EEG records of 25 s were segmented into ten 2.5 s intervals for each state. The mean and variance and power spectra (2.5 to 17.5 Hz at resolution of 2.5 Hz) were calculated for each segment of 2.5 s interval. Thus, the sequences of mean values, variances and power at each frequency component were obtained for 25 s epoch. The stationarity of these sequences were tested by the run test and the trend test. The stationarity of mean, variance and power spectra were not rejected for 25 s records by any of two tests. In the records of 50 s, about 10–20% of records tested were rejected. The nonstationarity of the EEG appeared for 50 s records. This means that the EEG during period the eyes were closed, or opened can be regarded as stationary over time period as long as at least 25 s.     

Decapitation in rats: latency to unconsciousness and the 'wave of death'

The question whether decapitation is a humane method of euthanasia in awake animals is being debated. To gather arguments in this debate, obsolete rats were decapitated while recording the EEG, both of awake rats and of anesthetized rats. Following decapitation a fast and global loss of power of the EEG was observed; the power in the 13-100 Hz frequency band, expressing cognitive activity, decreased according to an exponential decay function to half the initial value within 4 seconds. Whereas the pre-decapitation EEG of the anesthetized animals showed a burst suppression pattern quite different from the awake animals, the power in the postdecapitation EEG did not differ between the two groups. This might indicate that either the power of the EEG does not correlate well with consciousness or that consciousness is briefly regained in the anesthetized group after decapitation. Remarkably, after 50 seconds (awake group) or 80 seconds (anesthetized group) following decapitation, a high amplitude slow wave was observed. The EEG before this wave had more power than the signal after the wave. This wave might be due to a simultaneous massive loss of membrane potentials of the neurons. Still functioning ion channels, which keep the membrane potential intact before the wave, might explain the observed power difference. Two conclusions were drawn from this experiment. It is likely that consciousness vanishes within seconds after decapitation, implying that decapitation is a quick and not an inhumane method of euthanasia. It seems that the massive wave which can be recorded approximately one minute after decapitation reflects the ultimate border between life and death. This observation might have implications in the discussions on the appropriate time for organ donation.     

Gamma Power Is Phase-Locked to Posterior Alpha Activity

Neuronal oscillations in various frequency bands have been reported in numerous studies in both humans and animals. While it is obvious that these oscillations play an important role in cognitive processing, it remains unclear how oscillations in various frequency bands interact. In this study we have investigated phase to power locking in MEG activity of healthy human subjects at rest with their eyes closed. To examine cross-frequency coupling, we have computed coherence between the time course of the power in a given frequency band and the signal itself within every channel. The time-course of the power was calculated using a sliding tapered time window followed by a Fourier transform. Our findings show that high-frequency gamma power (30–70 Hz) is phase-locked to alpha oscillations (8–13 Hz) in the ongoing MEG signals. The topography of the coupling was similar to the topography of the alpha power and was strongest over occipital areas. Interestingly, gamma activity per se was not evident in the power spectra and only became detectable when studied in relation to the alpha phase. Intracranial data from an epileptic subject confirmed these findings albeit there was slowing in both the alpha and gamma band. A tentative explanation for this phenomenon is that the visual system is inhibited during most of the alpha cycle whereas a burst of gamma activity at a specific alpha phase (e.g. at troughs) reflects a window of excitability.     

Coupled oscillators for modeling and analysis of EEG/MEG oscillations.

This study presents three EEG/MEG applications in which the modeling of oscillatory signal components offers complementary analysis and an improved explanation of the underlying generator structures. Coupled oscillator networks were used for modeling. Parameters of the corresponding ordinary coupled differential equation (ODE) system are identified using EEG/MEG data and the resulting solution yields the modeled signals. This model-related analysis strategy provides information about the coupling quantity and quality between signal components (example 1, neonatal EEG during quiet sleep), allows identification of the possible contribution of hidden generator structures (example 2, 600-Hz MEG oscillations in somatosensory evoked magnetic fields), and can explain complex signal characteristics such as amplitude-frequency coupling and frequency entrainment (example 3, EEG burst patterns in sedated patients).     

Detection of simple and pattern regularity violations occurs at different levels of the auditory hierarchy

Auditory deviance detection in humans is indexed by the mismatch negativity (MMN), a component of the auditory evoked potential (AEP) of the electroencephalogram (EEG) occurring at a latency of 100-250 ms after stimulus onset. However, by using classic oddball paradigms, differential responses to regularity violations of simple auditory features have been found at the level of the middle latency response (MLR) of the AEP occurring within the first 50 ms after stimulus (deviation) onset. These findings suggest the existence of fast deviance detection mechanisms for simple feature changes, but it is not clear whether deviance detection among more complex acoustic regularities could be observed at such early latencies. To test this, we examined the pre-attentive processing of rare stimulus repetitions in a sequence of tones alternating in frequency in both long and middle latency ranges. Additionally, we introduced occasional changes in the interaural time difference (ITD), so that a simple-feature regularity could be examined in the same paradigm. MMN was obtained for both repetition and ITD deviants, occurring at 150 ms and 100 ms after stimulus onset respectively. At the level of the MLR, a difference was observed between standards and ITD deviants at the Na component (20-30 ms after stimulus onset), for 800 Hz tones, but not for repetition deviants. These findings suggest that detection mechanisms for deviants to simple regularities, but not to more complex regularities, are already activated in the MLR range, supporting the view that the auditory deviance detection system is organized in a hierarchical manner.     

Activation phase ensures kinematic efficacy in flight-steering muscles of Drosophila melanogaster

During tethered flight in Drosophila melanogaster, spike activity of the second basalar flight-control muscle (M.b2) is correlated with an increase in both the ipsilateral wing beat amplitude and the ipsilateral flight force. The frequency of muscle spikes within a burst is about 100 Hz, or 1 spike for every two wing beat cycles. When M.b2 is active, its spikes tend to occur within a comparatively narrow phase band of the wing beat cycle. To understand the functional role of this phase-lock of firing in the control of flight forces, we stimulated M.b2 in selected phases of the wing beat cycle and recorded the effect on the ipsilateral wing beat amplitude. Varying the phase timing of the stimulus had a significant effect on the wing beat amplitude. A maximum increase of wing beat amplitude was obtained by stimulating M.b2 at the beginning of the upstroke or about 1 ms prior to the narrow phase band in which the muscle spikes typically occur during flight. Assuming a delay of 1 ms between the stimulation of the motor nerve and muscle activation, these results indicate that M.b2 is activated at an instant of the stroke cycle that produces the greatest effect on wing beat amplitude.     

Effect of alexithymia on the event-related theta-synchronization of human EEG during reception of emotional visual stimuli

The 62-channel EEG was recorded while control non-alexithymic (n = 21) and alexithymic (n = 20) participants viewed sequentially presented neutral, pleasant and unpleasant pictures and subjectively rated them after each presentation. The event-related synchronization (ERS) to these stimuli was assessed in the theta-1 (4-6 Hr) and theta-2 (6-8 Hz) frequency bands. The obtained findings indicate that alexithymia influences perception of only emotional stimuli. In the upper theta over anterior cortical regions alexithymia vs control individuals in response to both pleasant and unpleasant stimuli manifested decreased left hemisphere ERS in the early test period of 0-200 ms along with enhanced ERS in response to negative vs positive and neutral stimuli in the right hemisphere at 200-600 ms after stimulus onset. The findings provide the first EEG evidence that alexithymia construct, associated with a cognitive deficit in initial evaluation of emotion, is indexed by disrupted early frontal synchronization in the upper theta band that can be best interpreted to reflect disregulation during appraisal of emotional stimuli.     

On the synchrony of steady state visual evoked potentials and oscillatory burst events

In this paper, we investigate the large-scale synchrony of EEG oscillatory bursts, during stimulation by a flickering square of light. Whereas most studies focus on averaged raw EEG responses, this study considers oscillatory events within EEG of single trials, which leads to various new insights. We recorded EEG signals before, during and after stimulation by a flickering square of light in medium (16 Hz) and high frequency (32 Hz) ranges. Similar oscillatory bursts, to those observed in spontaneous EEG, can be found in single-trial synchrony of steady state visual evoked potentials (SSVEP). These bursts are extracted from the EEG of single trials using bump modeling. Stochastic event synchrony method is applied to those events, which quantifies synchronies of oscillatory bursts on a large-scale basis. Those oscillatory patterns have a significantly higher degree of co-occurrence during SSVEP, uncorrelated with ongoing signal synchrony. It means that EEG oscillatory patterns are presumably an outcome of brain activity, rather than a mere side effect of ongoing EEG. They undergo a consistent reorganization during visual stimulation, preferentially along the visual pathway, depending on magno or parvo stimulations. Flickering stimuli may induce some cognitive side-effects depending on the stimulation frequency.

Modulation of the frequency of glucose-dependent bursts of electrical activity by HCO3/CO2 in rodent pancreatic B-cells: experimental and theoretical results

The burst pattern of electrical activity recorded from pancreatic B-cells in response to 11 mM glucose shows a large islet to islet variability. The relationship between burst frequency and glucose sensing (the threshold for electrical activity and the graded increase in electrical response to glucose, i.e. active phase %) has not been investigated within the same islet. In this work, we show that low HCO₃ (5 mM) Hepes buffered solutions reversibly reduce the frequency of bursts compared to control (25 mM) HCO₃ buffered solutions in the same islet. There was no change in the threshold or active phase (%). Using the mathematical model of Sherman et al. 1988, we explored mechanisms for a change in frequency independent of a change in active phase (%). Increased exchangeable calcium pool size and increased cell to cell coupling were the two theoretical treatments which could reproduce the experimental data. We conclude that burst frequency can be modulated independent of the active phase and that alteration of a calcium pool size best fits the experimental data. Offprint requests to: P. B. Carroll