Integer-ratio entrainment in mutually-coupled non-linear oscillators

In recent years entrainment conditions for mutually-coupled, non-linear oscillators have been studied for a number of biomedical applications and using different analytical methods. The emphasis has been on entrainment between oscillators of similar frequencies. In this paper entrainment conditions are considered for oscillators having intrinsic frequency ratio of about 3:1 and which exhibit integer-ratio synchronization. This condition has application in the study of blood pressure regulation particularly in relation to respiratory effects. Coupling has been observed between respiration and the vasomotor activity associated with the baroreceptor reflex, which has an intrinsic 0·1 Hz component. At normal breathing frequencies the frequency ratio of the respiratory and vasomotor components is in the region of 3:1 hence integer-ratio entrainment is feasible. Using a coupled van der Pol model the entrainment zones for different parameters are described. The parameters considered allow for varying amounts of output, output rate and delay in the intercoupling structure. In particular, it is shown that the entrainment regions are strongly affected by the nature of the coupling. Within these zones the harmonic balance method is developed to provide an analytical solution to frequency, amplitudes and phase conditions. The assumed solution is valid only for certain regions of the stability zones and the reason for this is demonstrated and the means whereby this can be overcome are indicated.     

Frequency modulation of theta volleys of septal neurons during rhythmic oligosynaptic stimulation in the rabbit

Activity of the neurones with stable theta-bursts was recorded extracellularly in intact and hippocampectomized septum of unanaesthetized chronic rabbits during low-frequency (3-17 Hz) stimulation of horizontal limb of the diagonal band or the lateral septal nucleus. Gradual entrainment and phase-locking of the spontaneous theta-cycles occurred. Two types of entrainment were observed: "entrainment by pause", where interburst interval was reset by the stimuli; and "entrainment by burst", where bursts were time-locked to the stimuli. Such reorganization of the spontaneous bursts occurred in a narrow frequency range of stimulation (from 4 Hz up to 9-12 Hz), with the best resonance following in the range of "basic" theta frequencies of the awake rabbit (5-6 Hz). With stimulation beyond the theta-range three phenomena occurred: shift of the burst frequencies to higher or lower harmonics of stimulation frequencies; complex interactions of basic background frequency with the rhythm of stimulation ("beating"); escape from the influence of the stimuli with return to background theta-burst frequency.     

Hi-Fi transmission of periodic signals amid cell-to-cell variability

Since information in intracellular calcium signaling is often frequency encoded, it is physiologically critical and experimentally useful to have reliable, convenient, and non-invasive methods to entrain it. Because of cell-to-cell variability, synchronization of intracellular signaling across a population of genetically identical cells can still be difficult to achieve. For intrinsically oscillatory signaling pathways, such as calcium, upon continuous stimulation, cell-to-cell variability is manifested as differences in intracellular response frequencies. Even with entrainment using periodic stimulation, cell-to-cell variability is manifested as differences in the fidelity with which extracellular inputs are converted into intracellular signals. Here we present a combined theoretical and experimental analysis that shows how to appropriately balance stimulation strength, duration, and rest intervals to achieve entrainment with high fidelity stimulation-to-response ratios for G-protein-coupled receptor-triggered intracellular calcium oscillations. We further demonstrate that stimulation parameters that give high fidelity entrainment are significantly altered upon changes in intracellular enzyme levels and cell surface receptor levels. Theoretical analysis suggests that, at key threshold values, even small changes in these protein concentrations or activities can result in precipitous changes in entrainment fidelity, with implications for pathophysiology.     

Frequency-dependent response of SI RA-class neurons to vibrotactile stimulation of the receptive field

Three types of experiment were carried out on anesthetized monkeys and cats. In the first, spike discharge activity of rapidly adapting (RA) SI neurons was recorded extracellularly during the application of different frequencies of vibrotactile stimulation to the receptive field (RF). The second used the same stimulus conditions to study the response of RA-I (RA) cutaneous mechanoreceptive afferents. The third used optical intrinsic signal (OIS) imaging and extracellular neurophysiological recording methods together, in the same sessions, to evaluate the relationship between the SI optical and RA neuron spike train responses to low- vs high-frequency stimulation of the same skin site. RA afferent entrainment was high at all frequencies of stimulation. In contrast, SI RA neuron entrainment was much lower on average, and was strongly frequency-dependent, declining in near-linear fashion from 6 to 200 Hz. Even at 200 Hz, however, unambiguous frequencyfollowing responses were present in the spike train activity of some SI RA neurons. These entrainment results support the "periodicity hypothesis" of Mountcastle et al. ( J Neurophysiol 32: 452-484, 1969) that the capacity to discriminate stimulus frequency over the range 5-50 Hz is attributable to the ability of SI RA pyramidal neurons to discharge action potentials in consistent temporal relationship to stimulus motion, and raise the possibility that perceptual frequency discriminative capacity at frequencies between 50 and 200 Hz might be accounted for in the same way. An increase in vibrotactile stimulus frequency within the range 6-200 Hz consistently resulted in an increase in RA afferent mean spike firing rate (M FR). SI RA neuron M FR also increased as frequency increased between 6 and 50 Hz, but declined as stimulus frequency was increased over the range 50-200 Hz. At stimulus frequencies > 100 Hz, and at positions in the RF other than the receptive field center (RF center ), SI RA neuron MFR declined sharply within 0.5-2s of stimulus onset and rebounded transiently upon stimulus termination. In contrast, when the stimulus was applied to the RF center, MFR increased with increasing frequency and tended to remain well maintained throughout the period of high-frequency stimulation. The evidence obtained in "combined" OIS imaging and extracellular microelectrode recording experiments suggests that SI RA neurons with an RF center that corresponds to the stimulated skin site occupy small foci within the much larger SI region activated by same-site cutaneous flutter stimulation, while for the RA neurons located elsewhere in the large SI region activated by a flutter stimulus, the stimulus site and RF center are different.     

A high-density EEG investigation into steady state binaural beat stimulation

Binaural beats are an auditory phenomenon that has been suggested to alter physiological and cognitive processes including vigilance and brainwave entrainment. Some personality traits measured by the NEO Five Factor Model have been found to alter entrainment using pulsing light stimuli, but as yet no studies have examined if this occurs using steady state presentation of binaural beats for a relatively short presentation of two minutes. This study aimed to examine if binaural beat stimulation altered vigilance or cortical frequencies and if personality traits were involved. Thirty-one participants were played binaural beat stimuli designed to elicit a response at either the Theta (7 Hz) or Beta (16 Hz) frequency bands while undertaking a zero-back vigilance task. EEG was recorded from a high-density electrode cap. No significant differences were found in vigilance or cortical frequency power during binaural beat stimulation compared to a white noise control period. Furthermore, no significant relationships were detected between the above and the Big Five personality traits. This suggests a short presentation of steady state binaural beats are not sufficient to alter vigilance or entrain cortical frequencies at the two bands examined and that certain personality traits were not more susceptible than others.     

Entrainment ranges of forced phase oscillators

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest–neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest–neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.     

Evidence for the entrainment of breathing by locomotor pattern in human

In human, it has been shown that interactions between locomotor and respiratory patterns may lead to locomotor-respiratory couplings termed entrainment. In order to prove that this coupling is really an entrainment, we tried to show that it obeys one of the expected rules, i.e. that it evolves and is not present for all imposed locomotor frequencies. For that purpose, seventeen healthy volunteers were asked to run on a treadmill at 14 different locomotor rates (instead of 2 or 3 in previous works) for 40 s. All the subjects did not exhibit the same coupling and different relationships could be obtained: the most commonly observed was 2:1 (2 locomotor activities for a respiratory one) but other forms could appear (4:1 and even 5:2 or 3:2). When the coupling evolution was followed in the same subject, it did not appear for all locomotor frequencies but only for locomotor periods close to harmonics of respiratory ones (absolute coordination). On both sides of these values, it progressively evolved to relative coordination and to the lack of coordination. When two forms of absolute coordination were observed in a same subject, the phase relationships followed the rules of the entrainment. Compared to data obtained in quadrupeds, these results suggest that the entrainment of breathing frequency by the locomotor activity is due to central interactions between the respiratory and locomotor pattern generators and does not depend on a chemical regulation avoided here by short locomotor sequences.     

Spatiotemporally controlled cardiac conduction block using high-frequency electrical stimulation

Background

Methods for the electrical inhibition of cardiac excitation have long been sought to control excitability and conduction, but to date remain largely impractical. High-amplitude alternating current (AC) stimulation has been known to extend cardiac action potentials (APs), and has been recently exploited to terminate reentrant arrhythmias by producing reversible conduction blocks. Yet, low-amplitude currents at similar frequencies have been shown to entrain cardiac tissues by generation of repetitive APs, leading in some cases to ventricular fibrillation and hemodynamic collapse in vivo. Therefore, an inhibition method that does not lead to entrainment – irrespective of the stimulation amplitude (bound to fluctuate in an in vivo setting) – is highly desirable.

Methodology/Principal Findings

We investigated the effects of broader amplitude and frequency ranges on the inhibitory effects of extracellular AC stimulation on HL-1 cardiomyocytes cultured on microelectrode arrays, using both sinusoidal and square waveforms. Our results indicate that, at sufficiently high frequencies, cardiac tissue exhibits a binary response to stimulus amplitude with either prolonged APs or no effect, thereby effectively avoiding the risks of entrainment by repetitive firing observed at lower frequencies. We further demonstrate the ability to precisely define reversible local conduction blocks in beating cultures without influencing the propagation activity in non-blocked areas. The conduction blocks were spatiotemporally controlled by electrode geometry and stimuli duration, respectively, and sustainable for long durations (300 s).

Conclusion/Significance

Inhibition of cardiac excitation induced by high-frequency AC stimulation exhibits a binary response to amplitude above a threshold frequency, enabling the generation of reversible conduction blocks without the risks of entrainment. This inhibition method could yield novel approaches for arrhythmia modeling in vitro, as well as safer and more efficacious tools for in vivo cardiac mapping and radio-frequency ablation guidance applications.     

Modulation of Hydra attenuata rhythmic activity. Photic stimulation.

We investigated in Hydra attenuata the possibility of altering more or less permanently and in different environmental conditions, the frequency of Contraction Pulse Trains (CPT's) associated with the rhythmic spontaneous contraction activity, by repetitive light stimuli of variable duration, frequency and amplitude. The CPT's activity of various pieces of Hydra has been also investigated in indisturbed conditions and under stimulation. The following observations have been performed. 1. A transient effect, consisting of an increase or a decrease of CPT's frequency, occurs respectively after an abrupt decrease or increase of the light level. 2. If Hydra is stimulated by repetitive light pulses of 0.5-10 sec duration, at a frequency different from the CPT's average one, the CPT's frequency modifies; if the stimulation frequency is included in a range not too much up or below that of CPT's the new CPT's frequency equals exactly that of stimulation; close to this range the CPT's frequency is a multiple or submultiple of that of stimulation. 3. No habituation to such repetitive stimulation was found. 4. The phase relation between CPT's at the new frequency and light stimuli is a function of the difference between CPT's and stimulation frequencies. 5. Stimulation with repetition of light and darkness periods of some minutes duration induces activity only or mainly during darkness. 6. Modification of CPT's frequency by means of repetitive light stimulation [of the type mentioned either in 2) or 5)] has been observed also with hypostomal preparations. 7. With cessation of the light stimulation, the new CPT's frequency of the whole animal lasts in darkness for a time (10-85 min) that is about 5-10 times longer than that necessary to obtain CPT's syncronization with stimulation. 8. The influence of the light intensity level on transient CPT's frequency variation (see 1), CPT's inhibition and stimulation, promptness of entrainment, range of entrainability, phase relation between entrained CPT's and stimuli, retention time of entrained rhythm has been examined, together with the influence of the reversal of polarity of light transitions on CPT's inhibition and entrainment.     

Vagal feedback in the entrainment of respiration to mechanical ventilation in sleeping humans.

We studied the capacity of four "normal" and six lung transplant subjects to entrain neural respiratory activity to mechanical ventilation. Two transplant subjects were studied during wakefulness and demonstrated entrainment indistinguishable from that of normal awake subjects. We studied four normal subjects and four lung transplant subjects during non-rapid eye movement (NREM) sleep. Normal subjects entrained to mechanical ventilation over a range of ventilator frequencies that were within +/-3-5 breaths of the spontaneous respiratory rate of each subject. After lung transplantation, during which the vagi were cut, subjects did demonstrate entrainment during NREM sleep; however, entrainment only occurred at ventilator frequencies at or above each subject's spontaneous respiratory rate, and entrainment was less effective. We conclude that there is no absolute requirement for vagal feedback to induce entrainment in subjects, which is in striking contrast to anesthetized animals in which vagotomy uniformly abolishes entrainment. On the other hand, vagal feedback clearly enhances the fidelity of entrainment and extends the range of mechanical frequencies over which entrainment can occur.     

Alpha-Band Rhythms in Visual Task Performance: Phase-Locking by Rhythmic Sensory Stimulation

Oscillations are an important aspect of neuronal activity. Interestingly, oscillatory patterns are also observed in behaviour, such as in visual performance measures after the presentation of a brief sensory event in the visual or another modality. These oscillations in visual performance cycle at the typical frequencies of brain rhythms, suggesting that perception may be closely linked to brain oscillations. We here investigated this link for a prominent rhythm of the visual system (the alpha-rhythm, 8–12 Hz) by applying rhythmic visual stimulation at alpha-frequency (10.6 Hz), known to lead to a resonance response in visual areas, and testing its effects on subsequent visual target discrimination. Our data show that rhythmic visual stimulation at 10.6 Hz: 1) has specific behavioral consequences, relative to stimulation at control frequencies (3.9 Hz, 7.1 Hz, 14.2 Hz), and 2) leads to alpha-band oscillations in visual performance measures, that 3) correlate in precise frequency across individuals with resting alpha-rhythms recorded over parieto-occipital areas. The most parsimonious explanation for these three findings is entrainment (phase-locking) of ongoing perceptually relevant alpha-band brain oscillations by rhythmic sensory events. These findings are in line with occipital alpha-oscillations underlying periodicity in visual performance, and suggest that rhythmic stimulation at frequencies of intrinsic brain-rhythms can be used to reveal influences of these rhythms on task performance to study their functional roles.     

Modulation of signal transmission by the septal neurons of the rabbit during action on the cholinergic system

The neurons of the medial septum-diagonal band complex (MS-DB) were examined extracellularly in chronic unanaesthetized rabbits. Low-frequency (3-21 Hz) electrical stimulation was applied to the medial forebrain bundle within the horizontal limb of DB. In a group of the MS-DB units with stable background theta bursts the typical response consisting of entrainment of the phase-locked theta cycles was changed neither by physostigmine, nor by cholinergic-blocking drugs (scopolamine and atropine). In major group of the MS-DB units (60%), physostigmine completely blocked effects of electrical stimulation. This occurred both in the units with entrainment of the theta cycles and in the units with the effects of primary suppression of activity or (to a lesser extent) with single-spike discharges following the repetitive stimuli up to high frequencies. The cholinergic-blocking drugs restored and sometimes increased the initial reactivity of the MS-DB units. It is suggested, that intraseptal cholinergic system exerts a powerful gating effect upon transmission of signals to the hippocampus, probably, by the mechanisms of presynaptic inhibition.     

Mutual entrainment of two pacemaker cells. A study with an electronic parallel conductance model

Repetitive firing of pacemaker cells was simulated with the use of an electronic Hodgkin-Huxley-like membrane model having a capacitance, a sodium-, a potassium- and a leakage conductance connected in parallel. The frequency of firing of the model cells was controlled by the leakage conductance. Two such model cells of sometimes widely different intrinsic frequencies were coupled through a coupling resistance R_c. Mutual synchronization was allowed to occur by decreasing R_c in small steps from high to low values.The lower R_c, the more the ratio of the mean frequencies of both cells approaches unity. Around certain R_c values stable entrainment occurs during which the mean frequencies are related as simple integral values m: n (e.g. 2:1, 3:2, 4:3, 1:1 for decreasing R_c). Within the range of a fixed m: n entrainment the phase difference between the oscillations of the two cells declines with R_c. When R_c is close to the range ofm: n entrainment, “almost entrainment” occurs in which m: n entrainment is periodic. But also further away from the range of m: n entrainment periodic variations in action potential interval occur.The model cells were adjusted in such a way that a depolarizing current pulse can only advance the first occurring and all subsequent action potentials. This property is used to explain effects of R_c on the mean frequency of the two cells both within and without the R_c ranges of entrainment. Furthermore, it is illustrated how in certain cases the synchronized frequency at R_c = 0 ω follows from simple electrical considerations of the properties of the uncoupled cells.Several of the entrainment phenomena observed are well known in the fields of electronic relaxation oscillators, clinical electrocardiography and circadian rhythms, though with different terminologies. The present model results exemplify the general nature of these phenomena and clarify entrainment phenomena, recently observed in coupling experiments with pacemaker cells from embryonic heart tissue.     

Electrical Brain Responses to an Auditory Illusion and the Impact of Musical Expertise

The presentation of two sinusoidal tones, one to each ear, with a slight frequency mismatch yields an auditory illusion of a beating frequency equal to the frequency difference between the two tones; this is known as binaural beat (BB). The effect of brief BB stimulation on scalp EEG is not conclusively demonstrated. Further, no studies have examined the impact of musical training associated with BB stimulation, yet musicians'' brains are often associated with enhanced auditory processing. In this study, we analysed EEG brain responses from two groups, musicians and non-musicians, when stimulated by short presentation (1 min) of binaural beats with beat frequency varying from 1 Hz to 48 Hz. We focused our analysis on alpha and gamma band EEG signals, and they were analysed in terms of spectral power, and functional connectivity as measured by two phase synchrony based measures, phase locking value and phase lag index. Finally, these measures were used to characterize the degree of centrality, segregation and integration of the functional brain network. We found that beat frequencies belonging to alpha band produced the most significant steady-state responses across groups. Further, processing of low frequency (delta, theta, alpha) binaural beats had significant impact on cortical network patterns in the alpha band oscillations. Altogether these results provide a neurophysiological account of cortical responses to BB stimulation at varying frequencies, and demonstrate a modulation of cortico-cortical connectivity in musicians'' brains, and further suggest a kind of neuronal entrainment of a linear and nonlinear relationship to the beating frequencies.     

Differential Entrainment of Neuroelectric Delta Oscillations in Developmental Dyslexia

Oscillatory entrainment to the speech signal is important for language processing, but has not yet been studied in developmental disorders of language. Developmental dyslexia, a difficulty in acquiring efficient reading skills linked to difficulties with phonology (the sound structure of language), has been associated with behavioural entrainment deficits. It has been proposed that the phonological ‘deficit’ that characterises dyslexia across languages is related to impaired auditory entrainment to speech at lower frequencies via neuroelectric oscillations (<10 Hz, ‘temporal sampling theory’). Impaired entrainment to temporal modulations at lower frequencies would affect the recovery of the prosodic and syllabic structure of speech. Here we investigated event-related oscillatory EEG activity and contingent negative variation (CNV) to auditory rhythmic tone streams delivered at frequencies within the delta band (2 Hz, 1.5 Hz), relevant to sampling stressed syllables in speech. Given prior behavioural entrainment findings at these rates, we predicted functionally atypical entrainment of delta oscillations in dyslexia. Participants performed a rhythmic expectancy task, detecting occasional white noise targets interspersed with tones occurring regularly at rates of 2 Hz or 1.5 Hz. Both groups showed significant entrainment of delta oscillations to the rhythmic stimulus stream, however the strength of inter-trial delta phase coherence (ITC, ‘phase locking’) and the CNV were both significantly weaker in dyslexics, suggestive of weaker entrainment and less preparatory brain activity. Both ITC strength and CNV amplitude were significantly related to individual differences in language processing and reading. Additionally, the instantaneous phase of prestimulus delta oscillation predicted behavioural responding (response time) for control participants only.     

The entrainment of breathing rhythm to stride frequency in the dik-dik antelope (Rhynchotragus kirki) with observations on the thermoregulatory consequences

1. Three adult dik-dik antelopes with an average weight of 4.5 kg were used to investigate the rate of oxygen consumption (VO2) during exercise and the entrainment of respiratory rate with stride frequencies at running velocities between 2-11 km/hr. 2. The results of VO2 and the metabolic cost of horizontal locomotion were found to agree with what would be expected of an animal of this size. 3. Respiratory frequencies recorded during the most strenuous exercise were approximately 50% below the value observed when dik-diks are exposed to an ambient temperature (Ta) of 42 degrees C. 4. Respiratory evaporative heat loss was estimated to account for only 4% of the total heat production during exercise. 5. Respiratory frequencies were found to be entrained quite strongly to stride frequencies. The thermoregulatory consequences of this entrainment is discussed.     

The Effects of Alpha (10-Hz) and Beta (22-Hz) “Entrainment” Stimulation on the Alpha and Beta EEG Bands: Individual Differences Are Critical to Prediction of Effects

Two different groups of normal college students were formed: One (the alpha group) received 10-Hz audiovisual (AV) stimulation for 8 minutes, and the other (beta) group received 22-Hz AV stimulation for 8 minutes. EEG power in the alpha (8-13 Hz) and beta (13-30 Hz) bands was FFT-extracted before, during, and for 24 minutes after stimulation. It was found that baseline (prestimulation) alpha and beta power predict the effects of stimulation, leading to individual differences in responsivity. High-baseline alpha participants showed either no entrainment or relatively prolonged entrainment with alpha stimulation. Low-baseline participants showed transient entrainment. Baseline alpha also predicted the direction of change in alpha with beta stimulation. Baseline beta and alpha predicted beta band response to beta stimulation, which was transient enhancement in some participants, inhibition in others. Some participants showed relatively prolonged beta enhancement with beta stimulation.     

Steady-state visual evoked potentials can be explained by temporal superposition of transient event-related responses

Background

One common criterion for classifying electrophysiological brain responses is based on the distinction between transient (i.e. event-related potentials, ERPs) and steady-state responses (SSRs). The generation of SSRs is usually attributed to the entrainment of a neural rhythm driven by the stimulus train. However, a more parsimonious account suggests that SSRs might result from the linear addition of the transient responses elicited by each stimulus. This study aimed to investigate this possibility.

Methodology/Principal Findings

We recorded brain potentials elicited by a checkerboard stimulus reversing at different rates. We modeled SSRs by sequentially shifting and linearly adding rate-specific ERPs. Our results show a strong resemblance between recorded and synthetic SSRs, supporting the superposition hypothesis. Furthermore, we did not find evidence of entrainment of a neural oscillation at the stimulation frequency.

Conclusions/Significance

This study provides evidence that visual SSRs can be explained as a superposition of transient ERPs. These findings have critical implications in our current understanding of brain oscillations. Contrary to the idea that neural networks can be tuned to a wide range of frequencies, our findings rather suggest that the oscillatory response of a given neural network is constrained within its natural frequency range.     

Clustered Desynchronization from High-Frequency Deep Brain Stimulation

While high-frequency deep brain stimulation is a well established treatment for Parkinson’s disease, its underlying mechanisms remain elusive. Here, we show that two competing hypotheses, desynchronization and entrainment in a population of model neurons, may not be mutually exclusive. We find that in a noisy group of phase oscillators, high frequency perturbations can separate the population into multiple clusters, each with a nearly identical proportion of the overall population. This phenomenon can be understood by studying maps of the underlying deterministic system and is guaranteed to be observed for small noise strengths. When we apply this framework to populations of Type I and Type II neurons, we observe clustered desynchronization at many pulsing frequencies.