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Saccadic oscillations threaten clear vision by causing image motion on the retina. They are either purely horizontal (ocular flutter) or multidimensional (opsoclonus). We propose that ion channel dysfunction in the burst cell membrane is the underlying abnormality. We have tested this hypothesis by simulating a neuromimetic computational model of the burst neurons. This biologically realistic model mimics the physiologic properties and anatomic connections in the brainstem saccade generator. A rebound firing after sustained inhibition, called post-inhibitory rebound (PIR), and reciprocal inhibition between premotor saccadic burst neurons are the key features of this conceptual scheme. PIR and reciprocal inhibition make the circuits that generate the saccadic burst inherently unstable and can lead to oscillations unless stabilized by external inhibition. Our simulations suggest that alterations in membrane properties that lead to an increase in PIR, a reduction in external glycinergic inhibition, or both can cause saccadic oscillations.

Saccadic eye movements rapidly orient the line of sight towards the object of interest. Pre-motor burst neurons (BNs) controlling saccades receive excitation from superior colliculus and cerebellum, but inhibition by omnipause neurons (OPNs) prevents saccades. When the OPNs pause, BNs begin to fire. It has been presumed that part of the BN burst comes from post-inhibitory rebound (PIR). We hypothesized that in the absence of prior inhibition from OPNs there would be no PIR, and thus the increase in initial firing rate of BNs would be reduced. Consequently, saccade acceleration would be reduced. We measured eye movements and showed that sustained eye closure, which inhibits the activity of OPNs and thus hypothetically should weaken PIR, reduced the peak velocity, acceleration, and deceleration of saccades in healthy human subjects. Saccades under closed eyelids also had irregular trajectories; the frequency of the oscillations underlying this irregularity was similar to that of high-frequency ocular flutter (back-to-back saccades) often seen in normal subjects during attempted fixation at straight ahead while eyes are closed. Saccades and quick phases of nystagmus are generated by the same pre-motor neurons, and we found that the quick-phase velocity of nystagmus was also reduced by lid closure. These changes were not due to a mechanical hindrance to the eyes, because lid closure did not affect the peak velocities or accelerations of the eyes in the “slow-phase” response to rapid head movements of comparable speeds to those of saccades. These results indicate a role for OPNs in generating the abrupt onset and high velocities of saccades. We hypothesize that the mechanism involved is PIR in pre-motor burst neurons.

Background

Essential tremor (ET) is the most common movement disorder and its pathophysiology is unknown. We hypothesize that increased membrane excitability in motor circuits has a key role in the pathogenesis of ET. Specifically, we propose that neural circuits controlling ballistic movements are inherently unstable due to their underlying reciprocal innervation. Such instability is enhanced by increased neural membrane excitability and the circuit begins to oscillate. These oscillations manifest as tremor.

Methods

Postural limb tremor was recorded in 22 ET patients and then the phenotype was simulated with a conductance-based neuromimetic model of ballistic movements. The model neuron was Hodgkin-Huxley type with added hyperpolarization activated cation current (Ih), low threshold calcium current (IT), and GABA and glycine mediated chloride currents. The neurons also featured the neurophysiological property of rebound excitation after release from sustained inhibition (post-inhibitory rebound). The model featured a reciprocally innervated circuit of neurons that project to agonist and antagonist muscle pairs.

Results

Neural excitability was modulated by changing Ih and/or IT. Increasing Ih and/or IT further depolarized the membrane and thus increased excitability. The characteristics of the tremor from all ET patients were simulated when Ih was increased to ~10× the range of physiological values. In contrast, increasing other membrane conductances, while keeping Ih at a physiological value, did not simulate the tremor. Increases in Ih and IT determined the frequency and amplitude of the simulated oscillations.

Conclusion

These simulations support the hypothesis that increased membrane excitability in potentially unstable, reciprocally innervated circuits can produce oscillations that resemble ET. Neural excitability could be increased in a number of ways. In this study membrane excitability was increased by up-regulating Ih and IT. This approach suggests new experimental and clinical ways to understand and treat common tremor disorders.

Field stimulation of the jejunum elicited successively an action potential of spike form, a slow excitatory depolarization, a slow inhibitory hyperpolarization, and a postinhibitory depolarization as a rebound excitation. The slow depolarization often triggered the spike. The inhibitory potential showed lower threshold than did the excitatory potential. Both the excitatory potentials were abolished by atropine and tetrodotoxin. Effective membrane resistance measured by the intracellular polarizing method was reduced during the peak of the excitatory potential, but the degree of reduction was smaller than that evoked by iontophoretic application of acetylcholine. Conditioning hyperpolarization of the muscle membrane modified the amplitude of the excitatory potential. The estimated reversal potential level for the excitatory potenialt was about 0 mv. No changes could be observed in the amplitude of the inhibitory potential when hyperpolarization was induced with intracellularly applied current. Low [K]o and [Ca]o blocked the generation of the excitatory potential but the amplitude of the inhibitory potential was enhanced in low [K]o. Low [Ca]o and high [Mg]o had no effect on the inhibitory potential.

It is generally believed that students learn best through activities that require their direct participation. By using simulations as a tool for learning neuroscience, students are directly engaged in the activity and obtain immediate feedback and reinforcement. This paper describes a series of biophysical models and computer simulations that can be used by educators and students to explore a variety of basic principles in neuroscience. The paper also suggests ‘virtual laboratory’ exercises that students may conduct to further examine biophysical processes underlying neural function. First, the Hodgkin and Huxley (HH) model is presented. The HH model is used to illustrate the action potential, threshold phenomena, and nonlinear dynamical properties of neurons (e.g., oscillations, postinhibitory rebound excitation). Second, the Morris-Lecar (ML) model is presented. The ML model is used to develop a model of a bursting neuron and to illustrate modulation of neuronal activity by intracellular ions. Lastly, principles of synaptic transmission are presented in small neural networks, which illustrate oscillatory behavior, excitatory and inhibitory postsynaptic potentials, and temporal summation.

Intrathalamic oscillations related to sleep and epilepsy depend on interactions between synaptic mechanisms and intrinsic membrane excitability. One intrinsic conductance implicated in the genesis of thalamic oscillations is the H current – a cationic current activated by membrane hyperpolarization. Activation of H current promotes rebound excitation of thalamic relay neurons and can thus enhance recurrent network activity.

We examined the effects of H current modulation on bicuculline-enhanced network oscillations (2-4 Hz) in rat thalamic slices. The adrenergic agonist norepinephrine, a known regulator of H current, caused an alteration of the internal structure of the oscillations – they were enhanced and accelerated as the interval between bursts was shortened. The acceleration was blocked by the β-adrenergic antagonist propranolol. The β agonist isoproterenol mimicked the effect of norepinephrine on oscillation frequency and truncated the responses suggesting that a β-adrenergic upregulation of H current modifies the internal structure (frequency) of thalamic oscillations. Consistent with this, we found that H channel blockade by Cs+ or ZD7288 could decelerate the oscillations and produce more robust (longer lasting) responses. High concentrations of either Cs+ or ZD7288 blocked the oscillations.

These results indicate that a critical amount of H current is necessary for optimal intrathalamic oscillations in the delta frequency range. Up- or downregulation of H current can not only alter the oscillation frequency but also retard or promote the development of thalamic synchronous oscillations. This conclusion has important implications regarding the development of epilepsy in thalamocortical circuits.

Postinhibitory rebound spiking is characteristic of several neuron types and brain regions, where it sustains spontaneous activity and central pattern generation. However, rebound spikes are rarely observed in the principal cells of the hippocampus under physiological conditions. We report that CA1 pyramidal neurons support rebound spikes mediated by hyperpolarization-activated inward current (Ih), and normally masked by A-type potassium channels (KA). In both experiments and computational models, KA blockage or reduction consistently resulted in a somatic action potential upon release from hyperpolarizing injections in the soma or main apical dendrite. Rebound spiking was systematically abolished by the additional blockage or reduction of Ih. Since the density of both KA and Ih increases in these cells with the distance from the soma, such “latent” mechanism may be most effective in the distal dendrites, which are targeted by a variety of GABAergic interneurons. Detailed computer simulations, validated against the experimental data, demonstrate that rebound spiking can result from activation of distal inhibitory synapses. In particular, partial KA reduction confined to one or few branches of the apical tuft may be sufficient to elicit a local spike following a train of synaptic inhibition. Moreover, the spatial extent and amount of KA reduction determines whether the dendritic spike propagates to the soma. These data suggest that the plastic regulation of KA can provide a dynamic switch to unmask postinhibitory spiking in CA1 pyramidal neurons. This newly discovered local modulation of postinhibitory spiking further increases the signal processing power of the CA1 synaptic microcircuitry.

Saccade-generating burst neurons (BN) are inhibited by omnipause neurons (OPN), except during saccades. OPN activity pauses before saccade onset and resumes at the saccade end. Microstimulation of OPN stops saccades in mid-flight, which shows that OPN can end saccades. However, OPN pause duration does not correlate well with saccade duration, and saccades are normometric after OPN lesions. We tested whether OPN were responsible for stopping saccades both in late-onset Tay–Sachs, which causes premature saccadic termination, and in individuals with cerebellar hypermetria. We studied gaze shifts between two targets at different distances aligned on one eye, which consist of a disjunctive saccade followed by vergence. High-frequency conjugate oscillations during the vergence movements that followed saccades were present in all subjects studied, indicating OPN silence. Thus, mechanisms other than OPN discharge (e.g., cerebellar caudal fastigial nucleus–promoting inhibitory BN discharge) must contribute to saccade termination.

Many neurons display post-inhibitory rebound (PIR), in which neurons display enhanced excitability following inhibition. PIR can strongly influence the timing of spikes on rebound from an inhibitory input. We studied PIR in the Lateral Pyloric (LP) neuron, part of the stomatogastric ganglion of the crab Cancer borealis. The LP neuron is part of the pyloric network, a central pattern generator that normally oscillates with a period of ~ 1 s. We used the dynamic clamp to create artificial rhythmic synaptic inputs of various periods and duty cycles in the LP neuron. Surprisingly, we found that the strength of PIR increased slowly over multiple cycles of synaptic input. Moreover, this increased excitability persisted for 10–20 s after the rhythmic inhibition was removed. These effects are considerably slower than the rhythmic activity typically observed in LP. Thus this slow postinhibitory rebound allows the neuron to adjust its level of excitability to the average level of inhibition over many cycles, and is another example of an intrinsic “short-term memory” mechanism.

During attempted visual fixation, saccades of a range of sizes occur. These “fixational saccades” include microsaccades, which are not apparent in regular clinical tests, and “saccadic intrusions”, predominantly horizontal saccades that interrupt accurate fixation. Square-wave jerks (SWJs), the most common type of saccadic intrusion, consist of an initial saccade away from the target followed, after a short delay, by a “return saccade” that brings the eye back onto target. SWJs are present in most human subjects, but are prominent by their increased frequency and size in certain parkinsonian disorders and in recessive, hereditary spinocerebellar ataxias. Here we asked whether fixational saccades showed distinctive features in various parkinsonian disorders and in recessive ataxia. Although some saccadic properties differed between patient groups, in all conditions larger saccades were more likely to form SWJs, and the intervals between the first and second saccade of SWJs were similar. These findings support the proposal of a common oculomotor mechanism that generates all fixational saccades, including microsaccades and SWJs. The same mechanism also explains how the return saccade in SWJs is triggered by the position error that occurs when the first saccadic component is large, both in the healthy brain and in neurological disease.

Neural circuits controlling fast movements are inherently unsteady as a result of their reciprocal innervation. This instability is enhanced by increased membrane excitability. Recent studies indicate that the loss of external inhibition is an important factor in the pathogenesis of several tremor disorders such as essential tremor, cerebellar kinetic tremor or parkinsonian tremor. Shaikh and colleagues propose a new conceptual scheme to analyze tremor disorders. Oscillations are simulated by changing the intrinsic membrane properties of burst neurons. The authors use a model neuron of Hodgkin-Huxley type with added hyperpolarization activated cation current (Ih), low threshold calcium current (It), and GABA/glycine mediated chloride currents. Post-inhibitory rebound is taken into account. The model includes a reciprocally innervated circuit of neurons projecting to pairs of agonist and antagonist muscles. A set of four burst neurons has been simulated: inhibitory agonist, inhibitory antagonist, excitatory agonist, and excitatory antagonist. The model fits well with the known anatomical organization of neural circuits for limb movements in premotor/motor areas, and, interestingly, this model does not require any structural modification in the anatomical organization or connectivity of the constituent neurons. The authors simulate essential tremor when Ih is increased. Membrane excitability is augmented by up-regulating Ih and It. A high level of congruence with the recordings made in patients exhibiting essential tremor is reached. These simulations support the hypothesis that increased membrane excitability in potentially unsteady circuits generate oscillations mimicking tremor disorders encountered in daily practice. This new approach opens new perspectives for both the understanding and the treatment of neurological tremor. It provides the rationale for decreasing membrane excitability by acting on a normal ion channel in a context of impaired external inhibition.

A depth electrode-brain interface (EBI) is formed once electrodes are implanted into the brain. We investigated the impact of the EBI on the crossing electric currents during both deep brain recording (DBR) and deep brain stimulation (DBS) over the acute, chronic and transitional stages post-implantation, in order to investigate and quantify the effect which changes at the EBI have on both DBR and DBS. We combined two complementary methods: (1) physiological recording of local field potentials via the implanted electrode in patients; and (2) computational simulations of an EBI model. Our depth recordings revealed that the physiological modulation of the EBI in the acute stage via brain pulsation selectively affected the crossing neural signals in a frequency-dependent manner, as the amplitude of the electrode potential was inversely correlated with that of the tremor-related oscillation, but not the beta oscillation. Computational simulations of DBS during the transitional period showed that the shielding effect of partial giant cell growth on the injected current could shape the field in an unpredictable manner. These results quantitatively demonstrated that physiological modulation of the EBI significantly affected the crossing currents in both DBR and DBS. Studying the microenvironment of the EBI may be a key step in investigating the mechanisms of DBR and DBS, as well as brain-computer interactions in general.

During fixation, the eyes are not still, but often exhibit microsaccadic movements. The function of microsaccades is controversial, largely because the neural mechanisms responsible for their generation are unknown. Here we show that the superior colliculus (SC), a retinotopically organized structure involved in voluntary-saccade target selection, plays a causal role in microsaccade generation. Neurons in the foveal portion of the SC increase their activity before and during microsaccades with sizes of only a few minutes of arc, and exhibit selectivity for the direction and amplitude of these movements. Reversible inactivation of these neurons significantly reduces microsaccade rate without otherwise compromising fixation. These results, coupled with computational modeling of SC activity, demonstrate that microsaccades are controlled by the SC, and explain the link between microsaccades and visual attention.

Chiari type II malformation is a congenital deformity of the hindbrain. Square wave jerks are horizontal involuntary saccades that interrupt fixation. Cerebellar disorders may be associated with frequent square wave jerks or saccadic oscillations such as ocular flutter. The effects of Chiari type II malformation on visual fixation are unknown. We recorded eye movements using an eye tracker in 21 participants with Chiari type II malformation, aged 8 to 19 years while they fixated a target for 1 minute. Thirty-eight age-matched healthy participants served as controls. Square wave jerks’ parameters were similar in the 2 groups. Saccadic oscillations were not seen. Chiari type II malformation is not associated with pathological square wave jerks or abnormal saccadic oscillations. The congenital nature of this deformity may permit compensation that preserves stable visual fixation. Alternatively, the deformity of Chiari type II malformation may spare parts of the cerebellum that usually cause fixation instability when damaged.

We study oscillation death (OD) in a well-known coupled-oscillator system that has been used to model cardiovascular phenomena. We derive exact analytic conditions that allow the prediction of OD through the two known bifurcation routes, in the same model, and for different numbers of coupled oscillators. Our exact analytic results enable us to generalize OD as a multiparameter-sensitive phenomenon. It can be induced, not only by changes in couplings, but also by changes in the oscillator frequencies or amplitudes. We observe synchronization transitions as a function of coupling and confirm the robustness of the phenomena in the presence of noise. Numerical and analogue simulations are in good agreement with the theory.

Our objective was to characterize the saccadic eye movements in patients with type 3 Gaucher disease (chronic neuronopathic) in relationship to neurological and neurophysiological abnormalities. For approximately 4 years, we prospectively followed a cohort of 15 patients with Gaucher type 3, ages 8–28 years, by measuring saccadic eye movements using the scleral search coil method. We found that patients with type 3 Gaucher disease had a significantly higher regression slope of duration vs amplitude and peak duration vs amplitude compared to healthy controls for both horizontal and vertical saccades. Saccadic latency was significantly increased for horizontal saccades only. Downward saccades were more affected than upward saccades. Saccade abnormalities increased over time in some patients reflecting the slowly progressive nature of the disease. Phase plane plots showed individually characteristic patterns of abnormal saccade trajectories. Oculo-manual dexterity scores on the Purdue Pegboard test were low in virtually all patients, even in those with normal cognitive function. Vertical saccade peak duration vs amplitude slope significantly correlated with IQ and with the performance on the Purdue Pegboard but not with the brainstem and somatosensory evoked potentials. We conclude that, in patients with Gaucher disease type 3, saccadic eye movements and oculo-manual dexterity are representative neurological functions for longitudinal studies and can probably be used as endpoints for therapeutic clinical trials.

Trial Registration

ClinicalTrials.gov NCT00001289

Tremor is commonly encountered in medical practice, but can be difficult to diagnose and manage. It is an involuntary rhythmic oscillation of a body part produced by reciprocally innervated antagonist muscles. Tremors vary in frequency and amplitude and are influenced by physiologic and psychological factors and drugs. Categorization is based on position, posture, and the movement necessary to elicit the tremor. A resting tremor occurs when the body part is in repose. A postural tremor occurs with maintained posture and kinetic tremor with movement. Various pathologic conditions are associated with tremors. Essential tremor, which is the most common, is postural and kinetic, with a frequency between 4 and 8 Hz, and involves mainly the upper extremities and head. Essential tremor responds to treatment with primidone, beta-blockers, and benzodiazepines. Parkinson's disease causes a 4- to 6-Hz resting tremor in the arms and legs that responds to the use of anticholinergics and a combination of carbidopa and levodopa. Tremor can also be a manifestation of Wilson's disease, lesions of the cerebellum and midbrain, peripheral neuropathy, trauma, alcohol, and conversion disorders. Treatment should be directed to the underlying condition. Stereotactic thalamotomy of thalamic stimulation is a last resort.

Intrinsic noise is a common phenomenon in biochemical reaction networks and may affect the occurence and amplitude of sustained oscillations in the states of the network. To evaluate properties of such oscillations in the time domain, it is usually required to conduct long-term stochastic simulations, using for example the Gillespie algorithm. In this paper, we present a new method to compute the amplitude distribution of the oscillations without the need for long-term stochastic simulations. By the derivation of the method, we also gain insight into the structural features underlying the stochastic oscillations. The method is applicable to a wide class of non-linear stochastic differential equations that exhibit stochastic oscillations. The application is exemplified for the MAPK cascade, a fundamental element of several biochemical signalling pathways. This example shows that the proposed method can accurately predict the amplitude distribution for the stochastic oscillations even when using further computational approximations.

PACS Codes: 87.10.Mn, 87.18.Tt, 87.18.Vf

MSC Codes: 92B05, 60G10, 65C30

Saccadic eye movements are driven by motor commands that are continuously modified so that errors created by eye muscle fatigue, injury, or—in humans—wearing spectacles can be corrected. It is possible to rapidly adapt saccades in the laboratory by introducing a discrepancy between the intended and actual saccadic target. Neurophysiological and lesion studies in the non-human primate as well as neuroimaging and patient studies in humans have demonstrated that the oculomotor vermis (lobules VI and VII of the posterior cerebellum) is critical for saccadic adaptation. We studied the effect of transiently disrupting the function of posterior cerebellum with repetitive transcranial magnetic stimulation (rTMS) on the ability of healthy human subjects to adapt saccadic eye movements. rTMS significantly impaired the adaptation of the amplitude of saccades, without modulating saccadic amplitude or variability in baseline conditions. Moreover, increasing the intensity of rTMS produced a larger impairment in the ability to adapt saccadic size. These results provide direct evidence for the role of the posterior cerebellum in man and further evidence that TMS can modulate cerebellar function.

In a typical short-term saccadic adaptation protocol, the target moves intra-saccadically either toward (gain-down) or away (gain-up) from initial fixation, causing the saccade to complete with an endpoint error. A central question is how the motor system adapts in response to this error: are the motor commands changed to bring the eyes to a different goal, akin to a remapping of the target, or is adaptation focused on the processes that monitor the ongoing motor commands and correct them midflight, akin to changes that act via internal feedback? Here, we found that in the gain-down paradigm, the brain learned to produce a smaller amplitude saccade by altering the saccade's trajectory. The adapted saccades had reduced peak velocities, reduced accelerations, shallower decelerations, and increased durations compared to a control saccade of equal amplitude. These changes were consistent with a change in an internal feedback that acted as a forward model. On the other hand, in the gain-up paradigm the brain learned to produce a larger amplitude saccade with trajectories that were identical to those of control saccades of equal amplitude. Therefore, whereas the gain-down paradigm appeared to induce adaptation via an internal feedback that controlled saccades midflight, gain-up induced adaptation primarily via target remapping. Our simulations explained that for each condition, the specific adaptation produced a saccade that brought the eyes to the target with the smallest motor costs.

Purpose

The vitreous humor liquefies with age and readily sloshes during eye motion. The objective was to develop a computational model to determine the effect of sloshing on intravitreal drug transport for transscleral and intra-vitreal drug sources at various locations

Methods

A finite element model based on a telescopic implicit envelope tracking scheme was developed to model drug dispersion. Flow velocities due to saccadic oscillations were solved for and were used to simulate drug dispersion.

Results

Saccades induced a three-dimensional flow field that indicates intense drug dispersion in the vitreous. Model results showed that the time scale for transport decreased for the sloshing vitreous when compared to static vitreous. Macular concentrations for the sloshing vitreous were found be much higher than that for the static vitreous. For low viscosities the position of the intravitreal source did not have a big impact on drug distribution.

Conclusion

Model results show that care should be taken when extrapolating animal data, which are mostly done on intact vitreous, to old patients whose vitreous might be a liquid. The decrease in drug transport time scales and changes in localized concentrations should be considered when deciding on treatment modalities and dosing strategies.

Cognitive control is required for correct performance on antisaccade tasks, including the ability to inhibit an externally driven ocular motor repsonse (a saccade to a peripheral stimulus) in favor of an internally driven ocular motor goal (a saccade directed away from a peripheral stimulus). Healthy humans occasionally produce errors during antisaccade tasks, but the mechanisms associated with such failures of cognitive control are uncertain. Most research on cognitive control failures focuses on post-stimulus processing, although a growing body of literature highlights a role of intrinsic brain activity in perceptual and cognitive performance. The current investigation used dense array electroencephalography and distributed source analyses to examine brain oscillations across a wide frequency bandwidth in the period prior to antisaccade cue onset. Results highlight four important aspects of ongoing and preparatory brain activations that differentiate error from correct antisaccade trials: (i) ongoing oscillatory beta (20–30Hz) power in anterior cingulate prior to trial initiation (lower for error trials), (ii) instantaneous phase of ongoing alpha-theta (7Hz) in frontal and occipital cortices immediately before trial initiation (opposite between trial types), (iii) gamma power (35–60Hz) in posterior parietal cortex 100 ms prior to cue onset (greater for error trials), and (iv) phase locking of alpha (5–12Hz) in parietal and occipital cortices immediately prior to cue onset (lower for error trials). These findings extend recently reported effects of pre-trial alpha phase on perception to cognitive control processes, and help identify the cortical generators of such phase effects.

Insects have evolved diverse and delicate morphological structures in order to capture the inherently low energy of a propagating sound wave. In mosquitoes, the capture of acoustic energy and its transduction into neuronal signals are assisted by the active mechanical participation of the scolopidia. We propose a simple microscopic mechanistic model of the active amplification in the mosquito species Toxorhynchites brevipalpis. The model is based on the description of the antenna as a forced-damped oscillator coupled to a set of active threads (ensembles of scolopidia) that provide an impulsive force when they twitch. This twitching is in turn controlled by channels that are opened and closed if the antennal oscillation reaches a critical amplitude. The model matches both qualitatively and quantitatively with recent experiments: spontaneous oscillations, nonlinear amplification, hysteresis, 2 : 1 resonances, frequency response and gain loss owing to hypoxia. The numerical simulations presented here also generate new hypotheses. In particular, the model seems to indicate that scolopidia located towards the tip of Johnston's organ are responsible for the entrainment of the other scolopidia and that they give the largest contribution to the mechanical amplification.

Cell-autonomous and self-sustained molecular oscillators drive circadian behavior and physiology in mammals. From rhythms recorded in cultured fibroblasts we identified the dominant cause for amplitude reduction as desynchronization of self-sustained oscillators. Here, we propose a general framework for quantifying luminescence signals from biochemical oscillators, both in populations and individual cells. Our model combines three essential aspects of circadian clocks: the stability of the limit cycle, fluctuations, and intercellular coupling. From population recordings we can simultaneously estimate the stiffness of individual frequencies, the period dispersion, and the interaction strength. Consistent with previous work, coupling is found to be weak and insufficient to synchronize cells. Moreover, we find that frequency fluctuations remain correlated for longer than one clock cycle, which is confirmed from individual cell recordings. Using genetic models for circadian clocks, we show that this reflects the stability properties of the underlying circadian limit-cycle oscillators, and we identify biochemical parameters that influence oscillator stability in mammals. Our study thus points to stabilizing mechanisms that dampen fluctuations to maintain accurate timing in peripheral circadian oscillators.

Spectral analysis of a tremor record can sometimes produce a spectrum with multiple components of significant amplitude. The problem is to determine whether the presence of several peaks represents the coexistence of separate tremor mechanisms or be a consequence of fluctuations in the frequency or amplitude of a single tremor. The spectrum of a tremor whose frequency or amplitude vary and are independent has the recognisable pattern of a central carrier frequency with sidebands of equal amplitudes distributed symmetrically around the carrier. However, if tremor amplitude and frequency fluctuate and are not independent, (frequency proportional to amplitude or frequency inversely proportional to amplitude), the spectrum has a pattern of sidebands which are asymmetrical in amplitudes and may resemble the spectrum of the combined signal from different independent oscillators. The investigation of sidebands in spectra has been neglected in tremor studies and multiple irregular peaks on a tremor spectrum have sometimes been used wrongly as evidence for the coexistence of multiple tremor mechanisms or frequency components assumed to be concurrent.