Efficient assessment of real-world dynamics of circadian rhythms in heart rate and body temperature from wearable data

Laboratory studies have made unprecedented progress in understanding circadian physiology. Quantifying circadian rhythms outside of laboratory settings is necessary to translate these findings into real-world clinical practice. Wearables have been considered promising way to measure these rhythms. However, their limited validation remains an open problem. One major barrier to implementing large-scale validation studies is the lack of reliable and efficient methods for circadian assessment from wearable data. Here, we propose an approximation-based least-squares method to extract underlying circadian rhythms from wearable measurements. Its computational cost is ∼ 300-fold lower than that of previous work, enabling its implementation in smartphones with low computing power. We test it on two large-scale real-world wearable datasets: [Formula: see text] of body temperature data from cancer patients and ∼ 184 000 days of heart rate and activity data collected from the 'Social Rhythms' mobile application. This shows successful extraction of real-world dynamics of circadian rhythms. We also identify a reasonable harmonic model to analyse wearable data. Lastly, we show our method has broad applicability in circadian studies by embedding it into a Kalman filter that infers the state space of the molecular clocks in tissues. Our approach facilitates the translation of scientific advances in circadian fields into actual improvements in health.

Impact of Light Schedules and Model Parameters on the Circadian Outcomes of Individuals

Key differences exist between individuals in terms of certain circadian-related parameters, such as intrinsic period and sensitivity to light. These variations can differentially impact circadian timing, leading to challenges in accurately implementing time-sensitive interventions. In this work, we parse out these effects by investigating the impact of parameters from a macroscopic model of human circadian rhythms on phase and amplitude outputs. Using in silico light data designed to mimic commonly studied schedules, we assess the impact of parameter variations on model outputs to gain insight into the different effects of these schedules. We show that parameter sensitivity is heavily modulated by the lighting routine that a person follows, with darkness and shift work schedules being the most sensitive. We develop a framework to measure overall sensitivity levels of the given light schedule and furthermore decompose the overall sensitivity into individual parameter contributions. Finally, we measure the ability of the model to extract parameters given light schedules with noise and show that key parameters like the circadian period can typically be recovered given known light history. This can inform future work on determining the key parameters to consider when personalizing a model and the lighting protocols to use when assessing interindividual variability.

Reducing chronic disease may just be a walk in the park

Wearable technology allows the collection of real-world granular data at scales that would be impossible using traditional collection methods. Master et al. demonstrate the power of this technology to estimate the risk of disease based on daily step counts.¹.

Consumer-grade wearables identify changes in multiple physiological systems during COVID-19 disease progression

Consumer-grade wearables are needed to track disease, especially in the ongoing pandemic, as they can monitor patients in real time. We show that decomposing heart rate from low-cost wearable technologies into signals from different systems can give a multidimensional description of physiological changes due to COVID-19 infection. We find that the separate physiological features of basal heart rate, heart rate response to physical activity, circadian variation in heart rate, and autocorrelation of heart rate are significantly altered and can classify symptomatic versus healthy periods. Increased heart rate and autocorrelation begin at symptom onset, while the heart rate response to activity increases soon after symptom onset and increases more in individuals exhibiting cough. Symptom onset is associated with a blunting of circadian variation in heart rate, as measured by the uncertainty in the phase estimate. This work establishes an innovative data analytic approach to monitor disease progression remotely using consumer-grade wearables.

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We separate wearable heart rate into cardiopulmonary, circadian, and other signals

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Parameters from different physiological systems enable disease tracking

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Individual signals change in distinct ways around COVID-19 symptom onset

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Together, the parameter changes can distinguish healthy from infection periods

Evolution of the repression mechanisms in circadian clocks

BACKGROUND

Circadian (daily) timekeeping is essential to the survival of many organisms. An integral part of all circadian timekeeping systems is negative feedback between an activator and repressor. However, the role of this feedback varies widely between lower and higher organisms.

RESULTS

Here, we study repression mechanisms in the cyanobacterial and eukaryotic clocks through mathematical modeling and systems analysis. We find a common mathematical model that describes the mechanism by which organisms generate rhythms; however, transcription's role in this has diverged. In cyanobacteria, protein sequestration and phosphorylation generate and regulate rhythms while transcription regulation keeps proteins in proper stoichiometric balance. Based on recent experimental work, we propose a repressor phospholock mechanism that models the negative feedback through transcription in clocks of higher organisms. Interestingly, this model, when coupled with activator phosphorylation, allows for oscillations over a wide range of protein stoichiometries, thereby reconciling the negative feedback mechanism in Neurospora with that in mammals and cyanobacteria.

CONCLUSIONS

Taken together, these results paint a picture of how circadian timekeeping may have evolved.

Distinct Circadian Assessments From Wearable Data Reveal Social Distancing Promoted Internal Desynchrony Between Circadian Markers

Mobile measures of human circadian rhythms (CR) are needed in the age of chronotherapy. Two wearable measures of CR have recently been validated: one that uses heart rate to extract circadian rhythms that originate in the sinoatrial node of the heart, and another that uses activity to predict the laboratory gold standard and central circadian pacemaker marker, dim light melatonin onset (DLMO). We first find that the heart rate markers of normal real-world individuals align with laboratory DLMO measurements when we account for heart rate phase error. Next, we expand upon previous work that has examined sleep patterns or chronotypes during the COVID-19 lockdown by studying the effects of social distancing on circadian rhythms. In particular, using data collected from the Social Rhythms app, a mobile application where individuals upload their wearable data and receive reports on their circadian rhythms, we compared the two circadian phase estimates before and after social distancing. Interestingly, we found that the lockdown had different effects on the two ambulatory measurements. Before the lockdown, the two measures aligned, as predicted by laboratory data. After the lockdown, when circadian timekeeping signals were blunted, these measures diverged in 70% of subjects (with circadian rhythms in heart rate, or CRHR, becoming delayed). Thus, while either approach can measure circadian rhythms, both are needed to understand internal desynchrony. We also argue that interventions may be needed in future lockdowns to better align separate circadian rhythms in the body.

A method for characterizing daily physiology from widely used wearables

Millions of wearable-device users record their heart rate (HR) and activity. We introduce a statistical method to extract and track six key physiological parameters from these data, including an underlying circadian rhythm in HR (CRHR), the direct effects of activity, and the effects of meals, posture, and stress through hormones like cortisol. We test our method on over 130,000 days of real-world data from medical interns on rotating shifts, showing that CRHR dynamics are distinct from those of sleep-wake or physical activity patterns and vary greatly among individuals. Our method also estimates a personalized phase-response curve of CRHR to activity for each individual, representing a passive and personalized determination of how human circadian timekeeping continually changes due to real-world stimuli. We implement our method in the "Social Rhythms" iPhone and Android app, which anonymously collects data from wearable-device users and provides analysis based on our method.

Monitoring beliefs and physiological measures in students at risk for COVID-19 using wearable sensors and smartphone technology: Protocol for a mobile health study

BACKGROUND

The COVID-19 pandemic has impacted lives significantly and greatly affected an already vulnerable population, college students, in relation to mental health and public safety. Social distancing and isolation have brought about challenges to student's mental health. Mobile health apps and wearable sensors may help to monitor students at risk for COVID-19 and support their mental well-being.

OBJECTIVE

Through the use of a wearable sensor and smartphone-based survey completion, this study aimed to monitor students at risk for COVID-19.

METHODS

We conducted a prospective study of students, undergraduate and graduate, at a public university in the Midwest. Students were instructed to download the Fitbit, Social Rhythms, and Roadmap 2.0 apps onto their personal mobile devices (Android or iOS). Subjects consented to provide up to 10 saliva samples during the study period. Surveys were administered through the Roadmap 2.0 app at five timepoints - at baseline, 1-month later, 2-months later, 3-months later, and at study completion. The surveys gathered information regarding demographics, COVID-19 diagnoses and symptoms, and mental health resilience, with the aim of documenting the impact of COVID-19 on the college student population.

RESULTS

This study enrolled 2,158 college students between September 2020 and January 2021. Subjects are currently being followed on-study for one academic year. Data collection and analysis are ongoing.

CONCLUSIONS

This study examined student health and well-being during the COVID-19 pandemic. It also assessed the feasibility of wearable sensor use and survey completion in a college student population, which may inform the role of our mobile health tools on student health and well-being. Finally, using wearable sensor data, biospecimen collection, and self-reported COVID-19 diagnosis, our results may provide key data towards the development of a model for the early prediction and detection of COVID-19.

CLINICALTRIAL

ClinicalTrials.gov NCT04766788.

Predicting circadian phase across populations: a comparison of mathematical models and wearable devices

From smart work scheduling to optimal drug timing, there is enormous potential in translating circadian rhythms research results for precision medicine in the real world. However, the pursuit of such effort requires the ability to accurately estimate circadian phase outside of the laboratory. One approach is to predict circadian phase non-invasively using light and activity measurements and mathematical models of the human circadian clock. Most mathematical models take light as an input and predict the effect of light on the human circadian system. However, consumer-grade wearables that are already owned by millions of individuals record activity instead of light, which prompts an evaluation of the accuracy of predicting circadian phase using motion alone. Here, we evaluate the ability of four different models of the human circadian clock to estimate circadian phase from data acquired by wrist-worn wearable devices. Multiple datasets across populations with varying degrees of circadian disruption were used for generalizability. Though the models we test yield similar predictions, analysis of data from 27 shift workers with high levels of circadian disruption shows that activity, which is recorded in almost every wearable device, is better at predicting circadian phase than measured light levels from wrist-worn devices when processed by mathematical models. In those living under normal living conditions, circadian phase can typically be predicted to within 1 hour, even with data from a widely available commercial device (the Apple Watch). These results show that circadian phase can be predicted using existing data passively collected by millions of individuals with comparable accuracy to much more invasive and expensive methods.

Monitoring Health Care Workers at Risk for COVID-19 Using Wearable Sensors and Smartphone Technology: Protocol for an Observational mHealth Study

BACKGROUND

Health care workers (HCWs) have been working on the front lines of the COVID-19 pandemic with high risks of viral exposure, infection, and transmission. Standard COVID-19 testing is insufficient to protect HCWs from these risks and prevent the spread of disease. Continuous monitoring of physiological data with wearable sensors, self-monitoring of symptoms, and asymptomatic COVID-19 testing may aid in the early detection of COVID-19 in HCWs and may help reduce further transmission among HCWs, patients, and families.

OBJECTIVE

By using wearable sensors, smartphone-based symptom logging, and biospecimens, this project aims to assist HCWs in self-monitoring COVID-19.

METHODS

We conducted a prospective, longitudinal study of HCWs at a single institution. The study duration was 1 year, wherein participants were instructed on the continuous use of two wearable sensors (Fitbit Charge 3 smartwatch and TempTraq temperature patches) for up to 30 days. Participants consented to provide biospecimens (ie, nasal swabs, saliva swabs, and blood) for up to 1 year from study entry. Using a smartphone app called Roadmap 2.0, participants entered a daily mood score, submitted daily COVID-19 symptoms, and completed demographic and health-related quality of life surveys at study entry and 30 days later. Semistructured qualitative interviews were also conducted at the end of the 30-day period, following completion of daily mood and symptoms reporting as well as continuous wearable sensor use.

RESULTS

A total of 226 HCWs were enrolled between April 28 and December 7, 2020. The last participant completed the 30-day study procedures on January 16, 2021. Data collection will continue through January 2023, and data analyses are ongoing.

CONCLUSIONS

Using wearable sensors, smartphone-based symptom logging and survey completion, and biospecimen collections, this study will potentially provide data on the prevalence of COVID-19 infection among HCWs at a single institution. The study will also assess the feasibility of leveraging wearable sensors and self-monitoring of symptoms in an HCW population.

TRIAL REGISTRATION

ClinicalTrials.gov NCT04756869; https://clinicaltrials.gov/ct2/show/NCT04756869.

INTERNATIONAL REGISTERED REPORT IDENTIFIER (IRRID)

DERR1-10.2196/29562.

M1-Type, but Not M4-Type, Melanopsin Ganglion Cells Are Physiologically Tuned to the Central Circadian Clock

Proper circadian photoentrainment is crucial for the survival of many organisms. In mammals, intrinsically photosensitive retinal ganglion cells (ipRGCs) can use the photopigment melanopsin to sense light independently from rod and cone photoreceptors and send this information to many brain nuclei such as the suprachiasmatic nucleus (SCN), the site of the central circadian pacemaker. Here, we measure ionic currents and develop mathematical models of the electrical activity of two types of ipRGCs: M1, which projects to the SCN, and M4, which does not. We illustrate how their ionic properties differ, mainly how ionic currents generate lower spike rates and depolarization block in M1 ipRGCs. Both M1 and M4 cells have large geometries and project to higher visual centers of the brain via the optic nerve. Using a partial differential equation model, we show how axons of M1 and M4 cells faithfully convey information from the soma to the synapse even when the signal at the soma is attenuated due to depolarization block. Finally, we consider an ionic model of circadian photoentrainment from ipRGCs synapsing on SCN neurons and show how the properties of M1 ipRGCs are tuned to create accurate transmission of visual signals from the retina to the central pacemaker, whereas M4 ipRGCs would not evoke nearly as efficient a postsynaptic response. This work shows how ipRGCs and SCN neurons' electrical activities are tuned to allow for accurate circadian photoentrainment.

Predicting circadian misalignment with wearable technology: validation of wrist-worn actigraphy and photometry in night shift workers

STUDY OBJECTIVES

A critical barrier to successful treatment of circadian misalignment in shift workers is determining circadian phase in a clinical or field setting. Light and movement data collected passively from wrist actigraphy can generate predictions of circadian phase via mathematical models; however, these models have largely been tested in non-shift working adults. This study tested the feasibility and accuracy of actigraphy in predicting dim light melatonin onset (DLMO) in fixed night shift workers.

METHODS

A sample of 45 night shift workers wore wrist actigraphs before completing DLMO in the laboratory (17.0 days ± 10.3 SD). DLMO was assessed via 24 hourly saliva samples in dim light (<10 lux). Data from actigraphy were provided as input to a mathematical model to generate predictions of circadian phase. Agreement was assessed and compared to average sleep timing on non-workdays as a proxy of DLMO. Model code and an open-source prototype assessment tool are available (www.predictDLMO.com).

RESULTS

Model predictions of DLMO showed good concordance with in-lab DLMO, with Lin's concordance coefficient of 0.70, which was twice as high as agreement using average sleep timing as a proxy of DLMO. The absolute mean error of the predictions was 2.88 h, with 76% and 91% of the predictions falling with 2 and 4 h, respectively.

CONCLUSION

This study is the first to demonstrate the use of wrist actigraphy-based estimates of circadian phase as a clinically useful and valid alternative to in-lab measurement of DLMO in fixed night shift workers. Future research should explore how additional predictors may impact accuracy.

Gene-set Enrichment with Mathematical Biology (GEMB)

BACKGROUND

Gene-set analyses measure the association between a disease of interest and a "set" of genes related to a biological pathway. These analyses often incorporate gene network properties to account for differential contributions of each gene. We extend this concept further-defining gene contributions based on biophysical properties-by leveraging mathematical models of biology to predict the effects of genetic perturbations on a particular downstream function.

RESULTS

We present a method that combines gene weights from model predictions and gene ranks from genome-wide association studies into a weighted gene-set test. We demonstrate in simulation how such a method can improve statistical power. To this effect, we identify a gene set, weighted by model-predicted contributions to intracellular calcium ion concentration, that is significantly related to bipolar disorder in a small dataset (P = 0.04; n = 544). We reproduce this finding using publicly available summary data from the Psychiatric Genomics Consortium (P = 1.7 × 10-4; n = 41,653). By contrast, an approach using a general calcium signaling pathway did not detect a significant association with bipolar disorder (P = 0.08). The weighted gene-set approach based on intracellular calcium ion concentration did not detect a significant relationship with schizophrenia (P = 0.09; n = 65,967) or major depression disorder (P = 0.30; n = 500,199).

CONCLUSIONS

Together, these findings show how incorporating math biology into gene-set analyses might help to identify biological functions that underlie certain polygenic disorders.

Macroscopic Models for Human Circadian Rhythms

Mathematical models have a long and influential history in the study of human circadian rhythms. Accurate predictive models for the human circadian light response have been used to study the impact of a host of light exposures on the circadian system. However, generally, these models do not account for the physiological basis of these rhythms. We illustrate a new paradigm for deriving models of the human circadian light response. Beginning from a high-dimensional model of the circadian neural network, we systematically derive low-dimensional models using an approach motivated by experimental measurements of circadian neurons. This systematic reduction allows for the variables and parameters of the derived model to be interpreted in a physiological context. We fit and validate the resulting models to a library of experimental measurements. Finally, we compare model predictions for experimental measurements of light levels and discuss the differences between our model’s predictions and previous models. Our modeling paradigm allows for the integration of experimental measurements across the single-cell, tissue, and behavioral scales, thereby enabling the development of accurate low-dimensional models for human circadian rhythms.

Macroscopic models for networks of coupled biological oscillators

The study of synchronization of coupled biological oscillators is fundamental to many areas of biology including neuroscience, cardiac dynamics, and circadian rhythms. Mathematical models of these systems may involve hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. This often contrasts with the low-dimensional dynamics exhibited on the collective or macroscopic scale for these systems. We introduce a macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we find in experimental measurements of circadian protein expression and several mathematical models for coupled biological oscillators. The observed structure in the collective amplitude of the oscillator population differs from the well-known Ott-Antonsen ansatz, but its emergence can be characterized through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate its emergence in networks of noisy heterogeneous oscillators with complex network connectivity. Applying this structure, we derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system.

Multiplexing Visual Signals in the Suprachiasmatic Nuclei

The suprachiasmatic nuclei (SCN), the site of the mammalian circadian (daily) pacemaker, contains thousands of interconnected neurons, some of which receive direct retinal input. Here, we study the fast (<1 s) responses of SCN neurons to visual stimuli with a large-scale mathematical model tracking the ionic currents and voltage of all SCN neurons. We reconstruct the SCN network connectivity and reject 99.99% of theoretically possible SCN networks by requiring that the model reproduces experimentally determined receptive fields of SCN neurons. The model shows how the SCN neuronal network can enhance circadian entrainment by sensitizing a population of neurons in the ventral SCN to irradiance. This SCN network also increases the spatial acuity of neurons and increases the accuracy of a simulated subconscious spatial visual task. We hypothesize that much of the fast electrical activity within the SCN is related to the processing of spatial information.

Testing frameworks for personalizing bipolar disorder

The hallmark of bipolar disorder is a clinical course of recurrent manic and depressive symptoms of varying severity and duration. Mathematical modeling of bipolar disorder holds the promise of an ability to personalize diagnoses, to predict future mood episodes, to directly compare diverse datasets, and to link basic mechanisms to behavioral data. Several modeling frameworks have been proposed for bipolar disorder, which represent competing hypothesis about the basic framework of the disorder. Here, we test these hypotheses with self-report assessments of mania and depression symptoms from 178 bipolar patients followed prospectively for 4 or more years. Statistical analysis of the data did not support the hypotheses that mood arises from a rhythmic process or multiple stable states (e.g., mania or depression) or that manic and depressive symptoms are highly anti-correlated. Alternatively, it is shown that bipolar disorder could arise from an inability for mood to quickly return to normal when perturbed. This latter concept is embodied by an affective instability model that can be personalized to the clinical course of any individual with chronic disorders that have an affective component.

Guidelines for Genome-Scale Analysis of Biological Rhythms

Genome biology approaches have made enormous contributions to our understanding of biological rhythms, particularly in identifying outputs of the clock, including RNAs, proteins, and metabolites, whose abundance oscillates throughout the day. These methods hold significant promise for future discovery, particularly when combined with computational modeling. However, genome-scale experiments are costly and laborious, yielding “big data” that are conceptually and statistically difficult to analyze. There is no obvious consensus regarding design or analysis. Here we discuss the relevant technical considerations to generate reproducible, statistically sound, and broadly useful genome-scale data. Rather than suggest a set of rigid rules, we aim to codify principles by which investigators, reviewers, and readers of the primary literature can evaluate the suitability of different experimental designs for measuring different aspects of biological rhythms. We introduce CircaInSilico, a web-based application for generating synthetic genome biology data to benchmark statistical methods for studying biological rhythms. Finally, we discuss several unmet analytical needs, including applications to clinical medicine, and suggest productive avenues to address them.

Responses to Spatial Contrast in the Mouse Suprachiasmatic Nuclei

A direct retinal projection targets the suprachiasmatic nucleus (SCN) (an important hypothalamic control center). The accepted function of this projection is to convey information about ambient light (irradiance) to synchronize the SCN’s endogenous circadian clock with local time and drive the diurnal variations in physiology and behavior [1, 2, 3, 4]. Here, we report that it also renders the SCN responsive to visual images. We map spatial receptive fields (RFs) for SCN neurons and find that only a minority are excited (or inhibited) by light from across the scene as expected for irradiance detectors. The most commonly encountered units have RFs with small excitatory centers, combined with very extensive inhibitory surrounds that reduce their sensitivity to global changes in light in favor of responses to spatial patterns. Other units have larger excitatory RF centers, but these always cover a coherent region of visual space, implying visuotopic order at the single-unit level. Approximately 75% of light-responsive SCN units modulate their firing according to simple spatial patterns (drifting or inverting gratings) without changes in irradiance. The time-averaged firing rate of the SCN is modestly increased under these conditions, but including spatial contrast did not significantly alter the circadian phase resetting efficiency of light. Our data indicate that the SCN contains information about irradiance and spatial patterns. This newly appreciated sensory capacity provides a mechanism by which behavioral and physiological systems downstream of the SCN could respond to visual images [5].

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Many SCN units have receptive fields optimized for spatial discrimination

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A large majority of SCN neurons track changes in spatial patterns

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Spatial patterns can enhance SCN-maintained firing, but not circadian phase resetting

Quantifying Human Circadian Pacemaker Response to Brief, Extended, and Repeated Light Stimuli over the Phototopic Range

The authors' previous models have been able to describe accurately the effects of extended (5 h) bright-light (>4000 lux) stimuli on the phase and amplitude of the human circadian pacemaker, but they are not sufficient to represent the surprising human sensitivity to both brief pulses of bright light and light of more moderate intensities. Therefore, the authors have devised a new model in which a dynamic stimulus processor ( Process L) intervenes between the light stimuli and the traditional representation of the circadian pacemaker as a self-sustaining limit-cycle oscillator ( Process P). The overall model incorporating Process L and Process P is intended to allow the prediction of phase shifts to photic stimuli of any temporal pattern (extended and brief light episodes) and any light intensity in the photopic range. Two time constants emerge in the Process Lmodel: the characteristic duration for necessary bright-light pulses to achieve their full effect (5-10 min) and the characteristic stimulus-free (dark) interval that can be tolerated without incurring an excessive penalty in phase shifting (30-80 min). The effect of reducing light intensity is incorporated in Process L as an extension of the time necessary for the light pulse to be fully realized (a power-law relation between time and intensity). This new model generates a number of new testable hypotheses, including the surprising prediction that 24-h cycles consisting of8hof darkness and 16 h of only 3.5 lux would be capable of entraining a large fraction of the adult population (45%). Experimental data on the response of the human circadian system to lower light intensities and briefer stimuli are needed to allow for further refinement and validation of the model proposed here.

A global quantification of "normal" sleep schedules using smartphone data

The influence of the circadian clock on sleep scheduling has been studied extensively in the laboratory; however, the effects of society on sleep remain largely unquantified. We show how a smartphone app that we have developed, ENTRAIN, accurately collects data on sleep habits around the world. Through mathematical modeling and statistics, we find that social pressures weaken and/or conceal biological drives in the evening, leading individuals to delay their bedtime and shorten their sleep. A country's average bedtime, but not average wake time, predicts sleep duration. We further show that mathematical models based on controlled laboratory experiments predict qualitative trends in sunrise, sunset, and light level; however, these effects are attenuated in the real world around bedtime. Additionally, we find that women schedule more sleep than men and that users reporting that they are typically exposed to outdoor light go to sleep earlier and sleep more than those reporting indoor light. Finally, we find that age is the primary determinant of sleep timing, and that age plays an important role in the variability of population-level sleep habits. This work better defines and personalizes "normal" sleep, produces hypotheses for future testing in the laboratory, and suggests important ways to counteract the global sleep crisis.

Neural Integration Underlying a Time-Compensated Sun Compass in the Migratory Monarch Butterfly

Migrating eastern North American monarch butterflies use a time-compensated sun compass to adjust their flight to the southwest direction. Although the antennal genetic circadian clock and the azimuth of the sun are instrumental for proper function of the compass, it is unclear how these signals are represented on a neuronal level and how they are integrated to produce flight control. To address these questions, we constructed a receptive field model of the compound eye that encodes the solar azimuth. We then derived a neural circuit model that integrates azimuthal and circadian signals to correct flight direction. The model demonstrates an integration mechanism, which produces robust trajectories reaching the southwest regardless of the time of day and includes a configuration for remigration. Comparison of model simulations with flight trajectories of butterflies in a flight simulator shows analogous behaviors and affirms the prediction that midday is the optimal time for migratory flight.

Characterizing and modeling the intrinsic light response of rat ganglion-cell photoreceptors

Intrinsically photosensitive retinal ganglion cells (ipRGCs) mediate both image-forming vision and non-image-forming visual responses such as pupillary constriction and circadian photoentrainment. Five types of ipRGCs, named M1-M5, have been discovered in rodents. To further investigate their photoresponse properties, we made multielectrode array spike recordings from rat ipRGCs, classified them into M1, M2/M4, and M3/M5 clusters, and measured their intrinsic, melanopsin-based responses to single and flickering light pulses. Results showed that ipRGC spiking can track flickers up to ∼0.2 Hz in frequency and that flicker intervals between 5 and 14 s evoke the most spikes. We also learned that melanopsin's integration time is intensity and cluster dependent. Using these data, we constructed a mathematical model for each cluster's intrinsic photoresponse. We found that the data for the M1 cluster are best fit by a model that assumes a large photoresponse, causing the cell to enter depolarization block. Our models also led us to hypothesize that the M2/M4 and M3/M5 clusters experience comparable photoexcitation but that the M3/M5 cascade decays significantly faster than the M2/M4 cascade, resulting in different response waveforms between these clusters. These mathematical models will help predict how each ipRGC cluster might respond to stimuli of any waveform and could inform the invention of lighting technologies that promote health through melanopsin stimulation.

Collective phase response curves for heterogeneous coupled oscillators

Phase response curves (PRCs) have become an indispensable tool in understanding the entrainment and synchronization of biological oscillators. However, biological oscillators are often found in large coupled heterogeneous systems and the variable of physiological importance is the collective rhythm resulting from an aggregation of the individual oscillations. To study this phenomena we consider phase resetting of the collective rhythm for large ensembles of globally coupled Sakaguchi-Kuramoto oscillators. Making use of Ott-Antonsen theory we derive an asymptotically valid analytic formula for the collective PRC. A result of this analysis is a characteristic scaling for the change in the amplitude and entrainment points for the collective PRC compared to the individual oscillator PRC. We support the analytical findings with numerical evidence and demonstrate the applicability of the theory to large ensembles of coupled neuronal oscillators.

GABA-mediated repulsive coupling between circadian clock neurons in the SCN encodes seasonal time

The mammalian suprachiasmatic nucleus (SCN) forms not only the master circadian clock but also a seasonal clock. This neural network of ∼10,000 circadian oscillators encodes season-dependent day-length changes through a largely unknown mechanism. We show that region-intrinsic changes in the SCN fine-tune the degree of network synchrony and reorganize the phase relationship among circadian oscillators to represent day length. We measure oscillations of the clock gene Bmal1, at single-cell and regional levels in cultured SCN explanted from animals raised under short or long days. Coupling estimation using the Kuramoto framework reveals that the network has couplings that can be both phase-attractive (synchronizing) and -repulsive (desynchronizing). The phase gap between the dorsal and ventral regions increases and the overall period of the SCN shortens with longer day length. We find that one of the underlying physiological mechanisms is the modulation of the intracellular chloride concentration, which can adjust the strength and polarity of the ionotropic GABAA-mediated synaptic input. We show that increasing day-length changes the pattern of chloride transporter expression, yielding more excitatory GABA synaptic input, and that blocking GABAA signaling or the chloride transporter disrupts the unique phase and period organization induced by the day length. We test the consequences of this tunable GABA coupling in the context of excitation-inhibition balance through detailed realistic modeling. These results indicate that the network encoding of seasonal time is controlled by modulation of intracellular chloride, which determines the phase relationship among and period difference between the dorsal and ventral SCN.

Characterizing spiking in noisy type II neurons

Understanding the dynamics of noisy neurons remains an important challenge in neuroscience. Here, we describe a simple probabilistic model that accurately describes the firing behavior in a large class (type II) of neurons. To demonstrate the usefulness of this model, we show how it accurately predicts the interspike interval (ISI) distributions, bursting patterns and mean firing rates found by: (1) simulations of the classic Hodgkin-Huxley model with channel noise, (2) experimental data from squid giant axon with a noisy input current and (3) experimental data on noisy firing from a neuron within the suprachiasmatic nucleus (SCN). This simple model has 6 parameters, however, in some cases, two of these parameters are coupled and only 5 parameters account for much of the known behavior. From these parameters, many properties of spiking can be found through simple calculation. Thus, we show how the complex effects of noise can be understood through a simple and general probabilistic model.

It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms

Biological rhythms, generated by feedback loops containing interacting genes, proteins and/or cells, time physiological processes in many organisms. While many of the components of the systems that generate biological rhythms have been identified, much less is known about the details of their interactions. Using examples from the circadian (daily) clock in three organisms, Neurospora, Drosophila and mouse, we show, with mathematical models of varying complexity, how interactions among (i) promoter sites, (ii) proteins forming complexes, and (iii) cells can have a drastic effect on timekeeping. Inspired by the identification of many transcription factors, for example as involved in the Neurospora circadian clock, that can both activate and repress, we show how these multiple actions can cause complex oscillatory patterns in a transcription–translation feedback loop (TTFL). Inspired by the timekeeping complex formed by the NMO–PER–TIM–SGG complex that regulates the negative TTFL in the Drosophila circadian clock, we show how the mechanism of complex formation can determine the prevalence of oscillations in a TTFL. Finally, we note that most mathematical models of intracellular clocks model a single cell, but compare with experimental data from collections of cells. We find that refitting the most detailed model of the mammalian circadian clock, so that the coupling between cells matches experimental data, yields different dynamics and makes an interesting prediction that also matches experimental data: individual cells are bistable, and network coupling removes this bistability and causes the network to be more robust to external perturbations. Taken together, we propose that the interactions between components in biological timekeeping systems are carefully tuned towards proper function. We also show how timekeeping can be controlled by novel mechanisms at different levels of organization.

Optimal schedules of light exposure for rapidly correcting circadian misalignment

Jet lag arises from a misalignment of circadian biological timing with the timing of human activity, and is caused by rapid transmeridian travel. Jet lag's symptoms, such as depressed cognitive alertness, also arise from work and social schedules misaligned with the timing of the circadian clock. Using experimentally validated mathematical models, we develop a new methodology to find mathematically optimal schedules of light exposure and avoidance for rapidly re-entraining the human circadian system. In simulations, our schedules are found to significantly outperform other recently proposed schedules. Moreover, our schedules appear to be significantly more robust to both noise in light and to inter-individual variations in endogenous circadian period than other proposed schedules. By comparing the optimal schedules for thousands of different situations, and by using general mathematical arguments, we are also able to translate our findings into general principles of optimal circadian re-entrainment. These principles include: 1) a class of schedules where circadian amplitude is only slightly perturbed, optimal for dim light and for small shifts 2) another class of schedules where shifting occurs along the shortest path in phase-space, optimal for bright light and for large shifts 3) the determination that short light pulses are less effective than sustained light if the goal is to re-entrain quickly, and 4) the determination that length of daytime should be significantly shorter when delaying the clock than when advancing it.

When our body's internal timekeeping system becomes misaligned with the time of day in the outside world, many negative effects can be felt, including decreased performance, improper sleep, and jet lag. When misalignment is prolonged, it can also lead to serious medical conditions, including cancer, cardiovascular disease, and possibly even late-onset diabetes. Rapid readjustment of our internal daily (circadian) clock by properly timed exposure to light, which is the strongest signal to our internal circadian clock, is therefore important to the large proportion of the population which suffers from misalignment, including transmeridian travelers, shift workers, and individuals with circadian disorders. Here we develop a methodology to determine schedules of light exposure which may shift the human circadian clock in the minimum time. By calculating thousands of schedules, we show how the human circadian pacemaker is predicted to be capable of shifting much more rapidly than previously thought, simply by adjusting the timing of the beginning and end of each day. Schedules are summarized into general principles of optimal shifting, which can be applied without knowledge of the schedules themselves.

Modeling the effects of the circadian clock on cardiac electrophysiology

An internal circadian clock regulates the electrical activity of cardiac myocytes controlling the expression of potassium channel interacting protein-2 (KChIP2), which is a key regulator of cardiac electrical activity. Here, we examine how the circadian rhythm of KChIP2 expression affects the dynamics of human and murine ventricular action potentials (APs), as well as the intervals in the equivalent electrocardiograms (ECGs) reflecting the duration of depolarization and repolarization phases of the cardiac ventricular APs (QRS and QT intervals), with mathematical modeling. We show how the internal circadian clock can control the shape of APs and, in particular, predict AP, QRS, and QT interval prolongation following KChIP2 downregulation, as well as shortening of AP, QRS, and QT interval duration following KChIP2 upregulation. Based on the circadian expression of KChIP2, we can accurately predict the circadian rhythm in cardiac electrical activity and suggest the transient outward potassium currents as the key current for circadian rhythmicity. Our modeling work predicts a smaller effect of KChIP2 on AP and QT interval dynamics in humans. Taken together, these results support the role of KChIP2 as the key regulator of circadian rhythms in the electrical activity of the heart; we provide computational models that can be used to explore circadian rhythms in cardiac electrophysiology and susceptibility to arrhythmia.

Causes and consequences of hyperexcitation in central clock neurons

Hyperexcited states, including depolarization block and depolarized low amplitude membrane oscillations (DLAMOs), have been observed in neurons of the suprachiasmatic nuclei (SCN), the site of the central mammalian circadian (~24-hour) clock. The causes and consequences of this hyperexcitation have not yet been determined. Here, we explore how individual ionic currents contribute to these hyperexcited states, and how hyperexcitation can then influence molecular circadian timekeeping within SCN neurons. We developed a mathematical model of the electrical activity of SCN neurons, and experimentally verified its prediction that DLAMOs depend on post-synaptic L-type calcium current. The model predicts that hyperexcited states cause high intracellular calcium concentrations, which could trigger transcription of clock genes. The model also predicts that circadian control of certain ionic currents can induce hyperexcited states. Putting it all together into an integrative model, we show how membrane potential and calcium concentration provide a fast feedback that can enhance rhythmicity of the intracellular circadian clock. This work puts forward a novel role for electrical activity in circadian timekeeping, and suggests that hyperexcited states provide a general mechanism for linking membrane electrical dynamics to transcription activation in the nucleus.

Ionic mechanism underlying optimal stimuli for neuronal excitation: role of Na+ channel inactivation

The ionic mechanism underlying optimal stimulus shapes that induce a neuron to fire an action potential, or spike, is relevant to understanding optimal information transmission and therapeutic stimulation in the nervous system. Here we analyze for the first time the ionic basis for stimulus optimality in the Hodgkin and Huxley model and for eliciting a spike in squid giant axons, the preparation for which the model was devised. The experimentally determined stimulus is a smoothly varying biphasic current waveform having a relatively long and shallow hyperpolarizing phase followed by a depolarizing phase of briefer duration. The hyperpolarizing phase removes a small degree of the resting level of Na⁺ channel inactivation. This result together with the subsequent depolarizing phase provides a signal that is energetically more efficient for eliciting spikes than rectangular current pulses. Sodium channel inactivation is the only variable that is changed during the stimulus waveform, other than the membrane potential, V. The activation variables for Na⁺ and K⁺ channels are unchanged throughout the stimulus. This result demonstrates how an optimal stimulus waveform relates to ionic dynamics and may have implications for energy efficiency of neural excitation in many systems including the mammalian brain.

Circadian regulation of sleep-wake behaviour in nocturnal rats requires multiple signals from suprachiasmatic nucleus

The dynamics of sleep and wake are strongly linked to the circadian clock. Many models have accurately predicted behaviour resulting from dynamic interactions between these two systems without specifying physiological substrates for these interactions. By contrast, recent experimental work has identified much of the relevant physiology for circadian and sleep-wake regulation, but interaction dynamics are difficult to study experimentally. To bridge these approaches, we developed a neuronal population model for the dynamic, bidirectional, neurotransmitter-mediated interactions of the sleep-wake and circadian regulatory systems in nocturnal rats. This model proposes that the central circadian pacemaker, located within the suprachiasmatic nucleus (SCN) of the hypothalamus, promotes sleep through single neurotransmitter-mediated signalling to sleep-wake regulatory populations. Feedback projections from these populations to the SCN alter SCN firing patterns and fine-tune this modulation. Although this model reproduced circadian variation in sleep-wake dynamics in nocturnal rats, it failed to describe the sleep-wake dynamics observed in SCN-lesioned rats. We thus propose two alternative, physiologically based models in which neurotransmitter- and neuropeptide-mediated signalling from the SCN to sleep-wake populations introduces mechanisms to account for the behaviour of both the intact and SCN-lesioned rat. These models generate testable predictions and offer a new framework for modelling sleep-wake and circadian interactions.

Optimal stimulus shapes for neuronal excitation

An important problem in neuronal computation is to discern how features of stimuli control the timing of action potentials. One aspect of this problem is to determine how an action potential, or spike, can be elicited with the least energy cost, e.g., a minimal amount of applied current. Here we show in the Hodgkin & Huxley model of the action potential and in experiments on squid giant axons that: 1) spike generation in a neuron can be highly discriminatory for stimulus shape and 2) the optimal stimulus shape is dependent upon inputs to the neuron. We show how polarity and time course of post-synaptic currents determine which of these optimal stimulus shapes best excites the neuron. These results are obtained mathematically using the calculus of variations and experimentally using a stochastic search methodology. Our findings reveal a surprising complexity of computation at the single cell level that may be relevant for understanding optimization of signaling in neurons and neuronal networks.

Computational neuroscience seeks to understand the mechanisms by which signals excite a neuron or a neuronal network. An important consideration in these studies is optimality, i.e., what signal most effectively causes excitation. Optimization of neuronal signaling is important for networks that need to minimize energy costs, for sensory neurons to selectively respond to specific stimulus features, and for therapeutic deep brain stimulators to maximize battery life. Here we show in a classic mathematical model of the action potential and in experiments on a single cell preparation that: 1) a single neuron can be highly discriminatory for the shape of low amplitude stimuli that elicit an action potential and 2) the shape of the optimal stimulus depends upon the overall state of inputs to the neuron. Our findings reveal a surprising complexity of computation at the single cell level that may be important for understanding physiological function of the nervous system.

Emergence of noise-induced oscillations in the central circadian pacemaker

Bmal1 is an essential transcriptional activator within the mammalian circadian clock. We report here that the suprachiasmatic nucleus (SCN) of Bmal1-null mutant mice, unexpectedly, generates stochastic oscillations with periods that overlap the circadian range. Dissociated SCN neurons expressed fluctuating levels of PER2 detected by bioluminescence imaging but could not generate circadian oscillations intrinsically. Inhibition of intercellular communication or cyclic-AMP signaling in SCN slices, which provide a positive feed-forward signal to drive the intracellular negative feedback loop, abolished the stochastic oscillations. Propagation of this feed-forward signal between SCN neurons then promotes quasi-circadian oscillations that arise as an emergent property of the SCN network. Experimental analysis and mathematical modeling argue that both intercellular coupling and molecular noise are required for the stochastic rhythms, providing a novel biological example of noise-induced oscillations. The emergence of stochastic circadian oscillations from the SCN network in the absence of cell-autonomous circadian oscillatory function highlights a previously unrecognized level of circadian organization.

Daily electrical silencing in the mammalian circadian clock

Neurons in the brain's suprachiasmatic nuclei (SCNs), which control the timing of daily rhythms, are thought to encode time of day by changing their firing frequency, with high rates during the day and lower rates at night. Some SCN neurons express a key clock gene, period 1 (per1). We found that during the day, neurons containing per1 sustain an electrically excited state and do not fire, whereas non-per1 neurons show the previously reported daily variation in firing activity. Using a combined experimental and theoretical approach, we explain how ionic currents lead to the unusual electrophysiological behaviors of per1 cells, which unlike other mammalian brain cells can survive and function at depolarized states.

Clustering predicted by an electrophysiological model of the suprachiasmatic nucleus

Despite the wealth of experimental data on the electrophysiology of individual neurons in the suprachiasmatic nuclei (SCN), the neural code of the SCN remains largely unknown. To predict the electrical activity of the SCN, the authors simulated networks of 10,000 GABAergic SCN neurons using a detailed model of the ionic currents within SCN neurons. Their goal was to understand how neuronal firing, which occurs on a time scale faster than a second, can encode a set phase of the circadian (24-h) cycle. The authors studied the effects of key network properties including: 1) the synaptic density within the SCN, 2) the magnitude of postsynaptic currents, 3) the heterogeneity of circadian phase in the neuronal population, 4) the degree of synaptic noise, and 5) the balance between excitation and inhibition. Their main result was that under a wide variety of conditions, the SCN network spontaneously organized into (typically 3) groups of synchronously firing neurons. They showed that this type of clustering can lead to the silencing of neurons whose intracellular clocks are out of circadian phase with the rest of the population. Their results provide clues to how the SCN may generate a coherent electrical output signal at the tissue level to time rhythms throughout the body.

Taking the lag out of jet lag through model-based schedule design

Travel across multiple time zones results in desynchronization of environmental time cues and the sleep–wake schedule from their normal phase relationships with the endogenous circadian system. Circadian misalignment can result in poor neurobehavioral performance, decreased sleep efficiency, and inappropriately timed physiological signals including gastrointestinal activity and hormone release. Frequent and repeated transmeridian travel is associated with long-term cognitive deficits, and rodents experimentally exposed to repeated schedule shifts have increased death rates. One approach to reduce the short-term circadian, sleep–wake, and performance problems is to use mathematical models of the circadian pacemaker to design countermeasures that rapidly shift the circadian pacemaker to align with the new schedule. In this paper, the use of mathematical models to design sleep–wake and countermeasure schedules for improved performance is demonstrated. We present an approach to designing interventions that combines an algorithm for optimal placement of countermeasures with a novel mode of schedule representation. With these methods, rapid circadian resynchrony and the resulting improvement in neurobehavioral performance can be quickly achieved even after moderate to large shifts in the sleep–wake schedule. The key schedule design inputs are endogenous circadian period length, desired sleep–wake schedule, length of intervention, background light level, and countermeasure strength. The new schedule representation facilitates schedule design, simulation studies, and experiment design and significantly decreases the amount of time to design an appropriate intervention. The method presented in this paper has direct implications for designing jet lag, shift-work, and non-24-hour schedules, including scheduling for extreme environments, such as in space, undersea, or in polar regions.

Traveling across several times zones can cause an individual to experience “jet lag,” which includes trouble sleeping at night and trouble remaining awake during the day. A major cause of these effects is the desynchronization between the body's internal circadian clock and local environmental cues. A well-known intervention to resynchronize an individual's clock with the environment is appropriately timed light exposure. Used as an intervention, properly timed light stimuli can reset an individual's internal circadian clock to align with local time, resulting in more efficient sleep, a decrease in fatigue, and an increase in cognitive performance. The contrary is also true: poorly timed light exposure can prolong the resynchronization process. In this paper, we present a computational method for automatically determining the proper placement of these interventional light stimuli. We used this method to simulate shifting sleep–wake schedules (as seen in jet lag situations) and design interventions. Essential to our approach is the use of mathematical models that simulate the body's internal circadian clock and its effect on human performance. Our results include quicker design of multiple schedule alternatives and predictions of substantial performance improvements relative to no intervention. Therefore, our methods allow us to use these models not only to assess schedules but also to interactively design schedules that will result in improved performance.

Keeping the beat in the rising heat

Circadian clocks use temperature compensation to keep accurate time over a range of temperatures, thus allowing reliable timekeeping under diverse environmental conditions. Mehra et al. (2009) and Baker et al. (2009) now show that phosphorylation-regulated protein degradation plays a key role in circadian temperature compensation.

A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons

The Hodgkin and Huxley (HH) model predicts sustained repetitive firing of nerve action potentials for a suprathreshold depolarizing current pulse for as long as the pulse is applied (type 2 excitability). Squid giant axons, the preparation for which the model was intended, fire only once at the beginning of the pulse (type 3 behaviour). This discrepancy between the theory and experiments can be removed by modifying a single parameter in the HH equations for the K+ current as determined from the analysis in this paper. K+ currents in general have been described by IK=gK(V-EK), where gK is the membrane's K+ current conductance and EK is the K+ Nernst potential. However, IK has a nonlinear dependence on (V-EK) well described by the Goldman-Hodgkin-Katz equation that determines the voltage dependence of gK. This experimental finding is the basis for the modification in the HH equations describing type 3 behaviour. Our analysis may have broad significance given the use of IK=gK(V-EK) to describe K+ currents in a wide variety of biological preparations.

Rate constants rather than biochemical mechanism determine behaviour of genetic clocks

Many biological systems contain both positive and negative feedbacks. These are often classified as resonators or integrators. Resonators respond preferentially to oscillating signals of a particular frequency. Integrators, on the other hand, accumulate a response to signals. Computational neuroscientists often refer to neurons showing integrator properties as type I neurons and those showing resonator properties as type II neurons. Guantes & Poyatos have shown that type I or type II behaviour can be seen in genetic clocks. They argue that when negative feedback occurs through transcription regulation and post-translationally, genetic clocks act as integrators and resonators, respectively. Here we show that either behaviour can be seen with either design and in a wide range of genetic clocks. This highlights the importance of parameters rather than biochemical mechanism in determining the system behaviour.

Modeling the electrophysiology of suprachiasmatic nucleus neurons

Neurons in the SCN act as the central circadian (approximately 24-h) pacemaker in mammals. Using measurements of the ionic currents in SCN neurons, the authors fit a Hodgkin-Huxley-type model that accurately reproduces slow (approximately 28 Hz) neural firing as well as the contributions of ionic currents during an action potential. When inputs of other SCN neurons are considered, the model accurately predicts the fractal nature of firing rates and the appearance of random bursting. In agreement with experimental data, the molecular clock within these neurons modulates the firing rate through small changes in the concentration of internal calcium, calcium channels, or potassium channels. Predictions are made on how signals from other neurons can start, stop, speed up, or slow down firing. Only a slow sodium inactivation variable and voltage do not reach equilibrium during the interval between action potentials, and based on this finding, a reduced model is formulated.

After hours keeps clock researchers CRYing Overtime

Three recent reports, including one in this issue of Cell, reveal that the circadian regulator CRY is targeted for degradation by the F box E3 ubiquitin ligase FBXL3 (Siepka et al., 2007; Busino et al., 2007; Godinho et al., 2007). These studies confirm the importance of targeted protein degradation as a key design feature of the mammalian circadian clock.

Reversible protein phosphorylation regulates circadian rhythms

Protein phosphorylation regulates the period of the circadian clock within mammalian cells. Circadian rhythms are an approximately 24-hour cycle that regulates key biological processes. Daily fluctuations of wakefulness, stress hormones, lipid metabolism, immune function, and the cell division cycle are controlled by the molecular clocks that function throughout our bodies. Mutations in regulatory components of the clock can shorten or lengthen the timing of the rhythms and have significant physiological consequences. The clock is formed by a negative feedback loop of transcription, translation, and inhibition of transcription. The precision of clock timing is controlled by protein kinases and phosphatases. Casein kinase Iepsilon is a protein kinase that regulates the circadian clock by periodic phosphorylation of the proteins PER1 and PER2, controlling their stability and localization. The role of phosphorylation in regulating PER function in the clock has been explored in detail. Quantitative modeling has proven to be very useful in making important predictions about how changes in phosphorylation alter the clock's behavior. Quantitative data from biological studies can be used to refine the quantitative model and make additional testable predictions. A detailed understanding of how reversible protein phosphorylation regulates circadian rhythms and a detailed quantitative model that makes clear, testable, and accurate predictions about the clock and how we may manipulate it can have important benefits for human health. Pharmacological manipulation of rhythms could mitigate stress from jet lag, shift work, and perhaps even seasonal affective disorder.

Noisy inputs and the induction of on-off switching behavior in a neuronal pacemaker

Neuronal oscillators can function as bistable toggle switches, flipping between quiescence and rhythmic firing in response to an input stimulus. In theory, such switching should be sensitive to small noisy inputs if the bistable states are in close proximity, which we test here using a perfused squid axon preparation. We find that small noisy stimulus currents induce a multitude of paths between two nearby stable states: repetitive firing and quiescence. The pattern of on-off switching of the pacemaker depends on the intensity, spectral properties, and phase angle of stimulus current fluctuations. Analysis by spike-triggered averaging of the stimulus currents near the transitions reveals that sinusoidal stimuli timed antiphase or in phase with repetitive firing correlates with switching of the pacemaker off or on, respectively. Our results reveal a distinct form of bistability in which noise can either silence pacemaker activity, trigger repetitive firing, or induce sporadic burst patterns similar to those recorded in a variety of normal and pathological neurons.

An opposite role for tau in circadian rhythms revealed by mathematical modeling

Biological clocks with a period of approximately 24 h (circadian) exist in most organisms and time a variety of functions, including sleep-wake cycles, hormone release, bioluminescence, and core body temperature fluctuations. Much of our understanding of the clock mechanism comes from the identification of specific mutations that affect circadian behavior. A widely studied mutation in casein kinase I (CKI), the CKIepsilon(tau) mutant, has been shown to cause a loss of kinase function in vitro, but it has been difficult to reconcile this loss of function with the current model of circadian clock function. Here we show that mathematical modeling predicts the opposite, that the kinase mutant CKIepsilon(tau) increases kinase activity, and we verify this prediction experimentally. CKIepsilon(tau) is a highly specific gain-of-function mutation that increases the in vivo phosphorylation and degradation of the circadian regulators PER1 and PER2. These findings experimentally validate a mathematical modeling approach to a complex biological function, clarify the role of CKI in the clock, and demonstrate that a specific mutation can be both a gain and a loss of function depending on the substrate.

Stochastic simulation of the mammalian circadian clock

Circadian (nearly 24-h) clocks are remarkably accurate at timing biological events despite the randomness of their biochemical reactions. Here we examine the causes of their immunity to molecular noise in the context of a detailed stochastic mathematical model of the mammalian circadian clock. This stochastic model is a direct generalization of the deterministic mammalian circadian clock model previously developed. A feature of that model is that it completely specifies all molecular reactions, leaving no ambiguity in the formulation of a stochastic version of the model. With parameters based on experimental data concerning clock protein concentrations within a cell, we find accurate circadian rhythms in our model only when promoter interaction occurs on the time scale of seconds. As the model is scaled up by proportionally increasing the numbers of molecules of all species and the reaction rates with the promoter, the observed variability scales as 1/n(0.5), where n is the number of molecules of any species. Our results show that gene duplication increases robustness by providing more promoters with which the transcription factors of the model can interact. Although PER2 mutants were not rhythmic in the deterministic version of this model, they are rhythmic in the stochastic version.

Comparison of amplitude recovery dynamics of two limit cycle oscillator models of the human circadian pacemaker

At an organism level, the mammalian circadian pacemaker is a two-dimensional system. For these two dimensions, phase (relative timing) and amplitude of the circadian pacemaker are commonly used. Both the phase and the amplitude (A) of the human circadian pacemaker can be observed within multiple physiological measures--including plasma cortisol, plasma melatonin, and core body temperature (CBT)--all of which are also used as markers of the circadian system. Although most previous work has concentrated on changes in phase of the circadian system, critically timed light exposure can significantly reduce the amplitude of the pacemaker. The rate at which the amplitude recovers to its equilibrium level after reduction can have physiological significance. Two mathematical models that describe the phase and amplitude dynamics of the pacemaker have been reported. These models are essentially equivalent in predictions of phase and in predictions of amplitude recovery for small changes from an equilibrium value (A = 1), but are markedly different in the prediction of recovery rates when A < 0.6. To determine which dynamic model best describes the amplitude recovery observed in experimental data; both models were fit to CBT data using a maximum likelihood procedure and compared using Akaike's Information Criterion (AIC). For all subjects, the model with the lower recovery rate provided a better fit to data in terms of AIC, supporting evidence that the amplitude recovery of the endogenous pacemaker is slow at low amplitudes. Experiments derived from model predictions are proposed to test the influence of low amplitude recovery on the physiological and neurobehavioral functions.

Development and validation of computational models for mammalian circadian oscillators

Circadian rhythms are endogenous rhythms with a cycle length of approximately 24 h. Rhythmic production of specific proteins within pacemaker structures is the basis for these physiological and behavioral rhythms. Prior work on mathematical modeling of molecular circadian oscillators has focused on the fruit fly, Drosophila melanogaster. Recently, great advances have been made in our understanding of the molecular basis of circadian rhythms in mammals. Mathematical models of the mammalian circadian oscillator are needed to piece together diverse data, predict experimental results, and help us understand the clock as a whole. Our objectives are to develop mathematical models of the mammalian circadian oscillator, generate and test predictions from these models, gather information on the parameters needed for model development, integrate the molecular model with an existing model of the influence of light and rhythmicity on human performance, and make models available in BioSpice so that they can be easily used by the general community. Two new mammalian models have been developed, and experimental data are summarized. These studies have the potential to lead to new strategies for resetting the circadian clock. Manipulations of the circadian clock can be used to optimize performance by promoting alertness and physiological synchronization.