Forger_2016_ Research Statement_1

Statement of Research Interests Daniel Forger Overview: I use theory, mathematics, modeling, and simulations to understand problems in biology and medicine. I study biological problems deeply to uncover unexpected results, and see how they can inspire new mathematical and computational tools. I aim for foundational collaborative papers, each of which could be split into several standard papers, with deep theory and new data in high-impact interdisciplinary journals. My publications include papers in Science (1), Proceedings of the National Academy of Science (PNAS) (7), Nature Communications (1), Science Advances (1), Cell Reports (1), PLoS Biology (2), Molecular Systems Biology (1), Molecular Cell (1) and Translational Psychiatry (1)). Interdisciplinary training: My formal training at Harvard and NYU was in mathematics, but I acquired parallel training in the biological sciences. My PhD, for instance, was partially funded by an NIH training grant. I also spent two years in an experimental molecular biology post-doc at NYU, and portions of eight summers at the Marine Biological Laboratory at Woods Hole. As a result of this interdisciplinary training and research, I now hold appointments in the University of Michigan’s mathematics Department, medical school, and many centers throughout campus. Moreover, most of the students in my classes are engineers. My lab: My undergraduates, graduate students, and post-docs are trained with strong theoretical interests to work with real experimental data. Often they acquire such data themselves though partner labs, or they collect it with our mobile technology, currently installed in nearly 200,000 phones. However, to maximize the impact of our work, most experimental data comes from other labs. This collaborative focus arises because one lab cannot build synthetic circuits in bacteria, record from neurons in the Monarch brain, study the molecular biology of timekeeping intricately, and simultaneously record sleep. As a result, I have established collaborations with a wide variety of different labs. This innovative approach has led to several publications in leading academic journals and has launched the careers of the majority of my graduate students and post-docs into top academic positions. Several have even gone on to win major prizes, such as being the inaugural winner of the Peter Smereka Memorial Prize or the Sumner-Myers Prize). Research areas: Most of my work has historically been in the field of biological rhythms (project 1). In recent years, however, this has grown into understanding the physiology of human behavior, a key part of neuroscience using large-scale computational models to bridge experimental data at many levels, and mobile technology to accurately test hypotheses from these models in the real world. Key biological problems are mammalian (human) circadian (daily) timekeeping and sleep (projects 2 and 4), mood disorders (project 3) and subconscious vision (project 5). Future goals: Models and apps that improve global health including: 1) A detailed personalized model of the mammalian circadian clock that can be linked to our Entrain app measuring sleep and circadian rhythms 2) A physiological model proposing the etiology of Bipolar Disorder linked to an app tracking and predicting mood in individuals with Bipolar Disorder 3) A detailed model of vision for individuals who have limited or no rod or cone function linked to apps that can process visual scenes to improve vision.

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Research Topic 1: Biological Rhythms Project 1: The Theory of Biological Timekeeping (Website under construction at mitpress.mit.edu) Overview: My work studies biological rhythms using techniques from many disciplines including mathematics, physics, systems biology, statistics, nonlinear dynamics, biomedical engineering, pharmacokinetics control theory, and chemical dynamics. I have laid out this innovative approach, showing others how to employ similar tools and techniques to study these problems in a book now in press at MIT Press. In this book I show how to:

1) Develop models of biological clocks that accurately reflect physiology 2) Discover how genes, proteins, or cells interact to time events accurately 3) Determine how biological clocks respond to their environmental 4) Derive the optimal signals to shift biological clocks

Examples of novel mathematical results include:

1) Several new intuitive proofs of results in the theory of biological clocks. For example, I prove the result by Mirollo and Strogatz (Mirollo and Strogatz, 1990)that pulse coupled oscillators synchronize with probability 1 with a new definition of the speed of an oscillator. Moreover, I show how the Poincaré-Bendixson theorem, which holds for monotonic models of biological clocks of arbitrary dimension (Mallet-Paret and Smith, 1990), can be seen through iterative maps. I have also worked on a global version of the Secant Condition, which determines when oscillations appear in biochemical feedback loops, publishing the findings in PNAS (Forger, 2011).

2) The existence and uniqueness of equations of biochemical oscillators given their

solutions. For a set of equations: !"!"= 𝑓(𝑠 𝑡 )−  𝑔(𝑥), for instance, if we are given

s(t) and x(t) over a time interval, and we assume f and g are continuous, how do we know if they could come from an equation as above? If they can, do they uniquely determine f and g? Neither of these statements is true in general. I demonstrate, in particular, how this is an extension of the principle of identifibility for the modeling of biological clocks. I also define how “completely determinable dynamic,” or systems where a brief measurement of the dynamics start at an initial state, can determine dynamics globally. I have published the details of these findings in the SIAM journal on Applied Mathematics with Jae Kyoung Kim (Kim and Forger, 2012b).

General Results: 1) Robust Rhythm Generation in Biochemical Feedback Loops: In a paper published in

Molecular Systems Biology with my student Jae Kyoung Kim (Kim and Forger, 2012a) we establish a new structure for robust rhythm generation in biochemical feedback loops. A common design principle for generating rhythms in biological systems is to have a fast positive feedback loop combined with a slower negative feedback loop. Here, however, we show that a double negative feedback loop structure can also give robust rhythms and additionally can generate rhythms with a tightly controlled period. At the heart of this mechanism is a transcription regulation mechanism

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where an activator of a gene is sequestered by a repressor, a motif found in most circadian clocks. Our paper has two stars on F1000.

2) Temperature Compensation: In a paper published in Molecular Cell (Zhou et al., 2015) I demonstrate a new understanding of how biological clocks are temperature compensated. Some biological clocks are “temperature compensated” in that as the temperature of the system increases, and thus, presumably, all the reaction rates of the system increase, the period remains unchanged. Ruoff conjectured that all models of biochemical oscillators can be temperature compensated. I, however, provide several counterexamples to this claim in my book. Moreover, in further joint experimental/modeling work with the Virshup lab at Duke NUS, I have also explored how the mammalian circadian clock is temperature compensated (Zhou et al., 2015). In this piece, we demonstrate how one of the key proteins in the mammalian circadian clock, PERIOD2, acts as a phosphoswitch where competing phosphorylation events cause PERIOD2 to become more stable at higher temperatures, thereby providing temperature compensation.

3) Computational Neuroscience: Many neuronal systems show bursts of neuronal firing interspersed with quiescence. I developed a mathematical theory for this phenomenon, together with Katarina Bodova (IST) and David Paydarfar (UT Austin), which predicts the statistics of these bursts (Bodova et al., 2015). With Paydarfar and John Clay (NIH), I have also determined the optimal signals to cause a neuron to fire (Forger et al., 2011), and updated the Hodgkin-Huxley equations to fix their erroneous prediction of repetitive firing (Clay et al., 2008).

4) Synthetic Biology: In a collaboration funded by an NIH R01 with Alex Ninfa (Michigan), we built synthetic switches and clocks in bacteria that are particularly robust to environmental changes based on a double negative feedback loop design (Chang et al., 2010).

Future work:

1) Optimal control: I am currently seeking to determine how neuronal firing can be optimally controlled. That is, we are attempting to understand how to induce or stop firing of a neuronal network with an electrical signal that carries the least amount of current.

2) Neuroscience: I am currently exploring and have already published with Eli Shlizerman (University of Washington) our preliminary findings on how a Monarch butterfly’s brain processes information about the sun’s position, in conjunction with its internal circadian clock, to navigate thousands of miles (Shlizerman et al., 2016)

3) Physics: I am conducting ongoing research, with current graduate student Kevin Hannay and Victoria Booth, on the validity of the Ott-Antonsen Anzatz, which gives a simple picture of the behavior of coupled oscillators.

Funding: This research has been funded by the Sloan Foundation, the National Institutes of Health and the Air Force Office of Scientific Research (AFOSR). Sample press coverage: 1) http://www.livescience.com/887-understanding-human-body-clock-reworked.html 2) bbc.com/news/science-environment-36046746

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Research Topic 2: Large Scale Computational Models of the Brain Project 2: A Mathematical Model of the Mammalian Circadian Timekeeping System (Website: www.clockmodel.com, under construction) Overview: My PhD work with Charles Peskin developed a mass action mathematical model of the biochemistry of circadian timekeeping within mammalian cells (Forger and Peskin, 2003; Forger and Peskin, 2004; Forger and Peskin, 2005) I continue to improve upon this model as more data becomes available. To date, it currently accounts for approximately 1,000 chemical reactions (Kim and Forger, 2012a). In particular, I show how a core pacemaker of about 10,000 neurons within the suprachiasmatic nuclei of the hypothalamus controls daily behavior in mammals (DeWoskin et al., 2014; DeWoskin et al., 2015; Myung et al., 2015).I have also developed a Hodgkin-Huxley type mathematical model for the electrical activity of these neurons (Belle et al., 2009; Diekman and Forger, 2009; Diekman et al., 2013). These models have been tested against data from over 20 laboratories, many of which I have collaborated with closely. To fully describe circadian timekeeping, we created a model which links the biochemistry of circadian timekeeping with the electrical activity of these cells in each of the 10,000 neurons. This model, described in detail in two back-to-back PNAS papers published in 2015, links events on very different timescales (milliseconds and days) (DeWoskin et al., 2015; Myung et al., 2015) Novel computational/mathematical methods: The real theoretical challenge in this project is one of numerical analysis, in that it requires the efficient simulation of such a large-scale model. Several advances have made these simulations possible. For instance, here is a highlight of a population density approach that I have worked on with one of my post-docs, Adam Stinchcombe: Consider a model for the noisy dynamics of a biological oscillator within a cell: 𝑑𝑥 =𝑣 𝑥, 𝑡 𝑑𝑡 + 𝜎𝑑𝑊. We then consider M coupled cells and wish to simulate their collective dynamics. In our models, x can have hundreds of variables, and M can easily be >1,000. The computation time for such models can be prohibitively large, particularly when the communication between cells is dense. We seek numerical methods to efficiently simulate these models. To do this, we employ a methodology that initially seems counterintuitive, but that, in the end, provides an efficient numerical method as described in (Stinchcombe and Forger, 2016). We take a population density approach by swapping the problem of thousands of oscillators for a partial differential equation (pde) for hundreds of variables while recognizing that pdes of more than a few variables can be difficult to solve. Our reason for doing this is because of a property that I have discovered about large-scale models of biological clocks, namely that their dynamics can often contract down to an effectively low, often two-dimensional manifold, quite quickly (Forger and Kronauer, 2002). As a result, to approximate the probability density by a particle method means that, although a very large dimensional space is explored, the particles can quickly converge to a low dimensional

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system that can be efficiently explored with a low (< 50) number of particles. Being more precise, our population density equation is given by:

𝜕𝜌𝜕𝑡 +  ∇ ∙ 𝑣  𝜌 + ∇ ∙ 𝐶  𝜌 + ∇ ∙ −𝐷∇𝜌 = 0

where C is the coupling function, for example 𝑐(𝑥,𝑦, 𝑡)𝜌 𝑦, 𝑡 𝑑𝑦!! . We approximate ρ

from particles where 𝜌 𝑥, 𝑡 =   𝑝! 𝑡 𝛿! 𝑥 − 𝑥! 𝑡!!!! where each of the N particles

has a position 𝑥! 𝑡 and strength 𝑝! 𝑡 . The particles evolve according to

!!!"= 𝑣 𝑥, 𝑡 + 𝐶 𝑡 −

!∇!

!  .

While this is a good idea, the key mathematics of this method remain to be determined, particularly, how to optimally reposition particles to give the most accurate yet fast estimates of C(t) and 𝐷∇𝜌 for the smallest number of particles.

General Results:

1) Neuroscience: In a paper published in Science, I developed a novel mechanism of signaling in the brain through depolarization block (Belle et al., 2009).

2) Systems Biology/Neuroscience/Physics: In a paper published in PLoS Biology, I, along with collaborators in the Takahashi, Kay and Welsh labs showed how that the mammalian circadian clock can generate rhythms from noise and coupled non-oscillating cells (Ko et al., 2010). This work received two stars on F1000.

3) Genetics/Sleep: In a collaboration with David Virshup (Duke-NUS,) we discovered the basis of an animal model of familiar advanced sleep disorder was the opposite of what was originally thought (Gallego et al., 2006). This work also received two starts on F1000.

4) Neuroscience: We are discovering how GABA signaling can multiplex and simultaneously code for different systems(DeWoskin et al., 2015; Myung et al., 2015)

Future work: We aim to use our new numerical methods to develop a next generation model of the SCN. Funding: This research has been funded by XSEDE, the Human Frontier Science Organization, and the AFOSR. I am also in the process of applying for further funding through the National Science Foundation, and plan to submit the grant application by 11/15/2016. Sample press coverage: 1) http://www.forbes.com/2009/10/14/circadian-rhythm-math-technology-breakthroughs-brain.html

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Project 3: A Mathematical Model of Subconscious Vision Overview: Vision was thought to be mediated by rod and cones in the retina. However, in the early 2000’s, a new type of photoreceptive cell in the retina, intrinsically photosensitive retinal ganglion cells (ipRGCs) was discovered. These cells control subconscious visual tasks including adjusting the circadian clock to the right time of day, controlling mood, pupal size and other potential tasks such as balance and gaze control. In collaborative work with Kwoon Wong (University of Michigan) funded by the Army Research Office, we have derived the first mathematical models of subconscious vision (Walch et al., 2015). Novel computational/mathematical methods: Our goal was to develop a mathematical model for subconscious vision. This model simulated ipRGCs as well as their inputs from other parts of the retina. It involved simulating ~1,000,000 cells. Adam Stinchcombe and I received a grant with David Cai (Courant/SJTU) and Douglas Zhou to develop faster numerical methods for simulating this phenomenon and other neuronal networks. We are currently working on exponential integrators and library methods to speed up our simulations. These fast simulations are needed for the large-scale simulations we do to fit the model and test it against different visual scenes. General results: Vision/Neuroscience: We have developed a mathematical model of subconscious vision that shows that this system contains much more information than previously thought, including the processing of spatial information. With the Lucas laboratory at the University of Manchester, we have been able to show that the Suprachismatic Nuclei (SCN) not only retains spatiotemporal information about the visual scene, but also enhances it. This is contrary to the long established view of the SCN’s function being solely a timekeeper sensing the average light levels over the visual scene. This work, submitted several weeks ago as two manuscripts to Neuron, suggests a new role for this important region of the brain. Future work: We are currently exploring the limits and acuity of subconscious vision with the Wong and Lucas laboratories. ipRGCs are considered one of the most important potential cures for blindness. If ipRGCs were to be able to affect conscious vision, we could also then use our mathematical models to preprocess visual scenes and build a device that would enhance vision in the blind. Funding: This research has been funded by the Biomathematics program at the Army Research Office (ARO), and the Shanghai Jiao Tong/University of Michigan (SJTU/UM) Joint Institute funding for Data Sciences.

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Research Topic 3: Understanding Human Behavior Through Smartphones, Wearables, and Big Data Project 4: Entrain, a Smartphone App Collecting Real World Data on Circadian Rhythms (Website: www.entrain.org) Overview: My mathematical model of the human circadian pacemaker is used widely in the academy, industry, and the military to determine how light affects the human circadian clock. It is used to determine schedules for shift workers and travelers to avoid jet-lag and circadian misalignment. A smartphone app, ENTRAIN, developed with my graduate student Olivia Walch, simulates this model and gives users optimal schedules to shift their circadian clock. This research app went viral and now has almost 200,000 installs. Novel computational/mathematical methods: With former student Kirill Serkh (Yale) we calculated the optimal schedules to shift humans from any timezone to any other with this model. This problem was originally considered intractable because of the stiffness of the equations. We began by showing the problem was bang-bang and revising an algorithm originally proposed for robotic control for this problem (Meier and Bryson, 1990; Serkh and Forger, 2014). Our paper on this topic has been highly read, with over 40,000 views. We also focused on rapid simulation of our model on smartphones with limited computation. General results: Collecting data from Entrain users in over 100 countries, we have run one of the world’s largest sleep studies based out of a mathematics department,. Some of our recently published results (Walch et al., 2016) include:

1) That women globally schedule about 30 more minutes of sleep then men per night. 2) That individuals sleep more uniformly as they age. 3) That society’s influence on sleep is greatest near bedtime.

Future work:

1) Optimal Control: Schedules were pre-computed in the original version of Entrain. While the full optimal control problem is too difficult to solve on a smartphone, we are developing new numerical methods to interpolate between the optimal schedules derived on smartphones in order to give more customizable schedules.

2) Data Sciences: Entrain 3.0 links with wearable data, so we now are now developing techniques to estimate circadian phase from steps and heart rate.

3) Big Data/Genomics: In partnership with the Health Intern Study at the University of Michigan (http://www.srijan-sen-lab.com/intern-health-study), a study of the mood, sleep, circadian behavior and genetics of thousands of medical residents, we are using our detailed models of circadian timekeeping (see Project 2 above) to predict differences in circadian behavior of individuals based on genetics. From this basic research we hope to be able to use our mobile technology to test our predictions.

Funding: This works has been funded through the (ARO), and (AFOSR). I have recently submitted two grants to the NSF/NIH QUBBD call and MIDAS. Sample press coverage: https://www.wired.com/2016/05/welcome-twenty-first-century-sleep-science/

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Project 5: A New, Personalized Diagnosis for Bipolar Disorder (website: www.prechterfund.org, www.moodmath.com under construction) Overview: Bipolar disorder (BD) affects 1-5% of the population and is characterized by episodes of mania (elevated mood) and depression interspersed with periods of euthymia (normal mood). Several hypothesizes about the basis of BD have been proposed including that it represents chaos of a neuronal system. With Amy Cochran, I have tested these hypotheses with data from the Prechter BP study, one of the largest longitudinal studies of BP, headed by our collaborator Melvin McInnis (University of Michigan). We have also developed a mathematical framework for personalizing BD. Novel Mathematical/Computational Methods: In our first study, currently submitted to Biological Psychiatry, we fit time-course data from almost 200 Bipolar patients with a Bayesian non-parametric hierarchical model with latent classes. From this, we determined how many classes Bipolar subjects should be classified into. We also developed a personalized model for Bipolar testing various mathematical frameworks and found a best fit with a Cox-Ingersoll-Ross (CIR) model (Cox et al., 1985):

𝑑𝐷!  = 𝑎!   𝑏! − 𝐷!   𝑑𝑡 + 2𝑎!  𝜎!  𝐷!   1− 𝜌!    𝑑𝑉! + 𝜌𝑑𝑊!  𝑑𝑀! = 𝑎!  (𝑏! −𝑀!  )𝑑𝑡 + 2𝑎!𝜎!𝑀!  𝑑𝑊!

 where 𝑉! and 𝑊! are independent Wiener processes. The seven parameters are assumed to be positive with 𝑏! ≥ 𝜎!  and 𝑏! ≥ 𝜎! to ensure affect is strictly positive. General Results:

1) Classification: Testing for different classes in Bipolar Disorder based on longitudinal mood, we found that Bipolar 1 subjects could be best classified into three groups. These groups have strong clinical significance in that the groupings are predictive when we tested their clinical phenotype. For example, one group has a much higher rate of suicide attempts. We recently published this work in in Translational Psychiatry (Cochran et al., 2016).

2) Personlizing predictions: Testing for basic hypotheses on Bipolar Disorder based on data from the Prechter study, we have not found any evidence for Bipolar Disorder being fundamentally rhythmic, one-dimensional, or multistable. From this, we deveiled an affective instability (CIR) model, in which we hypothesize that Bipolar Disorder is not generated by a clock, or a systems with stable manic or depressive states. We are currently in the process of submitting our affective instability (CIR) model to the journal Biological Psychiatry.

Future work:

1) In a collaboration with the Gamble group (University of Alabama at Birmingham Medical School) in press at Nature Communications, we show how GSK3, a key target of Lithium, regulates neuronal firing (Paul JR et al., 2016). We have also published two papers in Proceedings of the National Academy of Science (PNAS) arguing how GABA regulation affects seasonal affective disorder and mood (DeWoskin et al., 2015; Myung et al., 2015). Based on this initial research, we are now at a point where we can begin to build detailed computational models to test these hypotheses,

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specifically focusing our attention on the physiological basis of Bipolar Disorder. Finding the physiological basis of Bipolar Disorder is considered the leading problem in this field. 2) In a grant just submitted to the National Institute of Health (NIH), we are seeking ways to further personalize our model of Bipolar Disorder by measuring neuronal signatures of EEG which we think may be linked to the etiology of our classes of Bipolar disorder. The goal is to develop smartphone applications or web interfaces that track mood in Bipolar Patients, thereby enabling us to offer personalized recommendations for them. This individualized approach would revolutionize the treatment of Bipolar Disorder worldwide.

Funding: This works has been funded through a program grant from the HFSP. I have also recently submitted an R01 to the new Computational Psychiatry program at the National Institute of Mental Health. Sample press coverage: http://ns.umich.edu/new/releases/20354-how-the-brain-s-daily-clock-controls-mood-a-new-project

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(Selected) Bibliography Belle, M. D., Diekman, C. O., Forger, D. B., Piggins, H. D., 2009. Daily electrical silencing in

the mammalian circadian clock. Science 326, 281-4, doi:10.1126/science.1169657. Bodova, K., Paydarfar, D., Forger, D. B., 2015. Characterizing spiking in noisy type II

neurons. J Theor Biol 365C, 40-54, doi:10.1016/j.jtbi.2014.09.041. Chang, D. E., Leung, S., Atkinson, M. R., Reifler, A., Forger, D., Ninfa, A. J., 2010. Building

biological memory by linking positive feedback loops. Proc Natl Acad Sci U S A 107, 175-80, doi:10.1073/pnas.0908314107.

Clay, J. R., Paydarfar, D., Forger, D. B., 2008. A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons. J R Soc Interface 5, 1421-8, doi:10.1098/rsif.2008.0166.

Cochran, A. L., McInnis, M. G., Forger, D. B., 2016. Data-driven classification of bipolar I disorder from longitudinal course of mood. Transl Psychiatry 6, e912, doi:10.1038/tp.2016.166.

Cox, J. C., Ingersoll, J. E., Ross, S. A., 1985. A Theory of the Term Structure of Interest-Rates. Econometrica 53, 385-407, doi:Doi 10.2307/1911242.

DeWoskin, D., Geng, W., Stinchcombe, A. R., Forger, D. B., 2014. It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms. Interface Focus 4, 20130076, doi:10.1098/rsfs.2013.0076.

DeWoskin, D., Myung, J., Belle, M. D., Piggins, H. D., Takumi, T., Forger, D. B., 2015. Distinct roles for GABA across multiple timescales in mammalian circadian timekeeping. Proc Natl Acad Sci U S A 112, E3911-9, doi:10.1073/pnas.1420753112.

Diekman, C. O., Forger, D. B., 2009. Clustering predicted by an electrophysiological model of the suprachiasmatic nucleus. J Biol Rhythms 24, 322-33, doi:10.1177/0748730409337601.

Diekman, C. O., Belle, M. D., Irwin, R. P., Allen, C. N., Piggins, H. D., Forger, D. B., 2013. Causes and consequences of hyperexcitation in central clock neurons. PLoS Comput Biol 9, e1003196, doi:10.1371/journal.pcbi.1003196.

Forger, D. B., 2011. Signal processing in cellular clocks. Proc Natl Acad Sci U S A 108, 4281-5, doi:10.1073/pnas.1004720108.

Forger, D. B., Kronauer, R. E., 2002. Reconciling mathematical models of biological clocks by averaging on approximate manifolds. SIAM Journal on Applied Mathematics 62, 1281-1296.

Forger, D. B., Peskin, C. S., 2003. A detailed predictive model of the mammalian circadian clock. Proc Natl Acad Sci U S A 100, 14806-11, doi:10.1073/pnas.2036281100.

Forger, D. B., Peskin, C. S., 2004. Model based conjectures on mammalian clock controversies. J Theor Biol 230, 533-9, doi:10.1016/j.jtbi.2004.04.041.

Forger, D. B., Peskin, C. S., 2005. Stochastic simulation of the mammalian circadian clock. Proc Natl Acad Sci U S A 102, 321-4, doi:10.1073/pnas.0408465102.

Forger, D. B., Paydarfar, D., Clay, J. R., 2011. Optimal stimulus shapes for neuronal excitation. PLoS Comput Biol 7, e1002089, doi:10.1371/journal.pcbi.1002089.

Gallego, M., Eide, E. J., Woolf, M. F., Virshup, D. M., Forger, D. B., 2006. An opposite role for tau in circadian rhythms revealed by mathematical modeling. Proc Natl Acad Sci U S A 103, 10618-23, doi:10.1073/pnas.0604511103.

Kim, J. K., Forger, D. B., 2012a. A mechanism for robust circadian timekeeping via stoichiometric balance. Mol Syst Biol 8, 630, doi:10.1038/msb.2012.62.

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Kim, J. K., Forger, D. B., 2012b. On the existence and uniqueness of biological clock models matching experimental data. SIAM Journal on Applied Mathematics 72, 1842-1855.

Ko, C. H., Yamada, Y. R., Welsh, D. K., Buhr, E. D., Liu, A. C., Zhang, E. E., Ralph, M. R., Kay, S. A., Forger, D. B., Takahashi, J. S., 2010. Emergence of noise-induced oscillations in the central circadian pacemaker. PLoS Biol 8, e1000513, doi:10.1371/journal.pbio.1000513.

Mallet-Paret, J., Smith, H. L., 1990. The Poincare-Bendixson Theorem for Monotone Cyclic Feedback Systems. Journal of Dynamics and Differential Equations 2, 367-421.

Meier, E. B., Bryson, A. E., 1990. Efficient algorithm for time-optimal control of a two-link manipulator. Journal of Guidance, Control, and Dynamics 13, 859-966.

Mirollo, R. E., Strogatz, S. H., 1990. Synchronization of pulse-coupled biological oscillators. SIAM Journal on Applied Mathematics 50, 1645-1662.

Myung, J., Hong, S., DeWoskin, D., De Schutter, E., Forger, D. B., Takumi, T., 2015. GABA-mediated repulsive coupling between circadian clock neurons in the SCN encodes seasonal time. Proc Natl Acad Sci U S A, doi:10.1073/pnas.1421200112.

Paul JR, DeWoskin DA, Forger DB, KL, G., 2016. GSK3 regulation of persistent Na+ current encodes daily rhythms of excitability Nature Communications In Press.

Serkh, K., Forger, D. B., 2014. Optimal schedules of light exposure for rapidly correcting circadian misalignment. PLoS Comput Biol 10, e1003523, doi:10.1371/journal.pcbi.1003523.

Shlizerman, E., Phillips-Portillo, J., Forger, D. B., Reppert, S. M., 2016. Neural Integration Underlying a Time-Compensated Sun Compass in the Migratory Monarch Butterfly. Cell Rep, doi:10.1016/j.celrep.2016.03.057.

Stinchcombe, A. R., Forger, D. B., 2016. An efficient method for simulation of noisy coupled multi-dimensional oscillators. Journal of Computational Physics 321, 932-946, doi:10.1016/j.jcp.2016.05.025.

Walch, O. J., Cochran, A., Forger, D. B., 2016. A global quantification of "normal" sleep schedules using smartphone data. Sci Adv 2, e1501705, doi:10.1126/sciadv.1501705.

Walch, O. J., Zhang, L. S., Reifler, A. N., Dolikian, M. E., Forger, D. B., Wong, K. Y., 2015. Characterizing and modeling the intrinsic light response of rat ganglion-cell photoreceptors. J Neurophysiol, jn 00544 2015, doi:10.1152/jn.00544.2015.

Zhou, M., Kim, J. K., Eng, G. W., Forger, D. B., Virshup, D. M., 2015. A Period2 Phosphoswitch Regulates and Temperature Compensates Circadian Period. Mol Cell 60, 77-88, doi:10.1016/j.molcel.2015.08.022.