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COVID-19 and silent hypoxemia in a minimal closed-loop model of the respiratory rhythm generator

Casey Diekman, Peter Thomas, Christopher Wilson
Preprint | April, 2023


Silent hypoxemia, or `happy hypoxia', is a puzzling phenomenon in which patients who have contracted \covid~exhibit very low oxygen saturation (SaO2 < 80%) but do not experience discomfort in breathing. The mechanism by which this blunted response to hypoxia occurs is unknown. We have previously shown that a computational model (Diekman et al., 2017, J. Neurophysiol) of the respiratory neural network can be used to test hypotheses focused on changes in chemosensory inputs to the central pattern generator (CPG). We hypothesize that altered chemosensory function at the level of the carotid bodies and/or the \emph{nucleus tractus solitarii} are responsible for the blunted response to hypoxia. Here, we use our model to explore this hypothesis by altering the properties of the gain function representing oxygen sensing inputs to the CPG. We then vary other parameters in the model and show that oxygen carrying capacity is the most salient factor for producing silent hypoxemia. We call for clinicians to measure hematocrit as a clinical index of altered physiology in response to COVID-19 infection.

Inferring parameters of pyramidal neuron excitability in mouse models of Alzheimer's disease using biophysical modeling and deep learning

Soheil Saghafi, Timothy Rumbell, Viatcheslav Gurev, James Kozloski, Francesco Tamagnini, Kyle Wedgwood, Casey Diekman
Preprint | February, 2023


Alzheimer’s disease (AD) is believed to occur when abnormal amounts of the proteins amyloid beta and tau aggregate in the brain, resulting in a progressive loss of neuronal function. Hippocampal neurons in transgenic mice with amyloidopathy or tauopathy exhibit altered intrinsic excitability properties. We introduce a novel parameter inference technique, deep hybrid modeling (DeepHM), that combines deep learning with biophysical modeling to map experimental data recorded from hippocampal CA1 neurons in transgenic AD mice and age-matched wild-type littermate controls to the parameter space of a conductance-based CA1 model. Although mechanistic modeling and machine learning methods are by themselves powerful tools for approximating biological systems and making accurate predictions from data, when used in isolation these approaches suffer from distinct shortcomings: model and parameter uncertainty limit mechanistic modeling, whereas machine learning methods disregard the underlying biophysical mechanisms. DeepHM addresses these shortcomings by using conditional generative adversarial networks (cGANs) to provide an inverse mapping of data to mechanistic models that identifies the distributions of mechanistic modeling parameters coherent to the data. Here, we demonstrate that DeepHM accurately infers parameter distributions of the conductance-based model and outperforms a Markov chain Monte Carlo method on several test cases using synthetic data. We then use DeepHM to estimate parameter distributions corresponding to the experimental data and infer which ion channels are altered in the Alzheimer’s mouse models compared to their wildtype controls at 12 and 24 months. We find that the conductances most disrupted by tauopathy, amyloidopathy, and aging are delayed rectifier potassium, transient sodium, and hyperpolarization-activated potassium, respectively.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Emel Khan, Soheil Saghafi, Casey Diekman, Horacio G. Rotstein
Journal Paper Chaos, Volume 32, Article 063137 | June, 2022


Several distinct entrainment patterns can occur in the FitzHugh–Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

Beyond the limits of circadian entrainment: Non-24-hour sleep-wake disorder, shift work, and social jet lag

Casey Diekman, Amitabha Bose
Journal Paper Journal of Theoretical Biology, Volume 545, Article 111148 | May, 2022


While the vast majority of humans are able to entrain their circadian rhythm to the 24-h light-dark cycle, there are numerous individuals who are not able to do so due to disease or societal reasons. We use computational and mathematical methods to analyze a well-established model of human circadian rhythms to address cases where individuals do not entrain to the 24-h light-dark cycle, leading to misalignment of their circadian phase. For each case, we provide a mathematically justified strategy for how to minimize circadian misalignment. In the case of non-24-h sleep-wake disorder, we show why appropriately timed bright light therapy induces entrainment. With regard to shift work, we explain why reentrainment times following transitions between day and night shifts are asymmetric, and how higher light intensity enables unusually rapid reentrainment after certain transitions. Finally, with regard to teenagers who engage in compensatory catch-up sleep on weekends, we propose a rule of thumb for sleep and wake onset times that minimizes circadian misalignment due to this type of social jet lag. In all cases, the primary mathematical approach involves understanding the dynamics of entrainment maps that measure the phase of the entrained rhythm with respect to the daily onset of lights.

The E3 ubiquitin ligase adaptor Tango10 links the core circadian clock to neuropeptide and behavioral rhythms

Jongbin Lee, Chunghun Lim, Tae Hee Han, Tomas Andreani, Matthew Moye, Jack Curran, Eric Johnson, William Kath, Casey Diekman, Bridget Lear, Ravi Allada
Journal Paper PNAS, Volume 118, Article e2110767118 | November, 2021


In animals, the control of daily sleep–wake rhythms is mediated by discrete circadian clock neurons via their rhythmic activity–dependent release of neuropeptides and neurotransmitters. Here, we describe a gene, Tango10, critical for daily behavioral rhythms. TANGO10 functions as an adaptor with its partner, the E3 ubiquitin ligase CULLIN 3, to regulate protein ubiquitination, neuronal excitability via voltage-gated potassium channels, and neuropeptide accumulation. These studies define a mechanism for control of neuronal excitability and daily rhythmicity.

Daily electrical activity in the master circadian clock of a diurnal mammal

Beatriz Bano-Otalora, Matthew Moye, Timothy Brown, Robert Lucas, Casey Diekman, Mino Belle
Journal Paper eLife, Article 10:e68179 | November, 2021


Daily or circadian rhythms in mammals are orchestrated by a master circadian clock within the hypothalamic suprachiasmatic nuclei (SCN). Here, cell-autonomous oscillations in gene expression, intrinsic membrane properties, and synaptic communication shape the electrical landscape of the SCN across the circadian day, rendering SCN neurons overtly more active during the day than at night. This well-accepted hallmark bioelectrical feature of the SCN has overwhelmingly emerged from studies performed on a small number of nocturnal rodent species. Therefore, for the first time, we investigate the spontaneous and evoked electrical activity of SCN neurons in a diurnal mammal. To this end, we measured the electrical activity of individual SCN neurons during the day and at night in brain slices prepared from the diurnal murid rodent Rhabdomys pumilio and then developed cutting-edge data assimilation and mathematical modelling approaches to uncover the underlying ionic mechanisms. We find that R. pumilio SCN neurons were more excited in the day than at night, recapitulating the prototypical pattern of SCN neuronal activity previously observed in nocturnal rodents. By contrast, the evoked activity of R. pumilio neurons included a prominent suppressive response that is not present in the SCN of nocturnal rodents. Our computational modelling approaches reveal transient subthreshold A-type potassium channels as the primary determinant of the suppressive response and highlight a key role for this ionic mechanism in tuning excitability of clock neurons and optimising SCN function to accommodate R. pumilio’s diurnal niche.

Entrainment dynamics organised by global manifolds in a circadian pacemaker model

Jennifer Creaser, Casey Diekman, Kyle Wedgwood
Journal Paper Frontiers in Applied Mathematics and Statistics, Volume 7, Article 703359, Pages 1-18 | July, 2021


Circadian rhythms are established by the entrainment of our intrinsic body clock to periodic forcing signals provided by the external environment, primarily variation in light intensity across the day/night cycle. Loss of entrainment can cause a multitude of physiological difficulties associated with misalignment of circadian rhythms, including insomnia, excessive daytime sleepiness, gastrointestinal disturbances, and general malaise. This can occur after travel to different time zones, known as jet lag; when changing shift work patterns; or if the period of an individual’s body clock is too far from the 24 h period of environmental cycles. We consider the loss of entrainment and the dynamics of re-entrainment in a two-dimensional variant of the Forger-Jewett-Kronauer model of the human circadian pacemaker forced by a 24 h light/dark cycle. We explore the loss of entrainment by continuing bifurcations of one-to-one entrained orbits under variation of forcing parameters and the intrinsic clock period. We show that the severity of the loss of entrainment is dependent on the type of bifurcation inducing the change of stability of the entrained orbit, which is in turn dependent on the environmental light intensity. We further show that for certain perturbations, the model predicts counter-intuitive rapid re-entrainment if the light intensity is sufficiently high. We explain this phenomenon via computation of invariant manifolds of fixed points of a 24 h stroboscopic map and show how the manifolds organise re-entrainment times following transitions between day and night shift work.

Circadian rhythms of early afterdepolarizations and ventricular arrhythmias in a cardiomyocyte model

Casey Diekman, Ning Wei
Journal Paper Biophysical Journal, Volume 120, Pages 319-333 | January, 2021


Sudden cardiac arrest is a malfunction of the heart’s electrical system, typically caused by ventricular arrhythmias, that can lead to sudden cardiac death (SCD) within minutes. Epidemiological studies have shown that SCD and ventricular arrhythmias are more likely to occur in the morning than in the evening, and laboratory studies indicate that these daily rhythms in adverse cardiovascular events are at least partially under the control of the endogenous circadian timekeeping system. However, the biophysical mechanisms linking molecular circadian clocks to cardiac arrhythmogenesis are not fully understood. Recent experiments have shown that L-type calcium channels exhibit circadian rhythms in both expression and function in guinea pig ventricular cardiomyocytes. We developed an electrophysiological model of these cells to simulate the effect of circadian variation in L-type calcium conductance. In our simulations, we found that there is a circadian pattern in the occurrence of early afterdepolarizations (EADs), which are abnormal depolarizations during the repolarization phase of a cardiac action potential that can trigger fatal ventricular arrhythmias. Specifically, the model produces EADs in the morning, but not at other times of day. We show that the model exhibits a codimension-2 Takens-Bogdanov bifurcation that serves as an organizing center for different types of EAD dynamics. We also simulated a two-dimensional spatial version of this model across a circadian cycle. We found that there is a circadian pattern in the breakup of spiral waves, which represents ventricular fibrillation in cardiac tissue. Specifically, the model produces spiral wave breakup in the morning, but not in the evening. Our computational study is the first, to our knowledge, to propose a link between circadian rhythms and EAD formation and suggests that the efficacy of drugs targeting EAD-mediated arrhythmias may depend on the time of day that they are administered.

Entrainment dynamics of forced hierarchical circadian systems revealed by 2-dimensional maps

Guangyuan Liao, Casey Diekman, Amitabha Bose
Journal Paper SIAM Journal on Applied Dynamical Systems, Volume 19, Pages 2135-2161 | September, 2020


The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical system consisting of a central circadian oscillator that drives a peripheral circadian oscillator. We develop techniques to reduce the higher-dimensional phase space of the coupled system to derive a generalized two-dimensional entrainment map. Determining the nature of various fixed points, together with an understanding of their stable and unstable manifolds, leads to conditions for existence and stability of periodic orbits of the circadian system. We use the map to investigate how various properties of solutions depend on parameters and initial conditions including the time to and direction of entrainment. We show that the concepts of phase advance and phase delay need to be carefully assessed when considering hierarchical systems.

CikA, an input pathway component, senses the oxidized quinone signal to generate phase delays in the cyanobacterial circadian clock

Pyonghwa Kim, Brianna Porr, Tetsuya Mori, Yong-Sung Kim, Carl Johnson, Casey Diekman, Yong-Ick Kim
Journal Paper Journal of Biological Rhythms, Volume 35, Pages 227-234 | June, 2020


The circadian clock is a timekeeping system in most organisms that keeps track of the time of day. The rhythm generated by the circadian oscillator must be constantly synchronized with the environmental day/night cycle to make the timekeeping system truly advantageous. In the cyanobacterial circadian clock, quinone is a biological signaling molecule used for entraining and fine-tuning the oscillator, a process in which the external signals are transduced into biological metabolites that adjust the phase of the circadian oscillation. Among the clock proteins, the pseudo-receiver domain of KaiA and CikA can sense external cues by detecting the oxidation state of quinone, a metabolite that reflects the light/dark cycle, although the molecular mechanism is not fully understood. Here, we show the antagonistic phase shifts produced by the quinone sensing of KaiA and CikA. We introduced a new cyanobacterial circadian clock mixture that includes an input component in vitro. KaiA and CikA cause phase advances and delays, respectively, in this circadian clock mixture in response to the quinone signal. In the entrainment process, oxidized quinone modulates the functions of KaiA and CikA, which dominate alternatively at day and night in the cell. This in turn changes the phosphorylation state of KaiC—the central oscillator in cyanobacteria-- ensuring full synchronization of the circadian clock. Moreover, we reemphasize the mechanistic input functionality of CikA, contrary to other reports that focus only on its output action.

Magnesium regulates the circadian oscillator in cyanobacteria

Young Jeong, Cristiano Dias, Casey Diekman, Helen Brochon, Pyonghwa Kim, Manpreet Kaur, Yong-Sung Kim, Hye-In Jang, Yong-Ick Kim
Journal Paper Journal of Biological Rhythms, Volume 34, Pages 380-390 | August, 2019


The circadian clock controls 24-h biological rhythms in our body, influencing many time-related activities such as sleep and wake. The simplest circadian clock is found in cyanobacteria, with the proteins KaiA, KaiB, and KaiC generating a self-sustained circadian oscillation of KaiC phosphorylation and dephosphorylation. KaiA activates KaiC phosphorylation by binding the A-loop of KaiC, while KaiB attenuates the phosphorylation by sequestering KaiA from the A-loop. Structural analysis revealed that magnesium regulates the phosphorylation and dephosphorylation of KaiC by dissociating from and associating with catalytic Glu residues that activate phosphorylation and dephosphorylation, respectively. High magnesium causes KaiC to dephosphorylate, whereas low magnesium causes KaiC to phosphorylate. KaiC alone behaves as an hourglass timekeeper when the magnesium concentration is alternated between low and high levels in vitro. We suggest that a magnesiumbased hourglass timekeeping system may have been used by ancient cyanobacteria before magnesium homeostasis was established.

CikA modulates the effect of KaiA on the period of the circadian oscillation in KaiC phosphorylation

Manpreet Kaur, Amy Ng, Pyonghwa Kim, Casey Diekman, Yong-Ick Kim
Journal Paper Journal of Biological Rhythms, Volume 34, Pages 218-223 | April, 2019


Cyanobacteria contain a circadian oscillator that can be reconstituted in vitro. In the reconstituted circadian oscillator, the phosphorylation state of KaiC oscillates with a circadian period, spending about 12 hours in the phosphorylation phase and another 12 hours in the dephosphorylation phase. Although some entrainment studies have been performed using the reconstituted oscillator, they were insufficient to fully explain entrainment mechanisms of the cyanobacterial circadian clock due to the lack of input pathway components in the in vitro oscillator reaction mixture. Here, we investigate how an input pathway component, CikA, affects the phosphorylation state of KaiC in vitro. In general, CikA affects the amplitude and period of the circadian oscillation of KaiC phosphorylation by competing with KaiA for the same binding site on KaiB. In the presence of CikA, KaiC switches from its dephosphorylation phase to the phosphorylation phase prematurely, due to an early release of KaiA from KaiB as a result of competitive binding between CikA and KaiA. This causes hyperphosphorylation of KaiC and lowers the amplitude of the circadian oscillation. The period of the KaiC phosphorylation oscillation is shortened by adding increased amounts of CikA. A constant period can be maintained as CikA is increased by proportionally decreasing the amount of KaiA. Our findings give insight into how to reconstitute the cyanobacterial circadian clock in vitro by adding an input pathway component, which affects the circadian oscillation by directly interacting with the oscillator components.

Data assimilation methods for neuronal state and parameter estimation

Matthew Moye, Casey Diekman
Journal Paper Journal of Mathematical Neuroscience, Volume 8, Pages 1-38 | August 9, 2018


This tutorial illustrates the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductance-based neuronal models. Modern data assimilation (DA) techniques are widely used in climate science and weather prediction, but have only recently begun to be applied in neuroscience. The two main classes of DA techniques are sequential methods and variational methods. We provide computer code implementing basic versions of a method from each class, the Unscented Kalman Filter and 4D-Var, and demonstrate how to use these algorithms to infer several parameters of the Morris-Lecar model from a single voltage trace. Depending on parameters, the Morris-Lecar model exhibits qualitatively different types of neuronal excitability due to changes in the underlying bifurcation structure. We show that when presented with voltage traces from each of the various excitability regimes, the DA methods can identify parameter sets that produce the correct bifurcation structure even with initial parameter guesses that correspond to a different excitability regime. This demonstrates the ability of DA techniques to perform nonlinear state and parameter estimation and introduces the geometric structure of inferred models as a novel qualitative measure of estimation success. We conclude by discussing extensions of these DA algorithms that have appeared in the neuroscience literature.

Reentrainment of the circadian pacemaker during jet lag: East-west asymmetry and the effects of north-south travel

Casey Diekman, Amitabha Bose
Journal Paper Journal of Theoretical Biology, Volume 437, Pages 261-285 | January 21, 2018


The normal alignment of circadian rhythms with the 24-hour light-dark cycle is disrupted after rapid travel between home and destination time zones, leading to sleep problems, indigestion, and other symptoms collectively known as jet lag. Using mathematical and computational analysis, we study the process of reentrainment to the light-dark cycle of the destination time zone in a model of the human circadian pacemaker. We calculate the reentrainment time for travel between any two points on the globe at any time of the day and year. We construct one-dimensional entrainment maps to explain several properties of jet lag, such as why most people experience worse jet lag after traveling east than west. We show that this east-west asymmetry depends on the endogenous period of the traveler’s circadian clock as well as daylength. Thus the critical factor is not simply whether the endogenous period is greater than or less than 24 hours as is commonly assumed. We show that the unstable fixed point of an entrainment map determines whether a traveler reentrains through phase advances or phase delays, providing an understanding of the threshold that separates orthodromic and antidromic modes of reentrainment. Contrary to the conventional wisdom that jet lag only occurs after east-west travel across multiple time zones, we predict that the change in daylength encountered during north-south travel can cause jet lag even when no time zones are crossed. Our techniques could be used to provide advice to travelers on how to minimize jet lag on trips involving multiple destinations and a combination of transmeridian and translatitudinal travel.

Neuronal oscillations on an ultra-slow timescale: daily rhythms in electrical activity and gene expression in the mammalian master circadian clockwork

Mino Belle, Casey Diekman
Journal Paper European Journal of Neuroscience, Volume 48, Pages 2696-2717 | October, 2018


Neuronal oscillations of the brain, such as those observed in the cortices and hippocampi of behaving animals and humans, span across wide frequency bands, from slow delta waves (0.1 Hz) to ultra-fast ripples (600 Hz). Here, we focus on ultra-slow neuronal oscillators in the hypothalamic suprachiasmatic nuclei (SCN), the master daily clock that operates on interlocking transcription-translation feedback loops to produce circadian rhythms in clock gene expression with a period of near 24 h (< 0.001 Hz). This intracellular molecular clock interacts with the cell's membrane through poorly understood mechanisms to drive the daily pattern in the electrical excitability of SCN neurons, exhibiting an up-state during the day and a down-state at night. In turn, the membrane activity feeds back to regulate the oscillatory activity of clock gene programs. In this review, we emphasise the circadian processes that drive daily electrical oscillations in SCN neurons, and highlight how mathematical modelling contributes to our increasing understanding of circadian rhythm generation, synchronisation and communication within this hypothalamic region and across other brain circuits.

Experimental validation of a closed-loop respiratory control model using dynamic clamp

Casey Diekman, Peter Thomas, Christopher Wilson
Conference Paper Conf Proc IEEE Eng Med Biol Soc, Pages 5273-5276, doi: 10.1109/EMBC.2018.8513424 | July, 2018


We have previously introduced a model for closed-loop respiratory control incorporating an explicit conductance-based model of bursting pacemaker cells driven by hypoxia sensitive chemosensory feedback. Numerical solution of the model equations revealed two qualitatively distinct asymptotically stable dynamical behaviors: one analogous to regular breathing (eupnea), and a second analogous to pathologically rapid, shallow breathing (tachypnea). As an experimental test of this model, we created a hybrid in vitrolin silico circuit. We used Real Time eXperimental Interface (RTXI) dynamic clamp to incorporate a living pacemaker cell recorded in vitro into a numerical simulation of the closed-loop control model in real time. Here we show that the hybrid circuit can sustain the same bistable behavior as the purely computational model, and we assess the ability of the hybrid circuit to recover from simulated bouts of transient hypoxia.

Computational neuroscience: mathematical and statistical perspectives

Robert Kass, Shun-Ichi Amari, Kensuke Arai, Emery Brown, Casey Diekman, Markus Diesmann, Brent Doiron, Uri Eden, Adrienne Fairhall, Grant Fiddyment, Tomoki Fukai, Sonja Grün, Matthew Harrison, Moritz Helias, Hiroyuki Nakahara, Jun-nosuke Teramae, Peter Thomas, Mark Reimers, Jordan Rodu, Horacio Rotstein, Eric Shea-Brown, Hideaki Shimazaki, Shigeru Shinomoto, Byron Yu, Mark Kramer
Journal Paper Annual Review of Statistics and Its Application, Volume 5, Pages 183-214 | March, 2018


Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm.

Delayed cryptochrome degradation asymmetrically alters the daily rhythm in suprachiasmatic clock neuron excitability

Sven Wegner, Mino Belle, Alun Hughes, Casey Diekman, Hugh Piggins
Journal Paper Journal of Neuroscience, Volume 37, Pages 7824-7836 | August 16, 2017


The intracellular molecular clock drives changes in SCN neuronal excitability, but it is unclear how mutations affecting post-translational modification of molecular clock proteins influence the temporal expression of SCN neuronal state or intercellular communication within the SCN network. Here we show for the first time, that a mutation that prolongs the stability of key components of the intracellular clock, the cryptochrome proteins, unexpectedly increases in the expression of hypoexcited neuronal state in the ventral SCN at night and enhances hyperpolarization of ventral SCN neurons at this time. This is accompanied by increased GABAergic signaling and by enhanced responsiveness to a neurochemical mimic of the light input pathway to the SCN. Therefore, post-translational modification shapes SCN neuronal state and network properties.

Eupnea, tachypnea, and autoresuscitation in a closed-loop respiratory control model

Casey Diekman, Peter Thomas, Christopher Wilson
Journal Paper Journal of Neurophysiology, Volume 118, Issue 4, Pages 2194-2215 | October 1, 2017


How sensory information influences the dynamics of rhythm generation varies across systems, and general principles for understanding this aspect of motor control are lacking. Determining the origin of respiratory rhythm generation is challenging because the mechanisms in a central circuit considered in isolation may be different from those in the intact organism. We analyze a closed-loop respiratory control model incorporating a central pattern generator (CPG), the Butera-Rinzel-Smith (BRS) model, together with lung mechanics, oxygen handling, and chemosensory components. We show that 1) embedding the BRS model neuron in a control loop creates a bistable system; 2) although closed-loop and open-loop (isolated) CPG systems both support eupnea-like bursting activity, they do so via distinct mechanisms; 3) chemosensory feedback in the closed loop improves robustness to variable metabolic demand; 4) the BRS model conductances provide an autoresuscitation mechanism for recovery from transient interruption of chemosensory feedback; and 5) the in vitro brain stem CPG slice responds to hypoxia with transient bursting that is qualitatively similar to in silico autoresuscitation. Bistability of bursting and tonic spiking in the closed-loop system corresponds to coexistence of eupnea-like breathing, with normal minute ventilation and blood oxygen level and a tachypnea-like state, with pathologically reduced minute ventilation and critically low blood oxygen. Disruption of the normal breathing rhythm, through either imposition of hypoxia or interruption of chemosensory feedback, can push the system from the eupneic state into the tachypneic state. We use geometric singular perturbation theory to analyze the system dynamics at the boundary separating eupnea-like and tachypnea-like outcomes.

Entrainment maps: A new tool for understanding properties of circadian oscillator models

Casey Diekman, Amitabha Bose
Journal Paper Journal of Biological Rhythms, Volume 31, Issue 6, Pages 598-616 | December 2016


Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment, however PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a one-dimensional map which we construct for three different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.

A conserved bicycle model for circadian clock control of membrane excitability

Matthieu Flourakis, ..., Casey Diekman, Indira Raman, Ravi Allada
Journal Paper Cell, Volume 162, Issue 4, Pages 836-848 | August 13, 2015


Circadian clocks regulate membrane excitability in master pacemaker neurons to control daily rhythms of sleep and wake. Here, we find that two distinctly timed electrical drives collaborate to impose rhythmicity on Drosophila clock neurons. In the morning, a voltage-independent sodium conductance via the NA/NALCN ion channel depolarizes these neurons. This current is driven by the rhythmic expression of NCA localization factor-1, linking the molecular clock to ion channel function. In the evening, basal potassium currents peak to silence clock neurons. Remarkably, daily antiphase cycles of sodium and potassium currents also drive mouse clock neuron rhythms. Thus, we reveal an evolutionarily ancient strategy for the neural mechanisms that govern daily sleep and wake.

Network symmetry and binocular rivalry experiments

Casey Diekman, Martin Golubitsky
Journal Paper Journal of Mathematical Neuroscience, Volume 4, Issue 12, doi: 10.1186/2190-8567-4-12 | May 7, 2014


Hugh Wilson has proposed a class of models that treat higher-level decision making as a competition between patterns coded as levels of a set of attributes in an appropriately defined network (Cortical Mechanisms of Vision, pp. 399-417, 2009; The Constitution of Visual Consciousness: Lessons from Binocular Rivalry, pp. 281-304, 2013). In this paper, we propose that symmetry-breaking Hopf bifurcation from fusion states in suitably modified Wilson networks, which we call rivalry networks, can be used in an algorithmic way to explain the surprising percepts that have been observed in a number of binocular rivalry experiments. These rivalry networks modify and extend Wilson networks by permitting different kinds of attributes and different types of coupling. We apply this algorithm to psychophysics experiments discussed by Kovács et al. (Proc. Natl. Acad. Sci. USA 93:15508-15511, 1996), Shevell and Hong (Vis. Neurosci. 23:561-566, 2006; Vis. Neurosci. 25:355-360, 2008), and Suzuki and Grabowecky (Neuron 36:143-157, 2002). We also analyze an experiment with four colored dots (a simplified version of a 24-dot experiment performed by Kovács), and a three-dot analog of the four-dot experiment. Our algorithm predicts surprising differences between the three- and four-dot experiments.

Discovering functional neuronal connectivity from serial patterns in spike train data

Casey Diekman, Kohinoor Dasgupta, Vijay Nair, K.P. Unnikrishnan
Journal Paper Neural Computation, Volume 26, Issue 7, Pages 1263-1297 | July, 2014


Repeating patterns of precisely timed activity across a group of neurons (called frequent episodes) are indicative of networks in the underlying neural tissue. This letter develops statistical methods to determine functional connectivity among neurons based on nonoverlapping occurrences of episodes. We study the distribution of episode counts and develop a two-phase strategy for identifying functional connections. For the first phase, we develop statistical procedures that are used to screen all two-node episodes and identify possible functional connections (edges). For the second phase, we develop additional statistical procedures to prune the two-node episodes and remove false edges that can be attributed to chains or fan-out structures. The restriction to nonoverlapping occurrences makes the counting of all two-node episodes in phase 1 computationally efficient. The second (pruning) phase is critical since phase 1 can yield a large number of false connections. The scalability of the two-phase approach is examined through simulation. The method is then used to reconstruct the graph structure of observed neuronal networks, first from simulated data and then from recordings of cultured cortical neurons.

Irregular activity arises as a natural consequence of synaptic inhibition

David Terman, Jonathan Rubin, Casey Diekman
Journal Paper Chaos, Volume 23, Issue 4, doi: 10.1063/1.4831752 | November 20, 2013


Irregular neuronal activity is observed in a variety of brain regions and states. This work illustrates a novel mechanism by which irregular activity naturally emerges in two-cell neuronal networks featuring coupling by synaptic inhibition. We introduce a one-dimensional map that captures the irregular activity occurring in our simulations of conductance-based differential equations and mathematically analyze the instability of fixed points corresponding to synchronous and antiphase spiking for this map. We find that the irregular solutions that arise exhibit expansion, contraction, and folding in phase space, as expected in chaotic dynamics. Our analysis shows that these features are produced from the interplay of synaptic inhibition with sodium, potassium, and leak currents in a conductance-based framework and provides precise conditions on parameters that ensure that irregular activity will occur. In particular, the temporal details of spiking dynamics must be present for a model to exhibit this irregularity mechanism and must be considered analytically to capture these effects.

Causes and consequences of hyperexcitation in central clock neurons

Casey Diekman, Mino Belle, Robert Irwin, Charles Allen, Hugh Piggins, Daniel Forger
Journal Paper PLOS Computational Biology, Volume 9, Issue 8, doi: 10.1371/journal.pcbi.1003196 | August 22, 2013


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.

Modeling the neuroprotective role of enhanced astrocyte mitochondrial metabolism during stroke

Casey Diekman, Christopher Fall, James Lechleiter, David Terman
Journal Paper Biophysical Journal, Volume 104, Issue 8, Pages 1752-1763 | April 16, 2013


A mathematical model that integrates the dynamics of cell membrane potential, ion homeostasis, cell volume, mitochondrial ATP production, mitochondrial and endoplasmic reticulum Ca(2+) handling, IP3 production, and GTP-binding protein-coupled receptor signaling was developed. Simulations with this model support recent experimental data showing a protective effect of stimulating an astrocytic GTP-binding protein-coupled receptor (P2Y1Rs) following cerebral ischemic stroke. The model was analyzed to better understand the mathematical behavior of the equations and to provide insights into the underlying biological data. This approach yielded explicit formulas determining how changes in IP3-mediated Ca(2+) release, under varying conditions of oxygen and the energy substrate pyruvate, affected mitochondrial ATP production, and was utilized to predict rate-limiting variables in P2Y1R-enhanced astrocyte protection after cerebral ischemic stroke.

Derived patterns in binocular rivalry networks

Casey Diekman, Martin Golubitsky, Yunjiao Wang
Journal Paper Journal of Mathematical Neuroscience, Volume 3, Issue 6, doi: 10.1186/2190-8567-3-6 | May 8, 2013


Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Wilson proposed a class of neuronal network models that generalize rivalry to multiple competing patterns. The networks are assumed to have learned several patterns, and rivalry is identified with time periodic states that have periods of dominance of different patterns. Here, we show that these networks can also support patterns that were not learned, which we call derived. This is important because there is evidence for perception of derived patterns in the binocular rivalry experiments of Kovács, Papathomas, Yang, and Fehér. We construct modified Wilson networks for these experiments and use symmetry breaking to make predictions regarding states that a subject might perceive. Specifically, we modify the networks to include lateral coupling, which is inspired by the known structure of the primary visual cortex. The modified network models make expected the surprising outcomes observed in these experiments.

Spontaneous autoresuscitation in a model of respiratory control

Casey Diekman, Christopher Wilson, Peter Thomas
Conference Paper Conf Proc IEEE Eng Med Biol Soc, Pages 6669-6672, doi: 10.1109/EMBC.2012.6347524 | November 12, 2012


We introduce a closed-loop model of respiratory control incorporating a conductance-based central pattern generator (CPG), low-pass filtering of CPG output by the respiratory musculature, gas exchange in the lung, metabolic oxygen demand, and chemosensation. The CPG incorporates Butera, Rinzel and Smith (BRS)'s (1999) conditional pacemaker model. BRS model cells can support quiescent, bursting, or beating activity depending on the level of excitatory drive; we identify these activity modes with apnea (cessation of breathing), eupnea (normal breathing), and tachypnea (excessively rapid breathing). We demonstrate the coexistence of two dynamically stable behaviors in the closed-loop model, corresponding respectively to eupnea and tachypnea. The latter state represents a novel failure mode within a respiratory control model. In addition, the closed-loop system exhibits a form of autoresuscitation: conductances intrinsic to the BRS model buffer the CPG against brief episodes of hypoxia, steering the system away from catastrophic collapse as can occur with tachypnea.

Reduction and dynamics of a generalized rivalry network with two learned patterns

Casey Diekman, Martin Golubitsky, Tyler McMillen, Yunjiao Wang
Journal Paper SIAM Journal of Applied Dynamical Systems, Volume 11, Issue 4, Pages 1270-1309 | October 11, 2012


We use the theory of coupled cell systems to analyze a neuronal network model for generalized rivalry posed by H. Wilson. We focus on the case of rivalry between two patterns and identify conditions under which large networks of n attributes and $m$ intensity levels can reduce to a model consisting of two or three cells depending on whether or not the patterns have any attribute levels in common. (The two-cell reduction is equivalent to certain recent models of binocular rivalry.) Notably, these reductions can lead to large recurrent excitation in the reduced network even though the individual cells in the original network may have none. We also show that symmetry-breaking Takens-Bogdanov (TB) bifurcations occur in the reduced networks, and this allows us to further reduce much of the dynamics to a planar system. We analyze the dynamics of the quotient systems near the TB singularity, discussing how variation of the input parameter I organizes the dynamics. This variation leads to a degenerate path through the unfolding of the TB point. We also discuss how the network structure affects recurrent excitation in the reduced networks, and the consequences for the dynamics.

Daily electrical silencing in the mammalian circadian clock

Mino Belle, Casey Diekman, Daniel Forger, Hugh Piggins
Journal Paper Science, Volume 326, Issue 5950, Pages 281-284 | October 9, 2009


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

Casey Diekman, Daniel Forger
Journal Paper Journal of Biological Rhythms, Volume 24, Issue 4, Pages 322-333 | August, 2009


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.

Statistical significance of sequential firing patterns in multi-neuronal spike trains

Casey Diekman, P.S. Sastry, K.P. Unnikrishnan
Journal Paper Journal of Neuroscience Methods, Volume 182, Issue 2, Pages 279-284 | September 15, 2009


Sequential firings with fixed time delays are frequently observed in simultaneous recordings from multiple neurons. Such temporal patterns are potentially indicative of underlying microcircuits and it is important to know when a repeatedly occurring pattern is statistically significant. These sequences are typically identified through correlation counts. In this paper we present a method for assessing the significance of such correlations. We specify the null hypothesis in terms of a bound on the conditional probabilities that characterize the influence of one neuron on another. This method of testing significance is more general than the currently available methods since under our null hypothesis we do not assume that the spiking processes of different neurons are independent. The structure of our null hypothesis also allows us to rank order the detected patterns. We demonstrate our method on simulated spike trains.

Hybrid methods for automated diagnosis of breast tumors

Casey Diekman, Wei He, Nagabhushana Prabhu, Harvey Cramer
Journal Paper Analytical and Quantitative Cytology and Histology, Volume 25, Issue 4, Pages 183-190 | August, 2003


OBJECTIVE: To design and analyze a new family of hybrid methods for the diagnosis of breast tumors using fine needle aspirates. STUDY DESIGN: We present a radically new approach to the design of diagnosis systems. In the new approach, a nonlinear classifier with high sensitivity but low specificity is hybridized with a linear classifier having low sensitivity but high specificity. Data from the Wisconsin Breast Cancer Database are used to evaluate, computationally, the performance of the hybrid classifiers. RESULTS: The diagnosis scheme obtained by hybridizing the nonlinear classifier ellipsoidal multisurface method (EMSM) with the linear classifier proximal support vector machine (PSVM) was found to have a mean sensitivity of 97.36% and a mean specificity of 95.14% and was found to yield a 2.44% improvement in the reliability of positive diagnosis over that of EMSM at the expense of 0.4% degradation in the reliability of negative diagnosis, again compared to EMSM. At the 95% confidence level we can trust the hybrid method to be 96.19-98.53% correct in its malignant diagnosis of new tumors and 93.57-96.71% correct in its benign diagnosis. CONCLUSION: Hybrid diagnosis schemes represent a significant paradigm shift and provide a promising new technique to improve the specificity of nonlinear classifiers without seriously affecting the high sensitivity of nonlinear classifiers.