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 magnesium-based hourglass timekeeping system may have been used by ancient cyanobacteria before magnesium homeostasis was established.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.