Casey O. Diekman

Daily Electrical Silencing in the Mammalian Circadian Clock

Quiet Clock Many physiological processes have circadian rhythms driven by a biological clock in the suprachiasmatic nuclei (SCN) of the brain. Within the SCN, some neurons express the molecular components of the clock and others do not. Exactly how the clock mechanism is coupled to neuronal activity is not precisely understood. Investigation of the electrophysiological properties of SCN neurons by Belle et al. (p. 281) found that, contrary to the conventionally expected rapid firing rate of the cells during the day, clock-containing cells tended not to fire, despite being in an electrically excited state. Modeling and experimental characterization of changes in channel activity revealed unexpected electrophysiological properties of the SCN cells requiring a reassessment of how the circadian clock regulates activity of SCN neurons. Clock-containing neurons in the mouse brain display complex electrophysiology not seen in other brain cells. 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.

A Conserved Bicycle Model for Circadian Clock Control of Membrane Excitability

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.

Causes and Consequences of Hyperexcitation in Central Clock Neurons

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

Modeling the Neuroprotective Role of Enhanced Astrocyte Mitochondrial Metabolism during Stroke

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.

Clustering Predicted by an Electrophysiological Model of the Suprachiasmatic Nucleus

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

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

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.

Network Symmetry and Binocular Rivalry Experiments

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.

Derived Patterns in Binocular Rivalry Networks

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.

Irregular activity arises as a natural consequence of synaptic inhibition

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.

Reduction and Dynamics of a Generalized Rivalry Network with Two Learned Patterns

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.