1,5-disubstituted 1,2,3-triazolylated carbohydrates and nucleosides.

In general, 1,5-disubstituted 1,2,3-triazolyl moiety is much less common in the synthesis and applications in comparison to its regioisomeric counterpart. Moreover, the synthesis of 1,5-disubstituted 1,2,3-triazoles are not so straightforward as is the case for copper catalyzed strategy of 1,4-disubstituted 1,2,3-triazoles. The preparation of 1,5-triazolylated carbohydrates and nucleosides are even more complex because of the difficulties in accessing the appropriate starting materials as well as the compatibility of reaction conditions with the various protecting groups. 1,5-Disubstitution regioisomeric triazoles of carbohydrates and nucleosides were traditionally obtained as minor products through straightforward heating of the mixture of azides and terminal alkynes. However, the separation of isomers was tedious or in some cases futile. On the other hand, regioselective synthesis using ruthenium catalysis triggered serious concern of residual metal content in therapeutically important ingredients. Therefore, serious efforts are being made by several groups to develop non-toxic metal based or completely metal-free synthesis of 1,5-disubstituted 1,2,3-triazoles. This article strives to summarize the pre-Click era as well as the post-2001 reports on the synthesis and potential applications of 1,5-disubstituted 1,2,3-triazoles in biological systems.

Dynamic models for musical rhythm perception and coordination.

Rhythmicity permeates large parts of human experience. Humans generate various motor and brain rhythms spanning a range of frequencies. We also experience and synchronize to externally imposed rhythmicity, for example from music and song or from the 24-h light-dark cycles of the sun. In the context of music, humans have the ability to perceive, generate, and anticipate rhythmic structures, for example, "the beat." Experimental and behavioral studies offer clues about the biophysical and neural mechanisms that underlie our rhythmic abilities, and about different brain areas that are involved but many open questions remain. In this paper, we review several theoretical and computational approaches, each centered at different levels of description, that address specific aspects of musical rhythmic generation, perception, attention, perception-action coordination, and learning. We survey methods and results from applications of dynamical systems theory, neuro-mechanistic modeling, and Bayesian inference. Some frameworks rely on synchronization of intrinsic brain rhythms that span the relevant frequency range; some formulations involve real-time adaptation schemes for error-correction to align the phase and frequency of a dedicated circuit; others involve learning and dynamically adjusting expectations to make rhythm tracking predictions. Each of the approaches, while initially designed to answer specific questions, offers the possibility of being integrated into a larger framework that provides insights into our ability to perceive and generate rhythmic patterns.

Entrainment within hierarchical circadian oscillator networks.

Circadian rhythms are endogenous oscillations, widely found across biological species, that have the capability of entraining to the 24-h light-dark cycle. Circadian systems often consist of both central oscillators that receive direct light-dark input and peripheral oscillators that receive input from the central oscillators. In this paper, we address questions related to what governs the time to and pattern of entrainment of these hierarchical circadian systems after an abrupt switch in the light-dark phasing. For a network consisting of a single central oscillator coupled to a chain of N feed-forward peripheral oscillators, we introduce a systematic way to derive an N-dimensional entrainment map whose fixed points correspond to entrained solutions. Using the map, we explain that the direction of reentrainment can involve fairly complicated phase advancing and delaying behavior as well as reentrainment times that depend sensitively on the nature of the perturbation. We also study the dynamics of a hierarchical system in which the peripheral oscillators are mutually coupled. We study how reentrainment times vary as a function of the degree to which the oscillators are desynchronized at the time of the change in light-dark phasing. We show that desynchronizing the peripheral oscillators can, in some circumstances, speed up their ultimate reentrainment following perturbations.

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

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.

A biophysical counting mechanism for keeping time.

The ability to estimate and produce appropriately timed responses is central to many behaviors including speaking, dancing, and playing a musical instrument. A classical framework for estimating or producing a time interval is the pacemaker-accumulator model in which pulses of a pacemaker are counted and compared to a stored representation. However, the neural mechanisms for how these pulses are counted remain an open question. The presence of noise and stochasticity further complicates the picture. We present a biophysical model of how to keep count of a pacemaker in the presence of various forms of stochasticity using a system of bistable Wilson-Cowan units asymmetrically connected in a one-dimensional array; all units receive the same input pulses from a central clock but only one unit is active at any point in time. With each pulse from the clock, the position of the activated unit changes thereby encoding the total number of pulses emitted by the clock. This neural architecture maps the counting problem into the spatial domain, which in turn translates count to a time estimate. We further extend the model to a hierarchical structure to be able to robustly achieve higher counts.

A one-dimensional map to study multi-seasonal coffee infestation by the coffee berry borer.

The coffee berry borer (CBB, Hypothenemus hampei) is the most serious insect pest of coffee worldwide; understanding the dynamics of its reproduction is essential for pest management. The female CBB penetrates the coffee berry, eats the seed, and reproduces inside it. A mathematical model of the infestation progress of the coffee berry by the CBB during several coffee seasons is formulated. The model represents the interaction among five populations: uninfested, slightly infested, and severely infested coffee berries, and free and encapsulated CBBs. Coffee harvesting is also included in the model. A one-dimensional map is derived for tracking the population dynamics subject to certain coffee harvesting percentages over several seasons. Stability analysis of the map's fixed points shows that CBB infestation could be eliminated or controlled to a specific level over multiple seasons of coffee harvesting. However, the percent of coffee harvesting required is determined by the level of CBB infestation at the beginning of the first season and in some cases it is impossible to achieve that percentage.

Vinyl sulfone-modified carbohydrates: Michael acceptors and 2π partners for the synthesis of functionalized sugars and enantiomerically pure carbocycles and heterocycles.

Increasing demands for molecules with skeletal complexity, including those of stereochemical diversity, require new synthetic strategies. Carbohydrates have been used extensively as chiral building blocks for the synthesis of various complex molecules. On the other hand, the vinyl sulfone group has been identified as a unique functional group, which acts either as a Michael acceptor or a 2π partner in cycloaddition reactions. A combination of the high reactivity of the vinyl sulfone group and the in-built chiralities of carbohydrates has the potential to function as a powerful tool to generate a wide variety of enantiomerically pure reactive intermediates. Since CS bond formation in carbohydrates is easily achieved with regioselectivity, further synthetic manipulations of these thiosugars has led to the generation of a wide range of vinyl sulfone-modified furanosyl, pyranosyl, acyclic, and bicyclic carbohydrates. Several approaches have been studied to standardize the preparative methods for accessing vinyl sulfone-modified carbohydrates at least on a gram scale. Reactions of these modified carbohydrates with appropriate reagents afford a large number of new chemical entities primarily via (i) Michael addition reactions, (ii) desulfostannylation, (iii) Michael-initiated ring-closure reactions, and (iv) cycloaddition reactions. A wide range of desulfonylating reagents in the context of sensitive molecules such as carbohydrates have also been extensively studied. Applications of these strategies have led to the synthesis of (a) amino sugars and branched-chain sugars, (b) C-glycosides, (c) enantiomerically pure cyclopropanes, five- and six-membered carbocycles, (d) saturated oxa-, aza-, and thio-monocyclic heterocycles, (e) bi-and tricyclic saturated oxa and aza heterocycles, (f) enantiomerically pure and trisubstituted pyrroles, (g) 1,5-disubstituted 1,2,3-triazolylated carbohydrates and the corresponding triazole-linked di- and trisaccharides, (h) divinyl sulfone-modified carbohydrates and densely functionalized S,S-dioxothiomorpholines, and (i) modified nucleosides. Details of reaction conditions were incorporated as much as possible and mechanistic discussions were included wherever necessary.

Order-indeterminant event-based maps for learning a beat.

The process by which humans synchronize to a musical beat is believed to occur through error-correction where an individual's estimates of the period and phase of the beat time are iteratively adjusted to align with an external stimuli. Mathematically, error-correction can be described using a two-dimensional map where convergence to a fixed point corresponds to synchronizing to the beat. In this paper, we show how a neural system, called a beat generator, learns to adapt its oscillatory behavior through error-correction to synchronize to an external periodic signal. We construct a two-dimensional event-based map, which iteratively adjusts an internal parameter of the beat generator to speed up or slow down its oscillatory behavior to bring it into synchrony with the periodic stimulus. The map is novel in that the order of events defining the map are not a priori known. Instead, the type of error-correction adjustment made at each iterate of the map is determined by a sequence of expected events. The map possesses a rich repertoire of dynamics, including periodic solutions and chaotic orbits.

Short-term synaptic dynamics control the activity phase of neurons in an oscillatory network.

In oscillatory systems, neuronal activity phase is often independent of network frequency. Such phase maintenance requires adjustment of synaptic input with network frequency, a relationship that we explored using the crab, Cancer borealis, pyloric network. The burst phase of pyloric neurons is relatively constant despite a > two fold variation in network frequency. We used noise input to characterize how input shape influences burst delay of a pyloric neuron, and then used dynamic clamp to examine how burst phase depends on the period, amplitude, duration, and shape of rhythmic synaptic input. Phase constancy across a range of periods required a proportional increase of synaptic duration with period. However, phase maintenance was also promoted by an increase of amplitude and peak phase of synaptic input with period. Mathematical analysis shows how short-term synaptic plasticity can coordinately change amplitude and peak phase to maximize the range of periods over which phase constancy is achieved.

A neuromechanistic model for rhythmic beat generation.

When listening to music, humans can easily identify and move to the beat. Numerous experimental studies have identified brain regions that may be involved with beat perception and representation. Several theoretical and algorithmic approaches have been proposed to account for this ability. Related to, but different from the issue of how we perceive a beat, is the question of how we learn to generate and hold a beat. In this paper, we introduce a neuronal framework for a beat generator that is capable of learning isochronous rhythms over a range of frequencies that are relevant to music and speech. Our approach combines ideas from error-correction and entrainment models to investigate the dynamics of how a biophysically-based neuronal network model synchronizes its period and phase to match that of an external stimulus. The model makes novel use of on-going faster gamma rhythms to form a set of discrete clocks that provide estimates, but not exact information, of how well the beat generator spike times match those of a stimulus sequence. The beat generator is endowed with plasticity allowing it to quickly learn and thereby adjust its spike times to achieve synchronization. Our model makes generalizable predictions about the existence of asymmetries in the synchronization process, as well as specific predictions about resynchronization times after changes in stimulus tempo or phase. Analysis of the model demonstrates that accurate rhythmic time keeping can be achieved over a range of frequencies relevant to music, in a manner that is robust to changes in parameters and to the presence of noise.

Phase-locking and bistability in neuronal networks with synaptic depression.

We consider a recurrent network of two oscillatory neurons that are coupled with inhibitory synapses. We use the phase response curves of the neurons and the properties of short-term synaptic depression to define Poincaré maps for the activity of the network. The fixed points of these maps correspond to phase-locked modes of the network. Using these maps, we analyze the conditions that allow short-term synaptic depression to lead to the existence of bistable phase-locked, periodic solutions. We show that bistability arises when either the phase response curve of the neuron or the short-term depression profile changes steeply enough. The results apply to any Type I oscillator and we illustrate our findings using the Quadratic Integrate-and-Fire and Morris-Lecar neuron models.

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

The normal alignment of circadian rhythms with the 24-h 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 h 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.

A balance of outward and linear inward ionic currents is required for generation of slow-wave oscillations.

Regenerative inward currents help produce slow oscillations through a negative-slope conductance region of their current-voltage relationship that is well approximated by a linear negative conductance. We used dynamic-clamp injections of a linear current with such conductance, INL, to explore why some neurons can generate intrinsic slow oscillations whereas others cannot. We addressed this question in synaptically isolated neurons of the crab Cancer borealis after blocking action potentials. The pyloric network consists of a distinct pacemaker and follower neurons, all of which express the same complement of ionic currents. When the pyloric dilator (PD) neuron, a member of the pacemaker group, was injected with INL with dynamic clamp, it consistently produced slow oscillations. In contrast, all follower neurons failed to oscillate with INL To understand these distinct behaviors, we compared outward current levels of PD with those of follower lateral pyloric (LP) and ventral pyloric (VD) neurons. We found that LP and VD neurons had significantly larger high-threshold potassium currents (IHTK) than PD and LP had lower-transient potassium current (IA). Reducing IHTK pharmacologically enabled both LP and VD neurons to produce INL-induced oscillations, whereas modifying IA levels did not affect INL-induced oscillations. Using phase-plane and bifurcation analysis of a simplified model cell, we demonstrate that large levels of IHTK can block INL-induced oscillatory activity whereas generation of oscillations is almost independent of IA levels. These results demonstrate the general importance of a balance between inward pacemaking currents and high-threshold K+ current levels in determining slow oscillatory activity.NEW & NOTEWORTHY Pacemaker neuron-generated rhythmic activity requires the activation of at least one inward and one outward current. We have previously shown that the inward current can be a linear current (with negative conductance). Using this simple mechanism, here we demonstrate that the inward current conductance must be in relative balance with the outward current conductances to generate oscillatory activity. Surprisingly, an excess of outward conductances completely precludes the possibility of achieving such a balance.

Entrainment Maps.

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 1-dimensional map that we construct for 3 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.

The role of electrical coupling in generating and modulating oscillations in a neuronal network.

A simplified model of the crustacean gastric mill network is considered. Rhythmic activity in this network has largely been attributed to half center oscillations driven by mutual inhibition. We use mathematical modeling and dynamical systems theory to show that rhythmic oscillations in this network may also depend on, or even arise from, a voltage-dependent electrical coupling between one of the cells in the half-center network and a projection neuron that lies outside of the network. This finding uncovers a potentially new mechanism for the generation of oscillations in neuronal networks.

Effects of synaptic plasticity on phase and period locking in a network of two oscillatory neurons.

We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response curves of the neurons with the short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network. Fixed points of these maps correspond to the phase-locked modes of the network. These maps allow us to analyze the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, we show how various parameters of the model affect the existence and stability of phase-locked solutions. We find conditions on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the phase of locking between the neurons or vice versa. A generalization to cobwebbing for two-dimensional maps is also discussed.

The role of linear and voltage-dependent ionic currents in the generation of slow wave oscillations.

Neuronal oscillatory activity is generated by a combination of ionic currents, including at least one inward regenerative current that brings the cell towards depolarized voltages and one outward current that repolarizes the cell. Such currents have traditionally been assumed to require voltage-dependence. Here we test the hypothesis that the voltage dependence of the regenerative inward current is not necessary for generating oscillations. Instead, a current I NL that is linear in the biological voltage range and has negative conductance is sufficient to produce regenerative activity. The current I NL can be considered a linear approximation to the negative-conductance region of the current-voltage relationship of a regenerative inward current. Using a simple conductance-based model, we show that I NL , in conjunction with a voltage-gated, non-inactivating outward current, can generate oscillatory activity. We use phase-plane and bifurcation analyses to uncover a rich variety of behaviors as the conductance of I NL is varied, and show that oscillations emerge as a result of destabilization of the resting state of the model neuron. The model shows the need for well-defined relationships between the inward and outward current conductances, as well as their reversal potentials, in order to produce stable oscillatory activity. Our analysis predicts that a hyperpolarization-activated inward current can play a role in stabilizing oscillatory activity by preventing swings to very negative voltages, which is consistent with what is recorded in biological neurons in general. We confirm this prediction of the model experimentally in neurons from the crab stomatogastric ganglion.

A mathematical model towards understanding the mechanism of neuronal regulation of wake-NREMS-REMS states.

In this study we have constructed a mathematical model of a recently proposed functional model known to be responsible for inducing waking, NREMS and REMS. Simulation studies using this model reproduced sleep-wake patterns as reported in normal animals. The model helps to explain neural mechanism(s) that underlie the transitions between wake, NREMS and REMS as well as how both the homeostatic sleep-drive and the circadian rhythm shape the duration of each of these episodes. In particular, this mathematical model demonstrates and confirms that an underlying mechanism for REMS generation is pre-synaptic inhibition from substantia nigra onto the REM-off terminals that project on REM-on neurons, as has been recently proposed. The importance of orexinergic neurons in stabilizing the wake-sleep cycle is demonstrated by showing how even small changes in inputs to or from those neurons can have a large impact on the ensuing dynamics. The results from this model allow us to make predictions of the neural mechanisms of regulation and patho-physiology of REMS.

Co-existent activity patterns in inhibitory neuronal networks with short-term synaptic depression.

A network of two neurons mutually coupled through inhibitory synapses that display short-term synaptic depression is considered. We show that synaptic depression expands the number of possible activity patterns that the network can display and allows for co-existence of different patterns. Specifically, the network supports different types of n-m anti-phase firing patterns, where one neuron fires n spikes followed by the other neuron firing m spikes. When maximal synaptic conductances are identical, n-n anti-phase firing patterns are obtained and there are conductance intervals over which different pairs of these solutions co-exist. The multitude of n-m anti-phase patterns and their co-existence are not found when the synapses are non-depressing. Geometric singular perturbation methods for dynamical systems are applied to the original eight-dimensional model system to derive a set of one-dimensional conditions for the existence and co-existence of different anti-phase solutions. The generality and validity of these conditions are demonstrated through numerical simulations utilizing the Hodgkin-Huxley and Morris-Lecar neuronal models.

The influence of the A-current on the dynamics of an oscillator-follower inhibitory network.

The transient potassium A-current is present in almost all neurons and plays an essential role in determining the timing and frequency of action potential generation. We use a three-variable mathematical model to examine the role of the A-current in a rhythmic inhibitory network, as is common in central pattern generation. We focus on a feed-forward architecture consisting of an oscillator neuron inhibiting a follower neuron. We use separation of time scales to demonstrate that the trajectory of the follower neuron within each cycle can be tracked by analyzing the dynamics on a 2-dimensional slow manifold that as determined by the two slow model variables: the recovery variable and the inactivation of the A-current. The steady-state trajectory, however, requires tracking the slow variables across multiple cycles. We show that tracking the slow variables, under simplifying assumptions, leads to a one-dimensional map of the unit interval with at most a single discontinuity depending on g(A), the maximal conductance of the A-current, or other model parameters. We demonstrate that, as the value of g(A) is varied, the trajectory of the follower neuron goes through a set of bifurcations to produce n:m periodic solutions where the follower neuron becomes active m times for each n cycles of the oscillator. Using a generalized Pascal triangle, each n:m trajectory can be constructed as a combination of solutions from a higher level of the triangle.