A general linear framework for the comparison and evaluation of models of sensorimotor synchronization

Sensorimotor synchronization (SMS), the temporal coordination of a rhythmic movement with an external rhythm, has been studied most often in tasks that require tapping along with a metronome. Models of SMS use information about the timing of preceding stimuli and responses to predict when the next response will be made. This article compares the theoretical structure and empirical predictions of four two-parameter models proposed in the literature: Michon (Timing in temporal tracking, Van Gorcum, Assen, 1967), Hary and Moore (Br J Math Stat Psychol 40:109–124, 1987b), Mates (Biol Cybern 70:463–473, 1994a; Biol Cybern 70:475–484, 1994b), and Schulze et al. (Mus Percept 22:461–467, 2005). By embedding these models within a general linear framework, the mathematical equivalence of the Michon, Hary and Moore, and Schulze et al. models is demonstrated. The Mates model, which differs from the other three, is then tested empirically with new data from a tapping experiment in which the metronome alternated between two tempi. The Mates model predictions are found to be invalid for about one-third of the trials, suggesting that at least one of the model’s underlying assumptions is incorrect. The other models cannot be refuted as easily, but they do not predict some features of the data very accurately. Comparison of the models’ predictions in a training/test procedure did not yield any significant differences. The general linear framework introduced here may help in the formulation of new models that make better predictions.     

Thalamocortical transformations of periodic stimuli: the effect of stimulus velocity and synaptic short-term depression in the vibrissa–barrel system

Recent works on the response of barrel neurons to periodic deflections of the rat vibrissae have shown that the stimulus velocity is encoded in the corti cal spike rate (Pinto et al., Journal of Neurophysiology, 83(3), 1158–1166, 2000; Arabzadeh et al., Journal of Neuroscience, 23(27), 9146–9154, 2003). Other studies have reported that repetitive pulse stimulation produces band-pass filtering of the barrel response rate centered around 7–10 Hz (Garabedian et al., Journal of Neurophysiology, 90, 1379–1391, 2003) whereas sinusoidal stimulation gives an increasing rate up to 350 Hz (Arabzadeh et al., Journal of Neuroscience, 23(27), 9146–9154, 2003). To explore the mechanisms underlying these results we propose a simple computational model consisting in an ensemble of cells in the ventro-posterior medial thalamic nucleus (VPm) encoding the stimulus velocity in the temporal profile of their response, connected to a single barrel cell through synapses showing short-term depression. With sinusoidal stimulation, encoding the velocity in VPm facilitates the response as the stimulus frequency increases and it causes the velocity to be encoded in the cortical rate in the frequency range 20–100 Hz. Synaptic depression does not suppress the response with sinusoidal stimulation but it produces a band-pass behavior using repetitive pulses. We also found that the passive properties of the cell membrane eventually suppress the response to sinusoidal stimulation at high frequencies, something not observed experimentally. We argue that network effects not included here must be important in sustaining the response at those frequencies.

Dynamic formation of oriented patches in chondrocyte cell cultures

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.     

A Computational Analysis of Localized Ca<Superscript>2+</Superscript>-Dynamics Generated by Heterogeneous Release Sites

We investigate the role of heterogeneous expression of IP₃R and RyR in generating diverse elementary Ca²⁺ signals. It has been shown empirically (Wojcikiewicz and Luo in Mol. Pharmacol. 53(4):656–662, 1998; Newton et al. in J. Biol. Chem. 269(46):28613–28619, 1994; Smedt et al. in Biochem. J. 322(Pt. 2):575–583, 1997) that tissues express various proportions of IP₃ and RyR isoforms and this expression is dynamically regulated (Parrington et al. in Dev. Biol. 203(2):451–461, 1998; Fissore et al. in Biol. Reprod. 60(1):49–57, 1999; Tovey et al. in J. Cell Sci. 114(Pt. 22):3979–3989, 2001). Although many previous theoretical studies have investigated the dynamics of localized calcium release sites (Swillens et al. in Proc. Natl. Acad. Sci. U.S.A. 96(24):13750–13755, 1999; Shuai and Jung in Proc. Natl. Acad. Sci. U.S.A. 100(2):506–510, 2003a; Shuai and Jung in Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 67(3 Pt. 1):031905, 2003b; Thul and Falcke in Biophys. J. 86(5):2660–2673, 2004; DeRemigio and Smith in Cell Calcium 38(2):73–86, 2005; Nguyen et al. in Bull. Math. Biol. 67(3):393–432, 2005), so far all such studies focused on release sites consisting of identical channel types. We have extended an existing mathematical model (Nguyen et al. in Bull. Math. Biol. 67(3):393–432, 2005) to release sites with two (or more) receptor types, each with its distinct channel kinetics. Mathematically, the release site is represented by a transition probability matrix for a collection of nonidentical stochastically gating channels coupled through a shared Ca²⁺ domain. We demonstrate that under certain conditions a previously defined mean-field approximation of the coupling strength does not accurately reproduce the release site dynamics. We develop a novel approximation and establish that its performance in these instances is superior. We use this mathematical framework to study the effect of heterogeneity in the Ca²⁺-regulation of two colocalized channel types on the release site dynamics. We consider release sites consisting of channels with both Ca²⁺-activation and inactivation (“four-state channels”) and channels with Ca²⁺-activation only (“two-state channels”) and show that for the appropriate parameter values, synchronous channel openings within a release site with any proportion of two-state to four-state channels are possible, however, the larger the proportion of two-state channels, the more sensitive the dynamics are to the exact spatial positioning of the channels and the distance between channels. Specifically, the clustering of even a small number of two-state channels interferes with puff/spark termination and increases puff durations or leads to a tonic response.     

Role of CAMKII in reinforcement learning: a computational model of glutamate and dopamine signaling pathways

Timely release of dopamine (DA) at the striatum seems to be important for reinforcement learning (RL) mediated by the basal ganglia. Houk et al. (in: Houk et al (eds) Models of information processing in the basal ganglia, (1995) proposed a cellular signaling pathway model to characterize the interaction between DA and glutamate pathways that have a role in RL. The model simulation results, using GENESIS KINETIKIT simulator, point out that there is not only prolongation of duration as proposed by Houk et al. (1995), but also an enhancement in the amplitude of autophosphorylation of CaMKII. Further, the autophosphorylated form of CaMKII may form a basis for the “eligibility trace” condition required in RL. This simulation study is the first of its kind to support the comprehensive theoretical proposal of Houk et al. (1995).     

A PDE Model for Imatinib-Treated Chronic Myelogenous Leukemia

We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181–1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728–744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells. An imatinib treatment is also incorporated into the model. The simulations of the new PDE model are shown to qualitatively agree with the results that were obtained with the discrete-time (difference equation and agent-based) models. At the same time, for a quantitative agreement, it is necessary to adjust the values of certain parameters, such as the rates of imatinib-induced inhibition and degradation.     

A nonlinear dynamics model for simulating long range correlations of cognitive bistability

Simulation results of bistable perception due to ambiguous visual stimuli are presented which are obtained with a behavioral nonlinear dynamics model using perception–attention–memory coupling. This model provides an explanation of recent experimental results of Gao et al. (Cogn Process 7:105–112, 2006a) and it supports their speculation that the fractal character of perceptual dominance time series may be understood in terms of nonlinear and reentrant dynamics of brain processing. Percept reversals are induced by attention fatigue and noise, with an attention bias which balances the relative percept duration. Dynamical coupling of the attention bias to the perception state introduces memory effects leading to significant long range correlations of perceptual duration times as quantified by the Hurst parameter H > 0.5 (Mandelbrot, The fractal geometry of nature, 1991), in agreement with Gao et al. (Cogn Process 7:105–112, 2006a).     

The role of topography in promoting fractal patchiness in a carbonate shelf landscape

Satellite remote sensing has shown numerous aspects of coral reef seascapes to be fractal. That is, they display characteristics of scale-invariance and complexity. To date, an understanding of why reefscapes adopt this curious scaling has been lacking. This property was investigated using high-resolution terrain models built using bathymetric LiDAR soundings of the shallow Puerto Rico insular shelf. A computer-simulation model constructed using simple random processes was adequate to describe many of the intricacies of actual coral reef terrain. This model, based on fractional Brownian motion (fBm), produced surfaces that were visually and statistically indistinguishable from natural seabeds, at spatial scales of 0.001–25 km². The conformity between model and nature allowed us to ascertain the importance of topography as a driver for the fractal patchiness that has been shown to occur in plan-view maps of reefscapes (e.g. Purkis et al. J Sediment Res 75:861–876, 2005, J Geol 115:493–508, 2007). For the considered Puerto Rican shelf, the necessary Brownian-like seabed topography likely arose through karst erosion overprinted by several episodes of reef development. Communicated by Geology Editor Dr. Bernhard Riegl     

Evolution of dispersal distance

The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. Ezoe (J Theor Biol 190:287–293, 1998) and Levin and Muller-Landau (Evol Ecol Res 2:409–435, 2000) approached this problem using models that were essentially deterministic, and used calculus to find optimal dispersal parameters. Here we follow Hiebeler (Theor Pop Biol 66:205–218, 2004) and use a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems is done by simulation or nonrigorous methods such as pair approximation. Here, we use machinery developed by Cox et al. (Voter model perturbations and reaction diffusion equations 2011) to rigorously and explicitly compute evolutionarily stable strategies.     

Climate change and coral reefs: Trojan horse or false prophecy?

Maynard et al. (Coral Reefs 27:745–749, 2008a) claim that much of the concern about the impacts of climate change on coral reefs has been “based on essentially untested assumptions regarding reefs and their capacity to cope with future climate change”. If correct, this claim has important implications for whether or not climate change represents the largest long-term threat to the sustainability of coral reefs, especially given their ad hominem argument that many coral reef scientists are guilty of “popularising worst-case scenarios” at the expense of truth. This article looks critically at the claims made by Maynard et al. (Coral Reefs 27:745–749, 2008a) and comes to a very different conclusion, with the thrust and veracity of their argument being called into question. Contrary to the fears of Grigg (Coral Reefs 11:183–186, 1992), who originally made reference to the Cassandra syndrome due to his concern about the sensationalisation of science, the proposition that coral reefs face enormous challenges from climate change and ocean acidification has and is being established through “careful experimentation, long-term monitoring and objective interpretation”. While this is reassuring, coral reef ecosystems continue to face major challenges from ocean warming and acidification. Given this, it is an imperative that scientists continue to maintain the rigour of their research and to communicate their conclusions as widely and clearly as possible. Given the shortage of time and the magnitude of the problem, there is little time to spare.     

Stable periodicity and negative circuits in differential systems

We provide a counter-example to a conjecture of René Thomas on the relationship between negative feedback circuits and stable periodicity in ordinary differential equation systems (Kaufman et al. in J Theor Biol 248:675–685, 2007). We also prove a weak version of this conjecture by using a theorem of Snoussi.     

S100β Protein Expression: Gender- and Age-Related Daily Changes

S100β is a soluble protein released by glial cells mainly under the activation of the 5-HT1A receptor. It has been reported as a neuro-trophic and -tropic factor that promotes neurite maturation and outgrowth during development. This protein also plays a role in axonal stability and the plasticity underlying long-term potentiation in adult brains. The ability of S100β to rapidly regulate neuronal morphology raises the interesting point of whether there are daily rhythm or gender differences in S100β level in the brain. To answer this question, the S100β expression in adult female and male rats, as well as in adult female CD-21 and S100β −/− female mice, were investigated. Scintillation counting and morphometric analysis of the immunoreactivity of S100β, showed rhythmic daily expression. The female and male rats showed opposite cycles. Females presented the highest value at the beginning of the rest phase (5:00 h), while in males the maximum value appeared in the beginning of the motor activity period (21:00 h). These results confirm previous S100β evaluations in human serum and cerebrospinal fluid reporting the protein’s function as a biomarker for brain damage (Gazzolo et al. in Clin Chem 49:967–970, 2003; Clin Chim Acta 330:131–133, 2003; Pediatr Res 58:1170–1174, 2005), similar behavior was also observed for GFAP in relation to Alzheimer Disease (Fukuyama et al. in Eur Neurol 46:35–38, 2001). The data should be taken into account when considering S100β as a biomarker of health condition. In addition, the results raise questions on which structure or condition imposes these rhythms as well as on the physiological meaning of the observed gender differences.     

Estimation of Cell Proliferation Dynamics Using CFSE Data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57–89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1–26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.     

TGF-Mediated Dynamics in a System of Many Coupled Nephrons

This paper presents a mathematical model of a system of many coupled nephrons branching from a common cortical radial artery, and accompanying analysis of that system. This modeling effort is a first step in understanding how coupling magnifies the tendency of nephrons to oscillate owing to tubuloglomerular feedback. Central to the present work is the single nephron integral model (as in Pitman et al., The IMA Volumes in Mathematics and Its Applications, vol. 129, pp. 345–364, 2002 and in Zaritski, Ph.D. Dissertation, 1999) which is a simplification of the single nephron PDE model of Layton et al. (Am. J. Physiol. 261, F904–F919, 1991). A second principal idea used in the present model is a coupling of model nephrons, generalizing the work of Pitman et al. (Bull. Math. Biol. 66, 1463–1492, 2004) who proposed a model of two coupled nephrons. In this study, we couple nephrons through a nearest neighbor interaction. Speaking generally, our results suggest that a series of similar nephrons coupled to their nearest neighbors are more prone to be found in an oscillatory mode, relative to a single nephron with the same properties. More specifically, we show analytically that, for N coupled identical nephrons, the region supporting oscillatory solutions in the time delay–gain parameter plane increases with N. Numerical simulations suggest that, if N nephrons have gains and time delays that do not differ by much, the system is, again, more prone to oscillate, relative to a single nephron, and the oscillations tend to be approximately synchronous and in-phase. We examine the effect of parameters on bifurcation. We also examine alternative models of coupling; this analysis allows us to conclude that the increased propensity of coupled nephrons to oscillate is a robust finding, true for several models of nephron interaction.     

Using interspike intervals to quantify noise effects on spike trains in temperature encoding neurons

This paper examines how noise interacts with the non-linear dynamical mechanisms of neuronal stimulus. We study the spike trains generated by a minimal Hodgkin-Huxley type model of a cold receptor neuron. The distributions of interspike intervals(ISIs) of purely deterministic simulations exhibit considerable differences compared to the noisy ones. We quantify the effect of noise using ISI return plots and the ISI-distance recently proposed by Kreuz et al. (J Neurosci Meth, 165:151–161, 2007). It is shown that the spike trains of a cold receptor neuron are more strongly affected by noise for low temperatures than for high temperatures. This trend is also observed in both regimes of cold receptors: tonic firing(which occurs for low and high temperatures) and bursting (which occurs for intermediate temperatures).     

Forward shift from reverse replay

In a recent experimental paper Lee et al. (Neuron 51:639–650, 2006) showed that the firing patterns of CA1 complex-spike neurons gradually shifted forward across trials toward prospective goal locations within a recording session over multiple trials. Here we propose a simple model of this result based on the phenomenon of awake sequence reverse replay (Foster and Wilson, Nature 440(7084):615–617, 2006) which occurs when the animal pauses at the reward location. The model is based on the CA3-CA1 anatomy with modulation of CA3-CA1 synaptic plasticity by feedback from CA3 projecting CA1 interneurons. Sequence replays, which are generated in CA3 by removal of subcortical inhibition on CA1 interneurons, are recoded into the synaptic weights of individual CA1 cells. This produces spatially extended CA1 firing fields, whose response provides a value function on experienced paths toward goal locations. Simulations show that the CA1 firing fields show positive movement in center of mass toward reward locations over many trials with negative shift in first few trials, and development of positive skew.     

Use of weighted reference panels based on empirical estimates of ancestry for capturing untyped variation

Many association methods use a subset of genotyped single nucleotide polymorphisms (SNPs) to capture or infer genotypes at other untyped SNPs. We and others previously showed that tag SNPs selected to capture common variation using data from The International HapMap Consortium (Nature 437:1299–1320, 2005), The International HapMap Consortium (Nature 449:851–861, 2007) could also capture variation in populations of similar ancestry to HapMap reference populations (de Bakker et al. in Nat Genet 38:1298–1303, 2006; González-Neira et al. in Genome Res 16:323–330, 2006; Montpetit et al. in PLoS Genet 2:282–290, 2006; Mueller et al. in Am J Hum Genet 76:387–398, 2005). To capture variation in admixed populations or populations less similar to HapMap panels, a “cosmopolitan approach,” in which all samples from HapMap are used as a single reference panel, was proposed. Here we refine this suggestion and show that use of a “weighted reference panel,” constructed based on empirical estimates of ancestry in the target population (relative to available reference panels), is more efficient than the cosmopolitan approach. Weighted reference panels capture, on average, only slightly fewer common variants (minor allele frequency > 5%) than the cosmopolitan approach (mean r ² = 0.977 vs. 0.989, 94.5% variation captured vs. 96.8% at r ² > 0.8), across the five populations of the Multiethnic Cohort, but entail approximately 25% fewer tag SNPs per panel (average 538 vs. 718). These results extend a recent study in two Indian populations (Pemberton et al. in Ann Hum Genet 72:535–546, 2008). Weighted reference panels are potentially useful for both the selection of tag SNPs in diverse populations and perhaps in the design of reference panels for imputation of untyped genotypes in genome-wide association studies in admixed populations. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.