Correction: Understanding narwhal diving behaviour using Hidden Markov Models with dependent state distributions and long range dependence.

[This corrects the article DOI: 10.1371/journal.pcbi.1006425.].

Narwhals react to ship noise and airgun pulses embedded in background noise.

Anthropogenic activities are increasing in the Arctic, posing a threat to niche-conservative species with high seasonal site fidelity, such as the narwhal Monodon monoceros. In this controlled sound exposure study, six narwhals were live-captured and instrumented with animal-borne tags providing movement and behavioural data, and exposed to concurrent ship noise and airgun pulses. All narwhals reacted to sound exposure with reduced buzzing rates, where the response was dependent on the magnitude of exposure defined as 1/distance to ship. Buzzing rate was halved at 12 km from the ship, and whales ceased foraging at 7-8 km. Effects of exposure could be detected at distances > 40 km from the ship.At only a few kilometres from the ship, the received high-frequency cetacean weighted sound exposure levels were below background noise indicating extreme sensitivity of narwhals towards sound disturbance and demonstrating their ability to detect signals embedded in background noise. The narwhal's reactions to sustained disturbance may have a plethora of consequences both at individual and population levels. The observed reactions of the whales demonstrate their auditory sensitivity but also emphasize, that anthropogenic activities in pristine narwhal habitats needs to be managed carefully if healthy narwhal populations are to be maintained.

Understanding narwhal diving behaviour using Hidden Markov Models with dependent state distributions and long range dependence.

Diving behaviour of narwhals is still largely unknown. We use Hidden Markov models (HMMs) to describe the diving behaviour of a narwhal and fit the models to a three-dimensional response vector of maximum dive depth, duration of dives and post-dive surface time of 8,609 dives measured in East Greenland over 83 days, an extraordinarily long and rich data set. Narwhal diving patterns have not been analysed like this before, but in studies of other whale species, response variables have been assumed independent. We extend the existing models to allow for dependence between state distributions, and show that the dependence has an impact on the conclusions drawn about the diving behaviour. We try several HMMs with 2, 3 or 4 states, and with independent and dependent log-normal and gamma distributions, respectively, and different covariates to characterize dive patterns. In particular, diurnal patterns in diving behaviour is inferred, by using periodic B-splines with boundary knots in 0 and 24 hours.

Oscillating systems with cointegrated phase processes.

We present cointegration analysis as a method to infer the network structure of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience.

The Space-Clamped Hodgkin-Huxley System with Random Synaptic Input: Inhibition of Spiking by Weak Noise and Analysis with Moment Equations.

We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model system obtained by an Euler method, it is found that with excitation only, there is a critical value of the steady-state excitatory conductance for repetitive spiking without noise, and for values of the conductance near the critical value, small noise has a powerfully inhibitory effect. For a given level of inhibition, there is also a critical value of the steady-state excitatory conductance for repetitive firing, and it is demonstrated that noise in either the excitatory or inhibitory processes or both can powerfully inhibit spiking. Furthermore, near the critical value, inverse stochastic resonance was observed when noise was present only in the inhibitory input process. The system of deterministic differential equations for the approximate first- and second-order moments of the model is derived. They are solved using Runge-Kutta methods, and the solutions are compared with the results obtained by simulation for various sets of parameters, including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex. The mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels. In the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells, the assumptions underlying the moment equation approach are not satisfied. The simulation results show that noisy synaptic input of either an excitatory or inhibitory character or both may lead to the suppression of firing in neurons operating near a critical point and this has possible implications for cortical networks. Although suppression of firing is corroborated for the system of moment equations, there seem to be substantial differences between the dynamical properties of the original system of stochastic differential equations and the much larger system of moment equations.

Absorption and initial metabolism of 75Se-l-selenomethionine: a kinetic model based on dynamic scintigraphic data.

Selenomethionine (SeMet) is an important organic nutritional source of Se, but the uptake and metabolism of SeMet are poorly characterised in humans. Dynamic gamma camera images of the abdominal region were acquired from eight healthy young men after the ingestion of radioactive 75Se-l-SeMet (75Se-SeMet). Scanning started simultaneously to the ingestion of 75Se-SeMet and lasted 120 min. We generated time-activity curves from two-dimensional regions of interest in the stomach, small intestine and liver. During scanning, blood samples were collected at 10-min intervals to generate plasma time-activity curves. A four-compartment model, augmented with a delay between the liver and plasma, was fitted to individual participants' data. The mean rate constant for 75Se-SeMet transport was 2·63 h-1 from the stomach to the small intestine, 13·2 h-1 from the small intestine to the liver, 0·261 h-1 from the liver to the plasma and 0·267 h-1 from the stomach to the plasma. The delay in the liver was 0·714 h. Gamma camera imaging provides data for use in compartmental modelling of 75Se-SeMet absorption and metabolism in humans. In clinical settings, the obtained rate constants and the delay in the liver may be useful variables for quantifying reduced intestinal absorption capacity or liver function.

Parameter inference from hitting times for perturbed Brownian motion.

A latent internal process describes the state of some system, e.g. the social tension in a political conflict, the strength of an industrial component or the health status of a person. When this process reaches a predefined threshold, the process terminates and an observable event occurs, e.g. the political conflict finishes, the industrial component breaks down or the person dies. Imagine an intervention, e.g., a political decision, maintenance of a component or a medical treatment, is initiated to the process before the event occurs. How can we evaluate whether the intervention had an effect? To answer this question we describe the effect of the intervention through parameter changes of the law governing the internal process. Then, the time interval between the start of the process and the final event is divided into two subintervals: the time from the start to the instant of intervention, denoted by S, and the time between the intervention and the threshold crossing, denoted by R. The first question studied here is: What is the joint distribution of (S,R)? The theoretical expressions are provided and serve as a basis to answer the main question: Can we estimate the parameters of the model from observations of S and R and compare them statistically? Maximum likelihood estimators are calculated and applied on simulated data under the assumption that the process before and after the intervention is described by the same type of model, i.e. a Brownian motion, but with different parameters. Also covariates and handling of censored observations are incorporated into the statistical model, and the method is illustrated on lung cancer data.

Estimating latency from inhibitory input.

Stimulus response latency is the time period between the presentation of a stimulus and the occurrence of a change in the neural firing evoked by the stimulation. The response latency has been explored and estimation methods proposed mostly for excitatory stimuli, which means that the neuron reacts to the stimulus by an increase in the firing rate. We focus on the estimation of the response latency in the case of inhibitory stimuli. Models used in this paper represent two different descriptions of response latency. We consider either the latency to be constant across trials or to be a random variable. In the case of random latency, special attention is given to models with selective interaction. The aim is to propose methods for estimation of the latency or the parameters of its distribution. Parameters are estimated by four different methods: method of moments, maximum-likelihood method, a method comparing an empirical and a theoretical cumulative distribution function and a method based on the Laplace transform of a probability density function. All four methods are applied on simulated data and compared.

Synaptic inhibition and excitation estimated via the time constant of membrane potential fluctuations.

When recording the membrane potential, V, of a neuron it is desirable to be able to extract the synaptic input. Critically, the synaptic input is stochastic and nonreproducible so one is therefore often restricted to single-trial data. Here, we introduce means of estimating the inhibition and excitation and their confidence limits from single sweep trials. The estimates are based on the mean membrane potential, V, and the membrane time constant, τ. The time constant provides the total conductance (G = capacitance/τ) and is extracted from the autocorrelation of V. The synaptic conductances can then be inferred from V when approximating the neuron as a single compartment. We further employ a stochastic model to establish limits of confidence. The method is verified on models and experimental data, where the synaptic input is manipulated pharmacologically or estimated by an alternative method. The method gives best results if the synaptic input is large compared with other conductances, the intrinsic conductances have little or no time dependence or are comparably small, the ligand-gated kinetics is faster than the membrane time constant, and the majority of synaptic contacts are electrotonically close to soma (recording site). Although our data are in current clamp, the method also works in V-clamp recordings, with some minor adaptations. All custom made procedures are provided in Matlab.

The Morris-Lecar neuron model embeds a leaky integrate-and-fire model.

We show that the stochastic Morris-Lecar neuron, in a neighborhood of its stable point, can be approximated by a two-dimensional Ornstein-Uhlenbeck (OU) modulation of a constant circular motion. The associated radial OU process is an example of a leaky integrate-and-fire (LIF) model prior to firing. A new model constructed from a radial OU process together with a simple firing mechanism based on detailed Morris-Lecar firing statistics reproduces the Morris-Lecar Interspike Interval (ISI) distribution, and has the computational advantages of a LIF. The result justifies the large amount of attention paid to the LIF models.

Warning of a forthcoming collapse of the Atlantic meridional overturning circulation.

The Atlantic meridional overturning circulation (AMOC) is a major tipping element in the climate system and a future collapse would have severe impacts on the climate in the North Atlantic region. In recent years weakening in circulation has been reported, but assessments by the Intergovernmental Panel on Climate Change (IPCC), based on the Climate Model Intercomparison Project (CMIP) model simulations suggest that a full collapse is unlikely within the 21st century. Tipping to an undesired state in the climate is, however, a growing concern with increasing greenhouse gas concentrations. Predictions based on observations rely on detecting early-warning signals, primarily an increase in variance (loss of resilience) and increased autocorrelation (critical slowing down), which have recently been reported for the AMOC. Here we provide statistical significance and data-driven estimators for the time of tipping. We estimate a collapse of the AMOC to occur around mid-century under the current scenario of future emissions.

Using quantile regression and relative entropy to assess the period of anomalous behavior of marine mammals following tagging.

Tagging of animals induces a variable stress response which following release will obscure natural behavior. It is of scientific relevance to establish methods that assess recovery from such behavioral perturbation and generalize well to a broad range of animals, while maintaining model transparency. We propose two methods that allow for subdivision of animals based on covariates, and illustrate their use on N = 20 narwhals (Monodon monoceros) and N = 4 bowhead whales (Balaena mysticetus), captured and instrumented with Acousonde™ behavioral tags, but with a framework that easily generalizes to other marine animals and sampling units. The narwhals were divided into two groups based on handling time, short ( t < 58 min) and long ( t ≥ 58 min), to measure the effect on recovery. Proxies for energy expenditure (VeDBA) and rapid movement (jerk) were derived from accelerometer data. Diving profiles were characterized using two metrics (target depth and dive duration) derived from depth data. For accelerometer data, recovery was estimated using quantile regression (QR) on the log-transformed response, whereas depth data were addressed using relative entropy (RE) between hourly distributions of dive duration (partitioned into three target depth ranges) and the long-term average distribution. Quantile regression was used to address location-based behavior to accommodate distributional shifts anticipated in aquatic locomotion. For all narwhals, we found fast recovery in the tail of the distribution (<3 h) compared with a variable recovery at the median (∼1-10 h) and with a significant difference between groups separated by handling time. Estimates of bowhead whale recovery times showed fast median recovery (<3 h) and slow recovery at the tail (>6 h), but were affected by substantial uncertainty. For the diving profiles, as characterized by the component pair (target depth, dive duration), the recovery was slower (narwhals-long: t < 16 h; narwhals-short: t < 10 h; bowhead whales: <9 h) and with a difference between narwhals with short vs long handling times. Using simple statistical concepts, we have presented two transparent and general methods for analyzing high-resolution time series data from marine animals, addressing energy expenditure, activity, and diving behavior, and which allows for comparison between groups of animals based on well-defined covariates.

Stuck in a corner: Anthropogenic noise threatens narwhals in their once pristine Arctic habitat.

Niche-conservative species are especially susceptible to changes in their environment, and detecting the negative effects of new stressors in their habitats is vital for safeguarding of these species. In the Arctic, human disturbance including marine traffic and exploration of resources is increasing rapidly due to climate change-induced reduction of sea ice. Here, we show that the narwhal, Monodon monoceros, is extremely sensitive to human-made noise. Narwhals avoided deep diving (> 350 m) with simultaneous reduction of foraging and increased shallow diving activity as a response to either ship sound alone or ship sound with concurrent seismic airgun pulses. Normal behavior decreased by 50 to 75% at distances where received sound levels were below background noise. Narwhals were equally responsive to both disturbance types, hence demonstrating their acute sensitivity to ship sound. This sensitivity coupled with their special behavioral-ecological strategy including a narrow ecological niche and high site fidelity makes them thus especially vulnerable to human impacts in the Arctic.

Spontaneous neurotransmission at evocable synapses predicts their responsiveness to action potentials.

Synaptic transmission relies on presynaptic neurotransmitter (NT) release from synaptic vesicles (SVs) and on NT detection by postsynaptic receptors. Transmission exists in two principal modes: action-potential (AP) evoked and AP-independent, "spontaneous" transmission. AP-evoked neurotransmission is considered the primary mode of inter-neuronal communication, whereas spontaneous transmission is required for neuronal development, homeostasis, and plasticity. While some synapses appear dedicated to spontaneous transmission only, all AP-responsive synapses also engage spontaneously, but whether this encodes functional information regarding their excitability is unknown. Here we report on functional interdependence of both transmission modes at individual synaptic contacts of Drosophila larval neuromuscular junctions (NMJs) which were identified by the presynaptic scaffolding protein Bruchpilot (BRP) and whose activities were quantified using the genetically encoded Ca2+ indicator GCaMP. Consistent with the role of BRP in organizing the AP-dependent release machinery (voltage-dependent Ca2+ channels and SV fusion machinery), most active BRP-positive synapses (>85%) responded to APs. At these synapses, the level of spontaneous activity was a predictor for their responsiveness to AP-stimulation. AP-stimulation resulted in cross-depletion of spontaneous activity and both transmission modes were affected by the non-specific Ca2+ channel blocker cadmium and engaged overlapping postsynaptic receptors. Thus, by using overlapping machinery, spontaneous transmission is a continuous, stimulus independent predictor for the AP-responsiveness of individual synapses.

Classification of brain states that predicts future performance in visual tasks based on co-integration analysis of EEG data.

Electroencephalogram (EEG) is a popular tool for studying brain activity. Numerous statistical techniques exist to enhance understanding of the complex dynamics underlying the EEG recordings. Inferring the functional network connectivity between EEG channels is of interest, and non-parametric inference methods are typically applied. We propose a fully parametric model-based approach via cointegration analysis. It not only estimates the network but also provides further insight through cointegration vectors, which characterize equilibrium states, and the corresponding loadings, which describe the mechanism of how the EEG dynamics is drawn to the equilibrium. We outline the estimation procedure in the context of EEG data, which faces specific challenges compared with the common econometric problems, for which cointegration analysis was originally conceived. In particular, the dimension is higher, typically around 64; there is usually access to repeated trials; and the data are artificially linearly dependent through the normalization done in EEG recordings. Finally, we illustrate the method on EEG data from a visual task experiment and show how brain states identified via cointegration analysis can be utilized in further investigations of determinants playing roles in sensory identifications.

Applying historical data in a nonlinear mixed-effects model can reduce the number of control rats required for calculation of the relative potency of insulin analogues.

This paper examines how to reduce the number of control animals in preclinical hyperinsulemic glucose clamp studies if we make use of information on historical studies. A dataset consisting of 59 studies in rats to investigate new insulin analogues for diabetics, collected in the years 2000 to 2015, is analysed. A simulation experiment is performed based on a carefully built nonlinear mixed-effects model including historical information, comparing results (for the relative log-potency) with the standard approach ignoring previous studies. We find that by including historical information in the form of the mixed-effects model proposed, we can to remove between 23% and 51% of the control rats in the two studies looked closely upon to get the same level of precision on the relative log-potency as in the standard analysis. How to incorporate the historical information in the form of the mixed-effects model is discussed, where both a mixed-effect meta-analysis approach as well as a Bayesian approach are suggested. The conclusions are similar for the two approaches, and therefore, we conclude that the inclusion of historical information is beneficial in regard to using fewer control rats.

Allosteric stabilization of calcium and phosphoinositide dual binding engages several synaptotagmins in fast exocytosis.

Synaptic communication relies on the fusion of synaptic vesicles with the plasma membrane, which leads to neurotransmitter release. This exocytosis is triggered by brief and local elevations of intracellular Ca2+ with remarkably high sensitivity. How this is molecularly achieved is unknown. While synaptotagmins confer the Ca2+ sensitivity of neurotransmitter exocytosis, biochemical measurements reported Ca2+ affinities too low to account for synaptic function. However, synaptotagmin's Ca2+ affinity increases upon binding the plasma membrane phospholipid PI(4,5)P2 and, vice versa, Ca2+ binding increases synaptotagmin's PI(4,5)P2 affinity, indicating a stabilization of the Ca2+/PI(4,5)P2 dual-bound state. Here, we devise a molecular exocytosis model based on this positive allosteric stabilization and the assumptions that (1.) synaptotagmin Ca2+/PI(4,5)P2 dual binding lowers the energy barrier for vesicle fusion and that (2.) the effect of multiple synaptotagmins on the energy barrier is additive. The model, which relies on biochemically measured Ca2+/PI(4,5)P2 affinities and protein copy numbers, reproduced the steep Ca2+ dependency of neurotransmitter release. Our results indicate that each synaptotagmin engaging in Ca2+/PI(4,5)P2 dual-binding lowers the energy barrier for vesicle fusion by ~5 kBT and that allosteric stabilization of this state enables the synchronized engagement of several (typically three) synaptotagmins for fast exocytosis. Furthermore, we show that mutations altering synaptotagmin's allosteric properties may show dominant-negative effects, even though synaptotagmins act independently on the energy barrier, and that dynamic changes of local PI(4,5)P2 (e.g. upon vesicle movement) dramatically impact synaptic responses. We conclude that allosterically stabilized Ca2+/PI(4,5)P2 dual binding enables synaptotagmins to exert their coordinated function in neurotransmission.

For our brains and nervous systems to work properly, the nerve cells within them must be able to 'talk' to each other. They do this by releasing chemical signals called neurotransmitters which other cells can detect and respond to. Neurotransmitters are packaged in tiny membrane-bound spheres called vesicles. When a cell of the nervous system needs to send a signal to its neighbours, the vesicles fuse with the outer membrane of the cell, discharging their chemical contents for other cells to detect. The initial trigger for neurotransmitter release is a short, fast increase in the amount of calcium ions inside the signalling cell. One of the main proteins that helps regulate this process is synaptotagmin which binds to calcium and gives vesicles the signal to start unloading their chemicals. Despite acting as a calcium sensor, synaptotagmin actually has a very low affinity for calcium ions by itself, meaning that it would not be efficient for the protein to respond alone. Synpatotagmin is more likely to bind to calcium if it is attached to a molecule called PIP₂, which is found in the membranes of cells The effect also occurs in reverse, as the binding of calcium to synaptotagmin increases the protein's affinity for PIP₂. However, how these three molecules ­ synaptotagmin, PIP₂, and calcium ­ work together to achieve the physiological release of neurotransmitters is poorly understood. To help answer this question, Kobbersmed, Berns et al. set up a computer simulation of 'virtual vesicles' using available experimental data on synaptotagmin's affinity with calcium and PIP₂. In this simulation, synaptotagmin could only trigger the release of neurotransmitters when bound to both calcium and PIP₂. The model also showed that each 'complex' of synaptotagmin/calcium/PIP₂ made the vesicles more likely to fuse with the outer membrane of the cell ­ to the extent that only a handful of synaptotagmin molecules were needed to start neurotransmitter release from a single vesicle. These results shed new light on a biological process central to the way nerve cells communicate with each other. In the future, Kobbersmed, Berns et al. hope that this insight will help us to understand the cause of diseases where communication in the nervous system is impaired.

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process.

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein-Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein-Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.

Firing variability is higher than deduced from the empirical coefficient of variation.

A convenient and often used summary measure to quantify the firing variability in neurons is the coefficient of variation (CV), defined as the standard deviation divided by the mean. It is therefore important to find an estimator that gives reliable results from experimental data, that is, the estimator should be unbiased and have low estimation variance. When the CV is evaluated in the standard way (empirical standard deviation of interspike intervals divided by their average), then the estimator is biased, underestimating the true CV, especially if the distribution of the interspike intervals is positively skewed. Moreover, the estimator has a large variance for commonly used distributions. The aim of this letter is to quantify the bias and propose alternative estimation methods. If the distribution is assumed known or can be determined from data, parametric estimators are proposed, which not only remove the bias but also decrease the estimation errors. If no distribution is assumed and the data are very positively skewed, we propose to correct the standard estimator. When defining the corrected estimator, we simply use that it is more stable to work on the log scale for positively skewed distributions. The estimators are evaluated through simulations and applied to experimental data from olfactory receptor neurons in rats.

Distinguishing between parallel and serial processing in visual attention from neurobiological data.

Serial and parallel processing in visual search have been long debated in psychology, but the processing mechanism remains an open issue. Serial processing allows only one object at a time to be processed, whereas parallel processing assumes that various objects are processed simultaneously. Here, we present novel neural models for the two types of processing mechanisms based on analysis of simultaneously recorded spike trains using electrophysiological data from prefrontal cortex of rhesus monkeys while processing task-relevant visual displays. We combine mathematical models describing neuronal attention and point process models for spike trains. The same model can explain both serial and parallel processing by adopting different parameter regimes. We present statistical methods to distinguish between serial and parallel processing based on both maximum likelihood estimates and decoding the momentary focus of attention when two stimuli are presented simultaneously. Results show that both processing mechanisms are in play for the simultaneously recorded neurons, but neurons tend to follow parallel processing in the beginning after the onset of the stimulus pair, whereas they tend to serial processing later on.