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.

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.

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.

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.

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.

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.].

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.

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.

Some like it cold: Temperature-dependent habitat selection by narwhals.

The narwhal (Monodon monoceros) is a high-Arctic species inhabiting areas that are experiencing increases in sea temperatures, which together with reduction in sea ice are expected to modify the niches of several Arctic marine apex predators. The Scoresby Sound fjord complex in East Greenland is the summer residence for an isolated population of narwhals. The movements of 12 whales instrumented with Fastloc-GPS transmitters were studied during summer in Scoresby Sound and at their offshore winter ground in 2017-2019. An additional four narwhals provided detailed hydrographic profiles on both summer and winter grounds. Data on diving of the whales were obtained from 20 satellite-linked time-depth recorders and 16 Acousonde™ recorders that also provided information on the temperature and depth of buzzes. In summer, the foraging whales targeted depths between 300 and 850 m where the preferred areas visited by the whales had temperatures ranging between 0.6 and 1.5°C (mean = 1.1°C, SD = 0.22). The highest probability of buzzing activity during summer was at a temperature of 0.7°C and at depths > 300 m. The whales targeted similar depths at their offshore winter ground where the temperature was slightly higher (range: 0.7-1.7°C, mean = 1.3°C, SD = 0.29). Both the probability of buzzing events and the spatial distribution of the whales in both seasons demonstrated a preferential selection of cold water. This was particularly pronounced in winter where cold coastal water was selected and warm Atlantic water farther offshore was avoided. It is unknown if the small temperature niche of whales while feeding is because prey is concentrated at these temperature gradients and is easier to capture at low temperatures, or because there are limitations in the thermoregulation of the whales. In any case, the small niche requirements together with their strong site fidelity emphasize the sensitivity of narwhals to changes in the thermal characteristics of their habitats.

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.

Rapid regulation of vesicle priming explains synaptic facilitation despite heterogeneous vesicle:Ca2+ channel distances.

Chemical synaptic transmission relies on the Ca2+-induced fusion of transmitter-laden vesicles whose coupling distance to Ca2+ channels determines synaptic release probability and short-term plasticity, the facilitation or depression of repetitive responses. Here, using electron- and super-resolution microscopy at the Drosophila neuromuscular junction we quantitatively map vesicle:Ca2+ channel coupling distances. These are very heterogeneous, resulting in a broad spectrum of vesicular release probabilities within synapses. Stochastic simulations of transmitter release from vesicles placed according to this distribution revealed strong constraints on short-term plasticity; particularly facilitation was difficult to achieve. We show that postulated facilitation mechanisms operating via activity-dependent changes of vesicular release probability (e.g. by a facilitation fusion sensor) generate too little facilitation and too much variance. In contrast, Ca2+-dependent mechanisms rapidly increasing the number of releasable vesicles reliably reproduce short-term plasticity and variance of synaptic responses. We propose activity-dependent inhibition of vesicle un-priming or release site activation as novel facilitation mechanisms.

Cells in the nervous system of all animals communicate by releasing and sensing chemicals at contact points named synapses. The 'talking' (or pre-synaptic) cell stores the chemicals close to the synapse, in small spheres called vesicles. When the cell is activated, calcium ions flow in and interact with the release-ready vesicles, which then spill the chemicals into the synapse. In turn, the 'listening' (or post-synaptic) cell can detect the chemicals and react accordingly. When the pre-synaptic cell is activated many times in a short period, it can release a greater quantity of chemicals, allowing a bigger reaction in the post-synaptic cell. This phenomenon is known as facilitation, but it is still unclear how exactly it can take place. This is especially the case when many of the vesicles are not ready to respond, for example when they are too far from where calcium flows into the cell. Computer simulations have been created to model facilitation but they have assumed that all vesicles are placed at the same distance to the calcium entry point: Kobbersmed et al. now provide evidence that this assumption is incorrect. Two high-resolution imaging techniques were used to measure the actual distances between the vesicles and the calcium source in the pre-synaptic cells of fruit flies: this showed that these distances are quite variable ­ some vesicles sit much closer to the source than others. This information was then used to create a new computer model to simulate facilitation. The results from this computing work led Kobbersmed et al. to suggest that facilitation may take place because a calcium-based mechanism in the cell increases the number of vesicles ready to release their chemicals. This new model may help researchers to better understand how the cells in the nervous system work. Ultimately, this can guide experiments to investigate what happens when information processing at synapses breaks down, for example in diseases such as epilepsy.

Neural decoding with visual attention using sequential Monte Carlo for leaky integrate-and-fire neurons.

How the brain makes sense of a complicated environment is an important question, and a first step is to be able to reconstruct the stimulus that give rise to an observed brain response. Neural coding relates neurobiological observations to external stimuli using computational methods. Encoding refers to how a stimulus affects the neuronal output, and entails constructing a neural model and parameter estimation. Decoding refers to reconstruction of the stimulus that led to a given neuronal output. Existing decoding methods rarely explain neuronal responses to complicated stimuli in a principled way. Here we perform neural decoding for a mixture of multiple stimuli using the leaky integrate-and-fire model describing neural spike trains, under the visual attention hypothesis of probability mixing in which the neuron only attends to a single stimulus at any given time. We assume either a parallel or serial processing visual search mechanism when decoding multiple simultaneous neurons. We consider one or multiple stochastic stimuli following Ornstein-Uhlenbeck processes, and dynamic neuronal attention that switches following discrete Markov processes. To decode stimuli in such situations, we develop various sequential Monte Carlo particle methods in different settings. The likelihood of the observed spike trains is obtained through the first-passage time probabilities obtained by solving the Fokker-Planck equations. We show that the stochastic stimuli can be successfully decoded by sequential Monte Carlo, and different particle methods perform differently considering the number of observed spike trains, the number of stimuli, model complexity, etc. The proposed novel decoding methods, which analyze the neural data through psychological visual attention theories, provide new perspectives to understand the brain.

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.

Spatial and temporal patterns of sound production in East Greenland narwhals.

Changes in climate are rapidly modifying the Arctic environment. As a result, human activities-and the sounds they produce-are predicted to increase in remote areas of Greenland, such as those inhabited by the narwhals (Monodon monoceros) of East Greenland. Meanwhile, nothing is known about these whales' acoustic behavior or their reactions to anthropogenic sounds. This lack of knowledge was addressed by instrumenting six narwhals in Scoresby Sound (Aug 2013-2016) with Acousonde™ acoustic tags and satellite tags. Continuous recordings over up to seven days were used to describe the acoustic behavior of the whales, in particular their use of three types of sounds serving two different purposes: echolocation clicks and buzzes, which serve feeding, and calls, presumably used for social communication. Logistic regression models were used to assess the effects of location in time and space on buzzing and calling rates. Buzzes were mostly produced at depths of 350-650 m and buzzing rates were higher in one particular fjord, likely a preferred feeding area. Calls generally occurred at shallower depths (<100 m), with more than half of these calls occurring near the surface (<7 m), where the whales also spent more than half of their time. A period of silence following release, present in all subjects, was attributed to the capture and tagging operations, emphasizing the importance of longer (multi-day) records. This study provides basic life-history information on a poorly known species-and therefore control data in ongoing or future sound-effect studies.

Estimation of Synaptic Conductances in Presence of Nonlinear Effects Caused by Subthreshold Ionic Currents.

Subthreshold fluctuations in neuronal membrane potential traces contain nonlinear components, and employing nonlinear models might improve the statistical inference. We propose a new strategy to estimate synaptic conductances, which has been tested using in silico data and applied to in vivo recordings. The model is constructed to capture the nonlinearities caused by subthreshold activated currents, and the estimation procedure can discern between excitatory and inhibitory conductances using only one membrane potential trace. More precisely, we perform second order approximations of biophysical models to capture the subthreshold nonlinearities, resulting in quadratic integrate-and-fire models, and apply approximate maximum likelihood estimation where we only suppose that conductances are stationary in a 50-100 ms time window. The results show an improvement compared to existent procedures for the models tested here.

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.

Responses of Leaky Integrate-and-Fire Neurons to a Plurality of Stimuli in Their Receptive Fields.

A fundamental question concerning the way the visual world is represented in our brain is how a cortical cell responds when its classical receptive field contains a plurality of stimuli. Two opposing models have been proposed. In the response-averaging model, the neuron responds with a weighted average of all individual stimuli. By contrast, in the probability-mixing model, the cell responds to a plurality of stimuli as if only one of the stimuli were present. Here we apply the probability-mixing and the response-averaging model to leaky integrate-and-fire neurons, to describe neuronal behavior based on observed spike trains. We first estimate the parameters of either model using numerical methods, and then test which model is most likely to have generated the observed data. Results show that the parameters can be successfully estimated and the two models are distinguishable using model selection.

Preface.

This Special Issue of Mathematical Biosciences and Engineering contains 11 selected papers presented at the Neural Coding 2014 workshop. The workshop was held in the royal city of Versailles in France, October 6-10, 2014. This was the 11th of a series of international workshops on this subject, the first held in Prague (1995), then Versailles (1997), Osaka (1999), Plymouth (2001), Aulla (2003), Marburg (2005), Montevideo (2007), Tainan (2009), Limassol (2010), and again in Prague (2012). Also selected papers from Prague were published as a special issue of Mathematical Biosciences and Engineering and in this way a tradition was started. Similarly to the previous workshops, this was a single track multidisciplinary event bringing together experimental and computational neuroscientists. The Neural Coding Workshops are traditionally biennial symposia. They are relatively small in size, interdisciplinary with major emphasis on the search for common principles in neural coding. The workshop was conceived to bring together scientists from different disciplines for an in-depth discussion of mathematical model-building and computational strategies. Further information on the meeting can be found at the NC2014 website at https://colloque6.inra.fr/neural_coding_2014. The meeting was supported by French National Institute for Agricultural Research, the world's leading institution in this field. This Special Issue of Mathematical Biosciences and Engineering contains 11 selected papers presented at the Neural Coding 2014 workshop. The workshop was held in the royal city of Versailles in France, October 6-10, 2014. This was the 11th of a series of international workshops on this subject, the first held in Prague (1995), then Versailles (1997), Osaka (1999), Plymouth (2001), Aulla (2003), Marburg (2005), Montevideo (2007), Tainan (2009), Limassol (2010), and again in Prague (2012). Also selected papers from Prague were published as a special issue of Mathematical Biosciences and Engineering and in this way a tradition was started. Similarly to the previous workshops, this was a single track multidisciplinary event bringing together experimental and computational neuroscientists. The Neural Coding Workshops are traditionally biennial symposia. They are relatively small in size, interdisciplinary with major emphasis on the search for common principles in neural coding. The workshop was conceived to bring together scientists from different disciplines for an in-depth discussion of mathematical model-building and computational strategies. Further information on the meeting can be found at the NC2014 website at https://colloque6.inra.fr/neural_coding_2014. The meeting was supported by French National Institute for Agricultural Research, the world's leading institution in this field. Understanding how the brain processes information is one of the most challenging subjects in neuroscience. The papers presented in this special issue show a small corner of the huge diversity of this field, and illustrate how scientists with different backgrounds approach this vast subject. The diversity of disciplines engaged in these investigations is remarkable: biologists, mathematicians, physicists, psychologists, computer scientists, and statisticians, all have original tools and ideas by which to try to elucidate the underlying mechanisms. In this issue, emphasis is put on mathematical modeling of single neurons. A variety of problems in computational neuroscience accompanied with a rich diversity of mathematical tools and approaches are presented. We hope it will inspire and challenge the readers in their own research. We would like to thank the authors for their valuable contributions and the referees for their priceless effort of reviewing the manuscripts. Finally, we would like to thank Yang Kuang for supporting us and making this publication possible.