Dynamical estimation of neuron and network properties II: path integral Monte Carlo methods

Hodgkin–Huxley (HH) models of neuronal membrane dynamics consist of a set of nonlinear differential equations that describe the time-varying conductance of various ion channels. Using observations of voltage alone we show how to estimate the unknown parameters and unobserved state variables of an HH model in the expected circumstance that the measurements are noisy, the model has errors, and the state of the neuron is not known when observations commence. The joint probability distribution of the observed membrane voltage and the unobserved state variables and parameters of these models is a path integral through the model state space. The solution to this integral allows estimation of the parameters and thus a characterization of many biological properties of interest, including channel complement and density, that give rise to a neuron’s electrophysiological behavior. This paper describes a method for directly evaluating the path integral using a Monte Carlo numerical approach. This provides estimates not only of the expected values of model parameters but also of their posterior uncertainty. Using test data simulated from neuronal models comprising several common channels, we show that short (<50 ms) intracellular recordings from neurons stimulated with a complex time-varying current yield accurate and precise estimates of the model parameters as well as accurate predictions of the future behavior of the neuron. We also show that this method is robust to errors in model specification, supporting model development for biological preparations in which the channel expression and other biophysical properties of the neurons are not fully known.     

Estimating parameters and predicting membrane voltages with conductance-based neuron models

Recent results demonstrate techniques for fully quantitative, statistical inference of the dynamics of individual neurons under the Hodgkin–Huxley framework of voltage-gated conductances. Using a variational approximation, this approach has been successfully applied to simulated data from model neurons. Here, we use this method to analyze a population of real neurons recorded in a slice preparation of the zebra finch forebrain nucleus HVC. Our results demonstrate that using only 1,500 ms of voltage recorded while injecting a complex current waveform, we can estimate the values of 12 state variables and 72 parameters in a dynamical model, such that the model accurately predicts the responses of the neuron to novel injected currents. A less complex model produced consistently worse predictions, indicating that the additional currents contribute significantly to the dynamics of these neurons. Preliminary results indicate some differences in the channel complement of the models for different classes of HVC neurons, which accords with expectations from the biology. Whereas the model for each cell is incomplete (representing only the somatic compartment, and likely to be missing classes of channels that the real neurons possess), our approach opens the possibility to investigate in modeling the plausibility of additional classes of channels the cell might possess, thus improving the models over time. These results provide an important foundational basis for building biologically realistic network models, such as the one in HVC that contributes to the process of song production and developmental vocal learning in songbirds.     

On observation distributions for state space models of population survey data

1.?State space models are starting to replace more simple time series models in analyses of temporal dynamics of populations that are not perfectly censused. By simultaneously modelling both the dynamics and the observations, consistent estimates of population dynamical parameters may be obtained. For many data sets, the distribution of observation errors is unknown and error models typically chosen in an ad-hoc manner. 2.?To investigate the influence of the choice of observation error on inferences, we analyse the dynamics of a replicated time series of red kangaroo surveys using a state space model with linear state dynamics. Surveys were performed through aerial counts and Poisson, overdispersed Poisson, normal and log-normal distributions may all be adequate for modelling observation errors for the data. We fit each of these to the data and compare them using AIC. 3.?The state space models were fitted with maximum likelihood methods using a recent importance sampling technique that relies on the Kalman filter. The method relaxes the assumption of Gaussian observation errors required by the basic Kalman filter. Matlab code for fitting linear state space models with Poisson observations is provided. 4.?The ability of AIC to identify the correct observation model was investigated in a small simulation study. For the parameter values used in the study, without replicated observations, the correct observation distribution could sometimes be identified but model selection was prone to misclassification. On the other hand, when observations were replicated, the correct distribution could typically be identified. 5.?Our results illustrate that inferences may differ markedly depending on the observation distributions used, suggesting that choosing an adequate observation model can be critical. Model selection and simulations show that for the models and parameter values in this study, a suitable observation model can typically be identified if observations are replicated. Model selection and replication of observations, therefore, provide a potential solution when the observation distribution is unknown.     

A molecular dynamics study of gating in dioxolane-linked gramicidin A channels.

The gating transition of the RR and SS dioxolane ring-linked gramicidin A channels were studied with molecular dynamics simulations using a detailed atomic model. It was found that the probable reaction path, describing the transition of the ring from the exterior to the interior of the channel where it blocked the permeation pathway, involved several steps including the isomerization of the transpeptide plane dihedral angle of Val1. Reaction coordinates along this pathway were defined, and the transition rates between the stable conformers were calculated. It was found, in good accord with experimental observations, that the calculated blocking rate for the RR-linked channel was 280/s with a mean blocking time of 0.04 ms, whereas such blocking did not occur in the case of the SS-linked channel. An important observation is that the resulting lifetime for the blocked state of the RR-linked channel was in good accord with the experimental observations only when the calculations were performed in the presence of a potassium ion inside the channel.     

A new mathematical model of the dynamics of an age cohort population

A new generalized conception of an organism is given. Based on this conception, a new mathematical model of ontogenesis of an individual and the survival of the age cohort of population was proposed. By using real data on the dynamics of the survival of the age cohort of population, the model enables one to determine the parameters characterizing the relationship man-environment in the context of survival and calculate the dynamics (from birth to death) of the model variables of the state of the organism.     

Effective stimuli for constructing reliable neuron models

The rich dynamical nature of neurons poses major conceptual and technical challenges for unraveling their nonlinear membrane properties. Traditionally, various current waveforms have been injected at the soma to probe neuron dynamics, but the rationale for selecting specific stimuli has never been rigorously justified. The present experimental and theoretical study proposes a novel framework, inspired by learning theory, for objectively selecting the stimuli that best unravel the neuron's dynamics. The efficacy of stimuli is assessed in terms of their ability to constrain the parameter space of biophysically detailed conductance-based models that faithfully replicate the neuron's dynamics as attested by their ability to generalize well to the neuron's response to novel experimental stimuli. We used this framework to evaluate a variety of stimuli in different types of cortical neurons, ages and animals. Despite their simplicity, a set of stimuli consisting of step and ramp current pulses outperforms synaptic-like noisy stimuli in revealing the dynamics of these neurons. The general framework that we propose paves a new way for defining, evaluating and standardizing effective electrical probing of neurons and will thus lay the foundation for a much deeper understanding of the electrical nature of these highly sophisticated and non-linear devices and of the neuronal networks that they compose.     

Reconstructing Mammalian Sleep Dynamics with Data Assimilation

Data assimilation is a valuable tool in the study of any complex system, where measurements are incomplete, uncertain, or both. It enables the user to take advantage of all available information including experimental measurements and short-term model forecasts of a system. Although data assimilation has been used to study other biological systems, the study of the sleep-wake regulatory network has yet to benefit from this toolset. We present a data assimilation framework based on the unscented Kalman filter (UKF) for combining sparse measurements together with a relatively high-dimensional nonlinear computational model to estimate the state of a model of the sleep-wake regulatory system. We demonstrate with simulation studies that a few noisy variables can be used to accurately reconstruct the remaining hidden variables. We introduce a metric for ranking relative partial observability of computational models, within the UKF framework, that allows us to choose the optimal variables for measurement and also provides a methodology for optimizing framework parameters such as UKF covariance inflation. In addition, we demonstrate a parameter estimation method that allows us to track non-stationary model parameters and accommodate slow dynamics not included in the UKF filter model. Finally, we show that we can even use observed discretized sleep-state, which is not one of the model variables, to reconstruct model state and estimate unknown parameters. Sleep is implicated in many neurological disorders from epilepsy to schizophrenia, but simultaneous observation of the many brain components that regulate this behavior is difficult. We anticipate that this data assimilation framework will enable better understanding of the detailed interactions governing sleep and wake behavior and provide for better, more targeted, therapies.     

Neuronal firing sensitivity to morphologic and active membrane parameters

Both the excitability of a neuron's membrane, driven by active ion channels, and dendritic morphology contribute to neuronal firing dynamics, but the relative importance and interactions between these features remain poorly understood. Recent modeling studies have shown that different combinations of active conductances can evoke similar firing patterns, but have neglected how morphology might contribute to homeostasis. Parameterizing the morphology of a cylindrical dendrite, we introduce a novel application of mathematical sensitivity analysis that quantifies how dendritic length, diameter, and surface area influence neuronal firing, and compares these effects directly against those of active parameters. The method was applied to a model of neurons from goldfish Area II. These neurons exhibit, and likely contribute to, persistent activity in eye velocity storage, a simple model of working memory. We introduce sensitivity landscapes, defined by local sensitivity analyses of firing rate and gain to each parameter, performed globally across the parameter space. Principal directions over which sensitivity to all parameters varied most revealed intrinsic currents that most controlled model output. We found domains where different groups of parameters had the highest sensitivities, suggesting that interactions within each group shaped firing behaviors within each specific domain. Application of our method, and its characterization of which models were sensitive to general morphologic features, will lead to advances in understanding how realistic morphology participates in functional homeostasis. Significantly, we can predict which active conductances, and how many of them, will compensate for a given age- or development-related structural change, or will offset a morphologic perturbation resulting from trauma or neurodegenerative disorder, to restore normal function. Our method can be adapted to analyze any computational model. Thus, sensitivity landscapes, and the quantitative predictions they provide, can give new insight into mechanisms of homeostasis in any biological system.     

Slowness: an objective for spike-timing-dependent plasticity?

Our nervous system can efficiently recognize objects in spite of changes in contextual variables such as perspective or lighting conditions. Several lines of research have proposed that this ability for invariant recognition is learned by exploiting the fact that object identities typically vary more slowly in time than contextual variables or noise. Here, we study the question of how this "temporal stability" or "slowness" approach can be implemented within the limits of biologically realistic spike-based learning rules. We first show that slow feature analysis, an algorithm that is based on slowness, can be implemented in linear continuous model neurons by means of a modified Hebbian learning rule. This approach provides a link to the trace rule, which is another implementation of slowness learning. Then, we show analytically that for linear Poisson neurons, slowness learning can be implemented by spike-timing-dependent plasticity (STDP) with a specific learning window. By studying the learning dynamics of STDP, we show that for functional interpretations of STDP, it is not the learning window alone that is relevant but rather the convolution of the learning window with the postsynaptic potential. We then derive STDP learning windows that implement slow feature analysis and the "trace rule." The resulting learning windows are compatible with physiological data both in shape and timescale. Moreover, our analysis shows that the learning window can be split into two functionally different components that are sensitive to reversible and irreversible aspects of the input statistics, respectively. The theory indicates that irreversible input statistics are not in favor of stable weight distributions but may generate oscillatory weight dynamics. Our analysis offers a novel interpretation for the functional role of STDP in physiological neurons.     

Fast- or Slow-inactivated State Preference of Na+ Channel Inhibitors: A Simulation and Experimental Study

Sodium channels are one of the most intensively studied drug targets. Sodium channel inhibitors (e.g., local anesthetics, anticonvulsants, antiarrhythmics and analgesics) exert their effect by stabilizing an inactivated conformation of the channels. Besides the fast-inactivated conformation, sodium channels have several distinct slow-inactivated conformational states. Stabilization of a slow-inactivated state has been proposed to be advantageous for certain therapeutic applications. Special voltage protocols are used to evoke slow inactivation of sodium channels. It is assumed that efficacy of a drug in these protocols indicates slow-inactivated state preference. We tested this assumption in simulations using four prototypical drug inhibitory mechanisms (fast or slow-inactivated state preference, with either fast or slow binding kinetics) and a kinetic model for sodium channels. Unexpectedly, we found that efficacy in these protocols (e.g., a shift of the “steady-state slow inactivation curve”), was not a reliable indicator of slow-inactivated state preference. Slowly associating fast-inactivated state-preferring drugs were indistinguishable from slow-inactivated state-preferring drugs. On the other hand, fast- and slow-inactivated state-preferring drugs tended to preferentially affect onset and recovery, respectively. The robustness of these observations was verified: i) by performing a Monte Carlo study on the effects of randomly modifying model parameters, ii) by testing the same drugs in a fundamentally different model and iii) by an analysis of the effect of systematically changing drug-specific parameters. In patch clamp electrophysiology experiments we tested five sodium channel inhibitor drugs on native sodium channels of cultured hippocampal neurons. For lidocaine, phenytoin and carbamazepine our data indicate a preference for the fast-inactivated state, while the results for fluoxetine and desipramine are inconclusive. We suggest that conclusions based on voltage protocols that are used to detect slow-inactivated state preference are unreliable and should be re-evaluated.     

Functional organization and structure of the serotonergic neuronal network of terrestrial snail

The extension of knowledge how the brain works requires permanent improvement of methods of recording of neuronal activity and increase in the number of neurons recorded simultaneously to better understand the collective work of neuronal networks and assemblies. Conventional methods allow simultaneous intracellular recording up to 2-5 neurons and their membrane potentials, currents or monosynaptic connections or observation of spiking of neuronal groups with subsequent discrimination of individual spikes with loss of details of the dynamics of membrane potential. We recorded activity of a compact group of serotonergic neurons (up to 56 simultaneously) in the ganglion of a terrestrial mollusk using the method of optical recording of membrane potential that allowed to record individual action potentials in details with action potential parameters and to reveal morphology of the neurons rcorded. We demonstrated clear clustering in the group in relation with the dynamics of action potentials and phasic or tonic components in the neuronal responses to external electrophysiological and tactile stimuli. Also, we showed that identified neuron Pd2 could induce activation of a significant number of neurons in the group whereas neuron Pd4 did not induce any activation. However, its activation is delayed with regard to activation of the reacting group of neurons. Our data strongly support the concept of possible delegation of the integrative function by the network to a single neuron.     

Probability density functions useful for parametrization of heterogeneity in growth and allometry data

This paper addresses the problem of modelling heterogeneous individual characteristics in a population. A flexible unified approach for stochastic parametrization dynamics of the distribution in population data is proposed. To approximate data with multiple observations per individual, models based on Markov processes are constructed. The method can be applied to scalar or multivariate characteristics, and its application to growth and allometry data is considered. Different stochastic versions of known growth and allometry functions are developed, which enable wide applicability. Simple informative growth indices are calculated as the moments of distribution. The three-parameter Gompertz growth model for size-at-age data was reparametrized to a size-increment data model with two parameters. An erratum to this article is available at .     

Slowness: An Objective for Spike-Timing–Dependent Plasticity?

Our nervous system can efficiently recognize objects in spite of changes in contextual variables such as perspective or lighting conditions. Several lines of research have proposed that this ability for invariant recognition is learned by exploiting the fact that object identities typically vary more slowly in time than contextual variables or noise. Here, we study the question of how this “temporal stability” or “slowness” approach can be implemented within the limits of biologically realistic spike-based learning rules. We first show that slow feature analysis, an algorithm that is based on slowness, can be implemented in linear continuous model neurons by means of a modified Hebbian learning rule. This approach provides a link to the trace rule, which is another implementation of slowness learning. Then, we show analytically that for linear Poisson neurons, slowness learning can be implemented by spike-timing–dependent plasticity (STDP) with a specific learning window. By studying the learning dynamics of STDP, we show that for functional interpretations of STDP, it is not the learning window alone that is relevant but rather the convolution of the learning window with the postsynaptic potential. We then derive STDP learning windows that implement slow feature analysis and the “trace rule.” The resulting learning windows are compatible with physiological data both in shape and timescale. Moreover, our analysis shows that the learning window can be split into two functionally different components that are sensitive to reversible and irreversible aspects of the input statistics, respectively. The theory indicates that irreversible input statistics are not in favor of stable weight distributions but may generate oscillatory weight dynamics. Our analysis offers a novel interpretation for the functional role of STDP in physiological neurons.     

Response of sensorimotor cortex neurons to weak disturbances of the magnetic field in Wistar rats. Cytochemical study

The influence of weak disturbances (up to 300 microT) of natural magnetic field on the protein metabolism in neurons of sensomotor cortex (layers III and V) in Wistar rats upon learning in a complex maze was studied. It was found that sensomotor neurons were very sensitive to weak disturbances of magnetic field. The protein content increased, while the nucleus-cytoplasm ratio and osmotic state of neurons remained unchanged. The specificity of neuron's reaction manifested itself in a sharp increase of nucleus and cytoplasm dimensions. In associative neurons (layer III), both the nucleus and cytoplasm were involved in the response; in efferent neurons (layer V), only nuclear parameters changed. The variance coefficients of all parameters of protein metabolism in sensomotor neurons, independently of their functional properties, were much higher than in control, which resulted in a wide diversity of cytochemical response.     

Neuromodulation of vertebrate motor neuron membrane properties

The short-term function of motor neurons is to integrate synaptic inputs converging onto the somato-dendritic membrane and to transform the net synaptic drive into spike trains. A set of voltage-gated ion channels determines the electro-responsiveness and thereby the motor neuron's input-output function. In addition, several of the decisive ion channels are transmitter controlled, which results in a flexible control of the input-output relationship.     

Estimating the hidden learning representations

Successful adaptation relies on the ability to learn the consequence of our actions in different environments. However, understanding the neural bases of this ability still represents one of the great challenges of system neuroscience. In fact, the neuronal plasticity changes occurring during learning cannot be fully controlled experimentally and their evolution is hidden. Our approach is to provide hypotheses about the structure and dynamics of the hidden plasticity changes using behavioral learning theory. In fact, behavioral models of animal learning provide testable predictions about the hidden learning representations by formalizing their relation with the observables of the experiment (stimuli, actions and outcomes). Thus, we can understand whether and how the predicted learning processes are represented at the neural level by estimating their evolution and correlating them with neural data. Here, we present a bayesian model approach to estimate the evolution of the internal learning representations from the observations of the experiment (state estimation), and to identify the set of models' parameters (parameter estimation) and the class of behavioral model (model selection) that are most likely to have generated a given sequence of actions and outcomes. More precisely, we use Sequential Monte Carlo methods for state estimation and the maximum likelihood principle (MLP) for model selection and parameter estimation. We show that the method recovers simulated trajectories of learning sessions on a single-trial basis and provides predictions about the activity of different categories of neurons that should participate in the learning process. By correlating the estimated evolutions of the learning variables, we will be able to test the validity of different models of instrumental learning and possibly identify the neural bases of learning.     

Modeling the differentiation of A- and C-type baroreceptor firing patterns

The baroreceptor neurons serve as the primary transducers of blood pressure for the autonomic nervous system and are thus critical in enabling the body to respond effectively to changes in blood pressure. These neurons can be separated into two types (A and C) based on the myelination of their axons and their distinct firing patterns elicited in response to specific pressure stimuli. This study has developed a comprehensive model of the afferent baroreceptor discharge built on physiological knowledge of arterial wall mechanics, firing rate responses to controlled pressure stimuli, and ion channel dynamics within the baroreceptor neurons. With this model, we were able to predict firing rates observed in previously published experiments in both A- and C-type neurons. These results were obtained by adjusting model parameters determining the maximal ion-channel conductances. The observed variation in the model parameters are hypothesized to correspond to physiological differences between A- and C-type neurons. In agreement with published experimental observations, our simulations suggest that a twofold lower potassium conductance in C-type neurons is responsible for the observed sustained basal firing, where as a tenfold higher mechanosensitive conductance is responsible for the greater firing rate observed in A-type neurons. A better understanding of the difference between the two neuron types can potentially be used to gain more insight about pathophysiology and treatment of diseases related to baroreflex function, e.g. in patients with autonomic failure, a syndrome that is difficult to diagnose in terms of its pathophysiology.     

Calibration of a complex hydro-ecological model through Approximate Bayesian Computation and Random Forest combined with sensitivity analysis

An automated calibration method is proposed and applied to the complex hydro-ecological model Delft3D-BLOOM which is calibrated from monitoring data of the lake Champs-sur-Marne, a small shallow urban lake in the Paris region (France). This method (ABC-RF-SA) combines Approximate Bayesian Computation (ABC) with the machine learning algorithm Random Forest (RF) and a Sensitivity Analysis (SA) of the model parameters. Three target variables are used (total chlorophyll, cyanobacteria and dissolved oxygen concentration) to calibrate 133 parameters. ABC-RF-SA is first applied on a set of simulated observations to validate the methodology. It is then applied on a real set of high-frequency observations recorded during about two weeks on the lake Champs-sur-Marne. The methodology is also compared to standard ABC and ABC-RF formulations. Only ABC-RF-SA allowed the model to reproduce the observed biogeochemical dynamics. The coupling of ABC with RF and SA thus appears crucial for its application to complex hydro-ecological models.