A kinetic theory approach to capturing interneuronal correlation: the feed-forward case

We present an approach for using kinetic theory to capture first and second order statistics of neuronal activity. We coarse grain neuronal networks into populations of neurons and calculate the population average firing rate and output cross-correlation in response to time varying correlated input. We derive coupling equations for the populations based on first and second order statistics of the network connectivity. This coupling scheme is based on the hypothesis that second order statistics of the network connectivity are sufficient to determine second order statistics of neuronal activity. We implement a kinetic theory representation of a simple feed-forward network and demonstrate that the kinetic theory model captures key aspects of the emergence and propagation of correlations in the network, as long as the correlations do not become too strong. By analyzing the correlated activity of feed-forward networks with a variety of connectivity patterns, we provide evidence supporting our hypothesis of the sufficiency of second order connectivity statistics. Action Editor: Carson C. Chow     

The Effect of Correlated Neuronal Firing and Neuronal Heterogeneity on Population Coding Accuracy in Guinea Pig Inferior Colliculus

It has been suggested that the considerable noise in single-cell responses to a stimulus can be overcome by pooling information from a large population. Theoretical studies indicated that correlations in trial-to-trial fluctuations in the responses of different neurons may limit the improvement due to pooling. Subsequent theoretical studies have suggested that inherent neuronal diversity, i.e., the heterogeneity of tuning curves and other response properties of neurons preferentially tuned to the same stimulus, can provide a means to overcome this limit. Here we study the effect of spike-count correlations and the inherent neuronal heterogeneity on the ability to extract information from large neural populations. We use electrophysiological data from the guinea pig Inferior-Colliculus to capture inherent neuronal heterogeneity and single cell statistics, and introduce response correlations artificially. To this end, we generate pseudo-population responses, based on single-cell recording of neurons responding to auditory stimuli with varying binaural correlations. Typically, when pseudo-populations are generated from single cell data, the responses within the population are statistically independent. As a result, the information content of the population will increase indefinitely with its size. In contrast, here we apply a simple algorithm that enables us to generate pseudo-population responses with variable spike-count correlations. This enables us to study the effect of neuronal correlations on the accuracy of conventional rate codes. We show that in a homogenous population, in the presence of even low-level correlations, information content is bounded. In contrast, utilizing a simple linear readout, that takes into account the natural heterogeneity, even of neurons preferentially tuned to the same stimulus, within the neural population, one can overcome the correlated noise and obtain a readout whose accuracy grows linearly with the size of the population.     

Natural Selection and Age-Structured Populations

This paper studies the properties of a new class of demographic parameters for age-structured populations and analyzes the effect of natural selection on these parameters. Two new demographic variables are introduced: the entropy of a population and the reproductive potential. The entropy of a population measures the variability of the contribution of the different age classes to the stationary population. The reproductive potential measures the mean of the contribution of the different age classes to the Malthusian parameter. The Malthusian parameter is precisely the difference between the entropy and the reproductive potential. The effect of these demographic variables on changes in gene frequency is discussed. The concept of entropy of a genotype is introduced and it is shown that in a random mating population in Hardy-Weinberg equilibrium and under slow selection, the rate of change of entropy is equal to the genetic variance in entropy minus the covariance in entropy and reproductive potential. This result is an information theoretic analog of Fisher''s fundamental theorem of natural selection.     

CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains

Recent developments in electrophysiological and optical recording techniques enable the simultaneous observation of large numbers of neurons. A meaningful interpretation of the resulting multivariate data, however, presents a serious challenge. In particular, the estimation of higher-order correlations that characterize the cooperative dynamics of groups of neurons is impeded by the combinatorial explosion of the parameter space. The resulting requirements with respect to sample size and recording time has rendered the detection of coordinated neuronal groups exceedingly difficult. Here we describe a novel approach to infer higher-order correlations in massively parallel spike trains that is less susceptible to these problems. Based on the superimposed activity of all recorded neurons, the cumulant-based inference of higher-order correlations (CuBIC) presented here exploits the fact that the absence of higher-order correlations imposes also strong constraints on correlations of lower order. Thus, estimates of only few lower-order cumulants suffice to infer higher-order correlations in the population. As a consequence, CuBIC is much better compatible with the constraints of in vivo recordings than previous approaches, which is shown by a systematic analysis of its parameter dependence.     

Low dimensional model of bursting neurons

A computationally efficient, biophysically-based model of neuronal behavior is presented; it incorporates ion channel dynamics in its two fast ion channels while preserving simplicity by representing only one slow ion current. The model equations are shown to provide a wide array of physiological dynamics in terms of spiking patterns, bursting, subthreshold oscillations, and chaotic firing. Despite its simplicity, the model is capable of simulating an extensive range of spiking patterns. Several common neuronal behaviors observed in vivo are demonstrated by varying model parameters. These behaviors are classified into dynamical classes using phase diagrams whose boundaries in parameter space prove to be accurately delineated by linear stability analysis. This simple model is suitable for use in large scale simulations involving neural field theory or neuronal networks.     

Coding of time-dependent stimuli in homogeneous and heterogeneous neural populations

We compare the information transmission of a time-dependent signal by two types of uncoupled neuron populations that differ in their sources of variability: i) a homogeneous population whose units receive independent noise and ii) a deterministic heterogeneous population, where each unit exhibits a different baseline firing rate (’disorder’). Our criterion for making both sources of variability quantitatively comparable is that the interspike-interval distributions are identical for both systems. Numerical simulations using leaky integrate-and-fire neurons unveil that a non-zero amount of both noise or disorder maximizes the encoding efficiency of the homogeneous and heterogeneous system, respectively, as a particular case of suprathreshold stochastic resonance. Our findings thus illustrate that heterogeneity can render similarly profitable effects for neuronal populations as dynamic noise. The optimal noise/disorder depends on the system size and the properties of the stimulus such as its intensity or cutoff frequency. We find that weak stimuli are better encoded by a noiseless heterogeneous population, whereas for strong stimuli a homogeneous population outperforms an equivalent heterogeneous system up to a moderate noise level. Furthermore, we derive analytical expressions of the coherence function for the cases of very strong noise and of vanishing intrinsic noise or heterogeneity, which predict the existence of an optimal noise intensity. Our results show that, depending on the type of signal, noise as well as heterogeneity can enhance the encoding performance of neuronal populations.     

Efficient and robust coding in heterogeneous recurrent networks

Cortical networks show a large heterogeneity of neuronal properties. However, traditional coding models have focused on homogeneous populations of excitatory and inhibitory neurons. Here, we analytically derive a class of recurrent networks of spiking neurons that close to optimally track a continuously varying input online, based on two assumptions: 1) every spike is decoded linearly and 2) the network aims to reduce the mean-squared error between the input and the estimate. From this we derive a class of predictive coding networks, that unifies encoding and decoding and in which we can investigate the difference between homogeneous networks and heterogeneous networks, in which each neurons represents different features and has different spike-generating properties. We find that in this framework, ‘type 1’ and ‘type 2’ neurons arise naturally and networks consisting of a heterogeneous population of different neuron types are both more efficient and more robust against correlated noise. We make two experimental predictions: 1) we predict that integrators show strong correlations with other integrators and resonators are correlated with resonators, whereas the correlations are much weaker between neurons with different coding properties and 2) that ‘type 2’ neurons are more coherent with the overall network activity than ‘type 1’ neurons.     

On the Number of Neurons and Time Scale of Integration Underlying the Formation of Percepts in the Brain

All of our perceptual experiences arise from the activity of neural populations. Here we study the formation of such percepts under the assumption that they emerge from a linear readout, i.e., a weighted sum of the neurons’ firing rates. We show that this assumption constrains the trial-to-trial covariance structure of neural activities and animal behavior. The predicted covariance structure depends on the readout parameters, and in particular on the temporal integration window w and typical number of neurons K used in the formation of the percept. Using these predictions, we show how to infer the readout parameters from joint measurements of a subject’s behavior and neural activities. We consider three such scenarios: (1) recordings from the complete neural population, (2) recordings of neuronal sub-ensembles whose size exceeds K, and (3) recordings of neuronal sub-ensembles that are smaller than K. Using theoretical arguments and artificially generated data, we show that the first two scenarios allow us to recover the typical spatial and temporal scales of the readout. In the third scenario, we show that the readout parameters can only be recovered by making additional assumptions about the structure of the full population activity. Our work provides the first thorough interpretation of (feed-forward) percept formation from a population of sensory neurons. We discuss applications to experimental recordings in classic sensory decision-making tasks, which will hopefully provide new insights into the nature of perceptual integration.     

Bayesian Population Physiologically-Based Pharmacokinetic (PBPK) Approach for a Physiologically Realistic Characterization of Interindividual Variability in Clinically Relevant Populations

Interindividual variability in anatomical and physiological properties results in significant differences in drug pharmacokinetics. The consideration of such pharmacokinetic variability supports optimal drug efficacy and safety for each single individual, e.g. by identification of individual-specific dosings. One clear objective in clinical drug development is therefore a thorough characterization of the physiological sources of interindividual variability. In this work, we present a Bayesian population physiologically-based pharmacokinetic (PBPK) approach for the mechanistically and physiologically realistic identification of interindividual variability. The consideration of a generic and highly detailed mechanistic PBPK model structure enables the integration of large amounts of prior physiological knowledge, which is then updated with new experimental data in a Bayesian framework. A covariate model integrates known relationships of physiological parameters to age, gender and body height. We further provide a framework for estimation of the a posteriori parameter dependency structure at the population level. The approach is demonstrated considering a cohort of healthy individuals and theophylline as an application example. The variability and co-variability of physiological parameters are specified within the population; respectively. Significant correlations are identified between population parameters and are applied for individual- and population-specific visual predictive checks of the pharmacokinetic behavior, which leads to improved results compared to present population approaches. In the future, the integration of a generic PBPK model into an hierarchical approach allows for extrapolations to other populations or drugs, while the Bayesian paradigm allows for an iterative application of the approach and thereby a continuous updating of physiological knowledge with new data. This will facilitate decision making e.g. from preclinical to clinical development or extrapolation of PK behavior from healthy to clinically significant populations.     

Emergence of Bursting Activity in Connected Neuronal Sub-Populations

Uniform and modular primary hippocampal cultures from embryonic rats were grown on commercially available micro-electrode arrays to investigate network activity with respect to development and integration of different neuronal populations. Modular networks consisting of two confined active and inter-connected sub-populations of neurons were realized by means of bi-compartmental polydimethylsiloxane structures. Spontaneous activity in both uniform and modular cultures was periodically monitored, from three up to eight weeks after plating. Compared to uniform cultures and despite lower cellular density, modular networks interestingly showed higher firing rates at earlier developmental stages, and network-wide firing and bursting statistics were less variable over time. Although globally less correlated than uniform cultures, modular networks exhibited also higher intra-cluster than inter-cluster correlations, thus demonstrating that segregation and integration of activity coexisted in this simple yet powerful in vitro model. Finally, the peculiar synchronized bursting activity shown by confined modular networks preferentially propagated within one of the two compartments (‘dominant’), even in cases of perfect balance of firing rate between the two sub-populations. This dominance was generally maintained during the entire monitored developmental frame, thus suggesting that the implementation of this hierarchy arose from early network development.     

Topographic Electrophysiological Signatures of fMRI Resting State Networks

Background

fMRI Resting State Networks (RSNs) have gained importance in the present fMRI literature. Although their functional role is unquestioned and their physiological origin is nowadays widely accepted, little is known about their relationship to neuronal activity. The combined recording of EEG and fMRI allows the temporal correlation between fluctuations of the RSNs and the dynamics of EEG spectral amplitudes. So far, only relationships between several EEG frequency bands and some RSNs could be demonstrated, but no study accounted for the spatial distribution of frequency domain EEG.

Methodology/Principal Findings

In the present study we report on the topographic association of EEG spectral fluctuations and RSN dynamics using EEG covariance mapping. All RSNs displayed significant covariance maps across a broad EEG frequency range. Cluster analysis of the found covariance maps revealed the common standard EEG frequency bands. We found significant differences between covariance maps of the different RSNs and these differences depended on the frequency band.

Conclusions/Significance

Our data supports the physiological and neuronal origin of the RSNs and substantiates the assumption that the standard EEG frequency bands and their topographies can be seen as electrophysiological signatures of underlying distributed neuronal networks.     

Multivariate structural statistics in natural history

Some multivariate statistics usually used for structural purposes are inappropriate for this. The total variance and derived parameters, some new, measure multivariate variation. The two major properties of correlation must be generalized differently: the total correlation for joint dependence and the tightness. The term coregression refers to the trend of two mutually dependent variables. A generalization of information theory permits estimation of the redundance produced by correlations among many continuous variables. A new parameter, the modality, expresses the non-uniformity of a set of metric points. A suitable distance among populations can be determined in the general bivariate case and, approximately, in most realistic multivariate cases.     

Calcium imaging of epileptiform events with single-cell resolution

Epileptic discharges propagate through apparently normal circuits, although it is still unclear how this recruitment takes place. To understand the role of different classes of neurons in neocortical epilepsy, we have developed a novel imaging assay that detects which neurons participate in epileptiform discharges. Using calcium imaging of neuronal populations during bicuculline-induced spontaneous epileptiform events in slices from juvenile mouse somatosensory cortex, we find that fast calcium transients correlate with epileptiform field potentials and intracellular depolarizing shifts and can be used as an optical signature that a given neuron has participated in an epileptiform event. Our results demonstrate a novel method to characterize epileptiform events with single-cell resolution. In addition, our data are consistent with an important role for layer 5 in generating neocortical seizures and indicate that subgroups of neurons are particularly prone to epileptiform recruitment.     

Synaptic information transfer in computer models of neocortical columns

Understanding the direction and quantity of information flowing in neuronal networks is a fundamental problem in neuroscience. Brains and neuronal networks must at the same time store information about the world and react to information in the world. We sought to measure how the activity of the network alters information flow from inputs to output patterns. Using neocortical column neuronal network simulations, we demonstrated that networks with greater internal connectivity reduced input/output correlations from excitatory synapses and decreased negative correlations from inhibitory synapses, measured by Kendall’s τ correlation. Both of these changes were associated with reduction in information flow, measured by normalized transfer entropy (nTE). Information handling by the network reflected the degree of internal connectivity. With no internal connectivity, the feedforward network transformed inputs through nonlinear summation and thresholding. With greater connectivity strength, the recurrent network translated activity and information due to contribution of activity from intrinsic network dynamics. This dynamic contribution amounts to added information drawn from that stored in the network. At still higher internal synaptic strength, the network corrupted the external information, producing a state where little external information came through. The association of increased information retrieved from the network with increased gamma power supports the notion of gamma oscillations playing a role in information processing.     

Calcium imaging of epileptiform events with single‐cell resolution

Epileptic discharges propagate through apparently normal circuits, although it is still unclear how this recruitment takes place. To understand the role of different classes of neurons in neocortical epilepsy, we have developed a novel imaging assay that detects which neurons participate in epileptiform discharges. Using calcium imaging of neuronal populations during bicuculline‐induced spontaneous epileptiform events in slices from juvenile mouse somatosensory cortex, we find that fast calcium transients correlate with epileptiform field potentials and intracellular depolarizing shifts and can be used as an optical signature that a given neuron has participated in an epileptiform event. Our results demonstrate a novel method to characterize epileptiform events with single‐cell resolution. In addition, our data are consistent with an important role for layer 5 in generating neocortical seizures and indicate that subgroups of neurons are particularly prone to epileptiform recruitment. © 2001 John Wiley & Sons, Inc. J Neurobiol 48: 215–227, 2001     

HOMOGENEITY OF THE GENETIC VARIANCE-COVARIANCE MATRIX FOR ANTIPREDATOR TRAITS IN TWO NATURAL POPULATIONS OF THE GARTER SNAKE THAMNOPHIS ORDINOIDES

Quantitative genetic models of evolution rely on the genetic variance-covariance matrix to predict the phenotypic response to selection. Both prospective and retrospective studies of phenotypic evolution across generations rely on assumptions about the constancy of patterns of genetic covariance through time. In the absence of robust theoretical predictions about the stability of genetic covariances, this assumption must be tested with empirical comparisons of genetic parameters among populations and species. Genetic variance-covariance matrices were estimated for a suite of antipredator traits in two populations of the northwestern garter snake, Thamnophis ordinoides. The characters studied include color pattern and antipredator behaviors that interact to facilitate escape from predators. Significant heritabilities for all traits were detected in both populations. Genetic correlations and covariances were found among behaviors in both populations and between color pattern and behavior in one of the populations. Phenotypic means differed among populations, but pairwise comparisons revealed no heterogeneity of genetic parameters between the populations. The structure of the genetic variance-covariance matrix has apparently not changed significantly during the divergence of these two populations.     

Narrow endemic species Bellevalia webbiana shows significant intraspecific variation in tertiary CSR strategy

Insufficient attention has been devoted to inter-population and inter-individual variability of CSR strategy. Hence, we selected Bellevalia webbiana as a case study for answering the following questions: (1) is there a significant intraspecific variability in leaf traits determining a significant intraspecific variability in CSR parameters, even in a narrow endemic species with a relatively homogeneous habitat? (2) If yes, does this provide a significant intraspecific CSR strategy variation? For five populations, we calculated C, S, and R parameters using leaf area, leaf fresh weight, and leaf dry weight for 10 individuals each. We found that Bellevalia webbiana is a “CS” species considering single individuals, populations or the “whole species”. However, our data suggest a significant loss of information if the species is plotted as a single average point, and attest for population-based plasticity, highlighting the applicability of CSR theory at micro-evolutionary scale. When heterogeneity in coefficients of variation (CVs) is higher for the “whole species”, “deviant” populations should be retained as additional points. Hence, we advise to perform CSR analyses on more populations and verify the applicability of a single ternary set to represent the “whole species” using PERMANOVA, complemented by simple CV comparison for each parameter.     

A mathematical and computational approach for integrating the major sources of cell population heterogeneity

Several approaches have been used in the past to model heterogeneity in bacterial cell populations, with each approach focusing on different source(s) of heterogeneity. However, a holistic approach that integrates all the major sources into a comprehensive framework applicable to cell populations is still lacking.In this work we present the mathematical formulation of a cell population master equation (CPME) that describes cell population dynamics and takes into account the major sources of heterogeneity, namely stochasticity in reaction, DNA-duplication, and division, as well as the random partitioning of species contents into the two daughter cells. The formulation also takes into account cell growth and respects the discrete nature of the molecular contents and cell numbers. We further develop a Monte Carlo algorithm for the simulation of the stochastic processes considered here. To benchmark our new framework, we first use it to quantify the effect of each source of heterogeneity on the intrinsic and the extrinsic phenotypic variability for the well-known two-promoter system used experimentally by Elowitz et al. (2002). We finally apply our framework to a more complicated system and demonstrate how the interplay between noisy gene expression and growth inhibition due to protein accumulation at the single cell level can result in complex behavior at the cell population level.The generality of our framework makes it suitable for studying a vast array of artificial and natural genetic networks. Using our Monte Carlo algorithm, cell population distributions can be predicted for the genetic architecture of interest, thereby quantifying the effect of stochasticity in intracellular reactions or the variability in the rate of physiological processes such as growth and division. Such in silico experiments can give insight into the behavior of cell populations and reveal the major sources contributing to cell population heterogeneity.