Avalanches in a Stochastic Model of Spiking Neurons

Neuronal avalanches are a form of spontaneous activity widely observed in cortical slices and other types of nervous tissue, both in vivo and in vitro. They are characterized by irregular, isolated population bursts when many neurons fire together, where the number of spikes per burst obeys a power law distribution. We simulate, using the Gillespie algorithm, a model of neuronal avalanches based on stochastic single neurons. The network consists of excitatory and inhibitory neurons, first with all-to-all connectivity and later with random sparse connectivity. Analyzing our model using the system size expansion, we show that the model obeys the standard Wilson-Cowan equations for large network sizes ( neurons). When excitation and inhibition are closely balanced, networks of thousands of neurons exhibit irregular synchronous activity, including the characteristic power law distribution of avalanche size. We show that these avalanches are due to the balanced network having weakly stable functionally feedforward dynamics, which amplifies some small fluctuations into the large population bursts. Balanced networks are thought to underlie a variety of observed network behaviours and have useful computational properties, such as responding quickly to changes in input. Thus, the appearance of avalanches in such functionally feedforward networks indicates that avalanches may be a simple consequence of a widely present network structure, when neuron dynamics are noisy. An important implication is that a network need not be “critical” for the production of avalanches, so experimentally observed power laws in burst size may be a signature of noisy functionally feedforward structure rather than of, for example, self-organized criticality.     

Genetic network models and statistical properties of gene expression data in knock-out experiments

It is shown here how gene knock-out experiments can be simulated in Random Boolean Networks (RBN), which are well-known simplified models of genetic networks. The results of the simulations are presented and compared with those of actual experiments in S. cerevisiae. RBN with two incoming links per node have been considered, and the Boolean functions have been chosen at random among the set of so-called canalizing functions. Genes are knocked-out (i.e. silenced) one at a time, and the variations in the expression levels of the other genes, with respect to the unperturbed case, are considered. Two important variables are defined: (i) avalanches, which measure the size of the perturbation generated by knocking out a single gene, and (ii) susceptibilities, which measure how often the expression of a given gene is modified in these experiments. A remarkable observation is that the distributions of avalanches and susceptibilities are very robust, i.e. they are very similar in different random networks; this should be contrasted with the distribution of other variables that show a high variance in RBN. Moreover, the distribution of avalanches and susceptibilities of the RBN models are close to those observed in actual experiments performed with S. cerevisiae, where the changes in gene expression levels have been recorded with DNA microarrays. These findings suggest that these distributions might be "generic" properties, common to a wide range of genetic models and real genetic networks. The importance of such generic properties is discussed.     

Fixation patterns in simple choice reflect optimal information sampling

Simple choices (e.g., eating an apple vs. an orange) are made by integrating noisy evidence that is sampled over time and influenced by visual attention; as a result, fluctuations in visual attention can affect choices. But what determines what is fixated and when? To address this question, we model the decision process for simple choice as an information sampling problem, and approximate the optimal sampling policy. We find that it is optimal to sample from options whose value estimates are both high and uncertain. Furthermore, the optimal policy provides a reasonable account of fixations and choices in binary and trinary simple choice, as well as the differences between the two cases. Overall, the results show that the fixation process during simple choice is influenced dynamically by the value estimates computed during the decision process, in a manner consistent with optimal information sampling.     

Network theory and metapopulation persistence: incorporating node self‐connections

Network analysis is gaining increasing importance in conservation planning. However, which network metrics are the best predictors of metapopulation persistence is still unresolved. Here, we identify a critical limitation of graph theory‐derived network metrics that have been proposed for this purpose: their omission of node self‐connections. We resolve this by presenting modifications of existing network metrics, and developing entirely new metrics, that account for node self‐connections. Then, we illustrate the performance of these new and modified metrics with an age‐structured metapopulation model for a real‐world marine reserve network case study, and we evaluate the robustness of our findings by systematically varying particular features of that network. Our new and modified metrics predict metapopulation persistence much better than existing metrics do, even when self‐connections are weak. Existing metrics become good predictors of persistence only when self‐connections are entirely absent, an unrealistic scenario in the overwhelming majority of metapopulation applications. Our study provides a set of novel tools that can substantially enhance the extent to which network metrics can be employed to understand, and manage for, metapopulation persistence.     

Performance Evaluation of Shared Mesh Protection in WDM Networks

The efficient use of network capacity in shared restoration schemes strongly depends upon the path selection procedure. In this paper we propose and evaluate path selection algorithms for sharable and restorable connections in optical networks. Namely, two distributed path selection algorithms are proposed. The first approach maintains global information on network resource usage to determine link sharability and compute optimal shared paths. The second approach only relies upon local information maintained at each node. Subsequently, we present an analytical model to evaluate the performance of these path selection algorithms and show its accuracy through numerical examples. Results indicate that path selection algorithms that maximally exploit the use of reserved sharable channels do not necessarily result in fast restoration; rather these two optimality criteria can conflict most of the time. Additionally, algorithms that maximally exploit the sharability condition typically result in lower scalability and higher complexity.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. II. Input selectivity—symmetry breaking

Spike-timing-dependent plasticity (STDP) is believed to structure neuronal networks by slowly changing the strengths (or weights) of the synaptic connections between neurons depending upon their spiking activity, which in turn modifies the neuronal firing dynamics. In this paper, we investigate the change in synaptic weights induced by STDP in a recurrently connected network in which the input weights are plastic but the recurrent weights are fixed. The inputs are divided into two pools with identical constant firing rates and equal within-pool spike-time correlations, but with no between-pool correlations. Our analysis uses the Poisson neuron model in order to predict the evolution of the input synaptic weights and focuses on the asymptotic weight distribution that emerges due to STDP. The learning dynamics induces a symmetry breaking for the individual neurons, namely for sufficiently strong within-pool spike-time correlation each neuron specializes to one of the input pools. We show that the presence of fixed excitatory recurrent connections between neurons induces a group symmetry-breaking effect, in which neurons tend to specialize to the same input pool. Consequently STDP generates a functional structure on the input connections of the network.     

主动联想记忆神经网络模型

在信息编码能提高联想记忆的存贮能力和脑内存在主动活动机制的启发下,提出一个主动联想记忆模型。模型包括两个神经网络,其一为输入和输出网络,另一个为在学习时期能自主产生兴奋模式的主动网络。两个网络的神经元之间有突触联系。由于自主产生的兴奋模式与输入无关,并可能接近于相互正交,因此,本模型有较高的存贮能力。初步分析和计算机仿真证明：本模型确有比通常联想记忆模型高的存贮能力,特别是在输入模式间有高度相关情况下、最后,对提出的模型与双向自联想记忆和光学全息存贮机制的关系作了讨论。     

Neural network model of the primary visual cortex: From functional architecture to lateral connectivity and back

The role of intrinsic cortical dynamics is a debatable issue. A recent optical imaging study (Kenet et al., 2003) found that activity patterns similar to orientation maps (OMs), emerge in the primary visual cortex (V1) even in the absence of sensory input, suggesting an intrinsic mechanism of OM activation. To better understand these results and shed light on the intrinsic V1 processing, we suggest a neural network model in which OMs are encoded by the intrinsic lateral connections. The proposed connectivity pattern depends on the preferred orientation and, unlike previous models, on the degree of orientation selectivity of the interconnected neurons. We prove that the network has a ring attractor composed of an approximated version of the OMs. Consequently, OMs emerge spontaneously when the network is presented with an unstructured noisy input. Simulations show that the model can be applied to experimental data and generate realistic OMs. We study a variation of the model with spatially restricted connections, and show that it gives rise to states composed of several OMs. We hypothesize that these states can represent local properties of the visual scene. Action Editor: Jonathan D. Victor     

On variability in the density of corticocortical and thalamocortical connections

Variability is an important but neglected aspect of connectional neuroanatomy. The quantitative density of the 'same' corticocortical or thalamocortical connection may vary by over two orders of magnitude between different injections of the same tracer. At present, however, the frequency distribution of connection densities is unknown. Therefore, it is unclear what kind of sampling strategies or statistical methods are appropriate for quantitative studies of connectivity. Nor is it clear if the measured variability represents differences between subjects, or if it is simply a consequence of intra-individual differences resulting from experimental technique and the exact placement of tracers relative to local spatial and laminar variation in connectivity. We used quantitative measurements of the density of a large number of corticocortical and thalamocortical connections from our own laboratories and from the literature. Variability in the density of given corticocortical and thalamocortical connections is high, with the standard deviation of density proportional to the mean. The frequency distribution is close to exponential. Therefore, analysis methods relying on the normal distribution are not appropriate. We provide an appendix that gives simple statistical guidance for samples drawn from exponentially distributed data. For a given corticocortical or thalamocortical connection density, between-individual standard deviation is 0.85 to 1.25 times the within-individual standard deviation. Therefore, much of the variability reported in conventional neuroanatomical studies (with one tracer deposited per animal) is due to within-individual factors. We also find that strong, but not weak, corticocortical connections are substantially more variable than thalamocortical connections. We propose that the near exponential distribution of connection densities is a simple consequence of 'patchy' connectivity. We anticipate that connection data will be well described by the negative binomial, a class of distribution that applies to events occurring in clumped or patchy substrates. Local patchiness may be a feature of all corticocortical connections and could explain why strong corticocortical connections are more variable than strong thalamocortical connections. This idea is supported by the columnar patterns of many corticocortical but few thalamocortical connections in the literature.     

Optimal Information Representation and Criticality in an Adaptive Sensory Recurrent Neuronal Network

Recurrent connections play an important role in cortical function, yet their exact contribution to the network computation remains unknown. The principles guiding the long-term evolution of these connections are poorly understood as well. Therefore, gaining insight into their computational role and into the mechanism shaping their pattern would be of great importance. To that end, we studied the learning dynamics and emergent recurrent connectivity in a sensory network model based on a first-principle information theoretic approach. As a test case, we applied this framework to a model of a hypercolumn in the visual cortex and found that the evolved connections between orientation columns have a "Mexican hat" profile, consistent with empirical data and previous modeling work. Furthermore, we found that optimal information representation is achieved when the network operates near a critical point in its dynamics. Neuronal networks working near such a phase transition are most sensitive to their inputs and are thus optimal in terms of information representation. Nevertheless, a mild change in the pattern of interactions may cause such networks to undergo a transition into a different regime of behavior in which the network activity is dominated by its internal recurrent dynamics and does not reflect the objective input. We discuss several mechanisms by which the pattern of interactions can be driven into this supercritical regime and relate them to various neurological and neuropsychiatric phenomena.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. I. Input selectivity–strengthening correlated input pathways

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity. In this paper, we extend previous studies of input selectivity induced by (STDP) for single neurons to the biologically interesting case of a neuronal network with fixed recurrent connections and plastic connections from external pools of input neurons. We use a theoretical framework based on the Poisson neuron model to analytically describe the network dynamics (firing rates and spike-time correlations) and thus the evolution of the synaptic weights. This framework incorporates the time course of the post-synaptic potentials and synaptic delays. Our analysis focuses on the asymptotic states of a network stimulated by two homogeneous pools of “steady” inputs, namely Poisson spike trains which have fixed firing rates and spike-time correlations. The (STDP) model extends rate-based learning in that it can implement, at the same time, both a stabilization of the individual neuron firing rates and a slower weight specialization depending on the input spike-time correlations. When one input pathway has stronger within-pool correlations, the resulting synaptic dynamics induced by (STDP) are shown to be similar to those arising in the case of a purely feed-forward network: the weights from the more correlated inputs are potentiated at the expense of the remaining input connections.     

Statistical analyses support power law distributions found in neuronal avalanches

The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.     

Optimal Bayesian foraging policies and prey population dynamics-some comments on Rodriguez-Girones and Vasquez

In this paper we show the density-dependent harvest rates of optimal Bayesian foragers exploiting prey occurring with clumped spatial distribution. Rodríguez-Gironés and Vásquez (1997) recently treated the issue, but they used a patch-leaving rule (current value assessment rule) that is not optimal for the case described here. An optimal Bayesian forager exploiting prey whose distribution follows the negative binomial distribution should leave a patch when the potential (and not instantaneous) gain rate in that patch equals the best long-term gain rate in the environment (potential value assessment rule). It follows that the instantaneous gain rate at which the patches are abandoned is an increasing function of the time spent searching in the patch. It also follows that the proportion of prey harvested in a patch is an increasing sigmoidal function of the number of prey initially present. In this paper we vary several parameters of the model to evaluate the effects on the forager's intake rate, the proportion of prey harvested per patch, and the prey's average mortality rate in the environment. In each case, we study an intake rate maximizing forager's optimal response to the parameter changes. For the potential value assessment rule we find that at a higher average prey density in the environment, a lower proportion of the prey is taken in a patch with a given initial prey density. The proportion of prey taken in a patch of a given prey density also decreases when the variance of the prey density distribution is increased and if the travel time between patches is reduced. We also evaluate the effect of using predation minimization, rather than rate maximization, as the currency. Then a higher proportion of the prey is taken for each given initial prey density. This is related to the assumption that traveling between patches is the most risky activity. Compared to the optimal potential value assessment rule, the current value assessment rule performs worse, in terms of long-term intake rate achieved. The difference in performance is amplified when prey density is high or highly aggregated. These results pertain to the foraging patch spatial scale and may have consequences for the spatial distribution of prey in the environment.     

How social network structure affects decision-making in Drosophila melanogaster

Animals use a number of different mechanisms to acquire crucial information. During social encounters, animals can pass information from one to another but, ideally, they would only use information that benefits survival and reproduction. Therefore, individuals need to be able to determine the value of the information they receive. One cue can come from the behaviour of other individuals that are already using the information. Using a previous extended dataset, we studied how individual decision-making is influenced by the behaviour of conspecifics in Drosophila melanogaster. We analysed how uninformed flies acquire and later use information about oviposition site choice they learn from informed flies. Our results suggest that uninformed flies adjust their future choices based on how coordinated the behaviours of the informed individuals they encounter are. Following social interaction, uninformed flies tended either to collectively follow the choice of the informed flies or to avoid it. Using social network analysis, we show that this selective information use seems to be based on the level of homogeneity of the social network. In particular, we found that the variance of individual centrality parameters among informed flies was lower in the case of a ‘follow’ outcome compared with the case of an ‘avoid’ outcome.     

A Simple Method for Estimating Informative Node Age Priors for the Fossil Calibration of Molecular Divergence Time Analyses

Molecular divergence time analyses often rely on the age of fossil lineages to calibrate node age estimates. Most divergence time analyses are now performed in a Bayesian framework, where fossil calibrations are incorporated as parametric prior probabilities on node ages. It is widely accepted that an ideal parameterization of such node age prior probabilities should be based on a comprehensive analysis of the fossil record of the clade of interest, but there is currently no generally applicable approach for calculating such informative priors. We provide here a simple and easily implemented method that employs fossil data to estimate the likely amount of missing history prior to the oldest fossil occurrence of a clade, which can be used to fit an informative parametric prior probability distribution on a node age. Specifically, our method uses the extant diversity and the stratigraphic distribution of fossil lineages confidently assigned to a clade to fit a branching model of lineage diversification. Conditioning this on a simple model of fossil preservation, we estimate the likely amount of missing history prior to the oldest fossil occurrence of a clade. The likelihood surface of missing history can then be translated into a parametric prior probability distribution on the age of the clade of interest. We show that the method performs well with simulated fossil distribution data, but that the likelihood surface of missing history can at times be too complex for the distribution-fitting algorithm employed by our software tool. An empirical example of the application of our method is performed to estimate echinoid node ages. A simulation-based sensitivity analysis using the echinoid data set shows that node age prior distributions estimated under poor preservation rates are significantly less informative than those estimated under high preservation rates.     

模拟昆虫视觉-行为抉择的强化学习模型

视觉信息用于行为抉择的过程是一个极其复杂的脑信息处理过程,昆虫或动物对外界环境的学习是以价值来控制的,并可影响其行为抉择,研究这一过程对揭示人类自身脑运行机制有重要意义.文章在郭爱克研究小组果蝇实验提供的生物依据基础上,提出了一种模拟果蝇视觉-行为抉择的神经网络模型.该模型引入了价值和基于价值的强化学习算法,应用于输入视觉图像的强化学习,以此建立果蝇脑内多巴胺和蘑菇体对于抉择判断的价值体系.模拟的结果表明,该模型可以模拟果蝇视觉信息的学习和行为抉择过程,其结果与生物实验相符,同时也为机器人视觉信息控制行为抉择的应用提供了基础.     

Probability Steiner trees and maximum parsimony in phylogenetic analysis

The phylogenetic tree (PT) problem has been studied by a number of researchers as an application of the Steiner tree problem, a well-known network optimisation problem. Of all the methods developed for phylogenies the maximum parsimony (MP) method is a simple and commonly used method because it relies on directly observable changes in the input nucleotide or amino acid sequences. In this paper we show that the non-uniqueness of the evolutionary pathways in the MP method leads us to consider a new model of PTs. In this so-called probability representation model, for each site a node in a PT is modelled by a probability distribution of nucleotide or amino acid states, and hence the PT at a given site is a probability Steiner tree, i.e. a Steiner tree in a high-dimensional vector space. In spite of the generality of the probability representation model, in this paper we restrict our study to constructing probability phylogenetic trees (PPT) using the parsimony criterion, as well as discussing and comparing our approach with the classical MP method. We show that for a given input set although the optimal topology as well as the total tree length of the PPT is the same as the PT constructed by the classical MP method, the inferred ancestral states and branch lengths are different and the results given by our method provide a plausible alternative to the classical ones.     

Information capacity and its approximations under metabolic cost in a simple homogeneous population of neurons

We calculate and analyze the information capacity-achieving conditions and their approximations in a simple neuronal system. The input–output properties of individual neurons are described by an empirical stimulus–response relationship and the metabolic cost of neuronal activity is taken into account. The exact (numerical) results are compared with a popular “low-noise” approximation method which employs the concepts of parameter estimation theory. We show, that the approximate method gives reliable results only in the case of significantly low response variability. By employing specialized numerical procedures we demonstrate, that optimal information transfer can be near-achieved by a number of different input distributions. It implies that the precise structure of the capacity-achieving input is of lesser importance than the value of capacity. Finally, we illustrate on an example that an innocuously looking stimulus–response relationship may lead to a problematic interpretation of the obtained Fisher information values.     

A model of naturalistic decision making in preference tests

Decisions as to whether to continue with an ongoing activity or to switch to an alternative are a constant in an animal’s natural world, and in particular underlie foraging behavior and performance in food preference tests. Stimuli experienced by the animal both impact the choice and are themselves impacted by the choice, in a dynamic back and forth. Here, we present model neural circuits, based on spiking neurons, in which the choice to switch away from ongoing behavior instantiates this back and forth, arising as a state transition in neural activity. We analyze two classes of circuit, which differ in whether state transitions result from a loss of hedonic input from the stimulus (an “entice to stay” model) or from aversive stimulus-input (a “repel to leave” model). In both classes of model, we find that the mean time spent sampling a stimulus decreases with increasing value of the alternative stimulus, a fact that we linked to the inclusion of depressing synapses in our model. The competitive interaction is much greater in “entice to stay” model networks, which has qualitative features of the marginal value theorem, and thereby provides a framework for optimal foraging behavior. We offer suggestions as to how our models could be discriminatively tested through the analysis of electrophysiological and behavioral data.     

Optimal Bayesian Foraging Policies and Prey Population Dynamics—Some Comments on Rodríguez-Gironés and Vásquez

In this paper we show the density-dependent harvest rates of optimal Bayesian foragers exploiting prey occurring with clumped spatial distribution. Rodríguez-Gironés and Vásquez (1997) recently treated the issue, but they used a patch-leaving rule (current value assessment rule) that is not optimal for the case described here. An optimal Bayesian forager exploiting prey whose distribution follows the negative binomial distribution should leave a patch when the potential (and not instantaneous) gain rate in that patch equals the best long-term gain rate in the environment (potential value assessment rule). It follows that the instantaneous gain rate at which the patches are abandoned is an increasing function of the time spent searching in the patch. It also follows that the proportion of prey harvested in a patch is an increasing sigmoidal function of the number of prey initially present. In this paper we vary several parameters of the model to evaluate the effects on the forager's intake rate, the proportion of prey harvested per patch, and the prey's average mortality rate in the environment. In each case, we study an intake rate maximizing forager's optimal response to the parameter changes. For the potential value assessment rule we find that at a higher average prey density in the environment, a lower proportion of the prey is taken in a patch with a given initial prey density. The proportion of prey taken in a patch of a given prey density also decreases when the variance of the prey density distribution is increased and if the travel time between patches is reduced. We also evaluate the effect of using predation minimization, rather than rate maximization, as the currency. Then a higher proportion of the prey is taken for each given initial prey density. This is related to the assumption that traveling between patches is the most risky activity. Compared to the optimal potential value assessment rule, the current value assessment rule performs worse, in terms of long-term intake rate achieved. The difference in performance is amplified when prey density is high or highly aggregated. These results pertain to the foraging patch spatial scale and may have consequences for the spatial distribution of prey in the environment.