Probabilistic inference in general graphical models through sampling in stochastic networks of spiking neurons

An important open problem of computational neuroscience is the generic organization of computations in networks of neurons in the brain. We show here through rigorous theoretical analysis that inherent stochastic features of spiking neurons, in combination with simple nonlinear computational operations in specific network motifs and dendritic arbors, enable networks of spiking neurons to carry out probabilistic inference through sampling in general graphical models. In particular, it enables them to carry out probabilistic inference in Bayesian networks with converging arrows ("explaining away") and with undirected loops, that occur in many real-world tasks. Ubiquitous stochastic features of networks of spiking neurons, such as trial-to-trial variability and spontaneous activity, are necessary ingredients of the underlying computational organization. We demonstrate through computer simulations that this approach can be scaled up to neural emulations of probabilistic inference in fairly large graphical models, yielding some of the most complex computations that have been carried out so far in networks of spiking neurons.     

Ensembles of Spiking Neurons with Noise Support Optimal Probabilistic Inference in a Dynamically Changing Environment

It has recently been shown that networks of spiking neurons with noise can emulate simple forms of probabilistic inference through “neural sampling”, i.e., by treating spikes as samples from a probability distribution of network states that is encoded in the network. Deficiencies of the existing model are its reliance on single neurons for sampling from each random variable, and the resulting limitation in representing quickly varying probabilistic information. We show that both deficiencies can be overcome by moving to a biologically more realistic encoding of each salient random variable through the stochastic firing activity of an ensemble of neurons. The resulting model demonstrates that networks of spiking neurons with noise can easily track and carry out basic computational operations on rapidly varying probability distributions, such as the odds of getting rewarded for a specific behavior. We demonstrate the viability of this new approach towards neural coding and computation, which makes use of the inherent parallelism of generic neural circuits, by showing that this model can explain experimentally observed firing activity of cortical neurons for a variety of tasks that require rapid temporal integration of sensory information.     

Spike-based population coding and working memory

Compelling behavioral evidence suggests that humans can make optimal decisions despite the uncertainty inherent in perceptual or motor tasks. A key question in neuroscience is how populations of spiking neurons can implement such probabilistic computations. In this article, we develop a comprehensive framework for optimal, spike-based sensory integration and working memory in a dynamic environment. We propose that probability distributions are inferred spike-per-spike in recurrently connected networks of integrate-and-fire neurons. As a result, these networks can combine sensory cues optimally, track the state of a time-varying stimulus and memorize accumulated evidence over periods much longer than the time constant of single neurons. Importantly, we propose that population responses and persistent working memory states represent entire probability distributions and not only single stimulus values. These memories are reflected by sustained, asynchronous patterns of activity which make relevant information available to downstream neurons within their short time window of integration. Model neurons act as predictive encoders, only firing spikes which account for new information that has not yet been signaled. Thus, spike times signal deterministically a prediction error, contrary to rate codes in which spike times are considered to be random samples of an underlying firing rate. As a consequence of this coding scheme, a multitude of spike patterns can reliably encode the same information. This results in weakly correlated, Poisson-like spike trains that are sensitive to initial conditions but robust to even high levels of external neural noise. This spike train variability reproduces the one observed in cortical sensory spike trains, but cannot be equated to noise. On the contrary, it is a consequence of optimal spike-based inference. In contrast, we show that rate-based models perform poorly when implemented with stochastically spiking neurons.     

Network Plasticity as Bayesian Inference

General results from statistical learning theory suggest to understand not only brain computations, but also brain plasticity as probabilistic inference. But a model for that has been missing. We propose that inherently stochastic features of synaptic plasticity and spine motility enable cortical networks of neurons to carry out probabilistic inference by sampling from a posterior distribution of network configurations. This model provides a viable alternative to existing models that propose convergence of parameters to maximum likelihood values. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience, how cortical networks can generalize learned information so well to novel experiences, and how they can compensate continuously for unforeseen disturbances of the network. The resulting new theory of network plasticity explains from a functional perspective a number of experimental data on stochastic aspects of synaptic plasticity that previously appeared to be quite puzzling.     

Fading memory and kernel properties of generic cortical microcircuit models

It is quite difficult to construct circuits of spiking neurons that can carry out complex computational tasks. On the other hand even randomly connected circuits of spiking neurons can in principle be used for complex computational tasks such as time-warp invariant speech recognition. This is possible because such circuits have an inherent tendency to integrate incoming information in such a way that simple linear readouts can be trained to transform the current circuit activity into the target output for a very large number of computational tasks. Consequently we propose to analyze circuits of spiking neurons in terms of their roles as analog fading memory and non-linear kernels, rather than as implementations of specific computational operations and algorithms. This article is a sequel to [W. Maass, T. Natschl?ger, H. Markram, Real-time computing without stable states: a new framework for neural computation based on perturbations, Neural Comput. 14 (11) (2002) 2531-2560, Online available as #130 from: ], and contains new results about the performance of generic neural microcircuit models for the recognition of speech that is subject to linear and non-linear time-warps, as well as for computations on time-varying firing rates. These computations rely, apart from general properties of generic neural microcircuit models, just on capabilities of simple linear readouts trained by linear regression. This article also provides detailed data on the fading memory property of generic neural microcircuit models, and a quick review of other new results on the computational power of such circuits of spiking neurons.     

Spiking networks for Bayesian inference and choice

Systems neuroscience traditionally conceptualizes a population of spiking neurons as merely encoding the value of a stimulus. Yet, psychophysics has revealed that people take into account stimulus uncertainty when performing sensory or motor computations and do so in a nearly Bayes-optimal way. This suggests that neural populations do not encode just a single value but an entire probability distribution over the stimulus. Several such probabilistic codes have been proposed, including one that utilizes the structure of neural variability to enable simple neural implementations of probabilistic computations such as optimal cue integration. This approach provides a quantitative link between Bayes-optimal behaviors and specific neural operations. It allows for novel ways to evaluate probabilistic codes and for predictions for physiological population recordings.     

Bayesian inference for dynamic transcriptional regulation; the Hes1 system as a case study

MOTIVATION: In this study, we address the problem of estimating the parameters of regulatory networks and provide the first application of Markov chain Monte Carlo (MCMC) methods to experimental data. As a case study, we consider a stochastic model of the Hes1 system expressed in terms of stochastic differential equations (SDEs) to which rigorous likelihood methods of inference can be applied. When fitting continuous-time stochastic models to discretely observed time series the lengths of the sampling intervals are important, and much of our study addresses the problem when the data are sparse. RESULTS: We estimate the parameters of an autoregulatory network providing results both for simulated and real experimental data from the Hes1 system. We develop an estimation algorithm using MCMC techniques which are flexible enough to allow for the imputation of latent data on a finer time scale and the presence of prior information about parameters which may be informed from other experiments as well as additional measurement error.     

Learning Universal Computations with Spikes

Providing the neurobiological basis of information processing in higher animals, spiking neural networks must be able to learn a variety of complicated computations, including the generation of appropriate, possibly delayed reactions to inputs and the self-sustained generation of complex activity patterns, e.g. for locomotion. Many such computations require previous building of intrinsic world models. Here we show how spiking neural networks may solve these different tasks. Firstly, we derive constraints under which classes of spiking neural networks lend themselves to substrates of powerful general purpose computing. The networks contain dendritic or synaptic nonlinearities and have a constrained connectivity. We then combine such networks with learning rules for outputs or recurrent connections. We show that this allows to learn even difficult benchmark tasks such as the self-sustained generation of desired low-dimensional chaotic dynamics or memory-dependent computations. Furthermore, we show how spiking networks can build models of external world systems and use the acquired knowledge to control them.     

Hardware implementation of stochastic spiking neural networks

Spiking Neural Networks, the last generation of Artificial Neural Networks, are characterized by its bio-inspired nature and by a higher computational capacity with respect to other neural models. In real biological neurons, stochastic processes represent an important mechanism of neural behavior and are responsible of its special arithmetic capabilities. In this work we present a simple hardware implementation of spiking neurons that considers this probabilistic nature. The advantage of the proposed implementation is that it is fully digital and therefore can be massively implemented in Field Programmable Gate Arrays. The high computational capabilities of the proposed model are demonstrated by the study of both feed-forward and recurrent networks that are able to implement high-speed signal filtering and to solve complex systems of linear equations.     

Spiking behavior and epileptiform oscillations in a discrete model of cortical neural networks

Summary We investigate the phenomenon of epileptiform activity using a discrete model of cortical neural networks. Our model is reduced to the elementary features of neurons and assumes simplified dynamics of action potentials and postsynaptic potentials. The discrete model provides a comparably high simulation speed which allows the rendering of phase diagrams and simulations of large neural networks in reasonable time. Further the reduction to the basic features of neurons provides insight into the essentials of a possible mechanism of epilepsy. Our computer simulations suggest that the detailed dynamics of postsynaptic and action potentials are not indispensable for obtaining epileptiform behavior on the system level. The simulation results of autonomously evolving networks exhibit a regime in which the network dynamics spontaneously switch between fluctuating and oscillating behavior and produce isolated network spikes without external stimulation. Inhibitory neurons have been found to play an important part in the synchronization of neural firing: an increased number of synapses established by inhibitory neurons onto other neurons induces a transition to the spiking regime. A decreased frequency accompanying the hypersynchronous population activity has only occurred with slow inhibitory postsynaptic potentials.     

Bursting dynamics remarkably improve the performance of neural networks on liquid computing

Burst firings are functionally important behaviors displayed by neural circuits, which plays a primary role in reliable transmission of electrical signals for neuronal communication. However, with respect to the computational capability of neural networks, most of relevant studies are based on the spiking dynamics of individual neurons, while burst firing is seldom considered. In this paper, we carry out a comprehensive study to compare the performance of spiking and bursting dynamics on the capability of liquid computing, which is an effective approach for intelligent computation of neural networks. The results show that neural networks with bursting dynamic have much better computational performance than those with spiking dynamics, especially for complex computational tasks. Further analysis demonstrate that the fast firing pattern of bursting dynamics can obviously enhance the efficiency of synaptic integration from pre-neurons both temporally and spatially. This indicates that bursting dynamic can significantly enhance the complexity of network activity, implying its high efficiency in information processing.     

MCMC estimation of Markov models for ion channels

Ion channels are characterized by inherently stochastic behavior which can be represented by continuous-time Markov models (CTMM). Although methods for collecting data from single ion channels are available, translating a time series of open and closed channels to a CTMM remains a challenge. Bayesian statistics combined with Markov chain Monte Carlo (MCMC) sampling provide means for estimating the rate constants of a CTMM directly from single channel data. In this article, different approaches for the MCMC sampling of Markov models are combined. This method, new to our knowledge, detects overparameterizations and gives more accurate results than existing MCMC methods. It shows similar performance as QuB-MIL, which indicates that it also compares well with maximum likelihood estimators. Data collected from an inositol trisphosphate receptor is used to demonstrate how the best model for a given data set can be found in practice.     

Conjugate Gibbs sampling for Bayesian phylogenetic models.

We propose a new Markov Chain Monte Carlo (MCMC) sampling mechanism for Bayesian phylogenetic inference. This method, which we call conjugate Gibbs, relies on analytical conjugacy properties, and is based on an alternation between data augmentation and Gibbs sampling. The data augmentation step consists in sampling a detailed substitution history for each site, and across the whole tree, given the current value of the model parameters. Provided convenient priors are used, the parameters of the model can then be directly updated by a Gibbs sampling procedure, conditional on the current substitution history. Alternating between these two sampling steps yields a MCMC device whose equilibrium distribution is the posterior probability density of interest. We show, on real examples, that this conjugate Gibbs method leads to a significant improvement of the mixing behavior of the MCMC. In all cases, the decorrelation times of the resulting chains are smaller than those obtained by standard Metropolis Hastings procedures by at least one order of magnitude. The method is particularly well suited to heterogeneous models, i.e. assuming site-specific random variables. In particular, the conjugate Gibbs formalism allows one to propose efficient implementations of complex models, for instance assuming site-specific substitution processes, that would not be accessible to standard MCMC methods.     

Oscillations in Large-Scale Cortical Networks: Map-Based Model

We develop a new computationally efficient approach for the analysis of complex large-scale neurobiological networks. Its key element is the use of a new phenomenological model of a neuron capable of replicating important spike pattern characteristics and designed in the form of a system of difference equations (a map). We developed a set of map-based models that replicate spiking activity of cortical fast spiking, regular spiking and intrinsically bursting neurons. Interconnected with synaptic currents these model neurons demonstrated responses very similar to those found with Hodgkin-Huxley models and in experiments. We illustrate the efficacy of this approach in simulations of one- and two-dimensional cortical network models consisting of regular spiking neurons and fast spiking interneurons to model sleep and activated states of the thalamocortical system. Our study suggests that map-based models can be widely used for large-scale simulations and that such models are especially useful for tasks where the modeling of specific firing patterns of different cell classes is important.     

Spiking Neurons in a Hierarchical Self-Organizing Map Model Can Learn to Develop Spatial and Temporal Properties of Entorhinal Grid Cells and Hippocampal Place Cells

Medial entorhinal grid cells and hippocampal place cells provide neural correlates of spatial representation in the brain. A place cell typically fires whenever an animal is present in one or more spatial regions, or places, of an environment. A grid cell typically fires in multiple spatial regions that form a regular hexagonal grid structure extending throughout the environment. Different grid and place cells prefer spatially offset regions, with their firing fields increasing in size along the dorsoventral axes of the medial entorhinal cortex and hippocampus. The spacing between neighboring fields for a grid cell also increases along the dorsoventral axis. This article presents a neural model whose spiking neurons operate in a hierarchy of self-organizing maps, each obeying the same laws. This spiking GridPlaceMap model simulates how grid cells and place cells may develop. It responds to realistic rat navigational trajectories by learning grid cells with hexagonal grid firing fields of multiple spatial scales and place cells with one or more firing fields that match neurophysiological data about these cells and their development in juvenile rats. The place cells represent much larger spaces than the grid cells, which enable them to support navigational behaviors. Both self-organizing maps amplify and learn to categorize the most frequent and energetic co-occurrences of their inputs. The current results build upon a previous rate-based model of grid and place cell learning, and thus illustrate a general method for converting rate-based adaptive neural models, without the loss of any of their analog properties, into models whose cells obey spiking dynamics. New properties of the spiking GridPlaceMap model include the appearance of theta band modulation. The spiking model also opens a path for implementation in brain-emulating nanochips comprised of networks of noisy spiking neurons with multiple-level adaptive weights for controlling autonomous adaptive robots capable of spatial navigation.     

Normalization for probabilistic inference with neurons

Recently, there have been a number of proposals regarding how biologically plausible neural networks might perform probabilistic inference (Rao, Neural Computation, 16(1):1-38, 2004; Eliasmith and Anderson, Neural engineering: computation, representation and dynamics in neurobiological systems, 2003; Ma et?al., Nature Neuroscience, 9(11):1432-1438, 2006; Sahani and Dayan, Neural Computation, 15(10):2255-2279, 2003). To be able to repeatedly perform such inference, it is essential that the represented distributions be appropriately normalized. Past approaches have considered normalization mechanisms independently of inference, often leaving them unexplored, or appealing to a notion of divisive normalization that requires pooling across many neurons. Here, we demonstrate how normalization and inference can be combined into an appropriate connection matrix, eliminating the need for pooling or a division-like operation. We algebraically demonstrate that such a solution is available regardless of the inference being performed. We show that such a solution is relevant to neural computation by implementing it in a recurrent spiking neural network.     

A generative spiking neural-network model of goal-directed behaviour and one-step planning

In mammals, goal-directed and planning processes support flexible behaviour used to face new situations that cannot be tackled through more efficient but rigid habitual behaviours. Within the Bayesian modelling approach of brain and behaviour, models have been proposed to perform planning as probabilistic inference but this approach encounters a crucial problem: explaining how such inference might be implemented in brain spiking networks. Recently, the literature has proposed some models that face this problem through recurrent spiking neural networks able to internally simulate state trajectories, the core function at the basis of planning. However, the proposed models have relevant limitations that make them biologically implausible, namely their world model is trained ‘off-line’ before solving the target tasks, and they are trained with supervised learning procedures that are biologically and ecologically not plausible. Here we propose two novel hypotheses on how brain might overcome these problems, and operationalise them in a novel architecture pivoting on a spiking recurrent neural network. The first hypothesis allows the architecture to learn the world model in parallel with its use for planning: to this purpose, a new arbitration mechanism decides when to explore, for learning the world model, or when to exploit it, for planning, based on the entropy of the world model itself. The second hypothesis allows the architecture to use an unsupervised learning process to learn the world model by observing the effects of actions. The architecture is validated by reproducing and accounting for the learning profiles and reaction times of human participants learning to solve a visuomotor learning task that is new for them. Overall, the architecture represents the first instance of a model bridging probabilistic planning and spiking-processes that has a degree of autonomy analogous to the one of real organisms.     

神经起步点自发放电节律及节律转化的分岔规律

在神经起步点的实验中观察到了复杂多样的神经放电([Ca^2 ]o)节律模式,如周期簇放电、周期峰放电、混沌簇放电、混沌峰放电以及随机放电节律等。随着细胞外钙离子浓度的降低,神经放电节律从周期l簇放电,经过复杂的分岔过程(包括经倍周期分岔到混沌簇放电、混沌簇放电经激变到混沌峰放电、以及混沌峰放电经逆倍周期分岔到周期峰放电)转化为周期l峰放电。在神经放电理论模型——Chay模型中,调节与实验相关的参数(Ca^2 平衡电位),可以获得与实验相似的神经放电节律和节律转换规律。这表明复杂的神经放电节律之间存在着一定的分岔规律,它们是理解神经元信息编码的基础。