Fast maximum likelihood estimation using continuous-time neural point process models

A recent report estimates that the number of simultaneously recorded neurons is growing exponentially. A commonly employed statistical paradigm using discrete-time point process models of neural activity involves the computation of a maximum-likelihood estimate. The time to computate this estimate, per neuron, is proportional to the number of bins in a finely spaced discretization of time. By using continuous-time models of neural activity and the optimally efficient Gaussian quadrature, memory requirements and computation times are dramatically decreased in the commonly encountered situation where the number of parameters p is much less than the number of time-bins n. In this regime, with q equal to the quadrature order, memory requirements are decreased from O(np) to O(qp), and the number of floating-point operations are decreased from O(np2) to O(qp2). Accuracy of the proposed estimates is assessed based upon physiological consideration, error bounds, and mathematical results describing the relation between numerical integration error and numerical error affecting both parameter estimates and the observed Fisher information. A check is provided which is used to adapt the order of numerical integration. The procedure is verified in simulation and for hippocampal recordings. It is found that in 95 % of hippocampal recordings a q of 60 yields numerical error negligible with respect to parameter estimate standard error. Statistical inference using the proposed methodology is a fast and convenient alternative to statistical inference performed using a discrete-time point process model of neural activity. It enables the employment of the statistical methodology available with discrete-time inference, but is faster, uses less memory, and avoids any error due to discretization.National Science Foundation (U.S.) (NSF grant DMS-1042134

A unified approach to linking experimental, statistical and computational analysis of spike train data

A fundamental issue in neuroscience is how to identify the multiple biophysical mechanisms through which neurons generate observed patterns of spiking activity. In previous work, we proposed a method for linking observed patterns of spiking activity to specific biophysical mechanisms based on a state space modeling framework and a sequential Monte Carlo, or particle filter, estimation algorithm. We have shown, in simulation, that this approach is able to identify a space of simple biophysical models that were consistent with observed spiking data (and included the model that generated the data), but have yet to demonstrate the application of the method to identify realistic currents from real spike train data. Here, we apply the particle filter to spiking data recorded from rat layer V cortical neurons, and correctly identify the dynamics of an slow, intrinsic current. The underlying intrinsic current is successfully identified in four distinct neurons, even though the cells exhibit two distinct classes of spiking activity: regular spiking and bursting. This approach – linking statistical, computational, and experimental neuroscience – provides an effective technique to constrain detailed biophysical models to specific mechanisms consistent with observed spike train data.Published versio

{\sc CosmoNet}: fast cosmological parameter estimation in non-flat models using neural networks

We present a further development of a method for accelerating the calculation of CMB power spectra, matter power spectra and likelihood functions for use in cosmological Bayesian inference. The algorithm, called {\sc CosmoNet}, is based on training a multilayer perceptron neural network. We compute CMB power spectra (up to $\ell=2000$) and matter transfer functions over a hypercube in parameter space encompassing the $4\sigma$ confidence region of a selection of CMB (WMAP + high resolution experiments) and large scale structure surveys (2dF and SDSS). We work in the framework of a generic 7 parameter non-flat cosmology. Additionally we use {\sc CosmoNet} to compute the WMAP 3-year, 2dF and SDSS likelihoods over the same region. We find that the average error in the power spectra is typically well below cosmic variance for spectra, and experimental likelihoods calculated to within a fraction of a log unit. We demonstrate that marginalised posteriors generated with {\sc CosmoNet} spectra agree to within a few percent of those generated by {\sc CAMB} parallelised over 4 CPUs, but are obtained 2-3 times faster on just a \emph{single} processor. Furthermore posteriors generated directly via {\sc CosmoNet} likelihoods can be obtained in less than 30 minutes on a single processor, corresponding to a speed up of a factor of $\sim 32$. We also demonstrate the capabilities of {\sc CosmoNet} by extending the CMB power spectra and matter transfer function training to a more generic 10 parameter cosmological model, including tensor modes, a varying equation of state of dark energy and massive neutrinos. {\sc CosmoNet} and interfaces to both {\sc CosmoMC} and {\sc Bayesys} are publically available at {\tt www.mrao.cam.ac.uk/software/cosmonet}.Comment: 8 pages, submitted to MNRA

The Neural Particle Filter

The robust estimation of dynamically changing features, such as the position of prey, is one of the hallmarks of perception. On an abstract, algorithmic level, nonlinear Bayesian filtering, i.e. the estimation of temporally changing signals based on the history of observations, provides a mathematical framework for dynamic perception in real time. Since the general, nonlinear filtering problem is analytically intractable, particle filters are considered among the most powerful approaches to approximating the solution numerically. Yet, these algorithms prevalently rely on importance weights, and thus it remains an unresolved question how the brain could implement such an inference strategy with a neuronal population. Here, we propose the Neural Particle Filter (NPF), a weight-less particle filter that can be interpreted as the neuronal dynamics of a recurrently connected neural network that receives feed-forward input from sensory neurons and represents the posterior probability distribution in terms of samples. Specifically, this algorithm bridges the gap between the computational task of online state estimation and an implementation that allows networks of neurons in the brain to perform nonlinear Bayesian filtering. The model captures not only the properties of temporal and multisensory integration according to Bayesian statistics, but also allows online learning with a maximum likelihood approach. With an example from multisensory integration, we demonstrate that the numerical performance of the model is adequate to account for both filtering and identification problems. Due to the weightless approach, our algorithm alleviates the 'curse of dimensionality' and thus outperforms conventional, weighted particle filters in higher dimensions for a limited number of particles

Particle-filtering approaches for nonlinear Bayesian decoding of neuronal spike trains

The number of neurons that can be simultaneously recorded doubles every seven years. This ever increasing number of recorded neurons opens up the possibility to address new questions and extract higher dimensional stimuli from the recordings. Modeling neural spike trains as point processes, this task of extracting dynamical signals from spike trains is commonly set in the context of nonlinear filtering theory. Particle filter methods relying on importance weights are generic algorithms that solve the filtering task numerically, but exhibit a serious drawback when the problem dimensionality is high: they are known to suffer from the 'curse of dimensionality' (COD), i.e. the number of particles required for a certain performance scales exponentially with the observable dimensions. Here, we first briefly review the theory on filtering with point process observations in continuous time. Based on this theory, we investigate both analytically and numerically the reason for the COD of weighted particle filtering approaches: Similarly to particle filtering with continuous-time observations, the COD with point-process observations is due to the decay of effective number of particles, an effect that is stronger when the number of observable dimensions increases. Given the success of unweighted particle filtering approaches in overcoming the COD for continuous- time observations, we introduce an unweighted particle filter for point-process observations, the spike-based Neural Particle Filter (sNPF), and show that it exhibits a similar favorable scaling as the number of dimensions grows. Further, we derive rules for the parameters of the sNPF from a maximum likelihood approach learning. We finally employ a simple decoding task to illustrate the capabilities of the sNPF and to highlight one possible future application of our inference and learning algorithm

BAMBI: blind accelerated multimodal Bayesian inference

In this paper we present an algorithm for rapid Bayesian analysis that combines the benefits of nested sampling and artificial neural networks. The blind accelerated multimodal Bayesian inference (BAMBI) algorithm implements the MultiNest package for nested sampling as well as the training of an artificial neural network (NN) to learn the likelihood function. In the case of computationally expensive likelihoods, this allows the substitution of a much more rapid approximation in order to increase significantly the speed of the analysis. We begin by demonstrating, with a few toy examples, the ability of a NN to learn complicated likelihood surfaces. BAMBI's ability to decrease running time for Bayesian inference is then demonstrated in the context of estimating cosmological parameters from Wilkinson Microwave Anisotropy Probe and other observations. We show that valuable speed increases are achieved in addition to obtaining NNs trained on the likelihood functions for the different model and data combinations. These NNs can then be used for an even faster follow-up analysis using the same likelihood and different priors. This is a fully general algorithm that can be applied, without any pre-processing, to other problems with computationally expensive likelihood functions.Comment: 12 pages, 8 tables, 17 figures; accepted by MNRAS; v2 to reflect minor changes in published versio

Hierarchical Implicit Models and Likelihood-Free Variational Inference

Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure. Next, we develop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. Key to LFVI is specifying a variational family that is also implicit. This matches the model's flexibility and allows for accurate approximation of the posterior. We demonstrate diverse applications: a large-scale physical simulator for predator-prey populations in ecology; a Bayesian generative adversarial network for discrete data; and a deep implicit model for text generation.Comment: Appears in Neural Information Processing Systems, 201