Recent developments in multi-electrode recordings enable the simultaneous measurement of the spiking activity of many neurons. Analysis of such multineuronal data is one of the key challenges in computational neuroscience today. In this work, we develop a multivariate point-process model in which the observed activity of a network of neurons depends on three terms: 1) the experimentally-controlled stimulus; 2) the spiking history of the observed neurons; and 3) a latent noise source that corresponds, for example, to “common input ” from an unobserved population of neurons that is presynaptic to two or more cells in the observed population. We develop an expectation-maximization algorithm for fitting the model parameters; here the expectation step is based on a continuous-time implementation of the extended Kalman smoother, and the maximization step involves two concave maximization problems which may be solved in parallel. The techniques developed allow us to solve a variety of inference problems in a straightforward, computationally efficient fashion; for example, we may use the model to predict network activity given an arbitrary stimulus, infer a neuron’s firing rate given the stimulus and the activity of the other observed neurons, and perform optimal stimulus decoding and prediction. We present several detailed simulation studies which explore the strengths and limitations of our approach. 1

There are two basic problems in the statistical analysis of neural data. The “encoding” problem concerns how information is encoded in neural spike trains: can we predict the spike trains of a neuron (or population of neurons), given an arbitrary stimulus or observed motor response? Conversely, the “decoding ” problem concerns how much information is in a spike train: in particular, how well can we estimate the stimulus that gave rise to the spike train? This chapter describes statistical model-based techniques that in some cases provide a unified solution to these two coding problems. These models can capture stimulus dependencies as well as spike history and interneuronal interaction effects in population spike trains, and are intimately related to biophysically-based models of integrate-and-fire type. We describe flexible, powerful likelihood-based methods for fitting these encoding models and then for using the models to perform optimal decoding. Each of these (apparently quite difficult) tasks turn out to be highly computationally tractable, due to a key concavity property of the model likelihood. Finally, we return to the encoding problem to describe how to use these models to adaptively optimize the stimuli presented to the cell on a trial-by-trial basis, in order that we may infer the optimal model parameters as efficiently as possible.

Adaptively optimizing experiments has the potential to significantly reduce the number of trials needed to build parametric statistical models of neural systems. However, application of adaptive methods to neurophysiology has been limited by severe computational challenges. Since most neurons are high dimensional systems, optimizing neurophysiology experiments requires computing high-dimensional integrations and optimizations in real time. Here we present a fast algorithm for choosing the most informative stimulus by maximizing the mutual information between the data and the unknown parameters of a generalized linear model (GLM) which we want to fit to the neuron’s activity. We rely on important log-concavity and asymptotic normality properties of the posterior to facilitate the required computations. Our algorithm requires only low-rank matrix manipulations and a 2-dimensional search to choose the optimal stimulus. The average running time of these operations scales quadratically with the dimensionality of the GLM, making real-time adaptive experimental design feasible even for high-dimensional stimulus and parameter spaces. For example, we

Understanding how stimulus information is encoded in spike trains is a central problem in computational neuroscience. Decoding methods provide an important tool for addressing this problem, by allowing us to explicitly read out the information contained in spike responses. Here we introduce several decoding methods based on point-process neural encoding models (i.e. “forward ” models that predict spike responses to novel stimuli). These models have concave log-likelihood functions, allowing for efficient fitting via maximum likelihood. Moreover, we may use the likelihood of the observed spike trains under the model to perform optimal decoding. We present: (1) a tractable algorithm for computing the maximum a posteriori (MAP) estimate of the stimulus — the most probable stimulus to have generated the observed single- or multiple-spike train response, given some prior distribution over the stimulus; (2) a Gaussian approximation to the posterior distribution, which allows us to quantify the fidelity with which various stimulus features are encoded; (3) an efficient method for estimating the mutual information between the stimulus and the response; and (4) a framework for the detection of change-point times (e.g. the time at which the stimulus undergoes a change in mean or variance), by marginalizing over the posterior distribution of stimuli. We show several examples illustrating the performance of these estimators with simulated data. 1

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Draft- to be submitted Abstract. We describe a class of models that can be used to predict how the instantaneous firing rate of a neuron varies in response to a dynamic stimulus. These models are based on learned pointwise nonlinear transforms of the stimulus, followed by a temporal linear filtering operation on the transformed inputs. In one case, the transformation is the same for all lag-times. Thus, this “input nonlinearity ” converts the initial numerical representation of the stimulus (e.g. air pressure) to a new representation which is optimal as input to the subsequent linear model (e.g. decibels). We present algorithms for estimating both the input nonlinearity and the linear weights, including regularization techniques, and for quantifying the experimental uncertainty in these estimates. In another approach, the model is generalized to allow a potentially different nonlinear transform of the stimulus value at each lag-time. Although more general, this model is algorithmically more straightforward to fit. However, it contains many more degrees of freedom, and thus requires considerably more data for accurate and precise estimation. The feasibility of these new methods is demonstrated both on synthetic data, and on responses recorded from a neuron in rodent barrel cortex. The models are shown to predict responses to novel data accurately, and to recover several important neuronal response properties. 1

This paper presents a general framework based on reproducing kernel Hilbert spaces (RKHS) to mathematically describe and manipulate spike trains. The main idea is the definition of inner products to allow spike train signal processing from basic principles while incorporating their statistical description as point processes. Moreover, because many inner products can be formulated, a particular definition can be crafted to best fit an application. These ideas are illustrated by the definition of a number of spike train inner products. To further elicit the advantages of the RKHS framework, a family of these inner products, called the cross-intensity (CI) kernels, is further analyzed in detail. This particular inner product family encapsulates the statistical description from conditional intensity functions of spike trains. The problem of their estimation is also addressed. The simplest of the spike train kernels in this family provides an interesting perspective to other works presented in the literature, as will be illustrated in terms of spike train distance measures. Finally, as an application example, the presented RKHS framework is used to derive from simple principles a clustering algorithm for spike trains.

ABSTRACT As recent advances in calcium sensing technologies facilitate simultaneously imaging action potentials in neuronal populations, complementary analytical tools must also be developed to maximize the utility of this experimental paradigm. Although the observations here are fluorescence movies, the signals of interest—spike trains and/or time varying intracellular calcium concentrations—are hidden. Inferring these hidden signals is often problematic due to noise, nonlinearities, slow imaging rate, and unknown biophysical parameters. We overcome these difficulties by developing sequential Monte Carlo methods (particle filters) based on biophysical models of spiking, calcium dynamics, and fluorescence. We show that even in simple cases, the particle filters outperform the optimal linear (i.e., Wiener) filter, both by obtaining better estimates and by providing error bars. We then relax a number of our model assumptions to incorporate nonlinear saturation of the fluorescence signal, as well external stimulus and spike history dependence (e.g., refractoriness) of the spike trains. Using both simulations and in vitro fluorescence observations, we demonstrate temporal superresolution by inferring when within a frame each spike occurs. Furthermore, the model parameters may be estimated using expectation maximization with only a very limited amount of data (e.g., ~5–10 s or 5–40 spikes), without the requirement of any simultaneous electrophysiology or imaging experiments.

Probabilistic modeling of correlated neural population firing activity is central to understanding the neural code and building practical decoding algorithms. No parametric models currently exist for modeling multivariate correlated neural data and the high dimensional nature of the data makes fully non-parametric methods impractical. To address these problems we propose an energy-based model in which the joint probability of neural activity is represented using learned functions of the 1D marginal histograms of the data. The parameters of the model are learned using contrastive divergence and an optimization procedure for finding appropriate marginal directions. We evaluate the method using real data recorded from a population of motor cortical neurons. In particular, we model the joint probability of population spiking times and 2D hand position and show that the likelihood of test data under our model is significantly higher than under other models. These results suggest that our model captures correlations in the firing activity. Our rich probabilistic model of neural population activity is a step towards both measurement of the importance of correlations in neural coding and improved decoding of population activity.

We propose a novel paradigm for spike train decoding, which avoids entirely spike sorting based on waveform measurements. This paradigm directly uses the spike train collected at recording electrodes from thresholding the bandpassed voltage signal. Our approach is a paradigm, not an algorithm, since it can be used with any of the current decoding algorithms, such as population vector or likelihood based algorithms. Based on analytical results and an extensive simulation study, we show that our paradigm is comparable to, and sometimes more efficient than, the traditional approach based on well isolated neurons, and that it remains efficient even when all electrodes are severely corrupted by noise, a situation that would render spike sorting particularly difficult. Our paradigm will also save time and computational effort, both of which are crucially important for successful operation of real-time brain-machine interfaces. Indeed, in place of the lengthy spike sorting task of the traditional approach, it involves an exact expectation EM algorithm that is fast enough that it could also be left to run during decoding to capture potential slow changes in the states of the neurons. Encoding Regress yi on movement variables � � � Obtain estimates of neurons tuning curves � (v)