A common goodness-of-fit framework for neural population models using marked point process time-rescaling

A critical component of any statistical modeling procedure is the ability to assess the goodness-of-fit between a model and observed data. For spike train models of individual neurons, many goodness-of-fit measures rely on the time-rescaling theorem and assess model quality using rescaled spike times. Recently, there has been increasing interest in statistical models that describe the simultaneous spiking activity of neuron populations, either in a single brain region or across brain regions. Classically, such models have used spike sorted data to describe relationships between the identified neurons, but more recently clusterless modeling methods have been used to describe population activity using a single model. Here we develop a generalization of the time-rescaling theorem that enables comprehensive goodness-of-fit analysis for either of these classes of population models. We use the theory of marked point processes to model population spiking activity, and show that under the correct model, each spike can be rescaled individually to generate a uniformly distributed set of events in time and the space of spike marks. After rescaling, multiple well-established goodness-of-fit procedures and statistical tests are available. We demonstrate the application of these methods both to simulated data and real population spiking in rat hippocampus. We have made the MATLAB and Python code used for the analyses in this paper publicly available through our Github repository at https://github.com/Eden-Kramer-Lab/popTRT.This work was supported by grants from the NIH (MH105174, NS094288) and the Simons Foundation (542971). (MH105174 - NIH; NS094288 - NIH; 542971 - Simons Foundation)Published versio

Rescaling, thinning or complementing? On goodness-of-fit procedures for point process models and Generalized Linear Models

Generalized Linear Models (GLMs) are an increasingly popular framework for modeling neural spike trains. They have been linked to the theory of stochastic point processes and researchers have used this relation to assess goodness-of-fit using methods from point-process theory, e.g. the time-rescaling theorem. However, high neural firing rates or coarse discretization lead to a breakdown of the assumptions necessary for this connection. Here, we show how goodness-of-fit tests from point-process theory can still be applied to GLMs by constructing equivalent surrogate point processes out of time-series observations. Furthermore, two additional tests based on thinning and complementing point processes are introduced. They augment the instruments available for checking model adequacy of point processes as well as discretized models.Comment: 9 pages, to appear in NIPS 2010 (Neural Information Processing Systems), corrected missing referenc

The Computational Structure of Spike Trains

Neurons perform computations, and convey the results of those computations through the statistical structure of their output spike trains. Here we present a practical method, grounded in the information-theoretic analysis of prediction, for inferring a minimal representation of that structure and for characterizing its complexity. Starting from spike trains, our approach finds their causal state models (CSMs), the minimal hidden Markov models or stochastic automata capable of generating statistically identical time series. We then use these CSMs to objectively quantify both the generalizable structure and the idiosyncratic randomness of the spike train. Specifically, we show that the expected algorithmic information content (the information needed to describe the spike train exactly) can be split into three parts describing (1) the time-invariant structure (complexity) of the minimal spike-generating process, which describes the spike train statistically; (2) the randomness (internal entropy rate) of the minimal spike-generating process; and (3) a residual pure noise term not described by the minimal spike-generating process. We use CSMs to approximate each of these quantities. The CSMs are inferred nonparametrically from the data, making only mild regularity assumptions, via the causal state splitting reconstruction algorithm. The methods presented here complement more traditional spike train analyses by describing not only spiking probability and spike train entropy, but also the complexity of a spike train's structure. We demonstrate our approach using both simulated spike trains and experimental data recorded in rat barrel cortex during vibrissa stimulation.Comment: Somewhat different format from journal version but same conten

Estimating the number of neurons in multi-neuronal spike trains

A common way of studying the relationship between neural activity and behavior is through the analysis of neuronal spike trains that are recorded using one or more electrodes implanted in the brain. Each spike train typically contains spikes generated by multiple neurons. A natural question that arises is "what is the number of neurons $\nu$ generating the spike train?"; This article proposes a method-of-moments technique for estimating $\nu$. This technique estimates the noise nonparametrically using data from the silent region of the spike train and it applies to isolated spikes with a possibly small, but nonnegligible, presence of overlapping spikes. Conditions are established in which the resulting estimator for $\nu$ is shown to be strongly consistent. To gauge its finite sample performance, the technique is applied to simulated spike trains as well as to actual neuronal spike train data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS371 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Contributions to statistical analysis methods for neural spiking activity

With the technical advances in neuroscience experiments in the past few decades, we have seen a massive expansion in our ability to record neural activity. These advances enable neuroscientists to analyze more complex neural coding and communication properties, and at the same time, raise new challenges for analyzing neural spiking data, which keeps growing in scale, dimension, and complexity. This thesis proposes several new statistical methods that advance statistical analysis approaches for neural spiking data, including sequential Monte Carlo (SMC) methods for efficient estimation of neural dynamics from membrane potential threshold crossings, state-space models using multimodal observation processes, and goodness-of-fit analysis methods for neural marked point process models. In a first project, we derive a set of iterative formulas that enable us to simulate trajectories from stochastic, dynamic neural spiking models that are consistent with a set of spike time observations. We develop a SMC method to simultaneously estimate the parameters of the model and the unobserved dynamic variables from spike train data. We investigate the performance of this approach on a leaky integrate-and-fire model. In another project, we define a semi-latent state-space model to estimate information related to the phenomenon of hippocampal replay. Replay is a recently discovered phenomenon where patterns of hippocampal spiking activity that typically occur during exploration of an environment are reactivated when an animal is at rest. This reactivation is accompanied by high frequency oscillations in hippocampal local field potentials. However, methods to define replay mathematically remain undeveloped. In this project, we construct a novel state-space model that enables us to identify whether replay is occurring, and if so to estimate the movement trajectories consistent with the observed neural activity, and to categorize the content of each event. The state-space model integrates information from the spiking activity from the hippocampal population, the rhythms in the local field potential, and the rat's movement behavior. Finally, we develop a new, general time-rescaling theorem for marked point processes, and use this to develop a general goodness-of-fit framework for neural population spiking models. We investigate this approach through simulation and a real data application

Estimating the number of neurons in multi-neuronal spike trains

A common way of studying the relationship between neural activity and behavior is through the analysis of neuronal spike trains that are recorded using one or more electrodes implanted in the brain. Each spike train typically contains spikes generated by multiple neurons. A natural question that arises is "what is the number of neurons $\nu$ generating the spike train?"; This article proposes a method-of-moments technique for estimating $\nu$. This technique estimates the noise nonparametrically using data from the silent region of the spike train and it applies to isolated spikes with a possibly small, but nonnegligible, presence of overlapping spikes. Conditions are established in which the resulting estimator for $\nu$ is shown to be strongly consistent. To gauge its finite sample performance, the technique is applied to simulated spike trains as well as to actual neuronal spike train data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS371 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

A simple Hidden Markov Model for midbrain dopaminergic neurons

Poster presentation: Introduction Dopaminergic neurons in the midbrain show a variety of firing patterns, ranging from very regular firing pacemaker cells to bursty and irregular neurons. The effects of different experimental conditions (like pharmacological treatment or genetical manipulations) on these neuronal discharge patterns may be subtle. Applying a stochastic model is a quantitative approach to reveal these changes. ..

Advances in point process modeling: feature selection, goodness-of-fit and novel applications

The research contained in this thesis extends multivariate marked point process modeling methods for neuroscience, generalizes goodness-of-fit techniques for the class of marked point processes, and introduces the use of a general history-dependent point process model to the domain of sleep apnea. Our first project involves further development of a modeling tool for spiking data from neural populations using the theory of marked point processes. This marked point process model uses features of spike waveforms as marks in order to estimate a state variable of interest. We examine the informational content of geometric features as well as principal components of the waveforms at hippocampal place cell activity by comparing decoding accuracies of a rat's position along a track. We determined that there was additional information available beyond that contained in traditional geometric features used for decoding in practice. The expanded use of this marked point process model in neuroscience necessitates corresponding goodness-of-fit protocols for the marked case. In our second project, we develop a generalized time-rescaling method for marked point processes that produces uniformly distributed spikes under a proper model. Once rescaled, the ground process then behaves as a Poisson process and can be analyzed using traditional point process goodness-of-fit methods. We demonstrate the method's ability to detect quality and manner of fit through both simulation and real neural data analysis. In the final project, we introduce history-dependent point process modeling as a superior method for characterizing severe sleep apnea over the current clinical standard known as the apnea-hypopnea index (AHI). We analyze model fits using combinations of both clinical covariates and event observations themselves through functions of history. Ultimately, apnea onset times were consistently estimated with significantly higher accuracy when history was incorporated alongside sleep stage. We present this method to the clinical audience as a means to gain detailed information on patterns of apnea and to provide more customized diagnoses and treatment prescriptions. These separate yet complementary projects extend existing point process modeling methods and further demonstrate their value in the neurosciences, sleep sciences, and beyond