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 equivalence of information-theoretic and likelihood-based methods for neural dimensionality reduction

Stimulus dimensionality-reduction methods in neuroscience seek to identify a low-dimensional space of stimulus features that affect a neuron's probability of spiking. One popular method, known as maximally informative dimensions (MID), uses an information-theoretic quantity known as "single-spike information" to identify this space. Here we examine MID from a model-based perspective. We show that MID is a maximum-likelihood estimator for the parameters of a linear-nonlinear-Poisson (LNP) model, and that the empirical single-spike information corresponds to the normalized log-likelihood under a Poisson model. This equivalence implies that MID does not necessarily find maximally informative stimulus dimensions when spiking is not well described as Poisson. We provide several examples to illustrate this shortcoming, and derive a lower bound on the information lost when spiking is Bernoulli in discrete time bins. To overcome this limitation, we introduce model-based dimensionality reduction methods for neurons with non-Poisson firing statistics, and show that they can be framed equivalently in likelihood-based or information-theoretic terms. Finally, we show how to overcome practical limitations on the number of stimulus dimensions that MID can estimate by constraining the form of the non-parametric nonlinearity in an LNP model. We illustrate these methods with simulations and data from primate visual cortex

A Point-process Response Model for Spike Trains from Single Neurons in Neural Circuits under Optogenetic Stimulation

Optogenetics is a new tool to study neuronal circuits that have been genetically modified to allow stimulation by flashes of light. We study recordings from single neurons within neural circuits under optogenetic stimulation. The data from these experiments present a statistical challenge of modeling a high frequency point process (neuronal spikes) while the input is another high frequency point process (light flashes). We further develop a generalized linear model approach to model the relationships between two point processes, employing additive point-process response functions. The resulting model, Point-process Responses for Optogenetics (PRO), provides explicit nonlinear transformations to link the input point process with the output one. Such response functions may provide important and interpretable scientific insights into the properties of the biophysical process that governs neural spiking in response to optogenetic stimulation. We validate and compare the PRO model using a real dataset and simulations, and our model yields a superior area-under-the- curve value as high as 93% for predicting every future spike. For our experiment on the recurrent layer V circuit in the prefrontal cortex, the PRO model provides evidence that neurons integrate their inputs in a sophisticated manner. Another use of the model is that it enables understanding how neural circuits are altered under various disease conditions and/or experimental conditions by comparing the PRO parameters.Comment: 24 pages, 7 figures. R package pro implementing the proposed method is available on CRAN at https://CRAN.R-project.org/package=pro . Published by Statistics in Medicine at http://onlinelibrary.wiley.com/doi/10.1002/sim.6742/ful

Statistical modelling of Ca2+ oscillations in the presence of single cell heterogeneity

Intracellular calcium oscillations are a versatile signalling mechanism responsible for many biological phenomena including immune responses and insulin secretion. There is now compelling evidence that whole-cell calcium oscillations are stochastic, arising from random molecular interactions at the subcellular level. This poses a significant challenge for modelling. In this thesis, we develop a probabilistic approach that treats calcium oscillations as a stochastic point process. By employing an intensity function — a one dimension function over time which corresponds to the mean calcium spiking rate — we capture intrinsic cellular heterogeneity as well as inhomogeneous extracellular conditions, such as time-dependent stimulation. We adopt a Bayesian approach to infer the model parameters from calcium oscillations. Under this approach we need to be able to infer the intensity function. One method is to use a parametric model for the intensity function. For example we could assume the intensity function has the linear form x(t) = at + b. Then the intensity function is reduced to only needing to infer the two parameters a and b. However, parametric models suffer from strict assumptions, in this case, for the shape of the intensity function. Therefore, to lessen such assumptions, we utilise a non-parametric approach. This requires a prior distribution over the space of functions. We use two such priors, namely Gaussian processes and piecewise constant functions. We use Markov chain Monte Carlo (MCMC) techniques to sample from the posterior distribution to obtain estimates for our model parameters. Although advan- tageous — due to sampling from the true posterior distribution — MCMC algorithms can experience issues relating to their computational cost and imprecise samplers. We discuss the issues arising for our particular model and data and develop methods to improve the functionality of the MCMC algorithms in this case. For example we discuss the difficulty of inferring the length scale of the Gaussian process when fitted from calcium oscillations. An important mechanism of calcium oscillations is the refractory period, the min- imum amount of time before the next calcium oscillation. Thus, it may be beneficial to explicitly include the refractory period as part of the model. We investigate the advantages and disadvantages of including the refractory period. We fit the model to HEK293 cells and astrocytes challenged under a variety of stimulation protocols. We find that our model can accurately generate surrogate spike sequences with similar properties to those the model is fitted from. Therefore, the model can be used to cheaply create spike sequences that are synonymous to those found experimentally. Moreover, our model captures the similarity between calcium spike sequences obtained from step-change stimulus protocols and constant stimulus protocols. Combining intensity functions inferred from constant stimulus experiments closely follow the intensity function from a step change experiment. This implies it may be possible to build surrogate spike sequences for complex time-dependent stimulation protocols by combining results from simpler experiments. Of particular interest are patterns found in the intensity function which describes the heterogeneity in the calcium oscillations over time. Common patterns could help to understand the different time scales of the calcium response. Standard approaches often fail in grouping intensity functions with similar shape. Therefore, we develop an approach to cluster intensity functions based on their shape alone by utilising the Haar basis. In summary, we have developed novel statistical approaches based on the concept of stochastic point processes and non-standard MCMC techniques. We have successfully applied these new methodologies to gain a deeper understanding into the stochastic nature of intracellular calcium oscillations, in particular how different cell types respond to a variety of stimulation protocols. In turn, this brings us one step closer to unravel the complex dynamics of this pivotal intracellular messenger which controls life from its very beginning to its end

Statistical modelling of Ca2+ oscillations in the presence of single cell heterogeneity

Intracellular calcium oscillations are a versatile signalling mechanism responsible for many biological phenomena including immune responses and insulin secretion. There is now compelling evidence that whole-cell calcium oscillations are stochastic, arising from random molecular interactions at the subcellular level. This poses a significant challenge for modelling. In this thesis, we develop a probabilistic approach that treats calcium oscillations as a stochastic point process. By employing an intensity function — a one dimension function over time which corresponds to the mean calcium spiking rate — we capture intrinsic cellular heterogeneity as well as inhomogeneous extracellular conditions, such as time-dependent stimulation. We adopt a Bayesian approach to infer the model parameters from calcium oscillations. Under this approach we need to be able to infer the intensity function. One method is to use a parametric model for the intensity function. For example we could assume the intensity function has the linear form x(t) = at + b. Then the intensity function is reduced to only needing to infer the two parameters a and b. However, parametric models suffer from strict assumptions, in this case, for the shape of the intensity function. Therefore, to lessen such assumptions, we utilise a non-parametric approach. This requires a prior distribution over the space of functions. We use two such priors, namely Gaussian processes and piecewise constant functions. We use Markov chain Monte Carlo (MCMC) techniques to sample from the posterior distribution to obtain estimates for our model parameters. Although advan- tageous — due to sampling from the true posterior distribution — MCMC algorithms can experience issues relating to their computational cost and imprecise samplers. We discuss the issues arising for our particular model and data and develop methods to improve the functionality of the MCMC algorithms in this case. For example we discuss the difficulty of inferring the length scale of the Gaussian process when fitted from calcium oscillations. An important mechanism of calcium oscillations is the refractory period, the min- imum amount of time before the next calcium oscillation. Thus, it may be beneficial to explicitly include the refractory period as part of the model. We investigate the advantages and disadvantages of including the refractory period. We fit the model to HEK293 cells and astrocytes challenged under a variety of stimulation protocols. We find that our model can accurately generate surrogate spike sequences with similar properties to those the model is fitted from. Therefore, the model can be used to cheaply create spike sequences that are synonymous to those found experimentally. Moreover, our model captures the similarity between calcium spike sequences obtained from step-change stimulus protocols and constant stimulus protocols. Combining intensity functions inferred from constant stimulus experiments closely follow the intensity function from a step change experiment. This implies it may be possible to build surrogate spike sequences for complex time-dependent stimulation protocols by combining results from simpler experiments. Of particular interest are patterns found in the intensity function which describes the heterogeneity in the calcium oscillations over time. Common patterns could help to understand the different time scales of the calcium response. Standard approaches often fail in grouping intensity functions with similar shape. Therefore, we develop an approach to cluster intensity functions based on their shape alone by utilising the Haar basis. In summary, we have developed novel statistical approaches based on the concept of stochastic point processes and non-standard MCMC techniques. We have successfully applied these new methodologies to gain a deeper understanding into the stochastic nature of intracellular calcium oscillations, in particular how different cell types respond to a variety of stimulation protocols. In turn, this brings us one step closer to unravel the complex dynamics of this pivotal intracellular messenger which controls life from its very beginning to its end