Christian Donner: Bayesian inference of inhomogeneous point process models: Methodological advances and modelling of neuronal spiking data

BCCN Berlin / GRK 1589 / TU Berlin

Arrival times of airplanes, positions of car accidents or astronomical objects in space, locations of ecological crisis, spike times of neurons, etc. are all data that surround us and can be viewed as realisations of point processes. Nowadays, the modelling of these data becomes increasingly more important, when we attempt to draw meaningful conclusions from this ever expanding amount of data. Models describing the statistics of point process data have been proposed in the past. However, to extract the model parameters given the observation of point process data, is generally challenging. Point process likelihoods of the observed data given the model parameters are difficult to deal with in practice because of their functional form. For Bayesian inference, where we aim at a tractable posterior distribution over the model parameters given the data, the task is even more demanding. In the first part of this thesis we focus on a specific model class for point process data. The central object of point process models is the non–negative intensity function, which determines the likelihood of registering an event at any given position in the observed space. To enforce non–negativity, point process models have been proposed, where the intensity function depends non–linearly on the model parameters via a scaled sigmoidal link function. By the augmentation of latent variables we show, that the likelihood of this model class can be rendered into a novel favourable form enabling efficient and fast Bayesian inference schemes for a tractable posterior over the model parameters. We utilise this new augmented form of the likelihood to perform inference for a Poisson process model, where the intensity function depends on a Gaussian process. The resulting algorithms are one order of magnitude faster than state-of-the-art methods solving the same problem. Furthermore, we show that the same algorithms can be utilised for Bayesian density estimation, i.e. inferring a posterior over densities for an observed set of points. Concluding the first part, the inference problem for a Markov jump process model, namely the kinetic Ising model from statistical physics, is addressed using the new favourable representation of point process likelihoods.

The second part of the thesis is devoted to the statistical description of a specific instance of point process data – the cell-resolved spiking activity of neurons. These data are believed to control information processing in the brain, and are highly non–stationary. We address the problem of statistical modelling such non–stationary spiking data. First, we propose a continuous time model accounting for effective couplings and temporal changes of the neuronal dynamics. Deriving an efficient inference algorithm, we demonstrate that the model can capture activity structures of in–vivo recorded data, that are not related to any controlled variables of the experiment. Finally, we propose a model which attempts to minimise the gap to the underlying system, based on the assumption of observing a population of integrate–and–fire neurons receiving common non–stationary input. We demonstrate how to efficiently evaluate the model likelihood, such that subsequent inference can be performed given spiking data recorded from a neuronal population. The novel scalable inference algorithms for point process data, and the new description of non–stationary spiking data presented in this thesis expand our ability to investigate large and complex point process datasets and draw meaningful conclusions from these data.

Additional Information

PhD defense in the research training group GRK 1589, "Sensory Computation in Neural Systems".

Organized by

Manfred Opper / Robert Martin

Location

TU Berlin, MAR 6.004, Marchstr. 23, 10587 Berlin