Learning interpretable continuous-time models of latent stochastic dynamical systems

We develop an approach to learn an interpretable semi-parametric model of a latent continuous-time stochastic dynamical system, assuming noisy high-dimensional outputs sampled at uneven times. The dynamics are described by a nonlinear stochastic differential equation (SDE) driven by a Wiener process, with a drift evolution function drawn from a Gaussian process (GP) conditioned on a set of learnt fixed points and corresponding local Jacobian matrices. This form yields a flexible nonparametric model of the dynamics, with a representation corresponding directly to the interpretable portraits routinely employed in the study of nonlinear dynamical systems. The learning algorithm combines inference of continuous latent paths underlying observed data with a sparse variational description of the dynamical process. We demonstrate our approach on simulated data from different nonlinear dynamical systems

Learning interpretable continuous-time models of latent stochastic dynamical systems

We develop an approach to learn an interpretable semi-parametric model of a latent continuoustime stochastic dynamical system, assuming noisy high-dimensional outputs sampled at uneven times. The dynamics are described by a nonlinear stochastic differential equation (SDE) driven by a Wiener process, with a drift evolution function drawn from a Gaussian process (GP) conditioned on a set of learnt fixed points and corresponding local Jacobian matrices. This form yields a flexible nonparametric model of the dynamics, with a representation corresponding directly to the interpretable portraits routinely employed in the study of nonlinear dynamical systems. The learning algorithm combines inference of continuous latent paths underlying observed data with a sparse variational description of the dynamical process. We demonstrate our approach on simulated data from different nonlinear dynamical systems

Dynamical structure in neural population activity

The question of how the collective activity of neural populations in the brain gives rise to complex behaviour is fundamental to neuroscience. At the core of this question lie considerations about how neural circuits can perform computations that enable sensory perception, motor control, and decision making. It is thought that such computations are implemented by the dynamical evolution of distributed activity in recurrent circuits. Thus, identifying and interpreting dynamical structure in neural population activity is a key challenge towards a better understanding of neural computation. In this thesis, I make several contributions in addressing this challenge. First, I develop two novel methods for neural data analysis. Both methods aim to extract trajectories of low-dimensional computational state variables directly from the unbinned spike-times of simultaneously recorded neurons on single trials. The first method separates inter-trial variability in the low-dimensional trajectory from variability in the timing of progression along its path, and thus offers a quantification of inter-trial variability in the underlying computational process. The second method simultaneously learns a low-dimensional portrait of the underlying nonlinear dynamics of the circuit, as well as the system's fixed points and locally linearised dynamics around them. This approach facilitates extracting interpretable low-dimensional hypotheses about computation directly from data. Second, I turn to the question of how low-dimensional dynamical structure may be embedded within a high-dimensional neurobiological circuit with excitatory and inhibitory cell-types. I analyse how such circuit-level features shape population activity, with particular focus on responses to targeted optogenetic perturbations of the circuit. Third, I consider the problem of implementing multiple computations in a single dynamical system. I address this in the framework of multi-task learning in recurrently connected networks and demonstrate that a careful organisation of low-dimensional, activity-defined subspaces within the network can help to avoid interference across tasks

Unsupervised methods for large-scale, cell-resolution neural data analysis

In order to keep up with the volume of data, as well as the complexity of experiments and models in modern neuroscience, we need scalable and principled analytic programmes that take into account the scientific goals and the challenges of biological experiments. This work focuses on algorithms that tackle problems throughout the whole data analysis process. I first investigate how to best transform two-photon calcium imaging microscopy recordings – sets of contiguous images – into an easier-to-analyse matrix containing time courses of individual neurons. For this I first estimate how the true fluorescence signal gets transformed by tissue artefacts and the microscope setup, by learning the parameters of a realistic physical model from recorded data. Next, I describe how individual neural cell bodies may be segmented from the images, based on a cost function tailored to neural characteristics. Finally, I describe an interpretable non-linear dynamical model of neural population activity, which provides immediate scientific insight into complex system behaviour, and may spawn a new way of investigating stochastic non-linear dynamical systems. I hope the algorithms described here will not only be integrated into analytic pipelines of neural recordings, but also point out that algorithmic design should be informed by communication with the broader community, understanding and tackling the challenges inherent in experimental biological science