Training deep neural density estimators to identify mechanistic models of neural dynamics

Mechanistic modeling in neuroscience aims to explain observed phenomena in terms of underlying causes. However, determining which model parameters agree with complex and stochastic neural data presents a significant challenge. We address this challenge with a machine learning tool which uses deep neural density estimators-- trained using model simulations-- to carry out Bayesian inference and retrieve the full space of parameters compatible with raw data or selected data features. Our method is scalable in parameters and data features, and can rapidly analyze new data after initial training. We demonstrate the power and flexibility of our approach on receptive fields, ion channels, and Hodgkin-Huxley models. We also characterize the space of circuit configurations giving rise to rhythmic activity in the crustacean stomatogastric ganglion, and use these results to derive hypotheses for underlying compensation mechanisms. Our approach will help close the gap between data-driven and theory-driven models of neural dynamics

Probabilistic Models of Motor Production

N. Bernstein defined the ability of the central neural system (CNS) to control many degrees of freedom of a physical body with all its redundancy and flexibility as the main problem in motor control. He pointed at that man-made mechanisms usually have one, sometimes two degrees of freedom (DOF); when the number of DOF increases further, it becomes prohibitively hard to control them. The brain, however, seems to perform such control effortlessly. He suggested the way the brain might deal with it: when a motor skill is being acquired, the brain artificially limits the degrees of freedoms, leaving only one or two. As the skill level increases, the brain gradually "frees" the previously fixed DOF, applying control when needed and in directions which have to be corrected, eventually arriving to the control scheme where all the DOF are "free". This approach of reducing the dimensionality of motor control remains relevant even today. One the possibles solutions of the Bernstetin's problem is the hypothesis of motor primitives (MPs) - small building blocks that constitute complex movements and facilitite motor learnirng and task completion. Just like in the visual system, having a homogenious hierarchical architecture built of similar computational elements may be beneficial. Studying such a complicated object as brain, it is important to define at which level of details one works and which questions one aims to answer. David Marr suggested three levels of analysis: 1. computational, analysing which problem the system solves; 2. algorithmic, questioning which representation the system uses and which computations it performs; 3. implementational, finding how such computations are performed by neurons in the brain. In this thesis we stay at the first two levels, seeking for the basic representation of motor output. In this work we present a new model of motor primitives that comprises multiple interacting latent dynamical systems, and give it a full Bayesian treatment. Modelling within the Bayesian framework, in my opinion, must become the new standard in hypothesis testing in neuroscience. Only the Bayesian framework gives us guarantees when dealing with the inevitable plethora of hidden variables and uncertainty. The special type of coupling of dynamical systems we proposed, based on the Product of Experts, has many natural interpretations in the Bayesian framework. If the dynamical systems run in parallel, it yields Bayesian cue integration. If they are organized hierarchically due to serial coupling, we get hierarchical priors over the dynamics. If one of the dynamical systems represents sensory state, we arrive to the sensory-motor primitives. The compact representation that follows from the variational treatment allows learning of a motor primitives library. Learned separately, combined motion can be represented as a matrix of coupling values. We performed a set of experiments to compare different models of motor primitives. In a series of 2-alternative forced choice (2AFC) experiments participants were discriminating natural and synthesised movements, thus running a graphics Turing test. When available, Bayesian model score predicted the naturalness of the perceived movements. For simple movements, like walking, Bayesian model comparison and psychophysics tests indicate that one dynamical system is sufficient to describe the data. For more complex movements, like walking and waving, motion can be better represented as a set of coupled dynamical systems. We also experimentally confirmed that Bayesian treatment of model learning on motion data is superior to the simple point estimate of latent parameters. Experiments with non-periodic movements show that they do not benefit from more complex latent dynamics, despite having high kinematic complexity. By having a fully Bayesian models, we could quantitatively disentangle the influence of motion dynamics and pose on the perception of naturalness. We confirmed that rich and correct dynamics is more important than the kinematic representation. There are numerous further directions of research. In the models we devised, for multiple parts, even though the latent dynamics was factorized on a set of interacting systems, the kinematic parts were completely independent. Thus, interaction between the kinematic parts could be mediated only by the latent dynamics interactions. A more flexible model would allow a dense interaction on the kinematic level too. Another important problem relates to the representation of time in Markov chains. Discrete time Markov chains form an approximation to continuous dynamics. As time step is assumed to be fixed, we face with the problem of time step selection. Time is also not a explicit parameter in Markov chains. This also prohibits explicit optimization of time as parameter and reasoning (inference) about it. For example, in optimal control boundary conditions are usually set at exact time points, which is not an ecological scenario, where time is usually a parameter of optimization. Making time an explicit parameter in dynamics may alleviate this

Robust and efficient inference and learning algorithms for generative models

Generative modelling is a popular paradigm in machine learning due to its natural ability to describe uncertainty in data and models and for its applications including data compression (Ho et al., 2020), missing data imputation (Valera et al., 2018), synthetic data generation (Lin et al., 2020), representation learning (Kingma and Welling, 2014), robust classification (Li et al., 2019b), and more. For generative models, the task of finding the distribution of unobserved variables conditioned on observed ones is referred to as inference. Finding the optimal model that makes the model distribution close to the data distribution according to some discrepancy measures is called learning. In practice, existing learning and inference methods can fall short on robustness and efficiency. A method that is more robust to its hyper-parameters or different types of data can be more easily adapted to various real-world applications. How efficient a method is in regard to the size and the dimensionality of data determines at what scale the method can be applied. This thesis presents four pieces of my original work that improves these properties in generative models. First, I introduce two novel Bayesian inference algorithms. One is called coupled multinomial Hamiltonian Monte Carlo (Xu et al., 2021a); it builds on Heng and Jacob (2019), which is a recent work in unbiased Markov chain Monte Carlo (MCMC) (Jacob et al., 2019b) and has been found to sensitive to hyper-parameters and less efficient compared to normal, biased MCMC. These issues are solved by establishing couplings to the widely-used multinomial Hamiltonian Monte Carlo, leading to a statistically more efficient and robust method. The other method is called roulette-based variational expectation (RAVE; Xu et al., 2019) that applies amortised inference to a model family called Bayesian non-parametric models, in which the number of parameters are allowed to grow unbounded as the data gets more complex. Unlike previous sampling-based methods that are slow or variational inference methods that rely on truncation, RAVE combines the advantages of both to achieve flexible inference that is also computational efficient. Second, I introduce two novel learning methods. One is called generative ratio-matching (Srivastava et al., 2019) which is a learning algorithm that makes deep generative models based on kernel methods applicable to high-dimensional data. The key innovation of this method is learning a projection of the data to a lower-dimensional space in which the density ratio is preserved such that learning can be done in the lowerdimensional space where kernel methods are effective. The other method is called Bayesian symbolic physics that combines Bayesian inference and symbolic regression in the context of naïve physics—the study of how humans understand and learn physics. Unlike classic generative models for which the structure of the generative process is predefined or deep generative models where the process is represented by data-hungry neural networks, Bayesian-symbolic generative processes are defined by functions over a hypothesis space specified by a context-free grammar. This formulation allows these models to incorporate domain knowledge in learning, which gives highly-improved sample efficiency. For all four pieces of work, I provide theoretical analyses and/or empirical results to validate that the algorithmic advances lead to improvements in robustness and efficiency for generative models. Lastly, I summarise my contributions to free and open-source software on generative modelling. This includes a set of Julia packages that I contributed and are currently used by the Turing probabilistic programming language (Ge et al., 2018). These packages, which are highly reusable components for building probabilistic programming languages, together form a probabilistic programming ecosystem in Julia. An important package that is primarily developed by me is called ADVANCEDHMC.JL (Xu et al., 2020), which provides robust and efficient implementations of HMC methods and has been adopted as the backend of Turing. Importantly, the design of this package allows an intuitive abstraction to construct HMC samplers similarly to how they are mathematically defined. The promise of these open-source packages is to make generative modelling techniques more accessible to domain experts from various backgrounds and to make relevant research more reproducible to help advance the field