Bayesian Statistics seminars

In the current virtual journal club Dr Di Luca will present findings from a series of psychophysical investigations where he measured sensitivity and bias in the perception of the timing of stimuli. He will present how improved detection with longer sequences and biases in reporting isochrony can be accounted for by optimal statistical predictions. Among his findings was also that the timing of stimuli that occasionally deviate from a regularly paced sequence is perceptually distorted to appear more regular. Such change depends on whether the context these sequences are presented is also regular. Dr Di Luca will present a Bayesian model for the combination of dynamically updated expectations, in the form of a priori probability, with incoming sensory information. These findings contribute to the understanding of how the brain processes temporal information to shape perceptual experiences.

Over the past decade we have demonstrated that the fusion of subject-specific structural information of the human brain with mathematical dynamic models allows building biologically realistic brain network models, which have a predictive value, beyond the explanatory power of each approach independently. The network nodes hold neural population models, which are derived using mean field techniques from statistical physics expressing ensemble activity via collective variables. Our hybrid approach fuses data-driven with forward-modeling-based techniques and has been successfully applied to explain healthy brain function and clinical translation including aging, stroke and epilepsy. Here we illustrate the workflow along the example of epilepsy: we reconstruct personalized connectivity matrices of human epileptic patients using Diffusion Tensor weighted Imaging (DTI). Subsets of brain regions generating seizures in patients with refractory partial epilepsy are referred to as the epileptogenic zone (EZ). During a seizure, paroxysmal activity is not restricted to the EZ, but may recruit other healthy brain regions and propagate activity through large brain networks. The identification of the EZ is crucial for the success of neurosurgery and presents one of the historically difficult questions in clinical neuroscience. The application of latest techniques in Bayesian inference and model inversion, in particular Hamiltonian Monte Carlo, allows the estimation of the EZ, including estimates of confidence and diagnostics of performance of the inference. The example of epilepsy nicely underwrites the predictive value of personalized large-scale brain network models. The workflow of end-to-end modeling is an integral part of the European neuroinformatics platform EBRAINS and enables neuroscientists worldwide to build and estimate personalized virtual brains.

Mobile organisms must be capable of deciding both where and when to move in order to keep up with a changing environment; therefore, a strong sense of time is necessary, otherwise, we would fail in many of our movement goals. Despite this intrinsic link between movement and timing, only recently has research begun to investigate the interaction. Two primary effects that have been observed include: movements biasing time estimates (i.e., affecting accuracy) as well as making time estimates more precise. The goal of this presentation is to review this literature, discuss a Bayesian cue combination framework to explain these effects, and discuss the experiments I have conducted to test the framework. The experiments herein include: a motor timing task comparing the effects of movement vs non-movement with and without feedback (Exp. 1A & 1B), a transcranial magnetic stimulation (TMS) study on the role of the supplementary motor area (SMA) in transforming temporal information (Exp. 2), and a perceptual timing task investigating the effect of noisy movement on time perception with both visual and auditory modalities (Exp. 3A & 3B). Together, the results of these studies support the Bayesian cue combination framework, in that: movement improves the precision of time perception not only in perceptual timing tasks but also motor timing tasks (Exp. 1A & 1B), stimulating the SMA appears to disrupt the transformation of temporal information (Exp. 2), and when movement becomes unreliable or noisy there is no longer an improvement in precision of time perception (Exp. 3A & 3B). Although there is support for the proposed framework, more studies (i.e., fMRI, TMS, EEG, etc.) need to be conducted in order to better understand where and how this may be instantiated in the brain; however, this work provides a starting point to better understanding the intrinsic connection between time and movement

Efficient navigation requires animals to track their position, velocity and heading direction (HD). Some animals’ behavior suggests that they also track uncertainties about these navigational variables, and make strategic use of these uncertainties, in line with a Bayesian computation. Ring-attractor networks have been proposed to estimate and track these navigational variables, for instance in the HD system of the fruit fly Drosophila. However, such networks are not designed to incorporate a notion of uncertainty, and therefore seem unsuited to implement dynamic Bayesian inference. Here, we close this gap by showing that specifically tuned ring-attractor networks can track both a HD estimate and its associated uncertainty, thereby approximating a circular Kalman filter. We identified the network motifs required to integrate angular velocity observations, e.g., through self-initiated turns, and absolute HD observations, e.g., visual landmark inputs, according to their respective reliabilities, and show that these network motifs are present in the connectome of the Drosophila HD system. Specifically, our network encodes uncertainty in the amplitude of a localized bump of neural activity, thereby generalizing standard ring attractor models. In contrast to such standard attractors, however, proper Bayesian inference requires the network dynamics to operate in a regime away from the attractor state. More generally, we show that near-Bayesian integration is inherent in generic ring attractor networks, and that their amplitude dynamics can account for close-to-optimal reliability weighting of external evidence for a wide range of network parameters. This only holds, however, if their connection strengths allow the network to sufficiently deviate from the attractor state. Overall, our work offers a novel interpretation of ring attractor networks as implementing dynamic Bayesian integrators. We further provide a principled theoretical foundation for the suggestion that the Drosophila HD system may implement Bayesian HD tracking via ring attractor dynamics.

A central goal of vision science is to understand the principles underlying the perception and neural coding of the complex visual environment of our everyday experience. In the visual cortex, foundational work with artificial stimuli, and more recent work combining natural images and deep convolutional neural networks, have revealed much about the tuning of cortical neurons to specific image features. However, a major limitation of this existing work is its focus on single-neuron response strength to isolated images. First, during natural vision, the inputs to cortical neurons are not isolated but rather embedded in a rich spatial and temporal context. Second, the full structure of population activity—including the substantial trial-to-trial variability that is shared among neurons—determines encoded information and, ultimately, perception. In the first part of this talk, I will argue for a normative approach to study encoding of natural images in primary visual cortex (V1), which combines a detailed understanding of the sensory inputs with a theory of how those inputs should be represented. Specifically, we hypothesize that V1 response structure serves to approximate a probabilistic representation optimized to the statistics of natural visual inputs, and that contextual modulation is an integral aspect of achieving this goal. I will present a concrete computational framework that instantiates this hypothesis, and data recorded using multielectrode arrays in macaque V1 to test its predictions. In the second part, I will discuss how we are leveraging this framework to develop deep probabilistic algorithms for natural image and video segmentation.

Behavior relies on the ability of sensory systems to infer changing properties of the environment from incoming sensory stimuli. However, the demands that detecting and adjusting to changes in the environment place on a sensory system often differ from the demands associated with performing a specific behavioral task. This necessitates neural coding strategies that can dynamically balance these conflicting needs. I will discuss our ongoing theoretical work to understand how this balance can best be achieved. We connect ideas from efficient coding and Bayesian inference to ask how sensory systems should dynamically allocate limited resources when the goal is to optimally infer changing latent states of the environment, rather than reconstruct incoming stimuli. We use these ideas to explore dynamic tradeoffs between the efficiency and speed of sensory adaptation schemes, and the downstream computations that these schemes might support. Finally, we derive families of codes that balance these competing objectives, and we demonstrate their close match to experimentally-observed neural dynamics during sensory adaptation. These results provide a unifying perspective on adaptive neural dynamics across a range of sensory systems, environments, and sensory tasks.

To act effectively and flexibly in an imperfectly predictable environment with only incomplete and unreliable sensory information, animals must learn to form and compute with internal representations that reflect their necessarily uncertain beliefs about the state of the world. The optimal approach to handling uncertainty is rooted in Bayesian probability, and indeed humans and other animals often approach Bayes optimality with a degree of robustness and flexibility that continues to evade artificial systems. However, the question of how neural circuits organise to achieve this performance remains one of the fundamental mysteries of neuroscience. I will discuss a series of models built around the idea that distributional information is naturally encoded in a distributed fashion by neural population firing rates that converge on the mean values of non-linear functions of state. We will see that such representations emerge naturally in task-optimised systems, and also provide a simple and effective substrate for unsupervised learning. Finally, I will sketch ongoing work that links the emergence of such representations to the architecture of recurrent neural circuits. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2021-07-07. Recording duration: 00:52:28.

Animals possess the ability to effortlessly and precisely time their actions even though information received from the world is often ambiguous and is inadvertently transformed as it passes through the nervous system. With such uncertainty pervading through our nervous systems, we could expect that much of human and animal behavior relies on inference that incorporates an important additional source of information, prior knowledge of the environment. These concepts have long been studied under the framework of Bayesian inference with substantial corroboration over the last decade that human time perception is consistent with such models. We, however, know little about the neural mechanisms that enable Bayesian signatures to emerge in temporal perception. I will present our work on three facets of this problem, how Bayesian estimates are encoded in neural populations, how these estimates are used to generate time intervals, and how prior knowledge for these tasks is acquired and optimized by neural circuits. We trained monkeys to perform an interval reproduction task and found their behavior to be consistent with Bayesian inference. Using insights from electrophysiology and in silico models, we propose a mechanism by which cortical populations encode Bayesian estimates and utilize them to generate time intervals. Thereafter, I will present a circuit model for how temporal priors can be acquired by cerebellar machinery leading to estimates consistent with Bayesian theory. Based on electrophysiology and anatomy experiments in rodents, I will provide some support for this model. Overall, these findings attempt to bridge insights from normative frameworks of Bayesian inference with potential neural implementations for the acquisition, estimation, and production of timing behaviors.

Ideas about optimization are at the core of how we approach biological complexity. Quantitative predictions about biological systems have been successfully derived from first principles in the context of efficient coding, metabolic and transport networks, evolution, reinforcement learning, and decision making, by postulating that a system has evolved to optimize some utility function under biophysical constraints. Yet as normative theories become increasingly high-dimensional and optimal solutions stop being unique, it gets progressively hard to judge whether theoretical predictions are consistent with, or "close to", data. I will illustrate these issues using efficient coding applied to simple neuronal models as well as to a complex and realistic biochemical reaction network. As a solution, we developed a statistical framework which smoothly interpolates between ab initio optimality predictions and Bayesian parameter inference from data, while also permitting statistically rigorous tests of optimality hypotheses.

Bursts in gamma and other frequency ranges are thought to contribute to the efficiency of working memory or communication tasks. Abnormalities in bursts have also been associated with motor and psychiatric disorders. The determinants of burst generation are not known, specifically how single cell and connectivity parameters influence burst statistics and the corresponding brain states. We first present a generic mathematical model for burst generation in an excitatory-inhibitory (EI) network with self-couplings. The resulting equations for the stochastic phase and envelope of the rhythm’s fluctuations are shown to depend on only two meta-parameters that combine all the network parameters. They allow us to identify different regimes of amplitude excursions, and to highlight the supportive role that network finite-size effects and noisy inputs to the EI network can have. We discuss how burst attributes, such as their durations and peak frequency content, depend on the network parameters. In practice, the problem above follows the a priori challenge of fitting such E-I spiking networks to single neuron or population data. Thus, the second part of the talk will discuss a novel method to fit mesoscale dynamics using single neuron data along with a low-dimensional, and hence statistically tractable, single neuron model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous ‘pools’ of neurons, and modelling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to either optimize parameters by gradient ascent on the log-likelihood, or to perform Bayesian inference using Markov Chain Monte Carlo (MCMC) sampling. We illustrate this approach using an E-I network of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived. We show that both single-neuron and connectivity parameters can be adequately recovered from simulated data.