6 research outputs found
Tree-Structured Recurrent Switching Linear Dynamical Systems for Multi-Scale Modeling
Many real-world systems studied are governed by complex, nonlinear dynamics.
By modeling these dynamics, we can gain insight into how these systems work,
make predictions about how they will behave, and develop strategies for
controlling them. While there are many methods for modeling nonlinear dynamical
systems, existing techniques face a trade off between offering interpretable
descriptions and making accurate predictions. Here, we develop a class of
models that aims to achieve both simultaneously, smoothly interpolating between
simple descriptions and more complex, yet also more accurate models. Our
probabilistic model achieves this multi-scale property through a hierarchy of
locally linear dynamics that jointly approximate global nonlinear dynamics. We
call it the tree-structured recurrent switching linear dynamical system. To fit
this model, we present a fully-Bayesian sampling procedure using Polya-Gamma
data augmentation to allow for fast and conjugate Gibbs sampling. Through a
variety of synthetic and real examples, we show how these models outperform
existing methods in both interpretability and predictive capability
Nonlinear control in the nematode C. elegans
Recent whole-brain calcium imaging recordings of the nematode C. elegans have
demonstrated that neural activity is dominated by dynamics on a low-dimensional
manifold that can be clustered according to behavioral states. Despite progress
in modeling the dynamics with linear or locally linear models, it remains
unclear how a single network of neurons can produce the observed features. In
particular, there are multiple clusters, or fixed points, observed in the data
which cannot be characterized by a single linear model. We propose a nonlinear
control model which is global and parameterized by only four free parameters
that match the features displayed by the low-dimensional C. elegans neural
activity. In addition to reproducing the average probability distribution of
the data, long and short time-scale changes in transition statistics can be
characterized via changes in a single parameter. Some of these macro-scale
transitions have experimental correlates to single neuro-modulators that seem
to act as biological controls, allowing this model to generate testable
hypotheses about the effect of these neuro-modulators on the global dynamics.
The theory provides an elegant characterization of the neuron population
dynamics in C. elegans. Moreover, the mathematical structure of the nonlinear
control framework provides a paradigm that can be generalized to more complex
systems with an arbitrary number of behavioral states
Investigating naturalistic hand movements by behavior mining in long-term video and neural recordings
Recent technological advances in brain recording and artificial intelligence
are propelling a new paradigm in neuroscience beyond the traditional controlled
experiment. Rather than focusing on cued, repeated trials, naturalistic
neuroscience studies neural processes underlying spontaneous behaviors
performed in unconstrained settings. However, analyzing such unstructured data
lacking a priori experimental design remains a significant challenge,
especially when the data is multi-modal and long-term. Here we describe an
automated approach for analyzing simultaneously recorded long-term,
naturalistic electrocorticography (ECoG) and naturalistic behavior video data.
We take a behavior-first approach to analyzing the long-term recordings. Using
a combination of computer vision, discrete latent-variable modeling, and string
pattern-matching on the behavioral video data, we find and annotate spontaneous
human upper-limb movement events. We show results from our approach applied to
data collected for 12 human subjects over 7--9 days for each subject. Our
pipeline discovers and annotates over 40,000 instances of naturalistic human
upper-limb movement events in the behavioral videos. Analysis of the
simultaneously recorded brain data reveals neural signatures of movement that
corroborate prior findings from traditional controlled experiments. We also
prototype a decoder for a movement initiation detection task to demonstrate the
efficacy of our pipeline as a source of training data for brain-computer
interfacing applications. Our work addresses the unique data analysis
challenges in studying naturalistic human behaviors, and contributes methods
that may generalize to other neural recording modalities beyond ECoG. We
publicly release our curated dataset, providing a resource to study
naturalistic neural and behavioral variability at a scale not previously
available
GP-Tree: A Gaussian Process Classifier for Few-Shot Incremental Learning
Gaussian processes (GPs) are non-parametric, flexible, models that work well
in many tasks. Combining GPs with deep learning methods via deep kernel
learning (DKL) is especially compelling due to the strong representational
power induced by the network. However, inference in GPs, whether with or
without DKL, can be computationally challenging on large datasets. Here, we
propose GP-Tree, a novel method for multi-class classification with Gaussian
processes and DKL. We develop a tree-based hierarchical model in which each
internal node of the tree fits a GP to the data using the P\'olya Gamma
augmentation scheme. As a result, our method scales well with both the number
of classes and data size. We demonstrate the effectiveness of our method
against other Gaussian process training baselines, and we show how our general
GP approach achieves improved accuracy on standard incremental few-shot
learning benchmarks
Variational Dynamic Mixtures
Deep probabilistic time series forecasting models have become an integral
part of machine learning. While several powerful generative models have been
proposed, we provide evidence that their associated inference models are
oftentimes too limited and cause the generative model to predict mode-averaged
dynamics. Modeaveraging is problematic since many real-world sequences are
highly multi-modal, and their averaged dynamics are unphysical (e.g., predicted
taxi trajectories might run through buildings on the street map). To better
capture multi-modality, we develop variational dynamic mixtures (VDM): a new
variational family to infer sequential latent variables. The VDM approximate
posterior at each time step is a mixture density network, whose parameters come
from propagating multiple samples through a recurrent architecture. This
results in an expressive multi-modal posterior approximation. In an empirical
study, we show that VDM outperforms competing approaches on highly multi-modal
datasets from different domains
Graph Gamma Process Generalized Linear Dynamical Systems
We introduce graph gamma process (GGP) linear dynamical systems to model
real-valued multivariate time series. For temporal pattern discovery, the
latent representation under the model is used to decompose the time series into
a parsimonious set of multivariate sub-sequences. In each sub-sequence,
different data dimensions often share similar temporal patterns but may exhibit
distinct magnitudes, and hence allowing the superposition of all sub-sequences
to exhibit diverse behaviors at different data dimensions. We further
generalize the proposed model by replacing the Gaussian observation layer with
the negative binomial distribution to model multivariate count time series.
Generated from the proposed GGP is an infinite dimensional directed sparse
random graph, which is constructed by taking the logical OR operation of
countably infinite binary adjacency matrices that share the same set of
countably infinite nodes. Each of these adjacency matrices is associated with a
weight to indicate its activation strength, and places a finite number of edges
between a finite subset of nodes belonging to the same node community. We use
the generated random graph, whose number of nonzero-degree nodes is finite, to
define both the sparsity pattern and dimension of the latent state transition
matrix of a (generalized) linear dynamical system. The activation strength of
each node community relative to the overall activation strength is used to
extract a multivariate sub-sequence, revealing the data pattern captured by the
corresponding community. On both synthetic and real-world time series, the
proposed nonparametric Bayesian dynamic models, which are initialized at
random, consistently exhibit good predictive performance in comparison to a
variety of baseline models, revealing interpretable latent state transition
patterns and decomposing the time series into distinctly behaved sub-sequences.Comment: 36 pages, 10 figure