Dynamical Inference and Representations in Complex Biological Systems Open Access
Calderon, Josuan (Spring 2024)
Abstract
This thesis introduces a comprehensive analytical framework that combines Temporal Autoencoders for Causal Inference (TACI) with a new pipeline for discovering and analyzing behavioral states in complex dynamical systems. Although these two methodologies have different purposes, they both focus on comprehending time-dependent phenomena through the lens of time series and temporal interactions, which are omnipresent in the natural world.
We begin with TACI, a novel methodology designed to analyze time-varying causal interactions within complex dynamical systems. Traditional approaches to causality often fall short when faced with non-linear, non-stationary interactions between system variables. To address these challenges, TACI leverages a novel metric, the Comparative Surrogate Granger Index (CGSI), alongside a two-headed Temporal Convolutional Network (TCN) autoencoder architecture. Through tests on both synthetic and real-world datasets, we demonstrate TACI’s ability to accurately quantify dynamic causal interactions across a variety of systems. Our findings display the method’s effectiveness compared to existing approaches and also enhance our understanding of time-varying interactions in various domains, from physical to biological systems. Through this work, TACI emerges as a significant advancement in the field of causal inference, promising to deepen our comprehension of dynamic systems across a range of scientific disciplines.
The thesis also explores the intricate dynamics of behavioral states using a comprehensive analytical framework rooted in proven methods of dynamical systems and advanced computational techniques. By integrating wavelet transforms with autoencoders, followed by predictive modeling using Long Short-Term Memory (LSTM) networks and dimensionality reduction via t-distributed stochastic neighbor embedding (t-SNE), we offer novel insights into the temporal and spectral characteristics of behavior. The use of LSTM networks to model the temporal sequences of behavioral states aims to predict future states and identify stable points within the system's dynamics. These fixed points are then mapped into a two-dimensional space using t-SNE, creating a visual landscape of behavioral basins of attraction. This visualization not only simplifies the interpretation of behavioral dynamics but also reveals the underlying structure and transitions between states, highlighting areas of stability and potential pathways for state changes. Our findings highlight the stability and fluidity of behavioral states, providing insights into the mechanisms governing behavioral transitions. The identification of basins of attraction and the hierarchical organization of behaviors suggest that complex behaviors may be constructed from simpler, foundational actions.
The thesis successfully demonstrates how the TACI methodology and the behavioral states pipeline provide an extensive strategy to understand dynamical systems. Together, they offer novel insights into the behavior and causality within these systems, highlighting the fluidity and stability of behavioral states, and providing a deeper understanding of the mechanisms driving transitions. This unified approach not only advances our understanding of individual systems but also offers a broader perspective on the temporal interactions that shape the complexity of the natural world, as exemplified by its application to diverse datasets including climate patterns, neural activities in monkeys, and the behaviors of rats.
Table of Contents
Quantifying temporal dynamics in biological time series 1
Introduction 1
Causality 11
Challenges in Causal Inference 12
The Dynamics of Behavior 15
Ethology and Tinbergen’s Four Questions 15
Measuring Behavior 17
Stereotyped Behavior 18
Thesis Outline 19
Background Information 22
Introduction 22
Methods for Causal Inference Analysis 23
Correlation 23
Granger Causality 28
Information theory as a tool for causality detection 33
Cross Mappings 39
Causal Neural Networks 43
Neural Granger Causality 46
TCN Introduction 52
Temporal Causal Discovery Framework 60
Methods for Behavioral States Discovery 64
Dynamical systems 64
Fixed Points 66
Recurrent Neural Networks 67
Wavelet Transform 73
Autoencoders as a dimensionality reduction technique 75
Spatial embedding 76
Inferring time-varying coupling of dynamical systems with temporal convolutional network autoencoders 78
Introduction 78
Overview of Methodology 80
Comparative Surrogate Granger Index (CSGI) 80
Temporal Autoencoders for Causal Inference 82
Other Methods We Compare Against 86
Results 87
Artificial Test Systems 87
Jena Climate Dataset 95
Electrocorticography in Non-Human Primates 97
Discussion 100
Materials and methods 103
Architecture 104
Training and Prediction 106
TACI Network Parameters 108
Building emergent representation of behavioral states using dynamical models 109
Introduction 109
Methods 110
Recurrent Neural Networks Fixed Points 112
Data 115
Analysis Pipeline 116
Results 128
Embedded Space Dynamics 128
Transition Matrices 129
Predictability and Hierarchy 131
Discussion 132
Conclusion and Future Directions 134
Thesis Contributions 134
Summaries of Chapters 136
Chapter 1 summary 136
Chapter 2 summary 137
Chapter 3 summary 138
Chapter 4 summary 139
Limitations 140
TACI and causal inference 140
Fixed Points Pipeline 141
Future Directions 142
Brain connectivity of prairie voles during social bonding 142
Brain States 143
Bibliography 145
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