Learning Brain Dynamics With Coupled Low-Dimensional Nonlinear Oscillators and Deep Recurrent Networks

Abstract

Many natural systems, especially biological ones, exhibit complex multivariate nonlinear dynamical behaviors that can be hard to capture by linear autoregressive models. On the other hand, generic nonlinear models such as deep recurrent neural networks often require large amounts of training data, not always available in domains such as brain imaging; also, they often lack interpretability. Domain knowledge about the types of dynamics typically observed in such systems, such as a certain type of dynamical systems models, could complement purely data-driven techniques by providing a good prior. In this work, we consider a class of ordinary differential equation (ODE) models known as van der Pol (VDP) oscil lators and evaluate their ability to capture a low-dimensional representation of neural activity measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI, in different living organisms: larval zebrafish, rat, and human. We develop a novel and efficient approach to the nontrivial problem of parameters estimation for a network of coupled dynamical systems from multivariate data and demonstrate that the resulting VDP models are both accurate and interpretable, as VDP's coupling matrix reveals anatomically meaningful excitatory and inhibitory interactions across different brain subsystems. VDP outperforms linear autoregressive models (VAR) in terms of both the data fit accuracy and the quality of insight provided by the coupling matrices and often tends to generalize better to unseen data when predicting future brain activity, being comparable to and sometimes better than the recurrent neural networks (LSTMs). Finally, we demonstrate that our (generative) VDP model can also serve as a data-augmentation tool leading to marked improvements in predictive accuracy of recurrent neural networks. Thus, our work contributes to both basic and applied dimensions of neuroimaging: gaining scientific insights and improving brain-based predictive models, an area of potentially high practical importance in clinical diagnosis and neurotechnology.

Generative Models of Brain Dynamics

“…Interactions and connectivity can be observed in a wide set of settings from resting-state activity (Piccinini et al, 2021) to task-specific experiments (Pillai and Jirsa, 2017) by various imaging techniques including fMRI (Hutchison et al, 2013), EEG (Atasoy et al, 2018), MEG (Tait et al, 2021), and Calcium imaging (Abrevaya et al, 2021). In order to study segregation and integration of dynamics, networks of brain connectivity need to be constructed based on imaging data.…”

Generative Models of Brain Dynamics -- A review

“…Distinct dynamics is observed in a wide set of settings from resting-state activity [109] to task-specific experiments [116]. These insights are also useful in a wide set of modularities including fMRI [117], EEG [118], MEG [119], and Calcium imaging [120]. One common formulation is to build the dynamical graph models of the cortex based on the anatomical, functional, or effective connectivity as described in Table 1.…”

“…In addition to network science, another axis for interpreting neural data is based on well-established tools initially developed for parametrizing the time evolution of physical systems. Famous examples of these systems include spin-glass [135], different types of coupled oscillators [120,136], and multistable and chaotic many-body systems [109,114]. This type of modeling has already offered promising and intuitive results.…”

“…For more information about the Wilson-Cowan model, please see Subsection 1.3.2. Recently, [120] have used coupled VDP oscillators to model a low-dimensional representation of neural activity in different living organisms (larval zebrafish, rats, and humans) measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI. Besides proposing a method for inferring functional connectivity by using the coupling matrices of the fitted models, it was demonstrated that dynamical systems models could be a valuable resource of data augmentation for spatiotemporal deep learning problems.…”

“…In the last decade, RNN has been used for reconstructing neural dynamics via interpretable latent space in different recording modalities such as fMRI [157] and Calcium imaging [120]. Continuity of time is another extension that can make RNNs more compatible with various forms of sampling and thus neural dynamics from spikes to oscillations.…”