Artificial and biological neural networks: Using data to inform dynamics

Pake Melland, Math, SMU

Using dynamic models to describe physical processes, such as a ball swinging from a rope, can be traced back to Newton’s laws of motion.  Classically, governing equations derived from first principles allow researchers to make theoretical statements using analytic or numeric methods.  However, for many modern applications in the sciences the equations describing a system may be incomplete or entirely unknown.  The lack of explicit equations leaves researchers reliant on observations or measurements to provide information about the underlying system.  

We will discuss two applications of data-informed modeling and analysis in different scientific domains.  The first application embedded an artificial neural network (ANN) as a component in a 1D Lagrangian numerical scheme to attenuate non-physical oscillations that arise in shock problems.  The neural network training took place in an ‘online’ setting in which differentiable programming, a paradigm for differentiating through generic computer code, was used to calculate the gradients necessary for network parameter tuning. 

The second application invoked methods from data-driven dynamical systems and dimensionality reduction to study electrocorticography (ECoG) data from human subjects listening to a perceptually bistable auditory stimulus.  Here, approximated eigenfunctions of the Koopman operator provided timeseries features equal in length to the duration of the auditory stimulus (5-minutes).  A subset of these features exhibited modulations between apparent attracting states and provided neural evidence supporting attractor-based principles in computational models of perception.