Data-Driven Modeling for Neuronal Dynamics: Discovering Single-Neuron Models from Time-Series of Voltage

Abstract

The human brain functions on electrical signals, the fundamental unit of which is the action potential of a single neuron, which is a rapid change in its membrane voltage in response to an external current stimulus. This behavior of the biological neuron has been extensively studied and various models have been proposed to describe the underlying mechanisms that produce it, as well as predict its dynamics given input stimuli. These models range from detailed biophysical models to more abstract phenomenological models that perform better in fitting real neuronal data than their highly parameterized biophysical counterparts.

Recently, a new approach has emerged for model design, where sparse, mathematically tractable models that are amenable to interpretation are algorithmically identified from data. This thesis explores the application of data-driven modeling techniques to the problem of finding a predictive and interpretable model for neuronal firing. We first explore the possibility of learning parameters of the Fitzhugh-Nagumo and the classical Hodgkin-Huxley models by employing physics-informed neural networks. We then consider an idealized case where full-state measurements of the Hodgkin-Huxley model are available, and explore the application of the sparse identification of nonlinear dynamics on the simulated data. Finally, we rely on Takens' time delay embedding theorem to simultaneously discover hidden variables of neuronal firing models and a sparse representation of their dynamics using time delays of a voltage time series generated from the Fitzhugh-Nagumo model. We analyze the discovered models and explore their potential as excitable or oscillatory neuronal models.