Neurons and the synaptic connections between them underlie the computational power of the brain. We present numerical models of neural behavior and show how to tune these models based on experimental evidence. Though the basic principles behind the creation and propagation of action potentials are understood, it is experimentally feasible to measure only a small number of the quantities that go into our models, substantially increasing the difficulty of making accurate predictions. Additionally, because biologically motivated models are very often nonlinear, we will focus on tools and techniques which do not require linearity.
We present novel methods of using time series of measurements to determine the features of nonlinear systems and predict their future behavior. We show how time-delayed coordinates can substitute for additional measurements and provide us with a better estimation of the state and parameters of the underlying system. A general expression for our objective function as a path integral is derived from probabilistic considerations and methods for evaluating the expression are discussed.
We demonstrate how the techniques developed can be used to determine properties of a biophysical system from a realistic set of limited measurements. We examine experimental electrophysiological recordings of zebra finch neurons and use them to hone the predictive powers of our models for single cells. Then, moving beyond the single cell level, we demonstrate how our approach can be used to determine changes in network connectivity due to synaptic plasticity in ways that direct experiment cannot.