Challenges in Modeling Nonlinear Systems: A Worked Example

The interaction between nonlinear dynamics and statistics has been rather limited over the two decades of concentrated work in nonlinear systems by physicat and biological scientists. This chapter is meant to be a contribution to stimulating that interaction by presenting a discussion of a problem in biology which is addressed by tools of nonlinear dynamics and by posing, along the way, issues of statistical relevance not answered by the community of nonlinear dynamicists. The overall issue is that of characterizing and modeling nonlinear systems using observed data. Typically this is in the initial absence of a model for the source of the data, but that often is the goal of the analysis. Models derived from these data can be black box or analytic. Black box models typically consist of a set of numerical rules for prediction or control of the system in the absence of any fundamental knowledge of the physics or biology of the system. Analytic models attempt to incorporate knowledge from the observations and their analysis into sets of differential equations or maps embodying the properties of the measured processes. In this chapter we focus on the analysis of membrane voltage data from identified neurons of the stomatogastric ganglion of the California spiny lobster with the goal of modeling individual neurons and their oscillatory behavior in a variety of environmental circumstances. The membrane voltage dynamics of these neurons is typically low dimensional and chaotic. Hodgkin-Huxley models describing the ion currents which flow through the membrane are not sufficient to capture this behavior, but the addition of a slow background dynamics, which we attribute to the storage and release of calcium in the cell, permits an excellent description of the observations. We will describe the experiments, the analysis of the data, and the model building connected with these statements, and hopefully we’ll leave the reader with the sense that much has been done, but much more is required to transform what seems to be a working set of usable implements into a scientifically sharp collection of tools.