Spectral Analysis for Neural Signals

Introduction This chapter introduces concepts fundamental to spectral analysis and applies spectral analysis to characterize neural signals. Spectral analysis is a form of time series analysis and concerns a series of events or measurements that are ordered in time. The goal of such an analysis is to quantitatively characterize the relationships between events and measurements in a time series. This quantitative characterization is needed to derive statistical tests that determine how time series differ from one another and how they are related to one another. Time series analysis comprises two main branches: time-domain methods and frequency-domain methods. Spectral analysis is a frequency-domain method for which we will use In this chapter, we will focus on relationships within and between one and two time series, known as univariate and bivariate time series. Throughout this discussion, we will illustrate the concepts with experimental recordings of spiking activity and local field potential (LFP) activity. The chapter on Multivariate Neural Data Sets will extend the treatment to consider several simultaneously acquired time series that form a multivariate time series, such as in imaging experiments. The chapter on " Application of Spectral Methods to Representative Data Sets in Electrophysiology and Functional Neuro-imaging " will review some of this material and present additional examples. First we begin by motivating a particular problem in neural signal analysis that frames the examples in this chapter. Second, we introduce signal processing and the Fourier transform and discuss practical issues related to signal sampling and the problem of alias-ing. Third, we present stochastic processes and their characterization through the method of moments. The moments of a stochastic process can be characterized in both the time domains and frequency domains, and we will discuss the relation between these characterizations. Subsequently, we present the problem of scientific inference, or hypothesis testing , in spectral analysis through the consideration of error bars. We finish by considering an application of spectral analysis involving regression.