Spectral Analysis for Neural Signals Bijan Pesaran PhD Center for Neural Science New York University New York New York 2008 Pesaran 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 the multitaper framework Thomson 1982 Percival and Walden 1993 The treatment here also draws on other sources Brillinger 1978 Jarvis and Mitra 2001 Mitra and Bokil 2007 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 Neuroimaging 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 aliasing 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 Motivation When a microelectrode is inserted into the brain the main features that are visible in the extracellular potential it measures are the spikes and the rhythms they ride on The extracellular potential results from 2008 Pesaran currents flowing in the extracellular space which in turn are produced by transmembrane potentials in local populations of neurons These cellular events can be fast around 1 ms for the action potentials that appear as spikes and slow up to 100 ms for the synaptic potentials that predominantly give rise to the LFP How spiking and LFP activity encode the sensory motor and cognitive processes that guide behavior and how these signals are related are fundamental open questions in neuroscience Steriade 2001 Buzsaki 2006 In this chapter we will illustrate these analysis techniques using recordings of spiking and LFP activity in macaque parietal cortex during the performance of a delayed look and reach movement to a peripheral target Pesaran et al 2002 This example should not be taken to limit the scope of potential applications and other presentations will motivate other examples The Basics of Signal Processing Spiking and LFP activity are two different kinds of time series and all neural signals fall into one of these two classes LFP activity is a continuous process and consists of a series of continuously varying voltages in time xt Spiking activity is a point process and assuming that all the spike events are identical consists of a sequence of spike times The counting process Nt is the total number of events that occur in the process up to a time t The mean rate of the process is given by the number of spike events divided by the duration of the interval If we consider a sufficiently short time interval t 1 ms either a spike event occurs or it does not Therefore we can represent a point process as the time derivative of the counting process dNt which gives a sequence of delta functions at the precise time of each spike tn We can also represent the process as a sequence of times between spikes n tn 1 tn which is called an interval process These representations are equivalent but capture different aspects of the spiking process We will focus on the counting process dNt dNt 1 t when there is a spike and dNt t elsewhere Note that these expressions correct for the mean firing rate of the process As we will see despite the differences between point and continuous processes spectral analysis treats them in a unified way The following sections present some basic notions that underlie statistical signal processing and time series analysis Fourier transforms Time series can be represented by decomposing them into a sum of elementary signals One domain for signals which we call the time domain is simply each point in time The time series is represented by its amplitude at each time point Another domain is the frequency domain and consists of sinusoidal 3 Notes 4 Notes functions one for each frequency The process is represented by its amplitude and phase at each frequency The time and frequency domains are equivalent and we can transform signals between them using the Fourier transform Fourier transforming a signal that is in the time domain xt will give the values of the signal in the frequency domain x f The tilde denotes a complex number with amplitude and phase N x f exp 2 iftn t 1 Inverse Fourier transforming x f transforms it to the time domain To preserve all the features in the process these transforms need to be carried out over an infinite time interval However this is never realized in practice Performing Fourier transforms on finite duration data segments distorts features in the signal and as we explain below spectral estimation employs data tapers to limit these distortions Nyquist frequency sampling theorem and aliasing Both point and continuous processes can be represented in the frequency domain When we sample a process by considering a sufficiently short interval in time t and measuring the voltage or