Application of Spectral Methods to Representative Data Sets in Electrophysiology and Functional Neuroimaging David Kleinfeld PhD Department of Physics and Graduate Program in Neurosciences University of California La Jolla California 2008 Kleinfeld Application of Spectral Methods to Representative Data Sets in Electrophysiology and Functional Neuroimaging Background Experimental neuroscience involves the use of many different tools from physics genetics and other fields Proper data analysis is an integral part of this set of tools Used correctly analysis will help to define the magnitude and significance of a given neural effect used creatively analysis can help reveal new phenomena In this chapter we consider five example cases that introduce the utility and implementation of spectral methods 1 Deduction of variation in the power of a high frequency cortical oscillation from a human electrocorticogram ECoG This will illustrate frequency domain concepts such as the spectrogram 2 Deduction of synaptic connectivity between neurons in the leech swim network This will emphasize notions of spectral coherence and its associated confidence level 3 The discovery of neurons in rat vibrissa motor cortex that report the pitch of vibrissa movement This will illustrate the notion of the spectral power density as the sum of functions corresponding to pure tones and a slowly varying or pink spectrum 4 The denoising of imaging data in the study of calcium waves This will introduce the concept of singular value decomposition SVD in the time domain and illustrate the notion of spacetime correlation in multisite measurements 5 The delineation of wave phenomena in turtle cortex This will illustrate the concept of SVD in the frequency domain and further illustrate the notion of space time coherence Throughout this discussion the emphasis will be on explaining the analysis and not on the scientific questions per se Advantages of Working in the Frequency Domain Why work in the frequency domain One part of the answer is to delineate the number of degrees of freedom required to calculate confidence intervals The following factors are relevant Determining the number of degrees of freedom is complicated in the time domain where all but white noise processes lead to correlation between neighboring data points In contrast counting the number of degrees of freedom is straightforward when neighbor 2008 Kleinfeld ing data points are uncorrelated This occurs in the frequency domain when the amplitude of spectral power in the data varies only slowly on the scale of the bandwidth so that neighboring points in frequency are uncorrelated A second part of the answer is that some phenomena have a simpler representation in the frequency domain rather than the time domain This chapter builds on the discussion of the timebandwidth product and multitaper analysis Thomson 1982 in Spectral Analysis for Neural Signals presented earlier in this Short Course by Bijan Pesaran First we recall the time frequency uncertainty T f 2p where T is the total length of the time series of the data 2p is the number of degrees of freedom and defines the time frequency product with p 1 and f is the resultant full bandwidth The power is concentrated in the frequency interval f optimally so for the use of family of Slepian tapers employed to estimate spectra Thomson 1982 The maximum number of tapers denoted K that supports this concentration and which is employed throughout our presentation is as follows K 2p 1 Rosetta Stone The variables used in the Pesaran chapter herein and past reviews Mitra and Pesaran 1998 Mitra et al 1999 expressed in discrete normalized variables by writing 1 T N t where N is the number of data points in the time series and t 1 2fNyquist is the sample time and 2 2W t f where W is the half bandwidth Thus NW p In normalized variables time spans the interval 1 N rather than t T frequency spans the interval rather than fNyquist fNyquist and the integral is replaced by with t 1 Tools for the numerical calculations used in the examples below are part of the Matlab based Chronux package www chronux org The primary textbooks include those by Percival and Walden 1993 and Mitra and Bokil 2008 Case one We analyze a trace of human ECoG data defined as V t that was obtained in a study on ultra slow brain activity Nir et al 2008 Fig 1A The mean value is removed to form the following 23 Notes 24 Notes Our goal is to understand the spectral content of this signal with confidence limits The Fourier transform of this signal with respect to the kth taper is as follows where w k t is the kth Slepian taper whose length is also T We then compute the spectral power density whose units are amplitude2 Hz in terms of an average over tapers where we further average the results over all trials if appropriate The above normalization satisfies Parseval s theorem i e The spectrum in this example is featureless having a hint of extra power between 50 Hz and 100 Hz Fig 1B One possibility is that the spectrum is better defined on a short time scale but drifts Fig 1B insert In this case it is useful to compute the running spectrum or spectrogram denoted S f t which is a function of both frequency and time Here we choose a narrow interval of time compute the spectrum over that interval and then step forward in time and recalculate the spectrum For the example data this process reveals an underlying modulation in the power between 40 Hz and 90 Hz Fig 1C the so called band How do we characterize the band s variations in power We treat the logarithm of the power in a band as a new signal found by integrating the spectrogram over the frequency the new time series are called the second spectrum denoted S f for this example The above formula shows a number of spectral features Fig 1E and raises two general issues The first issue is the calculation of confidence intervals For variables with a Gaussian dependence on their individual spectral amplitudes the confidence limits may be estimated in various asymptotic limits However the confidence intervals may also be estimated directly by a jackknife Thomson and Chave 1991 where we compute the standard error in terms of delete one means In this procedure we calculate K different mean spectra in which one term is left out Estimating the standard error of S f requires an extra step since spectral amplitudes are defined on the interval 0 while Gaussian variables exist on The delete one estimates Ci n f were replaced