Spectral Analysis

Neural signals such as the local field potential are rich in oscillatory structure. Which rhythms are present, how strong they are, when they appear, and whether two brain regions oscillate together are all questions that are most naturally answered in the frequency domain rather than in the raw voltage trace.

This unit builds up the toolbox for spectral analysis across three connected sessions, moving from constructed toy signals to synthetic V1 and V4 LFP recordings of a stimulus-presentation experiment.

In the homework, you will get hands-on practice with the core methods by working entirely with toy signals you build yourself: sinusoids, chirps, and sums of them. Because you decide exactly which frequencies go into each signal, this is a good way to check that an analysis tool recovers what you expect before applying it to real data. You will compute power spectra with the Fourier transform, track time-varying frequency content with Morlet wavelets, and estimate the spectra of noisy signals with multitapering.

The first in-person session takes these tools further. You will see how the sampling rate and recording duration set the Nyquist frequency and the frequency resolution of a Fourier spectrum, use the power spectral density and Parseval’s theorem as a sanity check on your spectra, and reduce variance by averaging power across windows. Because the Fourier transform assumes stationary frequency content, you will then turn to the Morlet wavelet transform for time-resolved spectra, learn how the wavelet bandwidth trades frequency resolution against time resolution, and use the cone of influence to mark the edge-artifacts you should not trust. The session ends with wavelet spectrograms of real LFP data around stimulus onset.

The final session asks how to tell whether two brain regions are coupled. You will compute spectral coherence, a frequency- and lag-resolved analog of correlation built on the wavelet transform, use the Hilbert transform to extract a signal’s instantaneous amplitude and phase, and summarise how consistently two signals hold a fixed phase relationship with the phase locking value. Along the way you will examine how much data is needed for a reliable PLV estimate and apply these measures to the V1-V4 LFP recordings.

The aim of the unit is not a fully formal treatment of every method, but to build practical intuition for what each spectral tool can and cannot tell us, and for the choices, such as window length, wavelet bandwidth, and filter width, that shape the answer.