#### Transcript LFPs 1: spectral analysis

LFPs 1: Spectral analysis Kenneth D. Harris 11/2/15 Local field potentials • Slow component of intracranial electrical signal • Physical basis for scalp EEG Today we will talk about • Physical basis of the LFP • Current-source density analysis • Some math (signal processing theory, Gaussian processes) • Spectral analysis Physical basis of the LFP signal • Kirchoff’s current law: • Current flowing into any location balances current flowing out of it. Synaptic input • Extracellular space is resistive • Ohm’s law applied to return current: Return current 𝑑𝑉 = −𝜌𝐼𝐸 𝑑𝑧 Charging current (capacitive) plus leak current (resistive) • Assumes uniformity across x and y Intracellular current Linear probe recordings • Record 𝑉1 , 𝑉2 , … 𝑉𝑛 , spacing Δ𝑧 • Extracellular current 𝑑𝑉 𝑉𝑛+1 − 𝑉𝑛 𝐼𝐸 ∝ ≈− 𝑑𝑧 Δ𝑧 • Intracellular current 𝐼𝐼 ∝ −𝐼𝐸 • Current source density (CSD) 𝑑𝐼 𝑑 2 𝑉 𝑉𝑛+1 − 2𝑉𝑛 + 𝑉𝑛−1 ∝ 2 ≈− 𝑑𝑧 𝑑𝑧 Δ𝑧 2 Spatial interpolation • To make nicer figures, interpolate before taking second derivative. • Which interpolation method? • Linear? • Quadratic? • Cubic spline method fits 3rd-order polynomials between each “knot”, 1st and 2nd derivative continuous at knots. Current source density • Laminar LFP recorded in V1 • Triggered average on spikes of simultaneously recorded thalamic neuron • Getting the sign right • Remember current flows from V+ to V– • Local minimum of V(z) = Current sink =second derivative positive Jin et al, Nature Neurosci 2011 Current source density: potential problems • Assumption of (x,y) homogeneity • Gain mismatch • The CSD is orders of magnitude smaller than the raw voltage • If the gain of channels are not precisely equal, raw signal bleeds through • Sink does not always mean synaptic input • Could be active conductance • Can’t distinguish sink coming on from source going off • Because LFP data is almost always high-pass filtered in hardware • Plot the current too! (i.e. 1st derivative). This is easier to interpret, and less susceptible to artefacts. Signal processing theory Typical electrophysiology recording system Amplifier Filter A/D converter • Filter has two components • High-pass (usually around 1Hz). Without this, A/D converter would saturate • Low-pass (anti-aliasing filter, half the sample rate). Sampling theorem • Nyquist frequency is half the sampling rate • If a signal has no power above the Nyquist frequency, the whole continuous signal can be reconstructed uniquely from the samples • If there is power above the Nyquist frequency, you have aliasing Power spectrum and Fourier transform • They are not the same! • Power spectrum estimates how much energy a signal has at each frequency. • You use the Fourier transform to estimate the power spectrum. • But the raw Fourier transform is a bad estimate. • Fourier transform is deterministic, a way of re-representing a signal • Power spectrum is a statistical estimator used when you have limited data Discrete Fourier transform • Represents a signal as a sum of sine/cosine waves 1 𝑥 𝑡 = 𝑁 𝑁−1 𝑥(𝑓) 𝑒 2𝜋𝑖𝑓𝑡/𝑁 𝑓=0 𝑒 2𝜋𝑖𝑓𝑡/𝑁 = cos 2𝜋𝑓𝑡/𝑁 + 𝑖 sin 2𝜋𝑓𝑡/𝑁 • 𝑥 𝑡 is real, but 𝑥 𝑓 is complex. • Magnitude of 𝑥 is wave amplitude • Argument of 𝑥 is phase • Still only 𝑁 degrees of freedom: 𝑥 𝑁 − 𝑓 = 𝑥 𝑓 ∗ . Using the Fourier transform to estimate power • Noisy! Power spectra are statistical estimates • Recorded signal is just one of many that could have been observed in the same experiment • We want to learn something about the population this signal came from • Fourier transform is a faithful representation of this particular recording • Not what we want Continuous processes • A continuous process defines a probability distribution over the space of possible signals Probability density 0.000343534976 Sample space = all possible LFP signals Stationary Gaussian process • Time series (𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑁−1 ) = 𝐱 • Multivariate Gaussian distribution: 𝐱~𝑁(𝝁, 𝚺) • Stationary Gaussian process • 𝚺𝑖,𝑗 = 𝐴 𝑖 − 𝑗 . • 𝐴 Δ𝑡 is autocovariance function • 𝜇𝑖 is a constant, usually 0. Autocovariance • Autocovariance 𝐴 Δ𝑡 = E 𝑥 𝑡 𝑥 𝑡 + Δ𝑡 • It is a 2nd order statistic of 𝑥(𝑡) • 𝐴 0 = 𝑉𝑎𝑟 𝑥 Power spectrum estimation error 𝑃 𝑓 =𝐸 𝑥 𝑓 2 • Power spectrum is Fourier transform of 𝐴(Δ𝑡) • Also a second order statistic • For a Gaussian process, 𝑥 𝑓 2 is proportional to a 𝜒22 distribution. • Std Dev = Mean, however much data you have • That’s why estimating power spectrum as 𝑥 𝑓 2 is so noisy Power spectrum estimation • Need to average 𝑥 𝑓 2 to reduce estimation error • If you observe multiple instantiations of the data, average over them • E.g. multiple trials Tapering • Fourier transform assumes a periodic signal • Periodic signal is discontinuous => too much high-frequency power Welch’s method • Average the squared FFT over multiple windows • Simplest method, use when you have a long signal Welch’s method results (100 windows) Averaging in time and frequency • Shorter windows => more windows • Less noisy • Less frequency resolution • Averaging over multiple windows is equivalent to averaging over neighboring frequencies Multi-taper method • Only one window, but average over different taper shapes • Use when you have short signals • Taper shapes chosen to have fixed bandwidth Multitaper method (1 window) http://www.chronux.org/ Hippocampus LFP power spectra • Typical “1/f” shape • Oscillations seen as modulations around this • Usually small, broad peaks CA1 pyramidal layer Buzsaki et al, Neuroscience 2003 Connexin-36 knockout Buhl et al, J Neurosci 2003 Stimulus changes power spectrum in V1 • High-frequency broadband power usually correlates with firing rate • Is this a gamma oscillation? Henrie and Shapley J Neurophys 2005 Attention changes power spectrum in V1 Chalk et al, Neuron 2010