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