Post on 22-Dec-2015
Circular statisticsMaximum likelihood
Local likelihoodKenneth D. Harris
4/3/2015
Relationship of cells to oscillations
Klausberger & Somogyi, Science 2008
Relationship of cells to oscillations
O’Keefe & Recce, Hippocampus 1993
How do we quantify:
• A cell’s average phase of firing?
• How phase depends on other variables?
Computing instantaneous phase• Hilbert transform
• Peak fitting
Phase histogram
• Different cells prefer different phases
• How do we compute each cell’s mean phase?
Linear mean doesn’t work
Circular mean
• Treat angles as points on a circle
• The mean of these gives you• Circular mean • Vector strength
• If all angles are the same:• is this angle• is 1
• If angles are completely uniform• is 0• is meaningless.
𝑧=𝑒𝑖𝜃
𝑧=𝑅𝑒𝑖𝜃
𝜃R
von Mises distribution
• is central angle
• is concentration parameter• Larger -> more peaked• Zero -> uniform distribution
• is a Modified Bessel function• Needed to make probabilities integrate to 1.
Maximum likelihood estimation
• Given data set , how do we estimate parameters and ?
• Choose them to maximize
Maximum likelihood estimation
• Given data set , how do we estimate parameters and ?
• Log likelihood
• Solve and
Maximum likelihood von Mises
• , ( = circular mean).
• is the solution of , ( = vector strength).
• is another modified Bessel function.
How does phase depend on another variable?
• Don’t use linear regression!
Locally-weighted likelihood
• Log likelihood
• Locally-weighted likelihood • Spike phases occur at positions :
E.g.
Kernel smoother
Is local likelihood for a Gaussian
Place field estimation
Local likelihood estimation for a Poisson distribution
Phase field
• Local likelihood estimation for von Mises
Physical space
Abstract space
Harris et al Nature 2002
Confirmatory statistics:
• How to test that phase is independent of place?
• Shuffling method?
• Test statistic?