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?