Info on Snowflake diagrams for spike train analysis

[David C. Tam]

This is a brief summary of the information on "snowflake" diagrams in
Neurobiological Signal Analysis in reply to the request by Judea Pearl (via
kroger at cognet.ucla.edu).

Snowflake scatter diagram was one of the spike train analytical methods
introduced by Donald Perkel and George Gerstein et al in the 1970's to
analyze the correlation between firing intervals among 3 neurons.

Background:  Spike trains are time-series of action potentials recorded from
biological neurons.  Since the firing times of spikes by neurons vary in time
(i.e., they jitter in time), the analysis of the timing relationships between
the firing of neurons require specialized statistical methods which deals
with pulse-codes.  The most often used statistics is the correlation analysis
(which is also developed by Donald Perkel and George Gerstein et al earlier
in the 1960's to analyze spike train data).

Snowflake analysis and correlation analysis are similar in the following
ways:  Whereas correlation analysis establishes statistics for pair-wise
correlation between 2 spike trains (neurons), snowflake analysis establishes
statistics for 3-wise correlation among 3 neurons.  Whereas correlation
analysis establishes statistics for all higher-order firing intervals between
neurons, snowflake analysis establishes statistics for only first-order
intervals.

Snowflake diagram and joint-interval histogram are similar in the following
ways:  Whereas joint-interval scatter diagram has 2 orthogonal axes (in a 2-D
plane) for displaying the adjacent cross-interval between 2 neurons,
snowflake scatter diagram has 3-axes (each 120 degrees from each other) in a
2-D plane for displaying the adjacent cross-interval between 3 neurons.  They
both establish first-order interval statistics.

I have worked with Donald Perkel until he deceased, but George Gerstein is
still at Univ. of Penn.  I have worked on numerous spike train analytical
methods including snowflake diagram.  I have also developed other similar
spike train analysis techniques, so further detailed questions can be
directed to me (David Tam, e-mail: dtam at next-cns.neusc.bcm.tmc.edu) if
needed.

Related references:

Perkel, D.H., Gerstein, G.L., Smith, M.S. and Tatton, W.G. (1975)
  Nerve-impulse patterns: a quantitative display technique for three neurons,
  Brain Research.  100: 271-296.

Gerstein, G. L. and Perkel, D. H. (1972)  Mutual temporal relationships among
  neuronal spike trains,  Biophysical Journal. 12: 453-473.

Perkel, D.H., Gerstein, G.L. and Moore, G.P. (1967) Neuronal spike trains and
  stochastic point process. I. The single spike train. Biophysical Journal.
  7: 391-418.

Perkel, D.H., Gerstein, G.L. and Moore, G.P. (1967) Neuronal spike trains and
  stochastic point process. II. Simultaneous spike trains. Biophysical
  Journal.  7: 419-440.

Tam, D.C, Ebner, T.J. and Knox, C.K. (1987) Conditional cross-interval
  correlation analyses with applications to simultaneously recorded
  cerebellar Purkinje neurons. Journal of Neurosci. Methods.  23: 23-33.