Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 275

Fig. 8.7 Analysis of salamander retina data, from [ 75 ]. The estimated block probability versus the
observed block probability for all blocks from range 1 - 4 (coded by colors ), for N =4 neurons
with a model of range R =3 for pairs and triplets. We include the equality line y = x and the
confidence bounds ( black lines ) for each model, corresponding to π ( T ) ( w ) ± 3 σ w with σ w being
the standard deviation for each estimated probability given the total sample length T ∼ 3 · 10 5 .In
the figure, h t corresponds to h ,Eq.( 8.31 )

There is a wide literature in statistics dealing with the subject. In the realm of
spike train analysis an important reference is [ 48 ] and references therein. Here, we
point out two criteria for model comparison, used in this chapter as an illustration.
A first and straightforward criterion consists of computing the empirical proba-
bility of blocks of range 1 , 2 ,... and to compare it to the probability predicted by
the model. Of course, the number of blocks of range R increases like 2

NR ; moreover
the probability of large blocks is expected to decrease fast with the block range. So,
practically, one considers a subset of possible blocks. The challenge is in fact to
predict the probability of events which have not been included as constraints in the
Gibbs potential of the model. For example, does an Ising model well predict the
probability of occurrence of triplets, quadruplets, of non simultaneous spikes?
The typical representation of this criterion is a graph, with, on abscissa, the
observed probability of blocks and, on ordinate, the predicted probability. Thus,
to each block corresponds a point in this two-dimensional graph. A “good” model is
such that all points spread around the diagonal y = x . The distance to the diagonal
is expected to increase as the probability of the block decreases thanks to the central
limit theorem. Indeed, if the exact probability of a block is P , then the empirical
estimation of this probability is a random variable with a Gaussian distribution, of
mean P , and a variance that can be computed from the topological pressure. A usual
approximation of this variance is P (1 −

P ). Thus, in a similar way as in Fig. 8.6 ,the
set of points in the g raph spreads around the diagonal in a region delimited by the
curves

P (1 − P )
T

± 3 called “confidence bounds”. An example is given in Fig. 8.7 .