Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 277

8.4.1 Are Ganglion Cells Independent Encoders?

This question can be now reformulated in the context of Gibbs distributions. Inde-
pendence between neurons means that spike statistics is described by a potential,
possibly non-stationary, of the form:

N

φ n ( ω )= φ n,k ( ω k ) . (8.33)

k =1

This assumption can be stated independently on the memory depth, so we write
here φ n ( ω ) instead of φ n ( ω n− n ) to alleviate notations and to be as generic as
possible. In ( 8.33 ) ω k is the spike train

{ ω k ( l ) } l≤n produced by neuron k only. In
this way the transition probabilities of the global network are products of transition
probability for each neuron and the Gibbs distribution ( 8.5 ) is a product of marginal
distributions for one neuron. On the opposite, if one believes that spike correlations
play an important role in statistics one has to include them in the Gibbs potential.
Typically, spike correlations are characterized by monomials and the potential
takes the generic form ( 8.29 ). Obviously, there are many possible choices for this
potential, depending on the set of observables assumed to be relevant.
To compare different models one can use the criteria described in Sect. 8.3.2.12 .
Does an independent model predicts correctly the spike blocks occurring in the
observations? If not, which correlations has to be included? How evolves the
Kullback-Leibler divergence as the type of correlations (monomials) taken into
account growth?
However, the application of those criteria is delicate in experimental data, taking
into account the large number of cells, their different types and the relatively small
size of spike train samples obtained from experiments. For these reasons analysis of
retina data has been performed either for memory-less models where the number of
neurons can be up to 100 neurons, or to models with memory with small range and
a small number of neurons. Let us present some of those works.

8.4.2 Weak-Pairwise Correlations Imply Strongly Correlated

Network States in a Neural Population

A breakthrough in spike train analysis has been made by the seminal paper of E.
Schneidman and collaborator [ 60 ]. They have considered carefully instantaneous
pairwise spike correlations in experiments on the vertebrate retina. It appears that,
mostly, those correlations are weak . But are they significant ? In Sect. 8.3.2.4 we
have shown that weak correlations can be hardly interpretable without further
analysis. The maximal entropy principle provides a way to quantify the role of
those correlations. Take a statistical model (a Gibbs potential) without constraint