Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 278

on instantaneous pairwise spike correlations, and fix only constraints on rates. This
provides a Bernoulli model with a potential ( 8.24 ). Take another model where
pairwise correlations are taken into account (but no higher-order interactions). This
is an Ising model with potential ( 8.26 ). The authors compare those two models using
the comparison between estimated block probability versus the observed block
probability for spatial blocks, but also some other criteria such as the probability
of having n spiking G cells in a time window of 20 ms.
Their work convincingly shows that although pairwise correlations are weak ,
they are nevertheless necessary to characterize spike trains statistics. The Ising
model clearly does quite better than a Bernoulli model and the authors claim to
“predict” about 90 % of the multi spiking structure of a large G cells population
in salamander and guinea pig. Note however that they focus on spatial pattern.
This shows that weak correlations between pairs of neurons coexist with strongly
collective behavior in the responses of ten or more neurons. Similarly, Shlens
et al. [ 65 ]predict99 % of a complete ON and OFF parasol G cells population in
primates with a Ising model . This suggests that the neural code is dominated by
correlation effects.
On the other hand, this work does not directly unravel the importance of neural
correlation for carrying information. Moreover, in [ 54 ], it has been shown that
although Ising model is good for small populations, this is an artifact of the way data
is binned and of the small size of the system. Additionally, it might be questionable
whether more general forms of Gibbs distributions (e.g., involving more general
monomials) could improve the estimation and account for deviations to Ising-
model [ 42 , 66 , 71 ] and provide a better understanding of the neural code from the
point of view of the maximal entropy principle [ 27 ]. Very recently, Ganmor and
collaborators [ 19 ] have extended the maximal entropy principle introducing higher
order instantaneous spikes correlations. Triplets and so on are considered although
all spikes arise at the same time. This therefore still corresponds to a memory-less
model. These authors have convincingly shown that such model describes more
accurately retina responses to natural images than Ising model. In particular spatio
temporal patterns where well predicted: binary words of 10 retinal G cells over 10
time steps. Note however that those data are binned (Sect. 8.5.1 for a definition and
a discussion).
As a matter of fact, back to 1995, [ 36 ] already considered multi-unit syn-
chronizations and proposed several tests to understand the statistical significance
of spike synchronizations. A few years later, [ 35 ] generalized this approach to
arbitrary spatio-temporal spike patterns and compared this method to other existing
estimators of high-order correlations or Bayesian approaches. More recently, several
papers have pointed out the importance of temporal patterns of activity at the
network level [ 31 , 63 , 76 ]. As a consequence, a few authors [ 2 , 34 , 53 ] have attempted
to define time-dependent Gibbs distributions on the base of a Markovian approach,
but with one time step memory only, and under assumptions such as detailed
balance, [ 34 ] or conditional independence between neurons, see Eq. (1) in [ 52 ].
A more general method relying on Gibbs distributions has been proposed in [ 75 ],
and applied to the same data as [ 60 ], in [ 75 ]. The results shows a clear increase in