Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 279

the model accuracy (measured with the KL divergence ( 8.30 )) when adding spatio-
temporal constraints.
In some sense, it is clear that the more parameters (or constraints) the more the
model fits the data, up to some limit, fixed by the data and especially the raster
size, where models become indistinguishable. Is there a minimal statistical model?
A related question is: what does a statistical model teach us about the underlying
neural network (e.g., the retina) and about neural coding? Let us first present an
important work addressing the second question, before addressing the first one.

8.4.3 The Architecture of Functional Interaction Networks

in the Retina

A critical question to elucidating the neural code, at least from a theoretical point of
view, is to confront multi-electrodes real data against sophisticated statistical models
testing for the underlying neural structure involved. Since the connectivity of any
neural network can in principle growth exponentially as a function of the number
of neurons, classical numerical methods can rapidly become inefficient. The work
of Schneidman and collaborators [ 60 ] has nevertheless suggested that although the
number of possible activity patterns and underlying interactions is exponentially
large in a neural network, a pairwise-based Ising model gives a surprisingly accurate
description of neural population activity patterns. So, an economical assumption to
reducing a putative network dimensionality, is to use a pairwise correlation model
able to recover the main network structure involved in the encoding of a natural
movie [ 60 ].
More recently, searching for further reduction on the dimensionality of the
network structure, Ganmor et al. [ 18 ] have shown the presence of small groups
of neurons having strong correlated activity. As an outcome of their MEA spike
train analysis the authors introduce the notion of functional connectivity .This
corresponds to associating with the parameters in the Gibbs potential a network
of “effective” interactions ( β ij for spike pairs, β ijk for triplets, ... ).
The performance of their model is clearly higher than when assuming indepen-
dence between neurons. It is able to predict the neural activity, including synchrony,
for larger neural networks as in [ 60 ] and the contribution of small groups of
neurons to the complete population activity. To further reduce the structure of the
network to most critical neural interactions, the authors apply a nearest-neighbors
paradigm. An interesting result is that the reduction of close to 50 % of the original
pairwise interactions does not change the accuracy of predictions, and models most
functional groups of nearest neurons with an accuracy > 95 %. Additionally, small
functional overlapping units (10-20 neurons) seem to be a critical structure for the
encoding of natural movies stimulus.
This work, together with [ 19 ], shows that a Gibbs potential with a relatively small
number of parameters, corresponding to effective interactions between neurons, is