Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 286

coefficient β kj 's in a binned-Ising model with a binning of 10-20 ms somewhat
integrate the synaptic transmission effects and neurons pairwise interactions appear
as instantaneous. In this way, one looses an important part of the dynamics and of the
network structure. The “functional interactions” evoked in Sect. 8.4.3 corresponds
to an integration of physical interactions over the binning time scale. For example, in
the Ising model, the pairwise coefficient β kj integrates the effect of several circuits
connecting neuron j to neuron k , as well as dynamic-dependent effects. As a matter
of fact its interpretation is rather delicate.
This is however certainly not the end of the story and this aspect has to be still
investigated on the theoretical and experimental side.

8.5.2 Linear Potentials Versus Combinatorial Explosion

Experimental attempts to go “beyond Ising” [ 34 , 73 ] suggest that Markovian models
with increasing range should describe better and better the spike statistics. This is
also expected from the theoretical analysis summarized in Sect. 8.4.4 .However,this
raises several remarks and questions. First, it is evident that the more parameters, the
best is the model, but what do we learn from this plethora of parameters? Second,
one has to face the critical issue of an exponential increase of parameters, with the
potential range and with the number of neurons, so that numerical methods can
rapidly become inefficient. Moreover, the sample size required to determine those
coefficients is expected to increase also exponentially, ruining any hope to extract
reliable coefficients from empirical data. Finally, as outlined in the previous section,
the interpretation of the coefficients is difficult even for the simplest pairwise
interactions β kj .
Our point of view is that the linear potential approach, based on the maximal
entropy principle, is maybe inappropriate. On the opposite, non linear potentials of
the form ( 8.41 ), truncated to a finite memory depend on a number of parameters,
the physical parameters of the network, which increases only polynomially with
the number of neurons. Although, the number of blocks determining the potential
increases exponentially fast with the memory depth, it could well be that only a
small proportion of blocks are sufficient to extract most of the information about the
hidden parameters. Finally, the interpretation of parameters is here straightforward
and such a model allows to treat the non-stationary case. This may provide an
alternative to Ising-like models to study spike train statistics in experiments.

8.6 Outlook

In this section we would like to point out a few challenges for the next future, on the
theoretical and experimental sides.