Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 272

In the memory-less case D =0where only the statistics of instantaneous spiking
patterns is considered, the Gibbs distribution reads:

e H β ( β,ω (0))
ω (0) e H β ( β,ω (0)) .

μ β ( ω (0)) = (8.22)

In this case,

ζ β =
ω (0)

e H β ( β,ω (0)) .

(8.23)

This is a constant (it does not depend on the raster). It is called partition function in
statistical physics.

8.3.2.8 The Maximal Entropy Principle

Assume now that we want to approximate the exact (unknown) probability μ by an
approximated probability μ ap that matches the constraints ( 8.13 ).Theideaistotake
as a statistical model μ ap the Gibbs distribution of a function of the form ( 8.18 ),
corresponding to a set of constraints attached to observables

O k ,wherethe β k 's
are free parameters of the model. Thus, the statistical model is fixed by the set of
observables and by the value of β . We write then, from now on, μ β instead of μ ap .
Looking at the variational principle ( 8.17 ), we have to take the supremum over
all probabilities ν that matches ( 8.13 ), i.e., μ β [ O k ]= C k so that μ β [ H ] β is
a constant for fixed β . Therefore, in this case ( 8.13 ) reduces to maximizing the
entropy rate given the constraints ( 8.13 ). This the classical way of introducing Gibbs
distributions in physics courses. Then, the β k 's appear as Lagrange multipliers, that
have to be tuned to match ( 8.13 ). This can been done thanks to ( 8.21 ). Note that the
topological pressure is convex so that the solution of ( 8.21 ) is unique.
The important point is that procedure provides a unique statistical model defined
by the transition probabilities ( 8.20 ). Thus, we have solved the degeneracy problem
of Sect. 8.3.2.5 in the stationary case.

8.3.2.9 Range-1 Potentials

Let us now present a few examples used in the context of spike train analysis of
MEA data, among others.
The easiest examples are potentials with a zero memory depth, in the stationary
case, where therefore the spiking pattern ω (0) is independent of ω ( − 1).This
corresponds to range- 1 potentials .
Among them, the simplest potential has the form:

N

φ β ( ω (0)) = β k ω k (0) − log ( ζ β ) . (8.24)

k =1