Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina - Modeling in Computational Biology and Biomedicine - page 266

n − m +1 )valuesof ω m . For example the

where the sum holds on all possible (2

average value of f ( ω )= ω k ( n ) is given by μ [ ω k ( n )] = ω k ( n ) ω k ( n ) μ [ ω k ( n )]
where the sum holds on all possible values of ω k ( n ) (0 or 1). Thus, finally

μ [ ω k ( n )] = μ [ ω k ( n )=1] , (8.10)

which is the probability of firing of neuron k at time n . This quantity is called
the instantaneous firing rate . Likewise, the average value of ω k 1 ( n ) ω k 2 ( n ) is the
probability that neuron k 1

and k 2 fire at the same time n :thisisameasureof

pairwise synchronization at time n .

8.3.2.3 Empirical Averages

In experiments, raster plots have a finite duration T and one has only access to
a finite number of those rasters, denoted ω (1) ,...,ω ( N ) . From these data one

computes empirical averages of observables. Depending on the hypotheses made on
the underlying system there are several ways of computing those averages.
A classical (though questionable assumption as far as experiments are concerned)
is stationarity : The statistics of spike is time-translation invariant. In this case
the empirical average reduces to a time average . We denote π ( T ω [ f ] the time
average of the function f computed for the raster ω of T . For example, when
f ( ω )= ω k ( n ), π ( T )

N

T T − 1

n =0 ω k ( n ), which provides an estimation of the
firing rate of neuron k (it is independent of time from the stationarity assumption).
If f is a monomial ω k 1 ( n 1 ) ...ω k m ( n m ), 1 ≤

[ f ]=

ω

n 1 ≤ n 2 ≤ n m <T ,then

T −n m

π ( T )

1

n =0 ω k 1 ( n 1 + n ) ...ω k m ( n m + n ), and so on. Why using
the cumbersome notation π ( T ω [ f ]? This is to remind the reader that such empirical
averages are random variables . They fluctuate from one raster to another i.e.,
π ( T )

[ f ]=

ω

T − n m

ω (1) [ f ] = π ( T ) ω (2) [ f ] for distinct rasters ω (1) ,ω (2) . Moreover, those fluctuations

depend on T .
Assume now that all empirical rasters have all been generated by an hidden
Markov chain and additionally that this chain is ergodic with a Gibbs distribution μ .
Then, all those rasters obey π ( T )

,
whatever f : the time average converges (almost-surely) to the average with respect
to the hidden probability μ (this is one of the definitions of ergodicity). As a
consequence the fluctuations of the time-average about the exact mean μ [ f ] tends
to 0, typically like

ω ( r ) [ f ] →

μ [ f ], r =1 ,..., N ,as T → + ∞

K f

√ T ,where K f is a constant depending on f . This is the
celebrated central limit theorem stating moreover that fluctuations about the mean
are Gaussian [ 23 ]. We come back to this point in Sect. 8.3.2.4 .
The remarkable consequence of ergodicity (which implies stationarity) is that the
empirical average can be estimated from one raster only. Now, if we have rasters

available we can use them to enlarge artificially the sample size, e.g., computing

N