View source for Neuronal synchrony measures

=== Detecting the asynchronous state in experiments ===

The criterion for synchrony based on computing the synchrony measure
<math>\chi</math> for several values of <math>N</math> is difficult to
check directly in experimental systems since this requires reliable
measurements from networks with variable sizes.
An alternative criterion
is based on the behavior of population averages 
(Ginzburg and Sompolinsky 1994, Golomb et al. 2001).
Let us suppose that we can measure the means of
the local observables <math>x_i(t)</math> over a subpopulation of size <math>K\ ,</math> where <math>K << N\ ,</math>
yielding
:<math>\label{popav}
X_K (t) \equiv \frac{1}{K} \sum_i^K x_i (t) ~.
</math>

Asynchronous states can be distinguished from synchronous states according
to the <math>K</math> dependence of the variance of <math>X\ ,</math>
:<math>\label{var}
\Delta(K) \equiv \langle ( X_K (t) - \langle X_K \rangle_t )^2 \rangle_t ~.
</math>

In an asynchronous state the local variables are weakly correlated, hence
:<math>\label{varN}
\Delta(K) \propto \frac{1}{K}~~, 1 << K << N ~.
</math>

On the other hand, in synchronous states
:<math>\label{delo}
\Delta(K) = O(1)
</math>

even for large <math>K\ .</math>
The advantage of this criterion is that it does
not rely on the absolute scale of <math>\Delta\ ,</math> but on its dependence on <math>K\ ,</math>
which, unlike <math>N\ ,</math> can be varied experimentally. The limitation of this
criterion is that the sampling of the <math>x_i</math>'s and the value of <math>K</math> should
be such that the sums are not dominated by unusually strongly
correlated variables.