13th International Conference on Membrane Computing - MTA Sztaki

R. Pagliarini, O. Agrigoroaiei, G. Ciobanu, V. Manca
r x = r x+ : x → x +
r x + = r x : x + → x
r m = r m+ : m → m +
r m + = r m : m + → m
r c,m = r c,m + : c → m + m +
r m,x = r m,x + : m → x + x +
r m + ,x = r m + ,x + : m+ → x + x +
Table 2. The set of rules modelling correlative causality in the mitotic oscillation in
early amphibian embryos.
found a statistical significant cross-correlation, with τ = 3 minutes between c
and m, and between m and x. These results are in accordance with the cascade
model for mitotic oscillation in early amphibian embryos, and allowed us to obtain
the set R of rules modelling statistical cause-effect relationships reported in
Table 2.
We consider the multiset v = 2x + x + which corresponds to considering
the time-series x with doubled values together with the time-series x + . Since
x and x + are left hand sides for some of the rules in R it follows that for any
cause G we must have v\rhs(G) = 0 which implies v ≤ rhs(G). As reasoned
before, G can have at most three elements since a fourth element would not
contribute anything to the appearance of v. These elements must belong to the
set S = {r x , r x +, r m,x , r m + ,x} since their right hand side must have at least one
object in common with v. By analyzing all possibilities we obtain that the causes
of v are r x + + r m,x ; r x + + r m + ,x; r m,x + r m + ,x; 2r m,x ; 2r m + ,x and 2r x + + r x . This
corresponds, according with the biological point of view of the mitotic cascade,
to the time-series m and m + being direct causes for x and x + (which indicates
that x + is directly correlated with x).
6 Conclusions and Discussions
In this paper we introduced two approaches to analyze cause-effect relationships
in reaction models and we proposed a way to integrate them in order to
study causality in terms of multisets of objects and multisets of rules in presence
of non-determinism and parallelism. Our approach is based on the fact that statistical
analysis can be used to build up a transition P system in a polynomial
complexity time. In fact, the computation of the different correlation indexes
that we use has polynomial order on the number n of species. From a computational
point of view, this means that it can become time expensive for n of the
order of thousands. However, this problem can be circumvent by using a parallel
implementation of the procedure, but it is not the aim of the paper to analyse
this point.
An important point is that the inferred transition P systems can be analyzed
by means of quantitative causality. The statistical approach that we proposed
starts from the fact that dynamics changes induce variability in species con-
366