We introduce a new class of P systems, called networks of cells, with rules
allowing several cells to simultaneously interact with each other in order to produce
some new objects inside some other output cells. We define different types of behavior
for networks of cells by considering alternative strategies for the application of the rules:
sequential application, free parallelism, maximal parallelism, locally-maximal parallelism
and minimal parallelism. We devise a way for translating network of cells into place-
transition nets with localities (PTL-nets, for short) - a specific class of Petri nets. Then,
for such a construction, we show a behavioral equivalence between network of cells and
corresponding PTL-nets only in the case maximal parallelism, sequential execution, and
free parallelism, whereas we observe that, in the case of locally-maximal parallelism and
minimal parallelism, the corresponding PTL-nets are not always able to mimic the behavior of network of cells. Also, we address the reverse problem of finding a corresponding
network of cells for a given PTL-net by obtaining similar results concerning the relation-
ships between their semantics. Finally, we present network-of-cells-based models of two
classical synchronization problems: producer/consumer and dining philosophers
