The evolution of life forms on our planet has led to the generation of an enormous variety of living structures. How such patterns of organization emerge 
, how contingency 
and constraints 
shape them and how they acquire robustness 
are unanswered questions that have been at the forefront of biology for more than a century and are still open key questions. The research field encompassing these fundamental issues is referred to as evolutionary developmental biology or in short evo-devo 
. With the increasing capacity of mathematical modeling to provide fresh insight into the biological processes 
, computer simulations and experimental approaches in this field have recently reached common ground (see 
as recent reviews).
A major conceptual problem for the modeling approach to evo-devo is the mapping between genotype (hereditary genetic information) and phenotype (the physical characteristics of the resultant organism). It is the phenotype that determines the organism's chances of survival (fitness), as it is on it that natural selection acts. The set of all genotypes, their resultant phenotypes and associated fitness is called fitness landscape. Since Wright's pioneering idea in the early 30's 
that the hill-climbing process of population's adaptive evolution intimately depends on how smooth or rugged the fitness landscape is, numerous theoretical works have been contributing to what now can be considered as the theory of fitness landscapes 
. Moreover, empirical studies of fitness landscapes can nowadays be performed in the laboratory 
, revealing the real evolutionary paths undertaken by the organisms, and thus opening a previously-unavailable window on the actual evolution process.
The extensively studied theoretical case that has become the classic example of evolution in a fitness landscape is provided by RNA folding 
. Here the genotype is defined by the nucleotide sequence, whereas the phenotype consists of the secondary structure formed by the (planar) pattern of the base pairs. Within the RNA context, the existence of iso-phenotypic genotypes (or neutrality) has significant implications in evolution, in general 
and evo-devo, in particular 
. More precisely, neutrality is hypothesized to allow a more exhaustive search in the genotype space and consequently, better accessibility to diverse and potentially fitter phenotypes 
The neutrality feature has been encountered and studied in other works of similar nature to the RNA's, such as in the origin and complexification of the protein universe 
, or in tunable-neutrality models of abstract molecular species 
, but also in other fields of very different nature. An example is provided by a model of feed-forward signaling networks 
. Here, a minimal Boolean network receives a set of input signals, and computes the output. The genotype is defined by the wiring diagram (the network topology plus the weight of each interaction), whereas the phenotype is specified by the Boolean computation being performed. An example closer to the current study is a Boolean model of genetic networks 
, a study that inquires on the requirement of “genetic flexibility” or more precisely, of phenotype continuity in evolution, and the subsequent constraints it may pose to species evolution in a changing environment. In a more recent work, the same group developed an evolutionary model of network evo-devo 
that adds to the same approach as the current study, with the two works providing complementary clues on the evolution of minimal developmental modules. Again under the Boolean approach, Andreas Wagner's studies ranging from the “epigenetic stability” of developmental pathways 
to bridging robustness and evolvability by means of neutrality features in models of gene networks 
complete the framework in which the present work is formulated. Moreover, the present formulation constitutes a continuation of the model introduced in 
, as well as a Hawk's eye view of an isolated genetic sub-system. Its exhaustive study allows uncovering of features that are generally not accessible from statistical large-scale studies of similar nature. As far as we know, no parallel exhaustive analogies of Boolean approaches have been applied within the context of spatially-explicit evo-devo.
We have addressed here the role of neutrality and robustness in the evolution of minimal developmental modules. It is now apparent that the genetic networks responsible for major events in the development of organisms present significant robustness to a wide range of perturbations 
. Moreover, experimental works reveal that certain genes and their interactions are recurrently encountered in very diverse organisms (e.g. Homeobox genes 
), suggesting that minimal genetic modules may underlie fundamental developmental pathways. The current work is inspired by the pioneering theoretical and empirical analysis of developmental genetic regulatory networks in long-germ-band insects (Drosophila melanogaster
and references therein) and plant (Arabidopsis thaliana
) development 
. As anticipated by 
is a suitable model organism to inquire on small gene modules that control specific parts of the development process. The goal of the current work is not a precise explanation of a specific genetic module, but a description of possible underlying principles of network assemblage and evolution.
In this context, our guiding questions are: what classes of spatial expression patterns can possibly emerge from signals mediated by juxtacrine (intra or inter-cellular) interactions in a minimal genetic network? Are there intrinsically robust modules and what are their defining characteristics? Our approach addressing these questions is organized as follows. We introduce the model of gene interactions whose dynamics provides the gene expression pattern. We present the minimal set of genes producing a specific, biologically-relevant expression pattern, and the exhaustive analysis of all possible gene interactions and their associated expression patterns. Among all these topologies, we identify those providing a robust expression pattern, being thus the candidates for the developmental modules discussed above. Ultimately, an evolutionary study of populations of such networks conditioned on diversity is presented, revealing rapid evolution towards robust stripe-like expression patterns. We show that the structure of the encountered minimal robust networks relates to the phenomenon of lateral inhibition, a widespread mechanism of biological pattern formation, emphasizing thus the importance of these minimal development-driving modules.