Programming the emergence in morphogenetically architected complex systems.

Large sets of elements interacting locally and producing specific architectures reliably form a category that transcends the usual dividing line between biological and engineered systems. We propose to call them morphogenetically architected complex systems (MACS). While taking the emergence of properties seriously, the notion of MACS enables at the same time the design (or "meta-design") of operational means that allow controlling and even, paradoxically, programming this emergence.

Functional geometry of the horizontal connectivity in the primary visual cortex.

We present a geometrical model of the functional architecture of the primary visual cortex. In particular we describe the geometric structure of connections found both in neurophysiological and psychophysical experiments, modeling both co-axial and trans-axial excitatory connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures such as the medial axis of a shape.

The symplectic structure of the primary visual cortex.

We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation-dilation group. The total space is endowed with a natural symplectic structure neurally implemented by long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures, such as the medial axis of a shape.

Dynamical systems and cognitive linguistics: toward an active morphodynamical semantics.

We propose a novel dynamical system approach to cognitive linguistics based on cellular automata and spiking neural networks. How can the same relationship 'in' apply to containers as different as 'box', 'tree' or 'bowl'? Our objective is to categorize the infinite diversity of schematic visual scenes into a small set of grammatical elements and elucidate the topology of language. Gestalt-inspired semantic studies have shown that spatial prepositions such as 'in' or 'above' are neutral toward the shape and size of objects.

An introduction to the Mumford-Shah segmentation model.

Some papers of this special issue concern recent results on mathematical models of segmentation. As they are rather technical we propose here a pedagogical introduction for the non-mathematical reader. We briefly present the variational model of image segmentation proposed by David Mumford and we summarize some fundamental results of De Giorgi's school.

The neurogeometry of pinwheels as a sub-Riemannian contact structure.

We present a geometrical model of the functional architecture of the primary visual cortex (V1) and, more precisely, of its pinwheel structure. The problem is to understand from within how the internal "imminent" geometry of the visual cortex can produce the "transcendent" geometry of the external space. We use first the concept of blowing up to model V1 as a discrete approximation of a continuous fibration pi: R x P --> P with base space the space of the retina R and fiber the projective line P of the orientations of the plane.