In this article, we present a self-contained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1/f-noise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scale-free phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (power-law distributions, fractals and 1/f-noise) and of critical phenomena. We then review typical mathematical models exhibiting such properties: edge of chaos, cellular automata and self-organized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.

The notion of gating as a mechanism capable of controlling the flow of information from one set of neurons to another, has been studied in many regions of the central nervous system. In the nucleus accumbens, where evidence is especially clear, gating seems to rely on the action of bistable neurons, i.e., of neurons that oscillate between a quiescent “down” state and a firing “up” state, and that act as AND-gates relative to their entries. Independently from these observations, a growing body of evidence now indicates that bistable neurons are also quite abundant in the cortex, although their exact functions in the dynamics of the brain remain to be determined. Here, we propose that at least some of these bistable cortical neurons are part of circuits devoted to gating information flow within the cortex. We also suggest that currently available structural, electrophysiological, and imaging data support the existence of at least three different types of gating architectures. The first architecture involves gating directly by the cortex itself. The second architecture features circuits spanning the cortex and the thalamus. The third architecture extends itself through the cortex, the basal ganglia, and the thalamus. These propositions highlight the variety of mechanisms that could regulate the passage of action potentials between cortical neurons sets. They also suggest that gating mechanisms require larger-scale neural circuitry to control the state of the gates themselves, in order to fit in the overall wiring of the brain and complement its dynamics.

We study the properties of soliton solutions in an analogue of the Skyrme model in 2ϩ1 dimensions whose Lagrangian contains the Skyrme term and the mass term, but no usual kinetic term. The model admits a symmetry under area-preserving diffeomorphisms. We solve the dynamical equations of motion analytically for the case of spinning isolated baryon-type solitons. We take fully into account the induced deformation of the spinning Skyrmions and the consequent modification of its moment of inertia to give an analytical example of related numerical behavior found by Piette, Schroers, and Zakrzewski. We solve the equations of motion also for the case of an infinite, open string, and a closed annular string. In each case, the solitons are of finite extent, so called ''compactons,'' being exactly the vacuum outside a compact region. We end with indications on the scattering of baby Skyrmions, as well as some considerations as the properties of solitons on a curved space. ͓S0556-2821͑97͒03512-1͔

In this review we present a pedagogical introduction to recent, more mathematical developments in the Skyrme model. Our aim is to render these advances accessible to mainstream nuclear and particle physicists. We start with the static sector and elaborate on geometrical aspects of the de nition of the model. Then we review the instanton method which yields an analytical approximation to the minimum energy con guration in any sector of xed baryon number, as well as an approximation to the surfaces which join together all the low energy critical points. We present some explicit results for B = 2 . W e then describe the work done on the multibaryon minima using rational maps, on the topology of the con guration space and the possible implications of Morse theory. Next we turn to recent w ork on the dynamics of Skyrmions. We focus exclusively on the low energy interaction, speci cally the gradient o w method put forward by Manton. We illustrate the method with some expository toy models. We end this review with a presentation of our own work on the semi-classical quantization of nucleon states and low energy nucleon-nucleon scattering.

We study the time evolution of a neural network model as it learns the three stages of a visual delayed-matching-to-sample (DMS) task: identification of the sample, retention during delay, and matching of sample and target, ignoring distractors. We introduce a neurobiologically plausible, uncommitted architecture, comprising an "executive" subnetwork gating connections to and from a "working" layer. The network learns DMS by reinforcement: reward-dependent synaptic plasticity generates task-dependent behaviour. During learning, working layer cells exhibit stimulus specialization and increased tuning of their firing. The emergence of top-down activity is observed, reproducing aspects of prefrontal cortex control on activity in the visual areas of inferior temporal cortex. We observe a lability of neural systems during learning, with a tendency to encode spurious associations. Executive areas are instrumental during learning to prevent such associations; they are also fundamental for the "mature" network to keep passing DMS. In the mature model, the working layer functions as a short-term memory. The mature system is remarkably robust against cell damage and its performance degrades gracefully as damage increases. The model underlines that executive systems, which regulate the flow of information between working memory and sensory areas, are required for passing tests such as DMS. At the behavioural level, the model makes testable predictions about the errors expected from subjects learning the DMS.