Self-organized criticality occurs through a nonlinear feedback mechanism triggering transitions between different metastable states. These transitions take the form of intermittent avalanchelike events distributed according to a power law. We present the first and simplest fully continuous partial differential formalism of this phenomenon, based on the introduction of a subcritical dynamics. SOC is identified as the regime where diffusive relaxation is faster than the instability growth rate. In the other limit of slow diffusion, avalanches comparable to the system size become dominant. This provides a general correspondence between SOC and synchronization of threshold oscillators. [S0031-9007(96)00101-9] PACS numbers: 64.60. Ht, 05.40.+j, 05.45.+b, 05.70.Ln The possibility for driven dissipative extended systems to exhibit a spontaneous organization towards a kind of dynamical critical point has been dubbed self-organized criticality (SOC) [1]. This concept has been mostly illustrated using cellular automata models [2,3] and discrete space-time models [4]. They are characterized by a very slow driving and a threshold dynamics, i.e., a local stepwise unstability occurs when the field exceeds some critical value leading to a rapid relaxation on neighbors which may cascade to create large avalanches well differentiated in time (this is where the slow driving is important). Attempts have been pursued to develop continuous field theoretical approaches to this phenomenon, based on continuous anisotropic nonlinear driven diffusion equations with stochastic noise [5]. However, avalanches are not described and the origin of the self-organization is not explained. This is due to the fact that the threshold dynamics [6] is replaced by a "weak" perturbative nonlinear term. Furthermore, the driving occurs on a fast time scale (stochastic noise) in contrast with the very slow driving common to all SOC models, whereas the order parameter exhibits slow diffusionlike relaxations similar to critical slowing down [7] in opposition to the fast relaxation induced by the avalanches. A physical system which exemplifies these features is provided by earthquakes which relax, over time scales of tens of seconds, the stress accumulated over centuries.Our goal is to construct a fully continuous formulation in terms of partial differential equations, in the spirit of the Landau-Ginzburg theory of phase transitions, which takes full account of the nonperturbative nature of the threshold mechanism. Our hope is that a continuous field theory constructed on the basis of symmetry and parsimony will be both sufficiently simple and general to teach us something on SOC, as for thermal critical transitions. We do not describe a specific experimental system but rather aim at a general understanding that will provide a starting point for specific applications. However, for the sake of pedagogy, we formulate the problem in the sandpile language. It will turn out that, notwithstanding our neglecting of many details, the basic properties as well as th...

During early development, waves of activity propagate across the retina and play a key role in the proper wiring of the early visual system. During a particular phase of the retina development (stage II) these waves are triggered by a transient network of neurons, called Starburst Amacrine Cells (SACs), showing a bursting activity which disappears upon further maturation. The underlying mechanisms of the spontaneous bursting and the transient excitability of immature SACs are not completely clear yet. While several models have attempted to reproduce retinal waves, none of them is able to mimic the rhythmic autonomous bursting of individual SACs and reveal how these cells change their intrinsic properties during development. Here, we introduce a mathematical model, grounded on biophysics, which enables us to reproduce the bursting activity of SACs and to propose a plausible, generic and robust, mechanism that generates it. The core parameters controlling repetitive firing are fast depolarizing V-gated calcium channels and hyperpolarizing V-gated potassium channels. The quiescent phase of bursting is controlled by a slow after hyperpolarization (sAHP), mediated by calcium-dependent potassium channels. Based on a bifurcation analysis we show how biophysical parameters, regulating calcium and potassium activity, control the spontaneously occurring fast oscillatory activity followed by long refractory periods in individual SACs. We make a testable experimental prediction on the role of voltage-dependent potassium channels on the excitability properties of SACs and on the evolution of this excitability along development. We also propose an explanation on how SACs can exhibit a large variability in their bursting periods, as observed experimentally within a SACs network as well as across different species, yet based on a simple, unique, mechanism. As we discuss, these observations at the cellular level have a deep impact on the retinal waves description.

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We study the gravity induced instability of a liquid film formed below a plane grid which is used as a porous media in an original hydrodynamic experiment. The film is continuously supplied with a controlled flow rate. We give through a phase diagram the full spectrum of the different flow regimes and we investigate the dynamics of the observed structures. True secondary instabilities of a 2D periodic pattern are described. The control parameters are the flow rate and the viscosity.