The tasks of neural computation are remarkably diverse. To function optimally, neuronal networks have been hypothesized to operate near a nonequilibrium critical point. However, experimental evidence for critical dynamics has been inconclusive. Here, we show that the dynamics of cultured cortical networks are critical. We analyze neuronal network data collected at the individual neuron level using the framework of nonequilibrium phase transitions. Among the most striking predictions confirmed is that the mean temporal profiles of avalanches of widely varying durations are quantitatively described by a single universal scaling function. We also show that the data have three additional features predicted by critical phenomena: approximate power law distributions of avalanche sizes and durations, samples in subcritical and supercritical phases, and scaling laws between anomalous exponents.

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Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given. §1. Introduction. In Banach space theory as well as in Analysis on Banach spaces, it is important to study families of smooth real valued functions with bounded and nonempty supports on given spaces (smooth bump functions) (cf. e.g., [BF], [Dev 2], [DGZ]). Such functions are usually constructed by composing smooth norms with appropriate real valued functions on the real line. R. Haydon has recently shown that there are Banach spaces which admit Lipschitz, continuously Frechet differentiable bump functions and admit no Gateaux differentiable norms ([HI], [H2], [H3]). In some classes of spaces and for some kinds of smoothness, the existence of a smooth bump function on X already implies the existence of a norm with a smoothness property not worse than that of the bump function. This is the case, for example, of separable spaces and Frechet smooth bump functions ([LW]) or of bump functions with locally uniformly continuous Frechet derivative on spaces X that do not contain an isomorphic copy of c 0 ([FWZ], [Fl], [DF]).In the first case, the result is obtained (rather indirectly) by using the technique of rough norms and Kadec renorming of separable spaces by locally uniformly rotund norms ([Ka], [LW], cf. e.g., [DGZ]).In the second case, first a bump function with uniformly continuous Frechet derivative is constructed on X, by using the Bessaga-Pelczynski characterization of spaces not containing isomorphic copies of c 0 . The norm is then constructed by using level sets of a proper bump function on X. The latter method cannot, in general, work for nonuniformly differentiable bump functions, as shown in Example III. 10 below.The purpose of this paper is to present a new method for constructing norms with moduli of smoothness of power type from pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type for spaces with the Radon-Nikodym property. Our technique shows that if a Banach space X has the Radon-Nikodym property and admits a Frechet differentiable bump function, then X admits a norm in which each closed convex and bounded [MATHEMATIKA, 40 (1993), 305-321] 306 R. DEVILLE, G. GODEFROY AND V. ZIZLER set is an intersection of balls (the Mazur intersection property). Dual results on norms with moduli of rotundity of power type are also shown. Our results imply that none of the locally uniformly rotund norms on a separable nonsuperreflexive space have pointwise modulus of rotundity of power type. We obtain a new characterization of spaces isomorphic to Hilbert spaces as Banach spaces X for which both X and X* admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type ...

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).

Abstract. We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength -as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.