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We propose a novel phase based analysis with the purpose of quantifying the periodic bursts of activity observed in various neuronal systems. The way bursts are intiated and propagate in a spatial network is still insufficiently characterized. In particular, we investigate here how these spatiotemporal dynamics depend on the mean connection length. We use a simplified description of a neuron’s state as a time varying phase between firings. This leads to a definition of network bursts, that does not depend on the practitioner’s individual judgment as the usage of subjective thresholds and time scales. This allows both an easy and objective characterization of the bursting dynamics, only depending on system’s proper scales. Our approach thus ensures more reliable and reproducible measurements. We here use it to describe the spatiotemporal processes in networks of intrinsically oscillating neurons. The analysis rigorously reveals the role of the mean connectivity length in spatially embedded networks in determining the existence of “leader” neurons during burst initiation, a feature incompletely understood observed in several neuronal cultures experiments. The precise definition of a burst with our method allowed us to rigorously characterize the initiation dynamics of bursts and show how it depends on the mean connectivity length. Although presented with simulations, the methodology can be applied to other forms of neuronal spatiotemporal data. As shown in a preliminary study with MEA recordings, it is not limited to in silico modeling.

Cultures of dissociated neurons are an invaluable experimental tool in studying neuronal networks at an intermediate scale in an in vitro controlled physico-chemical environment. Moreover, current micro-fabrication techniques allow the design of a custom connectivity between subpopulations, which could make it possible to carry out computations with devices involving living cells. The quorum percolation (QP) model has been designed in the context of neurobiology to describe bursts of activity occurring in neuronal cultures from the point of view of collective phenomena rather than from a dynamical synchronization approach. Such a model is well suited to describe triggered activity in neuronal devices, and its generic character points at the necessity of heavily structured devices to go beyond collective bursting. We derive a continuous extension of the QP model, seen as information propagation on a non-metric directed graph, and discuss how its critical behavior might give insight on the connectivity of neuronal networks. The link with metric graphs, embedded in a two-dimensional space, is tackled by the introduction of a geometrical model based upon a random walk, where axon growth is constrained by obstacles such as walls and channels. This provides a starting point for the construction of neuronal devices in vitro capable of more complex behaviors. Lastly, we show how simulations of bursts with a dynamical adaptive integrate-and-fire model can be interpreted in terms of QP, confirming the robustness of this synchronized behavior.

We study the modifications induced in the behavior of the quorum percolation model on neural networks with Gaussian in-degree by taking into account an uncorrelated Gaussian thresholds variability. We derive a mean-field approach and show its relevance by carrying out explicit Monte Carlo simulations. It turns out that such a disorder shifts the position of the percolation transition, impacts the size of the giant cluster, and can even destroy the transition. Moreover, we highlight the occurrence of disorder independent fixed points above the quorum critical value. The mean-field approach enables us to interpret these effects in terms of activation probability. A finite-size analysis enables us to show that the order parameter is weakly self-averaging with an exponent independent on the thresholds disorder. Last, we show that the effects of the thresholds and connectivity disorders cannot be easily discriminated from the measured averaged physical quantities.

We start from a continuous extension of a mean field approach of the quorum percolation model, accounting for the response of in vitro neuronal cultures, to carry out a normal form analysis of the critical behavior. We highlight the effects of nonlinearities associated with this mean field approach even in the close vicinity of the critical point. Statistical properties of random networks with Gaussian in-degree are related to the outcoming links distribution. Finite size analysis of explicit Monte Carlo simulations enables us to confirm the relevance of the mean field approach on such networks and to show that the order parameter is weakly self-averaging; dynamical relaxation is investigated. Furthermore we derive a mean field equation taking into account the effect of inhibitory neurons and discuss the equivalence with a purely excitatory network.

We present a formalism which describes the spatio-temporal evolution of a gel submitted to an autocatalytic chemical reaction to which it is responsive. This theory is based on an extension of a hydrodynamical multi-diffusional approach of the gel dynamics, which is plunged into a chemically active mixture. Emergent and autonomous volume self-oscillation dynamics of the gel are obtained from the nonlinear coupling of the elastic deformation, the chemical kinetics and the transport phenomena, that take place in the system. We apply this formalism to a spherical bead of gel plunged in a Belouzov–Zhabotinsky oscillatory chemical reaction, for which Yoshida et al. have obtained numerous experimental results. The case of a gel immersed in an autocatalytic bistable chemical reaction is also considered. We show that such formalism describes the autonomous volume self-oscillation dynamics of the gel beads.

The Quorum Percolation model has been designed in the context of neurobiology to describe bursts of activity occurring in neuronal cultures from the point of view of statistical physics rather than from a dynamical synchronization approach. It is based upon information propagation on a directed graph with a threshold activation rule; this leads to a phase diagram which exhibits a giant percolation cluster below some critical value mC of the excitability. We describe the main characteristics of the original model and derive extensions according to additional relevant biological features. Firstly, we investigate the effects of an excitability variability on the phase diagram and show that the percolation transition can be destroyed by a sufficient amount of such a disorder; we stress the weakly averaging character of the order parameter and show that connectivity and excitability can be seen as two overlapping aspects of the same reality. Secondly, we elaborate a discrete time stochastic model taking into account the decay originating from ionic leakage through the membrane of neurons and synaptic depression; we give evidence that the decay softens and shifts the transition, and conjecture than decay destroys the transition in the thermodynamical limit. We were able to develop mean-field theories associated with each of the two effects; we discuss the framework of their agreement with Monte Carlo simulations. It turns out that the the critical point mC from which information on the connectivity of the network can be inferred is affected by each of these additional effects. Lastly, we show how dynamical simulations of bursts with an adaptive exponential integrateand- fire model can be interpreted in terms of Quorum Percolation. Moreover, the usefulness of the percolation model including the set of sophistication we investigated can be extended to many scientific fields involving information propagation, such as the spread of rumors in sociology, ethology, ecology.

We investigate the selection of structures generated by diffusive instabilities in asymmetric bistable systems. It is shown that the nonlinear coupling between the zero mode and the different spatial modes emanating from the two homogeneous states can give rise to self-parametric instabilities favoring the occurrence of stable resonant rhombic and superlattice patterns.

Attractive Bose-Einstein condensates are investigated with numerical continuation methods capturing stationary solutions of the Gross-Pitaevskii equation. The branches of stable (elliptic) and unstable (hyperbolic) solutions are found to meet at a critical particle number through a generic Hamiltonian saddle node bifurcation. The condensate decay rates corresponding to macroscopic quantum tunneling, two and three body inelastic collisions, and thermally induced collapse are computed from the exact numerical solutions. These rates show experimentally significant differences with previously published rates. Universal scaling laws stemming from the bifurcation are derived. [S0031-9007(99)08550-6]

The competition between patterns of different symmetries is studied for the generalized Swift-Hohenberg model in the presence of a large quadratic coupling in one dimension (1D) and 2D. It is shown that hexagons with different phase relations may coexist for some values of the control parameter. Numerical experiments exhibiting the effects of linear spatial ramps of the control parameter on this selection are presented. The analogy with recent patterns obtained experimentally in open chemical reactors is also discussed.

Physico-chemical systems driven away from thermal equilibrium usually undergo various types of instabilities leading to the formation of spatio-temporal patterns on macroscopic time and space scales. In two-dimensional geometries, patterns of different symmetries may be simultaneously stable. We study here the phase stability of hexagonal planforms in the framework of amplitude equations and reduced dynamical models close to the instability points.

In this contribution we present a formalism to describe the spatio-temporalevolution of a responsive gel submitted to some autocatalytic chemical reaction. This theory is based on an hydrodynamical multi-diffusional approach of a gel, which is plunged in a chemically active mixture. Emergent volume self-oscillation dynamics of the gel result from the nonlinear coupling of the elastic deformation, the chemical kinetics and the transport phenomenon, that take place in the system. We apply our formalism to the case of the Belouzov-Zhabotinsky oscillatory chemical reaction, for which Yoshida et al. (see in this volume) have obtained many experimental results. In particular we discuss some possible coupling between the gel and the chemical reaction.

From Sustained Oscillations to Stationary Reaction-Diffusion Patterns P. De Kepper et al.- Mechanochemical Instabilities in Active Gels R. Yoshida.- An Excursion in Theoretical Non Linear Chemistry: From Oscillations to Turing Patterns P. Borckmans and S. Metens.- Chemomechanics: Oscillatory Dynamics in Chemoresponsive Gels: J. Boissonade and P. De Kepper.- Structural Approaches on the Toughness in Double Network Hydrogels T. Tominaga et al.- Chemomechanical Dynamics of Responsive Gels S. Metens et al.- Autonomous Rhythmic Drug Delivery Systems Based on Chemical and Biochemomechanical Oscillators R.A. Siegel.- Structure Formation and Nonlinear Dynamics in Polyelectrolyte Responsive Gels A.R. Khokhov et al.- Nonlinear Chemical Dynamics in Synthetic Polymer Systems J.A. Pojman Sr.- Internal Stress as a Link Between Macroscale and Mesoscale Mechanics K. Sekimoto.- On Some Passive and Active Motion in Biology C. Misbah.

Some basic principles of theoretical nonlinear chemistry are introduced to explain an origin of temporal oscillations and spatial bistability, that have both been used in other chapters in conjunction with gels sensitive to such chemical behavior. Because the use of inert gels played such importance in their experimental discovery, elements of the theory of Turing stationary spatial periodic patterns are also discussed.

The imploding dynamics of {sup 85}Rb condensates is obtained in terms of the nonadiabatic dynamical characteristics of the homoclinic bifurcation giving rise to the collapse of the cloud. The agreement with the recent experimental observation of the imploding dynamics is excellent.

We describe the formation of spatial structures generated by diffusive instabilities in bistable systems. The coupling between the different spatial modes emanating from the two homogeneous steady states can then give rise to self-parametric instabilities favoring the occurrence of resonant rhombic or quasiperiodic structures such as superlattices or quasicrystalline patterns.

Non-Isotropic Attractive Bose-Einstein condensates are investigated with Newton and inverse Arnoldi methods. The stationary solutions of the Gross-Pitaevskii equation and their linear stability are computed. Bifurcation diagrams are calculated and used to find the condensate decay rates corresponding to macroscopic quantum tunneling, two-three body inelastic collisions and thermally induced collapse. Isotropic and non-isotropic condensates are compared. The effect of anisotropy on the bifurcation diagram and the decay rates is discussed. Spontaneous isotropization of the condensates is found to occur. The influence of isotropization on the decay rates is characterized near the critical point., Comment: revtex4, 11 figures, 2 tables. Submitted to Phys. Rev. A

Pattern selection in reaction-diffusion systems exhibiting bistability of homogeneous steady states is discussed. In agreement with recent experimental results, we obtain new bifurcation diagrams involving large amplitude structures that arise from the coupling of the spatial critical modes with a quasi-neutral homogeneous mode.

The stationary pattern selection problem in one and two dimensions is discussed on the generalized Swift-Hohenberg model, in the presence of a large quadratic coupling term.

The concentration profiles along the feeding direction of a one side fed gel reactor are analyzed for the iodate-arsenous acid reaction. Multiplicity of inhomogeneous stationary solutions is derived. It is also shown that such profiles may undergo oscillatory bifurcations under long range activation conditions. The bifurcation diagram is analyzed using a Galerkin approximation, the asymptotic validity of which is discussed.