Abstract:
The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product networks is considered. To this end we operate in the linear regime and expand the time dependent perturbation on a basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, associated to each of the individual networks that build the Cartesian product. The dispersion relation which controls the onset of the instability depends on a set of discrete wave- lenghts, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constituive sub-graphs. Multiplex networks are also obtained under specific prescriptions. In this case, the criteria for the instability reduce to compact explicit formulae. Numerical simulations carried out for the Mimura-Murray reaction kinetics confirm the adequacy of the proposed theory.

Abstract:
We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from vector fields with one fixed point and a supercritical Hopf bifurcation. For the Brusselator and the Ginzburg-Landau reaction-diffusion equations, we obtain the bifurcation diagrams associated with the transition between time periodic solutions and asymptotically stable solutions (Turing patterns). In two-component systems of reaction-diffusion equations, we show that the existence of Turing instabilities is neither necessary nor sufficient for the existence of Turing pattern type solutions. Turing patterns can exist on both sides of the Hopf bifurcation associated to the local vector field, and, depending on the initial conditions, time periodic and stable solutions can coexist.

Abstract:
Reaction-diffusion systems may lead to the formation of steady state heterogeneous spatial patterns, known as Turing patterns. Their mathematical formulation is important for the study of pattern formation in general and play central roles in many fields of biology, such as ecology and morphogenesis. In the present study we focus on the role of Turing patterns in describing the abundance distribution of predator and prey species distributed in patches in a scale free network structure. We extend the original model proposed by Nakao and Mikhailov by considering food chains with several interacting pairs of preys and predators. We identify patterns of species distribution displaying high degrees of apparent competition driven by Turing instabilities. Our results provide further indication that differences in abundance distribution among patches may be, at least in part, due to self organized Turing patterns, and not necessarily to intrinsic environmental heterogeneity.

Abstract:
As proposed by Alan Turing in 1952 as a ubiquitous mechanism for nonequilibrium pattern formation, diffusional effects may destabilize uniform distributions of reacting chemical species and lead to both spatially and temporally heterogeneous patterns. While stationary Turing patterns are broadly known, the oscillatory instability, leading to traveling waves in continuous media and also called the wave bifurcation, is rare for chemical systems. Here, we extend the analysis by Turing to general networks and apply it to ecological metapopulations of biological species with dispersal connections between habitats. Remarkably, the oscillatory Turing instability does not lead to wave patterns in networks, but to spontaneous development of heterogeneous oscillations and possible extinction of some species, even though they are absent for isolated populations. Furthermore, our theoretical analysis reveals that this instability is more common in ecological metapopulations than in chemical reactions. Indeed, we find the instabilities for all possible food webs with three predator or prey species, under various assumptions about the mobility of individual species and nonlinear interactions between them. Therefore, we suggest that the oscillatory Turing instability is generic and must play a fundamental role in metapopulation dynamics, providing a common mechanism for dispersal-induced destabilization of ecosystems.

Abstract:
We experimentally investigate the interplay of Turing and Faraday (modulational) instabilities in a bistable passive nonlinear resonator. The Faraday branch is induced via parametric resonance owing to a periodic modulation of the resonator dispersion. We show that the bistable switching dynamics is dramatically affected by the competition between the two instability mechanisms, which dictates two completely novel scenarios. At low detunings from resonance switching occurs between the stable stationary lower branch and the Faraday-unstable upper branch, whereas at high detunings we observe the crossover between the Turing and Faraday periodic structures. The results are well explained in terms of the universal Lugiato-Lefever model.

Abstract:
The Turing instability is a paradigmatic route to patterns formation in reaction-diffusion systems. Following a diffusion-driven instability, homogeneous fixed points can become unstable when subject to external perturbation. As a consequence, the system evolves towards a stationary, nonhomogeneous attractor. Stable patterns can be also obtained via oscillation quenching of an initially synchronous state of diffusively coupled oscillators. In the literature this is known as the oscillation death phenomenon. Here we show that oscillation death is nothing but a Turing instability for the first return map associated to the excitable system in its synchronous periodic state. In particular we obtain a set of closed conditions for identifying the domain in the parameters space that yields the instability. This is a natural generalisation of the original Turing relations, to the case where the homogeneous solution of the examined system is a periodic function of time. The obtained framework applies to systems embedded in continuum space, as well as those defined on a network-like support. The predictive ability of the theory is tested numerically, using different reaction schemes.

Abstract:
Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion equation. The time delay changes remarkably the oscillation frequency, the intrinsic wave vector, and the intensities of both Turing and Hopf modes. The application of appropriate time delay can control the competition between the Turing and Hopf modes. Analysis shows that individual or both feedbacks can realize the control of the transformation between the Turing and Hopf patterns. Two dimensional numerical simulations validate the analytical results.

Abstract:
We improve the results by Siegelmann & Sontag (1995) by providing a novel and parsimonious constructive mapping between Turing Machines and Recurrent Artificial Neural Networks, based on recent developments of Nonlinear Dynamical Automata. The architecture of the resulting R-ANNs is simple and elegant, stemming from its transparent relation with the underlying NDAs. These characteristics yield promise for developments in machine learning methods and symbolic computation with continuous time dynamical systems. A framework is provided to directly program the R-ANNs from Turing Machine descriptions, in absence of network training. At the same time, the network can potentially be trained to perform algorithmic tasks, with exciting possibilities in the integration of approaches akin to Google DeepMind's Neural Turing Machines.

Abstract:
In this paper, the partial recursive function is constructed by Hopfield neural networks. The partial recursive function is equivalent with Turing machine, the computability of Hopfield neural networks is therefore equivalent with Turing machine.

Abstract:
Cellular signaling networks display complex architecture. Defining the design principle of this architecture is crucial for our understanding of various biological processes. Using a mathematical model for three-node feed-forward loops, we identify that the organization of motifs in specific manner within the network serves as an important regulator of signal processing. Further, incorporating a systemic stochastic perturbation to the model we could propose a possible design principle, for higher-order organization of motifs into larger networks in order to achieve specific biological output. The design principle was then verified in a large, complex human cancer signaling network. Further analysis permitted us to classify signaling nodes of the network into robust and vulnerable nodes as a result of higher order motif organization. We show that distribution of these nodes within the network at strategic locations then provides for the range of features displayed by the signaling network.