Pattern formation from spatially heterogeneous reaction–diffusion systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

What Is Consciousness? -Artificial Intelligence, Real Intelligence, Quantum Mind, and Qualia

We approach the question, "What is Consciousness?'' in a new way, not as Descartes' "systematic doubt'', but as how organisms find their way in their world. Finding one's way involves finding possible uses of features of the world that might be beneficial or avoiding those that might be harmful. "Possible uses of X to accomplish Y'' are "Affordances''. The number of uses of X is indefinite, the different uses are unordered and are not deducible from one another. All biological adaptations are either affordances seized by heritable variation and selection or, far faster, by the organism acting in its world finding uses of X to accomplish Y. Based on this, we reach rather astonishing conclusions: 1) Strong AI is not possible. Universal Turing Machines cannot "find'' novel affordances. 2) Brain-mind is not purely classical physics for no classical physics system can be an analogue computer whose dynamical behavior can be isomorphic to "possible uses''. 3) Brain mind must be partly quantum - supported by increasing evidence at 6.0 sigma to 7.3 Sigma. 4) Based on Heisenberg's interpretation of the quantum state as "Potentia'' converted to "Actuals'' by Measurement, a natural hypothesis is that mind actualizes Potentia. This is supported at 5.2 Sigma. Then Mind's actualization of entangled brain-mind-world states are experienced as qualia and allow "seeing'' or "perceiving'' of uses of X to accomplish Y. We can and do jury-rig. Computers cannot. 5) Beyond familiar quantum computers, we consider Trans-Turing-Systems.

Bespoke Turing Systems

Reaction–diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there are many prototypical “Turing systems” available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider the reverse problem. Namely, suppose we know the parameter region associated with the reaction kinetics in which patterning is required—we present a constructive framework for identifying systems that will exhibit the Turing instability within this region, whilst in addition often allowing selection of desired patterning features, such as spots, or stripes. In particular, we show how to build a system of two populations governed by polynomial morphogen kinetics such that the: patterning parameter domain (in any spatial dimension), morphogen phases (in any spatial dimension), and even type of resulting pattern (in up to two spatial dimensions) can all be determined. Finally, by employing spatial and temporal heterogeneity, we demonstrate that mixed mode patterns (spots, stripes, and complex prepatterns) are also possible, allowing one to build arbitrarily complicated patterning landscapes. Such a framework can be employed pedagogically, or in a variety of contemporary applications in designing synthetic chemical and biological patterning systems. We also discuss the implications that this freedom of design has on using reaction–diffusion systems in biological modelling and suggest that stronger constraints are needed when linking theory and experiment, as many simple patterns can be easily generated given freedom to choose reaction kinetics.

Turing patterns are common but not robust

Turing patterns (TPs) underlie many fundamental developmental processes, but they operate over narrow parameter ranges, raising the conundrum of how evolution can ever discover them. Here we explore TP design space to address this question and to distill design rules. We exhaustively analyze 2- and 3-node biological candidate Turing systems: crucially, network structure alone neither determines nor guarantees emergent TPs. A surprisingly large fraction (>60%) of network design space can produce TPs, but these are sensitive to even subtle changes in parameters, network structure and regulatory mechanisms. This implies that TP networks are more common than previously thought, and evolution might regularly encounter prototypic solutions. Importantly, we deduce compositional rules for TP systems that are almost necessary and sufficient (≈96% of TP networks contain them, and ≈95% of networks implementing them produce TPs). This comprehensive network atlas provides the blueprints for identifying natural TPs, and for engineering synthetic systems.