Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Effect of dark matter halo substructures on galaxy rotation curves

The effect of halo substructures on galaxy rotation curves is investigated in this paper using a simple model of dark matter clustering. A dark matter halo density profile is developed based only on the scale free nature of clustering that leads to a statistically self-similar distribution of the substructures at galactic scale. Semi-analytical method is used to derive rotation curves for such a clumpy dark matter density profile. It is found that the halo substructures significantly affect the galaxy velocity field. Based on the fractal geometry of the halo, this self-consistent model predicts an NFW-like rotation curve and a scale free power spectrum of the rotation velocity fluctuations.Comment: 6 pages, 3 figures. Accepted for publication in ApJ. The definitive version will be available at http://iopscience.iop.org

Practical applications of small-angle neutron scattering.

Recent improvements in beam-line accessibility and technology have led to small-angle neutron scattering (SANS) becoming more frequently applied to materials problems. SANS has been used to study the assembly, dispersion, alignment and mixing of nanoscale condensed matter, as well as to characterise the internal structure of organic thin films, porous structures and inclusions within steel. Using time-resolved SANS, growth mechanisms in materials systems and soft matter phase transitions can also be explored. This review is intended for newcomers to SANS as well as experts. Therefore, the basic knowledge required for its use is first summarised. After this introduction, various examples are given of the types of soft and hard matter that have been studied by SANS. The information that can be extracted from the data is highlighted, alongside the methods used to obtain it. In addition to presenting the findings, explanations are provided on how the SANS measurements were optimised, such as the use of contrast variation to highlight specific parts of a structure. Emphasis is placed on the use of complementary techniques to improve data quality (e.g. using other scattering methods) and the accuracy of data analysis (e.g. using microscopy to separately determine shape and size). This is done with a view to providing guidance on how best to design and analyse future SANS measurements on materials not listed below

Fracton pairing mechanism for "strange" superconductors: Self-assembling organic polymers and copper-oxide compounds

Self-assembling organic polymers and copper-oxide compounds are two classes of "strange" superconductors, whose challenging behavior does not comply with the traditional picture of Bardeen, Cooper, and Schrieffer (BCS) superconductivity in regular crystals. In this paper, we propose a theoretical model that accounts for the strange superconducting properties of either class of the materials. These properties are considered as interconnected manifestations of the same phenomenon: We argue that superconductivity occurs in the both cases because the charge carriers (i.e., electrons or holes) exchange {\it fracton excitations}, quantum oscillations of fractal lattices that mimic the complex microscopic organization of the strange superconductors. For the copper oxides, the superconducting transition temperature $T_c$ as predicted by the fracton mechanism is of the order of $\sim 150$ K. We suggest that the marginal ingredient of the high-temperature superconducting phase is provided by fracton coupled holes that condensate in the conducting copper-oxygen planes owing to the intrinsic field-effect-transistor configuration of the cuprate compounds. For the gate-induced superconducting phase in the electron-doped polymers, we simultaneously find a rather modest transition temperature of $\sim (2-3)$ K owing to the limitations imposed by the electron tunneling processes on a fractal geometry. We speculate that hole-type superconductivity observes larger onset temperatures when compared to its electron-type counterpart. This promises an intriguing possibility of the high-temperature superconducting states in hole-doped complex materials. A specific prediction of the present study is universality of ac conduction for $T\gtrsim T_c$.Comment: 12 pages (including separate abstract page), no figure

Complexity Phenomena and ROMA of the Magnetospheric Cusp, Hydrodynamic Turbulence, and the Cosmic Web

Dynamic Complexity is a phenomenon exhibited by a nonlinearly interacting system within which multitudes of different sizes of large scale coherent structures emerge, resulting in a globally nonlinear stochastic behavior vastly different from that could be surmised from the underlying equations of interaction. The hallmark of such nonlinear, complex phenomena is the appearance of intermittent fluctuating events with the mixing and distributions of correlated structures at all scales. We briefly review here a relatively recent method, ROMA (rank-ordered multifractal analysis), explicitly constructed to analyze the intricate details of the distribution and scaling of such types of intermittent structures. This method is then applied to the analyses of selected examples related to the dynamical plasmas of the cusp region of the magnetosphere, velocity fluctuations of classical hydrodynamic turbulence, and the distribution of the structures of the cosmic gas obtained through large scale, moving mesh simulations. Differences and similarities of the analyzed results among these complex systems will be contrasted and highlighted. The first two examples have direct relevance to the geospace environment and are summaries of previously reported findings. The third example on the cosmic gas, though involving phenomena much larger in spatiotemporal scales, with its highly compressible turbulent behavior and the unique simulation technique employed in generating the data, provides direct motivations of applying such analysis to studies of similar multifractal processes in various extreme environments. These new results are both exciting and intriguing.Comment: 36 page

Practical applications of small-angle neutron scattering.

Recent improvements in beam-line accessibility and technology have led to small-angle neutron scattering (SANS) becoming more frequently applied to materials problems. SANS has been used to study the assembly, dispersion, alignment and mixing of nanoscale condensed matter, as well as to characterise the internal structure of organic thin films, porous structures and inclusions within steel. Using time-resolved SANS, growth mechanisms in materials systems and soft matter phase transitions can also be explored. This review is intended for newcomers to SANS as well as experts. Therefore, the basic knowledge required for its use is first summarised. After this introduction, various examples are given of the types of soft and hard matter that have been studied by SANS. The information that can be extracted from the data is highlighted, alongside the methods used to obtain it. In addition to presenting the findings, explanations are provided on how the SANS measurements were optimised, such as the use of contrast variation to highlight specific parts of a structure. Emphasis is placed on the use of complementary techniques to improve data quality (e.g. using other scattering methods) and the accuracy of data analysis (e.g. using microscopy to separately determine shape and size). This is done with a view to providing guidance on how best to design and analyse future SANS measurements on materials not listed below

Phase separation and self-assembly in vitrimers: hierarchical morphology of molten and semi-crystalline polyethylene/dioxaborolane maleimide systems

Vitrimers - a class of polymer networks which are covalently crosslinked and insoluble like thermosets, but flow when heated like thermoplastics - contain dynamic links and/or crosslinks that undergo an associative exchange reaction. These dynamic crosslinks enable vitrimers to have interesting mechanical/rheological behavior, self-healing, adhesive, and shape memory properties. We demonstrate that vitrimers can self-assemble into complex meso- and nanostructures when crosslinks and backbone monomers strongly interact. Vitrimers featuring polyethylene (PE) as the backbone and dioxaborolane maleimide as the crosslinkable moiety were studied in both the molten and semi-crystalline states. We observed that PE vitrimers macroscopically phase separated into dioxaborolane maleimide rich and poor regions, and characterized the extent of phase separation by optical transmission measurements. This phase separation can explain the relatively low insoluble fractions and overall crystallinities of PE vitrimers. Using synchrotron-sourced small-angle X-ray scattering (SAXS), we discovered that PE vitrimers and their linear precursors micro-phase separated into hierarchical nanostructures. Fitting of the SAXS patterns to a scattering model strongly suggests that the nanostructures - which persist in both the melt and amorphous fraction of the semi-crystalline state - may be described as dioxaborolane maleimide rich aggregates packed in a mass fractal arrangement. These findings of hierarchical meso- and nanostructures point out that incompatibility effects between network components and resulting self-assembly must be considered for understanding behavior and the rational design of vitrimer materials

Universal Scaling Relations in Scale-Free Structure Formation

A large number of astronomical phenomena exhibit remarkably similar scaling relations. The most well-known of these is the mass distribution $\mathrm{d} N/\mathrm{d} M\propto M^{-2}$ which (to first order) describes stars, protostellar cores, clumps, giant molecular clouds, star clusters and even dark matter halos. In this paper we propose that this ubiquity is not a coincidence and that it is the generic result of scale-free structure formation where the different scales are uncorrelated. We show that all such systems produce a mass function proportional to $M^{-2}$ and a column density distribution with a power law tail of $\mathrm{d} A/\mathrm{d} \ln\Sigma\propto\Sigma^{-1}$. In the case where structure formation is controlled by gravity the two-point correlation becomes $\xi_{2D}\propto R^{-1}$. Furthermore, structures formed by such processes (e.g. young star clusters, DM halos) tend to a $\rho\propto R^{-3}$ density profile. We compare these predictions with observations, analytical fragmentation cascade models, semi-analytical models of gravito-turbulent fragmentation and detailed "full physics" hydrodynamical simulations. We find that these power-laws are good first order descriptions in all cases.Comment: 12 pages, 6 figures, 2 tables, submitted to MNRA