Fractal Fluency: Processing of Fractal Stimuli Across Sight, Sound, and Touch.

People are continually exposed to the rich complexity generated by the repetition of fractal patterns at different size scales. Fractals are prevalent in natural scenery and also in patterns generated by artists and mathematicians. In this chapter, we will investigate the powerful significance of fractals for the human senses. In particular, we propose that fractals with mid-range complexity play a unique role in our visual experiences because the visual system has adapted to these prevalent natural patterns. This adaptation is evident at multiple stages of the visual system, ranging from data acquisition by the eye to processing of this data in the higher visual areas of the brain. Based on these results, we will discuss a fluency model in which the visual system processes mid-complexity fractals with relative ease. This fluency optimizes the observer's capabilities (such as enhanced attention and pattern recognition) and generates an aesthetic experience accompanied by a reduction in the observer's physiological stress levels. In addition to reviewing people's responses to viewing fractals, we will compare these responses to recent research focused on fractal sounds and fractal surface textures. We will extend our fractal fluency model to allow for stimuli across multiple senses.

Neuron arbor geometry is sensitive to the limited-range fractal properties of their dendrites.

Fractal geometry is a well-known model for capturing the multi-scaled complexity of many natural objects. By analyzing three-dimensional images of pyramidal neurons in the rat hippocampus CA1 region, we examine how the individual dendrites within the neuron arbor relate to the fractal properties of the arbor as a whole. We find that the dendrites reveal unexpectedly mild fractal characteristics quantified by a low fractal dimension. This is confirmed by comparing two fractal methods-a traditional "coastline" method and a novel method that examines the dendrites' tortuosity across multiple scales. This comparison also allows the dendrites' fractal geometry to be related to more traditional measures of their complexity. In contrast, the arbor's fractal characteristics are quantified by a much higher fractal dimension. Employing distorted neuron models that modify the dendritic patterns, deviations from natural dendrite behavior are found to induce large systematic changes in the arbor's structure and its connectivity within a neural network. We discuss how this sensitivity to dendrite fractality impacts neuron functionality in terms of balancing neuron connectivity with its operating costs. We also consider implications for applications focusing on deviations from natural behavior, including pathological conditions and investigations of neuron interactions with artificial surfaces in human implants.

Comparison of fractal and grid electrodes for studying the effects of spatial confinement on dissociated retinal neuronal and glial behavior.

Understanding the impact of the geometry and material composition of electrodes on the survival and behavior of retinal cells is of importance for both fundamental cell studies and neuromodulation applications. We investigate how dissociated retinal cells from C57BL/6J mice interact with electrodes made of vertically-aligned carbon nanotubes grown on silicon dioxide substrates. We compare electrodes with different degrees of spatial confinement, specifically fractal and grid electrodes featuring connected and disconnected gaps between the electrodes, respectively. For both electrodes, we find that neuron processes predominantly accumulate on the electrode rather than the gap surfaces and that this behavior is strongest for the grid electrodes. However, the 'closed' character of the grid electrode gaps inhibits glia from covering the gap surfaces. This lack of glial coverage for the grids is expected to have long-term detrimental effects on neuronal survival and electrical activity. In contrast, the interconnected gaps within the fractal electrodes promote glial coverage. We describe the differing cell responses to the two electrodes and hypothesize that there is an optimal geometry that maximizes the positive response of both neurons and glia when interacting with electrodes.

Investigating Fractal Analysis as a Diagnostic Tool That Probes the Connectivity of Hippocampal Neurons.

Many of nature's fractal objects benefit from the favorable functionality that results from their pattern repetition at multiple scales. Our recent research focused on the importance of fractal scaling in establishing connectivity between neurons. Fractal dimension D A of the neuron arbors was shown to relate to the optimization of competing functional constraints-the ability of dendrites to connect to other neurons versus the costs associated with building the dendrites. Here, we consider whether pathological states of neurons might affect this fractal optimization and if changes in D A might therefore be used as a diagnostic tool in parallel with traditional measures like Sholl analyses. We use confocal microscopy to obtain images of CA1 pyramidal neurons in the coronal plane of the dorsal rat hippocampus and construct 3-dimensional models of the dendritic arbors using Neurolucida software. We examine six rodent groups which vary in brain condition (whether they had lesions in the anterior thalamic nuclei, ATN) and experience (their housing environment and experience in a spatial task). Previously, we showed ATN lesions reduced spine density in hippocampal CA1 neurons, whereas enriched housing increased spine density in both ATN lesion and sham rats. Here, we investigate whether ATN lesions and experience also effect the complexity and connectivity of CA1 dendritic arbors. We show that sham rats exposed to enriched housing and spatial memory training exhibited higher complexity (as measured by D A ) and connectivity compared to other groups. When we categorize the rodent groups into those with or without lesions, we find that both categories achieve an optimal balance of connectivity with respect to material cost. However, the D A value used to achieve this optimization does not change between these two categories, suggesting any morphological differences induced by the lesions are too small to influence the optimization process. Accordingly, we highlight considerations associated with applying our technique to publicly accessible repositories of neuron images with a broader range of pathological conditions.

Controlled assembly of retinal cells on fractal and Euclidean electrodes.

Controlled assembly of retinal cells on artificial surfaces is important for fundamental cell research and medical applications. We investigate fractal electrodes with branches of vertically-aligned carbon nanotubes and silicon dioxide gaps between the branches that form repeating patterns spanning from micro- to milli-meters, along with single-scaled Euclidean electrodes. Fluorescence and electron microscopy show neurons adhere in large numbers to branches while glial cells cover the gaps. This ensures neurons will be close to the electrodes' stimulating electric fields in applications. Furthermore, glia won't hinder neuron-branch interactions but will be sufficiently close for neurons to benefit from the glia's life-supporting functions. This cell 'herding' is adjusted using the fractal electrode's dimension and number of repeating levels. We explain how this tuning facilitates substantial glial coverage in the gaps which fuels neural networks with small-world structural characteristics. The large branch-gap interface then allows these networks to connect to the neuron-rich branches.

Aesthetics and Psychological Effects of Fractal Based Design.

Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal 'global-forest' designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant wellbeing. These designs are composite fractal patterns consisting of individual fractal 'tree-seeds' which combine to create a 'global fractal forest.' The local 'tree-seed' patterns, global configuration of tree-seed locations, and overall resulting 'global-forest' patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal ('tree-seed') patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity 'global-forest' patterns consisting of 'tree-seed' components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant wellbeing.

How neurons exploit fractal geometry to optimize their network connectivity.

We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three-dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal-like behavior well-described by an 'effective' fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D-dependent variations in inter-neuron connectivity along with the associated costs, we propose that their D values reflect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes).

The Roles of an Aluminum Underlayer in the Biocompatibility and Mechanical Integrity of Vertically Aligned Carbon Nanotubes for Interfacing with Retinal Neurons.

Retinal implant devices are becoming an increasingly realizable way to improve the vision of patients blinded by photoreceptor degeneration. As an electrode material that can improve restored visual acuity, carbon nanotubes (CNTs) excel due to their nanoscale topography, flexibility, surface chemistry, and double-layer capacitance. If vertically aligned carbon nanotubes (VACNTs) are biocompatible with retinal neurons and mechanically robust, they can further improve visual acuity-most notably in subretinal implants-because they can be patterned into high-aspect-ratio, micrometer-size electrodes. We investigated the role of an aluminum (Al) underlayer beneath an iron (Fe) catalyst layer used in the growth of VACNTs by chemical vapor deposition (CVD). In particular, we cultured dissociated retinal cells for three days in vitro (DIV) on unfunctionalized and oxygen plasma functionalized VACNTs grown from a Fe catalyst (Fe and Fe + Pl preparations, where Pl signifies the plasma functionalization) and an Fe catalyst with an Al underlayer (Al/Fe and Al/Fe + Pl preparations). The addition of the Al layer increased the mechanical integrity of the VACNT interface and enhanced retinal neurite outgrowth over the Fe preparation. Unexpectedly, the extent of neurite outgrowth was significantly greater in the Al/Fe than in the Al/Fe+Pl preparation, suggesting plasma functionalization can negatively impact biocompatibility for some VACNT preparations. Additionally, we show our VACNT growth process for the Al/Fe preparation can support neurite outgrowth for up to 7 DIV. By demonstrating the retinal neuron biocompatibility, mechanical integrity, and pattern control of our VACNTs, this work offers VACNT electrodes as a solution for improving the restored visual acuity provided by modern retinal implants.

Relaxing Floors: Fractal Fluency in the Built Environment.

This year's cover artists are members of a newly formed team of designers and scientists known as the Science and Design Laboratory, along with flooring manufacturing experts from the Mohawk Group. This unique collab-oration creates patterns for installation on the floors of versatile commercial, public and private spaces including airports, hospitals, offices and homes. Their goal is to create human-centered designs based on psychology experiments that investigate the positive impacts of viewing fractal patterns. These include reduced physiological stress levels, enhanced cognitive skills, and heightened concentration. Here, the fractal construction process and the resulting fractal characteristics of these designs are explained.