How neurons exploit fractal geometry to optimize their network connectivity

Hippocampome.org 2.0 is a knowledge base enabling data-driven spiking neural network simulations of rodent hippocampal circuits

Hippocampome.org is a mature open-access knowledge base of the rodent hippocampal formation focusing on neuron types and their properties. Previously, Hippocampome.org v1.0 established a foundational classification system identifying 122 hippocampal neuron types based on their axonal and dendritic morphologies, main neurotransmitter, membrane biophysics, and molecular expression (Wheeler et al., 2015). Releases v1.1 through v1.12 furthered the aggregation of literature-mined data, including among others neuron counts, spiking patterns, synaptic physiology, in vivo firing phases, and connection probabilities. Those additional properties increased the online information content of this public resource over 100-fold, enabling numerous independent discoveries by the scientific community. Hippocampome.org v2.0, introduced here, besides incorporating over 50 new neuron types, now recenters its focus on extending the functionality to build real-scale, biologically detailed, data-driven computational simulations. In all cases, the freely downloadable model parameters are directly linked to the specific peer-reviewed empirical evidence from which they were derived. Possible research applications include quantitative, multiscale analyses of circuit connectivity and spiking neural network simulations of activity dynamics. These advances can help generate precise, experimentally testable hypotheses and shed light on the neural mechanisms underlying associative memory and spatial navigation.

Hippocampome.org v2.0: a knowledge base enabling data-driven spiking neural network simulations of rodent hippocampal circuits

Hippocampome.org is a mature open-access knowledge base of the rodent hippocampal formation focusing on neuron types and their properties. Hippocampome.org v1.0 established a foundational classification system identifying 122 hippocampal neuron types based on their axonal and dendritic morphologies, main neurotransmitter, membrane biophysics, and molecular expression. Releases v1.1 through v1.12 furthered the aggregation of literature-mined data, including among others neuron counts, spiking patterns, synaptic physiology, in vivo firing phases, and connection probabilities. Those additional properties increased the online information content of this public resource over 100-fold, enabling numerous independent discoveries by the scientific community. Hippocampome.org v2.0, introduced here, besides incorporating over 50 new neuron types, now recenters its focus on extending the functionality to build real-scale, biologically detailed, data-driven computational simulations. In all cases, the freely downloadable model parameters are directly linked to the specific peer-reviewed empirical evidence from which they were derived. Possible research applications include quantitative, multiscale analyses of circuit connectivity and spiking neural network simulations of activity dynamics. These advances can help generate precise, experimentally testable hypotheses and shed light on the neural mechanisms underlying associative memory and spatial navigation.

Fractal Electronics for Stimulating and Sensing Neural Networks: Enhanced Electrical, Optical, and Cell Interaction Properties

Imagine a world in which damaged parts of the body - an arm, an eye, and ultimately a region of the brain - can be replaced by artificial implants capable of restoring or even enhancing human performance. The associated improvements in the quality of human life would revolutionize the medical world and produce sweeping changes across society. In this chapter, we discuss several approaches to the fabrication of fractal electronics designed to interface with neural networks. We consider two fundamental functions - stimulating electrical signals in the neural networks and sensing the location of the signals as they pass through the network. Using experiments and simulations, we discuss the favorable electrical performances that arise from adopting fractal rather than traditional Euclidean architectures. We also demonstrate how the fractal architecture induces favorable physical interactions with the cells they interact with, including the ability to direct the growth of neurons and glia to specific regions of the neural-electronic interface.

Fractal Resonance: Can Fractal Geometry Be Used to Optimize the Connectivity of Neurons to Artificial Implants?

In parallel to medical applications, exploring how neurons interact with the artificial interface of implants in the human body can be used to learn about their fundamental behavior. For both fundamental and applied research, it is important to determine the conditions that encourage neurons to maintain their natural behavior during these interactions. Whereas previous biocompatibility studies have focused on the material properties of the neuron-implant interface, here we discuss the concept of fractal resonance - the possibility that favorable connectivity properties might emerge by matching the fractal geometry of the implant surface to that of the neurons.To investigate fractal resonance, we first determine the degree to which neurons are fractal and the impact of this fractality on their functionality. By analyzing three-dimensional images of rat hippocampal neurons, we find that the way their dendrites fork and weave through space is important for generating their fractal-like behavior. By modeling variations in neuron connectivity along with the associated energetic and material costs, we highlight how the neurons' fractal dimension optimizes these constraints. To simulate neuron interactions with implant interfaces, we distort the neuron models away from their natural form by modifying the dendrites' fork and weaving patterns. We find that small deviations can induce large changes in fractal dimension, causing the balance between connectivity and cost to deteriorate rapidly. We propose that implant surfaces should be patterned to match the fractal dimension of the neurons, allowing them to maintain their natural functionality as they interact with the implant.

Morphology and Fractal-Based Classifications of Neurons and Microglia in Two and Three Dimensions

Microglia and neurons live physically intertwined, intimately related structurally and functionally in a dynamic relationship in which microglia change continuously over a much shorter timescale than do neurons. Although microglia may unwind and depart from the neurons they attend under certain circumstances, in general, together both contribute to the fractal topology of the brain that defines its computational capabilities. Both neuronal and microglial morphologies are well-described using fractal analysis complementary to more traditional measures. For neurons, the fractal dimension has proved valuable for classifying dendritic branching and other neuronal features relevant to pathology and development. For microglia, fractal geometry has substantially contributed to classifying functional categories, where, in general, the more pathological the biological status, the lower the fractal dimension for individual cells, with some exceptions, including hyper-ramification. This chapter provides a review of the intimate relationships between neurons and microglia, by introducing 2D and 3D fractal analysis methodology and its applications in neuron-microglia function in health and disease.

Fractal Time Series: Background, Estimation Methods, and Performances

Over the past 40 years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.

Multiquanta flux jumps in superconducting fractal

We study the magnetic field response of millimeter scale fractal Sierpinski gaskets (SG) assembled of superconducting equilateral triangular patches. Directly imaged quantitative induction maps reveal hierarchical periodic filling of enclosed void areas with multiquanta magnetic flux, which jumps inside the voids in repeating bundles of individual flux quanta Φ0. The number Ns of entering flux quanta in different triangular voids of the SG is proportional to the linear size s of the void, while the field periodicity of flux jumps varies as 1/s. We explain this behavior by modeling the triangular voids in the SG with effective superconducting rings and by calculating their response following the London analysis of persistent currents, Js, induced by the applied field Ha and by the entering flux. With changing Ha, Js reaches a critical value in the vertex joints that connect the triangular superconducting patches and allows the giant flux jumps into the SG voids through phase slips or multiple Abrikosov vortex transfer across the vertices. The unique flux behavior in superconducting SG patterns, may be used to design tunable low-loss resonators with multi-line high-frequency spectrum for microwave technologies.

Hippocampal and Medial Prefrontal Cortex Fractal Spiking Patterns Encode Episodes and Rules

A central question in neuroscience is how the brain represents and processes information to guide behavior. The principles that organize brain computations are not fully known, and could include scale-free, or fractal patterns of neuronal activity. Scale-free brain activity may be a natural consequence of the relatively small subsets of neuronal populations that respond to task features, i.e., sparse coding. The size of the active subsets constrains the possible sequences of inter-spike intervals (ISI), and selecting from this limited set may produce firing patterns across wide-ranging timescales that form fractal spiking patterns. To investigate the extent to which fractal spiking patterns corresponded with task features, we analyzed ISIs in simultaneously recorded populations of CA1 and medial prefrontal cortical (mPFC) neurons in rats performing a spatial memory task that required both structures. CA1 and mPFC ISI sequences formed fractal patterns that predicted memory performance. CA1 pattern duration, but not length or content, varied with learning speed and memory performance whereas mPFC patterns did not. The most common CA1 and mPFC patterns corresponded with each region's cognitive function: CA1 patterns encoded behavioral episodes which linked the start, choice, and goal of paths through the maze whereas mPFC patterns encoded behavioral "rules" which guided goal selection. mPFC patterns predicted changing CA1 spike patterns only as animals learned new rules. Together, the results suggest that CA1 and mPFC population activity may predict choice outcomes by using fractal ISI patterns to compute task features.

Development of abnormalities at the neuromuscular junction in the SOD1-G93A mouse model of ALS: dysfunction then disruption of postsynaptic structure precede overt motor symptoms

Introduction

The ultimate deficit in amyotrophic lateral sclerosis (ALS) is neuromuscular junction (NMJ) loss, producing permanent paralysis, ultimately in respiratory muscles. However, understanding the functional and structural deficits at NMJs prior to this loss is crucial for therapeutic strategy design. Should early interventions focus on reversing denervation, or supporting largely intact NMJs that are functionally impaired? We therefore determined when functional and structural deficits appeared in diaphragmatic NMJs relative to the onset of hindlimb tremor (the first overt motor symptoms) in vivo in the SOD1-G93A mouse model of ALS.

Materials and methods

We employed electrophysiological recording of NMJ postsynaptic potentials for spontaneous and nerve stimulation-evoked responses. This was correlated with fluorescent imaging microscopy of the postsynaptic acetylcholine receptor (AChR) distribution throughout the postnatal developmental timecourse from 2 weeks to early symptomatic ages.

Results

Significant reduction in the amplitudes of spontaneous miniature endplate potentials (mEPPs) and evoked EPPs emerged only at early symptomatic ages (in our colony, 18-22 weeks). Reductions in mEPP frequency, number of vesicles per EPP, and EPP rise time were seen earlier, at 16weeks, but this reversed by early symptomatic ages. However, the earliest and most striking impairment was an inability to maintain EPP amplitude during a 20 Hz stimulus train, which appeared 6 weeks before overt in vivo motor symptoms. Despite this, fluorescent α-bungarotoxin labelling revealed no systematic, progressive changes in 11 comprehensive NMJ morphological parameters (area, shape, compactness, number of acetylcholine receptor, AChR, regions, etc.) with disease progression. Rather, while NMJs were largely normally-shaped, from 16 weeks there was a progressive and substantial disruption in AChR concentration and distribution within the NMJ footprint.

Discussion

Thus, NMJ functional deficits appear at least 6 weeks before motor symptoms in vivo, while structural deficits occur 4 weeks later, and predominantly within NMJs. These data suggest initial therapies focused on rectifying suboptimal NMJ function could produce effective relief of symptoms of weakness.

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

The structure and diversity of retinal ganglion cells in the masked greenling Hexagrammos octogrammus Pallas, 1814 (Pisces: Scorpaeniformes: Hexagrammidae)

The authors studied the structure and diversity of retinal ganglion cells (GC) in the masked greenling Hexagrammos octogrammus. In vivo labelling with horseradish peroxidase revealed GCs of various structures in retinal wholemounts. A total of 154 cells were camera lucida drawn, and their digital models were generated. Each cell was characterized by 17 structural and topological parameters. Using nine clustering algorithms, a variety of clusterings were obtained. The optimum clustering was found using silhouette analysis. It was based on a set of three variables associated with dendritic field size and dendrite stratification depth in the retina. A total of nine cell types were discovered. A number of non-parametric tests showed significant pair-wise between-cluster differences in at least four parameters with medium and large effect sizes. Three large-field types differed mainly in dendritic field size, total dendrite length, level of dendrite stratification in the retina and position of somata. Six medium- to small-field types differed mainly in the structural complexity of dendritic arbors and level of dendrite arborization. Cells similar and obviously homologous to types 1-4 were identified in many fish species, including teleosts. Potential homologues of type 5 cells were identified in fewer teleost species. Cells similar to types 6-9 in relative dendritic field size and dendrite arborization pattern were also described in several teleostean species. Nonetheless, their homology is more questionable as their stratification patterns do not match so well as they do in large types. Potential functional matches of the GC types were identified in a number of teleostean species. Type 1 and 2 cells probably match spontaneously active units with the large receptive field centre, so-called dimming and lightening detectors; type 4 may be a counterpart of changing contrast detectors with medium receptive field centre size preferring fast-moving stimuli. Type 3 (biplexiform) cells have no obvious functional matches. Probable functional matches of types 6, 8 and 9 belong to ON-centre elements with small receptive fields such as ON-type direction-selective cells, ON-type spot detectors or ON-type spontaneously active units. Type 5 and 7 cells may match ON-OFF type units, in particular, changing contrast detectors or orientation-selective units. Potential functional matches of GC types presently described are involved in a wide spectrum of visual reactions related to adaptation to gradual change in illumination, predator escape, prey detection and capture, habitat selection and social behaviour.

Fractal analysis of healthy and diseased vasculature in pediatric Moyamoya disease

BACKGROUND AND PURPOSE

Fractal dimension is an objective metric for the notion of structural complexity. We sought to investigate differences in structural complexity between healthy and affected territories of cerebral vasculature in moyamoya, as well as associated scalp vasculature and native transdural collaterals in patients with moyamoya by comparing their respective fractal dimensions.

METHODS

Our cohort consisted of 15 transdural collaterals from 12 patients with unilateral anterior circulation moyamoya. Frames of distal arterial vasculature from internal and external carotid angiograms were selected then automatically segmented and also manually annotated by a cerebrovascular surgeon. In the affected hemisphere, the region with transdural collateral supply was compared to the contralateral region. The resulting skeletonized angiograms were analyzed for their fractal dimensions.

RESULTS

We found the average fractal dimension (Df) of the moyamoya-side ICA was 1.82 with slightly different means for the anteroposterial (AP) and lateral views (mean  =  1.82; mean  =  1.81). The overall mean for healthy cerebral vasculature was also found to be 1.82 (AP: mean  =  1.83; lateral: mean  =  1.81). Mean Df of native transdural collaterals was found to be 1.82 (AP: mean  =  1.83; lateral: mean  =  1.81). The mean Df difference between autosegmented and manually segmented images was 0.013.

CONCLUSION

In accordance with the clinical understanding of moyamoya disease, the distal arterial structural complexity is not affected in moyamoya, and is maintained by transdural collaterals formed by vasculogenesis. Autosegmentation of cerebral vasculature is also shown to be accurate when compared to manual segmentation.

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.

Guidelines for a Morphometric Analysis of Prokaryotic and Eukaryotic Cells by Scanning Electron Microscopy

The invention of a scanning electron microscopy (SEM) pushed the imaging methods and allowed for the observation of cell details with a high resolution. Currently, SEM appears as an extremely useful tool to analyse the morphology of biological samples. The aim of this paper is to provide a set of guidelines for using SEM to analyse morphology of prokaryotic and eukaryotic cells, taking as model cases Escherichia coli bacteria and B-35 rat neuroblastoma cells. Herein, we discuss the necessity of a careful sample preparation and provide an optimised protocol that allows to observe the details of cell ultrastructure (≥ 50 nm) with a minimum processing effort. Highlighting the versatility of morphometric descriptors, we present the most informative parameters and couple them with molecular processes. In this way, we indicate the wide range of information that can be collected through SEM imaging of biological materials that makes SEM a convenient screening method to detect cell pathology.

Halogen Bonded Chiral Emitters: Generation of Chiral Fractal Architecture with Amplified Circularly Polarized Luminescence

Self-assembled chiroptical materials have attracted considerable attention due to their great applications in wide fields. During the chiral self-assembly, it remains unknown how achiral molecules can affect the assembly process and their final chiroptical performance. Herein, we report an achiral molecule directed chiral self-assembly via halogen bonds, exhibiting not only an unprecedented chiral fractal architecture but also significantly amplified circularly polarized luminescence (CPL). Two axially chiral emitters with halogen bond sites co-assemble with an achiral 1,4-diiodotetrafluorobenzene (F₄ DIB) and well-ordered chiral fractal structures with asymmetry amplification are obtained. The enhancement of the dissymmetry factors of the assemblies was up to 0.051 and 0.011, which was approximately 100 folds than those of the corresponding molecules. It was found that both the design of the chiral emitter and the highly directional halogen bond played an important role in hierarchically chirality transfer from chiral emitters to the micrometer scale chiral fractal morphology and amplified dissymmetry factors. We hope that this strategy can give a further insight into the fabrication of structurally unique featured highly efficient chiroptical materials.