Optimal Current Transfer in Dendrites

Topology recapitulates morphogenesis of neuronal dendrites

Branching allows neurons to make synaptic contacts with large numbers of other neurons, facilitating the high connectivity of nervous systems. Neuronal arbors have geometric properties such as branch lengths and diameters that are optimal in that they maximize signaling speeds while minimizing construction costs. In this work, we asked whether neuronal arbors have topological properties that may also optimize their growth or function. We discovered that for a wide range of invertebrate and vertebrate neurons the distributions of their subtree sizes follow power laws, implying that they are scale invariant. The power-law exponent distinguishes different neuronal cell types. Postsynaptic spines and branchlets perturb scale invariance. Through simulations, we show that the subtree-size distribution depends on the symmetry of the branching rules governing arbor growth and that optimal morphologies are scale invariant. Thus, the subtree-size distribution is a topological property that recapitulates the functional morphology of dendrites.

Controlling morpho-electrophysiological variability of neurons with detailed biophysical models

Variability, which is known to be a universal feature among biological units such as neuronal cells, holds significant importance, as, for example, it enables a robust encoding of a high volume of information in neuronal circuits and prevents hypersynchronizations. While most computational studies on electrophysiological variability in neuronal circuits were done with single-compartment neuron models, we instead focus on the variability of detailed biophysical models of neuron multi-compartmental morphologies. We leverage a Markov chain Monte Carlo method to generate populations of electrical models reproducing the variability of experimental recordings while being compatible with a set of morphologies to faithfully represent specifi morpho-electrical type. We demonstrate our approach on layer 5 pyramidal cells and study the morpho-electrical variability and in particular, find that morphological variability alone is insufficient to reproduce electrical variability. Overall, this approach provides a strong statistical basis to create detailed models of neurons with controlled variability.

Computational synthesis of cortical dendritic morphologies

Neuronal morphologies provide the foundation for the electrical behavior of neurons, the connectomes they form, and the dynamical properties of the brain. Comprehensive neuron models are essential for defining cell types, discerning their functional roles, and investigating brain-disease-related dendritic alterations. However, a lack of understanding of the principles underlying neuron morphologies has hindered attempts to computationally synthesize morphologies for decades. We introduce a synthesis algorithm based on a topological descriptor of neurons, which enables the rapid digital reconstruction of entire brain regions from few reference cells. This topology-guided synthesis generates dendrites that are statistically similar to biological reconstructions in terms of morpho-electrical and connectivity properties and offers a significant opportunity to investigate the links between neuronal morphology and brain function across different spatiotemporal scales. Synthesized cortical networks based on structurally altered dendrites associated with diverse brain pathologies revealed principles linking branching properties to the structure of large-scale networks.

Reconstruction of neocortex: Organelles, compartments, cells, circuits, and activity

We assembled a semi-automated reconstruction of L2/3 mouse primary visual cortex from ∼250 × 140 × 90 μm3 of electron microscopic images, including pyramidal and non-pyramidal neurons, astrocytes, microglia, oligodendrocytes and precursors, pericytes, vasculature, nuclei, mitochondria, and synapses. Visual responses of a subset of pyramidal cells are included. The data are publicly available, along with tools for programmatic and three-dimensional interactive access. Brief vignettes illustrate the breadth of potential applications relating structure to function in cortical circuits and neuronal cell biology. Mitochondria and synapse organization are characterized as a function of path length from the soma. Pyramidal connectivity motif frequencies are predicted accurately using a configuration model of random graphs. Pyramidal cells receiving more connections from nearby cells exhibit stronger and more reliable visual responses. Sample code shows data access and analysis.

A general principle of dendritic constancy: A neuron's size- and shape-invariant excitability

Reducing neuronal size results in less membrane and therefore lower input conductance. Smaller neurons are thus more excitable, as seen in their responses to somatic current injections. However, the impact of a neuron's size and shape on its voltage responses to dendritic synaptic activation is much less understood. Here we use analytical cable theory to predict voltage responses to distributed synaptic inputs in unbranched cables, showing that these are entirely independent of dendritic length. For a given synaptic density, neuronal responses depend only on the average dendritic diameter and intrinsic conductivity. This remains valid for a wide range of morphologies irrespective of their arborization complexity. Spiking models indicate that morphology-invariant numbers of spikes approximate the percentage of active synapses. In contrast to spike rate, spike times do depend on dendrite morphology. In summary, neuronal excitability in response to distributed synaptic inputs is largely unaffected by dendrite length or complexity.

Dendritic normalisation improves learning in sparsely connected artificial neural networks

Artificial neural networks, taking inspiration from biological neurons, have become an invaluable tool for machine learning applications. Recent studies have developed techniques to effectively tune the connectivity of sparsely-connected artificial neural networks, which have the potential to be more computationally efficient than their fully-connected counterparts and more closely resemble the architectures of biological systems. We here present a normalisation, based on the biophysical behaviour of neuronal dendrites receiving distributed synaptic inputs, that divides the weight of an artificial neuron's afferent contacts by their number. We apply this dendritic normalisation to various sparsely-connected feedforward network architectures, as well as simple recurrent and self-organised networks with spatially extended units. The learning performance is significantly increased, providing an improvement over other widely-used normalisations in sparse networks. The results are two-fold, being both a practical advance in machine learning and an insight into how the structure of neuronal dendritic arbours may contribute to computation.

Prediction of Neural Diameter From Morphology to Enable Accurate Simulation

Accurate neuron morphologies are paramount for computational model simulations of realistic neural responses. Over the last decade, the online repository NeuroMorpho.Org has collected over 140,000 available neuron morphologies to understand brain function and promote interaction between experimental and computational research. Neuron morphologies describe spatial aspects of neural structure; however, many of the available morphologies do not contain accurate diameters that are essential for computational simulations of electrical activity. To best utilize available neuron morphologies, we present a set of equations that predict dendritic diameter from other morphological features. To derive the equations, we used a set of NeuroMorpho.org archives with realistic neuron diameters, representing hippocampal pyramidal, cerebellar Purkinje, and striatal spiny projection neurons. Each morphology is separated into initial, branching children, and continuing nodes. Our analysis reveals that the diameter of preceding nodes, Parent Diameter, is correlated to diameter of subsequent nodes for all cell types. Branching children and initial nodes each required additional morphological features to predict diameter, such as path length to soma, total dendritic length, and longest path to terminal end. Model simulations reveal that membrane potential response with predicted diameters is similar to the original response for several tested morphologies. We provide our open source software to extend the utility of available NeuroMorpho.org morphologies, and suggest predictive equations may supplement morphologies that lack dendritic diameter and improve model simulations with realistic dendritic diameter.

Statistical Laws of Protein Motion in Neuronal Dendritic Trees

Across their dendritic trees, neurons distribute thousands of protein species that are necessary for maintaining synaptic function and plasticity and that need to be produced continuously and trafficked to their final destination. As each dendritic branchpoint splits the protein flow, increasing branchpoints decreases the total protein number downstream. Consequently, a neuron needs to produce more proteins to maintain a minimal protein number at distal synapses. Combining in vitro experiments and a theoretical framework, we show that proteins that diffuse within the cell plasma membrane are, on average, 35% more effective at reaching downstream locations than proteins that diffuse in the cytoplasm. This advantage emerges from a bias for forward motion at branchpoints when proteins diffuse within the plasma membrane. Using 3D electron microscopy (EM) data, we show that pyramidal branching statistics and the diffusion lengths of common proteins fall into a region that minimizes the overall protein need.

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Surface proteins are more efficient at reaching distal sites than soluble proteins

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Daughter radius optimization reduces the number of proteins needed to populate dendrites

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Ratios of daughter radii at branchpoints are cell type specific

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Highly diffusive proteins incur a smaller extra cost for non-optimized radii

Excess Neuronal Branching Allows for Local Innervation of Specific Dendritic Compartments in Mature Cortex

The connectivity of cortical microcircuits is a major determinant of brain function; defining how activity propagates between different cell types is key to scaling our understanding of individual neuronal behavior to encompass functional networks. Furthermore, the integration of synaptic currents within a dendrite depends on the spatial organization of inputs, both excitatory and inhibitory. We identify a simple equation to estimate the number of potential anatomical contacts between neurons; finding a linear increase in potential connectivity with cable length and maximum spine length, and a decrease with overlapping volume. This enables us to predict the mean number of candidate synapses for reconstructed cells, including those realistically arranged. We identify an excess of potential local connections in mature cortical data, with densities of neurite higher than is necessary to reliably ensure the possible implementation of any given axo-dendritic connection. We show that the number of local potential contacts allows specific innervation of distinct dendritic compartments.

Dissecting Sholl Analysis into Its Functional Components

Sholl analysis has been an important technique in dendritic anatomy for more than 60 years. The Sholl intersection profile is obtained by counting the number of dendritic branches at a given distance from the soma and is a key measure of dendritic complexity; it has applications from evaluating the changes in structure induced by pathologies to estimating the expected number of anatomical synaptic contacts. We find that the Sholl intersection profiles of most neurons can be reproduced from three basic, functional measures: the domain spanned by the dendritic arbor, the total length of the dendrite, and the angular distribution of how far dendritic segments deviate from a direct path to the soma (i.e., the root angle distribution). The first two measures are determined by axon location and hence microcircuit structure; the third arises from optimal wiring and represents a branching statistic estimating the need for conduction speed in a neuron.

Adult-Born Hippocampal Neurons Undergo Extended Development and Are Morphologically Distinct from Neonatally-Born Neurons

During immature stages, adult-born neurons pass through critical periods for survival and plasticity. It is generally assumed that by 2 months of age adult-born neurons are mature and equivalent to the broader neuronal population, raising questions of how they might contribute to hippocampal function in old age when neurogenesis has declined. However, few have examined adult-born neurons beyond the critical period or directly compared them to neurons born in infancy. Here, we used a retrovirus to visualize functionally relevant morphological features of 2- to 24-week-old adult-born neurons in male rats. From 2 to 7 weeks, neurons grew and attained a relatively mature phenotype. However, several features of 7-week-old neurons suggested a later wave of growth: these neurons had larger nuclei, thicker dendrites, and more dendritic filopodia than all other groups. Indeed, between 7 and 24 weeks, adult-born neurons gained additional dendritic branches, formed a second primary dendrite, acquired more mushroom spines, and had enlarged mossy fiber presynaptic terminals. Compared with neonatal-born neurons, old adult-born neurons had greater spine density, larger presynaptic terminals, and more putative efferent filopodial contacts onto inhibitory neurons. By integrating rates of cell birth and growth across the life span, we estimate that adult neurogenesis ultimately produces half of the cells and the majority of spines in the dentate gyrus. Critically, protracted development contributes to the plasticity of the hippocampus through to the end of life, even after cell production declines. Persistent differences from neonatal-born neurons may additionally endow adult-born neurons with unique functions even after they have matured.SIGNIFICANCE STATEMENT Neurogenesis occurs in the hippocampus throughout adult life and contributes to memory and emotion. It is generally assumed that new neurons have the greatest impact on behavior when they are immature and plastic. However, since neurogenesis declines dramatically with age, it is unclear how they might contribute to behavior later in life when cell proliferation has slowed. Here we find that newborn neurons mature over many months in rats and may end up with distinct morphological features compared with neurons born in infancy. Using a mathematical model, we estimate that a large fraction of neurons is added in adulthood. Moreover, their extended growth produces a reserve of plasticity that persists even after neurogenesis has declined to low rates.

Objective Morphological Classification of Neocortical Pyramidal Cells

A consensus on the number of morphologically different types of pyramidal cells (PCs) in the neocortex has not yet been reached, despite over a century of anatomical studies, due to the lack of agreement on the subjective classifications of neuron types, which is based on expert analyses of neuronal morphologies. Even for neurons that are visually distinguishable, there is no common ground to consistently define morphological types. The objective classification of PCs can be achieved with methods from algebraic topology, and the dendritic arborization is sufficient for the reliable identification of distinct types of cortical PCs. Therefore, we objectively identify 17 types of PCs in the rat somatosensory cortex. In addition, we provide a solution to the challenging problem of whether 2 similar neurons belong to different types or to a continuum of the same type. Our topological classification does not require expert input, is stable, and helps settle the long-standing debate on whether cell-types are discrete or continuous morphological variations of each other.

Exact solutions to cable equations in branching neurons with tapering dendrites

Neurons are biological cells with uniquely complex dendritic morphologies that are not present in other cell types. Electrical signals in a neuron with branching dendrites can be studied by cable theory which provides a general mathematical modelling framework of spatio-temporal voltage dynamics. Typically such models need to be solved numerically unless the cell membrane is modelled either by passive or quasi-active dynamics, in which cases analytical solutions can be reduced to calculation of the Green's function describing the fundamental input-output relationship in a given morphology. Such analytically tractable models often assume individual dendritic segments to be cylinders. However, it is known that dendritic segments in many types of neurons taper, i.e. their radii decline from proximal to distal ends. Here we consider a generalised form of cable theory which takes into account both branching and tapering structures of dendritic trees. We demonstrate that analytical solutions can be found in compact algebraic forms in an arbitrary branching neuron with a class of tapering dendrites studied earlier in the context of single neuronal cables by Poznanski (Bull. Math. Biol. 53(3):457-467, 1991). We apply this extended framework to a number of simplified neuronal models and contrast their output dynamics in the presence of tapering versus cylindrical segments.

Fractional cable equation for general geometry: A model of axons with swellings and anomalous diffusion

Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.

Morphological determinants of dendritic arborization neurons in Drosophila larva

Pairing in vivo imaging and computational modeling of dendritic arborization (da) neurons from the fruit fly larva provides a unique window into neuronal growth and underlying molecular processes. We image, reconstruct, and analyze the morphology of wild-type, RNAi-silenced, and mutant da neurons. We then use local and global rule-based stochastic simulations to generate artificial arbors, and identify the parameters that statistically best approximate the real data. We observe structural homeostasis in all da classes, where an increase in size of one dendritic stem is compensated by a reduction in the other stems of the same neuron. Local rule models show that bifurcation probability is determined by branch order, while branch length depends on path distance from the soma. Global rule simulations suggest that most complex morphologies tend to be constrained by resource optimization, while simpler neuron classes privilege path distance conservation. Genetic manipulations affect both the local and global optimal parameters, demonstrating functional perturbations in growth mechanisms.

Branching morphology determines signal propagation dynamics in neurons

Computational modeling of signal propagation in neurons is critical to our understanding of basic principles underlying brain organization and activity. Exploring these models is used to address basic neuroscience questions as well as to gain insights for clinical applications. The seminal Hodgkin Huxley model is a common theoretical framework to study brain activity. It was mainly used to investigate the electrochemical and physical properties of neurons. The influence of neuronal structure on activity patterns was explored, however, the rich dynamics observed in neurons with different morphologies is not yet fully understood. Here, we study signal propagation in fundamental building blocks of neuronal branching trees, unbranched and branched axons. We show how these simple axonal elements can code information on spike trains, and how asymmetric responses can emerge in axonal branching points. This asymmetric phenomenon has been observed experimentally but until now lacked theoretical characterization. Together, our results suggest that axonal morphological parameters are instrumental in activity modulation and information coding. The insights gained from this work lay the ground for better understanding the interplay between function and form in real-world complex systems. It may also supply theoretical basis for the development of novel therapeutic approaches to damaged nervous systems.

Universal features of dendrites through centripetal branch ordering

Dendrites form predominantly binary trees that are exquisitely embedded in the networks of the brain. While neuronal computation is known to depend on the morphology of dendrites, their underlying topological blueprint remains unknown. Here, we used a centripetal branch ordering scheme originally developed to describe river networks—the Horton-Strahler order (SO)–to examine hierarchical relationships of branching statistics in reconstructed and model dendritic trees. We report on a number of universal topological relationships with SO that are true for all binary trees and distinguish those from SO-sorted metric measures that appear to be cell type-specific. The latter are therefore potential new candidates for categorising dendritic tree structures. Interestingly, we find a faithful correlation of branch diameters with centripetal branch orders, indicating a possible functional importance of SO for dendritic morphology and growth. Also, simulated local voltage responses to synaptic inputs are strongly correlated with SO. In summary, our study identifies important SO-dependent measures in dendritic morphology that are relevant for neural function while at the same time it describes other relationships that are universal for all dendrites.

Similarly to river beds, dendritic trees of nerve cells form elaborate networks that branch out to cover extensive areas. In the 1940s, ecologist Robert E. Horton developed an ordering system for branches in river networks that was refined in the 1950s by geoscientist Arthur N. Strahler, the Horton-Strahler order (SO). Branches at the tips start with order 1 and increase their order in a systematic way when encountering new branches on the way to the root. SO relationships have recently become popular for quantifying dendritic morphologies. Various branching statistics can be studied as a function of SO. Here we describe that topological measures such as the number of branches, the branch bifurcation ratio and the size of subtrees exhibit stereotypical relations with SO in dendritic trees independently of cell type, mirroring universal features of binary trees. Other functionally more relevant features such as mean branch lengths, local diameters and simulated voltage responses to synaptic inputs directly correlate with SO in a cell type-specific manner, indicating the importance of SO for understanding dendrite growth as well as neural computation.

Cable equation for general geometry

The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.