How the optic nerve allocates space, energy capacity, and information

Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

Synaptic energy use and supply

Neuronal computation is energetically expensive. Consequently, the brain's limited energy supply imposes constraints on its information processing capability. Most brain energy is used on synaptic transmission, making it important to understand how energy is provided to and used by synapses. We describe how information transmission through presynaptic terminals and postsynaptic spines is related to their energy consumption, assess which mechanisms normally ensure an adequate supply of ATP to these structures, consider the influence of synaptic plasticity and changing brain state on synaptic energy use, and explain how disruption of the energy supply to synapses leads to neuropathology.

Mathematically modeling the involvement of axons in Leber's hereditary optic neuropathy

PURPOSE

Leber's hereditary optic neuropathy (LHON), a mitochondrial disease, has clinical manifestations that reflect the initial preferential involvement of the papillomacular bundle (PMB). The present study seeks to predict the order of axonal loss in LHON optic nerves using the Nerve Fiber Layer Stress Index (NFL-S(I)), which is a novel mathematical model.

METHODS

Optic nerves were obtained postmortem from four molecularly characterized LHON patients with varying degrees of neurodegenerative changes and three age-matched controls. Tissues were cut in cross-section and stained with p-phenylenediamine to visualize myelin. Light microscopic images were captured in 32 regions of each optic nerve. Control and LHON tissues were evaluated by measuring axonal dimensions to generate an axonal diameter distribution map. LHON tissues were further evaluated by determining regions of total axonal depletion.

RESULTS

A size gradient was evident in the control optic nerves, with average axonal diameter increasing progressively from the temporal to nasal borders. LHON optic nerves showed an orderly loss of axons, starting inferotemporally, progressing centrally, and sparing the superonasal region until the end. Values generated from the NFL-S(I) equation fit a linear regression curve (R(2) = 0.97; P < 0.001).

CONCLUSIONS

The quantitative histopathologic data from this study revealed that the PMB is most susceptible in LHON, supporting clinical findings seen early in the course of disease onset. The present study also showed that the subsequent progression of axonal loss within the optic nerve can be predicted precisely with the NFL-S(I) equation. The results presented provided further insight into the pathophysiology of LHON.

Critical pathogenic events underlying progression of neurodegeneration in glaucoma

Glaucoma is a common optic neuropathy with a complex etiology often linked to sensitivity to intraocular pressure. Though the precise mechanisms that mediate or transduce this sensitivity are not clear, the axon of the retinal ganglion cell appears to be vulnerable to disease-relevant stressors early in progression. One reason may be because the axon is generally thin for both its unmyelinated and myelinated segment and much longer than the thicker unmyelinated axons of other excitatory retinal neurons. This difference may predispose the axon to metabolic and oxidative injury, especially at distal sites where pre-synaptic terminals form connections in the brain. This idea is consistent with observations of early loss of anterograde transport at central targets and other signs of distal axonopathy that accompany physiological indicators of progression. Outright degeneration of the optic projection ensues after a critical period and, at least in animal models, is highly sensitive to cumulative exposure to elevated pressure in the eye. Stress emanating from the optic nerve head can induce not only distal axonopathy with aspects of dying back neuropathy, but also Wallerian degeneration of the optic nerve and tract and a proximal program involving synaptic and dendritic pruning in the retina. Balance between progressive and acute mechanisms likely varies with the level of stress placed on the unmyelinated axon as it traverses the nerve head, with more acute insult pushing the system toward quicker disassembly. A constellation of signaling factors likely contribute to the transduction of stress to the axon, so that degenerative events along the length of the optic projection progress in retinotopic fashion. This pattern leads to well-defined sectors of functional depletion, even at distal-most sites in the pathway. While ganglion cell somatic drop-out is later in progression, some evidence suggests that synaptic and dendritic pruning in the retina may be a more dynamic process. Structural persistence both in the retina and in central projection sites offers the possibility that intrinsic self-repair pathways counter pathogenic mechanisms to delay as long as possible outright loss of tissue.

Unravelling cerebellar pathways with high temporal precision targeting motor and extensive sensory and parietal networks

Increasing evidence has implicated the cerebellum in providing forward models of motor plants predicting the sensory consequences of actions. Assuming that cerebellar input to the cerebral cortex contributes to the cerebro-cortical processing by adding forward model signals, we would expect to find projections emphasising motor and sensory cortical areas. However, this expectation is only partially met by studies of cerebello-cerebral connections. Here we show that by electrically stimulating the cerebellar output and imaging responses with functional magnetic resonance imaging, evoked blood oxygen level-dependant activity is observed not only in the classical cerebellar projection target, the primary motor cortex, but also in a number of additional areas in insular, parietal and occipital cortex, including sensory cortical representations. Further probing of the responses reveals a projection system that has been optimized to mediate fast and temporarily precise information. In conclusion, both the topography of the stimulation effects and its emphasis on temporal precision are in full accordance with the concept of cerebellar forward model information modulating cerebro-cortical processing.

Why do axons differ in caliber?

CNS axons differ in diameter (d) by nearly 100-fold (∼0.1-10 μm); therefore, they differ in cross-sectional area (d(2)) and volume by nearly 10,000-fold. If, as found for optic nerve, mitochondrial volume fraction is constant with axon diameter, energy capacity would rise with axon volume, also as d(2). We asked, given constraints on space and energy, what functional requirements set an axon's diameter? Surveying 16 fiber groups spanning nearly the full range of diameters in five species (guinea pig, rat, monkey, locust, octopus), we found the following: (1) thin axons are most numerous; (2) mean firing frequencies, estimated for nine of the identified axon classes, are low for thin fibers and high for thick ones, ranging from ∼1 to >100 Hz; (3) a tract's distribution of fiber diameters, whether narrow or broad, and whether symmetric or skewed, reflects heterogeneity of information rates conveyed by its individual fibers; and (4) mitochondrial volume/axon length rises ≥d(2). To explain the pressure toward thin diameters, we note an established law of diminishing returns: an axon, to double its information rate, must more than double its firing rate. Since diameter is apparently linear with firing rate, doubling information rate would more than quadruple an axon's volume and energy use. Thicker axons may be needed to encode features that cannot be efficiently decoded if their information is spread over several low-rate channels. Thus, information rate may be the main variable that sets axon caliber, with axons constrained to deliver information at the lowest acceptable rate.

Visual system scaling in teleost fish

Teleost fish grow continuously throughout their lifespan, and this growth includes visual system components: eyes, optic nerves, and brain. As fish grow, the optic nerve lengthens and neural signals must travel increasing distances from the eye to the optic tectum along thousands of retinal ganglion cell (RGC) axons. Larger fish have better vision that enhances their ability to capture prey, but they are faced with the potential computational problem of changes in the relative timing of visual information arriving at the brain. Optic nerve conduction delays depend on RGC axon conduction velocities, and velocity is primarily determined by axon diameters. If axon diameters do not increase in proportion to body length, then absolute and relative conduction delays will vary with fish size. We have measured optic nerve lengths and axon diameter distributions in different sized zebrafish (Danio rerio) and goldfish (Carassius auratus) and find that, as both species of fish grow, axon diameters increase to reduce average conduction delays by about half and to keep relative delays constant. This invariance of relative conduction delays simplifies computational problems faced by the optic tectum.

The energetics of CNS white matter

The energetics of CNS white matter are poorly understood. We derive a signaling energy budget for the white matter (based on data from the rodent optic nerve and corpus callosum) which can be compared with previous energy budgets for the gray matter regions of the brain, perform a cost-benefit analysis of the energetics of myelination, and assess mechanisms for energy production and glucose supply in myelinated axons. We show that white matter synapses consume ≤0.5% of the energy of gray matter synapses and that this, rather than more energy-efficient action potentials, is the main reason why CNS white matter uses less energy than gray matter. Surprisingly, while the energetic cost of building myelin could be repaid within months by the reduced ATP cost of neuronal action potentials, the energetic cost of maintaining the oligodendrocyte resting potential usually outweighs the saving on action potentials. Thus, although it dramatically speeds action potential propagation, myelination need not save energy. Finally, we show that mitochondria in optic nerve axons could sustain measured firing rates with a plausible density of glucose transporters in the nodal membrane, without the need for energy transfer from oligodendrocytes.

Wiring economy and volume exclusion determine neuronal placement in the Drosophila brain

Wiring economy has successfully explained the individual placement of neurons in simple nervous systems like that of Caenorhabditis elegans [1-3] and the locations of coarser structures like cortical areas in complex vertebrate brains [4]. However, it remains unclear whether wiring economy can explain the placement of individual neurons in brains larger than that of C. elegans. Indeed, given the greater number of neuronal interconnections in larger brains, simply minimizing the length of connections results in unrealistic configurations, with multiple neurons occupying the same position in space. Avoiding such configurations, or volume exclusion, repels neurons from each other, thus counteracting wiring economy. Here we test whether wiring economy together with volume exclusion can explain the placement of neurons in a module of the Drosophila melanogaster brain known as lamina cartridge [5-13]. We used newly developed techniques for semiautomated reconstruction from serial electron microscopy (EM) [14] to obtain the shapes of neurons, the location of synapses, and the resultant synaptic connectivity. We show that wiring length minimization and volume exclusion together can explain the structure of the lamina microcircuit. Therefore, even in brains larger than that of C. elegans, at least for some circuits, optimization can play an important role in individual neuron placement.

Astrocytic energy metabolism and glutamate formation--relevance for 13C-NMR spectroscopy and importance of cytosolic/mitochondrial trafficking

Glutamate plays a double role in (13)C-nuclear magnetic resonance (NMR) spectroscopic determination of glucose metabolism in the brain. Bidirectional exchange between initially unlabeled glutamate and labeled α-ketoglutarate, formed from pyruvate via pyruvate dehydrogenase (PDH), indicates the rate of energy metabolism in the tricarboxylic acid (V(TCA)) cycle in neurons (V(PDH, n)) and, with additional computation, also in astrocytes (V(PDH, g)), as confirmed using the astrocyte-specific substrate [(13)C]acetate. Formation of new molecules of glutamate during increased glutamatergic activity occurs only in astrocytes by combined pyruvate carboxylase (V(PC)) and astrocytic PDH activity. V(PDH, g) accounts for ~15% of total pyruvate metabolism in the brain cortex, and V(PC) accounts for another ~10%. Since both PDH-generated and PC-generated pyruvates are needed for glutamate synthesis, ~20/25 (80%) of astrocytic pyruvate metabolism proceed via glutamate formation. Net transmitter glutamate [γ-aminobutyric acid (GABA)] formation requires transfer of newly synthesized α-ketoglutarate to the astrocytic cytosol, α-ketoglutarate transamination to glutamate, amidation to glutamine, glutamine transfer to neurons, its hydrolysis to glutamate and glutamate release (or GABA formation). Glutamate-glutamine cycling, measured as glutamine synthesis rate (V(cycle)), also transfers previously released glutamate/GABA to neurons after an initial astrocytic accumulation and measures predominantly glutamate signaling. An empirically established ~1/1 ratio between glucose metabolism and V(cycle) may reflect glucose utilization associated with oxidation/reduction processes during glutamate production, which together with associated transamination processes are balanced by subsequent glutamate oxidation after cessation of increased signaling activity. Astrocytic glutamate formation and subsequent oxidative metabolism provide large amounts of adenosine triphosphate used for accumulation from extracellular clefts of neuronally released K(+) and glutamate and for cytosolic Ca(2+) homeostasis.

Axonal transmission in the retina introduces a small dispersion of relative timing in the ganglion cell population response

Background

Visual stimuli elicit action potentials in tens of different retinal ganglion cells. Each ganglion cell type responds with a different latency to a given stimulus, thus transforming the high-dimensional input into a temporal neural code. The timing of the first spikes between different retinal projection neurons cells may further change along axonal transmission. The purpose of this study is to investigate if intraretinal conduction velocity leads to a synchronization or dispersion of the population signal leaving the eye.

Methodology/Principal Findings

We ‘imaged’ the initiation and transmission of light-evoked action potentials along individual axons in the rabbit retina at micron-scale resolution using a high-density multi-transistor array. We measured unimodal conduction velocity distributions (1.3±0.3 m/sec, mean ± SD) for axonal populations at all retinal eccentricities with the exception of the central part that contains myelinated axons. The velocity variance within each piece of retina is caused by ganglion cell types that show narrower and slightly different average velocity tuning. Ganglion cells of the same type respond with similar latency to spatially homogenous stimuli and conduct with similar velocity. For ganglion cells of different type intraretinal conduction velocity and response latency to flashed stimuli are negatively correlated, indicating that differences in first spike timing increase (up to 10 msec). Similarly, the analysis of pair-wise correlated activity in response to white-noise stimuli reveals that conduction velocity and response latency are negatively correlated.

Conclusion/Significance

Intraretinal conduction does not change the relative spike timing between ganglion cells of the same type but increases spike timing differences among ganglion cells of different type. The fastest retinal ganglion cells therefore act as indicators of new stimuli for postsynaptic neurons. The intraretinal dispersion of the population activity will not be compensated by variability in extraretinal conduction times, estimated from data in the literature.

Matching mitochondria to metabolic needs at nodes of Ranvier

Myelinated axons conduct nerve impulses at high speed using a unique mode of excitation, referred to as saltatory conduction, which is enabled structurally by the narrowing of the site of action potentials to a tiny gap in the axon called the node of Ranvier. With this structural specialization comes an interesting metabolic matching problem. How do mitochondria find and supply energy to these tiny nodes of Ranvier distributed sparsely along a myelinated axon? Does the intense Na(+) influx at the node, which is produced by the highest known sodium channel density in all excitable membranes, help guide where mitochondria stop? Evidence suggests that during excitation in the peripheral nervous system, Na(+) influx recruits mitochondria to the node by triggering Ca(2+) elevation and activating Na(+) pumps. Intriguingly, indirect evidence suggests that in the central nervous system, activity recruits mitochondria to the internode (myelin-covered portion of the axon). Metabolic dysfunction thus might produce spatially distinct lesions in PNS and CNS myelinated fibers. Future dissection of regional variation in mitochondrial biology in myelinated axons using live imaging will likely yield surprises about sites of vulnerability in demyelinating diseases and clues for therapeutic intervention strategy.

Retina is structured to process an excess of darkness in natural scenes

Retinal ganglion cells that respond selectively to a dark spot on a brighter background (OFF cells) have smaller dendritic fields than their ON counterparts and are more numerous. OFF cells also branch more densely, and thus collect more synapses per visual angle. That the retina devotes more resources to processing dark contrasts predicts that natural images contain more dark information. We confirm this across a range of spatial scales and trace the origin of this phenomenon to the statistical structure of natural scenes. We show that the optimal mosaics for encoding natural images are also asymmetric, with OFF elements smaller and more numerous, matching retinal structure. Finally, the concentration of synapses within a dendritic field matches the information content, suggesting a simple principle to connect a concrete fact of neuroanatomy with the abstract concept of information: equal synapses for equal bits.

Bound pool fractions complement diffusion measures to describe white matter micro and macrostructure

Diffusion imaging and bound pool fraction (BPF) mapping are two quantitative magnetic resonance imaging techniques that measure microstructural features of the white matter of the brain. Diffusion imaging provides a quantitative measure of the diffusivity of water in tissue. BPF mapping is a quantitative magnetization transfer (qMT) technique that estimates the proportion of exchanging protons bound to macromolecules, such as those found in myelin, and is thus a more direct measure of myelin content than diffusion. In this work, we combined BPF estimates of macromolecular content with measurements of diffusivity within human white matter tracts. Within the white matter, the correlation between BPFs and diffusivity measures such as fractional anisotropy and radial diffusivity was modest, suggesting that diffusion tensor imaging and bound pool fractions are complementary techniques. We found that several major tracts have high BPF, suggesting a higher density of myelin in these tracts. We interpret these results in the context of a quantitative tissue model.

Demyelination and axonal preservation in a transgenic mouse model of Pelizaeus-Merzbacher disease

It is widely thought that demyelination contributes to the degeneration of axons and, in combination with acute inflammatory injury, is responsible for progressive axonal loss and persistent clinical disability in inflammatory demyelinating disease. In this study we sought to characterize the relationship between demyelination, inflammation and axonal transport changes using a Plp1-transgenic mouse model of Pelizaeus-Merzbacher disease. In the optic pathway of this non-immune mediated model of demyelination, myelin loss progresses from the optic nerve head towards the brain, over a period of months. Axonal transport is functionally perturbed at sites associated with local inflammation and ‘damaged’ myelin. Surprisingly, where demyelination is complete, naked axons appear well preserved despite a significant reduction of axonal transport. Our results suggest that neuroinflammation and/or oligodendrocyte dysfunction are more deleterious for axonal health than demyelination per se, at least in the short term.

Design of a trichromatic cone array

Cones with peak sensitivity to light at long (L), medium (M) and short (S) wavelengths are unequal in number on the human retina: S cones are rare (<10%) while increasing in fraction from center to periphery, and the L/M cone proportions are highly variable between individuals. What optical properties of the eye, and statistical properties of natural scenes, might drive this organization? We found that the spatial-chromatic structure of natural scenes was largely symmetric between the L, M and S sensitivity bands. Given this symmetry, short wavelength attenuation by ocular media gave L/M cones a modest signal-to-noise advantage, which was amplified, especially in the denser central retina, by long-wavelength accommodation of the lens. Meanwhile, total information represented by the cone mosaic remained relatively insensitive to L/M proportions. Thus, the observed cone array design along with a long-wavelength accommodated lens provides a selective advantage: it is maximally informative.

Human color perception arises by comparing the signals from cones with peak sensitivities, at long (L), medium (M) and short (S) wavelengths. In dichromats, a characteristic distribution of S and M cones supports blue-yellow color vision: a few S and mostly M. When L cones are added, allowing red-green color vision, the S proportion remains low, increasing slowly with increasing retinal eccentricity, but the L/M proportion can vary 5-fold without affecting red-green color perception. We offer a unified explanation of these striking facts. First, we find that the spatial-chromatic statistics of natural scenes are largely symmetric between the L, M and S sensitivity bands. Thus, attenuation of blue light in the optical media, and chromatic aberration after long-wavelength accommodation of the lens, can give L/M cones an advantage. Quantitatively, information transmission by the cone array is maximized when the S proportion is low but increasing slowly with retinal eccentricity, accompanied by a lens accommodated to red light. After including blur by the lens, the optimum depends weakly on the red/green ratio, allowing large variations without loss of function. This explains the basic layout of the cone mosaic: for the resources invested, the organization maximizes information.

What is the optimal value of the g-ratio for myelinated fibers in the rat CNS? A theoretical approach

BACKGROUND

The biological process underlying axonal myelination is complex and often prone to injury and disease. The ratio of the inner axonal diameter to the total outer diameter or g-ratio is widely utilized as a functional and structural index of optimal axonal myelination. Based on the speed of fiber conduction, Rushton was the first to derive a theoretical estimate of the optimal g-ratio of 0.6 [1]. This theoretical limit nicely explains the experimental data for myelinated axons obtained for some peripheral fibers but appears significantly lower than that found for CNS fibers. This is, however, hardly surprising given that in the CNS, axonal myelination must achieve multiple goals including reducing conduction delays, promoting conduction fidelity, lowering energy costs, and saving space.

METHODOLOGY/PRINCIPAL FINDINGS

In this study we explore the notion that a balanced set-point can be achieved at a functional level as the micro-structure of individual axons becomes optimized, particularly for the central system where axons tend to be smaller and their myelin sheath thinner. We used an intuitive yet novel theoretical approach based on the fundamental biophysical properties describing axonal structure and function to show that an optimal g-ratio can be defined for the central nervous system (approximately 0.77). Furthermore, by reducing the influence of volume constraints on structural design by about 40%, this approach can also predict the g-ratio observed in some peripheral fibers (approximately 0.6).

CONCLUSIONS/SIGNIFICANCE

These results support the notion of optimization theory in nervous system design and construction and may also help explain why the central and peripheral systems have evolved different g-ratios as a result of volume constraints.

Receptive fields and functional architecture in the retina

Functional architecture of the striate cortex is known mostly at the tissue level--how neurons of different function distribute across its depth and surface on a scale of millimetres. But explanations for its design--why it is just so--need to be addressed at the synaptic level, a much finer scale where the basic description is still lacking. Functional architecture of the retina is known from the scale of millimetres down to nanometres, so we have sought explanations for various aspects of its design. Here we review several aspects of the retina's functional architecture and find that all seem governed by a single principle: represent the most information for the least cost in space and energy. Specifically: (i) why are OFF ganglion cells more numerous than ON cells? Because natural scenes contain more negative than positive contrasts, and the retina matches its neural resources to represent them equally well; (ii) why do ganglion cells of a given type overlap their dendrites to achieve 3-fold coverage? Because this maximizes total information represented by the array--balancing signal-to-noise improvement against increased redundancy; (iii) why do ganglion cells form multiple arrays? Because this allows most information to be sent at lower rates, decreasing the space and energy costs for sending a given amount of information. This broad principle, operating at higher levels, probably contributes to the brain's immense computational efficiency.