Fractal analysis of ganglion cell dendritic branching patterns of the rat and cat retinae

Neural activity of retinal ganglion cells under continuous, dynamically-modulated high frequency electrical stimulation

Objective.Current retinal prosthetics are limited in their ability to precisely control firing patterns of functionally distinct retinal ganglion cell (RGC) types. The aim of this study was to characterise RGC responses to continuous, kilohertz-frequency-varying stimulation to assess its utility in controlling RGC activity.Approach.We usedin vitropatch-clamp experiments to assess electrically-evoked ON and OFF RGC responses to frequency-varying pulse train sequences. In each sequence, the stimulation amplitude was kept constant while the stimulation frequency (0.5-10 kHz) was changed every 40 ms, in either a linearly increasing, linearly decreasing or randomised manner. The stimulation amplitude across sequences was increased from 10 to 300µA.Main results.We found that continuous stimulation without rest periods caused complex and irreproducible stimulus-response relationships, primarily due to strong stimulus-induced response adaptation and influence of the preceding stimulus frequency on the response to a subsequent stimulus. In addition, ON and OFF populations showed different sensitivities to continuous, frequency-varying pulse trains, with OFF cells generally exhibiting more dependency on frequency changes within a sequence. Finally, the ability to maintain spiking behaviour to continuous stimulation in RGCs significantly reduced over longer stimulation durations irrespective of the frequency order.Significance.This study represents an important step in advancing and understanding the utility of continuous frequency modulation in controlling functionally distinct RGCs. Our results indicate that continuous, kHz-frequency-varying stimulation sequences provide very limited control of RGC firing patterns due to inter-dependency between adjacent frequencies and generally, different RGC types do not display different frequency preferences under such stimulation conditions. For future stimulation strategies using kHz frequencies, careful consideration must be given to design appropriate pauses in stimulation, stimulation frequency order and the length of continuous stimulation duration.

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.

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).

Characterization of Retinal Functionality at Different Eccentricities in a Diurnal Rodent

Although the properties of the neurons of the visual system that process central and peripheral regions of the visual field have been widely researched in the visual cortex and the LGN, they have scarcely been documented for the retina. The retina is the first step in integrating optical signals, and despite considerable efforts to functionally characterize the different types of retinal ganglion cells (RGCs), a clear account of the particular functionality of cells with central vs. peripheral fields is still wanting. Here, we use electrophysiological recordings, gathered from retinas of the diurnal rodent Octodon degus, to show that RGCs with peripheral receptive fields (RF) are larger, faster, and have shorter transient responses. This translates into higher sensitivity at high temporal frequencies and a full frequency bandwidth when compared to RGCs with more central RF. We also observed that imbalances between ON and OFF cell populations are preserved with eccentricity. Finally, the high diversity of functional types of RGCs highlights the complexity of the computational strategies implemented in the early stages of visual processing, which could inspire the development of bio-inspired artificial systems.

Asymmetry between ON and OFF α ganglion cells of mouse retina: integration of signal and noise from synaptic inputs

KEY POINTS

Bipolar and amacrine cells presynaptic to the ON sustained α cell of mouse retina provide currents with a higher signal-to-noise power ratio (SNR) than those presynaptic to the OFF sustained α cell. Yet the ON cell loses proportionately more SNR from synaptic inputs to spike output than the OFF cell does. The higher SNR of ON bipolar cells at the beginning of the ON pathway compensates for losses incurred by the ON ganglion cell, and improves the processing of positive contrasts.

ABSTRACT

ON and OFF pathways in the retina include functional pairs of neurons that, at first glance, appear to have symmetrically similar responses to brightening and darkening, respectively. Upon careful examination, however, functional pairs exhibit asymmetries in receptive field size and response kinetics. Until now, descriptions of how light-adapted retinal circuitry maintains a preponderance of signal over the noise have not distinguished between ON and OFF pathways. Here I present evidence of marked asymmetries between members of a functional pair of sustained α ganglion cells in the mouse retina. The ON cell exhibited a proportionately greater loss of signal-to-noise power ratio (SNR) from its presynaptic arrays to its postsynaptic currents. Thus the ON cell combines signal and noise from its presynaptic arrays of bipolar and amacrine cells less efficiently than the OFF cell does. Yet the inefficiency of the ON cell is compensated by its presynaptic arrays providing a higher SNR than the arrays presynaptic to the OFF cell, apparently to improve visual processing of positive contrasts. Dynamic clamp experiments were performed that introduced synaptic conductances into ON and OFF cells. When the amacrine-modulated conductance was removed, the ON cell's spike train exhibited an increase in SNR. The OFF cell, however, showed the opposite effect of removing amacrine input, which was a decrease in SNR. Thus ON and OFF cells have different modes of synaptic integration with direct effects on the SNR of the spike output.

The functional asymmetry of ON and OFF channels in the perception of contrast

To fully understand the relationship between perception and single neural responses, one should take into consideration the early stages of sensory processing. Few studies, however, have directly examined the neural underpinning of visual perception in the lateral geniculate nucleus (LGN), only one synapse away from the retina. In this study we recorded from LGN parvocellular (P) ON-center and OFF-center neurons while monkeys either passively viewed or actively detected a full range of contrasts. We found that OFF neurons were more sensitive in detecting negative contrasts than ON neurons were in detecting positive contrasts. Also, OFF neurons had higher spontaneous activities, higher peak response amplitudes, and were more sustained than ON neurons in their contrast responses. Puzzlingly, OFF neurons failed to show any significant correlations with the monkeys' perceptual choices, despite their greater contrast sensitivities. If, however, choice probabilities were calculated from interspike intervals instead of spike counts (thus taking into account the higher firing rates of OFF neurons), OFF neurons but not ON neurons were significantly correlated with behavioral choices. Taken together, these results demonstrate in awake, behaving animals that: 1) the ON and OFF pathways do not simply mirror each other in their functionality but instead carry qualitatively different types of information, and 2) the responses of ON and OFF neurons can be correlated with perceptual choices even in the absence of physical stimuli and interneuronal correlations.

Fractal dimension of apical dendritic arborization differs in the superficial and the deep pyramidal neurons of the rat cerebral neocortex

Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones.

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.

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.

Stimulus size and intensity alter fundamental receptive-field properties of mouse retinal ganglion cellsin vivo

The receptive field (RF) of most retinal ganglion cells (RGCs) is comprised of an excitatory center and an antagonistic surround. Interactions between these RF elements shape most of the visual responses of RGCs. To begin to investigate center-surround interactions of mouse RGCs quantitatively, we characterized their responses in anin vivopreparation to a variety of spot and full-field stimuli. When RGCs were stimulated with a spot that matched the cell's RF center diameter (optimal spot), all RGCs could be categorized as either ON- or OFF-center. In all RGCs, full-field stimulation significantly reduced both the peak and the mean firing rates evoked with an optimal spot stimulus. Full-field stimulation revealed differences in other response properties between ON- and OFF-center RGCs. With a full-field stimulus, the duration of the OFF-center RGCs response was reduced making them more transient, while the duration of the ON-center RGCs increased making them more sustained. Of most interest, full-field stimulation altered the RF center response sign in approximately half of the OFF-center RGCs, which became either OFF/ON or ON only. In contrast, all ON-center and the other OFF-center cells conserved their RF response sign in the presence of the full-field stimulus. We propose that sign-altering OFF-center RGCs possess an additional RF surround mechanism that underlies this alteration in their response. Of general interest these results suggest that the sole use of full-field stimulation to categorize visual response properties of RGCs does not adequately reflect their RF organization and, therefore, is not an optimal strategy for their classification.

Morphometric analysis of neurons from the marginal and substantia gelatinosa layers of human spinal cord: Classification according to laminar organization

The main goal of morphometric analysis of neuronal images, except for getting information about their geometry and dendritic branching patterns, is their classification based on laminar organization. The majority of contemporary techniques for image analysis are based on the application of fractal theory, which has some limitations on results analysis. For that reasons, the new, mostly nonfractal techniques for image analysis had been designed in the past few years. This study shows the analysis of morphometry of the human spinal cord neurons from the marginal (lamina I) and substantia gelatinosa (laminae I-II). For the analysis of neuron images two techniques of morphometric analysis were used: box-counting method as a mainly used technique for fractal analysis, and circle-counting method as a newly designed technique for measuring the length of dendrites. The use of these methods for neurons of the mentioned regions of human spinal cord showed that circlecounting method had given more accurate results than fractal analysis method. When the proposed method was used for the analysis of neuronal images, it was possible to classify neurons according to their laminar position.

AUTOMATED MORPHOMETRIC ANALYSIS OF THE CAT RETINAL α/Y, β/X AND δ GANGLION CELLS USING WAVELET STATISTICAL MOMENT AND CLUSTERING ALGORITHMS

Computational morphological analysis comprises the development of measures (indicators) that describe different form attributes of a neuron and provides additional parameters for classification algorithms. Our work addressed the problem of small group sizes often encountered in neuromorphological and neurophysiological research, automated classification tasks (unsupervised learning) and introduced a new morphological measure: the wavelet statistical moment. We analysed cat alpha/Y, beta/X and delta Golgi-stained retinal ganglion cells using six different shape features (circularity, 2(nd) statistical moment and entropy of Gaussian blurred images, wavelet statistical moment, number of terminations and the fractal dimension). This allowed us to compare the sensitivity of the methods in uniquely describing morphological attributes of these cells.

Use of fractal theory in neuroscience: methods, advantages, and potential problems

Fractal analysis has already found widespread application in the field of neuroscience and is being used in many other areas. Applications are many and include ion channel kinetics of biological membranes and classification of neurons according to their branching characteristics. In this article we review some practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and estimation of the fractal dimensions from the data points, as well as the advantages and problems of fractal geometric analysis, are discussed.

Fractal dimensions in the occurrence of miniature end-plate potential in a vertebrate neuromuscular junction

1. Occurrence of miniature endplate potentials (MEPP) in the sartorius muscle of Rana catesbiana in high Mg2+ Ringer solution were observed in standard intracellular recording. Intervals and amplitudes of sequentially occurring MEPP were registered and analyzed. 2. Interval histograms of a time series of MEPP showed exponential-like pattern as reported in the classical study by Fatt and Katz (1952). The cumulative distribution of the intervals plotted in logarithmic axes showed two distinct phases. In shorter intervals (< 1s), curve along exponential decay was observed, and in longer intervals (> or = 1s) linear decay can be seen. The latter power-law relation gave dimensions of 4.111 +/- 0.812 (mean and S.D.). Self-similarity in longer range implies a time-scale invariant nature and may suggest fractal nature in restoration process of synaptic vesicles, while exponential decay in the short time interval range implies random release of transmitter packet from the readily releasable pool. 3. Fluctuation of amplitudes in sequentially occurred MEPP were analyzed according to Higuchi's cumulative route-length analysis. The estimates for sequential amplitude curve showed the power-law relation in a logarithmic plot whose inclination (= D) estimated with linear regression analysis was 1.996 +/- 0.007 (mean and S.D.). This results indicate that fluctuation in the amplitude of MEPP shows possible maximum complexity as a graphic curve in 2-D plane. Similar result was obtained for fluctuation of intervals of successively occurring MEPP.

Comparative fractal analysis of cultured glia derived from optic nerve and brain demonstrate different rates of morphological differentiation

O-2A progenitor cells derived from neonatal rat cerebral hemispheres or optic nerves, were induced to differentiate in culture into either oligodendrocytes or type 2 astrocytes. The fractal dimensions, a measure of morphological complexity, of the differentiating glial cells were measured over time. Analysis of the changes in fractal dimension (D) with respect to time revealed specific rates of growth for each glial phenotype and a specific final D. The time course of these changes is well fit by a simple mathematical model. While brain-derived oligodendrocytes matured faster than the astrocytes, they ultimately attained comparable levels of complexity, with similar maximum fractal dimensions. Oligodendrocytes from nerve also matured faster than nerve derived astrocytes, in contrast, however, they attained a greater morphological complexity than nerve astrocytes. While the brain-derived oligodendrocytes showed a faster rate of maturation than their optic nerve counterparts, astrocytes from both regions had similar rates of morphological differentiation. Self-similarity, a defining property of fractal objects was investigated, by determining the fractal dimension of cells over a range of magnifications. The calculated fractal dimension remained constant over a 10-fold range in optical magnification, illustrating that cultured glial cells exhibit this important characteristic of fractal objects. In addition, we analyzed the branching patterns of glial processes by the Sholl method and found that the results were not as interpretable or meaningful as those of fractal analysis.