Fig 1.
Top. Schematic diagrams of the four neuronal classes defined in the study. Red circles represent the somatodendritic domain. Blue lines in class 1, class 2 and class 3 schemes represent the main axon trunk and collateral branches. Blue circles in class 1, class 2 and class 4 represent focal terminal arborizations of the axon. Blue oval filled with dots in class 3 represents a large and sparse axonal arborization. Bottom. One representative example of each neuronal class extracted from the NeuroMorpho dataset (Drawings generated from morphologies AA0641, AA0002, AA0051, AA0771 from Neuromorpho.org). Axon is represented in blue whereas dendrites and soma are depicted in red. Dashed contours enclose the spatial extent of the axonal arborizations. Scale bar, 1 mm.
Fig 2.
Methods for length estimation.
A. Projections method. One tissue section containing several axon fibers (above). The method consists of measuring the axon length projected in the X-Y plane (represented below) and multiplying it by a factor to obtain the 3D actual length. B. Stereology with virtual spheres. This method is based on the estimation of axonal length from the intersections occurring between the axon segments and the surface of virtual spheres introduced in sampling boxes inside the tissue section thickness. Four sampling boxes inside a section are represented in black, with the virtual spheres inside (in grey). C. Stereology with virtual planes estimates axonal length by using the intersections between the axon segments (in red) and a set of parallel and isotropically oriented planes contained within the sampling boxes included in the tissue sections. Four sampling boxes (in black) with parallel virtual planes inside (in grey) are represented; r and d stand respectively for sphere radius and distance between planes.
Fig 3.
Reconstruction of the complete axonal (blue) and dendritic (red) trees of 3 individually-labelled thalamocortical neurons used for the practical implementation of the length estimation methods in the laboratory. A. Class 1 neuron from the lateral geniculate nucleus (LGN) arborizing focally in primary visual cortex (V1). B. Class 2 neuron from the posterior nucleus (Po) arborizing focally in 3 separate cortical areas: primary motor cortex (M1), primary somatosensory cortex (S1) and secondary somatosensory cortex (S2). C. Class 3 neuron from the lateral posterior nucleus (LP) arborizing over a large extension of the temporal association cortex (TeA). D-I. Brightfield microscope images of 50 mm thick coronal sections showing different parts of the labeled neurons after immunostaining for GFP and ABC-DAB-Nickel intensification. D. Cell body and dendrites of a Class 1 neuron. E. Axonal arborization in the cortex of a Class 1 neuron. F. High—magnification details of the axonal branches shown in E. G-H. Axonal arborizations of the Class 2 neuron in the motor cortex (G) and the primary somatosensory cortex (H). I. Axonal arborization in the temporal cortex of the Class 3 neuron. Note the sparse distribution of the axonal fragments (Arrowheads). Scale bars: A-C, 500 μm; D, E, G, H, I, 100 μm; F, 10 μm.
Fig 4.
Axon length correlations for projection-based stereology.
A. Correlation between 3D real length and 2D projection length, by axonal class. Red lines represent the mean estimated 3D real length (ordinary least squares), and light red shaded areas correspond to 95% confidence intervals (CI95%). Gray dots depict full samples comprising all planes measurements (XY, XZ and YZ, note the organization of the data in triplets). Global adjusted R-squared = 0.99. B. Estimated slopes (red dots) and CI95% (horizontal black bars) values for correlations in A. Slopes can be interpreted as coefficients (alpha) that multiply 2D projection values to obtain predictions of 3D real lengths. Note that all axon classes share a common set of potential alphas at a 95% confidence level (gray shaded area).
Fig 5.
Estimation error probability projection-based stereology.
A. Estimation error is distributed according to the shown probability densities (axon classes in columns; red, green, and blue for XY, XZ, and YZ planes respectively). The probability of getting a certain error during axon length estimation will be given by the area below the curves (dark and light colors for + 5% and + 10% estimation error respectively). B. Absolute estimation error vs. probability (given by the area below the curves in A). Dashed lines point the probabilities of estimating axon length with absolute errors of 5% and 10%, (colored areas in A). Insets. Absolute error distributions for planes XY, XZ, and YZ.
Fig 6.
Mean absolute error for axon length estimation by spheres-based stereology.
A. The mean absolute error (in % of the real axon length indicated by colors–see color bar) is shown for different values of the probe diameters and the step between sampling boxes, and for axon classes 1 to 4 (columns). Note that the color bar is the same for classes 1, 2, and 3. B. Accuracy of the mean error estimation as maximum error estimation at 95% confidence (in % of the real axon length indicated by colors; same color bar for classes 1, 2, and 3). C. Mean effort, quantized by the mean number of intersections, required to estimate the axon length with different values of the probe diameter and the step for axon classes 1 to 4. Colors denote the number of intersections (in logarithmic scale; same color bar for classes 1, 2, and 3). Mean intersection values were obtained through linear interpolation from estimation for diameter values in 10, 15, 20,…,50 μm and step values in 70, 80,…,150 μm.
Fig 7.
Mean absolute error for axon length estimation by planes-based stereology.
A. The mean absolute error (in % of the real axon length indicated by colors–see color bar) is shown for different values of the distance between planes and the step between sampling boxes, and for axon classes 1 to 4 (columns). Note that the color bar is the same for classes 1, 2, and 3. B. Accuracy of the mean error estimation as maximum error estimation at 95% confidence (in % of the real axon length indicated by colors; same color bar for classes 1, 2, and 3). C. Mean effort, quantized by the mean number of intersections, required to estimate the axon length with different values of the distance between planes and the step for axon classes 1 to 4. Colors denote the number of intersections (in logarithmic scale; color bar is the same for classes 1, 2, and 3). Mean intersection values were obtained through linear interpolation from estimation for distance values in 3, 6, 9,…,30 μm and step values in 70, 80,…,150 μm.
Fig 8.
Error probability distribution for axon length estimation by planes-based stereology.
A. Probability of ±5% (first row) and ±10% (second row) axon length estimation error for different combinations of distance and step, and for axon classes 1 to 4 (columns); probability is marked by color gradient, from 0.25 to 0.92 (see color bar). Probabilities for classes 1, 2, and 3 are plotted with the same colored-scale. White dots point to the cases for distance = 18 μ and step = 110 μ. Values were obtained through linear interpolation (Adjusted R-squared for 5% error: class 1 = 0.80, class 2 = 0.86, class 3 = 0.81, class 4 = 0.91; Adjusted R-squared for 10% error: class 1 = 0.83, class 2 = 0.83, class 3 = 0.85, class 4 = 0.93). B. Error probability distribution for the particular cases detailed in A. The areas below the curves are the probabilities shown in A and pointed with the white dots.
Table 1.
Differences between the three axonal length estimation methods after their practical implementation.
Table 2.
Comparison of methodological approaches to measure or estimate axonal length in single neurons.