Data acquisition and preparation
Tissue was obtained from the anterolateral temporal gyri (Brodmann’s areas 21 and 38; see Garey
1994) of patients with pharmaco-resistant temporal lobe epilepsy (Department of Neurosurgery, ‘Hospital de la Princesa’, Madrid, Spain). This brain tissue was removed as part of surgical treatment of five male patients (28–48 years and mean 36.6 years) and had been used in the previous studies (Kastanauskaite et al.
2009; Arion et al.
2006; Sola et al.
2004). The five patients used in this study had normal IQs and each had a different history of medications and treatment—they were treated with a variety of anti-epileptic drugs that affect GABAergic transmission and other neurotransmitter systems. Furthermore, the disease severity was variable (with daily, weekly, or twice monthly seizures) as was the disease duration (from 10 to 29 years). However, as described below, in all the cases, the neocortical tissue used in the present study was histologically normal and without abnormal spiking activity.
In each case, video-EEG recording from bilateral foramen ovale electrodes was used to localize the epileptic focus in mesial temporal structures. Subdural recordings with a 20-electrode grid (lateral neocortex) and with a 4-electrode strip (uncus and parahippocampal) were used at the time of surgery to further identify epileptogenic regions. After surgery, the lateral temporal neocortices of all patients and the mesial temporal structures from all patients except one were available for the standard neuropathological assessment. In the latter case, most mesial structures were absorbed during surgical removal and, therefore, could not be examined. The lateral neocortices were histologically normal in all the cases. However, alterations were found in the hippocampal formations of three out of the four patients that could be examined; these three patients showed hippocampal sclerosis, whereas no apparent alterations were found in the hippocampal formation of the remaining patient. Furthermore, only neocortical tissue that showed no abnormal spiking—as characterized by normal ECoG activity—was used in this study (see Arion et al.
2006).
Surgically resected tissue was immediately immersed in cold 4 % paraformaldehyde in 0.1 M phosphate buffer, pH 7.4 (PB). After 2−3 h, the tissue was cut into small blocks (0.5 × 8 × 8 mm) which were flattened (e.g., Welker and Woolsey
1974) and post-fixed in the same fixative for 24 h at 4 °C. Horizontal sections (250 microns) were obtained using a Vibratome. By relating these sections to coronal sections, we were able to identify, using cytoarchitectural differences, the section that contained each cortical layer, allowing the subsequent injection of cells (e.g., Elston and Rosa
1997). Sections were prelabeled with 4,6-diamidino-2-phenylindole (DAPI; Sigma, St Louis, MO), and a continuous current was used to inject individual cells with Lucifer yellow (8 % in 0.1; Tris buffer, pH 7.4; LY) in cytoarchitectonically identified layers III and V of the anterolateral temporal cortex (see “
Results” for further details). Neurons were injected until the individual dendrites of each cell could be traced to an abrupt end at their distal tips, and the dendritic spines were readily visible, indicating that the dendrites were completely filled. After the injection of the neurons, the sections were first processed with a rabbit antibody to Lucifer yellow produced at the Cajal Institute [1:400,000 in stock solution: 2 % BSA (A3425; Sigma); 1 % Triton X-100 (30632; BDH Chemicals); and 5 % sucrose in phosphate buffer (PB)] and then with a biotinylated donkey anti-rabbit secondary antibody (1:200 in stock solution, RPN1004; Amersham Pharmacia Biotech), followed by a biotin–horseradish peroxidase complex (1:200 in PB, RPN1051; Amersham). 3,3′-Diaminobenzidine (D8001; Sigma Chemical Co.) was used as the chromogen, allowing the visualization of the entire basal dendritic arbor of pyramidal neurons. Finally, sections were mounted in 50 % glycerol in PB.
Possible changes in the size of the sections due to processing of the tissue were evaluated by measuring the cortical surface and thickness in adjacent sections before and after intracellular injections and processing of the tissue, using Neurolucida 11.07 and StereoInvestigator 11.02.1 from MicroBrightField (MBF, VT, USA). We found no shrinkage in the surface area of the sections, and a decrease in the thickness of only approximately 7 % was observed. Therefore, no correction factors were included. Neurons were reconstructed in three dimensions using Neurolucida (MicroBrightField) as previously described in detail (for further methodological details, see Elston et al.
2001; Benavides-Piccione et al.
2006).
We refer to branch order of a branching angle as the number of branchings (including itself) that exist between the branching angle and the root of the dendrite. As an example, a branching angle with branch order 4 comes after three preceding branching angles from the root of the dendrite, which is the branch order 1. We refer to maximum branch order or tree order of a dendrite as the total amount of branch orders of a dendrite, or the branching angle at the highest order that can be found in the dendrite.
The data set included: 57, 37, and 87 cells from layer IIIAnt (1452 measurements), VPost (1328 measurements), and IIIPost (2430 measurements), respectively. More precisely, the data set for layer IIIPost contained measurements of seven branch orders (300, 477, 430, 198, 39, 5, and 3 from orders 1–7, respectively) extracted from a total of 57 neurons. The data set for layer VPost contained the measurements of eight branch orders (247, 381, 373, 226, 82, 14, 4, and 1 from orders 1–8, respectively) extracted from a total of 37 neurons. Finally, the data set for layer IIIAnt contained the measurements of seven branch orders (470, 732, 714, 375, 114, 24, and 1 from orders 1–7, respectively), extracted from a total of 87 neurons. In this data, branch orders above five suffer from a very low number of observations, and thus, we will restrict our analysis to the first five branch orders. The 3D reconstructions of these cells will be available in another publication (Benavides-Piccione, Kastanaukaite, Rojo, and DeFelipe, in preparation).
Univariate truncated von Mises distribution
The statistical analysis of branching angles requires directional statistics, as the conventional statistics do not address well the circular properties. In this field, the von Mises distribution (Mardia
1975) is the most known distribution and the analog of the Gaussian distribution in the line. This distribution has properties, such as symmetry and positive support in all the values in a circle (0°, 360°), that are necessary simplifications of the data in many case studies. As it is found that in neuroscience, such simplifications may hinder the accuracy and reliability of the complex behaviors it studies, we propose for the first time to use the truncated von Mises distribution, and a generalization that adds two parameters that restrict the interval, where the distribution has a density greater than 0, as a step forward in better modeling the data. The truncated von Mises is defined with a four parameter probability density function:
$$ f_{\text{tvM}} (\theta ;\mu ,\kappa ,a,b) = \left\{ {\begin{array}{*{20}c} {\frac{{e^{\kappa \cos (\theta - \mu )} }}{{\int_{a}^{b} {e^{\kappa \cos (\theta - \mu )} } {\text{d}}\theta }}} & {{\text{if}}\;\;\theta \in {\mathbb{O}}_{a,b} } \\ 0 & {{\text{if}}\;\;\theta \in {\mathbb{O}}_{b,a} } \\ \end{array} } \right. $$
where
µ ∈
\( {\mathbb{O}} \) is the location parameter,
κ > 0 the concentration parameter,
\( {\mathbb{O}} \) is the circular set of points,
\( {\mathbb{O}} \)
a,b ⊂
\( {\mathbb{O}} \) is the circular interval obtained by selecting the points in the circular path from
a ∈
\( {\mathbb{O}} \) to
b ∈
\( {\mathbb{O}} \) in the preferred direction (counterclockwise), and
\( {\mathbb{O}} \)
b,a is its counterpart with respect to
\( {\mathbb{O}} \).
Using the truncation parameters, the distribution can present multiple shapes (strictly increasing, strictly decreasing, one global maximum, one global minimum, etc) and even not contain the mode or location parameter among the positive support. From a sample
θ
1,
θ
2,…,
θ
n of angular values, the maximum likelihood estimators for parameters
a and
b are
$$ \begin{aligned} \hat{a} & = \hbox{min} \{ \theta_{1} , \ldots ,\theta_{n} \} \\ \hat{b} & = \hbox{max} \{ \theta_{1} , \ldots ,\theta_{n} \} . \\ \end{aligned} $$
The estimators of parameters µ and κ cannot be computed analytically, and numerical optimization techniques have to be used to approximate their value.
Bivariate-truncated von Mises distribution
This distribution accounts for pairs of dependent angular variables. It can be used to study events that are defined by two angular measurements (
θ
1,
θ
2). It is a nine parameter distribution on the torus (
\( {\mathbb{O}} \) ×
\( {\mathbb{O}} \) → R), where four of the parameters correspond to that of a univariate truncated distribution for
θ
1 and other four correspond to that of a univariate truncated distribution for
θ
2 and the parameter
λ ∈ R, that measures the correlation between
θ
1 and
θ
2, which in the circle is defined as
\( {\mathbb{E}} \)[sin(
θ
1 −
µ
1) sin(
θ
2 −
µ
2)]. The random variable (
θ
1,
θ
2) following this distribution has the probability density function:
$$ f_{\text{btvM}} (\theta_{1} ,\theta_{2} ;{\mathbf{W}}) = \frac{{e^{{\kappa_{1} \cos (\theta_{1} - \mu_{1} ) + \kappa_{2} \cos (\theta_{2} - \mu_{2} ) + \lambda \sin (\theta_{1} - \mu_{1} )\sin (\theta_{2} - \mu_{2} )}} }}{{\int_{{a_{1} }}^{{b_{1} }} {\int_{{a_{2} }}^{{b_{2} }} {e^{{\kappa_{1} \cos (\theta_{1} - \mu_{1} ) + \kappa_{2} \cos (\theta_{2} - \mu_{2} ) + \lambda \sin (\theta_{1} - \mu_{1} )\sin (\theta_{2} - \mu_{2} )}} } } {\text{d}}\theta_{2} {\text{d}}\theta_{1} }}\quad {\text{if}}\;\;\theta_{1} \in {\mathbb{O}}_{{a_{1} ,b_{1} }} ,\theta_{2} \in {\mathbb{O}}_{{a_{2} ,b_{2} }} $$
and 0 otherwise.
W = {
λ,
µ
1,
µ
2,
κ
1,
κ
2,
a
1,
b
1,
a
2,
b
2} is the parameter vector. For a sample of the form {(
θ
1i
,
θ
2i
)
i = 1,…,
n}, maximum likelihood estimators for parameters
a
1,
b
1 and
a
2,
b
2 are
$$ \begin{aligned} \hat{a}_{1} & = \hbox{min} \{ \theta_{11} , \ldots ,\theta_{1n} \} \\ \hat{b}_{1} & = \hbox{max} \{ \theta_{11} , \ldots ,\theta_{1n} \} \\ \hat{a}_{2} & = \hbox{min} \{ \theta_{21} , \ldots ,\theta_{2n} \} \\ \hat{b}_{2} & = \hbox{max} \{ \theta_{21} , \ldots ,\theta_{2n} \} . \\ \end{aligned} $$
The estimators of parameters µ
1, µ
2, κ
1, κ
2, and λ cannot be computed analytically, and like in the univariate case, numerical optimization techniques have to be used for value approximation.
Statistical tests
Test of goodness-
of-
fit a univariate truncated von Mises distribution We tested if the angular data, under different groupings, can be properly modeled with a truncated von Mises distribution. As considered in Mardia and Jupp (
2000), we transformed the data
θ
1,…,
θ
n
by means of the angular variable
U(θ
i
) =
2πF(θ
i
), where
F(θ) is the probability distribution function of the truncated von Mises distribution. Then, we tested circular uniformity (i.e., the circular distribution, where every observation is equally likely to occur) using a modified Rayleigh statistic (Cordeiro and De Paula Ferrari
1991) that distributes according to a
χ
2
2
distribution under the null hypothesis to obtain the final
p value for the fit. If the data distribute following a truncated von Mises distribution, the previous transformation generated a uniform distribution from the data.
Test of goodness-of-fit to a univariate von Mises distribution A similar procedure is used for the von Mises distribution. The difference between both the cases is the probability distribution function F(θ) that is used. In this case, F(θ) is the probability distribution function of the von Mises distribution, and therefore, the angular variable U(θ
i
) = 2πF(θ
i
) for this case is also different.
Two sample tests for common distribution (similarity) We tested the hypothesis of similarity between two data sets, i.e., if two data sets can be considered to be drawn from the same probability distribution. We used the non-parametric Watson’s two sample
U
2 test (Watson
1962) that does not assume any underlying probability distribution. This test was used to perform the comparisons between layer IIIPost and layer VPost, and layer IIIAnt and layer IIIPost. In addition, it was used to perform comparisons between humans, rats, and mice (see Supplementary Tables 9, 10, and 11). Another test, the energy test (Rizzo and Szekely
2014), for the similarity of distributions outside directional statistics, was also used for the comparisons between branching angles distribution data with the “complexity” of the dendritic arbor in humans that was evaluated using the number and distribution of their branching points (i.e., total number of nodes (branch points) contained in the dendritic tree) (see Supplementary Table 15).
Tests for mean comparison We use Watson’s large sample (where “large” stands for samples greater or equal to 25) non-parametric test (Watson
1983) to test the null hypothesis of the same mean direction. The test does not assume any underlying probability distribution. It was used with three different subgroups of the data, as we were interested in testing if the means of the data, grouped by branchings or branchings together with maximum branch order, follow any noticeable tendency. It was additionally used for comparisons between layers IIIPost and VPost, for the comparisons of branch order 1 mean values and for the comparisons between humans, rats, and mice (see Supplementary Tables 1, 2, 4, and 12).
Tests for the concentration comparison Wallraff’s test for common concentration (Wallraff
1979) was useful for comparisons between layer IIIPost vs. layer VPost and layer IIIAnt vs. layer IIIPost. It is a non-parametric test with no assumptions regarding data generating distributions (see Supplementary Table 4).
Tests of independence We used two different tests to verify or reject the hypothesis of independence (i.e., if positive or negative significant correlations between two random variables exists). First, we used a randomized version of Rothman’s test for independence (Rothman
1971), a test that does not assume any underlying probability distribution for the two tested data sets (see Supplementary Table 8). Finally, we used a permutations tests over the λ parameter (that we previously estimated using the maximum likelihood method from the data sets) which tested the null hypothesis of
λ =
0
Test-
based diagrams We used two different forms of visualization for the comparison of test results. The first type of diagram, the test-based diagram, was originally proposed in (Bielza et al.
2014) and consists of a space of nodes that are connected or not by edges depending on the non-rejection or rejection result of the test, respectively. In this diagram, every node that appears is pairwise tested with respect to all the other nodes. These diagrams are shown in Figs.
2d and
3. The second type of diagram, the test-based tree, is first proposed here as a form to easily visualize comparisons between two cortical brain layers or two data sets, whose data are organized in a tree-like structure that includes branch orders. It consists of trees, where the branch order in the graphic corresponds to the branch order of the conducted test. If the space between the branches is subdivided and labeled with a number, the number that labels each subdivided area indicates the maximum branch order of the data of the conducted test. Finally, the green color or red color of the area between the branches indicates the non-rejection or rejection of the hypothesis of the conducted test, respectively. These diagrams are shown in Figs.
4a, b and
5a, b.