### Selection of studies

The pool of all eligible neuroimaging experiments was retrieved from the BrainMap database (

http://brainmap.org/) (Fox et al.

2005; Fox and Lancaster

2002; Laird et al.

2009,

2005; Vanasse et al.

2018) using a Sleuth query. BrainMap is an online open access database that uses a systematic coding scheme which contains over 15,000 published human neuroimaging experimental results and reports over 120,000 brain locations in stereotactic space. The main division of this database is between voxel-based morphometry (VBM) and functional data. For our meta-analysis, both the VBM and functional data sets have been used. First, using the BrainMap software application ‘Sleuth 2.4’ we queried the VBM BrainMap database (January 2018) using the following search criteria:

1.
decreases: (experiments context is disease) AND (experiment contrast is gray matter) AND (experiments observed changes is controls > patients);

2.
increases: (experiments context is disease) AND (experiment contrast is gray matter) AND (experiments observed changes is patients > controls).

We retrieved 994 experiments (i.e., 994 sets of alteration stereotactic coordinates indicating the foci of significant case–control alterations). Then the retrieved data set was codified on the basis of the ICD-10 classification (World Health Organization

1992) by an expert researcher. In addition, all the eligible articles were analyzed by two expert researchers to ascertain that they satisfied the following inclusion criteria: (a) to be an original work published in a peer-reviewed English language journal; (b) to include a whole-brain VBM analysis; (c) to include a comparison between pathological sample and healthy control participants; (d) to report GM decrease/increase changes in pathological sample; (e) to adopt a specified VBM analysis; f) to report the locations of GM changes (specifically cartesian coordinates in a standardized 3D space) in a definite stereotactic space (i.e., Talairach/Tournoux or Montreal Neurological Institute). On the grounds of the aforementioned criteria, 793 articles were included (585 of GM decreases and 208 of GM increases), for a total of 1361 experiments (980 of GM decreases and 381 of GM increases) and 29,403 subjects. Descriptive information of interest was extracted from each full-text article. Since some of the foci coordinates were reported in MNI space while other in Talairach space, locations reported in MNI were converted into Talairach space using Lancaster’s icbm2tal transform, following the approach of Laird et al. (

2010) and of Lancaster et al. (

2007). The complete overview of the selection process is reported in Table

1. More detailed information about the description and distribution of the VBM data set disease-related included in our meta-analysis are viewable in the Supplementary Table S1. Table S2 shows the sample characteristics for the four most represented brain disorders in the BrainMap VBM database (i.e., SCZ, AD, BD and DD).

Table 1
Synopsis of the selection procedure with number of articles identified at each stage

Articles 994 ⇓ Additional records 0 | Articles 2376 ⇓ Additional records 0 | Abstract exclusions Eligibility for full-text lecture | Full-text exclusions Selected studies 793 VBM 2376 Functional | Selected studies 585 GM decrease 208 GM increase Sample (N) 29,403 ⇓ SCZ (114 studies) AD (55 studies) DD (54 studies) BD (46 studies) Others (524 studies) | Selected studies 2376 Selected experiments 13,148 Sample (N) 68,152 |

Phase 1 ⇓ data search | Phase 2 ⇓ data search | Phase 3 | Phase 4 | Phase 5 ⇓ data extraction | Phase 6 ⇓ data extraction |

Finally, we did a systematic search on the functional data set of BrainMap using the following search criteria:

(1) (experiments context is normal mapping) AND (experiments activation is activations only) AND (subjects diagnosis is normals).

We retrieved 2376 articles, for a total of 13,148 experiments, 110 paradigm classes and 68,152 subjects. All the retrieved experiments were used for the subsequent MHC (Mancuso et al.

2019) analysis (see also Table S3 in Supplementary Material)

, after the conversion of the coordinates in Talairach space.

Authors declare to have signed a written agreement with the BrainMap group and the University of Texas, San Antonio, USA, so as to have access to the BrainMap database.

We adopted the definition of meta-analysis accepted by the Cochrane Collaboration (Green et al.

2008) and performed the process of selecting eligible articles according to the ‘PRISMA Statement’ international guidelines (Liberati et al.

2009; Moher et al.

2009) [see Figure S1 (PRISMA flow chart) in the online Supplementary Material].

### Anatomical likelihood estimation and creation of the modeled activation map

We employed the anatomical likelihood estimation (ALE) (Eickhoff et al.

2009,

2012; Turkeltaub et al.

2012) so as to construct the maps to feed the PHAC and Patel’s algorithms. The ALE is a quantitative voxel-based meta-analysis that can provide information about the anatomical reliability of results. It compares the results with a sample of reference studies obtained from the existing literature. Every focus of each experiment is considered to be the central point of a three-dimensional Gaussian probability distribution:

The ALE is a quantitative voxel-based meta-analysis that can provide information about the anatomical reliability of results. It compares the results with a sample of reference studies obtained from the existing literature. Every focus of each experiment is considered to be the central point of a three-dimensional Gaussian probability distribution:

$$ p\left( d \right) = \frac{1}{{\sigma ^{3} \sqrt {\left( {2\pi } \right)^{3} } }}e^{{\frac{{ - d^{2} }}{{2\sigma ^{2} }}}} $$

(1)

in which

d represents the Euclidean distance between the voxels and the focus taken into account, whereas σ represents the spatial uncertainty. The standard deviation is calculated through the full-width at half-maximum (FWHM) with the following formula:

$$ \sigma = \frac{{FWHM}}{{\sqrt {8ln2 } }} $$

(2)

which results in different values of σ and thus in modeled activation or alteration (MA) maps with different size for each experiment according to their number of subjects.

The MA maps are derived from a Gaussian probabilistic cloud for each focus. If the focus is close to the brain median line, then the probabilistic cloud may extend for few millimeters in both the hemispheres, thus producing spurious co-alteration/coactivation results. To address this potential issue, we adjusted the offset values that were close to the median line. By taking into consideration the mean spatial uncertainty that is typical of these meta-analytic data (Eickhoff et al.

2009), we expected that on average the Gaussian cloud may extend around 12 mm, so we modeled a sphere having a mean radius of 12 mm and compensated for the probabilistic cloud extension an area of 12 mm both on the left and on the right of the median line; to do so, we applied a weight decreasing function with distance in millimeters between the median line

\(i\) and the voxel

\(j\) taken into account, proportional to

\(\frac{1}{{d_{{ij}} }}\), which attributes to the voxels major or minor activations according to their proximity to the median line.

### Maps of pathological homotopic anatomical co-alteration and the calculation of the conditional probability unbalance

To determine the PHAC maps we conceived a novel method allowing us to construct a map of the homotopic anatomical co-alterations using meta-analytic data. This method can identify if the anatomical alteration of a cerebral area statistically co-occurred with the alteration of its homologue in the contralateral hemisphere. With this analysis we can therefore construct a PHAC map, in which values are assigned proportionally to the statistical relationship between cerebral regions of one hemisphere and their contralateral homologues.

The brain has been symmetrically partitioned by means of an anatomical atlas obtained from the Talairach atlas extracted from the Talairach Daemon (Lancaster et al.

1997,

2000;

http://talairach.org/). The atlas was co-registered to the same 2 mm resolution GingerALE standard of the MAs maps (

http://brainmap.org/ale/colin_tlrc_2x2x2.nii.gz) using FLIRT from FSL (Smith et al.

2004;

http://www.fmrib.ox.ac.uk/fsl/). To produce symmetric maps of homotopic co-alteration, the atlas was subsequently symmetrized by substituting the left hemisphere with a copy of the right one flipped along the midline. To construct the PHAC map, we created an alteration matrix with the couples of homologous areas as nodes. In a

N ×

M matrix every row indicates an experiment, whereas every column indicates a node corresponding to an area of the brain; in our case, the numbers of experiments (functional and VBM data) × 1105 nodes constitute the matrix. For every experiment, a node was considered to be altered if the MA map (thresholded at

p = 0.05) of the experiment reported 20% or more of the voxels of interest (VOIs) within the node. As showed in Mancuso et al. (

2019), the arbitrary percentage of 20% of altered voxels, which is needed to consider a VOI as altered, does not bias the results and was showed to be a reasonable middle ground between 0%, which is obviously too liberal, and 40%, which can be argued to be too conservative.

From the

N ×

M matrix we obtained the co-alteration strength between the homotopic nodes using the Patel’s κ index (Patel et al.

2006), thus generating the probability distribution of joint alteration occurrences for every couple of nodes. Specifically, given two nodes (

a and

b), it is possible to calculate the probability of all the possible conditions: (i)

a and

b are both altered; (ii) neither

a nor

b is altered; (iii)

a is altered but not

b; (iv)

b is altered but not

a (Table

2). Frequencies of these cases throughout the experiments result in the following four states of probabilities:

$$ \theta _{1} = P\left( {a = 1,b = 1} \right) $$

$$ \theta _{2} = P\left( {a = 0,b = 1} \right) $$

$$ \theta _{3} = P\left( {a = 1,b = 0} \right) $$

$$ \theta _{4} = P\left( {a = 0,b = 0} \right) $$

Table 2
Marginal probabilities between altered and unaltered volumes of interest (VOIs)

These states of probabilities represent the conjoint state frequencies of a couple of nodes (a and b) in their four possible combinations. The following table illustrates the marginal probabilities:

Considering these four probabilities, we can apply the two indices κ and

τ of Patel et al. (

2006) for determining connectivity and directionality, respectively. These two indices have been shown to be effective with simulated data by Smith et al. (

2011). However, with regard to the Patel’s

τ, Wang et al. (

2017) have criticized its usefulness. It should be observed that the criticism by Wang et al. focuses on issues (i.e., deconvolution of the hemodynamic response and temporal resolution) that are associated with the application of empirical Bayesian techniques to fMRI data; however, this is not the case of the present study, which takes into account morphometric data derived from the scientific literature.

The Patel’s κ is capable of measuring the probability that two nodes (

a and

b) are co-altered with respect to the probability that

a and

b are independently altered. Patel’s κ is defined as follows:

$$ \kappa = \left( {\vartheta _{1} - E} \right)/\left[ {D\left( {\max \left( {\vartheta _{1} } \right) - E} \right) + \left( {1 - D} \right)\left( {E - \min \left( {\vartheta _{1} } \right)} \right)} \right] $$

(3)

where

$$ E = \left( {\vartheta _{1} + \vartheta _{2} } \right)\left( {\vartheta _{1} + \vartheta _{3} } \right) $$

$$ \max \left( {\vartheta _{1} } \right) = \min \left( {\vartheta _{1} + \vartheta _{2} ,\vartheta _{1} + \vartheta _{3} } \right) $$

$$ \min \left( {\vartheta _{1} } \right) = \max \left( {0,2\vartheta _{1} + \vartheta _{2} + \vartheta _{3} - 1} \right) $$

$$ D = \left\{ {\begin{array}{*{20}c} {\frac{{\theta _{1} - E}}{{2\left( {\max \left( {\theta _{1} } \right) - E} \right)}} + 0.5,\quad\,{\text{if}}\,\theta _{1} \ge E} \\ {0.5 - \frac{{\theta _{1} - E}}{{2\left( {E - \min \left( {\theta _{1} } \right)} \right)}},\quad{\text{otherwise}}} \\ \end{array} } \right. $$

In the fraction, the numerator is the difference between the probability that

a and

b are altered together and the expected probability that

a and

b are independently altered, whereas the denominator is a weighted normalizing constant.

\({\text{Min}}\left( {\vartheta _{1} } \right)\) stands for the maximum value of conjoint probability

\(P\left( {a,b} \right)\), given

\(P\left( a \right)\) and

\(P\left( b \right)\), while

\({\text{max}}\left( {\vartheta _{1} } \right)\) stands for the minimum value of

\(P\left( {a,b} \right)\), given

\(P\left( a \right)\) and

\(P\left( b \right)\). Patel’s κ values range from –1 and 1. A value of |κ| that is close to 1 indicates high connectivity. Patel’s κ statistical significance is evaluated by simulating with a Monte Carlo algorithm, a multinomial, generative model of data, which can consider the alterations of all the nodes. The Monte Carlo method obtains an estimate of posterior probability using the multinomial model as likelihood:

$$ p\left( {z|\theta } \right)\mathop \prod \limits_{{i = 1}}^{4} \theta _{i}^{{z_{i} }} $$

where

\(z_{i}\) are the sum of the respective

\(\theta _{i}\) of all experiments, that is, the number of times the given couple of nodes is co-altered, and a Dirichlet prior:

$$ p\left( {\theta |\alpha } \right) \propto \mathop \prod \limits_{{i = 1}}^{4} \theta _{{i_{i} }}^{{\alpha _{{j - 1}} }} $$

with

\(\theta _{i} \ge 0\) and

\(\sum\nolimits_{{i = 1}}^{4} {\theta _{i} } = 1. \) Then, the posterior distribution

p(ϑ|z) is a Dirichlet with parameter

\(\gamma _{i} = \alpha _{i} + z_{i} - 1\) with

\(i = 1 \ldots 4.\)
The Monte Carlo samples from the posterior Dirichlet distribution 5000 random values and calculate the proportion of the samples in which κ > e, where e is the threshold of significance, set to 0.01. If this proportion is superior to 0.95 (p = 0.05), the edge is considered to be significant. This calculation has been run independently for each data set. To validate the Patel’s κ beyond any reasonable doubt, in the supplementary material is present a simulation of an (extremely unlikely) case that could produce false positives, showing that our methodology holds true even in worst case scenarios. Once the Patel’s κ of a couple of areas was calculated, such value was assigned to all the voxels of those two areas to obtain a PHAC map.

The

τ(

a,

b) index, in turn, is capable of measuring how the alteration of node

a influences the alteration of node

b. The

τ is calculated only on those edges that reached the statistical significance during the κ calculations. Thus, if two nodes

a and

b are significantly co-altered, the Patel’s

τ indicates the directionality of the edge between them. Patel’s

τ values range from –1 to 1. Positive values denote the influence of

a over

b, whereas negative values denote the influence of b over a. The

τ index is defined as:

$$ \tau \left\{ {\begin{array}{*{20}c} {1 - \frac{{\left( {\vartheta _{1} + \vartheta _{3} } \right)}}{{\left( {\vartheta _{1} + \vartheta _{2} } \right)}},\quad\,{\text{if}}\,\vartheta _{2} \ge \vartheta _{3} } \\ {\frac{{\left( {\vartheta _{1} + \vartheta _{2} } \right)}}{{\left( {\vartheta _{1} + \vartheta _{3} } \right)}} - 1,\quad{\text{otherwise}}} \\ \end{array} } \right. $$

(4)

This index allows to obtain a value of directionality between two nodes and is thresholded using the threshold obtained before calculating the κ metrics. In other words, if we look at Table

2, the numerator and the denominator of

τ are the marginal probability of the altered condition of the node

a independently from the condition of the node

b, and the marginal probability of the altered condition of the node

b independently of the condition of the node

a. This ratio of marginal probabilities gives a measure of alterations of two nodes and allows to estimate the directionality of alterations’ distribution, on the basis of the hypothesis is that if node

a is the origin of a pathological spread toward node

b, then node

a is more likely to be found altered in many groups, both in co-alteration with node

b (presumably in the groups of patients with a more advanced pathological development), and on its own. In contrast, node

b may not be frequently altered in patients with an early pathological development and, when altered, it may almost always occur in co-alteration with node

a. The Patel’s

τ was used on the VBM and functional databases, to calculate two directed PHAC (dPHAC) maps, one for the decreases and one for the increases.

As explained before, we calculate the co-occurrence of alterations in every experiment, each at a time. If there are many foci distributed across different papers, for example because more studies are related to a specific pathology, this may improve the sensitivity of our method for this pathology and not produce false positives, as the permutation for the threshold of this part would consider the amount of data. If, on the contrary, the number of foci were greater on one side in the same paper, this would not bias the result, as the contingency table would have always the same value: 1. Let us make an example by considering two nodes (A and B), and two experiments, one of which has few foci in A and many foci in B, but both nodes have a significant (albeit different) ALE value. The contingency table would have 1 because both nodes are altered, though with different intensity. Even in the case of an experiment reporting a balanced number of foci and significant ALE values of A and B, the contingency table would have again 1. In other words, the calculus of the joint probabilities does not consider the intensity of the ALE values but is based only on the fact that an area results or not in being altered. It worth noting that such consideration applies when the number of foci is uneven between two homotopic regions, but not when both nodes have very few foci. In this case, the statistical power of our technique will drop, as noted in the paragraph “

Unbalances in the directionality of the conditional probability among pathological homologous areas” of the discussion section.

Finally, the reliability towards subsampling of the PHAC and dPHAC maps was tested through a bootstrap procedure with 5000 iterations (see the Supplementary Materials for a detailed explanation and Figure S4).