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Abstract and Figures

Fluid intelligence is arguably the defining feature of human cognition. Yet the nature of its relationship with the brain remains a contentious topic. Influential proposals drawing primarily on functional imaging data have implicated “multiple demand” frontoparietal and more widely distributed cortical networks, but extant lesion-deficit studies with greater causal power are almost all small, methodologically constrained, and inconclusive. The task demands large samples of patients, comprehensive investigation of performance, fine-grained anatomical mapping, and robust lesion-deficit inference, yet to be brought to bear on it. We assessed 165 healthy controls and 227 frontal or non-frontal patients with unilateral brain lesions on the best-established test of fluid intelligence, Raven’s Advanced Progressive Matrices, employing an array of lesion-deficit inferential models responsive to the potentially distributed nature of fluid intelligence. Non-parametric Bayesian stochastic block models were used to reveal the community structure of lesion deficit networks, disentangling functional from confounding pathological distributed effects. Impaired performance was confined to patients with frontal lesions ( F (2,387) = 18.491; p < .001; frontal worse than non-frontal and healthy participants p < .01; p <.001), more marked on the right than left ( F (4,385) = 12.237; p < .001; right worse than left and healthy participants p <.01; p <.001). Patients with non-frontal lesions were indistinguishable from controls and showed no modulation by laterality. Neither the presence nor the extent of multiple demand network involvement affected performance. Both conventional network-based statistics and non-parametric Bayesian stochastic block modelling heavily implicated the right frontal lobe. Crucially, this localisation was confirmed on explicitly disentangling functional from pathology-driven effects within a layered stochastic block model, prominently highlighting a right frontal network involving middle and inferior frontal gyrus, pre- and post-central gyri, with a weak contribution from right superior parietal lobule. Similar results were obtained with standard lesion-deficit analyses. Our study represents the first large-scale investigation of the distributed neural substrates of fluid intelligence in the focally injured brain. Combining novel graph-based lesion-deficit mapping with detailed investigation of cognitive performance in a large sample of patients provides crucial information about the neural basis of intelligence. Our findings indicate that a set of predominantly right frontal regions, rather than a more widely distributed network, is critical to the high-level functions involved in fluid intelligence. Further they suggest that Raven’s Advanced Progressive Matrices is a useful clinical index of fluid intelligence and a sensitive marker of right frontal lobe dysfunction.
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1
Graph lesion-deficit mapping of fluid intelligence
Lisa Cipolotti,
1,2*
James K Ruffle,
2,3
Joe Mole,
1,2
Tianbo Xu,
2
Harpreet Hyare,
2,3
Tim Shallice,
4,5
Edgar Chan
1,2
and Parashkev Nachev
2
1. Department of Neuropsychology, National Hospital for Neurology and Neurosurgery, London,
United Kingdom (*Author for correspondence: l.cipolotti@ucl.ac.uk: phone +4420 3448 4793)
2. Institute of Neurology, University College London, London, United Kingdom
3. Department of Radiology, University College London Hospitals NHS Foundation Trust, London NW1
2PG, United Kingdom
4. Institute of Cognitive Neuroscience, University College London, United Kingdom.
5. International School for Advanced Studies (SISSA-ISAS), Trieste, Italy
*Correspondence to: Prof. Lisa Cipolotti
Department of Neuropsychology, National Hospital for Neurology and Neurosurgery, Queen Square,
London. WC1 3BG
l.cipolotti@ucl.ac.uk
Running title: Graph lesion mapping of fluid intelligence
Keywords: Frontal lobes; executive functions; fluency; focal lesion; lesion-deficit mapping
Abbreviations: AH4-1 = Part 1 of the Alice Heim 4; APM = Advanced Progressive Matrices; Gf = Fluid
intelligence; GNT = Graded Naming Test; LF = left frontals; LNF = left non-frontals; MD = multiple-
demand network; NART = National Adult Reading Test; P-FIT = parieto-frontal integration theory;
PLSM = parcelbased lesion symptom mapping; RF = right frontals; RNF = right non-frontals; ROI =
regions of interest; WAIC = widely applicable information criterion; WAIS = Wechsler Adult
Intelligence Scale.
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2
Abstract
Fluid intelligence is arguably the defining feature of human cognition. Yet the nature of its relationship
with the brain remains a contentious topic. Influential proposals drawing primarily on functional
imaging data have implicated “multiple demand” frontoparietal and more widely distributed cortical
networks, but extant lesion-deficit studies with greater causal power are almost all small,
methodologically constrained, and inconclusive. The task demands large samples of patients,
comprehensive investigation of performance, fine-grained anatomical mapping, and robust lesion-
deficit inference, yet to be brought to bear on it.
We assessed 165 healthy controls and 227 frontal or non-frontal patients with unilateral brain
lesions on the best-established test of fluid intelligence, Raven’s Advanced Progressive Matrices,
employing an array of lesion-deficit inferential models responsive to the potentially distributed nature
of fluid intelligence. Non-parametric Bayesian stochastic block models were used to reveal the
community structure of lesion deficit networks, disentangling functional from confounding
pathological distributed effects
.
Impaired performance was confined to patients with frontal lesions (
F
(2,387) = 18.491;
p
< .001;
frontal worse than non-frontal and healthy participants
p
< .01;
p
<.001), more marked on the right
than left (
F
(4,385) = 12.237;
p
< .001; right worse than left and healthy participants
p
<.01;
p
<.001).
Patients with non-frontal lesions were indistinguishable from controls and showed no modulation by
laterality. Neither the presence nor the extent of multiple demand network involvement affected
performance. Both conventional network-based statistics and non-parametric Bayesian stochastic
block modelling heavily implicated the right frontal lobe. Crucially, this localisation was confirmed on
explicitly disentangling functional from pathology-driven effects within a layered stochastic block
model, prominently highlighting a right frontal network involving middle and inferior frontal gyrus,
pre- and post-central gyri, with a weak contribution from right superior parietal lobule. Similar results
were obtained with standard lesion-deficit analyses.
Our study represents the first large-scale investigation of the distributed neural substrates of
fluid intelligence in the focally injured brain. Combining novel graph-based lesion-deficit mapping with
detailed investigation of cognitive performance in a large sample of patients provides crucial
information about the neural basis of intelligence. Our findings indicate that a set of predominantly
right frontal regions, rather than a more widely distributed network, is critical to the high-level
functions involved in fluid intelligence. Further they suggest that Raven’s Advanced Progressive
Matrices is a useful clinical index of fluid intelligence and a sensitive marker of right frontal lobe
dysfunction.
.CC-BY 4.0 International licenseavailable under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted August 1, 2022. ; https://doi.org/10.1101/2022.07.28.501722doi: bioRxiv preprint
3
Introduction
Fluid intelligence (Gf) refers to the ability to solve challenging novel problems when prior learning or
accumulated experience are of limited use.
1
Gf ranks amongst the most important features of
cognition, correlates with many cognitive abilities (e.g. memory),
2
and predicts educational and
professional success,
3
social mobility,
4
health
5
and longevity.
6
It is thought to be a key mental capacity
involved in ‘active thinking’,
7
Gf declines dramatically in various types of dementia
8
and reflects the
degree of executive impairment in older patients with frontal involvement.
9
Despite the importance of
Gf in defining human behaviour, it remains contentious whether this is a single or a cluster of cognitive
abilities and the nature of its relationship with the brain.
10
Gf is traditionally measured with tests of novel problem-solving with non-verbal material that
minimize dependence on prior knowledge. Such tests are known to have strong Gf correlations in
large-scale factor analyses.
11, 14
Raven’s Advanced Progressive Matrices
12
(APM), a test widely adopted
in clinical practice and research,
13
contains multiple choice visual analogy problems of increasing
difficulty. Each problem presents an incomplete matrix of geometric figures with a multiple choice of
options for the missing figure. Less commonly, verbal tests of Gf such as Part 1 of the Alice Heim 4
(AH4-1)
15
are adopted. The Wechsler Adult Intelligence Scale (WAIS)
16
has also been used to estimate
Gf by averaging performance on a diverse range of sub-tests. However, several sub-tests (e.g.
Vocabulary) emphasize knowledge, disproportionately weighting measures of “crystallized”
intelligence,
17,18
whilst others (e.g. Picture Completion) have rather low Gf correlations.
19
Hence, it has
been argued that tests such as the APM are the most suitable for a theoretically-based investigation
of changes in Gf after brain injury.
20,21
Proposals regarding the neural substrates of Gf have suggested close links with frontal and
parietal functions. For example, Duncan and colleagues
22
have argued that a network of mainly frontal
and parietal areas, termed the ‘multiple-demand network’ (MD), is “the seat” of Gf. The highly
influential parieto-frontal integration theory (P-FIT), based largely on neuroimaging studies of healthy
subjects, posits that structural symbolism and abstraction emerge from sensory inputs to parietal
cortex, with hypothesis generation and problem solving arising from interactions with frontal cortex.
Once the best solution is identified, the anterior cingulate is engaged in response selection and
inhibition of alternatives
23, 24
. Despite its name, P-FIT also posits occipital and temporal involvement,
implying widely distributed substrates of Gf.
25
A modification to P-FIT proposes a closer connection
between frontal than parietal, regions and Gf-related processes,
26,27
with the frontal lobes mediating
high-Gf “domain-independent” executive processes whilst posterior areas, including the parietal
lobes, mediating low-Gf “domain-dependent” processing of spatial, object, or verbal information.
A meta-analysis of the functional imaging literature has implicated a network of modality-
independent regions involving the inferior and middle frontal and inferior parietal lobes, with additional
frontal eye field activation in non-verbal tasks, and anterior cingulate and left inferior frontal activation
in verbal tasks.
28
This fronto-parietal attention network,
29
requires expansion to account for the
separate neuronal substrate underpinning visuospatial/verbal analytical reasoning
30,31
.
An important caveat of the functional imaging findings is that they do not imply causal
efficacy.
32
For example, though neuropsychological data commonly lateralise language to the left
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hemisphere,
33
neuroimaging activation is often bilateral.
34
So, merely considering the presence or
absence of activation may hide lateralized functions. Hence, lesion studies offer an advantage in
furthering our understanding of the neurocognitive architecture underpinning Gf. So far, these studies
have been surprisingly sparse.
Lesion studies investigating performance on Gf tasks have mainly enrolled veterans with
penetrating head injury.
35-43
For example, Weinstein and Teuber
43
reported that veterans with left
temporo-parietal entrance wounds suffered a decline in Army General Classification Test scores.
Barbey and colleagues
44
investigated Veterans’ performance on three subtests from the WAIS (Matrix
Reasoning, Block Design and Picture Completion). The authors reported that performance was
associated with damage to the superior longitudinal/arcuate fasciculus. However, several of the tests
adopted are not considered Gf measures
45
; the lesion characterisation was rather basic and lacked
modelling of the diffuse axonal injury the traumatic aetiology implies.
46
In the most recent of these
studies, impaired WAIS’s performance, not a specific test of Gf, was associated with damage to left
fronto-parietal regions and white matter association tracts lesions.
47
Glascher and colleagues
49
reported that the left frontal pole was associated with performance on general intelligence (g) in a
large sample of patients with stroke, encephalitis, temporal lobectomy and TBI. Similarly, Browen and
colleagues
50
investigated performance in general intelligence using the WAIS in a sample of patients
with similar pathologies, and reported an association with white matter tracts deep to the left
temporo-parietal junction, including the arcuate fasciculus.
Studies investigating patients with lesions caused by brain tumours or stroke have generally
relied on WAIS as a measure of Gf, with inconclusive results. Some studies have associated WAIS non-
verbal scale performance with right posterior damage.
48,51
However, Tranel and colleagues
52
reported
no significant differences between frontal and non-frontal damage on a non-verbal subtest of the WAIS
analogous to the APM (Matrix Reasoning). Preserved performance on the WAIS has been documented
in frontal patients.
53,54
In contrast, the very few studies adopting tasks loading more heavily on Gf have
reported frontal deficits. Duncan and colleagues
20
documented a substantial discrepancy between
scores on Scale 2 of Cattell’s Culture Fair and the WAIS in 3 frontal but not in 5 non-frontal patients.
However, the very small sample limited generalizability and prevented investigation of laterality
effects.
In a recent study we documented lateralised frontal effects on APM and AH4-1.
55
Compared
with healthy participants, only right frontal damage significantly impaired APM performance, and only
left frontal damage impaired AH4-1 performance. The relatively small sample prevented investigation
of finer anatomical effects.
Lesion studies investigating the underlying behavioural and anatomical aspects of the widely
used APM are old, inconclusive, and lacking in anatomical analysis. Results have variously shown no
difference between right or left hemisphere patients;
56-61
or impairment in left hemisphere
patients
56,62,63
or right hemisphere patients.
56,57,64-65
. Villardita
67
reported no difference in the
performance of left or right hemisphere patients on the Coloured Progressive Matrices version of
APM. However, on set I, involving visuoperceptual factors, right performed worse than left hemisphere
patients. There is similar uncertainty about the influence of aphasia, thought by some to degrade
performance,
68,58
but not by others.
62,69,60,63,70
. Large samples of patients with focal unilateral lesions,
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5
thorough investigation of performance on the APM, fine-grained anatomical mapping, and robust
lesion-deficit inference are vital for definitive scientific conclusions.
Here we assessed the largest number of patients yet reported with focal, unilateral, right or
left, frontal or non-frontal lesions (n=227; 146 Frontal, 81 Non-Frontal, 165 healthy participants) on
APM. We investigated overall performance, item difficulty, and relation to MD involvement. Building
upon our novel multimodal methodology,
71
we employed an array of lesion-deficit models responsive
to the potentially distributed nature of Gf. We focused on modelling the anatomy of neural dependence
as a graph, where interactions between distributed areas are explicitly tested. This approach permits
delineation of distributed substrates. It also distinguishes functionally critical areas from those the
distinctive pathological structure of lesions renders spuriously correlated: a problem shown to corrupt
lesion-deficit maps based on simple mass univariate methods.
72
In our approach each brain locus—
intact or lesioned—is conceived as a node or vertex of a graph, with the relationships between loci—
functional or merely lesion-pathology driven—defining its edges. This permits us to model network-
dependence explicitly, disentangling functional and pathological effects to reveal the underlying
substrate.
Materials and methods
Participants
Data from 332 patients with unilateral, focal lesions who attended the Neuropsychology Department
of the National Hospital for Neurology and Neurosurgery was retrospectively screened. Inclusion
criteria were: presence of a stroke or tumour; ≥70% of the total lesion, segmented from MRI or CT
scans obtained during routine clinical care (see neuroimaging investigations), falling within either
frontal or non-frontal areas; age between 18-80 years; absence of gross perceptual impairments (no
neglect, >5
th
cut-off on the Incomplete Letters test),
73
language impairments (>5
th
%ile on the Graded
Naming Test, GNT);
74
psychiatric disorders, history of alcohol or substance abuse, or other
neurological disorders; and native English language proficiency. Age at assessment, gender and years
of education were recoded.
Application of these criteria yielded 227 patients, 146 Frontal (LF 69; RF 77), and 81 Non-Frontal (LNF
39; RNF 42; see Table 1). There was no significant difference between tumour and stroke patients for
mean time between injury and neuropsychological assessment (
p
= 0.12; table 1). 165 healthy control
participants, with no neurological or psychiatric history, were recruited to match patients as closely
as possible for age, gender, years of education and National Adult Reading Test scores (NART)
75
.
The study was approved by The National Hospital for Neurology and Neurosurgery & Institute
of Neurology Joint Research Ethics Committee and conducted in accordance with the
“Declaration of Helsinki”.
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6
n
Healthy
Control Mean
n
Frontal
Mean
n
Non-Frontal
Mean
Age (years)
(SD)
165
49.61
(15.19)
146
49.17
(15.76)
81 51.76
(14.93)
Gender (Male/Female) 77/88 81/65 45/36
Aetiology (stroke/tumour/abscess/AVM) 38/104/4/0 35/45/0/1
Chronicity (days)
(SD)
124
238.52
(640.41)
73 267.26
(740.93)
Education (years)
(SD)
148
14.57
(2.65)
135
14.38
(3.91)
78 14.83
(2.83)
Premorbid NART IQ
(SD)
165
107.02
(10.01)
146
105.48
(12.71)
81 108.19
(10.60)
GNT (Correct/30)
(SD)
131
20.91
(5.14)
121
19.69
(4.78)
71 20.23
(5.26)
IL (Correct /20)
(SD)
123
19.59
(0.59)
111
19.48
(0.74)
68 19.49
(0.76)
S Fluency (Overall performance)
(SD)
69 17.19
(4.79)
114
11.54a***
b***
(5.71)
55 15.04
(5.17)
Hayling suppression error scaled score (SD) 63 5.95
(0.89)
81 3.91 a*** b***
(2.70)
51 5.84
(2.08)
Table 1. Demographics and cognitive test scores. n = Number. SD = Standard Deviation. AVM = arteriovenous
malformation. NART= National Adult Reading Test. GNT = Graded Naming Test. IL=Incomplete letters. Scores
with significant p values are in bold. ***= p <0.001. a indicates significant difference between frontal and non-
frontal patients. b indicates significant difference between frontal and healthy control patients.
Behavioural investigations
Patients were assessed with tests administered and scored in the published standard manner. Due to
the retrospective nature of our study, certain data were unavailable for some participants.
Background tests
Premorbid optimal level of functioning was assessed using the NART,
perception and naming using
Incomplete Letters and GNT. Two widely used executive tasks, known to require processes distinct
from Gf were also administered.
9,76
Verbal generation was assessed using the phonemic fluency test.
77
The total number of words recalled excluding errors was recorded. Strategy formation/response
inhibition was assessed using the Hayling Sentence Completion Test. Suppression Errors in Section 2
were calculated.
78
Fluid intelligence
Gf was assessed using APM.
12
We analysed the following:
a) Overall performance.
The total number of correct responses in Set 1 (/12) was recorded and
converted into age-adjusted scaled scores based on published norms.
b) Item difficulty.
Based on visual inspection of the percent correct in healthy control performance, we
graded the 12 items from easiest to hardest. We then formed three variables: ‘easy group’, containing
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the 4 easiest items (1, 7, 2, 4), ‘medium group’, containing the next 4 items (3, 5, 6, 10) and ‘hard group’,
containing the 4 hardest items (12, 9, 8, 11). We calculated each patient’s score for the three variables
(0-4). We compared the performance of the LF, RF and Controls on the three groups to investigate
differences in performance based on item difficulty.
c) Multiple-demand network
We compared overall performance in patients with versus without MD
damage, controlling for age and NART (see neuroimaging investigations). We used both frequentist
and Bayesian linear regression to investigate whether extent of MD involvement predicted APM
performance, over and above that predicted by age and NART.
Neuroimaging investigations
Imaging data were available for 176 patients (n=173 MRI, n=3 CT; n=110 Frontal, n=66 Non-Frontal).
MRI scans were obtained on either a 3T or 1.5T Siemen scanners following a diversity of clinically-
determined protocols outside our control. CT studies were obtained on Toshiba or Siemens spiral
scanners. Note that since the input to the imaging models is
not
raw image data but comparatively
large, manually-traced, binary lesion masks, in keeping with established practice in the field we made
the assumption that the effect of variations in acquisition parameters is likely negligible and need not
be explicitly modelled. Lesions were traced and independently classified using MIPAV
(https://mipav.cit.nih.gov/) by JM, EC and checked by PN, who was blind to the study results. In tumour
patients, the segmented lesion included the surgical cavity. The lesion masks were non-linearly
normalised to Montreal Neurological Institute (MNI) stereotaxic space at 2x2x2mm resolution using
SPM-12 software (http://www.fil.ion.ucl.ac.uk
79
). The lesion distribution is displayed in Figure 1.
Involvement of the MD was established by comparing each patient’s normalised lesion mask with a
template of MD regions in MNI space kindly provided by Professor Duncan’s group.
80
For each patient,
we determined whether their lesion involved the MD and calculated extent of MD involvement (i.e. MD
lesion volume/total lesion volume x 100).
Figure 1. Voxel-wise sum of the 221 modelled lesions. The underlay is the SPM152 T1 template distributed with
MRIcroGL(https://www.nitrc.org/projects/mricrogl). The images are displayed in neurological convention.
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Behavioural analysis
All statistical analyses were conducted using SPSS version 25. Neuropsychological data were
assessed for skewness and kurtosis and tested for normality using the Shapiro-Wilk test.
One-way univariate analysis of variance (ANOVA), independent samples
t
-tests or chi-square
analyses were conducted for continuous and categorical data respectively to investigate differences
between Frontal, Non-Frontal and Control participants on age, gender, aetiology, chronicity, lesion
volume, years of education, and neuropsychological variables (NART IQ, GNT, Incomplete Letters, S
fluency and Hayling suppression errors). Following significant differences, post-hoc tests with
Bonferroni correction (0.05/3=p=0.016) compared Frontal versus Non-Frontal, Frontal versus Control
and Non-Frontal versus Control. LF and RF were also compared on all demographic and
neuropsychological variables using t-tests.
Standard
and
Lateralisation
analyses were performed on APM overall performance. In the
Standard
analysis we established the sensitivity and specificity of the APM to the frontal lobes. This
analysis was critical because, only if there was a significant frontal deficit compared to Controls the
subsequent
Lateralization
analysis was carried out to investigate unilateral left and/or right frontal
contributions to APM.
In the
standard analysis
, ANCOVA was used to compare Frontal vs Non-Frontal vs Control,
adjusting for age and NART. Following significant differences, post-hoc tests with Bonferroni
correction (0.05/3=p=0.016) compared Frontal vs Non-Frontal, Frontal vs Control and Non-Frontal vs
Control. In the
laterality analysis
, ANCOVA was used to compare LF vs RF vs LNF vs RNF vs Control,
adjusting for age and NART. Following significant differences, pairwise comparisons with Bonferroni
correction (0.05/4 yields p=0.0125) compared each patient group against Control (i.e. LF vs Control, RF
vs Control, LNF vs Control, RNF vs Control), LF with RF and LNF with RNF.
To investigate potential differences in performance according to item difficulty in Frontal
patients, we used a 3 X 3 ANCOVA with Difficulty (Easy, Medium, Hard) as the within group factor and
Group (LF, RF and Controls) as the between group factor, covaried for age and NART. Significant main
effects of Group were followed by simple effects analyses with Bonferroni correction.
To investigate the contribution of the MD to overall performance one-way ANCOVA was used
to compare patients with (
N
= 153) versus without (
N
= 23) MD lesions, while adjusting for age and
NART. We also performed a multiple linear regression analysis, using the enter method, entering APM
performance as the outcome variable and age, NART and extent of MD involvement as predictor
variables.
Neuroimaging analysis
Lesion-deficit inference is complicated by the presence of correlations across damaged voxels, not
just functionally—arising from a distributed neural substrate—but also pathologically—arising from
the structure of the underlying pathological process
71
. Without explicit modelling of regional
interactions within high-dimensional models that demand large-scale data, spatial inferences are
likely to be unquantifiably distorted. In the absence of an established approach applicable to the
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comparatively small-scale data regimes inevitable in neuropsychology, we applied multiple inferential
methods, focusing on the graph-based approach with the strongest theoretical foundations.
Parcel-based analysis
PLSM ANALYSES
were completed using the NiiStat toolbox for Matlab
(http://www.nitrc.org/projects/niistat). To increase statistical power the brain was regionally
parcellated following the JHU-MNI atlas
81
into 189 regions of interest (ROI) spanning both grey and
white matter. To assure statistical power, only ROIs with damage in >=10 patients were included. Three
Freedman-Lane permutations
82
were performed with age, NART and lesion volume always entered as
nuisance regressors. Permutation thresholding (5,000 permutations) was used to correct for multiple
comparisons and control the family-wise error rate. An alpha of 0.05 was the threshold for
significance.
To investigate the contribution of the MD to APM overall performance, PLSM analyses
were repeated with age, NART and proportion of MD involvement entered as nuisance regressors.
PLSM analyses were conducted on Hayling suppression errors (scaled scores), with age, NART and
lesion volume or age, NART and proportion of MD involvement entered as nuisance regressors.
Bayesian multivariate lesion-deficit modelling of MD dependence
To quantify the regional contribution of components of the MD, Bayesian multivariate regression
implemented in BayesReg v1.91 was performed with each connected component of the MD map
treated as a predictor variable, and age and NART added as nuisance covariates. A selection of
shrinkage priors (ridge, lasso, g, horseshoe, horseshoe+) and noise models (Gaussian, Laplace,
Student t distribution) were evaluated, choosing g and Student t based on the widely applicable
information criterion (WAIC): a standard interpretable metric for Bayesian model comparison.
83
The
posterior distributions of the regression coefficients were estimated with Markov chain Monte Carlo
sampling over 100000 samples with a 100000 burn-in interval and thinning set at 10, reporting the
means and standard deviations of the regression coefficients that survive a 95% Bayesian credibility
interval. The effective sample size was >97 for all models.
Graph lesion-deficit modelling
Where the neural support of a function is distributed across a set of connected regions, the optimal
way of identifying it is through explicit modelling of both anatomical locations and their interactions.
Even where the neural support is local, the structure of the lesion pathology used to reveal it need not
be, and itself requires modelling of distributed relations. By structure here is meant characteristic
spatial patterns of coincident damage dictated by the underlying pathological process, such as the
patterns of ischaemic damage the vascular tree enforces in stroke. The difficulty is amplified when
both the neural and the pathological are distributed, for the former must then be disentangled from
the latter: a problem for which there is no established solution. Here we adopt an approach based on
statistical models of graphs. The fundamental idea is to conceive the brain as a densely interconnected
graph, where each node is an anatomical location and each edge indexes the extent to which its
connected nodes share a set of properties. In the context of lesion-deficit mapping, the properties of
interest are the presence of damage, the associated deficit, and nuisance factors that could confound
their relations. First, we apply conventional network-based statistics, fitting a general linear model to
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APM scores, revealing a network of dependence driven jointly by functional anatomy and spatial
patterns of damage. Second, we exploit recent developments in Bayesian stochastic block modelling
to identify communities of voxels distinctively influenced by fluid intelligence, disentangled from the
incidental spatial structure of lesions.
Network based statistics
The non-linearly registered lesion masks were linearly resampled to a resolution of 12 mm
3
. This
resolution offers considerably finer anatomical detail than published parcellation schemes
84-87
and
avoids the potentially biasing effects of their structuring determinants. We provide a bar plot showing
the mean node volume (± 95% confidence interval) if our approaches compared with commonly used
parcellation schemes in Supplementary Figure 1.
A graph where voxels are the
nodes
and their adjacent neighbours the
edges
was created as an
adjacency matrix, labelling any edge that linked two lesioned nodes as 1 and all others as 0. This
process yielded a graph of order and size 1017 nodes and 516636 edges for each patient in which the
lesion was larger than a single 12mm
3
voxel (N=172). The choice of a 12mm
3
voxel size was constrained
by the tractability of the statistical model, in line with the practice of others in related domains.
88,23
We proceeded to model lesion adjacency matrices with the Network-Based Statistics
connectome toolbox (v1.2).
90
NBS is an established statistical framework for network analysis,
described in extensive detail elsewhere
91-92
. In brief, it implements a non-parametric approach to
mass-univariate statistical inference on the edges of large graphs, yielding family-wise error (FWER)-
corrected
p
-values for each edge via permutation testing
93
. The approach can be viewed as the graph
analogue of the mass-univariate voxel-wise methods familiar from functional imaging and voxel-
based morphometry. It has been widely applied to investigate the organisation of brain networks.
90-92,
94
Here the inputs were the lesion graphs of each patient, with APM as the predictor and NART
and age as nuisance covariates. The model was fitted with 50,000 permutations, with a criterion for
statistical significance set at family-wise error rate corrected
p
<0.05, yielding an inferred group-level
network significantly associated with Gf. We evaluated the community structure of this inferred
network—the presence of clusters of voxels defined by similar inferred connectivity—with a Bayesian,
weighted, non-parametric, hierarchical, generative stochastic block model,
95,96
with additional
simulated annealing to approximate the global optimum of the function (see below). Edges were
weighted by the significant t-statistic adjacency matrix from the NBS model. To examine the potential
influence of aetiology, we compared this NBS model to another identically configured except for the
addition of aetiology as a nuisance covariate (see Supplementary material).
To illustrate the relation between the inferred network and fluid intelligence, we created a set
of Bayesian regression models with a target of APM adjusted for NART, and predictors constructed—
across separate models—from the dichotomized overlap between a lesion and the NBS-identified
network, or from the number of nodes of each patient’s graph included in the NBS network. We also
ran a multivariate regression model with the lesion adjacency matrix as columns of predictors. We
evaluated all models with various prior shrinkage schemes, using the WAIC to select the most
appropriate prior distributions and model goodness-of-fit.
83,97-98
All regression models were
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11
implemented in BayesReg v1.2, and employed a burn-in of 50000, taking 100000 samples from the
posterior distribution within a single MCMC chain. Note these analyses are not independent and are
designed to be merely illustrative of the NBS model from which they are derived.
Bayesian hierarchical stochastic block modelling
The foregoing simple network-based statistical model is potentially confounded by the anatomical
structure of pathological damage. It is also tractable only at relatively coarse spatial resolutions. To
overcome these defects, we exploited a recent innovation in the statistical modelling of graphs:
Bayesian stochastic block models.
103
These are nonparametric probabilistic statistical models of the
network structure of graphs that enable robust inference to distinct patterns of connectivity arising as
network “blocks” or “communities” within them. In the context of a graph model of the lesioned brain,
such communities may be shaped by the neural substrate of the behaviour under study, the anatomical
patterns of damage, or an interaction between the two. The approach allows us to disentangle these
two distinct types of node connectivity, in our case isolating the neural dependents of APM
performance from the incidental structure of the lesions used to reveal them. We employ a specific
kind of Bayesian stochastic block model designed to incorporate layered, multiple attribute
properties.
103
The layered formulation enables robust inference to the separability of the two types of
connectivity by Bayesian model comparison of variants whose layered structure either respects or
ignores them
95
. Though a comparatively recent innovation, such models rely on well-established
principles of Bayesian inference and graph theory, and are underwritten by their theoretically proven
validity
95,100-102
.
Graph theory provides a powerful method of modelling complex systems that combines flexibility with
intelligibility.
92
It treats individual factors of interest as the “nodes” of a network, and their interactions
as the connections, or “edges”, between them. In the context of lesion deficit inference, the nodes
identify anatomical locations in brain, and the edges describe their pairwise relations. Two nodes may
be related by their association with a deficit when lesioned, or by their tendency to be involved in the
same lesion, regardless of the deficit. The former is the effect of interest, the latter is a potential
confounder we wish to eliminate. To disentangle the two forms of relation we create a layered,
weighted, undirected graph whose layers correspond to the two different kinds of association.
Confining each form of relation to its own layer compels the model to disentangle them in inferring
the community structure of the graph. We can compare a layered model of this kind to a null model
where the edges are randomized across layers, employing Bayesian model comparison based on the
minimum description length of the model. Finding the layered model superior to the null is evidence
of the successful separation of the structuring effects of APM and lesion co-occurrence we seek here.
A detailed exposition of the inferential approach is given in Supplementary material.
To model our data, each non-linearly registered lesion was resampled to 4mm
3
resolution, and
the lesion adjacency matrix constructed for each patient as before. This resolution is much finer for
than conventional parcellation schemes published in the wider literature schemes (Supplementary
Figure 1).
84-87
We then constructed an undirected, weighted graph combining all individual lesion
networks across all patients. This network comprised nodes corresponding to all voxels of the brain,
and edges between voxels adjacent. These edges were weighted by two variables: the count of the
number of times a voxel and an adjacent neighbour were damaged
together
—a
lesion co-occurrence
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12
weight
and the inverse of the patient’s APM score divided by NART—an adjusted
APM
weight.
Naturally, the graph was undirected, as the direction of any relationship between collaterally lesioned
areas is not informed by the data at hand.
We filtered edges to limit analysis to the top 50% connected nodes, removing edges with fewer
than ~3 connections, where sampling was too low to support robust inference, but permitting still full
brain coverage. This yielded a graph of order and size 27509 nodes and 285545 edges. There were no
node self-loops. We rescaled both lesion co-occurrence and APM edge weights to the range 0 to 1.
We proceeded to evaluate the community structure of this network with a non-parametric
Bayesian hierarchical weighted stochastic block model incorporating layered and attributed
properties
103,109
implemented in graph-tool (https://graph-tool.skewed.de).
95-96, 102
We began by fitting a
null model, with the two kinds of edge weight—adjusted APM and lesion co-occurrence—randomly
distributed across two layers. We then fitted a test model with each type of weight consistently
assigned to its own layer. Adjusted APM weights were modelled as Gaussian; lesion co-occurrence
weights as Poisson distributions. Having initialised a fit, we used simulated annealing to further
optimise it, with a default inverse temperature of 1 to 10.
96
We did not specify a finite number of draws,
rather we specified a wait step of 100 iterations for a record-breaking event, to ensure that
equilibration was driven by changes in the entropy criterion, instead of driven by a finite number of
iterations
102
We used model entropy to determine if the layered model fit was better than the null, indicating
that the inferred community structure distinguished APM and lesion co-occurrence effects. To
visualise the inferred communities, we backprojected the incident edge weights onto the brain,
deriving the mean and 95% credible intervals for comparison. To examine if modelling lesion co-
occurrence requires explicit consideration of aetiology, we replicated the model with the addition of
aetiology as a third layer, again conducting formal comparison against a randomised null (see
Supplementary material).
Synthetic ground truth evaluation
The substrate of a function is definitionally unknown: it is what we are seeking to infer. To examine the
comparative fidelity of a set of models we therefore need synthetic ground truths
72
of the complexity
likely to obtain in reality. Here we used the meta-analytic repository NeuroQuery
104
to create six
realistically complex and distributed ground truth maps across the domains of action, aversion,
language, mood, motor and sensation (see Supplementary materials). The intersection between each
lesion and each ground truth was then used to generate a hypothetical deficit for each patient and
each domain, and the stochastic block model was subsequently applied exactly as in the case of the
real data. The fidelity of the inferred maps was then quantified by their Dice score, and compared to a
standard mass-univariate voxel-based lesion-deficit mapping baseline (see Supplementary
materials).
Data and code availability
The data and code that support the findings of this study are available from the corresponding author,
LC, upon reasonable request.
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13
Results
Demographic and behavioural investigations
Frontal, Non-Frontal and Control patients were well-matched for age, gender, chronicity and years of
education (all
p
> 0.05). There was no significant difference in lesion volume between LF and RF or
between LNF and RNF. There was a significantly greater proportion of tumour patients in Frontal than
Non-Frontal (χ2 (1, N = 227) = 5.68, p < 0.05) and a significantly greater proportion of stroke patients in
Non-Frontal than Frontal (χ2 (1, N = 227) = 7.05, p < 0.01). However, there were no significant
differences in the proportion of tumour or stroke patients between LF and RF (all
p
> 0.05) or between
LNF and RNF (all
p
> 0.05).
There were no significant differences between Frontal, Non-Frontal and Control participants
for NART, IL or GNT scores (all
p
> 0.05; see Table 1). One-way ANOVAs found highly significant
differences between Frontal, Non-Frontal and Control participants for S fluency and Hayling
suppression errors (
F
(2,238) = 25.319;
p
< .001;
F
(2,192) = 21.266;
p
< .001, respectively). Post-hoc tests
showed Frontal performed significantly worse than Non-Frontal and Control on S fluency (
p<
.001;
p
<.001, respectively) and Hayling suppression errors (
p
<.001;
p
<.001, respectively). Pairwise
comparisons revealed that LF were significantly more impaired than RF on S fluency (
p
<.01), RF were
significantly more impaired than LF on Hayling suppression errors (
p
<.05; Table 1).
Overall performance
Standard analysis
: A one-way ANCOVA controlling for age and NART, found a highly significant
difference between Frontal, Non-Frontal and Control participants in overall performance (
F
(2,387) =
18.491;
p
< .001). Post-hoc tests showed that Frontal performed significantly worse than Non-Frontal
(
p
< .01) and Control (
p
<.001). There was no significant difference between Non-Frontal and Control
(corrected p =.185; Table 2)
Lateralization analysis:
A one way ANCOVA controlling for age and NART found a highly
significant difference between LF, RF, LNF, RNF and Control participants (
F
(4,385) = 12.237;
p
< .001).
Pairwise comparisons showed a significant difference between RF and Controls (
p
<.001) and LF and
Control (
p
<.01). Importantly, RF were significantly more impaired than LF (
p
<.01). There was no
significant difference between RNF and Control or LNF and Control; table 2). Notably, performance
fell <1.5 standard deviations below Control in 43% of RF but only in 22% of LF.
Performance Healthy
control
Frontal
Non-Frontal
Frontal Non-Frontal
Left
n
= 69
Right
n
= 77
Left
n
= 39
Right
n
= 42
Mean number
correct /12
(SD)
8.67
(2.41)
7.07
a
**
b
***
(2.78)
8.07
(2.09)
7.71
b
**
(2.49)
6.49
b
***
c
**
(2.93)
8.00
(1.92)
8.14
(2.26)
Table 2 Overall performance on APM. n = Number. SD = Standard Deviation. Scores with significant p values
are in bold. **= p <0.01; ***= p <0.001.
a
indicates significant difference from non-frontal patients.
b
indicates
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14
significant difference from healthy control participants.
c
indicates significant difference between left and right
frontal patients.
Item difficulty
A 3x3 ANCOVA, controlling for age and NART, revealed a significant main effect of difficulty (F(2,364) =
5.360; p=0.005) and Group (F(2,182) = 22.707; p < .001). Critically, there was also a significant interaction
between difficulty and group (F(2,364) = 10.822; p<0.001). Post-hoc pairwise comparisons showed
significant differences for the medium group between RF and Control (p<.001), LF and Control (p<.05),
and RF and LF (p<0.05). Thus, Frontal patients were worse than Control, and RF performed the
poorest. For the hardest group there were significant differences only between RF and Control
(p<.001), and LF and Control (p<.001). There were no significant differences for the easy group. Thus,
RF impairment on the APM appears to be driven by poor performance on the medium items. Closer
inspection revealed that three specific items (3, 5 and 6) were responsible for driving the RF poorer
performance than LF, with an accuracy decrement of more than 20% in RF.
Multiple-demand network
A one-way ANCOVA, controlling for age and NART, showed no significant
difference in overall performance between patients with vs without MD lesions (
F
(1, 172) = 1.88,
p =
.172). A linear regression analysis, with age, NART and extent of MD involvement entered as predictor
variables, significantly predicted APM performance (r2 = .28,
F
(3, 172) = 21.726, p < .001). However,
only age and NART (both p<.001) were significant predictors. Extent of MD involvement did not
significantly contribute (p = .410).
Parcel-based analysis and Bayesian multivariate analysis of MD
PLSM analyses with age, NART and lesion volume entered as nuisance regressors, revealed that
poorer overall performance was associated with right posterior middle frontal gyrus, pars opercularis,
precentral gyrus, superior corona radiata and external capsule lesions. When the proportion of MD
involvement was entered as a nuisance regressor instead of lesion volume the results remained
unchanged.
PLSM analyses on Hayling suppression errors, with age, NART and lesion volume entered as
nuisance regressors revealed that poorer performance was associated with right posterior middle
frontal gyrus and pars opercularis lesions. When the proportion of MD involvement was entered as a
nuisance regressor instead of lesion volume the results remained unchanged.
Bayesian multivariate modelling of individual MD components yielded as credibly predictive
only NART (posterior mean coefficient 0.484, 95% credibility interval 0.336 to 0.630), age (mean -0.321,
95%CI -0.455–-0.187), and a right-sided MD component falling within precentral and posterior medial
frontal gyrus (mean -0.310, 95% CI -0.587 to -0.021). The credibility intervals of the coefficients of other
MD components all crossed zero.
Network lesion-deficit modelling
Network-based statistics identified a distinct predominantly right frontal network associated with
reduced APM (FWER-corrected
p
<0.0001, t-thresh >3.1) (Figure 2). The regions with the greatest
number of significant nodes (in order of descending degree count) included the right superior frontal
gyrus (degree count 22), right middle frontal gyrus (17), right frontal pole (16), right anterior cingulate
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cortex (16), left superior frontal gyrus (6), right inferior frontal gyrus (4), left anterior cingulate cortex
(4), right caudate nucleus (3), right mid cingulate cortex (3), right precentral gyrus (2), right
juxtapositional lobule (1), right frontal operculum (1) and right anterior insula (1). A stochastic block
model partitioned the network into a structure with 3 clustered components, broadly encompassing
medial wall, superolateral cortical surface and a superior frontal gyrus-dominant component. The
addition of lesion aetiology to the list of covariates in the NBS model yielded a near-identical result
(test-statistic correlation between significant edges, r .99,
p
<0.0001) (Supplementary Figure 2).
Bayesian univariate regression analysis confirmed that patients with lesions overlapping with
the network exhibited significantly lower adjusted APM scores, (R
2
0.105, coefficient mean ± SD -0.265
± 0.068 (95% credible interval [CI] -0.396 to -0.129) (Figure 3). The extent of overlap, indexed by the
degree (i.e. number of nodes) shared between an individual lesion network and the inferred network
exhibited a strong log-linear relationship to adjusted APM (R
2
0.190, coefficient mean ± SD -0.002 ±
0.0005 (95% CI -0.00250 to -0.00041). Bayesian multivariate regression models of adjusted APM
predicted by the lesion network adjacency matrix yielded a fit with R
2
0.640. It is important to note that
these regression analyses are not independent of the NBS model: they do not provide further evidence
but rather qualify its fidelity.
Generative hierarchical stochastic block modelling of APM performance
The foregoing models inevitably conflate the distributed spatial structure of the underlying neural
dependence with that of the causal pathology. To disentangle the two, we need a network model
capable of separating the target effects of APM performance from the incidental effects of lesion co-
occurrence. This can be achieved with a layered nested stochastic block model, where adjusted APM
and lesion co-occurrence weights are distributed in two distinct layers, yielding layer-specific patterns
of community structure reflecting the distinct effect of each weight on the network. This model
achieved substantially lower entropy—881118.22 vs 1182697.66 nats—than a null model with weights
randomised across the two layers (Figure 4), providing inferential support for distinguishing adjusted
APM from co-occurrence effects. This translates to a posterior Odds ratio of the layered formulation
being e
301579
more likely than the non-layered null.
The community structure was composed of blocks dominated by adjusted APM, lesion co-
occurrence, or neither weight. The adjusted APM layer revealed a set of brain communities with high
edge incidence linking the right middle and inferior frontal gyrus (including pars triangularis), right
pre- and post-central gyri, and—weakly—the right superior parietal lobule. These communities were
sharply distinct from the lesion weight.
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16
Figure 2. Network-based statistic maps. A. Network-based statistics identify a significant network associated
reduced adjusted APM scores (FWER-
p
<0.0001). B. Radial graph of the community structure of the network
inferred from a stochastic block model of its statistics shows that the network clusters into three discrete
components encompassing the superolateral cortical surface, the medial (and inferior) wall and a superior
frontal gyrus dominant cluster. Nodes are colour-coded in accordance with their stochastic block model cluster.
Node size is proportional to node degree count. Edge width and colour is proportional to the t-statistic from the
model, with a thicker and more yellow line denoting a stronger link between a given network connection and a
reduced adjusted APM score. Abbreviations: ACG, anterior cingulate gyrus; L, left; IFG, inferior frontal gyrus;
IFG-pt, inferior frontal gyrus pars triangularis; MFG, middle frontal gyrus; OFC, orbitofrontal cortex; PreCg, pre-
central gyrus; PoCg, post-central gyrus; R, right; SFG, superior frontal gyrus; SPL, superior parietal lobule.
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17
Figure 3. Network-based statistic behavioural correlations. A. Violin plots of the adjusted APM scores of patients
whose lesions do or do not overlap with the inferred network illustrate significantly lower APM scores in the
former (R2 0.105, coefficient mean ± SD -0.265 ± 0.068 (95% credible interval [CI] -0.396 to -0.129). B. Scatter
and line plot shows that the degree count of the overlap of a lesion with the inferred network significantly
correlates with adjusted APM scores within a univariate Bayesian regression model (R
2
0.190, coefficient mean
± SD -0.002 ± 0.0005 (95% CI -0.00250 to -0.00041). C. Histogram of the edge t-statistics from the network model
illustrates the population of edges significantly associated with the APM after multiple comparisons correction.
D. Scatter and line plot shows the predictability of adjusted APM from the network adjacency matrix within a
multivariate Bayesian model (R
2
0.640).
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18
Figure 4. Stochastic block model lesion-deficit mapping. A. Radial graphs of stochastic block models with
adjusted APM and lesion co-occurrence layered (top), vs randomly distributed across layers (bottom). Edge
colour and width is proportional to the associated edge weight. Model entropy favoured the layered over the null
model. B. Radial graph illustrating the layered stochastic block model fit with edge colour and width proportional
to the lesion co-occurrence weight, and node colour and size proportional to the lesion-weight degree. This
demonstrates a community of highly interconnected voxels involving the bilateral frontal pole and orbitofrontal
cortex, right superior and inferior frontal gyrus and anterior cingulate gyrus. C. Radial graph illustrating the
layered stochastic block model fit with edge colour and width proportional to the adjusted APM weight, node
colour and size proportional to the APM-weight degree. This illustrates a characteristically different segregation
of brain communities, with high edge incidence linking the right middle and inferior frontal gyrus, (including pars
triangularis), right pre-central gyrus and right superior parietal lobule. Brain images are overlayed
corresponding to the posterior mean edge weight at these communities.
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Many of the spatial constraints on the configuration of lesion patterns are imposed by the basic
anatomy of the brain and will be shared across aetiologies; those that are not will arise as additional
heterogeneity the stochastic block model could theoretically absorb. To determine if aetiology has a
substantial structuring effect that merits explicit accounting, we reran the model with an additional,
third layer identifying the aetiology of each lesion. This model exhibited far greater description length
(2133947.48), 2.4x that of the above for an increase of this single feature, indicating a poorer fit to the
data and providing no grounds for preference over the simpler model. The anatomical pattern of APM-
sensitive communities was in any event very similar (Supplementary Figure 3).
Figure 5. Graph community properties A. Axial slices of the mean posterior edge weight for each block at the l1
aggregation, with more red-orange areas corresponding to a greater value and greater relation to adjusted APM.
B. Scatterplot illustrating the relationship between posterior mean edge weights at each community block, for
both the lesion weight (y-axis) and adjusted APM (x-axis), with brain reconstructions overlaying these findings.
Of note, bilateral frontal-based blocks depicted higher lesion-weight edges, with right fronto blocks more
implicating APM. Abbreviations: ACG, anterior cingulate gyrus; L, left; IFG, inferior frontal gyrus; IFG-pt, inferior
frontal gyrus pars triangularis; MFG, middle frontal gyrus; OFC, orbitofrontal cortex; PreCg, pre-central gyrus;
PoCg, post-central gyrus; R, right; SFG, superior frontal gyrus; SPL, superior parietal lobule.
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Synthetic ground truth evaluation
Bayesian model comparison showed all layered models to be more plausible than the null
(Supplementary Figure 4). Compared with VLSM, the stochastic block model achieved significantly
superior results across all domains (
p
=0.028) (see Supplementary materials) (Supplementary Figure
5).. These experiments also demonstrated qualitatively the tendency of VLSM to mislocalise in
response to the underlying lesion structure, and the ability of the stochastic block model to resist it.
Discussion
Our study represents the first large-scale investigation of the distributed neural substrates of fluid
intelligence in the focally injured brain. We investigated one of the most widely used Gf tests, the APM,
in the largest number of patients with single, focal, unilateral, right or left, frontal or non-frontal
lesions and controls. We analysed overall performance, item difficulty and the contribution of MD
involvement. For the first time, non-parametric Bayesian stochastic block models were used to reveal
the intricate community graph structure of lesion deficit networks, disentangling functional from
confounding pathological distributed effects.
Similar to other groups
21,105-107
and in-keeping with our previous studies
71
we adopted a mixed
aetiology approach. Previous comparison of a large frontal and non-frontal sample with different
aetiologies on the APM and other executive tests showed that aetiology was not a strong predictor of
frontal or non-frontal deficits.
9,108
Hence, different aetiologies do not result in more severe
impairments than others and combining across vascular and tumour pathologies is unlikely to
significantly distort neuropsychological performance.
76
Instead, focal lesions may relate more closely
to the region of damage rather than aetiology. Moreover, data from multiple aetiologies will tend to
attenuate distorting effects arising from pathologically driven characteristic patterns of lesion co-
occurrence that are widely recognized to bedevil both network and focal lesion-deficit studies. Indeed,
less spatial distortion caused by the structure of the pathology may be expected if multiple pathologies
differing in their spatial properties are used.
Though Gf is widely thought to be dependent on the integrity of the frontal lobes, only a handful
of focal lesion studies, based on modest samples, have found impairments in following frontal
lesions.
9,21,55
Applying an array of lesion-deficit models to large scale data, we found APM performance
to be specifically vulnerable to the integrity of the right frontal lobe, and largely resistant to damage
elsewhere. The left frontal lobe appears to make a contribution to APM performance, if a more modest
one. We found that the performance of the left frontal patients was significantly different from healthy
controls and non-frontal patients. However, the left frontal patients performed significantly better than
the right frontal patients did.
Our findings speak to the theories of non-frontal involvement in Gf. The proponents of P-FIT
have argued that impairment of Gf should follow lesions of the posterior and anterior regions that
putatively subserve it.
23,24
We found no evidence of such non-frontal causal dependence on APM
performance. It is possible that functional imaging findings merely reflect a correlation between Gf
and posterior areas non-critically engaged by the necessary perceptual input.
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21
Our results are also relevant for the MD proposal. Notably, Woolgar
et al
.
80
investigated 80
patients with cortical lesions with a Gf task (Cattell Culture Fair IQ test). Though the authors reported
a significant correlation between MD involvement and Gf performance overall, in the 44 patients with
purely frontal damage the relationship was not significant when non-MD lesion volume was taken into
account. So, as far as the frontal lobes are concerned, the authors’ theoretical claim was not strongly
empirically supported. In our study patients with or without MD damage did not differ significantly in
performance on the APM. Moreover, the extent of MD involvement did not contribute to performance.
These findings do not support the claim that MD is the seat of Gf, but neither do they exclude it:
absence of evidence is not evidence of absence. In this context, we note that the Woolgar et al
80
study
included 30 non-frontal patients. Of these only 7 were patients with unilateral parietal lesions, whilst
2 patients had biparietal lesion. In contrast, our non-frontal sample includes 81 patients, of which 20
have a unilateral parietal lesion. Hence, our study has far greater coverage of non-frontal MD areas
than Woolgar’s study. While we cannot rule out the possibility that lower power may be a factor
relevant for our conclusion regarding weak contributions from non-frontal MD lesions, our sample is
larger than that of the study, which produced the opposite conclusions. Moreover, the Woolgar et al.
study reported for their parietal patients that MD lesion volume was a significant predictor for
performance on fluid intelligence with and without non-MD lesion volume partialled out (r=-0.65,
p=0.042; r=-0.63, p=0.035 respectively). We were not able to replicate this effect in our larger group
(r=-0.063, p=0.811; r=-0.17; p=0.950).
Our findings of greater involvement of the right frontal lobe in APM performance were
complemented and extended by our neuroimaging analyses. Both conventional network statistics and
non-parametric Bayesian stochastic block modelling heavily implicated the right frontal lobe.
Crucially, this localisation was confirmed on explicitly disentangling—uniquely in the field of lesion-
deficit mapping—functional from pathology-driven effects within a layered stochastic block model,
prominently highlighting a right frontal network including the middle and the inferior frontal gyrus,
including pars triangularis, and pre- and post-central gyri, with a comparatively weak contribution
from superior parietal lobule. The marked structuring effects of lesion co-occurrence observed
highlight the importance of explicitly modelling them in lesion-deficit inference, whether in the context
of network or focal analysis.
Standard PLSM analyses, potentially confounded by lesion co-occurrence effects, suggested
that poorer performance was associated with damage to a right frontal network including posterior
middle frontal gyrus, pars opercularis, precentral gyrus, superior corona radiata and external capsule,
invariantly to the degree of MD involvement. That a similar set of RF regions were implicated in Hayling
suppression errors, a verbal test, suggests that function lateralization in the frontal lobes is not
explained by task sensitivity to language alone.
Behaviourally we found a highly significant interaction between item difficulty and frontal lesion
lateralisation. The asymmetry in performance was nearly three times greater for the middle than for
the highest level of difficulty, with neither population nearing the ceiling or floor. Why might that be?
While complete agreement is lacking, factor analyses of progressive matrices indicate at least
two material components. Dillon
et al
.
110
identify a factor related to “perceiving the progression of a
pattern” (p.1301), and another to “the addition and/or subtraction of elements”. Lynn
et al
.
111
offer
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22
respectively “Gestalt continuation”, following Van der Ven and Ellis
112
, and “verbal-analytic reasoning”.
It is apparent that the three medium items (3, 5, 6) of the APM showing the greatest lateralisation, are
all those where perceiving the progression of a pattern is an obvious approach. By contrast, the non-
ceiling items with the smallest lateralisation (10, 11, 12) are all those where addition or subtraction
come into play. Though the limited number of items precludes firm conclusions on factorisation, these
findings suggest the components of APM may lateralise in different ways. Inferences involving the
perception of a progressive pattern may be especially sensitive to the integrity of the right frontal
network.
In conclusion, our study represents the most robust investigation of the hitherto poorly
characterized Gf in patients with single, focal, unilateral lesions. Our approach of combining novel
graph-based lesion-deficit mapping with detailed investigation of APM performance in a large sample
of patients provides crucial information about the neural basis of fluid intelligence. We suggest that a
right frontal network, rather than a wide set of regions distributed across the brain, is critical to the
high-level inferences, based on perceiving pattern progression, involved in Gf. Our findings further
corroborate the clinical utility of APM in evaluating Gf and identifying right frontal lobe dysfunction.
Funding
This work was supported by a Welcome Trust Grant (089231/A/ 09/Z). This work was undertaken at
UCLH/UCL, which received a proportion of funding from the Department of Health’s National Institute
for Health Research Biomedical Research Centre’s funding scheme. JM was funded by the National
Brain Appeal. JKR was funded by the Guarantors of Brain.
Competing interests
The authors have nothing to disclose.
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Graph lesion-deficit mapping of fluid intelligence
Lisa Cipolotti,
1,2*
James K Ruffle,
2,3
Joe Mole,
1,2
Tianbo Xu,
2
Harpreet Hyare,
2,3
Tim Shallice,
4,5
Edgar
Chan
1,2
and Parashkev Nachev
2
Supplementary Material
Stochastic block models
A stochastic block model (SBM)1 is a generative model of the community structure of a graph composed of nodes,
divided into blocks with edges

between blocks and . The model can be framed hierarchically, where edge counts

form block multigraphs with nodes corresponding to individual blocks and edge counts arising as edge multiplicities
between block pairs, including self-loops. We seek to infer the most plausible partition
of the nodes, where
 
identifies the block membership of node in observed network , with maximisation of the posterior likelihood

. The result is a hierarchically organised community structure of nodes assigned into blocks that yields the
most compact representation of the graph, as indexed by its minimum description length2, . The general approach is
described in further detail elsewhere1.
An extension to the hierarchical SBM is its layered formulation, permitting modelling of a graph structure where edges
reflect disparate forms of interaction3. In the context of lesion deficit mapping, we can use a layered SBM to model
relations between voxels driven by two distinct effects: the underlying neural dependence of the function of interest and
the pathological structure of the lesions used to map it. Key here is formal comparison between models that encode
these effects separately, within their own layers, vs those where the distinction is not respected. In a Bayesian setting3,
the procedure for model selection amounts to finding the model maximising posterior likelihood as



  


,
where  denotes the shorthand for the model parameters. In our case,


, where nodes are divided
into blocks via the membership vector
 
, and the distribution of covariates in edges in groups and is
given by the edge counts

, with

corresponding to the former at a given layer.  is the prior probability on these
parameters, with 
 corresponding to the normalisation constant. The approach is further detailed by Peixoto3,
formulating the most succinct representation of the data as one with the minimum description length2, . Since the prior
probabilities are nonparametric, the procedure also becomes parameter-free.
Choosing the model with the smallest description length is the means of balancing model complexity and goodness of
fit2. We consider two candidate models throughout our experimental design: model !
"
,
where layers are true descriptors
corresponding to the weighted edges of our deficit of interest in one layer and the connection matrix of the set of lesions
in another layer, and a null model !
#
where the edges describing deficit and the lesion connectivity effects are randomly
interspersed across layers. The comparative magnitude of the description length of each model yields the following
posterior odds ratio:
$ 

%
!
%
!
%

&
!
&
!
&
,
simplifying to
$ '()*+
!
%
!
&
.
In this instance, 
! is the posterior according to a given hypothesis !, i.e., the true or null layered
formulation. ! is then the prior belief for hypothesis !, and ,
"
*
#
the difference in the model description
length for these hypotheses.
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Semi-synthetic evaluation of stochastic block model lesion-deficit mapping
The application of stochastic block modelling (SBM) to lesion-deficit inference is underwritten by the validity
of the underlying statistical framework. No purely empirical external validation is possible here because the
functional anatomy of the neural substrate is definitionally unknown—it is what we are using lesion-deficit
models to infer—and discriminative models of outcome need only identify the anatomical boundaries—jointly
defined by lesion and neural effects—that determine outcomes. But we can establish a semi-synthetic
validation framework, where an array of plausible, empirically-guided anatomical ground truths are
hypothetically posited, and the fidelity of candidate models of real lesions used to retrieve them is explicitly
quantified
4
. Here we derive ground truths from the large-scale meta-analytic repository NeuroQuery
5
,
extracting functional maps of terms selected to span a broad range of cognitive domains, and exhibit widely
distributed neuroanatomical patterns. The terms used were “action”, “aversion”, “language”, “mood”,
“motor”, and “sensation”. Each map was re-sampled into the same 4mm
3
isotropic space employed in the
SBM pipeline, and thresholded at a conservative Z score of >=4. The connected components of each mask
where identified, and clusters smaller than 27 voxels or sampled by fewer than 3 lesions were removed. Six
sets of hypothetical continuous lesion-deficit relations were then created by summing over the intersection
between each lesion segmentation and each corresponding ground truth, yielding a weighted “deficit score”
for each patient and each functional domain.
We proceeded to evaluate these semi-synthetic lesion-deficit relations with a non-parametric Bayesian
hierarchical weighted stochastic block model incorporating layered and attributed properties,
implemented
in graph-tool (https://graph-tool.skewed.de)
2,3,6-8
, exactly as in the main analysis. We began by fitting a null
model, with the two kinds of edge weight—the deficit score and the lesion co-occurrence—randomly
distributed across two layers. We then fitted a test model where each type of weight was consistently assigned
to its own layer. Deficit weights were modelled as Gaussian; lesion co-occurrence weights as Poisson
distributions. Having initialised a fit, we used simulated annealing to further optimise it, with 1000 iterations
and a default inverse temperature of 1 to 10. We used model entropy to determine if the layered model fit was
better than the null, indicating that the inferred community structure corresponded to the synthetic ground
truth and lesion co-occurrence effects. To visualise the inferred communities, we backprojected the incident
edge weights onto the brain, deriving the mean and 95% credible intervals. Bayesian model comparison based
on minimum description length was used to determine if the layered models were more plausible than the
null (Supplementary Figure 4). To show the relationship between inferred deficit and lesion co-occurrence
edge weights, we extracted the posterior means whose 95% credible interval did not cross zero in each test
model and plotted their correlation.
To compare the fidelity of our anatomical retrieval to that achievable with conventional lesion-deficit mapping,
we performed standard mass-univariate voxel-wise lesion-deficit mapping (VLSM) of the same data
implemented in SPM. The lesions were smoothed with a Gaussian kernel of 4mm full-width at half-maximum
to facilitate voxel-wise spatial inference within a random Gaussian fields framework, and entered into a voxel-
wise general linear model with the lesion intensity as the dependent variable and the deficit score as the
independent variable. The resultant statistical maps were thresholded at p<0.05 family-wise error corrected.
Models omitting the smoothing step yielded very similar maps (data not shown). The comparative fidelity of
SBM vs VLSM models of the same data was then quantified by the difference in the Dice score for each inferred
map relative to the ground truth. Note that the complex spatial structure of the ground truth maps employed
here, chosen to provide the most robust test of the retrieval of distributed neural substrates, precludes
interpretation of the absolute Dice score: the focus here is on the comparison with conventional topological
models.
The SBM models were quantitatively superior to VLSM across the entire set (
p
= 0.028, Supplementary Figure
5). The qualitative results, visualised in the same figure, demonstrate the vulnerability of VLSM models to
spatial biases driven by lesion morphology, and the ability of SBM models to resist them.
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Supplementary Figure 1. Bar plot of the mean spatial unit of analysis volumes of our stochastic block (SBM) and network
based statistics (NBS) models compared with alternative regional parcellation schemes. Note our volumes are
substantially smaller. The errors identify 95% confidence intervals. Note the ordinate is decimal log transformed.
Supplementary Figure 2. A. Scatter plot of significant edges derived from network-based statistics models including
(ordinate) and excluding (abscissa) a lesion aetiology covariate. Note all points lie very close to the diagonal, indicating a
minimal difference between the two. B. Visual connectome plots of the network-based statistics models excluding (top)
and including (bottom) the aetiology covariate.
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Supplementary Figure 3. A. Scatter plot of the relation between posterior means of community blocks of stochastic block
models with (ordinate) and without (abscissa) an aetiology layer. Note the former exhibits a narrower range of APM-
related variation, suggesting weaker disentanglement from co-occurrence. B. Visualisation of the posterior means of
the SBM without (top) and with (bottom) inclusion of aetiology in a separate layer. The description length of the more
complex model was ~2.4 times that of the simpler model (881118.22 vs 2133947.48 nats), a dramatic rise in description
length for only a single additional feature, suggesting the former is overparameterised.
Supplementary Figure 4. Scatter plots of the correlation between lesion co-occurrence (ordinate, red-yellow) and deficit
(abscissa, blue-light blue) posterior means from the blocks of the test SBM models in each of the six ground truth
experiments. Each individual point corresponds to a block at the l
0
hierarchical layer. Below each plot is a bar chart of
the corresponding test (dark grey) and null (light grey) model entropies in nats. Note the disentanglement of lesion co-
occurrence and deficit effects, and lower entropies for the layered models across the set, indicating superiority. The
entropy difference, x, between the layered and null formulation translates to a posterior odds ratio of e
x
for the layered
formulation over the non-layered alternative, as is further detailed in the supplementary methods.
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Supplementary Figure 5. A-F. Triplanar view of the comparison between VLSM (rows 2, 5, and 8) and SBM (rows 3, 6, and
9) inference across the six domains, with the ground truth outlined in black over the source NeuroQuery weights (rows
1, 4 and 7). Note that VLSM retained no significant results for identifying the motor ground truth. G. Dice scores for SBM
(black) and VLSM (grey) ground truth retrieval. H. Dice score boxplots across the set, with the p value for the difference.
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... A Bayesian graphical analysis was used to model the high dimensional interplay between imaging and other investigatory features in the formulation of a network [25][26][27][28][29][30] . In brief, this approach permits the allocation of patient imaging and clinical factors as individual 'nodes' , and the commonality between them as 'edges' . ...
... All patients had received standard chemotherapy to treat their underlying malignancy. 27 www.nature.com/scientificreports/ changes, previous infarction, cerebral microangiopathy, microhemorrhage, subdural hematoma or hygroma, pachymeningeal abnormalities secondary to lumbar puncture (LP), or a normal intracranial appearance (n = 10). ...
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