研究 - 麻省理工学院人工智能研究专业

研究

Network topology of symbolic and nonsymbolic
number comparison

Benjamin N. 康拉德 1,2, 埃里克·D. Wilkey

1,2,3, Darren J. 杨 1,2,4, and Gavin R. Price1,2

1Psychology and Human Development, Vanderbilt University, Nashville, TN, 美国
2Vanderbilt Brain Institute, Vanderbilt University, Nashville, TN, 美国
3Brain & Mind Institute, Western University, 伦敦, 在, 加拿大
4Division of Psychology, School of Social Sciences, Nanyang Technological University, 新加坡

关键词: Numerical cognition, Functional connectivity, 模块化, Community allegiance, Beta-
series correlation

开放访问

杂志

抽象的

Studies of brain activity during number processing suggest symbolic and nonsymbolic
numerical stimuli (例如, Arabic digits and dot arrays) engage both shared and distinct neural
mechanisms. 然而, the extent to which number format inﬂuences large-scale functional
network organization is unknown. In this study, 使用 7 Tesla MRI, we adopted a network
neuroscience approach to characterize the whole-brain functional architecture supporting
symbolic and nonsymbolic number comparison in 33 adults. Results showed the degree of
global modularity was similar for both formats. The symbolic format, 然而, elicited
stronger community membership among auditory regions, whereas for nonsymbolic, stronger
membership was observed within and between cingulo-opercular/salience network and
basal ganglia communities. The right posterior inferior temporal gyrus, left intraparietal
sulcus, and two regions in the right ventromedial occipital cortex demonstrated robust
differences between formats in terms of their community membership, supporting prior
ﬁndings that these areas are differentially engaged based on number format. 此外,
a uniﬁed fronto-parietal/dorsal attention community in the nonsymbolic condition was
fractionated into two components in the symbolic condition. 合在一起, these results
reveal a pattern of overlapping and distinct network architectures for symbolic and
nonsymbolic number processing.

作者总结

Previous studies of local activity levels suggest that both shared and distinct neural
mechanisms support the processing of symbolic (Arabic digits) and nonsymbolic (dot sets)
number stimuli, involving regions distributed across frontal, 颞, and parietal cortices.
Network-level characterizations of functional connectivity patterns underlying number
processing have gone unexplored, 然而. In this study we examined the whole-brain
functional architecture of symbolic and nonsymbolic number comparison. Stronger
community membership was observed among auditory regions during symbolic processing,
and among cingulo-opercular/salience and basal ganglia networks for nonsymbolic. A dual
versus uniﬁed fronto-parietal/dorsal attention community organization was observed for
symbolic and nonsymbolic formats, 分别. 最后, the inferior temporal gyrus and left
intraparietal sulcus, both thought to be preferentially involved in processing number
symbols, demonstrated robust differences in community membership between formats.

引文: 康拉德, 乙. N。, Wilkey, 乙. D .,
杨, D. J。, & Price, G. 右. (2020).
Network topology of symbolic and
nonsymbolic number comparison.
网络神经科学, 4(3), 714–745.
https://doi.org/10.1162/netn_a_00144

DOI:
https://doi.org/10.1162/netn_a_00144

支持信息:
https://doi.org/10.1162/netn_a_00144
https://osf.io/sb5v2/
https://github.com/conradbn/
CR7T_Connectivity

已收到: 27 九月 2019
公认: 8 可能 2020

利益争夺: 作者有
声明不存在竞争利益
存在.

通讯作者:
Gavin R. Price
gavinprice@gmail.com

处理编辑器:
Lucina Uddin

版权: © 2020
麻省理工学院
在知识共享下发布
归因 4.0 国际的
(抄送 4.0) 执照

麻省理工学院出版社

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Network topology of symbolic and nonsymbolic number comparison

Symbolic number processing:
Operating on numerical information
derived from a learned notation
系统, 例如, Arabic numerals or
数字词.

Nonsymbolic number processing:
Operating on numerical information
derived from sets of perceptual
元素, 例如, dot arrays or tone
序列.

介绍

The representation and manipulation of numerical information in the brain has received con-
siderable attention over the last 20 年. A longstanding debate in the ﬁeld regards the extent
to which symbolic (例如, Arabic digits) and nonsymbolic (例如, dot arrays) number formats en-
gage shared versus distinct neural mechanisms (Cohen Kadosh & Walsh, 2009; Knops, 2017;
Leibovich & Ansari, 2016; Reynvoet & Sasanguie, 2016). Regions of the bilateral intraparietal
sulcus (IPS) have been found to be involved in symbolic (Ansari et al., 2005; Eger, Sterzer, 拉斯,
Giraud, & 克莱因施密特, 2003; Holloway, Battista, 沃格尔, & Ansari, 2013; Kaufmannet al.,
2005; Pinel, 德阿内, Riviére, & LeBihan, 2001) and nonsymbolic (Ansari & 迪塔尔, 2006;
Cantlon, Brannon, Carter, & Pelphrey, 2006; Piazza, Izard, Pinel, Le Bihan, & 德阿内, 2004)
numerical processing during both adaptation and number comparison paradigms, 支持-
ing proposals that the IPS provides a substrate for quantity encoding shared between formats
(德阿内 & 科恩, 1995; 德阿内, Piazza, Pinel, & 科恩, 2003). A recent meta-analysis
of the extant functional magnetic resonance imaging (功能磁共振成像) literature supports the notion that
the parietal lobe subserves a shared representation of numerical magnitude across nonsym-
bolic and symbolic formats, detecting convergence of activation foci between symbolic and
nonsymbolic number processing in several bilateral parietal areas, as well as medial superior
frontal gyrus (Sokolowski, Fias, Mousa, & Ansari, 2016). Such ﬁndings suggest shared neural
resources for numerical processing across formats.

In addition to evidence for shared neural substrates across numerical formats, there is also
considerable evidence for distinct mechanisms between formats. Symbolic number compar-
ison has been shown, 例如, to engage left superior temporal gyrus (STG) (Holloway,
Price, & Ansari, 2010) as well as bilateral angular gyri, to a greater extent than nonsymbolic
number comparison, potentially reﬂecting increased reliance on verbal mechanisms during
number symbol processing (Ansari, Lyons, van Eimeren, & 徐, 2007; Dehaene et al., 2003;
他, Zuo, 陈, & Humphreys, 2014; Holloway et al., 2010). A meta-analytic contrast between
formats suggests a general asymmetry in parietal activations, such that Arabic digits preferen-
tially engage left inferior parietal cortex, whereas nonsymbolic stimuli more strongly engage
distributed areas of right superior parietal, right frontal, and insular cortices (Sokolowski et al.,
2016). Symbolic number processing has also been shown to engage ventral occipitotemporal
(vOT) areas involved in the decoding of visual symbols (德阿内 & 科恩, 1995), 例如
during active processing of Arabic digits (Grotheer, Herrmann, 等人。, 2016; 波拉克 & Price,
2019; Shum et al., 2013; 杨, Wilkey, & Price, 2017). 然而, it is unclear the extent to
which vOT areas are differentially engaged between symbolic and nonsymbolic number for-
mats. 例如, the meta-analytic contrast performed by Sokolowski et al. (2016) found no
clusters in the vOT. 总之, the current state of evidence suggests that symbolic and non-
symbolic number processing share neural resources in frontal and parietal regions, 但是也
engage distinct mechanisms distributed across the frontal, 顶叶, 和颞叶皮质.

While prior work has focused on the extent to which brain regions are similarly or differ-
entially activated across number formats, complementary questions remain in regards to how
these regions interact and participate in functional networks during number processing. 自从
the earliest theories of numerical cognition, there has been an implicit understanding that
multiple brain systems and subnetworks are involved in extracting and operating on numeri-
校准信息 (坎贝尔, 1994; 德阿内, 1992; McCloskey, 1992; McCloskey, Caramazza,
& Basili, 1985), yet to date, whole-brain network-level descriptions have gone unexplored.
We propose that description of symbolic and nonsymbolic number processing mechanisms in
terms of their widespread functional interactions among multiple brain areas and systems, 这样的

网络神经科学

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Network topology of symbolic and nonsymbolic number comparison

Beta-series correlation:
A method for measuring functional
connectivity in event-related fMRI
data that assesses how similarly two
regions respond to trials of a given
状况.

社区:
A subnetwork of nodes (例如, 地区)
that are more densely connected to
each than to other nodes of the
网络.

as those implicated in visual perception, 注意力, and cognitive control, will shed new light
on the format-dependent versus format-independent processing debate, and provide comple-
mentary insights into how numerical cognition is accomplished in the human brain.

Graph theory provides a methodological framework for characterizing complex networks,
such as those derived from whole-brain functional connectivity data (Bassett & Gazzaniga,
2011; Bassett & 斯波恩斯, 2017). One application that has been particularly fruitful in cogni-
tive neuroscience has been the use of graph topology–based clustering techniques to group
brain regions into functional subnetworks, which can then be associated with various cognitive
and behavioral functions (加西亚, Ashourvan, Muldoon, Vettel, & Bassett, 2018; Power et al.,
2011; 斯波恩斯 & 贝策尔, 2016). In the present study, 我们收集了 7 Tesla fMRI data in adults who
performed a numerical magnitude comparison task involving symbolic and nonsymbolic stim-
uli. We employed task-evoked functional connectivity (IE。, beta-series correlation; Rissman,
Gazzaley, & DEsposito, 2004) and modularity maximization–based clustering (Lancichinetti
& Fortunato, 2012) to assess the network topology, 或组织, of symbolic and nonsym-
bolic number comparison. We contrasted topologies between formats ﬁrst at the macroscale
(IE。, global level involving all regions), then the community level (IE。, involving groups of re-
祇翁), and ﬁnally the individual region level. Speciﬁcally, we asked the following questions:
(1) Is there a difference in the whole-brain functional connectivity structure (IE。, global mod-
独特性) between the symbolic and nonsymbolic formats? (2) What is the topographic layout
of functional communities in response to each format? (3) To what extent is the connectiv-
ity within and between each community signiﬁcantly different between formats? (4) Do any
regions demonstrate a signiﬁcant change in their whole-brain connectivity proﬁle between
formats?

Given that the same task was performed across all trials, we expected that multiple task-
level processes would be shared between formats, 例如, involving cognitive control
and response selection mechanisms. We thus predicted a similar macroscale network archi-
tecture would be observed for each format (克里宁, 杨, & 巴克纳, 2014), indicated by
an equivalent degree of global modularity, and the inclusion of a fronto-parietal network in-
volving IPS and prefrontal regions. We hypothesized, 然而, that differences would arise
among regional communities, with the symbolic format differentially engaging ventral visual
and temporoparietal pathways involved in object recognition and orthographic decoding, 和
the nonsymbolic format engaging an occipitoparietal pathway involved in object location,
individuation, and summation coding. 此外, if the posterior inferior temporal gyrus
(pITG) is preferentially involved in the recognition of Arabic numerals, we expected this region
to show increased interaction with parietal regions. 然而, if the pITG plays a nonspeciﬁc
role in number processing more generally, we expected a similar pattern between formats in
the functional interactions of this region.

结果

Behavioral Results

Descriptive statistics of behavioral performance are provided in Table 1. Mean reaction times
were computed from correctly answered trials only. Symbolic comparison trials elicited a sig-
niﬁcantly lower error rate than nonsymbolic trials using paired tests (t(32) = 2.20, p = 0.035).
A signiﬁcant difference in mean reaction time was also observed, whereby nonsymbolic trials
required approximately 45 ms more time to complete than symbolic trials (t(32) =
6.27, p =
7). 总共, participants were both faster and more accurate when solving symbolic
5.02
versus nonsymbolic comparisons.

10-

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Network topology of symbolic and nonsymbolic number comparison

桌子 1. Scanner task performance.

标准

Mean deviation Minimum Maximum

Symbolic condition accuracy (%)
Nonsymbolic condition accuracy (%)
Symbolic condition reaction time (多发性硬化症)
Nonsymbolic condition reaction time (多发性硬化症)

97.1
94.9
774.2
818.8

3.7
7.0
142.6
129.0

85.0
71.3
550.9
627.1

100.0
100.0
1,133.6
1,130.2

模块化 (问):
Indexes the quality ofa network
partition relative to a null model
和, when maximized, summarizes
the extent of segregation versus
integration among a network’s
社区.

Structural resolution parameter (C):
A free variable in the modularity
quality function that affects the
number and size of communities
detected.

Global Modularity Across Topological Scales

Our ﬁrst question was, is there a difference in overall modular organization between formats?
The modularity of a network is indexed by the statistic, Q*, and provides a summary measure of
the extent to which a network can be decomposed into nonoverlapping communities when
each node (IE。, 地区) is assigned to only one community (see Methods for algorithmic deﬁni-
的). A positive Q* value indicates that intracommunity connections are stronger than chance
因此, Q* summarizes the modular topology of a network in regards to the relative bal-
ance of segregated versus integrated connectivity structure (high versus low Q*, 分别)
(纽曼, 2006). For each subject, we extracted the max Q* observed over 100 迭代
for the symbolic and nonsymbolic networks separately, and performed this procedure at each
step of the structural resolution parameter (C) over the resolution range of interest (数字 1;
see Methods for more details on this range). A paired t test was performed at each step and
revealed no signiﬁcant difference between the max Q* in the symbolic versus nonsymbolic
状况 (p > 0.05 uncorrected). This result indicates that there was no consistent difference
between formats at any topological scale in the degree of global segregation/integration across
大脑.

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数字 1. No format differences in global modularity. The maximum global modularity (Q*) 价值
for each participant/format was extracted at every step along the resolution sweep (γ = 0.05–5.0,
with steps of 0.05). The mean lines are plotted along with shaded bars representing the standard
deviation across participants.

网络神经科学

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Network topology of symbolic and nonsymbolic number comparison

Community Topology

While our ﬁrst analysis indicated a similar modular structure between formats at the global
等级, similar macroscale characteristics can arise from differing underlying topologies, 麦-
ing it important to supplement global metrics with more ﬁne-grained assessment of speciﬁc
network elements (斯波恩斯, 2014). 所以, we investigated the subnetwork organization of
functional communities, looking at community topography for each format separately and dif-
ferences between formats in the degree of community allegiance.

The graph-theoretical concept of allegiance refers to the
Functional communities within format
probability of two regions being assigned to the same community across multiple partitions of
a network, and has been used extensively to study community organization during task perfor-
曼斯 (Bassett, 哪个, Wymbs, & Grafton, 2015; Bertolero, Thomas Yeo, & D’Esposito, 2015;
Braun et al., 2015; Chai, Mattar, 空白的, Fedorenko, & Bassett, 2016; Mattar, Cole, 汤普森-
Schill, & Bassett, 2015; Rizkallah et al., 2018; Telesford et al., 2016). The allegiance matrix
contains the probability for every region pair, 那是, the tendency for two brain regions to be
part of the same functional community, and thus describes region-to-region associations with
respect to shared community membership. Where functional connectivity asks of two regions,
“How much do they communicate?,” allegiance asks, “Do they belong to the same group?”
While the two metrics are related (and especially so in functional connectivity matrices due to
the transitive nature of time series correlations), they can in principle diverge in a number of
scenarios. 例如, two regions may exhibit relatively low connectivity to each other but,
due to stronger association with other members of the same community, demonstrate high
allegiance. In another case, a hub region with strong connectivity to two communities may
demonstrate relatively low allegiance to any one region, due to inconsistency in its commu-
nity assignment (IE。, it belongs equally to two communities). From these examples, one can
appreciate that allegiance and connectivity provide complementary information about rela-
tionships among nodes of a network. 重要的, 然而, an allegiance-based approach has
some advantages. As an allegiance matrix is derived from network partitions (which reduce
connectivity information down to singular community assignments), allegiance values are ro-
bust to (1) noise inherent in connectivity measurements, 也 (2) differences in raw con-
nectivity strengths and/or distributions across conditions or subjects, making them appealing
for studying functional reconﬁguration across task states (Bassett et al., 2015). 换句话说,
the allegiance matrix serves as a pure representation of a network’s modular topology. 的确,
an allegiance-based approach was shown to be more sensitive to differences in functional net-
work architecture compared with connectivity information alone (Bassett et al., 2015). 在这个
study we used group-level allegiance measures to assess community and region-level topology
during symbolic and nonsymbolic number comparison (参见方法).

To evaluate community-level organization, one must ﬁrst deﬁne the topological “resolution”
of interest, which relates to the size of communities detected via modularity maximization and
is modulated by the resolution parameter γ (Reichardt & Bornholdt, 2006). Since our goal was
to compare topologies between the symbolic and nonsymbolic networks, we sought a reso-
lution at which there was a maximal balance between similarities and differences in group-
level community structure between formats (Mattar et al., 2015). Based on this heuristic, 我们
found an optimal resolution at γ = 2.45 and used this value for subsequent community-level
分析 (see Methods for details on this procedure). Our group-level consensus clustering
procedure delineated 44 communities for symbolic (25 nonsingleton and 19 singleton, 在哪里
singleton refers to a community of just one region) 和 36 communities for nonsymbolic (23

718

Allegiance:
The extent to which two regions
belong to the same community,
measured by the probability of
shared community membership
across multiple partitions.

Network partition:
A decomposition of network (IE。,
clustering solution) in which each
node is assigned to exactly one
社区.

Modularity maximization:
An algorithmic process to determine
a partition that maximizes the
modularity quality index (问), IE。, 这
optimal clustering solution.

Consensus clustering:
An algorithm for determining a
representative network partition from
many partitions of the same network
or from multiple subjects.

网络神经科学

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Network topology of symbolic and nonsymbolic number comparison

nonsingleton, 13 singleton). We reduced the set of communities to only those that demon-
strated a strong modularity contribution (Q*c) at the subject level, ensuring that the commu-
nities of interest were (1) not driven by noise, 和 (2) not solely a feature of the group-level
网络, but biologically meaningful in individuals (Bassett, Owens, Daniels, & Porter, 2012;
Betzel et al., 2017; Guimerà, Sales-Pardo, & Amaral, 2004). 简单地说, we computed the Q*c for
each community in the subject-level connectivity matrices and compared against null models,
selecting for further analyses only those that exceeded the 99th percentile of the null distribu-
的 (see Supporting Information Methods and Supporting Information Results Figure S1), 在一个
approach adapted from Betzel et al. (2017). This procedure resulted in 10 的 25 社区
selected for symbolic and 10 的 23 communities selected for nonsymbolic (数字 2).

One challenge in interpreting functional brain network results based on data-driven cluster-
ing solutions is the assignment of labels to the resulting communities (Yeo et al., 2011). 使用
the widely adopted Power et al. (2011) network deﬁnitions, we took a quantitative approach in
which we calculated the voxel-wise overlap of our ﬁnal community assignments with the con-
sensus assignments from Power et al. (acquired from https://www.jonathanpower.net/2011-
neuron-bigbrain.html). In almost all cases where an analogous community existed in the Power
等人. partition, a clear “winner” was determined based on the relative percentage overlap ver-
sus other communities, including the default mode network (DMN), fronto-parietal task con-
trol network (FPN), dorsal attention network (DAN), sensorimotor network (SMN), visual (Vis),
auditory (Aud), hippocampus (希普), thalamus (Thal), and basal ganglia (BG) 社区.
In the case of our smaller communities, involving the dorsal caudate (CdD), ventral caudate
and nucleus accumbens (CdV/NAc), and precuneus (Prec), there was not a corresponding
Power et al. assignment, so we labeled these according to their anatomical description in the
Brainnetome atlas (see Methods and Supporting Information Table S1) (Fan et al., 2016). 一
nonsymbolic community, involving primarily bilateral insular regions, was composed equally
of voxels within the cingulo-opercular task control (一氧化碳, 40.1%) and salience network (SN,
39.9%) communities from Power et al. Given that a clear distinction could not be made in this
案件, we refer to this community by using the combined label CO/SN.

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Qualitative differences were observed between formats in the existence and topography of
individual communities (数字 2). The FPN community, 例如, was more distributed in
the nonsymbolic condition, including bilateral superior and inferior parietal regions as well as
a majority of prefrontal regions in the right hemisphere. For symbolic, the regions making up
the FPN in nonsymbolic were largely divided into a more ventral FPN and a DAN community,
with the DAN including superior parietal lobule (声压级) regions and a region of right pITG (看
next section and Supporting Information Figure S2 for further investigation of the dual versus
uniﬁed FPN/DAN network architecture between formats). 此外, a left-lateralized audi-
保守党 (Aud) 社区, including left STG and bilateral superior temporal sulcus regions, 曾是
only detected in the symbolic condition, and a CO/SN community, involving primarily bilat-
eral insular regions, was only detected in the nonsymbolic condition. 另一方面, 这
visual, sensorimotor, default mode, and subcortical communities were topographically com-
parable between formats. Supporting Information Table S1 includes labels and community
assignments for all regions of the Brainnetome Atlas.

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As a ﬁnal analysis of community topography, we sought to determine whether the observed
dual versus uniﬁed FPN/DAN network architecture in the symbolic and nonsymbolic con-
版本, 分别, was speciﬁc to the particular γ value employed in these analyses (IE。,
γ = 2.45), or was robust across γ values. The dual FPN/DAN network architecture in the sym-
bolic condition persisted across a relatively large range of γ (Supporting Information Figure S2),

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Network topology of symbolic and nonsymbolic number comparison

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数字 2. Functional communities by format. Group-level communities were determined for symbolic and nonsymbolic formats separately.
The depicted communities represent the reduced set that passed our selection criterion of showing strong modularity in subject-level con-
nectivity matrices (see Supporting Information Methods and Supporting Information Figure S1 for details on the selection procedure). Cortical
regions are plotted to a surface representation and subcortical regions are presented as 3D volumes. Community labels were determined
based on voxel-wise overlap with the networks delineated in Power et al. (2011), 或者, when an analogous network did not exist in Power et al.,
based on the anatomical region(s). Colors were chosen to match those networks where possible. Gray indicates regions that did not belong to
a selected community; black indicates regions excluded based on signal dropout (see Supporting Information Methods).

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whereas these communities were not consistently distinguished in the nonsymbolic condition,
suggesting that the observed difference in FPN/DAN integration between formats is robust
across γ values.

Differences in the degree of within- and between-community allegiance between formats While the
previous analysis delineated the functional communities present within each condition sep-
arately, the detection of a community is driven in part by the organization of the rest of the
网络. 像这样, the existence of a community in one condition but not the other does not
necessarily indicate that allegiance among these regions (IE。, within-community allegiance) 是
stronger in that condition per se. 此外, even if community assignments are similar be-
tween conditions, differences may exist in the hierarchical organization of the network and/or

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Network topology of symbolic and nonsymbolic number comparison

Monte Carlo permutation:
A technique used for nonparametric
hypothesis testing in which a null
distribution is constructed by
randomly resampling observed data.

the degree of integration across communities, properties that can be captured by the strength
of between-community allegiance. Allegiance values within or between communities provide
an indication of how strongly connected, or “allegiant,” the members of a given community
are to each other across subjects. A mean allegiance of 0.8 within a community (Figure 3C
和D), 例如, speciﬁes that the regions of this community were members of the same
community in 80% of participants (on average across region pairs). To quantitatively assess
whether statistically signiﬁcant differences existed at the community level, we extracted the
vector of region-to-region connections (allegiance values below the diagonal) from the two
group-level matrices (Figure 3A and B), for each pair of communities identiﬁed in the sym-
bolic and nonsymbolic conditions. To assess the difference between formats, a paired t test
was then performed on the vectors of allegiance values (例如, allegiance across regions of the
symbolic DAN community in the symbolic matrix versus the same regions in the nonsymbolic
矩阵). A Monte Carlo permutation procedure with 50,000 iterations was used to determine
the signiﬁcance of the observed t statistics, whereby subject-level partitions from each con-
dition were randomly relabeled, null group-level allegiance matrices were constructed, 和
allegiance values were compared for each community, constructing a null distribution for each
t statistic of interest. This represents a nonparametric approach to signiﬁcance testing that is
appropriate for repeated measures (Nichols & Holmes, 2004). Z scores for the true (IE。, ob-
µTnull )/σTnull, and p values were assigned based
服务) t statistics were computed as (Ttrue −
on the position of the value t statistic within the null distribution (Figure 3E and F). No com-
munity pairs survived FDR correction for multiple comparison across the 55 tests of within-
and between-community allegiance conducted for each set of communities (Figure 3E and F)
(Benjamini & Hochberg, 1995). 然而, given the data-driven, exploratory approach em-
ployed here assessing all pairwise associations, we did not expect effect sizes to be strong
enough to pass such a stringent criterion. The goal here was rather to look for underlying
patterns in the data that could then generate hypotheses for future research. In line with rec-
ommendations for exploratory research, we report the outcome of our multiple comparison
correction procedure but also present the observed effect sizes (z scores) as well as the uncor-
rected p values, with a strong caution that these ﬁndings must be conﬁrmed in future studies
(Althouse, 2016; Feise, 2002; Rothman, 1990; Streiner, 2015; 汤普森, 赖特, & Bissett,
2020).

Several differences were observed between formats in regards to within- and/or between-
community allegiance based on an uncorrected threshold of p < 0.05 (two-tailed) (Figure 3E and F). Of the communities identiﬁed in the symbolic condition, stronger allegiance was found for symbolic versus nonsymbolic among regions of the auditory (Aud) community (Aud: z score = 2.11, p = 0.019). This suggests that not only was an auditory community identi- ﬁed in the symbolic condition and not the nonsymbolic condition, but that the regions of this community are signiﬁcantly more functionally coupled to each other during symbolic trials. Furthermore, allegiance between the dorsal attention (DAN) and visual (Vis) communities was stronger in the symbolic condition (DAN–Vis: z score = 1.97, p = 0.015). Weaker allegiance was observed in the symbolic condition between basal ganglia (BG) and both the default mode 2.09, p = 0.010; BG– (DMN) and sensorimotor (SMN) communities (BG–DMN: z score = 1.70, p = 0.016), and between the hippocampal (Hipp) and thalamic (Thal) SMN: z score = − 1.97, p = 0.004). Of the communities identiﬁed in the communities (Hipp–Thal: z score = nonsymbolic condition, stronger allegiance (nonsymbolic > 象征性的) was found among re-
gions of BG and cingulo-opercular/salience network (CO/SN) 社区 (BG: z score = 2.02,
p = 0.021; CO/SN: z score = 2.50, p = 0.005). In nonsymbolic condition, the CO/SN and BG
communities were also more allegiant to each other (CO/SN–BG: z score = 1.94, p = 0.004),

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Network topology of symbolic and nonsymbolic number comparison

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数字 3. Comparison of community allegiance between formats. (A and B) Group-level allegiance matrices are presented, indicating at
each cell in the matrix the proportion of subjects in which two regions were assigned to the same community. Matrix cells are reordered based
on the community composition of the symbolic (A) and nonsymbolic (乙) formats, 分别, but the values are identical within each format.
(C and D) Mean allegiance across region pairs is depicted for each format and was computed using communities as deﬁned for symbolic
and nonsymbolic, 分别. Black bars represent standard error of the mean across region pairs. (E and F) Allegiance difference within and
between communities detected in each format. No p values survived correction for multiple comparisons (p < 0.05, FDR corrected for 55 tests along + below the diagonal). The CdD community was composed of two regions (right/left dorsal caudate) and thus contained only one region-to-region connection. In this case, since a paired t test was not possible, the z score for CdD represents that of the simple difference between formats. DMN = default mode network; Thal = thalamus; FPN = fronto-parietal network; Vis = visual; DAN = dorsal attention network; BG = basal ganglia; SMN = sensorimotor network; Hipp = hippocampus; CdV/NAc = ventral caudate/nucleus accumbens; Aud = auditory; Prec = precuneus; CO/SN = cingulo-opercular/salience network; CdD = dorsal caudate. Network Neuroscience 722 Network topology of symbolic and nonsymbolic number comparison and both more allegiant to the SMN community (CO/SN–SMN: z score = 2.00, p = 0.014; BG–SMN: z score = 1.73, p = 0.016). Additionally, greater allegiance between the Hipp and Thal communities was observed in the nonsymbolic condition (Hipp–Thal: z score = 1.98, p = 0.004). Weaker allegiance in the nonsymbolic condition was observed between the FPN 1.98, p = 0.022; FPN– and both the Vis and and CO/SN communities (FPN–Vis: z score = CO/SN: z score = 1.96, p = 0.021). − − × We additionally asked whether the differences between formats we observed in commu- nity allegiance could have been captured by a simpler analysis of raw connectivity strengths. We therefore assessed the relation between allegiance and connectivity values for each format separately, and then critically, whether connectivity information provided a similar pattern of statistically signiﬁcant differences between formats. These analyses were performed using the median group-level connectivity matrices, group-level allegiance matrices, and the commu- nities as deﬁned at γ = 2.45. We ﬁrst constructed two 10 10 matrices representing the mean allegiance and mean connectivity within/between the 10 communities, for each format separately. We then computed the correlation between these matrices, excluding the mirrored values above the diagonal. In other words, we asked, “before taking the difference between formats, are community-level allegiance and connectivity values related?” We found that (as expected) these values were highly related (Pearson r = 0.64, p < 0.001 in the symbolic data; Pearson r = 0.55, p < 0.001 in the nonsymbolic data). Next, to determine whether a similar pattern of statistically signiﬁcant differences could be observed using the connectiv- ity data, we correlated the values depicted in the matrices of Figure 3E and F with those in Supporting Information Figure S3 (excluding values above the diagonal). Neither set of re- sults were signiﬁcantly correlated (Pearson r = 0.07, p = 0.60 using symbolic communities; Pearson r = 0.23, p = 0.08 using nonsymbolic communities), though a marginal positive ef- fect was observed using the nonsymbolic communities. We found the effect sizes of the values within these matrices (i.e., z scores of difference between formats, determined via our Monte Carlo procedure) were relatively smaller in the connectivity data (Supporting Information Figure S3) compared with the allegiance data (Figure 3E and F). This analysis highlights the di- vergence between connectivity and allegiance metrics, and suggests that an allegiance-based approach is sensitive to changes in community topology that are not captured by simple con- nectivity differences. Region-level Allegiance Proﬁles Across Topological Scales To provide insights into how the functional roles of particular regions may differ based on number format, we investigated the extent to which node-level allegiance proﬁles changed between the symbolic and nonsymbolic conditions. In the community-level analysis, we used modularity maximization to deﬁne communities of interest and followed this up with quan- titative analysis of allegiance differences between formats. This process necessitated a partic- ular setting of the resolution parameter (γ = 2.45). Regions, on the other hand, make up the fundamental units of the network and were predetermined by the atlas parcellation, so region- level analyses need not be constrained to a particular topological scale. We reasoned that a more comprehensive assessment of regional allegiance proﬁles would be to summarize differ- ences between formats across a large range of resolutions (Betzel & Bassett, 2016). We thus performed the following analysis of allegiance proﬁle (dis)similarity for every resolution step across our range of interest, γ = 0.05–5, with steps of 0.05 (see Methods for more details on this range). Network Neuroscience 723 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / / 4 3 7 1 4 1 8 6 7 3 1 1 n e n _ a _ 0 0 1 4 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Network topology of symbolic and nonsymbolic number comparison The allegiance proﬁle of a region was deﬁned as the vector of group-level allegiance val- ues from one region to all others, which characterizes a region’s position in the network in relation to the network’s community structure. We computed the (dis)similarity of a region’s allegiance proﬁle between the symbolic and nonsymbolic conditions using Pearson correlation coefﬁcient converted to Fisher Z values, with lower values indicating greater dissimilarity. The signiﬁcance of dissimilarity was assessed using a similar Monte Carlo permutation approach as before, involving 10,000 random reshufﬂings of the subject-level partitions to construct a null distribution of group-level allegiance matrices (Nichols & Holmes, 2004). From these null matrices, a z score and p value for the observed dissimilarity coefﬁcient (inverse of Fisher Z) was computed for every region. To characterize the overall robustness of allegiance proﬁle differences across the resolution sweep, we counted the number of steps in which a region demonstrated a signiﬁcant dissimilarity in its allegiance proﬁle between formats (p < 0.05 one-tailed, uncorrected), and the counts are depicted as a histogram in Figure 4A. In order to reduce the set of regions to those that showed the greatest total count, we ﬁrst ﬁt a negative binomial curve to the observed distribution, which is commonly used to model overdispersed count data (Bliss & Fisher, 1953; Hartley, 1958; White & Bennetts, 1996). We then assessed the goodness of ﬁt of this theoretical distribution to the observed counts using a quantile-quantile plot (Figure 4B). In general, there was high overlap in the observed versus ﬁtted distribution, with the exception of ﬁve outlying regions (highlighted in yellow, Figure 4B): the (1) right caudal lingual gyrus, (2) right caudal cuneus gyrus, (3) right area 4 in precentral gyrus, (4) right ventrolateral area 37 in pITG, and (5) left intraparietal area 7 (hIP3). These ﬁve regions demonstrated higher counts than expected given the observed pattern, indicating that, relative to other regions, they showed robust differences in community allegiance between formats across a wide range of organizational scales. We highlight these regions in Figure 4C, includ- ing their anatomical description and location, as well as plots of their dissimilarity z scores across the full resolution sweep. The right pITG region sits posterior but immediately adjacent to the putative “Number Form Area” as described in recent literature. The volumetric location of this region relative to the peak coordinates reported across multiple studies, including an fMRI meta-analysis, is depicted in Figure 5 (Abboud, Maidenbaum, Dehaene, & Amedi, 2015; Amalric & Dehaene, 2016; Grotheer, Ambrus, et al. , 2016; Grotheer, Herrmann, et al., 2016; Hermes et al., 2017; Shum et al., 2013; Yeo et al., 2017). Both the right pITG and left intraparietal hIP3 regions were part of the DAN community—one part of the dual fronto-parietal networks identiﬁed in the symbolic condition (Figures 2 and 3; Supporting Information Table S1). As a supplementary analysis, we investigated the extent to which the observed differences in regional allegiance proﬁles related to differences in the raw connectivity proﬁles of each region, using the median group-level connectivity matrices in place of the group-level alle- giance matrices (Supporting Information Figure S4). We found that the count of signiﬁcant steps over the resolution sweep was positively related to the degree of connectivity dissim- ilarity (Spearman’s rho = 0.202, p = 0.004), indicating that, albeit a modest relation, there is overlap between allegiance and connectivity at the region level. However, the speciﬁc re- gions showing the greatest counts versus connectivity dissimilarity differed. For instance, three regions showing a high count from the allegiance analysis (right caudal lingual gyrus, right area 4, and left intraparietal hIP3) demonstrated connectivity dissimilarity z scores near 0. Along with Supporting Information Figure S3, this indicates that allegiance and connectivity information are largely independent and that allegiance-based analyses can detect changes in regional community membership regardless of strong and/or consistent changes in the region’s raw connectivity proﬁle. Network Neuroscience 724 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / / 4 3 7 1 4 1 8 6 7 3 1 1 n e n _ a _ 0 0 1 4 4 p d t . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Network topology of symbolic and nonsymbolic number comparison l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / / 4 3 7 1 4 1 8 6 7 3 1 1 n e n _ a _ 0 0 1 4 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 4. Regions showing robust differences in allegiance proﬁle between formats. (A) Histogram plot showing counts of signiﬁcant steps over the resolution sweep from all 202 regions, in terms of a signiﬁcant dissimilarity in allegiance proﬁle between the symbolic and nonsym- bolic conditions (p < 0.05, uncorrected). The red line indicates the ﬁtted negative binomial distribution, and pobs indicates the probability of the selected count, given the distribution of all counts observed. (B) Quantile-quantile (QQ) of the ﬁtted distribution (x-axis) with respect to the sample distribution (y-axis). Note the ﬁve regions highlighted in yellow that show considerable deviation from what was expected given the counts demonstrated by the rest of the regions. (C) Table of the ﬁve regions showing the most robust differences in allegiance, with z scores over the resolution sweep plotted in the rightmost column. The dashed magenta line indicates γ = 2.45, the setting for the community-level analysis (Figure 3) and the red dot indicates p < 0.05 at that step. Both occipital regions were members of the visual community in the symbolic and nonsymbolic conditions (at γ = 2.45), so are presented as blue. The precentral gyrus region was a part of the CO/SN community (magenta) in the nonsymbolic community (and not selected in the symbolic condition). The pITG and hIP3 regions were part of the DAN community in the symbolic condition (green). In the nonsymbolic condition, the hIP3 region was part of the FPN community while the pITG region was not selected (see Supporting Information Table S1 for the community assignments of all regions). Network Neuroscience 725 Network topology of symbolic and nonsymbolic number comparison Figure 5. Posterior ITG region (ventrolateral area 37) relative to reported “Number Form Area” coordinates. The ventrolateral area 37 (A37vl) Brainnetome region in the posterior ITG showed signiﬁcant differences in its allegiance proﬁle between the symbolic and nonsym- bolic conditions (see the region-level allegiance analysis in Figure 4). This region is depicted (red) along with spheres centered at the peak MNI coordinates (x,y,z) from several recent studies demonstrating Arabic numeral-selectivity versus other stimuli, using either fMRI or electro- corticography measurements, as well as from an fMRI meta-analysis of numeral versus other symbol processing (Abboud et al., 2015; Amalric & Dehaene, 2016; Fan et al., 2016; Grotheer, Ambrus, et al., 2016; Grotheer, Herrmann, et al., 2016; Hermes et al., 2017; Shum et al., 2013; Yeo et al., 2017). Where coordinates were reported in Talairach space, the tal2mni.m function was used to convert to MNI space via the Brett transform (http://imaging.mrc-cbu.cam.ac.uk/downloads/MNI2tal/tal2mni.m). The A37vl region sits immediately posterior to these coordinates, and was the only ITG region in the Brainnetome Atlas that survived our signal dropout threshold (the others are more anterior). DISCUSSION Little is currently known regarding the network architecture of brain systems supporting numer- ical cognition. While region-based analyses have provided important insights into the neural mechanisms of numerical cognition, a growing body of work outside the numerical cognition ﬁeld suggests that understanding the network topology of those mechanisms, as well as how this topology is reconﬁgured across task states, can provide additional meaningful insights (e.g., Braun et al., 2015; Cohen & D’Esposito, 2016; Cole, Bassett, Power, Braver, & Petersen, 2014; Hearne, Cocchi, Zalesky, & Mattingley, 2017; Shine et al., 2019). In this study, we used measurements of task-evoked functional connectivity and network-based analysis to assess the similarities and differences between symbolic and nonsymbolic numerical magnitude process- ing. Our ﬁndings provide evidence for a similar global topology across formats, but selective differences in community and region-level organization. No Difference in Global Modularity Between Formats In regards to functional connectivity networks, the modularity index has been considered as a measure of segregated versus integrated information processing among functional subsystems, with lower modularity indicating greater integration across the brain (Bertolero, Yeo, Bassett, & D’Esposito, 2018; Godwin, Barry, & Marois, 2015; Shine et al., 2016). Our results showed no evidence of a difference in global modularity in subject-level connectivity networks at any topological scale (Figure 1), suggesting that the balance of segregated versus integrated in- formation processing among subnetworks of the brain is similar between formats. It is worth noting, however, that the task employed here involved comparison of each stimulus to the ref- erence number 5, with symbolic and nonsymbolic trials intermixed across the experiment. We speculate that global topological differences between formats may be more pronounced when more than one stimulus must be processed, as is the case in tasks using simultaneous or se- quential presentation of two stimuli to be compared, and/or if comparison tasks are performed Network Neuroscience 726 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / / 4 3 7 1 4 1 8 6 7 3 1 1 n e n _ a _ 0 0 1 4 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Network topology of symbolic and nonsymbolic number comparison separately for each format. For instance, in separate tasks, global topological differences may emerge from the adoption of different strategies or attentional states (Bertolero et al., 2018; Cocchi et al., 2014; Cohen & D’Esposito, 2016; Heinzle, Wenzel, & Haynes, 2012), rather than a difference in symbolic versus nonsymbolic processing per se. Our use of intermixed stimuli prevented participants from anticipating one format versus another, thus our results are likely to reﬂect format-dependent processing differences, rather than strategic differences related to task context. Community Allegiance Versus Functional Connectivity The remaining analyses focused on a comparison of network architectures between number formats at the subnetwork (i.e., community) and region levels. Our approach leveraged the concept of community allegiance, which characterizes region-to-region associations in terms of shared membership among functional subnetworks (see Results section for a more thorough introduction). Intuitively, functional connectivity indexes the extent to which two regions di- rectly interact, whereas allegiance indexes the tendency for two regions to be associated with the same community. Compared with connectivity, allegiance-based analysis is in principle more robust to noise and intersubject variance, and has been shown to be particularly sen- sitive to reconﬁgurations in brain network architectures (Bassett et al., 2015). We found that while connectivity and allegiance measures were related (which is expected, given that al- legiance is derived from partitions of connectivity matrices), they also diverged considerably (see Results, as well as Supporting Information Figures S3 and S4). The present study indicates that an allegiance-based approach offers a unique and complimentary perspective on network organization, and suggests that future research assessing functional brain networks may beneﬁt from the methodology described herein. Fronto-parietal Uniﬁcation and Fractionation Between Formats In contrast to the global level, our community-level results revealed that the topography of fronto-parietal control systems differed considerably between formats. Speciﬁcally, during non- symbolic trials, the fronto-parietal network (FPN) was a unitary community composed of 33 regions, while during symbolic trials, these regions were largely fractionated into two distinct communities, a dorsal attention (DAN) community and more ventrally/anteriorly positioned FPN (Figure 2A and B). Distinctions have previously been made between dorsal and ventral attention systems (Corbetta & Shulman, 2002). Previous literature suggests the DAN is involved in the voluntary deployment of attentional control based on goals and expectations as well as ultimately linking stimuli to responses, whereas the ventral network is involved in reorienting of attention to relevant stimuli (Corbetta, Patel, & Shulman, 2008; Ptak, Schnider, & Fellrath, 2017). Through top-down signaling, the DAN in particular is thought to bias sensory areas to be responsive to appropriate features of inputs, and is often found to be functionally connected to visual areas in the ventral occipitotemporal (vOT) cortex (in line with the inclusion of the right pITG region in the symbolic DAN community in the current data; see further discussion below and Figure 2A) (Corbetta et al., 2008; Fox, Corbetta, Snyder, Vincent, & Raichle, 2006; Ptak, 2012; A. C. Vogel, Miezin, Petersen, & Schlaggar, 2012). While both formats may engage the DAN, we speculate that nonsymbolic dot arrays place greater demand on the (ventral) at- tentional reorienting system by containing multiple objects organized randomly across space. While we did vary the location of the digits across trials (e.g., see Figure 2), it may be that the perceptual regularity of the symbolic stimuli leads to reduced interactions between the DAN and more ventral/anterior components of the FPN during task performance (Vossel, Geng, & Fink, 2014). A potentially related ﬁnding came from a recent study by Dixon et al. (2018), Network Neuroscience 727 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / / 4 3 7 1 4 1 8 6 7 3 1 1 n e n _ a _ 0 0 1 4 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Network topology of symbolic and nonsymbolic number comparison who showed that an FPN subsystem (“FPCNB”), which has a qualitatively similar spatial pro- ﬁle to our symbolic FPN community, was integrated with the DAN across a range of task states involving externally directed attention and perception. The other component of the FPN, the “FPCNA,” was more integrated with the default mode network (DMN) and recruited during internally directed thought (Dixon et al., 2018). The authors suggest that the FPCNB represents task-relevant information (e.g., rules and response mappings) and exerts moment-to-moment, top-down control of the DAN, allowing the DAN to deploy this information in the guidance of spatial attention (e.g., saccades) and ultimately action initiation (Baldauf & Desimone, 2014; Bichot, Heard, DeGennaro, & Desimone, 2015; Dixon et al., 2018). The uniﬁed FPN/DAN community we observed in the nonsymbolic condition supports this account, possibly reﬂect- ing a stronger engagement of this distributed, visuospatial attentional control system when processing dot sets compared to single digits. It is important to note that while Power et al. (2011) and Yeo et al. (2011) distinguish sep- arate dorsal attention, ventral attention, and fronto-parietal networks, their boundaries (along with the cingulo-opercular network; Dosenbach, Fair, Cohen, Schlaggar, & Petersen, 2008) are not completely consistent across the literature, nor with our own data-driven community assignments. During task performance, multiple components of these networks are recruited and likely integrated to form a so-called “task-positive” network (Dwyer et al., 2014; Fox et al., 2005). The extent to which each of these putative systems separately contribute to cognitive and attentional control is an ongoing area of research (Petersen & Posner, 2012). In general, our ﬁndings indicate that symbolic and nonsymbolic comparison differentially engage large- scale association networks, and further, suggest that assessment of community-level topology may help to incorporate canonical mechanisms of cognitive control and attention into our understanding of number processing. Cingulo-Opercular/Salience Network and Basal Ganglia Allegiance Increases During Nonsymbolic Processing A second pattern of results at the community level involved stronger allegiance, that is, shared community membership, within and between the cingulo-opercular/salience (CO/SN) and basal ganglia (BG) communities in the nonsymbolic condition (Figure 3F). The CO network has been implicated in tonic alertness and task-set maintenance, and demonstrates sustained activity across trials of a cognitive task (Dosenbach et al., 2007; Sadaghiani & D’Esposito, 2015). As part of the dual control network architecture proposed by Dosenbach et al. (2008), the CO operates in parallel to the FPN, which is instead thought to be involved in phasic alertness and adaptive control on a trial-level basis. Our beta-series approach speciﬁcally fo- cused on coupling in trial-level responses, and thus is expected to be less sensitive to cou- pling extending across multiple trials. And even so, the same “task-set” should have been engaged regardless of format, given the trials were intermixed and the same comparison oper- ation was required. Thus, it is difﬁcult to interpret the difference we observed between formats involving CO/SN allegiance within the context of the dual control network framework. On the other hand, this community was a hybrid between the CO and SN communities in the Power et al. (2011) parcellation. The SN is involved in the detection of relevant stimuli, in- cluding both internal and external cues, and, for instance, is particularly engaged during odd- ball paradigms in response to surprising or deviant cues (Menon, 2015). It is implicated in a broad range of goal-directed cognitive functions and is thought to integrate processing among disparate functional subnetworks, potentially ﬁltering input to the ventral attention system (Chang, Yarkoni, Khaw, & Sanfey, 2013; Corbetta et al., 2008). The increased allegiance during nonsymbolic comparison within the CO/SN community in our data may be due to the fact that Network Neuroscience 728 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / / 4 3 7 1 4 1 8 6 7 3 1 1 n e n _ a _ 0 0 1 4 4 p d t . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Network topology of symbolic and nonsymbolic number comparison dot arrays simply involve more visual objects to detect and process, requiring stronger com- munication among regions in this network. However, this interpretation is highly speculative and further empirical investigation is clearly required to elucidate the mechanisms underlying this ﬁnding. We also observed signiﬁcantly increased allegiance among regions of the basal ganglia (BG) community, including bilateral putamen, globus pallidus, ventral caudate, and nucleus accumbens (Figure 2B) in the nonsymbolic versus symbolic condition (Figure 3F). BG struc- tures are generally implicated in motor initiation and control, including eye movements, but are also known to participate in multiple cognitive processes such as inhibition, working memory, decision-making, and learning (Helie, Chakravarthy, & Moustafa, 2013; Leisman, Braun-Benjamin, & Melillo, 2014; Middleton & Strick, 2000; Packard & Knowlton, 2002; Wei & Wang, 2016). Multiple parallel cortico-basal ganglia loops are thought to be required during a stimulus-response task, where some cognitive operation is performed (e.g., < or > 5)
and then linked to a motor output (例如, press left or right) (格思里, Leblois, Garenne, & Boraud,
2013). Since symbolic and nonsymbolic trials required the same motor responses, our ﬁnding
of increased integration among BG for nonsymbolic trials may relate to the longer response
times in the nonsymbolic condition (Yarkoni, Barch, Gray, Conturo, & Braver, 2009). Interest-
英利, BG regions are often shown to coactivate with the SN, and our observation of greater
BG–CO/SN allegiance in the nonsymbolic condition (Figure 3F) further suggests a mechanistic
link between these ﬁndings (Menon, 2015). Given the diverse roles of the BG, 一氧化碳, and SN
systems in cognition, 然而, the pattern of increased allegiance among these communities
is difﬁcult to interpret and requires future investigation.

Symbolic Processing Engages an Auditory Community

A ﬁnal result of interest from the community-level analysis was the observation of the auditory
(Aud) community for symbolic trials (人物 2 and 3E). The Aud community was left lateralized
and extended along the left superior and middle temporal gyri (STG/MTG; IE。, Brodmann area
22), involving a set of auditory cortex regions involved in the representation of phonemes and
phonological retrieval (Binder, 2015; Liebenthal, Binder, 斯皮策, Possing, & Medler, 2005). 左边
temporal regions overlapping with the Aud community found here have been both theoreti-
卡莉 (德阿内 & 科恩, 1997) and empirically (Prado, Mutreja, & 展位, 2014) 受到牵连
in verbal arithmetic fact retrieval, but their role in symbolic quantity processing has been less
well appreciated. The comparison task we employed here does not explicitly require a digit
to be mapped to its phonological representation, and thus we did not predict a priori the en-
gagement of a superior temporal network. 然而, our analysis indicates this system forms a
coherent functional community during only the symbolic condition, potentially indicating the
implicit binding of number symbols to their spoken word forms. These results are in line with
previous studies showing increased functional activation in the superior/middle temporal cor-
tex during symbolic compared with nonsymbolic number comparison (Holloway et al., 2010;
Castaldi, Vignaud, & Eger, 2019) and addition (Van Der Ven, Takashima, Segers, Fernández,
& Verhoeven, 2016). Our analysis suggests a functional network involving bilateral superior
temporal cortex regions is engaged during symbolic but not nonsymbolic number comparison.

Robust Differences in Community Allegiance of Visual Regions Between Formats

In our region-level analysis we asked, at each step across the resolution range, whether a
region showed a difference between formats in its proﬁle of community membership. 到
identify those regions that showed a robust effect (IE。, regardless of topological scale), 我们
counted the number of steps in which a signiﬁcant difference was observed. The two regions

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showing the highest count were located in the right ventromedial occipital cortex (数字 4).
While we controlled for total occupied area and luminance across formats, it is possible that
the visual processing streams engaged by the symbolic and nonsymbolic stimuli quickly di-
verge, and that this is reﬂected in differential patterns of community afﬁliation for primary
visual cortex as well as higher level visual regions, speciﬁcally observed here in the caudal
cuneus and lingual gyri, 分别. In a study of symbolic and nonsymbolic processing by
Holloway et al. (2010), a univariate contrast between formats found multiple occipital areas
more engaged by nonsymbolic stimuli, and this was after subtracting out activation associated
with control conditions that were matched in terms of total pixels. Our ﬁndings of allegiance
differences involving visual regions seem convergent, and could reﬂect engagement of ob-
ject location, individuation, and summation mechanisms thought to underlie nonsymbolic
number processing (Verguts & Fias, 2004). 的确, these stages are proposed to serve as the
inputs necessary for numerical magnitude representation in parietal cortex, and have been
shown to involve an occipitoparietal pathway including the inferior and middle occipital gyri
(Roggeman, Santens, Fias, & Verguts, 2011). Symbolic numerals may take an alternative route to
their associated magnitude representations, bypassing the multi-object normalization and ac-
cumulation mechanisms required to process dot sets (Bulthé et al., 2019; Santens, Roggeman,
Fias, & Verguts, 2010), and instead engage a ventral stream object-recognition pathway ex-
tending into the ITG (Hannagan, Amedi, 科恩, Dehaene-Lambertz, & 德阿内, 2015; 哈雷尔,
2016; Price & Devlin, 2011). We hesitate to make any mechanistic interpretations, but our data
at least indicate that the “position” of these visual regions with respect to community topology
is signiﬁcantly altered between formats. Taken together with the trend of greater allegiance
during symbolic trials between the Visual and DAN as well as FPN communities (Figure 3E
and F), our results build on prior univariate work, suggesting that number format inﬂuences
not only local activity but also the distributed interactions of occipital regions.

Robust Differences in Community Allegiance of the pITG and hIP3 Between Formats

A growing body of evidence suggests that number symbol processing engages a region in pos-
terior inferior temporal gyrus (pITG) that preferentially responds to digits compared to other
symbols (Grotheer, Herrmann, 等人。, 2016; Hermes et al., 2017; 波拉克 & Price, 2019; Shum
等人。, 2013). Given this literature, we hypothesized that if this region is preferentially engaged
during the recognition of Arabic numerals, it would show a greater tendency to join a func-
tional community involving parietal areas during the symbolic trials. Among the 202 地区
analyzed here, the right pITG (数字 5) was one of only ﬁve regions demonstrating robust dif-
ferences in its allegiance proﬁle between conditions (数字 4). Its afﬁliation with the symbolic
DAN community (数字 2 and Supporting Information Table S1) provides evidence that inter-
actions between the right pITG and superior parietal cortex and intraparietal sulcus (IPS) 是
particularly relevant for processing number symbols. This is in line with recent electrocorticog-
raphy and fMRI work showing coupling between lateral vOT and parietal areas during numeral
recognition and arithmetic tasks, including behavioral relevance of task-evoked coupling for
math ability and developmental increases over adolescence (Battista et al., 2018; Daitch et al.,
2016). Even at rest a similar pattern of robust connectivity is observed, speciﬁcally involving
pITG to IPS regions (Abboud et al., 2015; Nemmi, Schel, & Klingberg, 2018). Building on this
文学, we found that connectivity patterns of the pITG are differentiated in a task context,
and that selective coupling of the pITG with parietal regions during symbolic processing occurs
even while performing a simple magnitude comparison task. Our analysis further indicates that
the topological position of the pITG with respect to large-scale community organization is a
distinguishing feature between formats, suggesting the pITG plays a pivotal role in the sym-
bolic number processing network. The pITG to parietal coupling we observed here may reﬂect

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a mechanism by which visual number forms are “mapped” to magnitude representations in the
IPS. 或者, though not mutually exclusive, this interaction may represent a more general
process of recurrent exchange between neural populations in the vOT (tuned to learned sets
of visual features) and the DAN (involved in feature-based, attentional control), whereby pre-
sentation of a preferred stimulus (例如, a number symbol) elicits efﬁcient attentional capture,
allowing for identiﬁcation and further task-level processing. Such an argument has been made
for the role of the visual word form area in reading and why it shows similarly strong connec-
tivity to the DAN (A. C. Vogel et al., 2012; note the visual word form area is near the pITG
region described here, albeit in left vOT). Whatever the case, the present results, together with
other recent ﬁndings, suggest work looking into the mechanistic contributions of the pITG and
its functional interactions will be a fruitful avenue for future research in numerical cognition.

最后, we found that a region in the hIP3 subdivision of the left IPS demonstrated robust
differences in its allegiance proﬁle between formats. The IPS has been extensively implicated
in number processing tasks and thought to be involved in the representation of both sym-
bolic and nonsymbolic magnitude information (Ashkenazi, Henik, Ifergane, & Shelef, 2008;
Dehaene et al., 2003; Fias, Lammertyn, Reynvoet, Dupont, & Orban, 2003; Piazza et al.,
2004). Compared with the right IPS, the left IPS has been shown to be more sharply tuned to
Arabic numerals than to nonsymbolic arrays (Piazza, Pinel, Le Bihan, & 德阿内, 2007) 和
is also thought to be more affected by the ontogenetic process of learning number symbols
(Ansari, 2008; Cantlon et al., 2006; Rivera, Reiss, Eckert, & Menon, 2005; S. 乙. 沃格尔, Gofﬁn,
& Ansari, 2015). 有趣的是, Santens et al. (2010) reported a differential connectivity proﬁle
of the left IPS between symbolic and nonsymbolic processing, despite this region being equally
activated by both formats. Our ﬁndings are convergent, demonstrating that the left hIP3 takes
on a distinct pattern of community membership between formats, and that this distinction is
pronounced relative to other brain regions. This line of evidence indicates that, while both
formats may engage the left IPS, the functional pathways to and from this region are divergent
between symbolic and nonsymbolic processing, and highlights the additional level of insight
afforded by connectivity-based approaches. It is important to note that the analysis we have
performed here does not directly speak to the question of whether symbolic and nonsymbolic
number engage shared representations in the IPS, as has been a topic of intense study in recent
年 (例如, Bulthé, De Smedt, & Op de Beeck, 2015, 2018; Lyons, Ansari, & Beilock, 2015).
Our results instead provide complementary evidence that the whole-brain functional architec-
tures involving left hIP3 are different between formats, and suggest that a more comprehensive
understanding of hIP3 function during number processing will require a characterization of this
region’s role with respect to functional network dynamics. 此外, given previous reports
that symbolic processing in the left IPS changes over development, longitudinal studies look-
ing at the trajectory of this region’s connectivity proﬁle over learning may provide mechanistic
insights into the acquisition of symbolic number abilities.

Limitations

One limitation of this work is that our community-level analyses were conducted at a partic-
ular setting of the structural resolution parameter (γ = 2.45), which inﬂuences the size and
composition of the detected communities (例如, 参见方法) (Fortunato & Hric, 2016). 我们
determined this setting based on the heuristic of maximizing the balance across regions of
community stability and change between conditions (Mattar et al., 2015). The goal here was
to be sensitive to differences in community organization while avoiding an unrealistic scenario
in which a majority of regions change their community afﬁliation (IE。, assuming that there is

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some common functional architecture between formats). For other questions, such as “What
is the shared community structure across task states?,” using a heuristic where partitions are
maximally consistent between conditions may be more appropriate. At other settings of γ, 这
number of communities as well as regional allegiance proﬁles will change (例如, 参见方法),
and thus additional patterns of community-level differences between formats may have been
observed.

Another limitation to note is that the Brainnetome Atlas we used to deﬁne our nodes repre-
sents a parcellation of the brain at particular spatial scale (n = 246 地区) and was derived
based on structural-connectivity patterns. We chose this atlas because it has been shown to
capture areas with relatively homogenous functional connectivity proﬁles (Fan et al., 2016).
然而, node deﬁnitions at different spatial scales or based on different properties, 对于前任-
充足, anatomical, architectonic, or random subdivisions, may demonstrate different patterns
from what we observed (假如, 扎莱斯基, & Breakspear, 2013; 斯波恩斯, 2014; Zalesky et al.,
2010).

结论

总之, in this study we adopted a data-driven, network neuroscience approach to charac-
terize the whole-brain functional architecture supporting symbolic and nonsymbolic number
比较. Our ﬁndings provide several novel insights. A uniﬁed fronto-parietal control
network in the nonsymbolic condition was fractionated into FPN and DAN communities during
symbolic number processing. We found evidence for stronger allegiance among a left-lateralized
Auditory network in the symbolic condition, potentially reﬂecting access to phonological rep-
resentations of number word forms. The right pITG, proximal to the putative “Number Form
Area,” joined the DAN system in the symbolic condition only, supporting its proposed role in
the processing of number symbols. Both the right pITG and the hIP3 subdivision of the left
IPS demonstrated robust differences in their community allegiance proﬁles between formats.
合在一起, these results reveal a pattern of overlapping and distinct network architectures
for symbolic and nonsymbolic number processing, and highlight the additional level of insight
provided by analysis of functional network topology.

方法

参加者

Forty neurologically healthy, right-handed subjects participated in the study for undergraduate
course credit. Of those recruited, 6 participants were excluded from analyses due to high levels
of motion and 1 was excluded for signiﬁcant signal dropout (exclusion criteria detailed below),
resulting in a ﬁnal sample of 33 参与者 (19.5
0.9 年, 20 女性). All subjects had
normal or corrected-to-normal vision. Informed consent was obtained from each subject in
accordance with the Vanderbilt University Institutional Review Board policy.

fMRI Tasks

Participants completed two consecutive runs of an event-related number comparison paradigm.
For each trial, participants judged whether an Arabic digit (象征性的) or dot array (nonsym-
bolic) was less or more than ﬁve by pressing a button with either their right index or right
middle ﬁnger, 分别, as quickly and accurately as possible. A total of 160 试验是
提出, composed of 80 symbolic trials and 80 nonsymbolic trials that were intermixed and
pseudo-randomly ordered (IE。, 不多于 3 consecutive trials were of the same number
or format). Stimulus numerosity was either 2, 4, 6, 或者 8, which occurred at equal probability
and was counterbalanced across formats. Stimuli were created using the MATLAB package

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ﬁrst described by Piazza et al. (2004). Nonsymbolic stimuli were controlled for total surface
area across numerosities by reducing dot size with increasing numerosity. 此外, stim-
uli were controlled for total occupied area and luminance across formats (IE。, 一般, dot
sets contained the same number of pixels as numerals) in an effort to control for nonnumerical
visual parameters across trials. Stimulus duration was 500 ms and interstimulus intervals (ISI)
ranged from 3,300–7,300 ms, in 1,000-ms increments, with an average of 5,300 多发性硬化症. ISI was
counterbalanced across numerosities and formats.

Imaging

240

Imaging was performed using a 7 Tesla (7时间) Philips Achieva scanner with a 32-channel head
coil. An MP2RAGE (Marques et al., 2010) image was acquired for anatomical reference,
aligned to the anterior/posterior commissures, with the following parameters: TR = 4.315 多发性硬化症,
= 1.92 多发性硬化症, ﬂip angle = 7, 240 coronal slices, voxel size = 1 mm3, imaging matrix = 240
192, acquisition time = 1,010 s. These images were corrected for B1-ﬁeld inhomo-
×
geneities, as well as proton density and T2* effects according to the procedure described by
Marques et al. (2010). For the event-related experiment, functional T2*-weighted images were
acquired over two runs of 243 volumes each, with the following parameters: TR = 2, 000 多发性硬化症,
= 25 多发性硬化症, ﬂip angle = 63, 46 axial slices (with no interslice gap), voxel size = 2.5 mm3,
46, acquisition time = 500 s per run (16 米 40 s of functional data
imaging matrix = 96
全部的).

预处理

fMRI data was preprocessed in AFNI (考克斯, 1996) using the afni_proc.py program. Preprocess-
ing involved slice-time and motion correction, coregistration, normalization to the MNI152
2009c template using afﬁne registration, smoothing with a 4-mm kernel, and scaling. All trans-
formations of the raw functional data were concatenated and applied in one step to reduce
multiple resampling and interpolation errors. Head-motion/outlier censoring was performed by
removing volumes from the activation model (described below) that demonstrated between-
volume movement of >0.3-mm Euclidean norm distance or if >5% of voxels in a volume
(within a brain mask) were determined to be outliers. Across our ﬁnal sample, the percentage
of censored volumes was 4.0

4.1%.

Beta-Series Estimation

To assess task-related functional connectivity, we used a beta-series correlation method (BSC)
(Rissman et al., 2004). This method allows for estimation of condition-speciﬁc, task-evoked
connectivity by looking at correlated ﬂuctuations in coactivity patterns across multiple stim-
ulus events. Compared with an alternative, commonly used methodology of standard or gen-
eralized psychophysiological interaction analysis (PPI/gPPI) (Friston et al., 1997; McLaren,
Ries, 徐, & 约翰逊, 2012), BSC has been shown to be more powerful for detecting task-
evoked connectivity in event-related designs that feature many trials, short stimulus durations,
and short ISIs, such as we have implemented here (Cisler, 衬套, & Steele, 2014). To com-
pute trial-wise beta (IE。, activation) 地图, we employed the “least-squares separate” (LS-S)
method proposed by Mumford et al. (2012) using AFNI’s 3dLSS function (Geib, 斯坦利,
丹尼斯, Woldorff, & Cabeza, 2017; Mumford Turner, Ashby, & Poldrack, 2012). 比较的
with the original implementation of BSC in which each trial is included as a separate regressor
in a single general linear model (GLM), the LS-S method includes a regressor for the trial of inter-
est and then a nuisance regressor with all other trials of interest combined. This method is more
computationally efﬁcient and, 重要的是, was found to produce more accurate estimates of

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trial-speciﬁc activations in simulations (Mumford Turner, Ashby, & Poldrack, 2012). Also in-
cluded in these subject-level GLMs were the following nuisance regressors: 6 motion param-
埃特斯, 6 motion derivatives, 0th- to 4th-order Legendre polynomials to model low-frequency
drifts (per run), the ﬁrst ﬁve principal components in white matter voxels, and ﬁrst ﬁve princi-
pal components in the lateral ventricles. Tissue masks were created using the Computational
in SPM12 (https://www.ﬁl.ion.ucl.ac.uk/spm/software/spm12/)
Anatomy Toolbox (CAT12)
(Gaser & Dahnke, 2016). The ventricle region of interest (ROI) was created by masking the
cerebrospinal ﬂuid segmentation with the lateral ventricle regions from the Neuromorphome-
tric Atlas included in CAT12. The inclusion of these tissue-based principal component regres-
索尔斯, known as the aCompCor method, was intended to account for nonneural physiological
ﬂuctuations as well as residual motion-related noise and has been shown to improve con-
nectivity estimates in resting-state studies (Caballero-Gaudes & Reynolds, 2017; Ciric et al.,
2017; Muschelli et al., 2014). The preprocessed 4D functional data (after smoothing and scal-
英) served as the inputs into the 3dLSS function for beta-series estimation. Censoring was
32 volumes (IE。, obser-
performed at this stage and resulted in GLMs with a mean of 933
vations) 在......之外 972 collected volumes. These models included trials/volumes from a separate
identiﬁcation task (not analyzed here) performed at the beginning of the scanning session. 这
inclusion of all acquired data into one model ensured more accurate estimates of nuisance
effects. For each trial a separate GLM was constructed that included a total of 42 nuisance
regressors plus a regressor for the trial of interest and one for the sum of all other trial re-
响应. The output of this process involved 3D beta maps for every trial indicating the model
coefﬁcient at each voxel, serving as trial-level activation maps that could then be sorted by
formats.

Within-Format Normalization

To ensure that differences in voxel-wise activity levels and/or variance between formats did
not confound our estimates of connectivity, prior to extraction of average beta series from
ROIs, we implemented a within-format normalization procedure in which the beta maps were
ﬁrst separated by format and concatenated, then each voxel-wise beta series was normalized
by subtracting the voxel-wise mean and dividing by the voxel-wise standard deviation. 这
procedure was adapted from the multivariate pattern analysis literature, in which normalization
of beta values is commonly employed prior to classiﬁer training (Misaki, Kim, Bandettini, &
Kriegeskorte, 2010). With this procedure, we could be more certain our connectivity estimates
were derived from covariance in the pattern of trial-level activation, rather than biased by the
magnitude of activations, 为他自己.

Beta-Series Extraction and Scrubbing

We extracted the average beta series in each of 202 cortical and subcortical ROIs from the
Brainnetome Atlas (Fan et al., 2016), which represented the reduced set of ROIs (起初
246) after accounting for signal dropout (数字 6; see Supporting Information Methods). 我们
chose the Brainnetome Atlas as it a connectivity-based parcellation which provides a bio-
logically plausible set of ﬁne-grained regions in both cortex and subcortex (Fan et al., 2016).
Since censoring (IE。, exclusion of high-motion/outlier volumes) was performed in subject-level
GLMs, some trials’ convolved response contained fewer volumes in the ﬁnal model. 注意
the hemodynamic response to a single trial lasted approximately 13.1 s and spanned either
six or seven volumes of data at our 2-s sampling rate (TR). We expected less reliable beta
estimates for trials in which the response proﬁle contained censored volumes. 所以, 我们
implemented a “beta-scrubbing” approach similar to Ray et al. (2017), in which we excluded
trials during which two or more volumes were censored (IE。, the associated beta was removed

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数字 6. Functional connectivity and consensus clustering pipeline. The data processing pipeline is depicted, starting from trial-level beta
map estimation to ﬁnal group-level allegiance matrices and community assignments. Gray arrows indicate the stepwise progression from one
stage of processing to the next. All plots portray representative data from the analysis.

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from the series before running correlations). Across the original sample of 40 subjects, 这
median number of retained trials was 147 (91.9%). We determined six subjects were outliers
in the number of retained trials based on a threshold of 3.5
the median absolute deviation
(MAD), a metric for robust outlier detection (Leys, Ley, 克莱因, Bernard, & Licata, 2013). 这些
subjects thus demonstrated a signiﬁcant degree of motion compared with the rest of the sample
and were excluded from further analyses, along with one additional subject who demonstrated
signiﬁcant signal dropout (see Supporting Information Methods). In the ﬁnal sample of 33 子-
项目, 在......之外 80 trials presented for each format, there was an average of 74.2
±
7.3%) trials used for symbolic and 74.6
6.4%) used for nonsymbolic. A paired
5.2 (93.2
t test indicated the number of retained trials was not signiﬁcantly different between formats
(t(32) =

1.08, p = 0.29)

5.8 (92.8

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Network topology of symbolic and nonsymbolic number comparison

Functional Connectivity

To construct subject-level connectivity matrices, we computed the Pearson correlation be-
tween all region pairs by using the average beta-series vector from each region, yielding a
202 matrix that was symmetric about the diagonal (数字 6). Matrices were constructed
202
for symbolic and nonsymbolic trials separately. Correlation coefﬁcients were Fisher Z trans-
3 ), where N is the
formed and converted to z scores via dividing by the standard error (
number of observations (IE。, betas) retained after the scrubbing procedure (Fisher, 1915). 这
z-score matrices served as the ﬁnal subject-level matrices for network analyses.

1
√N

Modularity Maximization and Consensus Clustering

To characterize network topology during the symbolic and nonsymbolic conditions, we em-
ployed the graph-theoretical approach of modularity maximization. This data-driven cluster-
ing procedure attempts to partition a network into nonoverlapping communities where each
node is assigned to exactly one community, such that there is high within-community and low
between-community connectivity (纽曼, 2006). The modularity quality function quanti-
ﬁes the degree to which this is achieved given a partition of matrix w. Speciﬁcally, we em-
ployed a generalized Louvain algorithm (Blondel, Guillaume, 兰比奥特, & Lefebvre, 2008)
that maximized a variant of the standard Newman–Girvan quality function, termed Q*, 哪个
incorporates an asymmetric treatment of negative weights, as proposed by Rubinov & 斯波恩斯
(2011; Dwyer et al., 2014). This modiﬁed treatment of negative weights is recommended for
functional networks because of differing neurobiological signiﬁcance of positive and negative
correlations in BOLD timeseries (for further discussion, see Rubinov & 斯波恩斯, 2011) 并且是
calculated as follows:

Q∗ =

1
v+ ∑ij (w+

ij −

γe+

ij )δMi Mj −

1
v+ + v−

∑ij (w−ij −

γe−ij )δMi Mj

where each brain region i is assigned to community Mi and region j is assigned to community
Mj; δMi Mj = 1 if Mi = Mj and equals 0 否则; e±ij is the within-community connection
strength expected by chance for either positive or negative weights; v+ and v− denote the sum
of all positive and negative weights in the network, 分别; 和, γ is the structural resolu-
tion parameter, which inﬂuences the size of the detected communities (Reichardt & Bornholdt,
2006) (see below for details of a data-driven approach to choosing a value for this parameter).

The Louvain algorithm uses a two-step procedure in which ﬁrst every node is assigned to its
own community, and iterative moves are made assigning a node to its neighboring community
until a maximum Q* is reached. In a second step, the resulting communities are combined
into a smaller network that sums the node weights within each community, and this simpliﬁed
network gets submitted back to the ﬁrst step for a second pass, 等等, until no changes
can be made in step 1 to improve Q* (Blondel et al., 2008). Critically, the resulting community
structure is dependent on the random order of moves considered at each pass. 原则,
there is some “optimal” partition of the network, 那是, a partition or set of partitions exists that
returns the highest possible Q* for that matrix. 在实践中, the optimal partition of a given net-
work is unknown and multiple iterations of modularity maximization result in distinct, 尽管
often similar, solutions, referred to as the so-called degeneracy of modular solutions (好的,
De Montjoye, & Clauset, 2010). Degeneracy can be leveraged through the use of consensus
clustering methods, which aggregate across many partitions of the same network to deﬁne a
consensus partition (Lancichinetti & Fortunato, 2012). We implemented a multistep approach
that involved modularity maximization and consensus clustering at the subject level, then con-
sensus clustering at the group level (数字 6).

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Network topology of symbolic and nonsymbolic number comparison

Each fully weighted, subject-level matrix was ﬁrst partitioned using the generalized com-
munity Louvain algorithm (Blondel et al., 2008) (via the community_louvain.m function of the
Brain Connectivity Toolbox (BCT), http://www.brain-connectivity-toolbox.net, with the “neg-
ative_asym” ﬂag). The algorithm was run 103 次, with each iteration resulting in a single
partition of the 202 regions into communities (IE。, a vector of values indicating the commu-
nity assignment for each region). From these partitions, A 202
202 allegiance matrix was
constructed which represents, at each cell, the percentage of iterations in which two regions
were assigned to the same community (Bassett et al., 2015) (via the agreement_weighted.m
function of the BCT). The subject-level allegiance matrix was then further clustered using a
consensus clustering procedure to obtain a ﬁnal, consensus partition of regional communities
(Lancichinetti & Fortunato, 2012) (via the consensus_und.m function of the BCT; note the “tau”
parameter was set to 0.5 for all analyses herein, which thresholds the allegiance matrix to re-
move elements weaker than 0.5, IE。, when there is less than 50% agreement across partitions).
The ﬁnal subject-level partitions were then used to create a group-level allegiance matrix that
代表, at each cell, the percentage of subjects in which two regions were assigned to
the same community. The same consensus clustering procedure was then used to determine
the ﬁnal group-level community assignments (数字 6). Because assigned community labels
are arbitrary, labels were maximally matched between the ﬁnal partitions of the symbolic and
nonsymbolic conditions based on community overlap (via the pair_labeling.m function of the
Network Community Toolbox, http://commdetect.weebly.com). As a ﬁnal step, permutation
testing was performed to reduce the set of communities to only those demonstrating signiﬁ-
cance at the subject level (see Supporting Information Methods).

Deﬁning the Structural Resolution Parameter

When applying modularity maximization, an important consideration is the deﬁnition of the
free parameter, C, referred to as the structural resolution parameter. The effect of γ is such
that lower values result in fewer and larger communities, and higher values result in more and
smaller communities, which provides some constraint on the topological scale of the resulting
partitions. We fully sampled this parameter space in Figure 7A and B, varying γ from 0.05 到 15.
At the highest settings, the algorithm discovered as many communities as there were regions
(IE。, all singletons, which refers to regions that are assigned a community unto themselves).
To further reﬁne this space to a range of interest, we determined the point at which there was
relatively little change in community structure. We used a sliding window approach in which
we calculated average normalized variation of information, a measure of partition distance,
over community partitions within a window size of 0.35 (Meilˇa, 2007). This procedure revealed
that at approximately γ = 5, there was little change in community structure, suggesting that a
stable set of strongly connected (nonsingleton) communities existed at this topological scale,
and that beyond this point, nodes from these communities progressively became singletons
and no new community organization was discovered (Figure 7B and C). We therefore adopted
the range from γ = 0.05 到 5 as our resolution range of interest for all subsequent analyses.

With the range of interest deﬁned, we still needed to determine the ideal setting of γ for
the community-level analysis (note that for the global- and region-level analyses, we present
the results across the full range of interest; 人物 1 和 4). Though setting γ to an arbitrary
value of 1 is common practice, this is not necessarily optimal. A more principled approach
involves “sweeping” across parameter space and using some heuristic to determine the op-
timal value, 例如, where community deﬁnitions are most consistent across repeated
迭代 (贝策尔 & Bassett, 2016; Betzel et al., 2017; Gu et al., 2015). Since we were inter-
ested here in a comparison of network architecture between the symbolic and nonsymbolic

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数字 7. Resolution parameter sweep. (A and B) Results of a parameter “sweep,” in which the structural resolution parameter (C) was varied
从 0 到 15 in increments of 0.05, 和 100 iterations of the multistep consensus clustering approach at each level of γ. The mean for each
format is plotted in red and blue, with shaded regions indicating the SD across iterations. (C) Mean partition distance over sliding window
3 脚步 (window size = 0.35 C) is plotted for each format, measured using the normalized variation of information (Meilˇa, 2007). 这
的
plot indexes the change in community structure over incremental steps in γ, highlighting the point in parameter space in which no new scale
5. We used this result to deﬁne the range of interest for our analyses of format differences
of community organization is discovered, at γ
across resolutions (IE。, γ = 0.05–5). (D) The red dashed line indicates the γ at which the variability (标清) of ﬂexibility was maximal, 和
ﬂexibility deﬁned as the number of regions that changed their community assignment between the symbolic and nonsymbolic conditions.
The maximum SD would result from half of all regions changing community assignments and the other half maintaining their assignment. 这
distribution at each gamma value represents the result of all pairwise comparisons of the 100 group-level partitions generated for each format
at that step. The mean of this distribution (black circle) was used to determine the optimal gamma setting for further analyses (IE。, γ = 2.45).

≈

状况, we wanted to conduct our investigation at a resolution that would be sensitive
to both similarities and differences between formats. To achieve this, we employed a heuris-
tic, used previously by Mattar et al. (2015), where the optimal γ value is that which maxi-
mizes variability in the ﬂexibility coefﬁcient across regions. Flexibility is deﬁned simply as the

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Network topology of symbolic and nonsymbolic number comparison

proportion of (任务) conditions in which a region changes its assigned community (calculated
via the ﬂexibility.m function of the Network Community Toolbox). With only two conditions,
a region’s ﬂexibility can be 1 或者 0, and the variability of ﬂexibility is the standard deviation of
the vector of 1’s and 0’s (one value for each of the 202 地区). By maximizing the variability of
ﬂexibility across brain regions, we are looking for a γ at which there is a balance between sta-
ble and changing community assignments between conditions (IE。, is maximal if exactly half
of regions change their assignment between conditions). 重要的, the procedure does not
ensure the differences are signiﬁcant within subjects (which we assess via permutation testing),
just that some differences and some similarities exist between the ﬁnal group-level partitions
of each condition. This approach is thus orthogonal to the prospect of ﬁnding reliable within-
subject effects. To determine the optimal γ setting for further investigation, we performed the
multistep group partitioning procedure 100 times at each value of γ, 范围从 0.05 到 5
in increments of 0.05 (for this analysis, subject-level matrices were partitioned 100 times with
community_louvain.m). For each iteration, we calculated the ﬂexibility of nodal assignments
between formats in the ﬁnal group-level partitions (Figure 7D). At low levels of γ there are
fewer communities (Figure 7A) and the region assignments are stable between formats, 因此
resulting in many regions with a ﬂexibility of 0, 那是, low variability across regions. At high
levels of γ, there are many communities, including an increasing number of singletons. 后
2.8, the number of nonsingleton communities detected drops, due to the fact that
γ values
a few previously detected communities break apart into singleton communities. 最终,
we found a maximum in the mean variability (标清) of ﬂexibility at an intermediate value of
γ = 2.45. The subsequent community-level analyses involved modularity maximization using
this value.

≈

致谢

We thank all the volunteers who participated in this study.

SUPPORTING INFORMATION

Supporting Information for this article is available at https://doi.org/10.1162/netn_a_00144.
Data and code for this project have been made publicly available at https://osf.io/sb5v2/ and
https://github.com/conradbn/CR7T_Connectivity, 分别 (Conrad et al., 2020A, 2020乙).

作者贡献

Benjamin N Conrad: 概念化; 数据管理; 形式分析; 方法; Soft-
器皿; 可视化; Writing – Original Draft; Writing – 审查 & Editing. Eric D Wilkey:
概念化; 数据管理; 资金获取; 调查; 方法; 项目
行政; 资源; Writing – 审查 & Editing. Darren J Yeo: 概念化; 数据
curation; 调查; 方法; Writing – 审查 & Editing. Gavin R Price: Conceptual-
化; 数据管理; 资金获取; 方法; 项目管理; 资源;
监督; Writing – 审查 & Editing.

资金信息

Gavin R. Price, Peabody College Small Research Grant. Gavin R. Price, National Science Foun-
日期 (美国国家科学基金会) Division of Research on Learning in Formal and Informal Settings (http://dx.doi.org/
10.13039/100000173), 奖项ID: 1660816. Gavin R. Price, National Science Foundation
(美国国家科学基金会) Division of Research on Learning in Formal and Informal Settings (http://dx.doi.org/10.

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Network topology of symbolic and nonsymbolic number comparison

13039/100000173), 奖项ID: 1750213. 埃里克·D. Wilkey is the recipient of a BrainsCAN Post-
doctoral Fellowship at Western University, funded by the Canada First Research Excellence
基金 (CFREF) and Banting Postdoctoral Fellowship funded by the Natural Sciences and Engi-
neering Research Council (NSERC) of Canada. Darren J. Yeo is supported by the Humanities,
艺术, and Social Sciences International PhD Scholarship, co-funded by Nanyang Technological
University and the Government of Singapore: Ministry of Education.

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