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Edited by: Philippe Chassy, Liverpool Hope University, UK

Reviewed by: Robert Whelan, University College Dublin, Ireland; Michael Anthony Keane, Dublin City University, Ireland

This article was submitted to the journal Frontiers in Human Neuroscience.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

As enhanced fronto-parietal network has been suggested to support reasoning ability of math-gifted adolescents, the main goal of this EEG source analysis is to investigate the temporal binding of the gamma-band (30–60 Hz) synchronization between frontal and parietal cortices in adolescents with exceptional mathematical ability, including the functional connectivity of gamma neurocognitive network, the temporal dynamics of fronto-parietal network (phase-locking durations and network lability in time domain), and the self-organized criticality of synchronizing oscillation. Compared with the average-ability subjects, the math-gifted adolescents show a highly integrated fronto-parietal network due to distant gamma phase-locking oscillations, which is indicated by lower modularity of the global network topology, more “connector bridges” between the frontal and parietal cortices and less “connector hubs” in the sensorimotor cortex. The time domain analysis finds that, while maintaining more stable phase dynamics of the fronto-parietal coupling, the math-gifted adolescents are characterized by more extensive fronto-parietal connection reconfiguration. The results from sample fitting in the power-law model further find that the phase-locking durations in the math-gifted brain abides by a wider interval of the power-law distribution. This phase-lock distribution mechanism could represent a relatively optimized pattern for the functional binding of frontal–parietal network, which underlies stable fronto-parietal connectivity and increases flexibility of timely network reconfiguration.

In the fields of education and psychology, exceptional logical reasoning and visual-spatial imagery abilities are regarded as the main characteristics of mathematically gifted adolescents. Numerous neuroscience studies have reached an agreement that the gifted mathematical thinking abilities are supported by a cooperative fronto-parietal network (O’Boyle et al.,

The parieto-frontal integration theory (P-FIT) on individual differences in reasoning competence emphasizes the crucial process of information communication between association cortices within the parietal and frontal brain regions (Jung and Haier,

On the other hand, the network with dynamic binding not only depends on the transient coupling between neural assembles, but also requires the timely reconfiguration of connections to adapt to external stimuli and inner perturbation. As a representation of functional coupling strength between adjacent or distant brain areas, the synchronization between neuronal assembles is actually operated in a metastable dynamic system (Werner,

Through EEG source analysis of the gamma cortical network, the present study aims to find the giftedness-related capacity of functional binding in the crucial fronto-parietal network of reasoning, by assessing the task-related functional connectivity and adaptive network reconfiguration. The study first compared the basic cortical network topologies constituted by gamma phase-locking oscillations in math-gifted and average-ability adolescents while they were performing a deductive reasoning task. Furthermore, at a neural-mechanistic level of analysis, the study investigated the temporal dynamics of the fronto-parietal network, including the phase-locking intervals/durations (PLI) and the lability of fronto-parietal network reorganization. Then, the parameter fitting of the PLIs in the power-law model was conducted to assess the criticality of phase-locking durations, which could construct an association between the functional connectivity and adaptive reconfiguration of fronto-parietal network. After that, the relationship among the enhanced fronto-parietal connectivity, the extensive reorganization of fronto-parietal connections, and the high criticality of PLIs in the math-gifted brain was analyzed and discussed.

Two groups of subjects were enrolled in this study. The math-gifted group included 11 adolescents (eight males and three females) aged 15–18 years (mean ± SD: 16.3 ± 0.6), who were from the Science and Engineering Experimental Class at Southeast University (Nanjing, China). The class was composed of adolescents who had been recruited through a special college entrance examination aiming at gifted students under 15 years old with exceptional abilities in mathematics and natural sciences. Three criteria were employed to select math-gifted subjects from the class according to the definition of “school giftedness” (Renzulli,

The exclusion criteria adopted included left handedness, medical, neurological or psychiatric illness, and history of brain injury or surgery. To avoid the long-term training effect on the human brain activity, students who had received special training course of Mathematical Olympiad were excluded from this experiment. All the subjects were given informed consent and the study was approved by the Academic Committee of the Research Center for Learning Science, Southeast University, China. The subjects received financial compensation for their participation.

As the essential mathematical skill and the standard type of deductive reasoning, a categorical syllogism task of analytic type (verbal–logical way) was adopted in this study. Categorical syllogism is constituted by a major premise, a minor premise, and a conclusion. The actual reasoning process has been considered to emerge during the presentation of the minor premise and remain active until the validation of the conclusion (Fangmeier et al.,

The syllogistic sentences without specific content include three basic items: “S,” “M,” and “P.” “M” is the medium item and is presented in both the major premise and the minor premise. “S” and “M” constitute the major premise, and “M” and “P” the minor premise. From the two premises, the inferred relationship between “S” and “P” forms the conclusion (Figures

The experiment adopted a three-block paradigm, included a valid block (32 trials), an invalid block (32 trials), and a baseline block (40 trials). The combinations of syllogistic sentences following the true logical rules constituted a valid block, which employed the logic expressions proposed by Evans et al. (

The trials of the three blocks were presented in a random order, which was performed by the E-Prime 2.0 experimental procedure. The stimuli presentations of all the trials took about 30 min. The major premise, minor premise, and conclusion were presented sequentially along the timeline (Figure

Before the formal experiment, a practice session including five trials was conducted by each subject. After that, they decided whether to practice again or enter the following formal procedure. The sentences included Chinese characters and English letters, which were white on black background to avoid visual fatigue.

The EEG data were recorded using the Neuroscan system at sampling rate 1000 Hz, with 60 scalp electrodes placed according to the international 10–20 system (Figure

By using the Scan 4.3 data preprocessing software, the continuous EEG signals with correct responses were band-pass filtered between 1 and 100 Hz. The epoch of each trial was extracted using a time window of 9500 ms (500 ms pre-stimulus and 9000 ms post-stimulus), and was baseline-corrected according to the pre-stimulus time interval. Ocular artifacts were removed according to the simultaneously recorded EOG signals. After the artifact rejection with the thresholds ranging from 50 to 75 μV, the blink and electrocardiogram noises were excluded. Finally, 18–22 trials were retained for each math-gifted subject and 15–25 trials were retained for each control subject. In addition, the independent component analysis (ICA) in the EEGLAB Toolbox was used to further clear the visible artifacts, such as the components of possible ocular and muscle movements. Since the emergence of the minor premise in the syllogistic sentence was viewed as the beginning of the actual reasoning process, the time interval 3000–9000 ms (presentation time of the minor premise and conclusion) of the artifact-free EEG signal was selected as the event-related time window. Because of the individual differences in response speed and completion time of each trial, the interval 4000–8000 ms was further extracted as the time window for data analysis.

Task-induced response at the gamma frequency of the human brain activity was first assessed in each EEG channel by calculating event-related synchronization/desynchronization (ERS/ERD), which was expressed as the percentage of power increase/decrease relative to the baseline resting state:

There is a limitation that the EEG-based brain connectivity analysis was influenced by the volume conduction, which was caused by the variation of the electrical conductivity among the different head layers (Langer et al.,

To quantify the strength of connectivity, the cortical currents were followed by a phase-locking value (PLV) calculation between each pair of the nodes. PLV is a representative method of PS through obtaining a statistical quantification of the frequency-specific synchronization strength between two neuroelectric signals (Lachaux et al., _{x}_{y}_{x}_{y}_{x}_{y}

After calculating the PLV matrix of size 248 × 248 for all the cortical vertices, a fixed connection density was employed to acquire the adjacency matrix. The connection density of the network was set to

In the following definitions of the graph-theoretical measures based on an adjacency matrix [_{i}_{,}_{j}_{i}_{,}_{j}_{i}_{,}_{j}

_{uv}

_{i}_{ij}_{i}_{→}_{j}_{ij}

_{i}_{i}

_{hj}_{hj}

_{hk}_{hk}

Since PLV is the temporal statistic of the intermittent phase-locking durations in a specified time interval, the PLIs between frontal and parietal cortical signals were extracted to further quantify the distribution characteristic of the continuous synchronizations. PLI is the period of time when two oscillators maintain the synchronization activity in their phase difference within a limited range, i.e., Δ_{xy}_{x}_{y}

On the other hand, to measure the coordinated change of functional coupling states of the synchronizing network during reasoning task, the fronto-parietal lability was calculated in the selected nodes ranging from frontal, sensorimotor to parietal cortices. The lability is quantified by the total number of phase-locking pairs of signals in a dynamic network that can change over time. The number of signal pairs that were phase-locked at any time points can be acquired according to the following preset condition of phase difference:

The lability of a synchronizing network is defined as
^{2}(

For all the trials, the scattergrams were constituted by the samples with the feature distribution of mean fronto-parietal PLI and mean lability of fronto-parietal network in 10, 15, 20, and 25 ms time intervals. Linear discriminant analysis (LDA) (Webb,

To construct an association between PLI and functional reorganization of network, critical dynamics of the fronto-parietal synchronization is assessed by fitting the PLIs in the “power-law” model. The PLI distributed in a critical interval indicates that a “metastable” synchronization is in effect, which implies the synchronizing state would access “neuronal avalanche” and adaptive reorganization by synaptic interaction in the face of endogenous perturbation and external event (Werner,

Playing the role of functional integration between posterior parietal and frontal cortices in reasoning (Jung and Haier,

In this study, the parameter fitting method proposed by Clauset et al. was applied to the PLIs set. The method has been proven valid on various datasets from the natural phenomenon with power-law distribution characteristic (Clauset et al., _{min} can fit in the power-law distribution with less bias. While the data are drawn from a distribution that follows a power-law exactly for _{min}, the scaling parameter _{min} = 1, the maximum likelihood estimator (MLE) used for appropriate estimation of _{min} > 1, the zeta function is replaced by the generalized zeta _{min}, _{min}, and _{min}. The optimal estimation of _{min} is the one that gives the minimum value of _{i}

The single-trial analysis results obtained from 215 samples of the math-gifted group and 252 samples of the control group were examined statistically using the one-way analysis of variance (ANOVA) in the Matlab Statistics Toolbox, with group (gifted/control subjects) serving as the between-subjects factor. At the nodal level of the graph-theoretical analysis, clustering coefficient and node betweenness centrality of each cortical vertex were statically tested by the one-way ANOVA. Moreover, edge betweenness centrality was tested as well for 30 × 30 links connecting frontal–parietal nodes. The Bonferroni Corrections were used in the multiple statistical tests, with significance level set to 0.05. At the global level of the functional network, the ANOVA was conducted on modularity and characteristic path length, respectively. Additionally, the relevant fitting parameters of PLIs in the power-law model from the single-trial analytical results were statistically compared between the two groups. For the behavioral data, the AVOVA tests were used to identify the group difference in task performances in terms of accuracy and response time.

In the deductive reasoning task, the math-gifted group has outperformed the control group in average response accuracy (mean ± SD: 75.14 ± 12.58% in the math-gifted group and 68.20 ± 15.29% in the control group). Regarding the reaction time of correct response, significant group difference (

The ERS/ERD based brain topological maps show that the gamma-band response induced by the deductive reasoning task is mainly distributed in the prefrontal, frontal, sensorimotor, parietal, and occipital regions. The math-gifted group in particular has higher gamma-band ERS in the central sensorimotor regions as compared with the average-ability subjects (Figure

From the graph-theoretical analysis results of the gamma cortical network, the basic neurocognitive network topologies of the two groups are primarily composed of the prefrontal, frontotemporal, parietal, occipital, and fronto-parietal modules. With the same connection density employed in the two groups, the gamma synchronization network in the math-gifted group shows an expanded fronto-parietal module that integrates more cortical vertices in frontal, parietal, and sensorimotor regions and the relatively shrinking frontotemporal modules, by using the Louvain method for functional community detection (Blondel et al.,

In the math-gifted brain, the expanded fronto-parietal functional module and enhanced connectivity of the frontal–parietal network are associated with the emergence of more connections between structurally separated frontal and parietal cortical vertices. The ANOVA results indicate that some frontal–parietal links show significantly higher edge betweenness centrality in the cortical network of the math-gifted subjects (adjusted

Besides, the ANOVA analysis of the global network further demonstrates that the math-gifted adolescents have significantly lower modularity in the global network topology as compared to the average-ability subjects (Figure

Source | SS | df | MS | |||
---|---|---|---|---|---|---|

Modularity | Group | 0.0131 | 1 | 0.0131 | 11.09 | |

Error | 0.5486 | 465 | 0.0012 | |||

Total | 0.56169 | 466 | ||||

Characteristic path length | Group | 0.0852 | 1 | 0.0852 | 3.91 | |

Error | 10.1389 | 465 | 0.0218 | |||

Total | 10.2241 | 466 |

From the result of PLI analysis, the increased inter-module connections of fronto-parietal network can be attributed to stable phase dynamics of synchronization oscillation between distant brain regions (Thatcher et al.,

Although too long phase-locking duration has been surmised to lead to the lack of flexibility of neural activity (Thatcher et al.,

The coordination relationship in functional binding of fronto-parietal network can be explained by the power-law distribution of PLIs. Based on a plenty of PLI samples from the trial concatenation for each subject (the sample size ^{6}) (Table _{e} < 0.5%), which is manifested as an exponential fall-off. It is notable that the obvious difference between the two groups is presented in the distribution tail that represents large but rare synchronization and critical behavior as well (Clauset et al.,

_{max}, maximum PLI; _{tail} = [_{max}]; _{e}, standard deviation of estimated values

_{max} |
_{tail} |
_{e} (×10^{−2}) |
|||||
---|---|---|---|---|---|---|---|

01 | 311242 | 20 | 316 | 36 | 2.89 | 280 | 0.31 |

02 | 316268 | 21 | 438 | 37 | 2.83 | 401 | 0.26 |

03 | 325099 | 22 | 332 | 37 | 2.86 | 295 | 0.27 |

04 | 205294 | 23 | 376 | 33 | 2.87 | 343 | 0.33 |

05 | 335295 | 21 | 355 | 37 | 2.80 | 318 | 0.23 |

06 | 201462 | 20 | 347 | 35 | 2.82 | 312 | 0.25 |

07 | 268759 | 19 | 334 | 33 | 2.92 | 301 | 0.29 |

08 | 260084 | 20 | 265 | 36 | 2.9 | 229 | 0.29 |

09 | 334978 | 21 | 432 | 35 | 2.92 | 397 | 0.30 |

10 | 390596 | 21 | 456 | 37 | 2.86 | 419 | 0.21 |

11 | 568511 | 21 | 401 | 36 | 2.85 | 365 | 0.28 |

Mean value | 319781 | 21 | 368 | 37 | 2.86 | 333 | 0.27 |

01 | 318080 | 18 | 258 | 31 | 2.90 | 227 | 0.32 |

02 | 322137 | 19 | 296 | 35 | 2.94 | 261 | 0.30 |

03 | 252469 | 18 | 290 | 31 | 2.95 | 259 | 0.32 |

04 | 313898 | 17 | 261 | 31 | 2.90 | 230 | 0.36 |

05 | 307197 | 20 | 292 | 36 | 2.85 | 256 | 0.28 |

06 | 321789 | 19 | 355 | 34 | 2.92 | 321 | 0.27 |

07 | 312976 | 21 | 319 | 36 | 2.81 | 283 | 0.25 |

08 | 383190 | 19 | 289 | 34 | 2.92 | 255 | 0.32 |

09 | 297221 | 17 | 221 | 29 | 2.93 | 192 | 0.41 |

10 | 401429 | 20 | 316 | 34 | 2.88 | 282 | 0.33 |

11 | 396750 | 19 | 281 | 33 | 2.93 | 248 | 0.32 |

12 | 362526 | 19 | 249 | 34 | 2.94 | 215 | 0.26 |

13 | 275928 | 19 | 331 | 33 | 2.91 | 298 | 0.31 |

Mean value | 328122 | 19 | 289 | 33 | 2.9 | 255 | 0.31 |

The basic parameters of the power-law fitting from the single-trial data provide statistic evidence for the difference between the two groups. Corresponding to the higher maximum PLI values, the math-gifted subjects show wider power-law interval of PLIs distribution, i.e., the critical interval, and lower power-law exponent (Figures

Source | SS | df | MS | |||
---|---|---|---|---|---|---|

Mean phase-locking duration | Group | 261.8558 | 1 | 261.8558 | 142.1430 | |

Error | 856.6231 | 465 | 1.8422 | |||

Total | 1.1185e + 003 | 466 | ||||

Maximum of phase-locking duration | Group | 1.3654e + 005 | 1 | 1.3654e + 005 | 41.4677 | |

Error | 1.5311e + 006 | 465 | 3.2927e + 003 | |||

Total | 1.6677e + 006 | 466 | ||||

Power-law exponent | Group | 0.1901 | 1 | 0.1901 | 131.0518 | |

Error | 0.6746 | 465 | 0.0015 | |||

Total | 0.8647 | 466 | ||||

Power-law distribution interval | Group | 1.1235e + 005 | 1 | 1.1235e + 005 | 35.5266 | |

Error | 1.4705e + 006 | 465 | 3.1624e + 003 | |||

Total | 1.5829e + 006 | 466 |

Critical synchronization can be compatible with the rapid network reorganization in response to temporary perturbation and stimulus, which promotes the adaptive ability of a functional network in spatial reconfiguration of connections (Bassett et al.,

The paradigms used in the previous studies on math-gifted adolescents or children mostly involved visuospatial imagery tasks that were related to mathematical thinking ability, such as RAPM test and mental rotation. As an essential mathematical skill, a cognitive task of the analytic type (verbal–logical way) was designed in this study for determining whether the previous research results were specific to the mathematical thinking or just the general attributions of problem solving. The logical syllogism used in this experiment is viewed as a basic form of mathematically logical thinking and fills the void of the experimental paradigm in neuroscience studies of mathematical giftedness.

To the best of our knowledge, this is the first time that the individual difference between math-gifted and average-level abilities is investigated through EEG dynamic network analysis. With the highest criticality in the fractal networks of the human brain, the cortical network at the classic gamma frequency is assessed by transforming the scalp-recorded EEG signals into the cortical dipoles. According to the results obtained from the graph-theoretical analysis, the math-gifted adolescents demonstrate a highly integrated fronto-parietal network that is supported by the prolonged gamma binding-by-synchrony activity among discrete neuronal assembles, which is in line with the results of the previous fMRI studies and the P-FIT model of reasoning. Furthermore, as the prolonged periods of phase-locking are more likely to occur between the processes within the same functional module (Kitzbichler et al.,

Functional connectivity of the phase coherent network is positively related to the phase-locking duration and stability of phase dynamics. In the context of temporally stable fronto-parietal connectivity in the math-gifted brain, the theory of critical dynamics is applied to the realistic data from the high-order cognitive task through the analysis of single-trial samples, which constructs an association between the enhanced functional connectivity and the highly adaptive reconfiguration of the fronto-parietal network in the math-gifted brain. From the perspective of criticality, the existence of power-law distribution of PLIs in the brain puts the large synchronization on a “metastable island”; that is, the longer the PLI is, the higher the desynchronization possibility will be (Werner,

The optimized synchronization pattern of the fronto-parietal network also plays a key role in information processing. The prolonged fronto-parietal phase-locking durations distributed in a wider critical interval indicate that some optimizations of information processing would occur simultaneously. Firstly, the generally prolonged phase-locking durations enhance the global synchronization of the gamma network through a widespread stability of phase dynamics, which could increase the capacity of information storage of the network. Secondly, the phase-locking duration at a critical state supports effective information communication between neuronal assembles because the long synchronization leads to efficient information transmission. Finally, when the synchronizing activity is maintained at a critical state, it would decrease the stability of the connection but increase the adaptiveness of the network for timely reorganization of connections. In conclusion, the optimizations of the fronto-parietal synchronization enhance the information processing of the math-gifted brain during the deductive reasoning task, and further support the exceptional logical thinking ability of math-gifted adolescents.

John Q. Gan and Haixian Wang designed the research and provided analytic tools; Li Zhang conducted experiment, analyzed data, and wrote the paper; John Q. Gan and Haixian Wang improved the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported in part by the National Natural Science Foundation of China under Grant 31130025 and Grant 61375118, the Natural Science Foundation of Jiangsu Province under Grant BK2011595, and the Program for New Century Excellent Talents in University of China under Grant NCET-12-0115.