Fig 1.
Neurodevelopmental assumptions and overview of the in silico model.
The figure illustrates the assumptions regarding neurogenesis that were varied in the in silico model. The spatial growth of the cortical sheet of a single hemisphere was modelled in three possible ways: First, planar growth, in which the neurons comprising a cortical area develop at the same time and the cortical sheet expands as more areas materialize. Second, radial growth, in which neurons across the entire extent of the final cortical sheet develop at the same time, and the final complement of neurons is reached by gradual growth of neurons at a constant rate. Third, no growth, that is, a static cortical sheet on which the final complement of neurons is already present from the onset. Regarding the gradients of architectonic differentiation, we considered three possible relationships between the time at which an area was formed (time of neurogenesis) and its architectonic differentiation, approximated by neuron density. First, areas could be more differentiated the later in ontogenesis they were formed (increasingly differentiated). This scenario corresponds to the realistically oriented density gradient we incorporated in the in silico model. Second, areas could be less differentiated the later their time of origin was (decreasingly differentiated). This scenario corresponds to the inversely oriented density gradient in the in silico model. Third, there could be no gradient of differentiation aligned with neurogenetic timing, that is, the neuron density of newly formed areas varied randomly throughout ontogenesis. As a third factor that determined the spatiotemporal growth trajectory of the cortical sheet, we considered the number of neurogenetic origins. There could either be a single origin, such that more recently formed areas occupied the fringes of the cortical sheet, or there could be two or three origins. In this case, recently formed areas would be interleaved with areas that were formed earlier, as the neurogenetic origins were moved apart by addition of areas around them. From these assumptions on neurogenetic processes shaping the cortical sheet, we set up different variants of an in silico model in which axons grew randomly across the developing cortical sheet and stochastically formed connections. We translated the resulting neuron-level connectivity to area-level connectivity and extracted structural measurements from the simulated cortical sheet. As in previous studies of mammalian connectomes, we considered the difference in architectonic differentiation between areas and their spatial distance. Thus, we simulated sets of measures which we could then analyse in the same way as the empirical data, and compared the results to empirical findings. Specifically, we used simulated architectonic differentiation and spatial distance to classify whether a connection existed in the final simulated network; we probed whether there was an association between simulated architectonic differentiation and the number of connections maintained by an area; and we used a classifier trained on the simulated data to predict connection existence in two sets of empirical connectivity data, from the cat and the macaque cortex.
Table 1.
Growth layouts.
Fig 2.
Developmental trajectories of growth layouts.
The figure illustrates the spatiotemporal growth trajectory for different growth layouts. The successive population of the cortical sheet with neurons is shown for the first three growth events. For static growth, all neurons grow simultaneously, hence only one growth event is shown. Here, all growth layouts of growth mode 1D 2 rows are shown. See S1 Fig for an illustration of the developmental trajectories of all 21 growth layouts.
Fig 3.
Validation procedure for measures of simulation-to-empirical classification performance.
The figure illustrates the general procedure for assessing the performance of the classification of empirical data from the cat and macaque cortex by classifiers that were trained on simulated data; see main text for details. We computed median measures of classification performance for each growth layout and compared these measures against chance performance, as assessed by a permutation analysis. Specifically, for each of the 21 growth layouts shown in Fig 8 and Table 4, 100 instances were simulated. For each instance, classification was performed using 10 different classification threshold probabilities. For each threshold probability, a simulation-trained classifier assigned labels to the empirical data, resulting in an accuracy value Athr. Additionally, a distribution of chance performance accuracies, Achance, was generated by classifying 100 times from randomly permuted non-sensical labels. A z-test quantified the probability that Athr was an element of the distribution of Achance. The corresponding p-value pthr was used for further calculations. For each simulation instance, classification performance from all 10 threshold probabilities was averaged, resulting in one mean accuracy value and one median value of pthr per instance, thus amounting to a total of 100 values each per growth layout. Fig 8 shows the distribution of mean accuracy values from these 100 instances, and indicates the median accuracy. The indication of significance in Fig 8 refers to the p-value obtained from a sign-test which assessed whether the median of the distribution of median values of pthr was larger than the chosen significance threshold αz-test of 0.05 (with a small value of psign-test indicating that pthr was very unlikely to be larger than αz-test). Table 4 includes the median accuracy, median z-test p-value and the result of the sign-test. Shown here for accuracy, the procedure was analogous for the Youden index J, which is shown in Fig 9 and Table 4.
Table 2.
Summary of correspondence between simulation results and empirical observations.
Fig 4.
(A) Percentage of connected areas, shown as the fraction of possible connections that are present in the final simulated network. (B) Total number of connections among all areas. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 3 for a summary. Abbreviations and background colours as in Table 1.
Table 3.
Summary connectivity statistics, correlation with relative projection frequency, classification performance logistic regression, and correlation with area degree.
Fig 5.
Correlation of distance and absolute density difference with relative connection frequency.
Spearman rank correlation coefficients are provided for the correlation between relative connection frequency and distance (blue) or absolute density difference (green). A sign test was used to test whether the distribution of associated Spearman rank correlation p-values had a median value smaller than α = 0.05. The result of the sign test is indicated on top; black star: median p < 0.05, red circle: median p > = 0.05. See S2 Fig for representative plots of the correlation for individual simulation instances. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 3 for a summary. Abbreviations and background colours as in Table 1.
Fig 6.
Logistic regression performance for classification of simulation data from simulation data.
Within each growth layout, a logistic regression was performed to classify connection existence from three sets of factors: distance (blue), absolute density difference (green), or distance as well as absolute density difference simultaneously (purple). To assess whether classification performance was better than chance, McFadden’s Pseudo R2 was computed against performance of a null-model, where a constant was the only factor included in the logistic regression. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 3 for a summary. Abbreviations and background colours as in Table 1.
Fig 7.
Correlation of area degree with neuron density.
Spearman rank correlation coefficients for the correlation between area degree (number of connections) and area neuron density. A sign test was used to test whether the distribution of associated Spearman rank correlation p-values had a median value smaller than α = 0.05. The result of the sign test is indicated on top; black star: median p < 0.05, red circle: median p > = 0.05. See S3 Fig for representative plots of the correlation for individual simulation instances. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 3 for a summary. Abbreviations and background colours as in Table 1.
Fig 8.
Classification accuracy for prediction of empirical connection existence from simulation data.
A classifier was trained to predict connection existence of a simulated network from the associated distance and absolute density difference. Classification accuracy for predicting existence of connections in two species (macaque, blue; cat, green) by this classifier is shown. Accuracy was determined at each classification threshold (see Methods); here, we show mean accuracy across thresholds 0.750 to 0.975. Whether classification accuracy was better than chance was assessed by a permutation test, where classification accuracy was calculated for prediction from randomly permuted labels and a z-test was performed. A sign test was used to test whether the distribution of associated z-test p-values had a median value smaller than α = 0.05. The result of the sign test is indicated on top; black star: performance better than chance with median p < 0.05, red circle: performance not better than chance with median p > = 0.05. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 4 for a summary. Abbreviations and background colours as in Table 1.
Fig 9.
Youden index for prediction of empirical connection existence from simulation data.
A classifier was trained to predict connection existence of a simulated network from the associated distance and absolute density difference. Youden index J for predicting existence of connections in two species (macaque, blue; cat, green) by this classifier is shown. Youden index J was determined at each classification threshold (see Methods); here, we show mean J across thresholds 0.750 to 0.975. Whether the Youden index was better than chance was assessed by a permutation test, where J was calculated for prediction from randomly permuted labels and a z-test was performed. A sign test was used to test whether the distribution of associated z-test p-values had a median value smaller than α = 0.05. The result of the sign test is indicated on top; black star: performance better than chance with median p < 0.05, red circle: performance not better than chance with median p > = 0.05. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 4 for a summary. Abbreviations and background colours as in Table 1.
Fig 10.
Percentage of empirical connectivity data that were classified from simulation data.
A classifier was trained to predict connection existence of a simulated network from the associated distance and absolute density difference. This classifier was then used to predict connection existence in two species (macaque, blue; cat, green). Here, we show which fraction of the empirical data set was classified. This fraction differs across classification thresholds (see Methods); here, we show the mean fraction across thresholds 0.750 to 0.975. Box plots show distribution across 100 simulation instances per growth layout, indicating median (line), interquartile range (box), data range (whiskers) and outliers (crosses, outside of 2.7 standard deviations). See Table 4 for a summary. Abbreviations and background colours as in Table 1.
Table 4.
Summary classification of empirical connectivity from simulated connectivity.
Table 5.
Anova on classification performance of realistically oriented density gradient growth layouts.
Table 6.
Post-hoc comparisons for classification performance of realistically oriented density gradient growth layouts.
Fig 11.
Number and relation of neurogenetic and architectonic gradients.
A synthesis of all the results presented here indicates that the presence of two origins of neurogenesis, resulting in two neurogenetic (temporal) and architectonic gradients is necessary for the closer correspondence of the in silico model to the empirical relations between connectivity and architectonic differentiation. Importantly, the empirically observed relations are replicated in silico only if the less-to-more differentiated architectonic gradients align with early-to-late ontogenetic gradients. Hence, the suggested mechanism is consistent with correspondence of neurogenesis and architectonic differentiation [36, 37, 40] and a dual origin of the cerebral cortex [38, 71].