Reply to: Eigenmodes of the brain: revisiting connectomics and geometry

16 In Pang, Aquino et al. (2023)1, we presented multiple lines of evidence to indicate that brain 17 geometry plays a previously under-appreciated role in shaping dynamics. Mansour et al. raise 18 concerns about one specific analysis, in which we showed that eigenmodes derived from the 19 geometry of the human cortex can reconstruct diverse activity maps generated with functional 20 magnetic resonance imaging (fMRI) better than eigenmodes derived from connectomes 21 estimated with diffusion MRI (dMRI). Here, we address their concerns and show how our 22 findings and conclusions remain valid. 23 24

25 Mansour et al. motivate their work by claiming that our findings have “been widely interpreted 26 to mean that geometry imposes stronger constraints on cortical dynamics than connectivity”. 27 This interpretation rests on an artificial and inaccurate dichotomy between brain geometry and 28 connectivity, as detailed in the extensive supplementary material (Section S8) of our original 29 article1 showing the precise mathematical relation between geometry and connectivity, and our 30 follow-up piece further explaining why the dichotomy is incorrect2. To clarify, geometric 31 eigenmodes assume a specific form of connectivity in which cortical locations are coupled 32 through an isotropic, distance-dependent kernel, such that connectivity between any two points 33 decays as an approximately exponential function of their physical separation. This assumption 34 follows the well-known exponential distance rule (EDR) demonstrated empirically to dominate 35 the connectomes of diverse species3–8. Geometric eigenmodes thus account for the effects of 36 both cortical geometry and EDR-like connectivity. In contrast, connectome eigenmodes do not 37 directly account for geometry. Connectomes are dominated by EDR-like connectivity4,9, but 38 they also account for the effects of topologically complex connections that are not incorporated 39 in an EDR approximation4,10. 40 41 The stronger performance of geometric eigenmodes in our original analysis1 indicated that 42 geometric eigenmodes, and the connectivity approximation that they entail, are sufficient to 43 account for diverse fMRI activity maps, despite the simplicity of the model. We thus concluded 44 that geometric eigenmodes “provide a more compact, accurate, and parsimonious 45 .CC-BY-NC-ND 4.0 International licensemade available under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is The copyright holder for this preprintthis version posted August 23, 2024. ; https://doi.org/10.1101/2024.08.21.608487doi: bioRxiv preprint 2 representation of [the brain’s] macroscale activity than alternative connectome-based models” 46 and that “the comparatively poor performance of connectome eigenmodes indicates that 47 topologically complex connections that exist beyond a simple EDR afford minimal further 48 benefit in obtaining eigenmodes that can accurately explain spatiotemporal patterns of cortical 49 activity as measured with fMRI” (p572)1. 50 51 Mansour et al. focus almost exclusively on our specific comment about the “comparative 52 performance of connectome eigenmodes”, citing it multiple times. They report a thorough set 53 of analyses to address this “comparatively poor performance” by showing that, when dMRI 54 data are processed using a specific pipeline of “state-of-the-art connectome reconstruction 55 techniques”, the accuracy of the connectome and geometric models becomes approximately 56 equal. We applaud their efforts to identify a dMRI processing regime that improves the 57 accuracy of the connectome model. However, we disagree with their conclusion that the 58 “evidence presented to support the comparative proposition that “eigenmodes derived from 59 brain geometry represent a more fundamental anatomical constraint on dynamics than the 60 connectome” may require reconsideration.” Below, we explain why their findings only 61 strengthen our original claims. 62 63 Geometric eigenmodes are more parsimonious than connectome eigenmodes 64 Mansour et al. consider five dMRI processing steps that can mitigate “biases and inaccuracies” 65 associated with “high-resolution connectome mapping ”. Each of these steps requires 66 investigator-dependent choices between alternative approaches that substantially influence 67 connectivity estimates and model accuracy, as thoroughly detailed in Mansour et al.’s 10 multi-68 panel supplementary figures (their Figs. S1–10). For instance, their two different methods for 69 gyral bias correction yield connectomes with quite different architectures (Figs. 1–2 and their 70 Fig. S1), neither of which completely address the gyral bias that was suggested to contaminate 71 our original connectome (Fig. 2b). This simple methodological variation is sufficient to reduce 72 the correlation between the two resulting connectivity matrices to 0.46, despite the source data 73 being otherwise identical. This major discrepancy in connectivity weights, caused by a single 74 processing choice, underscores the fragility of the resulting connectomes (as suggested in our 75 own original analysis1) and poses a difficult challenge for the construction of a reliable basis 76 set, particularly given the priority that Mansour et al. assign to the analysis of weighted 77 connectomes. 78 79 Notably, this critical choice of gyral bias correction is only one in a sequence of decisions 80 required in their connectome reconstruction pipeline. These decisions yield a parameter space 81 comprising many plausible processing steps that can result in very different connectomes11,12. 82 It is difficult to ascertain how one should choose between these different processing options. 83 Mansour et al. use the geometric eigenmodes as a benchmark for optimizing their set of 84 choices. This approach is vulnerable to overfitting the data to obtain a desired result. It is 85 unclear how well this optimized set of choices generalizes to independent data. 86 87 Geometric eigenmodes obviate the need to navigate this analytic “garden of forking paths ”. 88 They are derived using robust and widely accepted pipelines for reconstructing cortical 89 surfaces from T1-weighted scans13. In fact, the end point of the geometric eigenmode pipeline 90 is the starting point of the connectome eigenmode pipeline, which requires acquisition of an 91 additional MRI modality and the construction of a complex sequence of processing steps based 92 on numerous choices. Mansour et al.’s analysis shows that, at best, a specific subset of these 93 choices yields a basis set that matches, but never substantially surpasses, the accuracy of the 94 geometric eigenmodes (their Figs 1, S2, S4–10, and S12). The fact that equivalent performance 95 .CC-BY-NC-ND 4.0 International licensemade available under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is The copyright holder for this preprintthis version posted August 23, 2024. ; https://doi.org/10.1101/2024.08.21.608487doi: bioRxiv preprint 3 is the best result for the connectome approach, despite the large parameter space explored by 96 Mansour et al., is strong evidence for the superior parsimony of the geometric basis set. Their 97 analysis thus strengthens our conclusion that geometric eigenmodes offer a sufficient and 98 parsimonious account of fMRI data . Future innovations in dMRI processing may further 99 improve the performance of connectome eigenmodes, but the gain in model performance 100 should be of a sufficient magnitude to justify the added complexity and problematic reliability 101 of the connectome model. 102 103 Topologically complex connections make a negligible contribution to reconstruction 104 accuracy 105 The ability to account for the specificity of brain connectivity, and particularly of the 106 topologically complex, often long-range, connections that cannot be explained by an EDR , is 107 the primary motivation for using connectome eigenmodes. However, Mansour et al.’s own 108 findings show that removal of longer-range connections (i.e., >16 mm) has a minimal effect on 109 the reconstruction accuracy of connectome eigenmodes, whereas removal of short -range 110 connections has a much larger impact (their Fig. S12). This result, when taken with their finding 111 of approximately equivalent performance of the geometric and “state-of-the art” connectome 112 eigenmodes, strengthens our original claim that an isotropic, EDR -like connectivity 113 approximation is sufficient to reconstruct diverse activity patterns mapped with fMRI and that 114 topologically complex connectivity (i.e., connectivity not readily captured by an EDR -like 115 approximation) makes a minimal contribution. Mansour et al.’s demonstration that removal of 116 short-range connections has a larger impact on reconstruction accuracy also supports our 117 original claim that local, short -range connections, which facilitate continuous propagation of 118 waves of excitation through the cortex, are important for understanding fMRI activity1. As we 119 acknowledged in our original article 1, topologically complex connections undoubtedly play 120 important roles for brain function but their effects may not be easily revealed by traditional 121 fMRI paradigms. 122 123 “State-of-the-art” dMRI processing increases the similarity between connectome and 124 geometric eigenmodes 125 If long-range connections make a negligible contribution to model accuracy, what drives the 126 performance improvement of “state-of-the-art” connectome eigenmodes? The answer to this 127 question is found in Figs 2 and S11b of Mansour et al., which show that their processing 128 strategy increases the alignment of the connectome eigenmodes to the geometric eigenmodes. 129 This increased similarity occurs because their additional processing steps accentuate the local 130 homogeneity and EDR -like dependence of the connectome ––the two key elements of the 131 connectivity approximation upon which the geometric approach relies. For instance, their Figs 132 S3f–g show how local homogeneity and the EDR dependence are dramatically enhanced by 133 connectome spatial smoothing. In Fig. 3a, we additionally demonstrate that their gyral bias 134 correction truncates the tails of the vertex connectivity strength distribution relative to the 135 uncorrected case, yielding an approximately Gaussian shape in which there is a low probability 136 of finding vertices with substantially higher or lower connectivity than the mean . Thus, 137 regardless of any debate over whether gyral bias correction improves the biological plausibility 138 of vertex -wise connectivity estimates, it ultimately increases the homogeneity of those 139 estimates. Figure 3b also shows that the processing steps applied by Mansour et al. exaggerate 140 the distance-dependent decay of connectivity edge weight s by attenuating precisely those 141 longer-range connections that deviate from an EDR footprint (see also Fig. 1b). 142 143 These considerations indicate that the performance of connectome eigenmodes approaches 144 geometric eigenmodes when the former are made to look like the latter . This is done through 145 .CC-BY-NC-ND 4.0 International licensemade available under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is The copyright holder for this preprintthis version posted August 23, 2024. ; https://doi.org/10.1101/2024.08.21.608487doi: bioRxiv preprint 4 data processing steps that increase the convergence between empirical connectivity estimates 146 and the isotropic, EDR-like connectivity approximation that underpins the geometric model. 147 The findings of Mansour et al. thus align with our original results and support the fundamental 148 role of geometric eigenmodes and the connectivity approximation that they entail in shaping 149 fMRI activity maps. 150 151

152 Mansour et al. show that: (i) best-estimate connectome eigenmodes only ever match, but never 153 substantially surpass, the reconstruction accuracy of the geometric model ; (ii) long-range 154 connections make a minimal contribution to the performance of the connectome model ; and 155 (iii) the performance improvements of the connectome model are attributable to processing 156 steps that increase the similarity between empirical connectome s and the connectivity 157 approximation inherent in the geometric approach. These findings beg the question: why use 158 the more complex connectome model if it offers little added value beyond the simpler 159 geometric approach? 160 161 The answer to this question depends on the goals of the investigator, since both geometric and 162 connectome eigenmodes have distinct strengths and limitations. Geometric eigenmodes offer 163 a simple and parsimonious account of brain function as they are formally linked with the 164 biophysical model of dynamics provided by neural field theory 14–17. This link allows one to 165 associate each eigenmode with characteristic spatial and temporal frequencies, which can be 166 used to understand how different types of stimuli drive cortical dynamics18. However, a 167

of the geometric approach is that it is not straightforward to integrate cortical and 168 subcortical eigenmodes into a single model. Connectome eigenmodes can account for both 169 cortical and subcortical connectivity and facilitate mappings between specific eigenmodes and 170 task activation patterns in specific circumstances (see Fig . S13 of Mansour et al.). However, 171 connectome eigenmodes are susceptible to myriad data processing choices and include 172 overlapping contributions from topologically complex connections , EDR -like connectivity, 173 and geometry that can be difficult to disentangle without additional analysis. Future work will 174 benefit from understanding the strengths and limitations of the two approaches, and developing 175 approaches to unify them. 176 177 178 Figure 1. Original Pang, Aquino et al. and reprocessed Mansour et al. connectomes . (a) Panels from 179 left to right show the original connectome used in our work , our connectome after performing Mansour et 180 al.’s gyral bias correction via regression, and the connectome preferred by Mansour et al. (b) Same as panel 181 .CC-BY-NC-ND 4.0 International licensemade available under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is The copyright holder for this preprintthis version posted August 23, 2024. ; https://doi.org/10.1101/2024.08.21.608487doi: bioRxiv preprint 5 a but the vertices are reordered according to their spatial locations from posterior to anterior. This reordering 182 shows how the connectome preferred by Mansour et al. attenuates long-range connectivity, thus converging 183 on the distance-dependent connectivity approximation inherent in the geometric approach. 184 185 186 Figure 2. Spatial maps of v ertex-level connectivity strength s for the original and reprocessed 187 connectomes. (a) Spatial map of the vertex connectivity strengths as depicted by Mansour et al. (b) Same 188 as panel a but the values are zscored , which attenuates the effect of outlying values and allows a clearer 189 visualization of the spatial structure in the map. For panels a and b, panels from left to right show results 190 using our original connectome, our connectome after performing Mansour et al.’s gyral bias correction via 191 regression, and Mansour et al.’s preferred connectome. Panel b clearly shows that gyral bias is still 192 evident in the reprocessed connectomes and that the two correction strategies by Mansour et al. can lead 193 to highly divergent maps . These visualizations are preferable to the regression slopes and linear 194 correlations shown in Fig. S1b of Mansour et al., which will be biased by the outliers, heteroscedasticity, 195 and non -monotonicity evident in their scatterplots of the association between vertex connectivity 196 strength and curvature. 197 198 199 Figure 3. Vertex connectivity strength distributions and edge weight distance-dependencies for the 200 original and reprocessed connectomes. (a) Distribution of vertex connectivity strengths. (b) Edge weight 201 vs euclidean distance between vertices in semilogarithmic scale. For panels a and b, left to right shows results 202 using our original connectome, our connectome after performing Mansour et al.’s gyral bias correction via 203 regression, and Mansour et al.’s preferred connectome. The processing steps proposed by Mansour et al. 204 truncate the tails of the vertex connectivity strength distribution (panel a ) and attenuate the weights of 205 .CC-BY-NC-ND 4.0 International licensemade available under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is The copyright holder for this preprintthis version posted August 23, 2024. ; https://doi.org/10.1101/2024.08.21.608487doi: bioRxiv preprint 6 medium-to-long-range connections (between 50 and 100 mm) (panel b; see red arrows ). The processing 206 steps of Mansour et al. thus enforce a structure on the empirical data that more closely resembles the locally 207 isotropic, EDR-like connectivity approximation of the geometric model. 208 209

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