PNAs: Geometric renormalization reveals the self similarity of multi-scale human connectome

The structural connectivity of the brain is usually studied by reducing its observation to a single spatial resolution . However , The brain has a rich framework of organizations connected to each other on multiple scales . We explored the multi-scale organization of the human connectome using five different resolution reconstructed healthy subject data sets . We found that , When the resolution of observation decreases gradually with the grading and coarsening of anatomical regions , The structure of the human brain is still self similar . Here's the striking thing , A geometric network model whose distance is not Euclidean predicts the multiscale characteristics of the connection group , Including self similarity . The model relies on geometric renormalization (GR) Application of protocol , The protocol reduces the resolution by coarse granularity and averaging over short similar distances .

Our results show that , Simple organizing principles are the basis of the multi-scale architecture of the human brain structure network , The same connection rule determines the short and long distance connections of different brain regions at different resolutions . The impact is diverse , There may be some fundamental arguments of great significance , For example, whether the brain works near a critical point , And the application of advanced tools that simplify the digital reconstruction and Simulation of the brain .

1. sketch

In this study , We are in two different data sets , Reconstructed with five anatomical resolutions 84 Multiscale humans with healthy subjects (MH) Connectome . First , We measured the network properties of connectors at each scale , It is found that with the gradual decrease of observation resolution , Their structures remain self similar . secondly , Get the hyperbola of the highest resolution layer of each connector , application GR Carry out multi-scale expansion . On every scale , We found a striking agreement between the empirical observations and the predictions given by the model . Third , We discuss the influence of the damage of the geometric attributes of the connector on the self similarity and navigation . All in all , Our results show that , The same rules explain the formation of short-range and long-range connections in the brain within the length scale covered by the data set , And support GR As an effective prototype model of multi-scale structure of human brain .

2. result

2.1MH Empirical evidence of connectome self similarity

We used two different datasets , All in all 84 Healthy human subjects . The first data set is contained in UL Scanned 40 MR diffusion spectrum data of subjects . The nerve fibers track the connection area through the maximum diffusion direction . The second data set is used for cross validation results , Contains 44 Multiscale connectome of healthy subjects who tested retest subsamples . The fiber orientation distribution function in voxel is calculated by constrained spherical deconvolution (fODFs) Estimate fiber bundles . All connectors of two data sets using deterministic streamline fiber tracing method , And get the multi-scale segmentation of the cortex . Even if UL Data set than HCP Much sparser , But similar results were found in both queues .

For each subject , We reconstructed a five layer multiscale connector with different anatomical resolutions . The nodes of each layer correspond to the cortex and subcortical regions ( Excluding the brain stem ), The connection indicates that there are fibers between them . Multiscale segmentation is anatomically hierarchical , From... By iteration l=0 The coarsening operation at the beginning of the layer , To produce a subsequent layer with reduced resolution . This technique groups two or three adjacent brain regions , Create a new brain area , And recalculate the connection density between each pair of brain regions . These layers contain approximately 1014;462;233;128;82 Nodes ( These figures fluctuated slightly between subjects ), They are marked as l =0, 1, 2, 3, 4. The coarsening of the hierarchical structure determines the length scale sequence characterizing the multi-scale connectome . As resolution decreases , Each node corresponds to a larger brain region , And the average fiber length of the connection calculated from the streamline fiber tracer also increased , Because coarse-grained partitions absorb short-distance connections ( chart 1).

chart 1 UL Subjects 10 and HCP Subjects 15 Different resolutions MH The average fiber length connected in the connector

For each layer of each subject , Measure the following characteristics ： Complementary cumulative degree distribution ; The normalized average degree of the nearest neighbor is used for degree correlation ; Degree dependent clustering coefficient ; The rich club coefficient ; Average degree ; Average clustering coefficient . These quantities are calculated as a function of the degree of scaling to explain the change in the average degree of each layer . chart 2 Shows UL The results of a typical subject in the data set . chart 2A-D The overlap of curves represents the cross layer self similarity distribution 、 Degree of relevance , clustering , The rich club effect ( Pay attention to the picture 2D The value corresponding to the height threshold is omitted , To avoid confusing the diagram , Because the corresponding subgraph is usually small , therefore , It's messy ).

chart 2B,C The insertion section shows MH The average clustering coefficient across five layers of connectome 、 Average degree . As the resolution increases （l from 0-4）, The increase was first slight , Then it becomes more obvious , This explains the figure 2B Corresponding to the movement of the curve . On the other hand , In the 3 Tier and tier 4 The layer first increases slightly , Consistent with an almost constant average , Then significantly reduce .MH The last two layers of the connector are more susceptible to the finite size effect , Because they have fewer nodes , It is also affected by the higher variability of the surface area of the anatomical region , This may lead to a deviation in the flow line determination .

Last , use Louvain Methods infer the community partition . The modularity of the monitored partition Qemp(l) Pictured 2E, Accompanied by 0 The community partition and modularity detected by layer are Qemp(l,0) The layer l The layer caused by the community partition in 0 Adjusted mutual information between community partitions . The overlap between communities at different resolutions is still important , Even if the modularity is slightly weakened , Especially on the last two floors , The effect of finite size is stronger due to its reduction .

All in all , Our results strongly support each MH The self similarity of connectors in data sets . Besides , We found that the connectors of different subjects in each dataset are similar to each other . chart 2F-J Shows UL Each subject in the data set was in the 0 Properties of layer measurements , chart 2 AE Subjects used in are highlighted . About UL Information from other layers in the dataset can be found in SI Find... In the figure in the appendix .S10 and S11, SIAppendix, chart S29 and S30 by HCP All layers of the dataset provide results . Evaluate each connector at l =0 The statistical test results of the average consistency between time and cohort further support the homogeneity among subjects in the data set . For each connector , We compare the degree of nodes 、 The sum of degrees of adjacent nodes, the number of triangles each node participates in and the corresponding queue average . For each brain region , By calculating the standard deviation and mean value of these characteristics of all subjects, the average value of the queue is obtained .

chart 2 MH Self similarity of connectome at different resolutions

2.2 Geometric reconstruction of human body connector

We now prove that , Observed reality MH The scale invariance of connectors can be explained by geometric network model , The distance is not Euclidean , This includes a renormalization protocol .

2.2.1 Geometric description of the connector

The connector graph is based on S1 Of the network model . Every brain area i Characterized by two random variables ： A degree of concealment ki, Quantify its popularity and set its connection size , Angular position in a one-dimensional sphere θi; Or similar space , Aggregate all other attributes that regulate connection possibilities , Including but not limited to Euclidean physiological three-dimensional brain (3D) The embedded .

S1 The connectors in the model are paired , The probability is in the form of the law of gravity ：

therefore , The likelihood of a link between two nodes increases with the product of their hiding degrees , As their angular distance decreases ( So as their similarity increases ). Parameters μ Average degree of synthetic connector generated by control model ,β Control the clustering level , And the coupling strength between the topology of the network and its geometry . Angular distance ： The radius of joint similar subspace R Give a similar distance . The model assigns implicit variables to nodes , Extract the hidden degree from some heterogeneous distributions , So as to produce a small world at the same time 、 Highly aggregated 、 Heterogeneous degree distribution and rich club network . The formula 1 An important feature of , It simultaneously encodes the possibility of long-distance and short-distance connections over all distances , Therefore, there is no need for different mechanisms to describe . Another related property of the model is , node i The expectation and concealment of Ki In direct proportion to .

Similarity captures the affinity between brain regions , When two brain regions are close together in a similar space , They are more similar , More likely to form a connection . One result is , Compared to other parts of the network , Groups of nodes that are close to each other in similar spaces tend to have stronger correlations . therefore , Infer that the angular position of the brain region is similar to the community structure information provided by the subspace, and analyze the relevant information of connectome and neuroanatomy , Areas that belong to the same neuroanatomy as nodes are strongly concentrated in a narrow angular part similarity space . Please note that , Euclidean distance is certainly an important factor , But it is not the only factor that determines the similar distance , It is also negatively correlated with the measurement of homophobic attraction used in the brain generation model .

S1 The model has an isomorphic pure geometric form H2 Model , The model converts the implicit degree into radial coordinates , The popularity and similarity dimensions are combined into a single distance on the hyperbolic plane . The popularity and similarity coordinates of nodes in the real connector , Hyperbolic mapping , Coordinates can be found by using statistical reasoning , To reproduce the structure of the real connector of our geometric model as much as possible .

UL Subjects no.10 The embedding of is shown in the figure 3. chart 3A Shows l =0 Time map , according to l =4 Layer of 82 A coarse-grained region shades the nodes . The left and right hemispheres are naturally separated by the red and green boxes on the edge of the disc , Nodes belonging to the same brain region gather at nearby angular positions . This is consistent with the previous results . For testing embedded accuracy , Use formula 1 produce 100 A composite network as a whole . We compare the topological properties of the whole with those measured on real connectors . chart 3B-D The complementary cumulative degree distribution is shown 、 Degree dependent clustering coefficient 、 The results of nearest neighbor average degree and rich people's Club coefficient . chart 3E-G It shows the local characteristics of the real connector and the composite whole —— degree 、 The sum of neighbors' degrees 、 Good consistency between the trigonometric numbers of nodes involved . It turns out that , The resulting network can reproduce the topological properties with remarkable accuracy .

chart 3 UL Subjects 10 Hyperbolic connectome graph of

2.2.2 GR Transformation

Considering that the connector can pass through S1 The model is well described , The connector is mapped to l =0 The similarity distance at allows the use of renormalization techniques to systematically study their characteristics at different resolutions . Given connector map , reference 19 Introduced in GR Transformation by capturing longer connections between coarse-grained groups of nodes , A low resolution self similar 、 Reduced copy , Thus, the average length of the connections that determine the length scale in the similarity space increases .

Get in l =0 After the layer is embedded ,GR The transformation is carried out by defining the size in the similar circle as r =2 Non overlapping blocks of continuous nodes of , Coarsening to form super nodes . Assign an angular coordinate to the supernode in the region of the similar subspace defined by the nodes in the block , Keep the original angle sorting . second , If and only if i At least one node in the block is connected to the original layer j When there is at least one node in the block , There are two super nodes in the new layer i and j Connected to a . If the implicit variable of the supernode is shown in the new graph as 2, Then the result layer has an S1 A highly consistent geometric description of the model

2.2.3 Multiscale of human connectome GR shell

We will GR The transformation was applied to each subject's second 0 Floor connector diagram . Through four iterations , Five multi-scale shells of each connector are obtained . Due to the first 0 The layer contains approximately 1014 Nodes , therefore GR Each layer produced has 507;254;127; and 64 Nodes , And MH In the connectome layer 462;233;128;82 Compare nodes .

Pictured 4B,C, Distribution p(Δθs) yes MH and GR Graph similarity . In two cases , All distributions reach their peaks near the low mean angle separation , Even if MH Distribution can reach large values . When we compare the angular distribution of any layer of child nodes with l =4 When comparing the angular distributions of the super nodes corresponding to the layer , On average, , The angular spacing of child nodes in the super node is also small ( chart 4b, illustrations ). Preservation of low average angle separation in coarse-grained anatomical regions , namely MH Preservation of map similarity , It shows that the inferred coordinates are consistent in scale , And encoded the important information of the hierarchical anatomical structure of the connector . Even if each layer is embedded independently .

Because super nodes are generated by coarse-grained neighboring nodes in similar spaces , So the original sort is kept , therefore GR Flow well reproduces this feature .

We're going to figure 2 Shown MH Topological properties of connectome and their corresponding GR The topological properties of each layer in the shell are compared ; See the picture 4 D-H. Whether the model can also increase the fiber length with the increase of the observed resolution ( chart 1), We calculate that at each upper layer we leave the outer layer of the supernode 0 The average fiber length of the connection ( chart 4 G). The two curves fit very well , This means that the length scale range covered by the real multi-scale connector is consistent with the length scale range in the model . chart 4H yes GR Normalized mutual information of shell modularity and adjustment . In the process of flow , The module structure is preserved to a great extent , The adjusted mutual information value is consistent with MH The similarity measured in the connector . We also reported MH The overlap between topological communities at each level of the connector , And in 0 Measured in the projection on the layer GR flow . here ,GR The shell well approximates the community structure of the real connector .

chart 4 MH Hyperbolic graph of connectome and GR flow

Last , chart 5 given MH Euclidean distance of connector ( The three-dimensional distance between the center of the area ) and GR Empirical connection probability of effective hyperbolic distance in shell . Despite the picture 5A Scaling of connection probability in , The Euclidean distance itself does not contain enough information to explain MH Connection properties of connector . Pictured 5 C and D Shown , The model based on Euclidean distance can not reproduce the empirical observations . Euclidean distance is certainly an important factor , But not the only reliable representation MH The factors that determine the similarity distance required by the topological properties of the connector . As a contrast ,GR The foundation of technology —— Based on similarity distance S1 The model of you and is very good .

chart 5 Connecting empirical and theoretical probabilities

in summary ,GR Can predict naturally and accurately MH Scale invariance and self similarity of connectome . in fact ,GR Provides a level of readjustment , It simulates a larger scale brain structure statistically , Only structural information measured at a single resolution is used . Let us emphasize once again , stay GR In the process of normalization , From one resolution to another , Not used about MH New information about anatomical coarsening of brain regions in connectome ; We just extrapolate a geometry from the highest resolution empirical data , In this space, continuous nodes are used to generate the structure of each renormalization layer .

2.2.4 Self similarity and navigability （navigability）

Hyperbolic network graph maintains effective navigation , This remarkable finding also applies to the brain .

To check the navigability of the connector at different resolutions , We implement greedy routing , This is a decentralized communication protocol , In this Agreement , A source node sends a message along its nearest neighbor in its metric space . The performance of greedy routing is determined by the success rate ps And the average stretch of the successful greedy path s bar measurement . We study the navigation of Euclidean embedding in anatomy , An individual on a hyperbolic plane MH Embedded collection , And all UL The subject's GR shell ; The result is shown in Fig. 6 Shown .

chart 6 Different resolutions MH Connection navigability and GR shell

3. Discuss The structure of the human brain spans a series of length scales , This magnifies its complexity , Otherwise, it will be limited by the overall pattern . We found that self similarity is a pattern in the multi-scale structure of human connectome , Paradoxically , It takes simplicity as an organizing principle . In our work , Simplicity has a very precise meaning . It points out that , The structure of connectome and the basic connection rules for explaining this structure are independent of the scale of observation ( At least within the scope of this work ). let me put it another way , There is no need to set specific rules for each scale .