PAB: Markov Chain Defined for Stochastic and Disynaptic Connectivity Maps in the N=2 P2 Matrix
In Supplementary Data 3–5 these maps are defined using Pab instead of Wab to allow comparison with the probability maps for disynaptic pathways.
The normalization is with respect to the number of cells ({|B|}) of type B, because the connectivity map is defined as the average number of synapses received by a B cell from A cells, if the origin of the coordinate system is placed at the B cell.
It is necessary to tell you where cell a is in the hexagonal lattice and how many synapses there are from cell a to cell b. Then the monosynaptic connectivity map from cell type A to cell b is
Similarly, the matrix PAB can be interpreted as the Markov chain defined by a ‘backwards’ random walk on cell types. At each step, the walker crosses the synep at random to reach the presynaptic cell type in the present cell type. Then PAB denotes the probability of stepping from cell type B to cell type A.
A non-parametric estimation of receptive field size and length using lattice point projections on hexagonal manifolds
The orthogonal directions (h, p⟂ and q⟂) point to next-nearest neighbours, which are (\sqrt{3}) lattice constant away. The projections grouped hexels with equal q − p, q or p. It was in units of lattice constant.
The ellipse approximation gives a parametric estimate of receptive field size. A non-parametric estimate was made using projections onto directions defined on a hexagonal lattice. Each hexel was given coordinates (p, q), with the origin placed at the anchor location used for alignment.
The above has implicitly defined (2\sigma ) as the width of a 1D Gaussian distribution, for which ({\sigma }^{2}) is the variance. This is the full-width at ({e}^{-1/2}\approx 0.6) of the maximum. The width could be estimated by the full- width, (sigma 2sqrt2mathrm)ln. For either estimate, the width is proportional to (\sigma ). I stick with the simpler estimate (2\sigma ), which can be readily scaled by any multiplicative factor of the reader’s preference.
The first term of the covariance matrix C effectively regards the probability distribution as a weighted combination of Dirac delta functions located at the lattice points. The second term is when a uniform distribution is used to replace each delta function. This replacement makes biological sense because a column receives visual input from a non-zero solid angle. Without the correction, the length and width would vanish if the image consists of a single hexel concentrated at a single delta function. With the correction, the length and width of an image with a single non-zero hexel become (s\sqrt{5/3}=a\sqrt{5}/3), in which (a=s\sqrt{3}) is the lattice constant. The correction becomes relatively minor when the length and width of the image are large.
I say the 2 2 identity matrix and the length of a hexagon side. The length and width of the hexel image are defined as (2{\sigma }{\max }) and (2{\sigma }{\min }), in which ({\sigma }{\max }^{2}) and ({\sigma }{\min }^{2}) are the larger and smaller eigenvalues of the covariance matrix. The approximating ellipse is centred at the image centroid, and oriented along the principal eigenvector of the covariance matrix.
Neuronal Reconstruction by Human Proofreading, Part III: Comparing Dm3, TmY4, and MY9 Cell Types
As detailed elsewhere16,61, neurons were reconstructed by human proofreading of a 3D electron micrograph that was automatically segmented using convolutional nets. The partners were assigned sonians using the machine learning techniques. The accuracy is state-of-the-art, judging from the comparisons with the neuronal wiring diagrams. I haven’t tried to untangle biological and technical sources of cell variability, but I can use some statistical models.
The type-to-type connectivity matrix of equation (5) was the starting point for clustering the cell types. The normalized rows and columns were used to input and output fractions for each cell type, as described in the text following equation (12) and this is yet another way of computing type centres. Feature vectors included only dimensions corresponding to cell types intrinsic to the optic lobe. Then, the average linkageHierarchical clustering was used to yield a dendrogram. The flat clustering was produced by thresholding the dendrogram.
The companion paper released with the Annotations of Dm3, TmY4 and TmY9 cell types, did not mention any of them. The present work shows the connectomic properties of these cell types for the first time.
Strausfeld describes line amacrine cells in his Golgi studies. Strausfeld also mentioned unpublished observations of line amacrine cells in Locusta and Apis1. Dm3 was the name of the line amacrine cells in the Golgi study.
Light microscopy with multicolour stochastic labelling3 went beyond Golgi studies by splitting Dm3 into two types with dendrite at orthogonal orientations. Dm3p and Dm3q were shown to have different transcriptomes prior to adulthood. The names Dm 3a and Dm3b were used. Immunostaining showed that Dm3q expresses Bifid, whereas Dm3p does not. The results of the reconstruction showed Dm3q and Dm3p prefer to scythe onto each other, foreshadowing what is to come in this work.
TmY4 and TmY9 were described before. The two TmY9 types can be distinguished by the tangential directions of their neurites (Fig. 2b,c), or by their connectivity (Fig. 3a, and Extended Data Figs. 4 and 5). Their stratification profiles are slightly different. TmY9q⟂ stratifies in layers 1 and 2 of the lobula plate, whereas TmY9q stratifies only in layer 1. TmY9q is more frequently monostratified than bistratified in layers 5 and 6 of the lobula.
LC10 cells project from the lobula to the anterior and have been linked to courtship behaviors. Four LC10 types were previously identified using GAL4 transgenic lines, on the basis of their stratification in the lobula13. Using the connectomic approach described in a companion paper4, I identified a fifth type (LC10e), which stratifies in layer 6 of the lobula. The two groups were further classified on the basis of their internet connections. The groups cover the same part of the body.
My conjecture that LC10e detects a corner or T-junction is specific to the ventral variant, which receives strong input from TmY9q and TmY9q⟂. The ventral visual field is expected to be more important for form vision, assuming that the fly is above the landmarks or objects to be seen.
FlyWire provides predictions of neurotransmitter identity that are based on the electron micrographs67. Dm3 is predicted to be a drug of dependence, while TmY4 and TmY9 are predicted to be a drug of dependency. The same inferences can be drawn by examining expression of neurotransmitter synthesis and transport genes19,68.
Whether a neurotransmitter has an excitatory or inhibitory effect on the postsynaptic neuron depends on the identity of the postsynaptic receptor. Acetylcholine is excitatory when the postsynaptic receptor is nicotinic, which is generally the case in the fly brain69. Glutamate is inhibitory in Drosophila when the postsynaptic receptor is GluClα70.
The hexagonal cell type of Drosophila: I. GluCl and Dm3p, DmY4 and LPi14
According to transcriptomic data19,68, Dm3 expresses GluClα. There are published data indicating that TmY4 and TmY9 also express personal communication. It should be noted that transcriptomic information so far exists for Dm3p and Dm3q, but not Dm3v.
LPi14 and LPi15 are predicted to be GABAergic on the basis of electron micrographs16,67, and presumed to be inhibitory. LPi07 cells are predicted to be GABAergic, glutamatergic or uncertain on the basis of electron micrographs, and are presumed to be inhibitory.
Hexagonal lattices are drawn in the figures as if they were perfectly uniform. The drawings are intended to portray only the nearest-neighbour relations of cells and columns, and do not accurately represent distances. More geometrically accurate representations of the lattices were constructed in ref. 29, which quantitatively characterized how lattice properties vary in space for the left optic lobe of the same electron microscopy dataset used in this study, and for many Drosophila eyes29. Visual acuity also varies across the retina in flies and other insects52,77.
Cell types that occur once per cartridge or column are said to be modular64, and are in one-to-one correspondence with hexels. I defined hexel cell types to be those that are modular with receptive fields that are one-ommatidium wide. Tm1, Tm2, Tm9, Mi1, Mi4 and Mi9 are included because the full-width at half-maximum of their receptive fields ranges from 6° to 8°, roughly equivalent to the angular spacing between ommatidia30. L1 to L5 are also included, on the basis of observed receptive fields33. (L3 turns out to be the main contributor to the disynaptic pathways studied.) This list of hexel types is provisional because receptive fields of modular types have not yet been quantified exhaustively.
The v762 reconstruction included 745 Tm1, 746 Tm2, 716 Tm9, 796 Mi1, 749 Mi4, 730 Mi9, 785 L1, 763 L2, 609 L3, 721 L4 and 776 L5 cells. These numbers are smaller than the total number of cells proofread in v783 (ref. 4), but the deficit is generally less than 10%.
All Mi1 cells were semi-automatically assigned to hexagonal lattice points. L cells were placed in one-to-one correspondence with Mi 1 cells using a Hungarian technique to assign their locations. The locations of other hexel types were assigned by placing them in one-to-one correspondence with L cells, again with the Hungarian algorithm.
The locations of hexel types are given in Supplementary Data 2. All three axes of the hexagonal lattice point are following the convention defined in ref. 29 The vertical axis is directed dorsally. The p and q axes are directed in the anterodorsal and posterodorsal directions in the medulla, respectively. Note that for Drosophila the hexagons of the lattice are oriented with flat sides at the top and bottom, and pointy tips at the left and right. The relation is more complex than shown in the first figure due to the medulla being curved rather than flat.
The figures depict the columns. The lattice is left–right inverted due to the optic chiasm, so it’s an excellent example of what a similar lattice can be used for. It is front-to-back motion on the medulla lattice that is behind the back-to-front motion on the retina. In other words, the p and q axes are swapped in the eye relative to the medulla. The p and q axes are in close proximity of the medulla, which has an anterior–posterior direction. The p and q axes are close together, which is closer to the shape of a hexagonal lattice.
The hexel coordinates approximation: a test of discrimination in neuronal morphology. Supplementary Data 4
Suppose I ran from 1 to N points of a hexagonal lattice, with hexel values (h_i) at the coordinates of the image. Normalizing the image yields a probability distribution. Get the coordinates of the image centroid.
This is how the mathopsum works: limits_i=1Np_i left.
We found that it was sufficient for feature dimensions to include only intrinsic types (T = 227). Alternatively, feature dimensions can be defined as including both intrinsic and boundary types (T > 700), and this yields similar results (data not shown).
The sums are all over the brain. If neuron i is a cell intrinsic to one optic lobe, the only nonvanishing terms in the sums are due to the intrinsic and boundary neurons for that optic lobe.
There is a cell type discriminator that takes the conjunction of the result and input and output fractions. A set of dimensions and threshold values are included in the search for a discriminator. To simplify the search, we require that the cell type be discriminated only from other types in the same neuropil family, rather than from all other types. It is almost always enough to use two dimensions of the normalized featureVector under these conditions.
Discriminators for all types are provided in Supplementary Data 4. Not all discriminations are accurate. Both intrinsic and boundary types are included as discriminative features.
We can see the Pm family in two-dimensional space of C3 input fraction and TmY3 output fraction. The Pm cells in this space can be compared to each other by the input fraction and output fraction, and can be discriminated with 100% accuracy. This conjunction of two features is a more accurate discriminator than either feature by itself.
The clustering of type T predicates: precision, recall and F-score. The problem re-opened and a tool to help find partners of all types
We also measured the drop in the quality of predicates if excluding boundary types (where the predicates are allowed to contain intrinsic types only). As with clustering metrics, the impact on predicates is marginal.
All possible combinations of input types will be looked for when computing the predicates, in order to find their precision, recall and F-score. A few optimization techniques are used to speed up this computation, by calculating minimum precision and recall thresholds from the current best candidate predicate and pruning many tuples early.
Recall of a predicate for type T is the ratio of true positive predictions (cells matching the predicate) to the total number of true positives (cells of type T). It measures the predicate’s ability to identify all positive instances of a given type.
The total number of predictions made by the logical predicate is greater than the true positive predictions that are of type T.
Once we arrived at the final list of types, we estimated the ‘centre’ of each type using the element-wise trimmed mean. Next, we computed the nearest type centre for every cell. For 98% of the cells, the nearest type centre coincided with the assigned type. We reviewed the disagreements manually. The human annotators had made some errors, most of which were of inattention, but the program was correct in most cases. Some of the remaining cases were due to errors. There were also cases in which type centres had been contaminated by human-misassigned cells (see the ‘Morphological variation’ section), which in turn led to more misassignment by the algorithm. After addressing these issues, we applied the automatic corrections to all but 0.1% of cells, which were rejected using distance thresholds.
Programmatic tools were created to help with searching for cells of the same type. One important script traced partners-of-partners that is source celldownstream partners and their upstream partners. This was based on the assumption that cells of the same type will probably synapse with the same target cells, which often turned out to be true. The tool could either look for partners-of-all-partners or partners-of-any-partners. The resulting lists of cells were very lengthy and were subject to filters that included cells that already had been identified or segments with low ID numbers. Another tool created from lobula plate tangential cells (for example, HS, VS, H1) aided definition of layers in the lobula plate. It enabled identification of many cell types, especially T4 and T5.
Citizen scientists created farms in FlyWire with all of the found cells visible. Farms showed visually where cells still remained to be found. If they found a bald spot, a popular method to find missing cells was to move the 2D plane in that place and add segments to the farm one after another in search of cells of the correct type. Farms also helped with identifying cells near to the edges of neuropils, where neurons are usually deformed. It was possible to observe how a cell at the edge should look from all the other cells.
Citizen scientists created a comprehensive guide with text and screen captures that expanded on the visual guide. They looked at all the publicly available scientific literature regarding the part of the body that doesn’t have hair. As of October 10, 2023, there were over 2,500 posts at discuss.flywire.ai. Community managers shared findings from the scientific literature with citizen scientists, as well as consulted flies specialists on FlyWire.
An environment for sharing ideas and information between community members was fostered by additional community resources. Community managers answered questions, provided resources, shared updates, and worked to organize the community activity. The discussion board/forums gave project progress information including the number of annotations submitted per individual. Live interaction, demonstrations and communal problem solving occurred during weekly Twitch video livestreams led by a community manager. The environment created by these resources allowed citizen scientists to self-organize in several ways: community driven information sharing, programmatic tools and ‘farms’.
Source: Neuronal parts list and wiring diagram for a visual system
Connectomic cell approach for the study of optic lobe synapses in the Eyewire9 citizen scientists and implications for synaptic and hemibral connectivity
The optic lobes are divided into five regions (neuropils): lamina of the compound eye (LA); medulla (ME); accessory medulla (AME); lobula (LO); lobula plate (LOP). All non-photoreceptor cells with synapses in these regions are split into two groups: optic lobe intrinsic neurons and boundary neurons.
The top 100 Eyewire9 players were invited to review their work. They were encouraged to label neurons when they were confident after 3 months of having their eyes checked. Most citizen scientists did a mixture of annotation and proofreading. Sometimes they annotated cells after proofreading, and other times searched for cells of a particular type to proofread.
Our connectomic cell approach to typing is initially seeded with some set of types, to define the feature vectors for cells (Fig. 2a), after which the types are refined by computational methods. For the initial seeding, we relied on the time-honoured approach of morphological cell typing, sometimes assisted by computational tools that analysed connectivity. Morphology is a misconception, because it only refers to shape. Orientation and position are actually more fundamental properties because of their influence on stratification in neuropil layers. Thus, ‘single-cell anatomy’ would be more accurate than morphology, although the latter is the standard term.
The connection might not exist as a result of other studies. T1 cells lack output. Therefore, in our analyses, we typically regarded the few outgoing T1 synapses in our data as false positives and discarded them.
Extreme asymmetrical in the matrix is one of the things to look for. The latter connection may be spurious if the number of connections is larger than the number of connections from A to B. The reason is that the strong connection from A to B means the contact area between A and B is large, which means more opportunity for false-positive synapses from B to A. False-positive rates for synapses are estimated in the flagship paper24.
In the central brain, most cell types have a mirror twin in the opposite hemisphere. In the hemibrain, the cardinality is typically reduced to one. There is not enough data currently to determine if there is a connection between cell type A and cell type B. It makes sense to set the threshold to a high value if false positives are to be avoided.
A matrix of connections was provided by the seven column reconstruction. This shows good agreement with our data (Methods and Extended Data Fig. 9), providing a check on the accuracy of our reconstruction in the optic lobe. This validation complements the estimates of reconstruction accuracy in the central brain that are provided in the flagship paper24.
The 7 column reconstruction that we agreed with stated that the Tm21, Dm2 and TmY5a are not modular. The definition ofModularity was violated by some of our types, which contain more than 800 proofread cells. This agrees with the seven column reconstruction28, which stated that T3 and T2a were modular, and T4 and Tm3 were not. T4 is an unusual case, as T4c is above 800 while the other T4 types are below 800. It should be noted that all of the above cell numbers could still creep upward with further proofreading.
Intrinsic neuronal parts list and wiring diagram for a visual system. Part 1: Cell classification and complexity of the flywire connectome
84% of neurons are intrinsic to the brain, meaning that their projections are fully contained in the brain volume3. Central brain neurons are fully contained in the central brain, while optic lobe intrinsic neurons are fully contained in the optic lobes. Visual projection neurons have inputs in the optic lobes and outputs in the central brain. Visual centrifugal neurons have inputs in the central brain and outputs in the optic lobe. The sensory neurons come from the centre of the brain and are divided into different classes. Refer to our companion paper for more details on the classification criteria5. The FlyWire community gave us labels that we used.
The inner and outer R7 and R8 are the cells that correspond to the flywire connectome. These numbers are not inconsistent with modularity because the under-recovery in this dataset is higher than usual and the numbers are difficult to read.
tmin left(x-t)
The weighted Jaccard distance is one minus the weighted Jaccard similarity. The quantities are nonnegative since the feature vectors are negative. We have found in our cell typing work that when features are sparse, Jaccard similarity works better than cosine similarity.
This cost function is convex, as d is a metric satisfying the triangle inequality. The cost function has a minimum. We used various approximate methods to minimize the cost function.
Source: Neuronal parts list and wiring diagram for a visual system
Autocorrection of Type Assignments for Neuronal Parts List and Wiring Diagram for a Visual System: A Coordinate Descent Approach
For auto-correction of type assignments, the trimmed mean was used. We found empirically that this gave good robustness to noise from false synapse detections. To minimize the cost function we used a coordinate descent approach. The loop did not include all of the i that were non-zero. The loop has this in it within a few times.
Each cluster is usually made up of a mixture of types from different families. The clustering noted above at the largest distances may be indicative of such mixing as a result of the noisiness. The closest types tend to be from the same family. But plenty of dendrogram merges between types of different families happen at intermediate distances rather than the largest distances. Some of the mixes from different families seems to be related to biology.
The wiring diagrams were drawn by us with Cytoscape 81. Organic layout was used for Figs. 3 and 7c, and hierarchical layout was used for the others. The hierarchical layout tries to make arrows point downwards. After Cytoscape generated a diagram, the nodes were shifted manually to make sure there were no obstructions.
Source: Neuronal parts list and wiring diagram for a visual system
Class, family and type: Axons and Narrow-Field Amacrine Cells in the Sm and Sm Families
Boundary neurons are those with at least 5% (and less than 95%) of synapses in the optic lobe regions, and are either visual projection, visual centrifugal or heterolateral neurons.
In the main text (in the ‘Class, family and type’ section), we used the term ‘axon’. Axon is a portion of the brain with a high ratio of presynapses to postsynapses. This ratio might be high in an absolute sense. Or the ratio in the axon might only be high relative to the ratio elsewhere in the neuron (the dendrite). In either case, the axon is typically not a pure output element, but has some postsynapses as well as presynapses. For many types it is obvious if there is an axon, and for a few we made judgement calls. Even without looking at the scepter, the axon can be seen from the presence of presynaptic boutons. The opposite of an axon is a dendrite, which has a high ratio of postsynapses to presynapses.
The axon–dendrite distinction does not apply when it comes to amacrine cells because the ratio of presynapses and postsynapses is the same throughout. The branches of amacrine Cells are often called dendrites, but the neutral term nerite is better for avoiding confusion.
Mi was defined as projecting from the center of the medulla to the south. Mi has both a lot and a lot of types. We identified five (Mi1, 2, 4, 9, 10) of the dozen Mi types originally defined6, and three (Mi13, 14, 15) types uncovered by EM reconstruction27. Mi1, Mi4, and Mi9 are consistent with the classical definition, but Mi13 projects from proximal to distal medulla. The term narrow-field amacrine may be more accurate than columnar for Mi types. Nevertheless we will adhere to the convention that they are columnar. In the Sm family, narrow-field amacrine cells exist.
There is a neuron projecting from the medulla to the side of the body. We will refer to this type as MLt1, and have discovered more types of the same family, MLt2 to MLt8. Mlt1 and Mlt2 dendrites span both distal and proximal medulla, and Mlt3 dendrites are in the distal medulla, so MLt1 to MLt3 receive L input (Supplementary Data 2 and 5). The Mlt4 dendrites are found inside the medulla. The Mlt8 to Mlt5 are connected with many Sm types that are to be discussed later in the data. Interaction between MLt types is fairly weak, with the exception of MLt7 to MLt5 (Supplementary Data 5). MLt7 and MLt8 are restricted to the dorsal and dorsal rim areas.
Source: Neuronal parts list and wiring diagram for a visual system
A Family of LPi Types for Increased Average Cell Volume and a Possible Correspondence between Two Cross-Brain Types
Pm1, 1a and 26 were each split into two types. Pm3 and 4 remain as previously defined85. We identified six new Pm types, numbered Pm01 to Pm14, in order to increase average cell volume. The new names can be distinguished from the old ones by the presence of leading zeros. All are predicted to have a calming effect. Pm1 was divided into Pm06 and Pm04, Pm1a into Pm05 and Pm03, and Pm2 into Pm03 and Pm08.
We discovered a type that was projected from the top of the structure to the bottom. This is not a family.
Silimetry is no longer enough for naming now that there are moreLPi types. Adding letters to distinguish between different kinds of cells could be used to rebuild the naming system. When that means full-field and’s’ means small, there are two names for it. For simplicity and brevity, we instead chose the names LPi01 to LPi15, in order of increasing average cell volume. Correspondences with old stratification-based names are detailed in Codex.
Here we show you two new families of cross- brain types, one is amacrine and the other is tangential. Along with two new tangential families that only contain single types, and the already known Am and CT1 types, there are a total of 21 cross-neuropil types that are non-columnar. Each of the new types (except PDt with 6 cells) contains between 10 and 100 cells.
Dm8 cells were divided into two types depending on whether or not they express DIP51, 53. Physiological studies demonstrated that yDm8 and pDm8 have differing spectral sensitivities89. The dendrites were found to have a connection with R7 in yellow and pale columns. Our Dm 8a, which is connected to Tm5a51, 53, may have some correspondence with yDm8 as well. It is not clear if there is a true one-to-one correspondence between pDm8 and yDm8. It is the case that Dm8a and Dm8b strongly prefer to synapse onto Tm5a and Tm5b, respectively. Tm5a and Tm5b are not in letters with yellow and pale columns. There are two main branches of Tm 5a, one of which is specific to yellow columns, and the other one of which is only found in pale columns. The ratio of yellow to pale columns is similar to the Dm8a and Dm8b cells, which are roughly equal in number. It is still speculative to have a correspondence with Dm8a and Dm8b. The yellow/pale issue may be reexamined in the future if accurate photoreceptorsynaptics become available.
Global Clustering Coefficient and Directional Connection Probabilities for SCCs: A Comparison of Null Models with Directed ER Models
SCCs are defined as subnetworks in which all neurons are mutually reachable through directed pathways63. The directionality of connections is not taken into account in the WCCs which are a relaxed criterion.
For a given neuron i, the in-degree ({d}{i}^{+}) is the number of incoming synaptic partners the neuron has and the out-degree ({d}{i}^{-}) is the number of outgoing synaptic partners the neuron has. The total degree of a neuron is the sum of in and out degrees.
We compared the statistics of the wiring diagram G(V,E) with the statistics of various null models. The ER model is directed where all edges are drawn independently at random, and the connection probability is set so that the number of edges in the ER model equals. The connection probability is constant for any nodes.
The (global) clustering coefficient is the probability that for three neurons α, β and γ, given that neurons α and β are connected and neurons α and γ are connected (regardless of directionality), neurons β and γ are connected:
We computed these metrics both across the whole-brain and within-brain-region (neuropil) subnetworks. We also systematically quantified the occurrence of distinct directed three-node motifs within the network, ensuring that duplicates are eliminated: any subgraph involving three unique nodes is counted only once in our analysis. We assumed the neurons connect independent of neurotransmitter and compared the relevant neurotransmitter probabilities for the motifs of interest. The expectation was compared to the true frequencies of these neurotransmitter combinations.
We made a generalized ER model to retain the expected number of reciprocal edges as the wiring diagram was over-represented. The generalized ER model ({\mathcal{G}}(V,{p}^{{\rm{uni}}},{p}^{{\rm{bi}}})) has two parameters, unidirectional connection probability puni and bidirectional connection probability pbi, both of which are set to match the wiring diagram. To do this, we defined the sets of unidirectional and bidirectional edges as:
Source: Network statistics of the whole-brain connectome of Drosophila
Random rewiring of graphs with the same degree sequences through a directed configuration model: A switch-and-hold algorithm 65
$$\begin{array}{c}{E}^{{\rm{uni}}}\,:= \,{(i,j){\rm{| }}(i,j)\in E\wedge (j,i)\notin E},\ {E}^{{\rm{bi}}}\,:= \,{(i,j){\rm{| }}(i,j)\in E\wedge E.endarray$$
$$\begin{array}{l}P\left[{i}{\nleftarrow }^{\to }\,j\right]=P\left[{i}{ \nrightarrow }^{\leftarrow }\,j\right]={p}^{{\rm{uni}}}=\frac{\left|{E}^{{\rm{uni}}}\right|}{\left|V\right|\left(\left|V\right|-1\right)},\ P\left[{i}{\to }^{\leftarrow }\,j\right]={p}^{{\rm{bi}}}=\frac{\left|{E}^{{\rm{bi}}}\right|}{\left|V\right|\left(\left|V\right|-1\right)},\ \left[{i}{ \nrightarrow }^{\nleftarrow }\,j\right]=1-2{p}^{{\rm{uni}}}-{p}^{{\rm{bi}}}.\end{array}$$
Consistent with previous work12,49, we also used a directed configuration model (CFG), ({\mathcal{G}}(V,{{d}{i}^{+}},\,{{d}{i}^{-}})), which preserves degree sequences during random rewiring. We sampled 1,000 random graphs uniformly from a configuration space of graphs with the same degree sequences as the observed graph by applying the switch-and-hold algorithm65, where we randomly select two edges in each iteration and swap their target endpoints (switch), or else keep them unchanged (hold), under the conditions that doing so does not introduce self-loops or multiple edges. With these conditions, this CFG model is mathematically equivalent to the Maslov–Sneppen edge-swapping null model66,67,68.
The NPC model was developed to give a more tractable spatial null model. The degree-corrected model is called the DC-SBM69. We assigned each neuron to one of the 78 ‘neuropil blocks’ based on the neuropil in which the neuron has the most outgoing synapses. The inter- and intra-neuropil connection probabilities are preserved during random rewiring. Moreover, like in the CFG model, we keep the degree sequences unchanged during randomization and prohibit self-loops and multiple edges. The interneuropil connection densities implicitly contain mesoscale spatial information. The total number of internal edges in each neuropil is still the same after reshuffling.
The identity matrix is where Pi and I are. The reverse Markov chain is similar to the mathcalLrmrev. The eigen-spectra of ({\mathcal{L}}) and ({{\mathcal{L}}}^{{\rm{rev}}}) are shown in Extended Data Fig. 1f,g, respectively. Conductance properties of the graph can be seen through the gaps between eigenvalues.
Pi -frac12 left
Source: Network statistics of the whole-brain connectome of Drosophila
Denseness of rich-club neurons in a neural network model and small-worldness quantification of the whole-brain fly connectome
The rich-club coefficients were computed in three different ways, by sweeping by total degree, in degree and out degree. As we observed, when the total degrees of the remaining nodes surpass 37, the network becomes denser compared with randomized networks. The peak occurs at total degree = 75. The model predicted the denseness of 38.9% of the neurons, but they are more dense than that. The network becomes sparse as the minimal total degree gets close to 93. We thought rich club neurons with total degrees above 37 were a sub network because of their denser connections. In terms of in-degree, the range for denser-than-random connectivity is between 10 and 54. Considering out-degree alone did not reveal any specific onset or offset threshold for rich-club behaviour, as the subnetwork always remains sparser than random. The rich-club coefficients are compared to the NPC model. The null model was computed similarly with 100 samples.
The standard method of determining the rich-club threshold is to look for values of k for which a s.d. of CFG and n is not specified. As the sample’s s.d. is small near the rich club coefficient, we decided to define the threshold of the rich club as 1.01.
We quantified the small-worldness of the connectome by comparing it to an ER graph. The average undirected path length in the ER graph, denoted as ({{\ell }}{{\rm{rand}}}), is estimated to be 3.57 hops, similar to the observed average path length in the fly brain’s WCC (({{\ell }}{{\rm{obs}}}=3.91)). The ER graph has a small clustering coefficients, which is much smaller than the observed clustering coefficients. (Table 2). The whole-brain fly connectome has a small-worldness coefficients.
To identify broadcaster neurons, we filtered the intrinsic rich-club neurons (dtot > 37) for those that had an out-degree that was at least five times higher than their in-degree:
We ran the model with subsets of sensory cells as seeds, including olfactory receptor cells, gustatory receptors cells, mechanosensory cells, head and neck cortex cells, thermosensory cells, and hygro sensory cells. The model was run with a set of input neurons as seed cells. All neurons in the brain were then ranked by their traversal distance from each set of starting neurons, and this ranking was normalized to return a percentile rank. The visual projection neurons are a proxy for visual input to the central brain, but this is not a true sensory population.