Connectal Coding

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**An Introduction to Graph Statistics**<br>
Joshua Vogelstein | 
{[BME](https://www.bme.jhu.edu/),[CIS](http://cis.jhu.edu/), [KNDI](http://kavlijhu.org/)}@[JHU](https://www.jhu.edu/)

.foot[[jovo@jhu.edu](mailto:j1c@jhu.edu) | <http://neurodata.io/talks> | [@neuro_data](https://twitter.com/neuro_data)]

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## .center[https://neurodata.io/graspy/]

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## Outline

- Background
- Statistical Models of Connectomes 
- Statistical Models of Populations of Connectomes
- Applications
- Discussion

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## .center[.k[Background]]

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### What is a Connectome?

- .r[Network] of a brain, at a spatiotemporal precision & extent
- .r[Nodes] are distinct biophysical entities 
- .r[Edges] indicate the presence of a connection/communication between nodes
- .r[Attributes] of the network, nodes, or edges are possible

<br>

- Example nodes: cells, cellular compartment, cellular ensembles
- Example edges: synapses, gap junction, fiber bundles

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## Many adjacency matrices, one graph

Drosophila larva connectome, Eichler et al. 2017
<img src="images/multi-adj-dros.png"  style="width: 700px;"/>

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### Definitions

.r[Neural (activity) coding]: inferring the relationships between neural *activity* and past, present, or future events, states, or traits

.r[Connectal coding]: inferring the relationships between neural *connectivity* and past, present, or future events, states, or traits

.s[events]: genetic, developmental, experiential (stimuli/behavior)

.s[traits]: IQ, sex, personality, learning disabled

.s[states]: happy, resting, manic

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### Implications

- A brain could have many different connectomes, at different times and/or resolutions, and measured in different ways (e.g. structural or functional)
- The definition of node can mean different things at different scales
- We measure properties of the brain  to .r[estimate] connectomes
- Estimates are always .r[noisy]  -->

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### Why Use Statistical Models?
 
- Connectome estimates are noisy
- Connectomes, and their relationships to events, states, or traits, can be complicated 
- We wish to 
  - quantify uncertainty
  - incorporate domain knowledge to the extent possible
  - summarize natural phenomena in a "simple" way
  - understand limitations of analyses

### Implications
- All inferences about population .r[depend on model]

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### Connectome Analysis Styles

- Bag of edges
- Bag of features
- Bag of parameters

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### Bag of Edges

- Treat each edge as independent
- Implicit model: independent edge model

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### Bag of Features

- Choose $m$ features and compute them per graph
- Characterize connectome with this set of "parameters"
- Implicit model: exponential random graph model

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### Bag of Parameters

- Build a **statistical parametric model** of brain network
- Can encode some domain knowledge explicitly 
- Can model edges, nodes, communities
- Implicit model: latent structure model

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### Limitations of approaches

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### Limitations of Bag of Edges

- Completely ignores graph structure of data 
- Too simple for many questions

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#### Sometimes the signal is in the node

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#### Edge-wise stats show no significance

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#### Node-wise test finds the signal

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### Limitations of Bag of Features

- how do I choose which features (hint: arbitrary)? 
- how many features are possible given a graph with $n$ nodes (hint: many)?
- do these features characterize the brain (hint: no)?
- can we make causal claims using these features (hint: no)?
- are these features independent (hint: no)?
- least well understood of the approaches, but very common

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### Same Stats, Different Graphs
<div>
<img src="images/same_stats_diff_graphs.png" style="height: 400px;" align="right"/>
</div>
- num vertices = 12
- num edges = 21 
- number of triangles = 10
- global clustering coefficient = 0.5

.foot[[Chen et al.](https://link.springer.com/chapter/10.1007/978-3-030-04414-5_33)]

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### Distribution of Features, n=10

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### Condition on "close" to base graph

.footnote[(edges=31, threshold=3, n=200k)]

<!-- Model of 1 graph
  - for each of the 3 approaches (IE, ERGM, LSM)
    - define different approaches
    - give examples
  - demonstrate pro's and con's of different approaches
    - prob with ER: too simple
    - prob with IE: sometimes signal is in the nodes
    - prob with ERGM: j1c's thingy
    - prob with latent positions: less intuitive
    - value of latent positions:
      - characterize the data

- model of \geq 2 graphs
    - correlated ER
    - COSIE
    - MRDPG
  - applications of multigraph models
      - test for equality, mouse example
      - test for indepedence, bear example
      - test for heritability
      - graph matching
  - mic drop

-->

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### Limitations of Bag of Parameters

- Conceptually less intuitive

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## .center[.k[Statistical Models of Connectomes]]

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### Erdos-Renyi (ER)

- akin to assuming a neuron's spike rate is Poisson with a fixed rate.
- all edges independent
- all edges sampled from identical distribution
- only 1 parameter: prob of an edge
- $\mathbb{P}[A_{i,j}] = p$

Notes
- <b> Simplest random graph model; lacks descriptive power </b>

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### Drosophila Connectome ER

<br>

<img src="images/dros_er_model.png" style="width: 800px;" />
- p = 0.166

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### Degree Corrected Erdos-Renyi (DCER)

- edges are independent 
- edges are sampled from .r[different] distributions
- .r[n+1 parameters]: degree correction for each node
- $\mathbb{P}[A_{i,j}] = \theta_i\theta_jp$

Notes 
- n+1 paramers is much larger than 1
- still ignores structure

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### Drosophila Connectome DCER

<br>

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### Stochastic Block Model (SBM)

- akin to assuming a neuron's are in different states, which determine Poisson rate.
- edges are .r[conditionally] independent 
- each node has a class assignment
- $\mathbb{P}[A_{i,j}]$ = $B$(class i, class j)

Notes
- simplest >2 parameter model

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### Drosophila Connectome SBM

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### Degree-corrected Stochastic Block Model (DCSBM)

- edges are .r[conditionally] independent
- each node has a class assignment
- $\mathbb{P}[A_{i,j}]$ = $\theta_i\theta_jB$(class i, class j)

Notes
- simplest >2 parameter model

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## Drosophila Connectome DCSBM

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### Random Dot Product Graphs (RDPG)

- akin to latent state models in population coding
- edges are conditionally independent 
- each node has a .r[latent position in d-dimensions]
- $\mathbb{P}[A_{i,j}]$ = f(latent position i, latent position j)
- for example, $\mathbb{P}[A_{i,j}]$ is the dot product of latent positions

Notes
- generalizes previous models

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### Latent positions allow for more general relationships

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### Drosophila Connectome RDPGs

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### Model considerations/extensions

- Directed extensions exist
- Loopy extensions exist
- Weighted extensions exist (mostly)

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### What can we do with parameters?

- Node clustering or classification
- Bootstrap? -->

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## .center[.k[Statistical Models of] .r[Populations of] .k[ Connectomes]]

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## Population Graph Models

- Joint RDPG
- Common Subspace Independent Edge Graph (COSIE) -->

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### Joint Random Dot Product Graphs

- All nodes have a latent position in d-dimensional space
- Each graph has a latent position matrix

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### Common Subspace Independent Edge Graph (COSIE)

- Common latent position matrix shared across all graphs
- Individual graphs are transformation of the common matrix

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## .center[.k[Application of Population Models]]

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### Mouse Connectomes From Same Genotype are Similar

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### Structural Connectomes are Heritable

- MRDPG model

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### COSIE Model Can Recover Bilateral Separation

- HNU1 Dataset

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### COSIE Model Can Recover Clusters of Same Test-Retest Scans

- HNU1 Dataset

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### COSIE Model Can Perfectly Classify of Subjects

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## .center[.k[Discussion]]

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## Summary and Next Steps

- Connectomes are the  mechanistic link:

.center[.r[genotype --> phenotype]]

- Extend ideas from coding theory to support these analyses
- Connectomes, genetic and phenotypic data are available

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### References

- Connectal Coding [[1]](https://doi.org/10.1016/j.conb.2019.04.005)
- Description of GraSPy [[2]](https://arxiv.org/abs/1904.05329)
- Statistics on RDPG [[3]](https://dl.acm.org/citation.cfm?id=3242083)
- Two-sample hypothesis testing for RDPG [[4]](https://arxiv.org/abs/1403.7249)
- Two-sample hypothesis testing for two random graphs [[5]](https://projecteuclid.org/euclid.bj/1489737619)
- COSIE model and estimation [[6]](https://arxiv.org/abs/1906.10026)
- Omnibus Embedding for JRDPG estimation [[7]](https://ieeexplore.ieee.org/document/8215766)
- Mouse Connectome Heritability [[8]](https://www.biorxiv.org/content/10.1101/701755v1)
- Connectome smoothing [[9]](https://ieeexplore.ieee.org/document/8570772)

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### Acknowledgements

<div class="small-container">
  <img src="faces/jovo.png" />
  <div class="centered">Joshua Vogelstein</div>
</div>

<div class="small-container">
  <img src="faces/jaewon.jpg" />
  <div class="centered">Jaewon Chung</div>
</div>

<div class="small-container">
  <img src="faces/pedigo.jpg"/>
  <div class="centered">Ben Pedigo</div>
</div>

<div class="small-container">
  <img src="faces/ebridge.jpg"/>
  <div class="centered">Eric Bridgeford</div>
</div>

<div class="small-container">
  <img src="faces/hayden.png"/>
  <div class="centered">Hayden Helm</div>
</div>

<div class="small-container">
  <img src="faces/jesus.jpg"/>
  <div class="centered">Jesus Arroyo</div>
</div>

<div class="small-container">
  <img src="faces/ronak.jpg"/>
  <div class="centered">Ronak Mehta</div>
</div>

<div class="small-container">
  <img src="faces/cep.png"/>
  <div class="centered">Carey Priebe</div>
</div>

<div class="small-container">
  <img src="faces/randal.jpg"/>
  <div class="centered">Randal Burns</div>
</div>

<div class="small-container">
  <img src="faces/mim.jpg"/>
  <div class="centered">Michael Miller</div>
</div>

<div class="small-container">
  <img src="faces/dtward.jpg"/>
  <div class="centered">Daniel Tward</div>
</div>

<div class="small-container">
  <img src="faces/vikram.jpg"/>
  <div class="centered">Vikram Chandrashekhar</div>
</div>

<div class="small-container">
  <img src="faces/drishti.jpg"/>
  <div class="centered">Drishti Mannan</div>
</div>

<div class="small-container">
  <img src="faces/jesse.jpg"/>
  <div class="centered">Jesse Patsolic</div>
</div>

<div class="small-container">
  <img src="faces/falk_ben.jpg"/>
  <div class="centered">Benjamin Falk</div>
</div>

<div class="small-container">
  <img src="faces/loftus.jpg"/>
  <div class="centered">Alex Loftus</div>
</div>

<div class="small-container">
  <img src="faces/bcaffo.jpg"/>
  <div class="centered">Brian Caffo</div>
</div>

<div class="small-container">
  <img src="faces/avanti.jpg"/>
  <div class="centered">Avanti Athreya</div>
</div>

<div class="small-container">
  <img src="faces/vince.jpg"/>
  <div class="centered">Vince Lyzinski</div>
</div>

<div class="small-container">
  <img src="faces/dpmcsuss.jpg"/>
  <div class="centered">Daniel Sussman</div>
</div>

<div class="small-container">
  <img src="faces/youngser.jpg"/>
  <div class="centered">Youngser Park</div>
</div>

<div class="small-container">
  <img src="faces/cshen.jpg"/>
  <div class="centered">Cencheng Shen</div>
</div>

<div class="small-container">
  <img src="faces/shangsi.jpg"/>
  <div class="centered">Shangsi Wang</div>
</div>

<div class="small-container">
  <img src="faces/ronan.jpg"/>
  <div class="centered">Ronan Perry</div>
</div>

<div class="small-container">
  <img src="faces/vivek.jpg"/>
  <div class="centered">Vivek Gopalakrishnan</div>
</div>

<div class="small-container">
  <img src="faces/tommy_athey.jpg"/>
  <div class="centered">Tommy Athey</div>
</div>

<div class="small-container">
  <img src="faces/patsolic_heather.jpg"/>
  <div class="centered">Heather Patsolic</div>
</div>

<div class="small-container">
  <img src="faces/bijan.jpg"/>
  <div class="centered">Bijan Varjavand</div>
</div>

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## .center[.k[Additional Information]]

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## Adjacency Spectral Embedding

- Method for estimating parameter for RDPG model (for single graph)
- $\hat{X} = UD^{1/2}$ where $U, D, V = SVD(A)$.

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## Omnibus Embedding

- Method for estimating parameters for Joint RDPG model

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## Multiple Adjacency Spectral Embedding (MASE)

- Method for estimating parameters for COSIE model

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### Genotype, Phenotype, Connectotype

- .r[Phenotype]: a description of an individual's  properties with regard to a phenomenon of interest 
- .r[Genotype]: a set of genes and associated variants associated with that phenotype
- .r[Connectotype]: a set of nodes, edges, and their properties that are associated with that phenotype

.center[Genotype --> Connectotype --> Phenotype]

**Connectotypes are the implementation-level mechanisms linking genotypes to phenotypes**

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