Connectal Coding

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**Connectal Coding: Discovering the Mechanism Linking Cognitive Phenotypes to Individual Histories**<br>
Joshua T. Vogelstein | 
{[BME](https://www.bme.jhu.edu/),[ICM](https://icm.jhu.edu/),[CIS](http://cis.jhu.edu/),[KNDI](http://kavlijhu.org/)}@[JHU](https://www.jhu.edu/)

.foot[[jovo@jhu.edu](mailto:jovo@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
- Example Connectomes
- Connectal Coding 
- Applications
- Discussion

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## .k[Background]

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### Two definitions

.r[Neural (activity) coding]: inferring the relationships between neural activity and stimuli or responses

.r[Connectal coding]: inferring the relationships between *neural circuit structure* and *individual histories*

- .r[Individual histories]: genetic, developmental, experiential 
- .r[Neural circuit structure (aka, a connectome)]:  a network of a brain

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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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### Connectome in Literature

- Sporns et al.  *PLoS CB* (2005)
- Hagmann (2005)
- PubMed (circa 2018): .r[3000+] hits.

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### Connectome according to jovo

- .r[Network] of a brain, at a spatiotemporal precision & extent
- .r[Nodes] are distinct biophysical entities 
- .r[Edges] are  *structural*  objects connecting a pair of nodes
- .r[Attributes] of the network, nodes, or edges are possible

<br>

- Example nodes: neurons, neural compartment, neural ensembles
- Example edges: synapses, gap junction, fiber bundles

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### Implications

- A brain could have many .r[different] connectomes, at different times and/or resolutions
- We measure properties of the brain  to .r[estimate] connectomes
- Those measurements can be .r[structural] or .r[functional]
- Estimates are always .r[noisy]

### Some caveats

- connectomes are not comprehensive, unlike genomes
- nodes can be abstractions, eg, all neurons type A 
- attributes can be arbitrary, eg, distribution of synaptic vesicles 
- connectomics is the study of neural circuitry

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### Connectomes for Connectal Coding

<br>

Hhypothesis generation not hypothesis testing

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## .k[Example Connectomes]

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### C Elegans

- directed, multi

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###  Drosophila Mushroom Body

- directed

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### Mouse Ex Vivo Diffusion MRI

- undirected

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### Human  MRI

- undirected, multi (semi-dense)

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## .k[Connectal Coding]

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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
- does this completely ignores graph structure of data (hint: yes)?
- requires multiple hypothesis correction for valid tests
- does anybody know a good way to correct (hint: no)?
      - BH: way under-conservative  (false positives)
      - Bonferroni: way over-conservative (false negatives)
      - Network based statistics: no theoretical guarantees

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### Bag of Features

- choose m features and compute them per node or graph 
- how do i choose m? how do i choose which features (hint: arbitrary)? 
- how many features are possible given a graph with n nodes (hint: many)?
- does these features characterize the brain (hint: no)?
- can we make causal claims using these features (hint: no)?
- least well known of the approaches

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### Bag of Parameters

- build a statistical parametric model of brain network
- does it treat edges independently (hint: no)?
- do we have ways of choosing model and model complexity (hint: kind of)?
- do these models characterize the brain (hint: yup!)?
- can we make causal claims (hint: kind of)?

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## What is a Code?

A code is a system of (potentially stochastic) rules that translate from one representation of information into another.

.pull-left[
Examples: 
- Morse: one-to-one
- genetic: deterministic
- neural: stochastic
]

.pull-right[
<img src="https://upload.wikimedia.org/wikipedia/commons/b/b5/International_Morse_Code.svg"  style="height: 400px;"/>
]

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## Statistical Coding

- Random variables: X, Y
- Distributions: X ~ P, Y ~ P 
- Conditionals:  P[X | Y], P[Y | X]

Formally, codes are conditional distributions.

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## Connectal Coding Model

Random variables of interest include:

- P: phenotypes 
- C: connectomes 
- G: genome  
- E: environment

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## Abstract Connectome Codes

- Pr[C | G]: prob of connectome, given a genome 
- Pr[P | C]: prob of phenotype, given a connectome
- Pr[P | C, G, E]: probability of phenotype given connectome, genome, and environment

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## .k[Statistical Models of Connectomes]

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### Independent & Identical Edges

Erdos-Renyi (ER): akin to assuming a neuron's spike rate is Poisson with a fixed rate.

- edges are binary
- all edges independent
- all edges sampled from identical distribution 
- $\Rightarrow$ only 1 parameter: prob of an edge

Notes
- directed vs. undirected
- loopy vs. no loops
- Simplest random graph model 
- lacks sufficient complexity/descriptive power for most questions

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### Independent & Identical Edges

.r[Weighted] Erdos-Renyi: akin to Poisson model using a bigger bin width.

- edges can take  .r[any value]
- edges are independent 
- edges are sampled from identical distribution 
- $\Rightarrow$ can still only be 1 parameter: expected weight  of an edge

Notes
- directed vs. undirected
- loopy vs. no loops
- simplest .r[weighted] random graph model 
- lacks sufficient complexity/descriptive power for most questions

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### Independent & Identical Edge

.r[Zero-Inflated] Weighted Erdos-Renyi: akin to assuming a bursty neuron, modeling both probability of burst and expected number of spikes in each burst

- edges can take any value
- edges are independent 
- edges are sampled from identical distribution 
- .r[2 parameters]: prob of edge, and expected weight of edge.

Notes
- directed vs. undirected
- loopy vs. no loops
- simplest sparse weighted random graph model 
- can provide useful/interesting description of a connectome
 
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### IIE Models of Connectomes

<br>

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### Independent Edge Model

- edges are binary 
- edges are independent 
- edges are sampled from .r[differnt] distribuions
- .r[n*n parameters]: prob of edge between each pair. 
- P[ A(i,j) ] = p(i,j)

Notes 
- same generalizations as above apply here as well 
- n*n paramers is much larger than 1,
- still ignores structure
- can't fit without lots of samples or further assumptions/restrictions

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## Categorical Conditionally Independent Edge Models

Stochastic Block Model (SBM): akin to assuming a neuron's are in different states, which determine Poisson rate.

- edges are binary
- edges are .r[conditionally] independent 
- each node has a class assignment
- P[ A(i,j) ] = B(class i, class j)

Notes
- directed vs. undirected
- loopy vs. no loops
- simplest >2 parameter model

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## Connectome SBMs

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## Generalized SBMs

Weighted Stochastic Block Model (SBM)

- edges are .r[weighted]
- edges are conditionally independent 
- each node has a class assignment
- P[ A(i,j) ] = B(class i, class j) is expected weight of connection

Notes
- directed vs. undirected
- loopy vs. no loops

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## Generalized SBMs

Zero-Inflated Weighted Stochastic Block Model (SBM)

- edges are weighted
- edges are conditionally independent 
- each node has a class assignment
- P[ A(i,j) ] = is defined by a matrix of probabilities of connection, and a matrix of expected weights

Notes
- directed vs. undirected
- loopy vs. no loops

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## Continuous Conditionally Independent Edge Models

Random Dot Product Graphs (RDPG): akin to latent state models in population coding

- edges are binary
- edges are conditionally independent 
- each node has a .r[latent position in d-dimensions]
- P[ A(i,j) ] = f(latent position i, latent position j)
- for example, P[ A(i,j) ] is the product of latent positions

Notes
- directed vs. undirected
- no loops is ickier
- generalizes previous models

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## Connectome RDPGs

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## Generalized RDPG

- Weighted RDPG: edges have weights 
- Zero-Inflated Weighted RDPG: edges have probabilities and expected weights

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## Latent Structure Models

- Special case of RDPG, where latent positions are organized into .r[structures]
- Examples
  - each node class has a distribution of latent positions, eg, Gaussian 
  - latent positions are hierarchical, eg, multiscale atlas 
  - repeated motif, eg, cortical columns 
  - latent positions are curved

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## Drosophila Mushroom Bodies

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## Population Graph Models

- Mixture of RDPG 
- Joint Heterogeneous RDPG

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## .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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### Acknowledgements

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  <div class="centered">Eric Bridgeford</div>
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  <div class="centered">Ben Pedigo</div>
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  <div class="centered">Daniel Tward</div>
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  <div class="centered">Benjamin Falk</div>
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  <div class="centered">Vince Lyzinski</div>
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  <div class="centered">Daniel Sussman</div>
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  <div class="centered">Youngser Park</div>
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  <div class="centered">Cencheng Shen</div>
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  <div class="centered">Shangsi Wang</div>
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  <div class="centered">Tyler Tomita</div>
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</div><span style="font-size:200%; color:red;">♥, 🦁, 👪, 🌎, 🌌</span>

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