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Outline
- 1. Neural connectivity
- 2. Graphical models
- 3. Gaussian Graphical models
- 4. Functional connectivity in Macaque V4
Graphical models for Neuroscience Part I Giuseppe Vinci - - PowerPoint PPT Presentation
Graphical models for Neuroscience Part I Giuseppe Vinci - - PowerPoint PPT Presentation
Graphical models for Neuroscience Part I Giuseppe Vinci Department of Statistics Rice University 1/1 Outline 1. Neural connectivity 2. Graphical models 3. Gaussian Graphical models 4. Functional connectivity in Macaque V4 2/1 Neural
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Neural connectivity
How does the brain work? Through concerted ac- tions of many connected neurons. We need to know these connections to:
- Understand causes of brain disorders.
E.g. Parkinson, Alzheimer.
- Interface prosthetic devices.
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Neural connectivity
Two different perspectives
- structural connectivity = physical connections
- functional connectivity = neurons activity dependence
→ Understanding brain functions under differing experimental conditions. → It needs to be inferred from (in vivo) experimental data recordings, i.e. we need to see how neurons behave.
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Functional Connectivity = neurons activity dependence
Dependence between brain areas (macroscopic scale) fMRI, EEG, ECOG, LFP
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Functional Connectivity = neurons activity dependence
Dependence between neurons (microscopic scale)
video link
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Dependence
- Let X = (X1, ..., Xd) ∼ P be neural signals
- How could we describe the dependence structure of X?
- Analytically
- Contour density plots, spectral analysis
- Graphs
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Dependence Graph
Graph G = {N, E} where N = {1, ..., d} = nodes and E = [Eij] = edges with Eij ∈ {0, 1}.
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Markov Random Field
- A random vector X forms a Markov Random Field with respect to a graph G
if it satisfies the Markov properties
- 1. Pairwise: no edge ⇔ indep. conditionally on all other variables
- 2. Local: indep. conditionally on neighbors
- 3. Global: indep. conditionally on separators
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Gaussian Graphical Model
- A Gaussian Graphical model is a Markov Random Field
- The p.d.f. of X ∼ N(0, Θ−1), with Θ = [θij] ≻ 0, is
pΘ(x) = (2π)−d/2√ det Θ exp
- −1
2x′Θx
- =
exp −
d
- i,j=1
1 2θijxixj + const.
- θij = 0 ⇔ no edge between Xi, Xj ⇔ cond. independence
- Note that partial correlation ρij = −
θij
√
θiiθjj ∈ [−1, 1] is a more interpretable
measure of conditional dependence.
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Gaussian Graphical Model
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GGM - Estimation
- We need to estimate Θ.
- Suppose we collect data vectors Data = {X 1, ..., X n} from n repeat experiments.
- To estimate Θ, we start by defining the likelihood function
L(Θ; Data) =
n
- r=1
pΘ(X r)
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GGM - MLE
- ΘMLE = arg max
Θ≻0
L(Θ; Data)
- Simple but can have undesirable statistical properties because:
- Major problem: number of parameters in Θ is d +
d
2
- = d(d+1)
2
and sample size may be small in some applications. E.g. d = 100 ⇒ 5050 parameters and n = 200
- Minor problem:
ΘMLE does not give exact zeros.
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GGM - Penalized MLE
- Penalized MLE
- Θ(λ) = arg max
Θ≻0
log L(Θ; Data) − Pλ(Θ) where Pλ ≥ 0, and λ is chosen in some way (CV, BIC, Stability). E.g. Pλ(Θ) = λΘq
q (usually excluding diagonal elements)
- q = 0 penalizes number of edges.
- q = 1 is the graphical lasso1.
- q = 2 is the ridge regularization.
1Yuan and Lin (2007), Friedman et al.(2008), Rothman et al.(2008), Fan et al.(2009), Ravikumar et al.(2011), Mazumder and Hastie (2012), Hsieh et al.(2014).
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GGM - Bayesian regularization
- Penalized MLE can be seen as the maximum of the posterior distribution
π(Θ | Data) ∝ L(Θ; Data)
- likelihood
× π(Θ | λ)
- prior
where π(Θ | λ) ∝ e−Pλ(Θ)I(Θ ≻ 0).
- Note that we can also compute posterior means instead of max.
- The parameter λ can be selected by imposing a hyper prior π(λ).
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What if data is not Gaussian?
- Transform the data: e.g. √spike-count can be approximately Gaussian
- Hierarchical model: Y ∼ PX, and X ∼ N(0, Θ−1).
- Use another graphical model: e.g. Poisson graphs
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