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Causal Dynamic Networks : ODE Network Modeling of fMRI
Xi (Rossi) LUO
Brown University
Department of Biostatistics
Center for Statistical Sciences
Computation in Brain and Mind
Brown Institute for Brain Science
Brown Data Science Initiative
ABCD Research Group
CMStatistics, Pisa Italy
December 16, 2018
Funding: NIH R01EB022911, P20GM103645, P01AA019072, P30AI042853; NSF/DMS (BD2K) 1557467
Co-Authors
Xuefei Cao
Brown Applied Math
Björn Sandstede
Brown Applied Math
Slides viewable on web:
bit.ly /cmstat18
fMRI Experiments
Task fMRI: performs tasks under brain scanning
Randomized stop/go task:
press button if "go";
withhold pressing if "stop"
Resting-state: "do nothing" during scanning
Goal: infer task-related brain activation and connectivity
Like this talk: want to know your brain processed
fMRI data : blood-oxygen-level dependent (BOLD) signals from each
cube /voxel (~millimeters),
$10^5$ ~ $10^6$ voxels in total.
Lego brain of real brain
red cube is a voxel
track activity from each cube at each time vector
dimension: address temporal separately
Conceptual Model with Stimulus
Sci Goal : infer intrinsic connections,
"Go" -task related connections,
"Stop" connections
Some Existing Methods and Limitions
Functional (nondirectional) connectivity:
Correlations
PCA, independent component analysis (ICA) Calhoun, Guo, and colleagues
Graphical models (inverse covariance)
Bayesian methods Bownman, Guindani, Vannucci, Zhang, and colleagues
Effective (directional) connectivity:
Granger causality, autoregressive modelsDing, Hu, Ombao, and colleagues
Structural equation models
Limitations:biophysical interpretability $\propto$ scalability${}^{-1}$
Fail to model task-depend connections/activations
Connections unlikely to be causal/neuronal
Some are hard to scale to large networks
Dynamic Causal Modeling (DCM)
Proposed by Friston et al, 2003 (> 3000 citations)
System approach to address previous limitations:
Latent neuronal states: a network ODE model
From neuronal states to observed BOLD signals: another ODE
(Bayesian) priors for model parameters
Bayes factors for comparing a few candidate models
DCM essentially unchanged for the past 15 yrs Friston et al, 17
DCM: Advantages and Limitations
Advantages:
Task -dependent, directional connections
Neuronal/causal connections
Model brain activations (and non-stationary time series)
Limitations:
Computationally expensive
Bayesian model comparison over exponentially many models
Model performance depends on priors Frassle et al, 15
Hard to scale (~10 nodes), some successes for simplified models
Mostly for hypothesis validation, not data driven
Causal Dynamic Networks
A two-level model
1. DCM neuronal state model (latent $\bm{x}$, stimulus $\bm{u}$):
$$\frac{d\bm{x}(t)}{dt}=\bm{A}\bm{x}(t)+\sum_{j}u_j(t)\bm{B_j}\bm{x}(t)+\bm{C}\bm{u}(t)$$
2. BOLD data model (data $\bm{y}$, noise $\bm{\epsilon}$) at discrete $t_i$:
$$ \bm{y}(t_i) =\int h(s)\bm{x}(t_i-s) ds + \bm{\epsilon} (t_i) $$
$h$ hemodynamic response function
$\bm{A}$ intrinsic connection matrix, $\bm{B}$ task-dependent connection tensor, $\bm{C}$ stimulus activation matrix
Functional/Dynamic Data Analysis
Usually, observed data model
$$ y(t) = x(t) + \epsilon(t) $$
and latent $x(t)$ follows an ODE model of interest
Various approaches for estimating the ODE parameters: nonlinear least squares Xue, Miao, Wu, 10 , two-stage smoothing Varah, 82 , principal differential analysis Ramsey, 96 , Bayesian Girolami, 08 , EcoG Zhang et al, 15
The observed data model not applicable to fMRI
For example, two-stage smoothing approaches not directly applicable to BOLD convolutions :
$$ y(t) = \int x(t-u)h(u) du + \epsilon(t) $$
Hemodynamic Response Function (HRF)
fMRI responses last long (~30 seconds) after neural activities
"Smooth" BOLD far from neuronal activity
Method
An optimization-based approach
Minimize the following
\[\begin{multline*} \scriptstyle
l(\bm{x},\bm{\theta})=\sum_{t_i} \| \bm{y}(t_i)-h \star \bm{x}(t_i)) \|^2 \\
\scriptstyle +\lambda\int \left \| \frac{d \bm{x} (t) } {dt} - (A \bm{x}(t)+\sum_{j} u_j(t) \bm{B_j} \bm{x}(t)+
\bm{C}\bm{u}(t)) \right\|^2 dt
\end{multline*} \]
Balancing data fitting errors and ODE fitting errors
Plug in basis-expansion of $\bm{x}(t) = \bm{\Gamma} \bm{\Phi}(t)$
Allows convolution (vs two-stage smooth approach)
Computationally fast to allow Bootstrap inference
Algorithm
Prove conditional convexity of $O(J d^2)$ parameters
Iterative block coordinate descent algorithm
Prove explicit update formulas (no numerical optimization algorithms needed)
Special Case: Resting-state fMRI
Set our parameter $\bm{B}$ and $\bm{C}$ to zero
Only fit intrinsic connection $\bm{A}$
Can fit much larger networks
Simulation: vs GCA/VAR
Our CDN yields higher network recovery accuracy than Granger Causality Analysis (GCA, aka vector autoregressive models)
Simulation: vs DCM
Our CDN yields higher accuracy using only a fraction of the computation time of DCM
Uncovering Neuronal States
Decent recovery of (latent) neuronal states
Task fMRI and Resting-state fMRI
Stop/Go fMRI
Brain activations and instrinsic connections between regions
Task Specific Connections
"Go" connections
"Stop" connections
Better understanding of brain mechanisms
Resting-state Connections
Ours (A) close to DCM (C), different from correaltions (B)
Real Data: 264 Brain Regions
CDN uncovers a large-scale brain network
Discussion
Joint optimization method for infer ODE networks
Flexible models for observations from causal ODEs
Computationally efficient for large-scale modeling
PyPI pacakge: cdn-fmri
Thank you!
Comments? Questions?
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