A Hierarchical State-Space Model with Gaussian Process ...rahuln/pdf/2019-neurips-lmrl-post… · A...

A Hierarchical State-Space Model with Gaussian Process Dynamics for Functional Connectivity EstimationRahul Nadkarni (rahuln@cs.washington.edu), Nicholas J. Foti (nfoti@uw.edu), Adrian KC Lee (akclee@uw.edu), Emily B. Fox (ebfox@uw.edu)

Goal

• We present an approach for learningdirected, dynamic functionalinteractions between brain regionsacross multiple subjects frommagnetoencephalography (MEG) data

• We pool subjects’ data to learn agroup-level connectivity structure whilealso modeling interesting subject-specific differences

• Our method incorporates Gaussianprocesses to model functionalconnections that are smoothly time-varying while capturing estimationuncertainty

Problem setup

• Obtain MEG sensor recordings ! frommultiple subjects performing a task

• Observation model – which includesforward model (lead field matrix) " andsensor noise covariance # – is known(i.e., can be calculated) for each subject

• Goal is to infer time-varying, directedfunctional interactions between aspecified set of brain regions of interest(ROIs) using sensor recordings andobservation model for each subject

Results

• Hierarchical model is able to learnglobal dynamic connectivity structure(top left) as well as subject-specificvariation (other subplots)

• Bayesian inference allows the model tocapture uncertainty in the time-varyingconnectivity (blue band shows 95% CI)

Future directions

• Application to real MEG data recordedfrom multiple subjects performing anauditory attention task (analysisongoing)

• Incorporating subject-specific auxiliarydata (e.g., behavioral and cognitiveassessments, audiological test results,subjective questionnaire answers)

Description of model

• Sensor values !$(&) and observation model "(&), #(&) known for each subject

• ROI dynamics ((&)()), ((*+,-.+)()) and noise / unknown, inferred via Gibbs sampling• Entries of (()) indicate strength of directed interaction between regions over time

Simulation experiments

• Sensor data simulated from hierarchical model for 3 subjects and 2 ROIs (above), using real MEG

forward model "(&) and sensor noise

covariance #(&) for each subject

• Simulated dynamics ( *+,-.+ )

constructed to include a time-varying interaction from one ROI (blue) directed towards the other (red)

12

is a Gaussian process with mean function and covariance function . We use the squared exponential kernel:

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m<latexit sha1_base64="DEhp+DyYvHI4WO3F1aJPm+sIa5g=">AAAB6HicbZDLSgMxFIbP1Fsdb1WXboJFcFVmFNGNWHTjsgV7gXYomTTTxiaZIckIZegTuHGhiFt9GPduxLcxbV1o6w+Bj/8/h5xzwoQzbTzvy8ktLC4tr+RX3bX1jc2twvZOXcepIrRGYh6rZog15UzSmmGG02aiKBYhp41wcDXOG3dUaRbLGzNMaCBwT7KIEWysVRWdQtEreROhefB/oHjx7p4nb59upVP4aHdjkgoqDeFY65bvJSbIsDKMcDpy26mmCSYD3KMtixILqoNsMugIHVini6JY2ScNmri/OzIstB6K0FYKbPp6Nhub/2Wt1ERnQcZkkhoqyfSjKOXIxGi8NeoyRYnhQwuYKGZnRaSPFSbG3sa1R/BnV56H+lHJPy6dVL1i+RKmysMe7MMh+HAKZbiGCtSAAIV7eIQn59Z5cJ6dl2lpzvnp2YU/cl6/ATdnkDU=</latexit>

k<latexit sha1_base64="VE7fMrsTtUpC0ToJYouNmnwqX4Y=">AAAB6HicbZBNS8NAEIYn9avGr6pHL8EieCqJInoRi148tmA/oA1ls520azebsLsRSugv8OJBEa/6Y7x7Ef+N29aDtr6w8PC+M+zMBAlnSrvul5VbWFxaXsmv2mvrG5tbhe2duopTSbFGYx7LZkAUciawppnm2Ewkkijg2AgGV+O8cYdSsVjc6GGCfkR6goWMEm2s6qBTKLoldyJnHrwfKF682+fJ26dd6RQ+2t2YphEKTTlRquW5ifYzIjWjHEd2O1WYEDogPWwZFCRC5WeTQUfOgXG6ThhL84R2Ju7vjoxESg2jwFRGRPfVbDY2/8taqQ7P/IyJJNUo6PSjMOWOjp3x1k6XSaSaDw0QKpmZ1aF9IgnV5ja2OYI3u/I81I9K3nHppOoWy5cwVR72YB8OwYNTKMM1VKAGFBDu4RGerFvrwXq2XqalOeunZxf+yHr9BjRfkDM=</latexit>

k(ti, tj) = �2exp(�d(ti � tj)

2)

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model

Inference

• Gibbs sampling with iterative approximation methods for efficient

sampling from posterior of ((&)())• Conjugate gradient (CG):

Σ, x → approximation to Σ45x

used for 67 = 9[(;<(&) ⋅ ∣ ?@A)]

• Lanczos algorithm:Σ45, x → approximation to Σ5/DxUsed to sample from E(0, Σ)

• Given Σ745 = Cov (;<& ⋅ ?@A)

45:

CG to construct 67Lanczos to generate J7 ~E 0, Σ7Construct sample 67 + J7

A

(global)

ij (·) ⇠ GP(Iij , k0)

A

(s)ij (·) ⇠ GP(A(global)

ij (·), k1

), s = 1, . . . ,# subjects

x

(s)t+1

= A

(s)(t) x(s)t + ✏

(s)t , ✏

(s)t ⇠ N (0, Q)

y

(s)t = C

(s)x

(s)t + ⌘

(s)t , ⌘

(s)t ⇠ N (0, R(s))

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← ROIs

← sensors

Example of global time-varying connection with subject-specific variation