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A Hierarchical State-Space Model with Gaussian Process Dynamics for Functional Connectivity EstimationRahul Nadkarni ([email protected]), Nicholas J. Foti ([email protected]), Adrian KC Lee ([email protected]), Emily B. Fox ([email protected])
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:
GP(m, k)<latexit sha1_base64="SQ2d7PrCkQ3rqqT9qEm5VOIsRuo=">AAAB+3icbVDLSsNAFJ3UV62vWJduhhahopREEV0WXeiygn1AE8pkOmmHziRhZiKGkL9w7caFIm79EXf9GydtF9p6YOBwzr3cM8eLGJXKsiZGYWV1bX2juFna2t7Z3TP3y20ZxgKTFg5ZKLoekoTRgLQUVYx0I0EQ9xjpeOOb3O88EiFpGDyoJCIuR8OA+hQjpaW+WXY4UiOMWHrbzGr8FI6P+2bVqltTwGViz0m1UXFOnieNpNk3v51BiGNOAoUZkrJnW5FyUyQUxYxkJSeWJEJ4jIakp2mAOJFuOs2ewSOtDKAfCv0CBafq740UcSkT7unJPKlc9HLxP68XK//KTWkQxYoEeHbIjxlUIcyLgAMqCFYs0QRhQXVWiEdIIKx0XSVdgr345WXSPqvb5/WLe93GNZihCA5BBdSADS5BA9yBJmgBDJ7AC3gD70ZmvBofxudstGDMdw7AHxhfP+S+lqs=</latexit>
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)
<latexit sha1_base64="JcAd+jU4xbiSD78ZTaaxY6KcoPQ=">AAACGXicbVDLSgNBEJyNrxhfqx69DAYhARN2o6IXIejFYwTzgCQus5NJMmb2wUyvGJb8hhd/xYsHRTzqyb9xNslBEwsaiqpuurvcUHAFlvVtpBYWl5ZX0quZtfWNzS1ze6emgkhSVqWBCGTDJYoJ7rMqcBCsEUpGPFewuju4TPz6PZOKB/4NDEPW9kjP511OCWjJMa1BDhx+iMG5y+Nz3FK855HbEm6xhzBX6OCcdjguJD7Oaz3vmFmraI2B54k9JVk0RcUxP1udgEYe84EKolTTtkJox0QCp4KNMq1IsZDQAemxpqY+8Zhqx+PPRvhAKx3cDaQuH/BY/T0RE0+poefqTo9AX816ifif14yge9aOuR9GwHw6WdSNBIYAJzHhDpeMghhqQqjk+lZM+0QSCjrMjA7Bnn15ntRKRfuoeHJ9nC1fTONIoz20j3LIRqeojK5QBVURRY/oGb2iN+PJeDHejY9Ja8qYzuyiPzC+fgDzoZx/</latexit>
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