Cross references:  Red Nucleus    Red Nucleus Cerebellum    

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Cerebellum (Wiki) 
    "The cerebellum (Latin for little brain) is a region of the brain that plays an important role in motor control."  

Drawing of the human brain, showing cerebellum and pons

The cerebellum does not initiate movement, but it contributes to coordination, precision, and accurate timing. It receives input from sensory systems of the spinal cord and from other parts of the brain, and integrates these inputs to fine tune motor activity.[2]

File:Cerebellum animation small.gif
Full length article available for free.  From the abstract: 
    "... we delineate the role of the cerebellum in several nonmotor systems simultaneously and in the same subjects using resting state functional connectivity MRI. Independent component analysis was applied to resting state data from two independent datasets to identify common cerebellar contributions to several previously identified intrinsic connectivity networks (ICNs) involved in executive control, episodic memory/self-reflection, salience detection, and sensorimotor function. We found distinct cerebellar contributions to each of these ICNs. The neocerebellum participates in
    (1) the right and left executive control networks (especially crus I and II),
    (2) the salience network (lobule VI), and
    (3) the default-mode network (lobule IX).
Little to no overlap was detected between these cerebellar regions and the sensorimotor cerebellum (lobules V-VI). Clusters were also located in pontine and dentate nuclei, prominent points of convergence for cerebellar input and output, respectively. The results suggest that the most phylogenetically recent part of the cerebellum, particularly crus I and II, make contributions to parallel cortico-cerebellar loops involved in executive control, salience detection, and episodic memory/self-reflection. The largest portions of the neocerebellum take part in the executive control network implicated in higher cognitive functions such as working memory."