Understanding of eye movements is aided by knowledge of their evolutionary development, and readers are strongly encouraged to look at the section on the evolution of eye movements.

1. Supranuclear pathways:

The supranuclear (ie, above the oculomotor nuclei) ocular motor system is principally concerned with bilateral eye movements and as such causes bilateral eye movement deficiencies such as gaze preference or palsy; however, supranuclear dysfunction is also a frequent cause of ocular misalignment and binocular diplopia1.
The supranuclear ocular motor system has rich afferent connections, including projections from the cerebral hemispheres (frontal eye fields and basal ganglia), cerebellum (for calibration and modulation of vestibular and otolith systems), and all portions of the brainstem ( vestibular and otolith projections) which ultimately converge on the ocular motor nuclei and serve to govern the distinct classes of eye movements, including saccades, pursuit, the vestibuloocular reflex (VOR), gaze holding, fixation, optokinetic nystagmus, and vergence (convergence and divergence).

The same motor neurons and extraocular muscles are active for all types of eye movements (eg, saccades, pursuit and vergence), indicating that the systems generating these different movement types share a final common path2

2. Midbrain centers for Saccades and Gaze Holding:

The center for vertical/torsional saccades is the rostral interstitial nucleus of the medial longitudinal fasciculus (riMLF), and the center for vertical gaze-holding function is the interstitial nucleus of Cajal (INC).
Clinically, this means that an isolated vertical saccadic paresis or isolated vertical gaze evoked nystagmus would suggest a midbrain lesion. 
The INC projects to the ocular motoneurons, mainly via the posterior commissure3.

3. Pontine and pontomedullary centers for Saccades and Gaze Holding:

The connections of the horizontal semicircular canal to the medial vestibular nucleus in the medulla activate second order neurons in the vestibular nucleus, which project to the contralateral abducens nucleus.
The abducens nucleus projects to:
  -Ipsilateral lateral rectus (the 3rd neuron in the three neuron arc of the vestibulo-ocular reflex)
  -Contralateral medial rectus via the medial longitudinal fasciculus.

Pons
The center for horizontal saccades is the paramedian pontine reticular formation (PPRF)
Clinically, this means that isolated horizontal saccadic palsy indicates a pontine lesion, and a unilateral PPRF lesion will result in saccadic disturbances on the side of the lesion. 

Medulla

The medulla contains most of the vestibular nuclei. Other major medullary structures involved in the control of eye movements include the perihypoglossal nuclei, including the nucleus prepositus hypoglossi (NPH), the paramedian nuclei and tracts, the inferior olivary nuclei, and the inferior cerebellar peduncles.

Components of the neural integrator in the medulla

The center for horizontal gaze-holding function is the NPH, which acts in concert with the medial vestibular nucleus of the medulla, and the vestibulocerebellum, together forming the “neural integrator.”

Disturbances of the medulla are best described in relation to a stroke of the dorsolateral medulla, a Wallenberg syndrome.

   

 
Vestibular nuclei
The vestibular nuclei receive information about the vestibular function of the opposite ear through the vestibular commissure which carries excitatory and inhibitory
signals from the vestibular systems, left and right. Multimodal integration of information at the level of the vestibular nuclei takes place allowing vision and proprioception
to guide vestibular responses.

Multimodal input to Vestibular Nuclei

   

Input to the vestibular nuclei comes from the three paired semicircular canals, the otolith organs, and from the cerebellum.  There is also vestibular nerve input directly to
the cerebellum (flocculo-nodular lobe)(the vestibular nuclei can be viewed as ectopic cerebellar nuclei).
  Projections to the vestibular nuclei from the vestibular apparatus derive from:
  -The semicircular canals predominantly to the rostral portions of the vestibular nuclei complex (medial and superior vestibular nuclei).
  -The otolith organs extend predominantly to the caudal portions of the vestibular nuclei complex (lateral and inferior vestibular nuclei).  

    Output from the vestibular nuclei (superior) runs in the medial longitudinal fasciculus, connecting the oculomotor nuclei together (3rd, 4th and 6th).
Similarly, output from the vestibular nuclei in the form of the vestibulospinal tracts passes to the brain stem and spinal cord, and brings information about the body's position in space
to the antigravity muscles.

See: Vestibular Neuron & Vestibular Nuclei

4. Cerebellar centers:

The cerebellum performs critical coordination and calibration functions for the ocular motor system, particularly the vestibulocerebellum (flocculus, paraflocculus, nodulus, and uvula), vermis, and fastigial nuclei1.

The flocculus and paraflocculus are involved in smooth pursuit, gaze holding, and calibration of the VOR.
The vermis and fastigial nuclei are involved in saccadic and pursuit control. The nodulus and uvula participate in modulation of the vestibular system.

Cerebellar lesions are often accompanied by clinically easily identifiable oculomotor disturbances: 

Figure 1. Brainstem centres for gaze holding and saccade generation

Figure redrawn from Clinical Examination of the Ocular Motor and Cerebellar Ocular Motor System. Strupp, M. https://www.youtube.com/watch?v=meXAjVoQdCI

 

 

References

  1. Eggenberger ER. Supranuclear eye movement abnormalities. Continuum (Minneap Minn). 2014 Aug;20(4 Neuro-ophthalmology):981-92. doi: 10.1212/01.CON.0000453308.50604.52.
  2. Ramat S, Leigh RJ, Zee DS, Optican LM. What clinical disorders tell us about the neural control of saccadic eye movements. Brain. 2007;130(Pt 1):10-35.
  3. Kheradmand A, Colpak AI, Zee DS. Eye movements in vestibular disorders. Handb Clin Neurol. 2016;137:103-17.