Microsaccades, Drifts, Hopf Bundle and Neurogeometry

Area of Science:

• Computational Neuroscience
• Computer Vision
• Mathematical Modeling

Background:

• Early vision image processing involves static and dynamic conditions, including eye movements.
• Donders' and Listing's laws describe eye movement kinematics but lack a unified theoretical framework.

Purpose of the Study:

• To interpret Donders' and Listing's laws using Hopf fibration of the 3-sphere over the 2-sphere.
• To model the configuration space of the eyeball and saccades using geometric principles.
• To analyze fixation eye movements (drift, microsaccades) and their role in vision.

Main Methods:

• Review of image processing in static and dynamic vision.
• Mathematical interpretation of Donders' and Listing's laws via Hopf fibration.
• Geometric modeling of the eyeball's configuration space (Listing hemisphere) and saccades as geodesics.
• Analysis of fixation eye movements within the proposed geometric framework.

Main Results:

• The eyeball's configuration space (head fixed) is identified as the 2D hemisphere SL+, the Listing hemisphere.
• Saccades are mathematically described as geodesic segments on the Listing hemisphere.
• A model for fixation eye movements (drift, microsaccades) is proposed, explaining presaccadic receptive field shifts.

Conclusions:

• Hopf fibration provides a novel mathematical framework for understanding eye movement laws.
• The geometric model unifies static and dynamic aspects of eye movements, including fixation.
• The proposed model offers insights into the functional role of fixation eye movements in visual perception.