Geometrical structures determined by the functional order in nervous nets.

Area of Science:

• Neuroscience
• Information Theory
• Computational Neuroscience

Background:

• Functional order in neural systems is defined by signal covariances, differing from externally observable geometrical order.
• Previous work established that covariances can construct a partially ordered set representing functional order.
• This order is intrinsically available to the neural system itself.

Purpose of the Study:

• To demonstrate that functional order, derived from signal covariances, can be isomorphic with geometrical entities under specific constraints.
• To explore the logical possibility of neural pathways, like the optic nerve, carrying signals with intrinsic dimensionality.
• To show that neural signal dimensionality can be determined from multi-unit recordings without relying on anatomical data.

Main Methods:

• Defining functional order via the total covariance of signals within a neural collection.
• Constructing a partially ordered set from these covariances to represent functional order.
• Investigating mathematical constraints for isomorphism between the functional order set and geometrical structures (e.g., triangulations, hypersphere overlaps).
• Utilizing cross-correlation analysis of multi-unit recordings to determine the dimension of neural modalities.

Main Results:

• Demonstrated that under certain constraints, the partially ordered set representing functional order is isomorphic to geometrical entities like triangulations.
• Showed that the overlap relations of hyperspheres in n-dimensional space can mirror the signal order in nerves.
• Established the logical possibility of the optic nerve carrying two-dimensional signals, independent of retinal geometry.
• Confirmed that the dimension of a neural modality can be derived from cross-correlation analysis of neural recordings.

Conclusions:

• Functional order, derived from neural signal covariances, offers an intrinsic system perspective distinct from external geometrical order.
• The mathematical framework allows for the representation of functional order as geometrical structures, suggesting potential underlying dimensional properties of neural information processing.
• Cross-correlation analysis of multi-unit recordings provides a method to ascertain the dimensionality of neural signals without anatomical reference.