Differential Geometry Methods for Constructing Manifold-Targeted Recurrent Neural Networks.

Area of Science:

• Computational Neuroscience
• Machine Learning
• Dynamical Systems Theory

Background:

• Neural computations are dynamical processes where network structure reflects function.
• Linking network connectivity, dynamics, and computation is crucial but challenging.
• Existing methods for engineering recurrent neural networks (RNNs) with specific dynamics are limited by high-dimensional state spaces.

Purpose of the Study:

• To simplify the computation of tangent vectors for engineering RNNs with targeted dynamics.
• To leverage topology and differential geometry for designing on-manifold dynamics.
• To broaden the applicability of network engineering approaches in neuroscience and machine learning.

Main Methods:

• Computing tangent vectors on low-dimensional topological manifolds.
• Embedding these vectors into high-dimensional state spaces.
• Utilizing concepts from differential geometry to map onto neural dynamics.

Main Results:

• A simplified procedure for computing tangent vectors for manifold-targeted RNNs.
• Facilitation of the creation of RNNs with task-solving, on-manifold dynamics.
• Demonstration of differential geometry's utility in describing neural dynamics.

Conclusions:

• The proposed method simplifies the engineering of RNNs with specific dynamics.
• This approach enhances the design of task-solving neural network dynamics.
• Differential geometry offers a powerful framework for understanding neural dynamics and computation.