Manifold representation in continuous attractor neural networks: a general constructive approach

Low dimensional attractor dynamics have now been established as a core mechanism by which neural populations represent, remember, and compute with continuous-valued external variables. Arriving at this state in the field has relied on the intuition-guided construction of neural circuit models with continuous attractor dynamics that led to population-level predictions about dynamics, followed by validation from large-scale recordings. Recent deep learning approaches require less intuition but extensive training to generate networks with continuous attractor and integrator dynamics, however extracting the essential components of the circuit that support their functionality (interpretibility) is challenging. Here we consider whether it is possible to take an alternative approach to these methods, and directly construct neural circuits that generate continuous attractor dynamics and velocity integration functionality appropriate for a given external variable with arbitrary topology.
We first provide a general theory characterizing which variables and topologies can be faithfully mapped onto which manifolds and topologies. Second, we provide a direct and generic method for building neural networks whose dynamics lie on an arbitrary desired manifold, which we call the MADE (Manifold Attractor Direct Engineering) method. Third, we show how neural circuits can be directly and generically structured by MADE to perform path integration on the manifold. Just as the Hopfield prescription showed how to embed prespecified discrete states as discrete fixed points in a neural circuit, this work provides a prescription for how to embed prespecified continuous states as a continuous attractor manifold in a neural circuit.
Thus, our theoretical work furthers our understanding of the relationship between neural manifold topologies and computation. The MADE general modeling framework provides a tool for directly building clearly interpretable and predictive circuit models of how the brain might represent and integrate arbitrary continuous variables for which we did not yet possess models.