A no-go theorem for one-layer feedforward networks

TL;DR

This paper proves that one-layer feedforward neural networks can realize all coarse combinatorial neural codes, challenging the idea that recurrent connections are necessary for shaping neural response patterns.

Contribution

It establishes a no-go theorem showing limitations of feedforward networks in shaping neural codes, using combinatorial topology tools.

Findings

Large class of codes cannot be formed by feedforward networks alone

All maximal pattern codes can be realized by one-layer feedforward networks

Recurrent or multi-layer architectures are not needed for coarse code shaping

Abstract

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely from feedforward processing. We set out to find such codes in the context of one-layer feedforward networks, and identified a large class of combinatorial codes that indeed cannot be shaped by the feedforward architecture alone. However, these codes are difficult to distinguish from codes that share the same sets of maximal activity patterns in the presence of noise. When we coarsened the notion of combinatorial neural code to keep track only of maximal patterns, we found the surprising result that all such codes can in fact be realized by one-layer feedforward networks. This suggests that recurrent or many-layer feedforward architectures are not necessary for shaping the (coarse) combinatorial features of neural codes. In particular, it is not possible to infer a computational role for recurrent connections from the combinatorics of neural response patterns alone. Our proofs use mathematical tools from classical combinatorial topology, such as the nerve lemma and the existence of an inverse nerve. An unexpected corollary of our main result is that any prescribed (finite) homotopy type can be realized by removing a polyhedron from the positive orthant of some Euclidean space.