A No-Go Theorem for One-Layer Feedforward Networks

Abstract

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 a subset of the form R n ≥0 \ P, where P is a polyhedron.

On Open and Closed Convex Codes

“…A simple corollary of Lemma 2.4 and the nerve lemma is the following observation (which first appeared in [5]) that provides a class of 'local' obstructions to being an open (or closed) convex code. Proposition 2.6.…”

“…Assume the converse, then there exists x ∈ i∈σ cl(U i ) and r > 0 such that denote the corresponding atom of cl(U ). If A U σ = ∅, then using (15) and (5) we conclude that…”

“…The proof is given in the Appendix, Section 5.1. A different example, [5] , was originally considered in [10], where it was proved that it is not open convex and possesses a realization by a good open cover (Figure 2.2b), thus does not have any "local obstructions" to convexity. However, it turns out that this code is closed convex (see a closed realization in Figure 2.2a).…”

Algebraic signatures of convex and non-convex codes

“…Thus A-4 applies, and so C 2 is not convex. 6 Alternatively, we can see σ ∪ {j} ∈ C|σ∪τ algebraically, by considering ρ = xσ i∈τ \j (1 − xi). We have ρ(1 − xj) ∈ CF(JC) while ρ / ∈ JC, by minimality of elements in CF(JC), and so Lemma 3.4 guarantees ρxj = xσxj i∈τ \j (1 − xi) / ∈ JC.…”

Error-correcting codes and neural networks

“…Then, dynamic statistical characteristics of this trial and error attempts below the asymptotic boundary will show signatures of criticality: cf. [7,8,22].…”