Combinatorial threshold-linear networks (CTLNs) are a special class of neural networks whose dynamics are tightly controlled by an underlying directed graph. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics [1], and we conjectured that these are the only stable fixed points allowed [2]. In this paper we prove that the conjecture holds in a variety of special cases, including for graphs with very strong inhibition and graphs of size n ≤ 4. We also provide further evidence arXiv:1909.02947v1 [q-bio.NC] 27 Aug 2019 def = {1, . . . , n} which have positive firing rate (see Section 2.1 for more details). In prior work [1], we proved a series a graph rules showing that we can rule in and rule out fixed points of W solely from features of the corresponding graph. We also conjectured that the supports of stable fixed points of CTLNs precisely correspond to target-free cliques. These are subsets σ ⊆ [n] that are cliques (all-to-all bidirectionally connected in G) that do not have targets. A node k is a target of σ if k / ∈ σ and i → k for all i ∈ σ. Conjecture 1.3 ([2]). Let W = W (G, ε, δ) be a CTLN on n nodes with graph G, and let σ ⊆ [n]. Then σ is the support of a stable fixed point if and only if σ is a target-free clique.

Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to the stabilizer of the generating subspace. Finally, with a linkage construction larger, and longer, constant dimension codes can be derived from cyclic orbit codes without compromising the distance.

Abstract. Neural codes allow the brain to represent, process, and store information about the world.

For a growing number of applications such as cellular, peer-to-peer, and sensor networks, efficient error-free transmission of data through a network is essential. Toward this end, Kötter and Kschischang propose the use of subspace codes to provide error correction in the network coding context. The primary construction for subspace codes is the lifting of rank-metric or matrix codes, a process that preserves the structural and distance properties of the underlying code. Thus, to characterize the structure and error-correcting capability of these subspace codes, it is valuable to perform such a characterization of the underlying rank-metric and matrix codes. This paper lays a foundation for this analysis through a framework for classifying rank-metric and matrix codes based on their structure and distance properties.To enable this classification, we extend work by Berger on equivalence for rank-metric codes to define a notion of equivalence for matrix codes, and we characterize the group structure of the collection of maps that preserve such equivalence. We then compare the notions of equivalence for these two related types of codes and show that matrix equivalence is strictly more general than rank-metric equivalence. Finally, we characterize the set of equivalence maps that fix the prominent class of rank-metric codes known as Gabidulin codes. In particular, we give a complete characterization of the rank-metric automorphism group of Gabidulin codes, correcting work by Berger, and give a partial characterization of the matrix-automorphism group of the expanded matrix codes that arise from Gabidulin codes.