What makes a neural code convex?

Transcription

1 What makes a neural code convex? Carina Curto, Elizabeth Gross, Jack Jeffries,, Katherine Morrison 5,, Mohamed Omar 6, Zvi Rosen 7,, Anne Shiu 8, and Nora Youngs 6 November 9, 05 Department of Mathematics, The Pennsylvania State University, University Park, PA 680 Department of Mathematics, San José State University, San José, CA 959 Department of Mathematics, University of Utah, Salt Lake City, UT 8 Department of Mathematics, University of Michigan, Ann Arbor, MI School of Mathematical Sciences, University of Northern Colorado, Greeley, CO Department of Mathematics, Harvey Mudd College, Claremont, CA 97 7 Department of Mathematics, University of California, Berkeley, Berkeley, CA Department of Mathematics, Texas A&M University, College Station, TX 778 Abstract Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? Using tools from combinatorics and commutative algebra, we uncover a variety of signatures of convex and non-convex codes. In many cases, these features are sufficient to determine convexity, and reveal bounds on the minimal dimension of the underlying Euclidean space. Keywords: neural coding, convex codes, neural ideal, simplicial complexes, links, Nerve lemma, Hochster s formula Contents Introduction. Convex neural codes Preliminaries Summary of main results Local obstructions to convexity 8. Receptive field relationships Local obstructions From local obstructions to mandatory codewords 9. Link lemmas and the proof of Theorem Computing mandatory codewords algebraically Alexander duality, the Stanley-Reisner ideal, and Hochster s formula Using free resolutions to compute M H ( ) Combinatorial signatures of convex codes 5. Classification of convex codes on n neurons Codes with non-overlapping maximal codewords Linear codes and convexity Algebraic signatures of local obstructions 8 5. Algebraic description of neural codes

2 5.. The neural ideal J C The canonical form CF(J C ) Comparison to other combinatorial ideals The canonical form CF(J C ) vs. receptive field relationships RF(C) Algebraic characterization of intersection-complete codes Algebraic signatures of convex and non-convex codes Example codes illustrating algebraic signatures Bounds on the minimal embedding dimension of convex codes 5 6. Embedding dimension and d-representability Bounds from Helly s theorem Bounds from the Fractional Helly theorem Comparison of dimension bounds Appendix 9 7. Proof of C-7 in Theorem Table and figures for codes on n neurons J H = J C for linear codes Introduction. Convex neural codes Cracking the neural code is one of the central challenges of neuroscience. Typically, this has been understood as finding the relationship between the activity of neurons and the stimuli they represent. More generally, neural codes must also reflect relationships between stimuli, such as proximity between locations in an environment []. Convex codes, comprised of activity patterns for neurons with classical receptive fields, may be the brain s solution to this problem. These codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus. Hubel and Wiesel s discovery in 959 of orientation-tuned neurons in the primary visual cortex was perhaps the first example of convex coding in the brain []. This was followed by O Keefe s discovery of hippocampal place cells in 97 [], showing that convex codes are also used in the brain s representation of space. Both discoveries were groundbreaking for neuroscience, and were later recognized with Nobel Prizes in 98 [] and 0 [5], respectively. Our motivating example of a convex code is, in fact, the hippocampal place cell code. A place cell is a neuron that acts as a position sensor, exhibiting a high firing rate when the animal s location lies inside the cell s preferred region of the environment its place field. Figure displays the place fields of four place cells recorded while a rat explored a two-dimensional environment. Each place field is an approximately convex subset of R. Taken together, the set of all activity patterns that can arise in a population of place cells comprise a convex code for the animal s position in space. place field of neuron # place field of neuron # place field of neuron # place field of neuron # Figure : Place fields of four CA pyramidal neurons (place cells) in rat hippocampus, recorded while the animal explored a.5m.5m square box environment. Red areas correspond to regions of space where the corresponding place cells exhibited high firing rates, while dark blue denotes near-zero activity. Place fields were computed from data provided by the Pastalkova lab, as described in [6].

3 Like the above example, the definition of a convex code is extrinsic: a code is convex if it can be realized by an arrangement of convex open sets in some Euclidean space. How can we characterize convex codes intrinsically? In other words, in terms of neural activity patterns alone, what makes a neural code convex? If a code is not convex, how can we prove this? If a code is convex, what is the minimal dimension needed for the corresponding open sets? Identifying intrinsic features of convex and non-convex codes will enable us to infer coding properties from population recordings of neural activity, without needing a priori knowledge of the stimuli being encoded. Understanding the structure of convex codes is also essential to uncovering the basic principles of how neural networks are organized in order to learn, store, and process information. In this work, we tackle the above questions of convexity and dimension for combinatorial neural codes, building on mathematical ideas first introduced in [7] and [8]. After reviewing the necessary background in Section., we give an overview of the main results in Section.. In Section we introduce local obstructions, our main tool for proving that a code is not convex, and in Section we explain how they can be detected using commutative algebra. In particular, we find that free resolutions and Hochster s formula enable us to easily compute a set of mandatory codewords that must be included in order for a given code to be convex. In Section we classify all convex and nonconvex codes on n neurons, and find special conditions guaranteeing that a code is convex. We also consider linear codes, and describe the relationship between linearity and convexity. In Section 5 we review the neural ideal J C and its canonical form CF(J C ), algebraic objects that can be associated to any combinatorial neural code [7], in order to prove a key theorem that allows us to detect local obstructions algebraically. Finally, in Section 6 we explore dimension bounds for convex codes stemming from Helly s theorem and d-representability of simplicial complexes.. Preliminaries Here we briefly review some frequently used definitions. See [7, Section ] for additional details. A binary pattern on n neurons is a string of 0s and s of length n, with a for each active neuron and a 0 denoting silence; equivalently, it is a subset of (active) neurons σ [n] def = {,..., n}. We will abuse notation and consider 0/ strings of length n and subsets of [n] interchangeably. For example, 0 and 000 are also denoted {,, } and {}, respectively. A neural code on n neurons is a collection of binary patterns C [n]. In other words, it is a binary code of length n where we interpret each binary digit as the on or off state of a neuron. This type of code is also known in the neuroscience literature as a combinatorial code []. The elements of a code are called codewords. For convenience, we will always assume a neural code C includes the all-zeros codeword, 00 0 C; the presence or absence of this codeword has no effect on the code s convexity or minimal embedding dimension (see Definition., below). To any code C, we can associate a simplicial complex: (C) def = {σ [n] σ c for some c C}. This is the smallest abstract simplicial complex on [n] that contains all elements of C. When referring to the subsets that comprise a simplicial complex, we will use the terms elements and faces interchangeably. The dimension of a face σ is defined to be σ. Faces that are maximal under inclusion are referred to as facets. Note that any facet of (C) must be a codeword of C, and so these facets are also referred to as maximal codewords. The hollow simplex on n vertices is the simplicial complex containing all proper subsets of [n] i.e., the simplex missing only the top-dimensional face, [n]. The restriction of to σ is the simplicial complex def σ = {ω ω σ}.

4 Another important simplicial complex we will make use of is the link, Lk σ ( ). For any σ, the link of σ inside is Lk σ ( ) def = {ω σ ω = and σ ω }. Let X be a topological space. A collection of open sets U = {U,..., U n }, where each U i X, is called an open cover. If every non-empty intersection, U σ def = i σ U i for σ [n], is contractible, then we say that U is a good cover. Given an open cover U, the code of the cover is the neural code C(U) def = { σ [n] U σ \ U j }. j [n]\σ Each codeword in C(U) corresponds to a region that is defined by the intersections of open sets in U. By convention, the empty intersection U = i U i equals X, so that C(U) if and only if i [n] U i X. We also assume i [n] U i X, so that 00 0 C (i.e., C), in agreement with our convention. Note that there always exists U = {U,..., U n } such that C = C(U) [7, Lemma.], but it may be impossible to choose the U i s to all be convex. We thus have the following definitions: Definition.. Let C be a neural code on n neurons. If there exists an open cover U = {U,..., U n } such that C = C(U) and the U i s are all convex subsets of R d, then we say that C is convex. The smallest d such that this is possible is called the minimal embedding dimension, d(c). If a code C is not convex, our convention is to set d(c) =. The following example depicts a convex code with minimal embedding dimension d(c) =. Example.. Consider the open cover U illustrated in Figure a. The corresponding code, C = C(U), has 0 codewords. C is a convex code with d(c) =. The simplicial complex (C) is depicted in Figure b. a. U U U b U Figure : (a) An arrangement U = {U,..., U } of convex open sets. Black dots mark regions corresponding to distinct codewords. From left to right, the codewords are 0000, 000, 00, 00, 0, 00, 000, 0, 00, and 000. (b) The simplicial complex (C) for the code C = C(U) defined in (a). The two facets, and, correspond to the two maximal codewords, 0 and 0, respectively. It is important to note that C(U) is not the same as the nerve of the cover: N (U) def = {σ [n] U σ }. It is more common to write Lk (σ) or link (σ), instead of Lk σ( ) (see, for example, [9]). However, because we will often fix σ and consider its link inside different simplicial complexes, such as σ τ, it is more convenient to put σ in the subscript. A set is contractible if it is homotopy-equivalent to a point [0].

5 In fact, N (U) = (C(U)), and C(U) N (U). The nerve of any cover U such that C = C(U) can thus be recovered directly from the code as (C), without reference to a specific cover. The celebrated Nerve lemma tells us that N (U) carries a surprising amount of topological information about the underlying space covered by the sets in U. The following statement is a direct consequence of [0, Corollary G.]. Lemma. (Nerve lemma). If U is a good cover, then the covered space Y = n i= U i is homotopyequivalent to N (U). In particular, Y and N (U) have exactly the same homology groups. Because an open cover comprised of convex sets is always a good cover, the Nerve lemma tells us that if C = C(U) for a collection of convex open sets U, then (C) must match the homotopy type of n i= U i. This fact was previously exploited to extract topological information about the represented space from hippocampal place cell activity []. The code C(U), however, contains additional information about U that is not captured by the nerve alone (see [7, Section..]). In particular, a non-empty intersection U σ may itself be covered by some of the other open sets, so that U σ i τ U i, but this information is not present in N (U). For example, in Figure a we have U U U and U U U. Note that if U σ i τ U i, the collection of open sets {U σ U i } i τ forms an open cover of U σ. The nerve of this cover is related to the nerve of the original cover via the link: if = N (U), then N ({U σ U i } i τ ) = Lk σ ( σ τ ). We will make use of this fact when we discuss local obstructions in Section... Summary of main results To prove that a neural code is convex, it suffices to exhibit a convex realization. Our strategy for proving that a code is not convex is to show that it has a local obstruction to convexity. A local obstruction arises when the combinatorial data of the code C dictate that any convex realization as C = C(U) would have a cover U that violates the Nerve lemma, and is thus not a good cover (contradicting convexity). We will define local obstructions more precisely in Section. We say that a code is locally good if it has no local obstructions. The following lemma is a direct consequence of Lemma. in Section.. Lemma.. If C is a convex code, then C is locally good. This was also observed in [8], using slightly different language. Although we previously conjectured that the converse also holds, this in fact is not true. In recent work [], it was shown that the code C = {5,,, 5,,,,, 5,, } is not convex despite having no local obstructions. Our first result states that C is locally good if and only if σ C for any non-contractible link Lk σ ( (C)). We denote the elements of a simplicial complex exhibiting non-contractible links as M( ) def = {σ Lk σ ( ) non-contractible}. Note that M( ) always contains all facets of, since the link of a facet is the empty set, which by convention is non-contractible. In fact, all elements of M( ) are intersections of facets. Theorem.5. A code C is locally good if and only if M( (C)) C. If σ M( ), then σ is an intersection of facets of. The elements of M( (C)) may be thought of as mandatory codewords: they must be included in order for a code C with simplicial complex (C) to be convex. While it may be difficult to compute M( ) in general, we denote the subset of faces with non-contractible links that can be detected via homology as M H ( ) def = {σ dim H i (Lk σ ( )) > 0 for some i}. We omit the coefficients for homology, as they do not affect the results. The H i denote reduced homology groups. 5

6 Clearly, M H ( ) M( ), so if any element of M H ( (C)) is missing from the code C, then C is not locally good and hence non-convex. In Section, we give the proof of Theorem.5, and describe how to compute M H ( ) using Hochster s formula. These computations can be carried out using existing computational algebra software. Theorem.5 also tells us that all mandatory codewords correspond to intersections of facets. In order to guarantee that a code is locally good, irrespective of the topology of the links Lk σ ( ), it thus suffices for the code to contain all intersections of maximal codewords, ensuring that M( (C)) C. We refer to such codes as max intersection-complete codes, and write C is max -complete. Strengthening this definition we obtain intersection-complete ( -complete) codes, which have the property that for any pair of codewords ω, ω C, we must have ω ω C. We provide an algebraic characterization of -complete codes in Section 5.. It was recently proven that all -complete codes are convex []. This is not, however, a necessary condition for convexity. The following example shows that a code need not be even max -complete in order to be convex. Example.6. Consider the simplicial complex shown in Figure a. The facets are,, and 5, and their intersections yield the faces,, and. Note that the links Lk ( ) = {, } and Lk ( ) = {, 5} are non-contractible, while Lk ( ) is contractible (see Figure b). The set of faces with non-contractible links is thus M( ) = {,,,, 5}. The code C = {,,,, 5,,, 5}, with (C) =, is max -complete, and thus locally good, but it is not -complete because it is missing = () (5). Now consider the code Ĉ = \ {}, missing only the vertex of. Ĉ is not max -complete. However, M( (Ĉ)) Ĉ, and so by Theorem.5 we know that Ĉ is locally good. Ĉ is also, in fact, convex (see Figure c). a b c Figure : (a) A simplicial complex. The vertex is an intersection of facets, but / M( ) because Lk ( ) is contractible. (b) Lk ( ). (c) A convex realization of the code Ĉ from Example.6. Collecting the above consequences of Theorem.5, together with several additional results, we obtain the following theorem. Theorem.7. Let C be a code on n neurons. Each of the combinatorial signatures in rows C-,..., C-9 in Table implies the corresponding property of C. In rows C- and C-5, the implications are bidirectional. Rows C-, C-, C-, and C- of Table are all immediate consequences of Theorem.5. C-5 corresponds to Theorem., which is presented in Section. together with a classification of convex codes for n. C-6 was recently proven in [], while C-7 follows from a result of Tancer, which we summarize in Appendix 7.. C-8 corresponds to Proposition.6 of Section., and C-9 corresponds to Proposition.9 of Section.. Although C- of Theorem.7 tells us only that max -complete codes are locally good, C-5, C-6, C-7, C-8, and C-9 all provide special cases of max -complete codes that are, in fact, also convex. This motivates the following conjecture: Conjecture.8. If C is a max -complete code, then C is convex. Note that the converse to this conjecture does not hold for n 5. Example.6 is convex, despite the fact that it is not max -complete. In particular, the code Ĉ of 6

7 Combinatorial signature of C Property of C C- M( (C)) C locally good C- M H ( (C)) C non-convex C- Lk σ ( (C)) is non-contractible for some σ (C) \ C non-convex C- C is max -complete locally good C-5 C is max -complete, and n convex (for n ) C-6 C is -complete convex C-7 C = (C) (for n > ) convex, d(c) n C-8 C, or all facets of (C) are disjoint convex, d(c) C-9 C is max -complete and also a linear code convex, d(c) Table : Combinatorial signatures of convex, non-convex, and locally good codes. The next theorem provides algebraic signatures that allow us to determine whether or not a given code is locally good. Note that every neural code C has an associated neural ideal J C, which has a canonical form CF(J C ). These notions were first defined in [7] and will be reviewed in Section 5.. The proof of this theorem is given in Section 5.. Theorem.9. Let J C denote the neural ideal associated to the code C, and CF(J C ) = CF (J C ) CF (J C ) its canonical form. For any x σ i τ ( x i) CF (J C ), if Lk σ ( σ τ ) is not contractible then C is not a convex code. In particular, if J C exhibits any of the algebraic signatures in rows A-, A-, or A- of Table, then C is not convex. If, alternatively, A- or A-5 is satisfied, then C is locally good. Codes that satisfy signature A-5 are -complete, and hence convex. Algebraic signature of J C Property of C A- x σ ( x i )( x j ) CF (J C ) s.t. x σ x i x j J C non-convex A- x σ i τ ( x i) CF (J C ) s.t. G C (σ, τ) is disconnected non-convex A- x σ i τ ( x i) CF (J C ) s.t. x σ x τ CF (J C ) non-convex A- x σ i τ ( x i) CF (J C ), x σ x τ / J C locally good A-5 x σ i τ ( x i) CF (J C ), τ = complete (convex) Table : Algebraic signatures of convex, non-convex, and locally good codes. G C (σ, τ) is the simple graph on vertex set τ with edge set {(ij) τ τ x σ x i x j / J C }. Finally, in Section 6 we discuss lower bounds on the minimal embedding dimension d(c) obtained from the d-representability of (C) and Helly s theorem. We derive an additional bound from the Fractional Helly theorem, and show that depending on (C) it can be either stronger or weaker than the bound from Helly s theorem alone. In Section 6 we also show that the presence or absence of a single codeword, namely the all-ones word, can cause the dimension to jump from very small to very large: Proposition.0. Let C be a code on n neurons, and suppose that for some k with k < n, (C) contains all k-dimensional faces. If C, then d(c) ; otherwise, d(c) > k. Unfortunately, all our results on dimension rely only on information about the code that is present in the simplicial complex (C). In our classification of convex codes for n, however, it is clear 7

8 that the presence or absence of specific codewords can affect d(c), even if (C) remains unchanged (see Table 7 in Appendix 7.). The problem of how to use this additional information about a code in order to improve the bounds on d(c) remains wide open. Open Questions. The results summarized above suggest several open questions. First, what are the non-local obstructions to convexity, and how can we detect them? Can non-local obstructions arise for max -complete codes, or does Conjecture.8 hold? Related to this, can we characterize max -complete codes algebraically, as we did for -complete codes? Finally, can we further improve the bounds on the minimal embedding dimension d(c), using information about the code that goes beyond the simplicial complex data in (C)? Local obstructions to convexity We cannot, in general, determine whether or not a neural code C is convex from data in its simplicial complex (C) alone. This is because for any simplicial complex, there exists a convex cover U such that = N (U) []. Obstructions to convexity must therefore emerge from information in the code that goes beyond what is reflected in (C). As was shown in [7], this additional information is precisely the receptive field relationships, which we turn to now.. Receptive field relationships For a code C on n neurons, let U = {U,..., U n } be any collection of open sets such that C = C(U). A receptive field relationship (RF relationship) of C is a pair (σ, τ) corresponding to the set containment U σ i τ U i, where σ, σ τ =, and U σ U i for all i τ. If τ =, then the relationship (σ, ) simply states that U σ =. Note that relationships of the form (σ, ) reproduce the information in (C), while those of the form (σ, τ) for τ reflect additional structure in the code. The set of all RF relationships {(σ, τ)} for a given code C is denoted RF(C). A minimal RF relationship is one such that no single neuron can be removed from σ or τ without destroying the containment. It is important to note that RF(C) is well defined, independent of the choice of open sets U. Moreover, RF(C) can be inferred directly from the code without appealing to a realization as C(U) [7], as will be explained later in Section 5.. Example. (Example. continued). The code C = C(U) from Example. has RF relationships RF(C) = {({, }, ), ({,, }, ), ({,, }, ), ({,,, }, ), ({}, {, }), ({}, {,, }), ({, }, {})}. Of these, the pairs ({, }, ), ({}, {, }), and ({, }, {}), corresponding to U U =, U U U, and U U U, are the minimal RF relationships.. Local obstructions Local obstructions are our main tool for proving that a code is not convex. We begin with a simple example that illustrates how a code can fail to have a convex realization. Lemma.. Let C = C(U). If for some RF relationship U σ U i U j we have that U σ U i U j =, then C is not a convex code. Proof. If (σ, τ = {i, j}) is a RF relationship, then U σ U i U j, U σ U i, and U σ U j. Since by assumption U σ U i U j =, the sets V i = U σ U i and V j = U σ U j are disjoint open sets that each intersect U σ, and such that U σ V i V j. It follows that U σ is disconnected in any open cover U such that C = C(U), and so C cannot have a convex realization. 8

9 The idea illustrated here is that U σ is covered by a collection of sets whose topology does not match that of U σ. We can generalize this problem via the definition of a local obstruction to convexity. Local obstructions arise when U σ i τ U i, for nonempty σ, τ [n] with σ τ =, but the nerve of the corresponding cover of U σ by the sets {U i } i τ is not contractible. This violates the Nerve lemma (Section.) if we assume the U i s are all convex, allowing us to conclude that C cannot be a convex code. Recalling from Section. that where = N (U), we have the following definition: N ({U σ U i } i τ ) = Lk σ ( σ τ ), Definition.. Let (σ, τ) RF(C), and let = (C). We say that (σ, τ) is a local obstruction of C if τ and Lk σ ( σ τ ) is not contractible. Note that τ implies σ / C and U σ, as the definition of a RF relationship requires that U σ U i for all i τ. Any local obstruction (σ, τ) must therefore have σ (C) \ C and Lk σ ( σ τ ) nonempty. The motivation for the above definition comes from the following simple consequence of the Nerve lemma, which was previously observed in [8]. Lemma.. If C has a local obstruction, then C is not a convex code. Proof. We will assume that C is a convex code with a local obstruction (σ, τ), and obtain a contradiction. Let U = {U,..., U n } be a collection of convex open sets such that C = C(U), and let = (C). Observe that the intersections U σ and U σ U i are also convex. It follows that {U σ U i } i τ is a good cover of U σ. By the Nerve lemma, N ({U σ U i } i τ ) must be contractible because U σ is contractible. However, N ({U σ U i } i τ ) = Lk σ ( σ τ ), contradicting the fact that (σ, τ) is a local obstruction. Recall from Section. that a code is locally good if it has no local obstructions. Lemma. now immediately follows as a corollary. The following simple condition on RF relationships guarantees that a code is locally good. Lemma.5. Let C = C(U). If for each (σ, τ) RF(C) we have U σ U τ, then C is locally good. Proof. U σ U τ implies Lk σ ( σ τ ), for = (C), is the full simplex on the vertex set τ, which is contractible. If this is true for every RF relationship, then none can give rise to a local obstruction. Note that if C, then U σ U τ for any pair σ, τ [n], so C is locally good. We thus obtain as a corollary a weaker version of C-8 in Theorem.7. Corollary.6. If C contains the all-ones codeword, then C is locally good. From local obstructions to mandatory codewords In the previous section, a local obstruction was defined as a pair (σ, τ) RF(C) such that the corresponding link, Lk σ ( σ τ ), is non-contractible (where = (C)). This suggests that in order to show that a given code is locally good, one might need to check the contractibility of all links corresponding to pairs (σ, τ) RF(C). Nevertheless, Theorem.5 states that a code C is locally good if and only if M( (C)) C, where M( ) def = {σ Lk σ ( ) non-contractible}. It thus suffices to check only the contractibility of links inside the full simplicial complex, rather than inside all restricted sub-complexes σ τ. The key to the proof is the observation that for any 9

10 pair (σ, τ) for which Lk σ ( σ τ ) is non-contractible, there exists a σ σ such that Lk σ ( ) is also non-contractible. Moreover, if σ C, then (σ, τ) cannot be a local obstruction. Proofs of these facts, together with the proof of Theorem.5, are in Section. below. As noted in Section., M( ) is a set of mandatory codewords that must be included in any convex code with simplicial complex. If any element of M( ) is missing, then the code cannot be convex. Conversely, a code that contains all elements of M( ) has no local obstructions, and is thus locally good. Although simpler than finding all local obstructions, M( ) may still be difficult to compute in general. For this reason, we have also defined M H ( ) in Section., as the subset of non-contractible links Lk σ ( ) that can be detected via homology. It turns out that M H ( ) can be computed algebraically in one shot, via a minimal free resolution for an ideal built from. Specifically, the elements of M H ( ) can be read off from the Betti numbers of a free resolution as a consequence of Hochster s formula. Moreover, the free resolution can be computed using existing computational algebra software, such as Macaulay [5]. In Section., we describe in detail how to compute M H ( ) algebraically, and provide an explicit example.. Link lemmas and the proof of Theorem.5 As we saw in Section, local obstructions are detected via non-contractible links of the form Lk σ ( σ τ ). Figure displays all possible links that can arise for τ. Non-contractible links are highlighted in red. In this section, we present some useful lemmas pertaining to the contractibility of links. We then use these lemmas to prove Theorem.5. In what follows, the notation cone v ( ) def = {{v} ω ω } denotes the cone of v over, where v is a new vertex not contained in. Any simplicial complex that can be expressed as a cone over a sub-complex is automatically contractible. In Figure, the only contractible link that is not a cone is L. This is the same link that appeared in Figure b of Example.6. Lemma.. Let be a simplicial complex on [n], σ, and v [n] such that v / σ. Lk σ {v} ( ) Lk σ ( [n]\{v} ), and Then Lk σ ( ) = Lk σ ( [n]\{v} ) cone v (Lk σ {v} ( )). Proof. The proof follows from the definition of the link. First, observe that Lk σ {v} ( ) = {ω [n] v / ω, ω σ =, and ω σ {v} } = {ω [n] \ {v} ω σ =, and ω σ {v} } {ω [n] \ {v} ω σ =, and ω σ [n]\{v} } = Lk σ ( [n]\{v} ), which establishes that Lk σ {v} ( ) Lk σ ( [n]\{v} ). Next, observe that cone v (Lk σ {v} ( )) = {{v} ω ω Lk σ {v} ( )} = {{v} ω ω [n], v / ω, ω σ =, and ω {v} σ } = {τ [n] v τ, τ σ =, and τ σ } = {ω Lk σ ( ) v ω}, and Lk σ ( [n]\{v} ) = {ω Lk σ ( ) v / ω}. From here the second statement is clear. 0

11 L L L L L5 L6 L7 L8 L9 L0 L L L L L5 L6 L7 L8 L9 L0 L L L L L5 L6 L7 L8 Figure : All simplicial complexes on up to vertices, up to symmetry. These can each arise as links, Lk σ ( σ τ ), for τ. Red labels correspond to non-contractible complexes. Note that L is the only simplicial complex on n vertices that is contractible but not a cone.

12 Corollary.. Assume v / σ. If Lk σ {v} ( ) is contractible, then Lk σ ( ) and Lk σ ( [n]\{v} ) are homotopy-equivalent. Proof. The above lemma shows that Lk σ ( ) can be obtained from Lk σ ( [n]\{v} ) by coning off the subcomplex Lk σ {v} ( ). If this subcomplex is contractible, then the homotopy type is preserved. Another useful corollary follows from the one above by simply setting = σ τ {v} and [n] = σ τ {v}. We immediately see that if both Lk σ ( σ τ {v} ) and Lk σ {v} ( σ τ {v} ) are contractible, then Lk σ ( σ τ ) is contractible. Stated another way: Corollary.. Assume v / σ, and σ τ =. If Lk σ ( σ τ ) is not contractible, then Lk σ ( σ τ {v} ) and/or Lk σ {v} ( σ τ {v} ) is not contractible. This corollary can be extended to show that for every non-contractible link Lk σ ( σ τ ), there exists a non-contractible big link Lk σ ( ) for some σ σ. This is because vertices outside of σ τ can be added one by one to either σ or its complement, preserving the non-contractibility of the new link at each step. In other words, we have the following lemma. Lemma.. Suppose σ τ =, and Lk σ ( σ τ ) is not contractible. Then there exists σ σ such that σ τ = and Lk σ ( ) is not contractible. The next results show that only intersections of facets (maximal faces under inclusion) can possibly yield non-contractible links. For any σ, we denote by ρ σ the intersection of all facets of containing σ. In particular, σ = ρ σ if and only if σ is an intersection of facets of. It is also useful to observe that a simplicial complex is a cone if and only if the common intersection of all its facets is non-empty. (Any element of that intersection can serve as a cone point, and a cone point is necessarily contained in all facets.) Lemma.5. Let σ. Then σ = ρ σ Lk σ ( ) is not a cone. Proof. Recall that Lk σ ( ) is a cone if and only if all facets of Lk σ ( ) have a non-empty common intersection ν. This can happen if and only if σ ν ρ σ. Note that since ν Lk σ ( ), we must have ν σ = and hence Lk σ ( ) is a cone if and only if σ ρ σ. Furthermore, it is easy to see that every simplicial complex that is not a cone can in fact arise as the link of an intersection of facets. For any that is not a cone, simply consider = cone v ( ); v is an intersection of facets of, and Lk v ( ) =. The above lemma immediately implies the following corollary: Corollary.6. If σ ρ σ, then Lk σ ( ) is a cone and hence contractible. In particular, if Lk σ ( ) is not contractible, then σ must be an intersection of facets of. Using the above facts about links, we can now prove Theorem.5 which states that: (i) C is locally good if and only if M( (C)) C, and (ii) if σ M( ), then σ is an intersection of facets of. Recall that M( ) is the set of σ such that Lk σ ( ) is not contractible. Theorem (Theorem.5). A code C is locally good if and only if M( (C)) C. If σ M( ), then σ is an intersection of facets of. Proof. The second statement is a direct consequence of Corollary.6. We now focus on the first statement. ( ) We prove the contrapositive. Suppose there exists σ M( (C)) such that σ / C. Then for = (C), Lk σ ( ) is not contractible and thus (σ, σ) is a local obstruction. It follows that C is not locally good. ( ) We again prove the contrapositive. Suppose C is not locally good, and let (σ, τ) be any local obstruction of C. This means that σ τ and Lk σ ( σ τ ) is not contractible. By Lemma., there exists σ σ such that σ τ = and Lk σ ( ) is not contractible. This means σ M( (C)). Moreover, U σ U σ i τ U i with σ τ =, which implies σ / C. It follows that M( (C)) C.

13 . Computing mandatory codewords algebraically Recall from C- of Theorem.7, that a code C is non-convex if M H ( (C)) C, where M H ( ) denotes the elements of with non-contractible links that can be detected via homology: M H ( ) def = {σ dim H i (Lk σ ( )) > 0 for some i}. If there exists an element σ M H ( ) such that σ / C, then we can conclude that C is not convex. Otherwise, all we can say is that C has no local obstructions that are detectable via the dimensions of homology groups. In this section, we explain how to compute M H ( ) algebraically using free resolutions of monomial ideals. This allows us to quickly identify all elements in M H ( ) using existing computational algebra software, such as Macaulay [5]. The key to computing M H ( ) algebraically is an application of Hochster s formula, which we review in Section... In Section.., we explain how this can be used to find M H ( ) and illustrate the method with an example... Alexander duality, the Stanley-Reisner ideal, and Hochster s formula For any simplicial complex on vertex set [n], the Alexander dual is the related simplicial complex: def = { τ τ / }, where τ = [n] \ τ denotes the complement of τ in [n]. Note that ( ) =. Any also has an associated ideal known as the Stanley-Reisner ideal: I def = { i σ x i σ / }. Alexander duality relates the reduced homology of a simplicial complex to the cohomology of its Alexander dual, H i ( ) = H n i ( ), while Hochster s formula relates the nonzero Betti numbers from a minimal free resolution of the Stanley-Reisner ideal to the reduced cohomology of restricted simplicial complexes [9]. Specifically, β i,σ (I ) = β i,σ (S/I ) = dim H σ i ( σ ), where S = k[x,..., x n ] and β i,σ refer always to Betti numbers for a minimal free resolution. The following link lemma can be used to derive the dual version of Hochster s formula, which is also well known [9, Corollary.0]. The dual formulation is more useful to us, as it allows us to compute the dimensions of all non-trivial homology groups for all links, Lk σ ( ), from a single free resolution. Lemma.7. Lk σ ( ) = ( σ ). Proof. First, observe that σ = {τ τ σ and τ }. The dual is thus ( σ ) = { σ \ τ τ σ and τ } = {ω ω σ and σ \ ω / } = {ω ω σ = and σ ω } = Lk σ ( ). Lemma.8 (Hochster s formula, dual version). dim H i (Lk σ ( )) = β i+, σ (S/I ). Proof. Using (in order) Lemma.7, Alexander duality, and the original version of Hochster s formula (above), we obtain: dim H i (Lk σ ( )) = dim H i (( σ ) ) = dim H σ i ( σ ) = β i+, σ (S/I ). This is all that can be concluded because M H( ) is only a subset of the mandatory codewords M( ) that are necessary for a code to be locally good (Theorem.5). The reason M H( ) M( ) in general is that we do not measure torsion, and moreover a homologically trivial simplicial complex also needs trivial fundamental group in order to be contractible. For example, consider the -skeleton of a triangulation of the Poincare homology sphere: this simplicial complex has all-vanishing reduced homology groups, but is non-contractible due to its non-trivial fundamental group [6].

14 It is important to note that if σ is a facet of, then Lk σ ( ) =, which is non-contractible due to nontrivial homology in degree. Hochster s formula thus detects facets of via the nonzero Betti numbers β, σ, as these correspond to σ such that dim H (Lk σ ( )) > 0. Note also that Lk ( ) =, so if itself has nontrivial reduced homology in degree i, this will be detected as a nonzero Betti number β i+,[n], where σ = [n] is the complement of σ =... Using free resolutions to compute M H ( ) In the last section we saw that for any simplicial complex, with Alexander dual, Hochster s formula (Lemma.8) relates the homology of the links Lk σ ( ) to the Betti numbers β i, σ (S/I ) of a minimal free resolution of the Stanley-Reisner ring S/I. 5 The set of all σ such that Lk σ ( ) has nontrivial homology is thus given by: M H ( ) = {σ β i, σ (S/I ) > 0 for some i > 0}. As illustrated in the following example, the nonzero β i, σ (S/I ) can be read off of a minimal free resolution for the S-module S/I. This process can also be automated using standard computational algebra software such as Macaulay [5]. Example.9. Let be the simplicial complex L5 in Figure. The Stanley-Reisner ideal is given by I = x x x, x x x, and its Alexander dual is I = x, x, x x, x, x = x x, x, x. A minimal free resolution of S/I is: 0 S/I [ x x x x ] S( ) S( ) x x 0 x x 0 x 0 x x x x x S( ) x x S( ) S( ) 0 The Betti number β i,σ (S/I ) is the dimension of the module in multidegree σ at step i of the resolution, where S/I is step 0 and the steps increase as we move from left to right. At step 0, the total degree is always 0. For the above resolution, the multidegrees at S( ) S( ) (step ) are 00, 000, and 000; at S( ) S( ) (step ), we have 0, 0, and 00; and at S( ) (step ) the multidegree is. This immediately gives us the nonzero Betti numbers: β 0,0000 (S/I ) =, β,00 (S/I ) =, β,000 (S/I ) =, β,000 (S/I ) =, β,0 (S/I ) =, β,0 (S/I ) =, β,00 (S/I ) =, β, (S/I ) =. Recalling that the multidegrees correspond to complements σ of faces in, we can now immediately read off the elements of M H ( ) from the above β i, σ for i > 0 as: M H ( ) = {00, 0, 0, 000, 000, 00, 0000} = {,,,,,, }, where we have used binary strings and subsets of [n] interchangeably to represent faces. Note that the first three elements of M H ( ) in Example.9, obtained from the Betti numbers β, in step of the resolution, are precisely the facets of. The next three elements, 000, 000, and 00, are non-maximal codewords that must be included for a code with simplicial complex to be convex. These all correspond to pairwise intersections of facets, and are obtained from the Betti 5 See [9, Chapter ] for more details about free resolutions of monomial ideals.

15 numbers β, at step of the resolution; this is consistent with the fact that the corresponding links are all disconnected, resulting in non-trivial H 0 (Lk σ ( )). The last element, 0000, reflects the fact that Lk ( ) =, and dim H ( ) = for = L5. By convention, however, we always include the all-zeros codeword in our codes (see Section.). Using Macaulay [5], the Betti numbers for the simplicial complex in Example.9 can be computed through the following sequence of commands (suppressing outputs except for the Betti tally at the end): i : kk = ZZ/; i : S = kk[x,x,x,x, Degrees => {{,0,0,0},{0,,0,0},{0,0,,0},{0,0,0,}}]; i : I = monomialideal(x*x*x,x*x*x); i : Istar = dual(i); i5 : M = S^/Istar; i6 : Mres = res M; [comment: this step computes the minimal free resolution] i7 : peek betti Mres o7 = BettiTally{(0, {0, 0, 0, 0}, 0) => } (, {0, 0, 0, }, ) => (, {0,, 0, 0}, ) => (, {, 0,, 0}, ) => (, {0,, 0, }, ) => (, {, 0,, }, ) => (, {,,, 0}, ) => (, {,,, }, ) => Each line of the BettiTally displays (i, {σ}, σ ) β i,σ. This yields (in order): β 0,0000 =, β,000 =, β,000 =, β,00 =, β,00 =, β,0 =, β,0 =, β, =, which is the same set of nonzero Betti numbers we previously obtained in Example.9. Recalling, again, that the multidegrees correspond to complements σ, this output immediately gives us M H ( ), exactly as in Example.9. Combinatorial signatures of convex codes In Sections and, we showed how local obstructions can be used to prove that neural codes are not convex. In this section we present a collection of combinatorial signatures that allow one to draw the opposite conclusion: that a code is convex. In Section., we classify all convex codes on n neurons, and show that they correspond precisely to max -complete codes; in Section., we prove that codes with non-overlapping maximal codewords are all convex and low-dimensional; and in Section., we show that any code that is both linear and max -complete is necessarily convex. Along the way, we give our remaining proofs for Theorem.7, corresponding to rows C-5, C-8, and C-9.. Classification of convex codes on n neurons For n = or n =, all codes are convex. The first non-convex codes appear for n =. Using our convention that all codes include the all-zeros codeword, there are a total of 0 permutation-inequivalent codes on neurons [7]. Of these, only 6 are non-convex. These can all be detected and proven to be non-convex via local obstructions. Proposition.. There are 6 non-convex codes on n neurons, up to permutation equivalence. They are the codes shown in Table. 5

16 label code C (C) B 000, 00, 00, 0, 0 L6 B5 000, 00, 0, 0 L6 B6 000, 0, 0 L6 E 000, 00, 00, 0, 0, 0 L7 E 000, 00, 0, 0, 0 L7 E 000, 0, 0, 0 L7 Table : All non-convex codes on n = neurons, up to permutation equivalence. Code labels are the same as in [7], and simplicial complex labels are as in Figure 8. Proof. Codes B, B5, and B6 all have simplicial complex L6, with facets {} and {}, but are missing the codeword 00, corresponding to σ = {} (C). For each of these codes, the pair (σ, τ) = ({}, {, }) is a local obstruction because Lk ( ) is the simplicial complex L, which is not contractible. Codes E, E, and E all have simplicial complex L7, with facets {}, {}, and {}, but are missing the codeword 00, corresponding to σ = {} (C). For each of these codes, the pair (σ, τ) = ({}, {, }) is a local obstruction because Lk ( ) is the simplicial complex L, which is not contractible. All remaining codes for n = neurons were shown to be convex in [7], via explicit convex realizations in two dimensions. We now turn to n. In Figure 8 in Appendix 7., we display all simplicial complexes on n neurons, up to permutation equivalence, and highlight all max intersections (i.e., intersections of facets). We show that every link corresponding to a max intersection is not contractible, so for n the set of mandatory codewords M( ) is always equal to the set of max intersections. Lemma.. If n, then Lk ρ ( ) is not contractible for any ρ that is the intersection of two or more facets of. Proof. See Figure 8 in Appendix 7., with intersections of two or more facets highlighted in red and orange. One can easily check that all corresponding links are non-contractible. Corollary.. If n, then C is locally good if and only if C is max -complete. Proof. The backward direction is an immediate consequence of Theorem.5. For the forward direction, consider the contrapositive and observe that if C is not max -complete, then for any intersection of facets ρ / C, Lk ρ ( ) is not contractible by Lemma., so C is not locally good. In fact, for n we have exhaustively checked that all codes with no local obstructions are in fact convex. Table 7 in Appendix 7. classifies convex and non-convex codes according to the corresponding simplicial complexes that can arise as (C). Figure 9 in Appendix 7. shows explicit convex realizations for most of the simplicial complexes L-L8. The omitted complexes L-L5, L9-L0, and L all have obvious convex realizations, while L5 and L6 are obvious given the realizations for L7 and L8, respectively. We thus have the following theorem, which is equivalent to C-5 of Theorem.7. Theorem.. If n, then C is convex if and only if C is max -complete. As we saw in Example.6, for n 5 there exist convex codes that are not max -complete, so this theorem cannot be extended to n >. The n = 5 case is studied in detail, including a complete enumeration of connected simplicial complexes and their mandatory codewords M( ), in []. 6

17 . Codes with non-overlapping maximal codewords We begin with an example of a code that contains the all-ones codeword, and illustrate how it can be embedded in R. Example.5. For the code C = {, 0, 0, 00, 00, 000, 000, 0000}, Figure 5 depicts the construction of a convex realization in R, which is described more generally in the proof of Proposition.6. Here C has a unique maximal codeword, the all-ones word U U U U 0 Figure 5: A convex realization in R of the code C = {, 0, 0, 00, 00, 000, 000, 0000}. If each nonzero codeword is contained in a unique facet of (C), the facets provide a partition of the code into non-overlapping parts. The above construction can be repeated in parallel to obtain the same dimension bound, giving us the following proposition which is equivalent to C-8 of Theorem.7. Proposition.6. Let C {0, } n be a neural code. If the facets of (C) are all disjoint (i.e., all maximal codewords of C are non-overlapping), then C is convex and d(c). Proof. Let ρ,..., ρ k [n] be the disjoint facets of (C), and define C ρ to be the restricted code consisting of the codewords of C whose supports are contained in ρ, excluding the all-zeros word, Note that C \00 0 is precisely the disjoint union of these restricted codes C ρ,..., C ρk, because every nonzero codeword of C is contained in a unique facet. We can thus construct a realization of C in R by realizing each C ρi with sets {U j } j ρi separately, and then taking the union of these disjoint realizations. The all-zeros codeword, 00 0 C, is assigned to the region of R not covered by these realizations. To realize C ρi, let m i = (C ρi ), the number of non-maximal codewords in C ρi. Inscribe a regular m i -gon P i in a circle, so that there are m i sectors surrounding P i (as in Figure 5). If m i <, inscribe a triangle into the circle and leave any unnecessary sectors unlabeled. To each sector assign a distinct non-maximal codeword c j C ρi, and to the polygon P i assign the maximal codeword ρ i. For each j ρ i, set U j to be the union of the polygon P i and all sectors whose assigned codeword contains j in its support. Each U j is open and convex, and C ρi = C({U j } j ρi ). Finally, taking U = k i= {U j} j ρi R results in C = C(U), so we can conclude that d(c). As an immediate consequence, we see that if (C) contains the all-ones word, and thus has a unique facet, then C has a convex realization in R. Corollary.7. If C, then C is convex and d(c).. Linear codes and convexity We now turn our attention briefly to linear codes, which are the primary codes of interest in classical coding theory. A code is called linear if it forms a subspace over its ground field. Restricting to binary codes, whose ground field is F, a code is linear if and only if every sum of codewords is also a codeword. (Note that + = 0 in F ; so, for example, = 0.) Linear codes are particularly useful in 7

18 engineering applications because they have a compact representation that allows for efficient storage and simplified computations [7]. Given the importance of linear codes in coding theory, it is natural to ask how the properties of linearity and convexity interact. Using Theorem. (C-5 of Theorem.7), we can determine whether or not a code of length n is convex by simply checking if it is max -complete. Table shows all linear codes for n (up to permutation-equivalence and having no trivial coordinates), organized by their corresponding simplicial complexes. There are both convex and non-convex linear codes, indicating that these properties do not have a straightforward relationship. Note that every convex code in Table contains the all-ones word, and is thus known to satisfy d(c) by Corollary.7 (C-8 of Theorem.7). This observation generalizes. possible # linear (C) codes linear codes convex? L C = 00, yes C = 00, 0, 0, yes L7 C = 000, 0, 0, 0 no C = 000, yes L8 C = 000, 00, 0, yes C = {0, } yes L5 C = 0000, 0, 0, 00 no L6 C = 0000, 0, 0, 0, 00, 00, 00, 000 no C = 0000, yes C = 0000, 000, 0, yes L8 6 C = 0000, 000, 000, 00, 00, 0, 0, yes C = 0000, 00, 00, yes C 5 = 0000, 00, 00, 00, 00, 00, 00, yes C 6 = {0, } yes Table : All linear codes for n, up to permutation equivalence and with no trivial coordinates (i.e. no coordinates are 0 in all codewords). For each possible simplicial complex, all corresponding linear codes are listed in the third column. A yes or no in the final column indicates whether or not a code is convex. Lemma.8. If C is a linear max -complete code, then C has a unique maximal codeword. Proof. Let C be a linear max -complete code, and suppose that C contains at least two distinct maximal codewords: ρ and ρ. Since C is max -complete, ρ ρ C. Now consider ρ ρ = ρ +ρ +ρ ρ. Because C is linear, ρ ρ C, but this contradicts the assumption that ρ and ρ are maximal and distinct. We conclude that C has a unique maximal codeword. As an immediate corollary we obtain the following proposition, which is equivalent to C-9 of Theorem.7. Proposition.9. If C is a linear max -complete code, then C is convex and d(c). Proof. Lemma.8 tells us that (C) has a unique facet. Applying Proposition.6 (C-8 of Theorem.7), we conclude that C is convex and d(c). 5 Algebraic signatures of local obstructions We saw in Section. that tools from commutative algebra such as Hochster s formula and minimal free resolutions can be used to determine that a code is not convex. The goal of this section is to prove Theorem.9, which provides algebraic signatures of convex and non-convex codes in terms of the neural ideal J C and its canonical form CF(J C ), objects first introduced in [7]. In Sections 5. and 5. we 8

19 review the necessary background on J C and CF(J C ), and highlight the connection between CF(J C ) and the receptive field relationships RF(C) defined in Section.. Next, in Section 5., we give an algebraic characterization of -complete codes in terms of CF(J C ). In Section 5. we use observations from the previous two sections to prove Proposition 5.7 and Theorem 5.0, which together imply Theorem.9. We end by illustrating the application of these algebraic signatures to example codes in Section Algebraic description of neural codes To capture the full combinatorial data of C algebraically, we turn to two ideals, I C and J C, that were first introduced in [7]. Here we give only a brief review of the most relevant definitions; for more details, see [7, Section ]. 5.. The neural ideal J C In order to represent a neural code algebraically, it is useful to consider binary patterns of length n as elements of F n, where F is the finite field of two elements: 0 and. Given a code C F n, perhaps the most natural ideal to consider is the set of all polynomials that vanish on all codewords: I C def = {f F [x,..., x n ] f(c) = 0 for all c C}, where F [x,..., x n ] is a polynomial ring with coefficients in F. Note that a polynomial f F [x,..., x n ] can be evaluated on a binary pattern of length n by simply replacing each indeterminate x i with the 0/ value of the i th neuron. In this way, it can be viewed as a function f : F n F. For example, if f = x x ( x ) F [x,..., x ], then f(0) = and f(00) = 0. The ideal I C always contains a set of Boolean relations B = x x,..., x n x n that carries no information about the code C, simply because of the binary nature of codewords. In fact, I C can be expressed as the sum I C = J C + B, where J C is the neural ideal, generated by indicator functions, J C def = χ v v F n \ C, χ v def = x i {i v i =} ( x j ). {j v j =0} The functions χ v generating J C are examples of pseudo-monomials, which are polynomials f F [x,..., x n ] that can be written in the form f = x σ ( x j ), where x σ def = i σ x i and σ, τ [n] with σ τ =. Pseudo-monomials come in three types: Type : x σ, for σ, j τ Type : x σ i τ ( x i), for σ, τ, σ τ =, and Type : i τ ( x i), for τ. For any ideal J F [x,..., x n ], a pseudo-monomial f J is called minimal if there does not exist another pseudo-monomial g J with deg(g) < deg(f) such that f = hg for some h F [x,..., x n ]. 9

20 5.. The canonical form CF(J C ) If J is an ideal generated by a set of pseudo-monomials, we define the canonical form of J to be the set of all minimal pseudo-monomials of J: CF(J) def = {f J f is a minimal pseudo-monomial}. In particular, for any neural code C, the neural ideal J C is generated by pseudo-monomials. We will make frequent use of its canonical form, CF(J C ). Since our convention is to assume 00 0 C, J C and CF(J C ) contain no Type pseudo-monomials. We denote the Type and Type pseudo-monomials of CF(J C ) by CF (J C ) and CF (J C ), respectively, so that: CF(J C ) = CF (J C ) CF (J C ). Note that the canonical form CF(J C ) can be computed algorithmically, starting from the code C. In [7, Section.5], one such algorithm was described that used the primary decomposition of pseudomonomial ideals. This algorithm has since been improved [8], and software for computing CF(J C ) has been made publicly available [9]. Example 5. (Example. continued). Recall the code C = C(U) from Example.. For this code, there are six non-codewords: 000, 00, 00, 0, 0, and. The neural ideal J C is thus generated by the corresponding pseudo-monomials χ v : J C = x ( x )( x )( x ), x x ( x )( x ), x x ( x )( x ), x x x ( x ), x x x ( x ), x x x x. The canonical form is CF(J C ) = {x x, x ( x )( x ), x x ( x )}, with CF (J C ) = {x x } and CF (J C ) = {x ( x )( x ), x x ( x )}. 5.. Comparison to other combinatorial ideals Recall from Section.. that to any simplicial complex we can associate the Stanley-Reisner ideal I = {x σ σ / }. It was shown in [7, Section.] that the neural ideal J C can be viewed as a generalization of the Stanley-Reisner ideal of (C). In fact, I (C) = CF (J C ), and so CF (J C ) contains the additional information about the code not present in its simplicial complex. In the case of linear codes, described in Section., it is natural to associate a parity check ideal, J H, based on the code s representation as the null space of a parity check matrix H. The ideal J H is naturally generated by linear polynomials, whereas the neural ideal J C for the corresponding code is defined in terms of pseudo-monomials. Nevertheless, it is not difficult to show that J H = J C for linear codes (see Appendix 7.). 5. The canonical form CF(J C ) vs. receptive field relationships RF(C) Here we review some key results from [7], relating the neural ideal J C to receptive field relationships RF(C) (described in Section.). The following lemma is essentially equivalent to [7, Lemma.]. Lemma 5.. Let C = C(U) be a neural code on n neurons, with neural ideal J C. For σ, τ [n] with σ τ =, x σ x i ) J C U σ i τ( U i. i τ 0

21 In the previous section, we saw that elements of the form x σ i τ ( x i), with σ τ =, are called pseudo-monomials and naturally break up into three types. Lemma 5. indicates that each type corresponds to a different kind of RF relationship: Type : x σ J C U σ =. Type : x σ i τ ( x i) J C U σ i τ U i. Type : i τ ( x i) J C X i τ U i. While the Type pseudo-monomials generate the Stanley-Reisner ideal I (C), the Type pseudomonomials capture information about the code that is not contained in (C). Specifically, for any cover U such that C = C(U), the corresponding Type relationships encode intersections U σ that are fully covered by other sets, a feature that is not reflected in the nerve N (U) = (C). Recall that, by convention, we always assume 00 0 C and thus eliminate the Type pseudomonomials. The canonical form can then be written as: CF(J C ) = CF (J C ) CF (J C ), where CF (J C ) and CF (J C ) are disjoint subsets of minimal Type and Type pseudo-monomials, respectively (Section 5.). The following useful fact is a direct consequence of [7, Theorem.], and allows us to interpret the elements of CF(J C ) as minimal RF relationships. Lemma 5.. x σ i τ ( x i) CF(J C ) (σ, τ) is a minimal RF relationship. Example 5. (Example. continued). Recall the code C = C(U) from Example.. The canonical form is CF(J C ) = {x x, x ( x )( x ), x x ( x )}. Using Lemma 5. we can immediately read off the minimal RF relationships: U U =, U U U, and U U U (cf. Figure and Example.). 5. Algebraic characterization of intersection-complete codes Here we characterize -complete codes algebraically. These codes were defined combinatorially in Section.: C is -complete if ω, ω C implies ω ω C. Because -complete codes are always max -complete, they are locally good (see C- of Theorem.7). Moreover, they have also been shown to be convex (see C-6 of Theorem.7, which was proven in []). We begin with the special case of simplicial complex codes, where C = (C), which are obviously -complete since for these codes, ω C implies σ C for any σ ω. If a code is a simplicial complex, then the only elements of CF(J C ) will be Type pseudo-monomials. The converse is also true, giving us an easy algebraic characterization of these codes: Lemma 5.5. C = (C) if and only if CF(J C ) contains only pseudo-monomials x σ i τ ( x i), with τ = 0 (i.e., only monomials x σ appear in CF(J C )). As a natural generalization of the above description for simplicial complexes, we find -complete codes can also be characterized via the canonical form. Proposition 5.6. C is -complete if and only if CF(J C ) only contains pseudo-monomials of the form x σ i τ ( x i), with τ. In other words, -complete codes are precisely the codes for which only terms of the form x σ and x σ ( x i ) appear in CF(J C ). In the proof of Proposition 5.6, we will use the notation C σ for the code obtained from C by restricting all codewords to the neurons in σ. We also use the obvious fact that if C is -complete, then C σ is -complete for any σ [n].

22 Proof. ( ) Suppose C is -complete, and consider an element x σ i τ ( x i) CF(J C ). To obtain a contradiction, assume τ >. For each i τ, there exists a codeword ω i C σ τ such that σ i ω i, but j / ω i for any j τ \ {i}. (If there were a j τ \ {i} such that j ω i for all possible ω i, then i could be removed from τ and x σ k τ\i ( x k) J C, contradicting the minimality of x σ k τ ( x k) CF(J C ).) Since C, and hence C σ τ, is -complete, it follows that σ C σ τ. But this contradicts the fact that σ can only appear in a codeword together with some i τ, as indicated by the fact that x σ i τ ( x i) CF(J C ). We conclude that τ. ( ) Suppose CF(J C ) only contains pseudo-monomials with τ. Consider a pair of codewords ω, ω C and let σ = ω ω. Then x σ / J C, since x ω, x ω / J C. To obtain a contradiction, suppose σ / C. This implies x σ i τ ( x i) J C for some τ (since x σ / J C ). It follows that x σ i τ ( x i) must be a multiple of a Type pseudo-monomial in the canonical form, say x σ ( x i ) CF(J C ), where i τ and σ σ is nonempty. In particular, x σ ( x i ) J C, which implies that any codeword containing σ also contains i. This contradicts the fact that σ = ω ω. We conclude that σ C, and hence C is -complete. 5. Algebraic signatures of convex and non-convex codes In this section we prove Theorem.9, and relate algebraic signatures in CF(J C ) to receptive field relationships in RF(C). The first statement of Theorem.9 is precisely the following proposition, while the rest of the theorem follows directly from Theorem 5.0, below. Proposition 5.7. If there exists x σ i τ ( x i) CF (J C ) such that Lk σ ( σ τ ) is not contractible, then C is not a convex code. Proof. x σ i τ ( x i) CF (J C ) implies (σ, τ) RF(C), by Lemma 5., and τ. If Lk σ ( σ τ ) is not contractible, then (σ, τ) is a local obstruction and hence C is not a convex code. The converse, unfortunately, is not true. In other words, there exist non-convex codes with local obstructions despite the fact that the links Lk σ ( σ τ ) are contractible for every pair (σ, τ) such that x σ i τ ( x i) CF (J C ). These local obstructions correspond to RF relationships that are not minimal. The smallest example occurs for n = neurons: Example 5.8. Consider the code C = {0000, 0, 0, 0, 0}. This code has (C) = L7 (see Figure 8), and canonical form CF(J C ) = {x x x x } {x i ( x j )( x k ) i, j, k [] and distinct}. The RF relationships (σ, τ) = ({i}, {j, k}) that are detected by the canonical form all have corresponding links Lk i ( {i,j,k} ) that are equivalent to L, and hence contractible. C is not, however, a convex code. To see this, note that links such as Lk ( ) and Lk ( ), corresponding to missing codewords, are non-contractible. Thus, C has local obstructions and is therefore not convex. On the other hand, for elements of CF (J C ) having τ =, we can safely say that no higher-order local obstructions arise. Lemma 5.9. Suppose x σ ( x i ) CF(J C ). Then there are no local obstructions (σ, τ) for any σ σ and any τ such that i τ. Proof. Suppose x σ ( x i ) CF(J C ), and (σ, τ) RF(C) for some σ σ and τ such that i τ. We will show that (σ, τ) is not a local obstruction by proving that Lk σ ( σ τ ) is contractible, where = (C). Note that x σ ( x i ) CF(J C ) implies that i / σ, but for every ω σ, i ω. Thus, for any ω σ σ, we have i ω, and so i ω for any ω Lk σ ( σ τ ). The link Lk σ ( σ τ ) is thus a cone, and hence contractible.

23 Despite the fact that not all local obstructions can be detected directly from the canonical form CF(J C ), the consequences of Proposition 5.7 allow us to catalog a rich set of algebraic signatures for detecting whether or not a code is locally good. These are summarized in the following theorem. Note that signature A- uses the notation G C (σ, τ) for the simple graph on vertex set τ with edges {(ij) τ τ x σ x i x j / J C }, and G U (σ, τ) for the simple graph on vertex set τ with edge set {(ij) τ τ U σ U i U j }. Theorem 5.0. If C has any of the algebraic signatures in rows A-, A-, or A- of Table, then C is not a convex code. If, alternatively, C has signature A- or A-5, then C is locally good. Codes with signature A-5 are -complete, and hence convex. Each algebraic signature has a corresponding RF condition, which also implies the given property of C. A- A- A- A- A-5 Algebraic signature Receptive field condition Property of C x σ ( x i )( x j ) CF (J C ) (σ, {i, j}) RF(C) and non-convex s.t. x σ x i x j J C U σ U i U j = x σ i τ ( x i) CF (J C ) (σ, τ) RF(C) and non-convex s.t. G C (σ, τ) is disconnected G U (σ, τ) is disconnected x σ i τ ( x i) CF (J C ) (σ, τ) RF(C) and U σ U τ =, non-convex s.t. x σ x τ CF (J C ) but U σ U τ τ τ x σ i τ ( x i) CF (J C ), (σ, τ) RF(C), U σ U τ locally good x σ x τ / J C x σ i τ ( x i) CF (J C ), (σ, τ) RF(C), complete (convex) τ = U σ U i for some i τ Table 5: Algebraic signatures and receptive field conditions for convex, non-convex, and locally good codes. Proof of Theorem 5.0. For each signature, A-,..., A-5, we prove that algebraic signature RF condition property of C. (A-) Clearly, this signature implies (σ, {i, j}) RF(C), and thus U σ U i U j. By Lemma 5., x σ x i x j J C implies U σ U i U j =. Applying Lemma., we see that C is non-convex. (A-) It is easy to see that if C = C(U), then G C (σ, τ) = G U (σ, τ), which equals the -skeleton of Lk σ ( σ τ ). Since we assume this is disconnected, it follows that Lk σ ( σ τ ) is not contractible, and hence C is non-convex by Proposition 5.7. (A-) x σ x τ CF (J C ) implies that U σ U τ =, and U σ U τ for all τ τ. This means Lk σ ( σ τ ) is a hollow simplex, which is not contractible. By Proposition 5.7, C is non-convex. (A-) Note that, by Lemma 5., x σ x τ / J C implies U σ U τ. Since we assume this is true for all (σ, τ) RF(C), we can apply Lemma.5 and conclude that C is locally good. (A-5) If the only elements of CF (J C ) are of the form x σ ( x i ), then we must have that for any RF relationship, U σ j τ U j, there is an i τ such that U σ U i. The converse is also true, since CF (J C ) reflects only minimal RF relationships. Thus, the algebraic signature is equivalent to the corresponding RF condition. It follows from Proposition 5.6 that signature A-5 is equivalent to C being -complete, thus completing the proof. We end this section by noting that the graph G C (σ, τ), appearing in signature A-, can be constructed directly from the canonical form. This is because the condition x σ x i x j / J C is easy to check from knowledge of CF (J C ), provided we already know that x σ k τ ( x k) CF (J C ).

24 Lemma 5.. Suppose x σ k τ ( x k) CF (J C ). Then for any i, j τ, x σ x i x j J C x σx i x j CF (J C ) for some σ σ. Proof. The backwards direction ( ) is obvious. To see the forward direction ( ), suppose x σ x i x j J C. Then it is a multiple of some monomial x ρ CF (J C ). We have four possibilities: () ρ σ. Then x ρ divides x σ k τ ( x k), contradicting that x σ k τ ( x k) CF(J C ). () ρ = σ {i} for some σ σ. Then x σ x i k τ\{i} ( x k) J C, and hence x σ x i k τ\{i} ( x k ) + x σ ( x k ) = x σ k τ k τ\{i} ( x k ) J C, contradicting that x σ k τ ( x k) CF(J C ). () ρ = σ {j}. This argument is identical to the previous one, leading to a contradiction. Thus, the only possibility left is: () ρ = σ {i, j}, so there must be some σ σ with x σ x i x j CF(J C ). 5.5 Example codes illustrating algebraic signatures In this section, we provide examples to illustrate how one might use Theorem.9 (equivalently, Theorem 5.0) to detect whether or not a neural code is convex. Given a code C, we first calculate its canonical form CF(J C ). This can be done via the algorithm described in [8], with software available at [9]. Once CF(J C ) is obtained, we partition it into its two components: CF (J C ), consisting of monomials x σ, and CF (J C ), consisting of pseudo-monomials x σ i τ ( x i) with τ. This allows us to check whether any of the signatures A- A-5 of Theorem.9 (equivalently, Theorem 5.0) apply. Table 6 gives five example codes, together with their partitioned canonical forms. Code CF (J C ) CF (J C ) C = 000, 0, 0, 0 x x x {x i ( x j )( x k ) i, j, k =,, } 00, 00, and x x x, x x x 5 {x ( x )( x )( x )( x ) i i i i i i5... i 5 [5]} C = 00000, 00, 00 x x x, x x x 5 all words with two s x x x 5, x x x 5 C = 0000, 0, 0, 0 {x x x x x i ( x )( x j ) i, j =,, } 0, 00, 00, 00 x ( x )( x )( x ) C = 000, 00, 00, 0, 0, x ( x )( x ), x x ( x ) C 5 = 00000, 0, 0, 0 00, 00, 00, 000 x x x 5 {x i ( x j ) i [5]; j =, } Table 6: Example codes and their canonical forms. element of CF(J C ) must always be distinct. Note that the indices that appear in a single Code C in Table 6 has three elements in CF (J C ): x ( x )( x ), x ( x )( x ) and x ( x )( x ). For x ( x )( x ), we have x x x CF (J C ) and hence x x x J C. Thus signature A- applies, and we can conclude that C is not convex. For code C, we see that A- does not apply since every element in CF (J C ) has τ >, and so we turn to signature A-. Consider x ( x )( x )( x )( x 5 ) CF (J C ), where σ = {} and τ = {,,, 5}. We will construct the graph G = G C (σ, τ) whose vertices are precisely the elements of τ. By definition, whenever x σ x i x j / J C for i, j τ, (ij) is an edge in G. Using CF (J C ), we immediately see that (), (5), () and (5) are not edges in G. Applying Lemma 5., we see () and (5) are edges in G. Thus G consists only of two disjoint edges, and is hence disconnected. Therefore, signature A- is satisfied and C is not convex.

25 Next consider the code C. Since there are three elements of CF (J C ) of the form x i ( x j )( x k ), we might first try to apply A-. However, since x x x x CF (J C ), and is thus minimal, x i x j x k / J C for any i, j, k {,,, }. On the other hand, if we consider x ( x )( x )( x ) CF (J C ), we see that x x x x CF (J C ), and so A- applies. Thus C is not convex. For C, we see there are no monomials in CF (J C ), and thus no monomials in J C, and so none of the signatures A- A- can apply. Meanwhile, we see that A- is satisfied since for both (σ, τ) pairs arising from CF (J C ), we have x σ x τ / J C. Thus, C is locally good. Finally, consider the code C 5. We immediately see that all elements of CF (J C5 ) satisfy τ =, and so signature A-5 applies. Thus, C 5 is -complete and hence convex. 6 Bounds on the minimal embedding dimension of convex codes We now turn to the problem of determining the minimal embedding dimension d(c) of a convex code C, as defined in Section.. There is no general method for computing d(c), though bounds can be obtained from the information present in the simplicial complex (C). In this section, we review known results on d-representability and Helly s theorem, and apply them to obtain lower bounds on d(c). We also use Helly s theorem to prove Proposition.0, from Section.. Finally, we obtain an additional bound on d(c) from the Fractional Helly theorem, and examine how it compares to the Helly s theorem bound. Our dimension bounds all rely solely on features of the code captured by (C), and do not take into account the finer structure of the code. Nevertheless, as we have seen in Section., the presence or absence of a single codeword can have a significant effect on d(c), even if the simplicial complex = (C) is fixed (e.g., see Table 7 in Appendix 7. for L8, L, L6, etc.). It remains an open question how to use this additional information in order to improve the bounds on d(c). 6. Embedding dimension and d-representability The problem of determining d(c) for a convex code C has not been directly addressed in the literature. However, the related problem of determining when a simplicial complex can be realized as the nerve N (U) of a cover U has received considerable attention (see [, 0] and references therein). A simplicial complex is said to be d-representable if there exists a collection of convex (not necessarily open) sets U = {U,..., U n }, with U i R d, such that = N (U). Note that for such a U, the corresponding code C(U) need not be equal to, though it is always true that (C(U)) =. While d-representability of (C) does not tell us the value of d(c), it does provide a bound. Since a code C can be embedded in R d(c) via some collection of convex open sets U, it follows that (C) = N (U) must be d-representable. This motivates us to define the nerve dimension d N (C) of a code C to be the minimal d such that (C) is d-representable. It immediately follows that d(c) d N (C). In other words, lower bounds on d-representability for (C) automatically give lower bounds on d(c). Unfortunately, d N (C) may be difficult to compute in general. In contrast, we can obtain a bound from Helly s theorem that is simple to read off from the canonical form CF(J C ). 6. Bounds from Helly s theorem One common tool used for addressing d-representability of simplicial complexes is Helly s theorem. Theorem 6. (Helly s theorem []). Let U = {U,..., U n } be a collection of convex open sets in R d. If for every d + sets in U, the intersection is non-empty, then the full intersection n i= U i is non-empty. 5

26 Helly s theorem implies that if is d-representable and contains all possible d-dimensional faces, then must be the full simplex. On the other hand, if contains all possible d-dimensional faces but is not the full simplex, then it is not d-representable. This immediately yields a proof for Proposition.0, illustrating that the presence or absence of a single codeword can have a large effect on d(c). Proof of Proposition.0. In the first case, where C, the fact that C is convex and d(c) follows from C-8 of Theorem.7. For the second case, where / C, suppose C is realizable as a convex code in R d for some d k, so that C = C(U) for some collection of convex open sets U = {U, U,..., U n }, with each U i R d. Since, by hypothesis, (C) contains all k-dimensional faces, it also contains all d-dimensional faces, and so the intersection of every collection of d + subsets in U is non-empty. Thus, by Helly s Theorem, the full intersection of all sets in U is non-empty, and so C. This contradicts the fact that / C; hence, we must have d(c) > k. We can also apply Helly s theorem to every subcollection {U i,..., U im } U, or equivalently to the induced subcomplex on elements i,..., i m, to see that if all the d-dimensional faces of this subcomplex are present, then the top-dimensional face must also be present in order for to be d- representable. This leads us to the following definitions. A simplicial complex is said to contain an induced k-dimensional simplicial hole if it contains k + vertices such that the induced subcomplex on those vertices is isomorphic to a hollow simplex []. We define the Helly dimension 6 of C, denoted d H (C), to be the dimension of the largest induced simplicial hole of (C): d H (C) def = max{k (C) has a k-dimensional induced simplicial hole}. Clearly, d H (C) d(c). The Helly dimension is particularly useful because it can be detected from the canonical form CF(J C ). Lemma 6.. Let C be a neural code with canonical form CF(J C ). Then d H (C) = max σ. x σ CF(J C ) Proof. Let σ [n] satisfy x σ CF(J C ). Then, by the minimality of generators in CF(J C ), we have that σ / (C) but all subsets of σ are in (C). Thus, (C) has an induced simplicial hole of dimension σ, and so d H (C) = max xσ CF(J C ) σ. The above expression for d H (C) in terms of CF(J C ) was previously observed to give a lower bound for d(c) in [7]. 6. Bounds from the Fractional Helly theorem The Fractional Helly theorem is a well-known extension of Helly s theorem that provides new bounds on d(c), though they are not always better. Theorem 6. (Fractional Helly theorem 7, Theorem 6.7 of []). Let α > 0, and U = {U, U,..., U n } be a collection of convex open sets in R d such that at least α ( n d+) of the (d + )-tuples of sets in U have non-empty intersections. Then there exists σ [n] such that σ > α d+ n, and i σ U i is non-empty. The Fractional Helly theorem indicates that if a code C can be embedded in R d, and the simplicial complex (C) has many d-dimensional faces, then (C) must have some sufficiently high-dimensional face. The following lemma quantifies these observations in our context. 6 The closely-related notion of Helly number for a simplicial complex was previously introduced in the literature. Specifically, the Helly number of (C) is d H(C) +. 7 In [], Kalai gave a sharp version of Theorem 6.: If α ( n d+) of the (d + )-tuples have non-empty intersections, then there exists σ with σ > ( ( α) /(d+) )n such that the corresponding intersection is non-empty. The precise value of β for which σ > βn is not essential to the results in the remainder of our paper. Therefore, for ease of notation and computation, we prefer β = α/(d + ) as in Theorem 6., while keeping in mind that a tighter bound exists. 6

27 Lemma 6.. Let be a k-dimensional simplicial complex on n elements, and let f d ( ) be the number of d-dimensional faces in for d < n. If is d-representable, then k + > f d ( )/ ( ) n d. Proof. By definition of d-representable, we have = N (U) for some U = {U, U,..., U n }, where each U i R d. Since each d-dimensional face of corresponds to an intersection of (d + ) of the U i s, we have that f d ( (C)) of the (d + )-tuples have non-empty intersections. By the Fractional Helly theorem, there is some σ [n] with σ > α d+ n such that i σ U i, where α = f d( (C)). Since (C) is k-dimensional, it follows that σ k +, and so k + > α d+ n = f d( (C))/ ( n d ( d+) n This leads us to the following definition. Let C be a code on n neurons with a k-dimensional simplicial complex (C), and let f d ( (C)) be the number of d-dimensional faces in (C) for d < n. The Fractional Helly dimension d F H (C) of C is given by: { ( ) } d F H (C) def n = + max d f d ( (C)) (k + ), d < n. d As with the Helly dimension, d F H (C) can be computed from the canonical form CF(J C ), but with a bit more effort because we must compute the f-vector f d ( (C)). Recall from Section 5., that the ideal generated by the Type relations of CF(J C ) is the Stanley-Reisner ideal I (C) F [x,..., x n ]; thus, CF(J C ) allows for a direct computation of I (C). It is well known that f d ( (C)) is the (d + ) st coefficient in the Z 0 -graded Hilbert polynomial of the Stanley-Reisner ring F [x,..., x n ]/I (C) [9, Section.]. This provides a procedure for computing f d ( (C)) directly from the Type relations in CF(J C ), using standard computational algebra software such as Macaulay [5]. 6. Comparison of dimension bounds How do the Helly and Fractional Helly dimensions d H (C) and d F H (C) compare to each other and to the nerve dimension d N (C)? First, note that although a simplicial complex cannot be represented in any dimension less than its Helly dimension d H ( ), Helly s theorem does not guarantee that is d H - representable. Thus we have d H (C) d N (C) d(c). By the same reasoning, d F H (C) d N (C) d(c). The next examples show that we can have d H (C) < d N (C) and d F H (C) < d N (C); i.e., the nerve dimension can provide a strictly stronger lower bound. Example 6.5. Consider any code C such that (C) is the simplicial complex in Figure 6a. We obtain d H (C) = because the two maximal induced simplicial holes of (C) arise from the subsets {, } and {, }, which both have dimension. Although (C) is contractible, the induced subcomplex on {,,, } is a -cycle, so (C) is at best -representable and d N (C). Thus, d N (C) > d H (C). Figure 6b shows that d(c) =. The following example shows not only that we can have d F H (C) < d N (C), but also that it is possible for d F H (C) < d H (C) or for d H (C) < d F H (C), depending on the code. So although d N (C) is always the strongest of the three bounds, neither of the easier-to-compute d H (C) and d F H (C) bounds is universally stronger than the other. Example 6.6. (a) Let n =, and consider the code C = {000, 0, 0, 0}, whose simplicial complex (C ) is the empty triangle, as shown in Figure 7a. This is the boundary of a -simplex, so we have d H (C ) =. In contrast, a quick computation yields d F H (C ) =, and so d H (C ) > d F H (C ). (b) Let n = r, where r, and suppose (C ) = K r,r, the complete bipartite graph on r vertices, as shown in Figure 7b. This graph contains no triangles, so the largest induced simplicial holes result from missing edges in (C ), which have dimension. Thus, d H (C ) =. To compute d F H (C ), we first find the f-vector for (C ). Observe that f 0 ( (C )) = n, f ( (C )) = r, and f i ( (C )) = 0 for ). 7

28 a b U 5 U U 5 U U Figure 6: (a) A simplicial complex, with facets 5, 5, 5, 5. For any C such that (C) =, the Helly dimension d H (C) =, while the nerve dimension d N (C) =. (b) A convex realization of C in R, showing that d(c) =. a b Figure 7: (a) The simplicial complex of code C from Example 6.6. (b) The simplicial complex of code C from Example 6.6, for r =. i < n. Note also that k =, since (C ) is -dimensional. Since f d ( (C )) = 0 for d, the inequality f d ( (C )) (k + ) (n ) d is not satisfied for these values of d. However, we have ( ) n f ( (C )) = r (r ) = (n ) = (k + ), with the inequality being valid for all r. Thus, directly from the definition, we find that d F H (C ) =. Hence, d F H (C ) > d H (C ). Putting together the two inequalities resulting from the Helly-type theorems, we obtain the following proposition about the minimal embedding dimension of a code. Proposition 6.7. Let C be a convex neural code on n neurons. The minimal embedding dimension satisfies: d(c) max{d H (C), d F H (C)}, where d H (C) = max x σ CF(J C ) d F H (C) = + max Acknowledgments σ, { d f d ( (C)) (k + ) ( n d ) }, d < n This work began at a 0 AMS Mathematics Research Community, Algebraic and Geometric Methods in Applied Discrete Mathematics, which was supported by NSF DMS-79. CC was supported by NSF DMS-5666/DMS-578 and an Alfred P. Sloan Research Fellowship; EG was supported by NSF DMS-067; and AS was supported by NSF DMS-0080 and NSF DMS-7. We thank Chad Giusti, Vladimir Itskov, Carly Klivans, William Kronholm, Keivan Monfared, and Yan X Zhang for numerous discussions, and Eva Pastalkova for providing the data used to create Figure.. 8

29 7 Appendix 7. Proof of C-7 in Theorem.7 In the case of codes that are themselves simplicial complexes, C = (C), we can apply a construction described by Tancer [] to show that C is a convex code with dimension d(c) n. Thus, we can prove row C-7 in Theorem.7. Proof of C-7 in Theorem.7. In [, Section.], Tancer describes a construction that realizes any simplicial complex on [n] as the intersection patterns of an arrangement of closed convex sets in R n (for n > ). It is straightforward to check that open neighborhoods of these closed sets form a convex realization for the code C =. Thus, any code on n > neurons with C = (C) is convex, and satisfies d(c) n. Note that for simplicial complex codes, the upper bound d(c) n is tight. For any n, if C = where is the hollow simplex on n vertices, then we must have d(c) n. 7. Table and figures for codes on n neurons Figure 8, Table 7, and Figure 9 are referred to in Section., in connection to the proof of Theorem.. 9

30 L L o o o o L L L5 o d = d = d = d = L6 L7 + L8 o d = d = d = d =, L9 o L0 o L Lo d = d = d = d = L L L5+ o L6 d = d =, d = d =, L7 L8 L9 + L0 + d = d =, d = d = L L L+ L L5 d = d =, d = d =, L6 L7+ L8o d = d = d = d =, Figure 8: Classification of convex codes for n. L-L8 are the 8 possible simplicial complexes = (C) that can arise, up to symmetry. For each simplicial complex, faces highlighted in color correspond to pairwise (red) and triple (orange) intersections of facets (maximal faces); by Lemma., these are exactly the mandatory codewords M( ). By Theorem., any code C with simplicial complex is convex if and only if C includes all codewords corresponding to the colored faces. All other non-maximal faces are optional codewords, whose inclusion or exclusion does not affect convexity. Simplicial complexes with no optional codewords are labeled +, while those with no (non-maximal) mandatory codewords are labeled. When both + and labels apply, as in L, L, and L, we simply use. The possible minimal embedding dimensions d = d(c) that can arise for convex codes are shown in the lower right corners. 0

31 # non maximal mandatory optional convex codewords (facets) codewords codewords of d(c) picture notes L 0 none none L 0 0, 0 none none L 0 none all () L 0 00, 00, 00 none none L5 0 0, 00 none all () L6 0, , 00 L6-series L7 + 0, 0, 0 all () none L7-series convex C = (C) L8 0 none all (6), L8-series d = L , , 000 none none L0 0 00, 000, 000 none all () L 00, 00, , 000 L6-series L 0 00, 00 none all () L 7 00, 00, , , 000 L6-series L 00, 00, , 000, 000, L6-series d = L5 + 00, 00 00, 000 all () none like L7 convex C = (C) L6 0 0, 000 none all (6), like L8 d = L7 0 00, , , L7-series d = C contains L8 0 0, , 000, 000, L8-series {000, 000}, or 00, 00, 00 {00, 00, 00} L , 00 00, 00 all () none L6-series convex C = (C) L , 00, 00 00, 00 all () none L7-series convex C = (C) L 68 0, 00, , , , 00 L8-series 000, 000, 000 d =, L 6 0, , 00, 00, L-series or {00, 00} C, 00, 00 or {00, 00} C L + 00, 00, 00 00, 00, 00 all () none L7-series convex C = (C) L 6 0, , 000 L8-series d = no 00, 00, 00, 00, , 000 d= codes optional codewords L5 68 0, 0, , , 000, , 00, 00 L-series L6 07 0, 0, 0 000, , 000, , 00 00, 00, 00 L-series L , 0 0, 0 all (0) none d= codes convex C = (C) L8 0 none all (), L-series d = Table 7: Convexity and dimension for codes on n neurons. For each, the second column is the number of non-convex codes C such that (C) =, up to permutation equivalence and including the all-zeros codeword, while the sixth column d(c) displays the possible minimal embedding dimensions for convex codes only. Mandatory codewords are non-maximal elements of that must be included in order for C to be convex. Optional codewords do not affect whether or not C is convex, though they may alter the minimal embedding dimension d(c). When all non-maximal codewords are mandatory or optional, their total number is given in parentheses. The picture column indicates the groupings used for the convex realizations in Figure 9. In the notes column, indicates that the set of optional codewords in C can not form a -chain. A collection of codewords forms a chain if we can completely order the respective sets by containment so {, 00, 000} is a chain, but {0, 000, 0} is not. A collection of codewords can form a -chain if it can be partitioned into two sets (possibly empty) which are both chains. + and are the same as in Figure 8.

32 L6 - series L6 L7 - series L7 L8 - series L8 L L - series L L7 L8 d= codes Lb L L0 L5 L L7 L6 L L8 L La L9 Figure 9: Convex realizations for codes on n neurons.