Algebraic signatures of convex and non-convex codes

07/08/2018 ∙ by Carina Curto, et al. ∙ University of Northern Colorado 0

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.

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Abstract

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.

Keywords: neural coding, convex codes, neural ideal, local obstructions, simplicial complexes, links

1 Introduction

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space (see Section 1.1 for a precise definition and example). Convex codes have been experimentally observed in sensory cortices [8] and hippocampus [11], where they arise from convex receptive fields; this connection has previously been described in detail in [2, 3, 5]. Given their relevance to neuroscience, it is valuable to further understand the intrinsic structure of convex codes. In particular, how can we detect if a neural code is convex?

We have previously found combinatorial constraints that must be satisfied by any code that is convex [3]. In this work, we further address the question of convexity via an algebraic object known as the neural ideal , first introduced in [5], which is a generalization of the well-studied Stanley-Reisner ideal. We first present conditions, which we refer to as algebraic signatures, on and its standard generating set the canonical form that detect that a code is not convex. We also connect these signatures to precise conditions on the arrangement of sets that prevent a code from being convex. Finally, we also provide algebraic signatures of certain combinatorial families of convex codes, including intersection-complete codes, first introduced in [1].

In Section 1.1, we provide some background on the algebra of neural codes, convexity of codes, receptive field relationships, and local obstructions to convexity. Next, Section 1.2 highlights the main results of the paper. Specifically, Theorem 1.7 provides algebraic signatures of two classes of local obstructions; codes satisfying these signatures are thus guaranteed to be non-convex. Theorem 1.9 gives an algebraic signature for the class of intersection-complete codes, which have been proven to be convex. Section 1.3 illustrates these main results through a series of example codes satisfying these algebraic signatures.

The remainder of the paper is organized as follows: Section 2.1 formalizes the notion of local obstruction, and Section 2.2 provides further results on detecting local obstructions algebraically, including the proof of Theorem 1.7. Section 3 focuses on algebraic signatures guaranteeing convexity, and includes the proof of Theorem 1.9. Finally, Section 4 collects all the algebraic signatures presented in this paper and provides additional examples of codes satisfying these signatures.

1.1 Background

In this paper, we develop algebraic tools for analyzing neural codes, which are collections of binary patterns. A binary pattern on neurons is a string of s and s of length , with a for each active neuron and a denoting silence. We can also view a binary pattern as the subset of active neurons , so that precisely when there is a in the th entry of the binary pattern; thus, we will consider 0/1 strings of length and subsets of interchangeably. For example, and are also denoted and , respectively.

A neural code on neurons, , is a collection of binary patterns. Such a code is also referred to as a combinatorial code in the neuroscience literature [4]. The elements of a code are called codewords. For convenience, we will always assume a neural code includes the all-zeros codeword, ; the presence or absence of the all-zeros codeword has no effect on the code’s convexity (see Definition 1.5, below), which is the main focus of this paper.

Algebra of neural codes

In order to represent a neural code algebraically, it is useful to consider binary patterns of length as elements of , where is the finite field of two elements: and . Polynomials can be evaluated on a binary pattern of length by evaluating each indeterminate at the 0/1 value of the neuron. For example, if , then and .

It is natural to then consider the ideal

of polynomials that vanish on a neural code . However, this ideal contains extraneous Boolean relations that do not capture any information specific to the code. Thus we turn instead to the neural ideal , first introduced in [5], which captures all the information in

that is specific to the code, thus omitting the Boolean relations. More precisely, the neural ideal can be defined in terms of characteristic functions of non-codewords:

where is the characteristic function

(1)

Note that the variety of both and is precisely the code [5].

The characteristic functions used to define the neural ideal are examples of pseudo-monomials, polynomials that can be written in the form

where and with . Pseudo-monomials in come in two types222There is a third type (see [5]), but this is eliminated by our convention that .:

  • Type 1: , for , and

  • Type 2: , for .

For any ideal , a pseudo-monomial is called minimal if there does not exist another pseudo-monomial with such that for some . If is an ideal generated by a set of pseudo-monomials, the canonical form of is the set of all minimal pseudo-monomials of :

For any neural code , the neural ideal is generated by pseudo-monomials, and thus has a canonical form .333Furthermore, every ideal generated by pseudo-monomials is actually the neural ideal of some neural code [9]. We denote the Type 1 and Type 2 pseudo-monomials of by and , respectively, so that:

Example 1.1.

Consider the code . The neural ideal is given by

which has canonical form , where

Note that is a generating set for , as every pseudo-monomial of is a multiple of an element in . Furthermore, generates the ideal of monomials in , which is precisely the Stanley-Reisner ideal of the associated simplicial complex , where

is the smallest abstract simplicial complex on that contains all elements of [5]. In particular, if is a simplicial complex, then is precisely the Stanley-Reisner ideal of . Note that the facets of , which are maximal elements of the simplicial complex under inclusion, correspond to the maximal codewords of .

The canonical form of a code can be computed algorithmically; for example, [5, Section 4.5] provides an algorithm using primary decompositions of pseudo-monomial ideals. A more efficient algorithm has since been proposed in [12], with software publicly available [14]. Supplemental Text S1 gives full details for computing the canonical form of an example code by hand; for information on using software to compute , see [12].

The code of a cover

Let be a topological space. A collection of non-empty open sets , where each , is called an open cover. Given an open cover , the code of the cover is the neural code

where . We say that a code is realized by if . Observe that is subdivided into regions defined by intersections of the open sets in . Each codeword in then corresponds to a non-empty intersection that is not covered by other sets in (see Example 1.2). By convention, the empty intersection equals , so that if and only if . We will assume , so that (i.e., ), in agreement with our convention.

It is important to note that is not the same as the nerve of the cover, which consists of all non-empty intersections, regardless of whether the intersection region is covered by other sets:

In fact, , the simplicial complex of the code [5]. The nerve of any cover such that can thus be recovered directly from the code as , without reference to a specific cover. The code , however, contains additional information about that is not captured by the nerve alone (see [5, Section 2.3.2]).

Example 1.2.

Consider the configuration of sets shown in Figure 1. The code of the cover is Note that from alone, we can detect that any realization must have , since every codeword with a 1 in the 4th position has a 1 in the 1st or 2nd position as well. However, this containment information is not available from the nerve .

Figure 1: Code of the cover .
RF relationships and the neural ideal.

Any realization of a code by an open cover will satisfy relationships among the that are intrinsic to the code itself. Because of the neuroscience motivation, where the model receptive fields, we call these receptive field relationships [5].

Definition 1.3.

For with and , we say that is a receptive field (RF) relationship of a code if

for any where . denotes the collection of RF relationships of .

It is important to note that the receptive field relationships are strictly a function of the code itself and do not depend on any particular realization of as . Specifically, RF relationships correspond to pseudo-monomials in as shown in Table 1, and thus are detectable algebraically without reference to a specific cover [5].

Relation type   Pseudo-monomial RF condition
     Type 1 hi
     Type 2 hi
Table 1: Types of pseudo-monomials in and the corresponding conditions on receptive fields. Note that the presence of a Type 2 pseudo-monomial is not sufficient to guarantee that is actually an RF relationship. Such a pseudo-monomial ensures the covering relationship , but to guarantee that for we must also have for all .

The RF relationships of the form capture when , and thus , yielding a complete description of . In contrast, the RF relationships for capture when an intersection is covered so that despite , thus measuring how deviates from its simplicial complex.

A RF relationship is called minimal if no neuron can be removed from or without destroying the containment . The following useful fact is a direct consequence of [5, Theorem 4.3], which allows us to interpret the elements of as minimal RF relationships.

Lemma 1.4.

The pseudo-monomial if and only if is a minimal RF relationship of .

Thus, the canonical form gives a compact description of that captures all the minimal intersection and containment relations that must exist among sets that give rise to the code.

Convex codes

When the open cover is contained in for some , the sets may (for some codes) be chosen to all be convex. If this is possible, we say that the code is convex:

Definition 1.5.

Let be a neural code on neurons. If there exists an open cover such that and every is a convex subset of for a fixed , then we say that is convex.

Note that the code in Example 1.2 is convex since it can be realized via the convex sets shown in Figure 1. In contrast, the code from Example 1.1 is not convex, as the following example shows.

Example 1.6.

Recall the code from Example 1.1. Neuron 1 always co-fires with neuron 2 or neuron 3 since a 1 only occurs in the first entry when it is accompanied by a 1 in the second or third entry. This forces the RF relationship to hold in any realization of the code. But neurons 1, 2, and 3 never co-fire, so . Thus is the disjoint union of non-empty open sets and , and so is disconnected. Since any convex set is connected, we conclude that cannot be convex, and thus is not convex.

This topological mismatch between the underlying set and its cover by and is an example of a local obstruction [3, 6]; we define local obstructions precisely in Section 2.1. Notice that this local obstruction is immediately identifiable from the canonical form seen in Example 1.1: the RF relationship is detectable from and the RF relationship is captured by .

1.2 Summary of main results

Detecting non-convex codes.

Example 1.6 shows that some local obstructions to convexity can be detected algebraically from the neural ideal of a code. In particular, any code satisfying the algebraic signature and is guaranteed to be non-convex. This is because is forced to be disconnected since it is the disjoint union of the nonempty sets and .

Theorem 1.7 gives two additional algebraic signatures of local obstructions that force a code to be non-convex. The first signature captures more generally when the nerve of a cover of is disconnected, thus forcing to be disconnected and non-convex. The second signature captures cases when the nerve is a hollow simplex, thus forcing to contain a hole. In other words, these signatures capture when the nerve of the cover of has a nontrivial 0th homology group and nontrivial top homology group, respectively. It remains an open question to identify algebraic signatures that can detect when a relevant nerve has an intermediate homology group that is nontrivial.

Theorem 1.7.

Let be a code with neural ideal and canonical form , and let be the simple graph on vertex set with edge set . The following algebraic signatures imply that is not convex.

Algebraic signature of Property of
(i) s.t. is disconnected non-convex
(ii) s.t. non-convex
Table 2: Algebraic signatures of non-convex codes.

It is important to note that although signature (i) in Table 2 requires the construction of a graph based on the absence of pseudo-monomials from all of , this condition can actually be checked in a straightforward manner from alone (see Lemma 2.5 in Section 1.7). The signatures of local obstructions in Theorem 1.7 can thus be directly detected from the canonical form of the code. The proof of Theorem 1.7 is given in Section 2.2.

Our previous work has given an alternative method of identifying the full set of local obstructions; however, the recasting of those local obstructions in terms of RF relationships is less well understood. A characterization of the full set of local obstructions of a code is given in Theorem 1.3 of [3]. In general, however, the absence of local obstructions does not guarantee that is convex [10]. Thus, it is essential to have other methods of identifying convexity.

Detecting convex codes.

Currently the only known method for proving a code is convex is to produce a convex realization or establish that it belongs to a combinatorial family of codes for which a construction of a convex realization is known. In the following, we give algebraic signatures for identifying when a code belongs to any of four combinatorial families of codes for which convex constructions are known.

The simplest algebraic signatures of families of convex codes are or . Since captures minimal subsets missing from , the signature implies is the full simplex, and so must contain the all-ones word. Convex realizations of such codes were given in [3]. When contains the all-ones word (), has a single facet, and this fact is exploited in the construction of convex realizations of these codes. More generally, if has disjoint facets this same construction can be employed in parallel for each facet, ensuring these codes are also convex [3]. These codes can also be detected algebraically, but the signature is more complicated, so we save the statement and proof of the signature for Section 3.

On the other hand, implies that is a simplicial complex, which is guaranteed to have a convex realization [3, 13]. These codes can be generalized to a broader family of codes known as intersection-complete codes, which are also known to be convex [1].

Definition 1.8.

A code is intersection-complete (-complete) if every intersection of codewords is also a codeword in ; i.e.  implies that .

The algebraic signature for -complete codes is given in the following theorem, whose proof appears in Section 3.

Theorem 1.9.

A code is -complete if and only if every pseudo-monomial has . If is -complete, then is convex.

Note that if for all elements of , then , which is the signature for simplicial complex codes. Using Table 1, the algebraic signature in Theorem 1.9 can be reinterpreted in terms of receptive fields as follows: for any realization of an -complete code , every intersection for is minimally covered by a single set for some .

The families of codes presented above, for which we have algebraic signatures, are special cases of max -complete codes: codes for which every intersection of a collection of facets of is also a codeword in . In [1], convex realizations of max -codes were constructed, guaranteeing their convexity.

Theorem 1.10.

[1, Theorem 4.4] If a code is max -complete, then is convex.

Finding an algebraic signature of max -complete codes remains an open question. Given that these codes generalize -complete codes, one might hope to generalize the algebraic signature of -complete codes to obtain a signature for this broader class. One natural generalization is the class of codes for which every pseudo-monomial has . Unfortunately, Example 1.11 (below) shows that codes with this property need not be max -complete and vice versa. In particular, the code in Example 1.11(b) has the property, but is not even convex.

Example 1.11.

(a) Consider the code

with maximal codewords 1110 and 1011. This code is max -complete because it would in fact be a simplicial complex except that it is missing 1000, which is not an intersection of maximal codewords. However, does not satisfy , since .

(b) Consider the code

with maximal codewords (i.e. facets ). is not max -complete since it does not contain the triple intersection of facets . However, satisfies for all since

Interestingly, this code is not convex, although it has no local obstructions [10].

Note that the code from Example 1.6 also satisfies and is not convex, but it has a local obstruction. Thus, the signature does not ensure convexity or provide guarantees about the presence/absence of local obstructions.

1.3 Examples illustrating main results

This section gives examples of codes satisfying each of the algebraic signatures presented in Theorems 1.7 and 1.9 together with an analysis of the implications of these signatures for RF relationships.

We begin with an example of a code on neurons that satisfies the first signature in Theorem 1.7.

Example 1.12 (Theorem 1.7, signature (i)).

Consider the code

This code has and

where all the indices in the pseudo-monomials of are distinct. Consider

where and . We will construct the graph whose vertices are precisely the elements of . By definition, whenever for , then is an edge in . Using , we immediately see that and are not edges in , and that and are edges in (see Lemma 2.5). Thus consists only of two disjoint edges, and is disconnected. (Note that this implies that and are disjoint, and so is disconnected, as it is covered by the disjoint union of nonempty open sets.) Therefore, signature (i) of Theorem 1.7 is satisfied and is not convex.

The next example gives a code on neurons satisfying the second signature of Theorem 1.7.

Example 1.13 (Theorem 1.7, signature (ii)).

Consider . Then

Since and , we see that signature (ii) of Theorem 1.7 applies. Thus is not convex.

To see the obstruction to convexity here, note that since we have from Table 1 that is minimally covered by . Also, since , the full intersection is empty, but the minimality of elements in guarantees that every other intersection is non-empty. This forces to contain a hole (see Figure 2), and so cannot be convex, and hence cannot be convex.

Figure 2: For the code in Example 1.13, the set is the union of the shaded regions shown since it is covered by . must contain a hole since the covering sets all pairwise intersect, but the full intersection is missing.

Finally, the following example shows how to use the neural ideal to detect that a code is -complete, and thus convex.

Example 1.14 (Theorem 1.9).

Consider . This code has

We immediately see that all elements of satisfy , and so the signature from Theorem 1.9 applies. Thus, is -complete.

2 Detecting local obstructions

The primary method for showing that a code is not convex is to show that it has a local obstruction. Section 2.1 defines local obstructions and connects them to links of certain restricted simplicial complexes. Section 2.2 shows how to detect certain classes of local obstructions via and and provides the proof of Theorem 1.7.

2.1 Local obstructions

Recall that the code in Example 1.6 failed to have a convex realization because the receptive field was covered by a pair of disjoint nonempty open sets and , and thus no realization of could have as a convex set. In this case, the restricted cover of by and had a nerve that was disconnected and thus, if were convex, there would be a topological mismatch between and the nerve of its restricted cover. This topological mismatch is an example of a local obstruction. Specifically, the Nerve Lemma [7, Corollary 4G.3] guarantees that if is a convex open cover (and thus a “good cover”), then must have the same homotopy type as whenever is non-empty and covered by a union of sets , i.e. whenever . In particular, since is the intersection of convex sets, it must be convex and hence contractible444A set is contractible if it is homotopy-equivalent to a point, and every convex set is contractible [7]., and thus must also be contractible. Thus, if the nerve of such a restricted cover is not contractible, then a local obstruction is present. This restricted nerve has an alternative combinatorial formulation; specifically,

where is the restricted simplicial complex

and the link is given by

This alternative characterization of the nerve yields the following formal definition of local obstruction. For more details about local obstructions, see [3, Section 3].

Definition 2.1.

Let be a code on neurons with simplicial complex .
For with , we say that is a local obstruction of if and the link is not contractible.

As an immediate consequence of the Nerve Lemma, as described above, we obtain Lemma 2.2.

Lemma 2.2.

[3, Lemma 1.3] If has a local obstruction, then is not a convex code.

2.2 Algebraic detection of local obstructions

In general, the presence of a pseudo-monomial is not sufficient to guarantee that is a RF relationship (see Table 1), and thus a possible candidate for a local obstruction. This is because we cannot guarantee that for all . However, when is minimal, i.e. when , these conditions are guaranteed and . Thus, we focus on the canonical form to algebraically detect local obstructions.

Lemma 2.3.

For a code , if there exists such that and is not contractible, then is not convex.

With this result, we can now prove Theorem 1.7. Specifically, we prove Theorem 2.4, a broader result that also characterizes relevant RF conditions corresponding to these signatures.

Theorem 2.4.

If has any of the algebraic signatures in rows A-1, A-2, A-3, or A-4 of Table 3, then is not convex. More precisely, each algebraic signature corresponds to a RF condition (as illustrated in Figure 3), which implies that is not convex.

Algebraic signature Receptive field condition Property of
A-1 and non-convex
s.t.

A-2
and non-convex
s.t. is disconnected is disconnected

A-3
and non-convex
s.t. but

A-4
, and non-convex
s.t. but
but

Table 3: Algebraic signatures and receptive field conditions for non-convex codes. is the simple graph on vertex set with edge set . The graph has vertex set and edge set .
Figure 3: Illustrations of the RF conditions implied by signatures A-1 through A-4 in Theorem 2.4 (see Table 3). In each picture, is the union of the shaded regions; thus is not contractible and hence not convex.
Proof of Theorem 2.4.

(A-1) By Lemma 1.4, implies , and thus and both and are non-empty. Recall from Table 1 that implies . Thus, is the disjoint union of non-empty open sets and , and so is disconnected. Thus, is not convex, and so some is not convex. Hence is not convex.

(A-2) By Table 1 if , then since precisely when Furthermore, this graph is precisely the -skeleton555The 1-skeleton of a simplicial complex is the subcomplex consisting of all faces of dimension at most 1, i.e. the vertices and edges of the simplicial complex; thus the 1-skeleton is the underlying graph of the simplicial complex (see e.g. [7]). of . Since we assume this is disconnected, it follows that is not contractible, and hence is non-convex by Lemma 2.3. Alternatively, disconnected implies that is disconnected, and hence cannot be convex.

(A-3) The signature for A-3 is a special case of that for A-4 since guarantees for all by minimality of the elements in the canonical form. Thus, we prove non-convexity of these codes via the following proof of A-4.

(A-4) Note that implies that by Table 1 and thus as well. Thus, and so . For every , we have , since if it were in then some factor of it must be in , but for every and . Thus, for all , we have and so ; equivalently for all . This means is a simplex missing only the top dimensional face (i.e. a hollow simplex), and so is homotopy-equivalent to a sphere, and thus is not contractible. At the level of RF relationships, this implies that is not contractible since it must contain a hole. Thus, is non-convex.

As the proof of Theorem 2.4 illustrates, signature A-1 captures cases where is disconnected by a pair of sets. Signature A-2 generalizes A-1 and detects all cases where is minimally covered by a collection of sets for in a way that forces to be disconnected. Note that A-2 is signature (i) from Theorem 1.7 in the main results (Section 1.2).

Signature A-3 captures a particular case when is minimally covered by a collection of sets for and is a hollow simplex. Specifically, in the case of A-3, is the minimal missing intersection in that for all , we have ; thus everywhere outside of , has a non-empty intersection with each subcollection of sets from . More generally, signature A-4 captures all cases when is a hollow simplex. Specifically, the signature for A-4 does not require the minimality of the empty intersection , and so there may be a such that , in particular we may have . All that is required is that every intersection of with each proper subcollection of sets in is non-empty, which is guaranteed by ensuring that , for all