The Neural Ring: An Algebraic Tool for Analyzing the Intrinsic Structure of Neural Codes

Abstract

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can—in principle—be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode the full combinatorial data of a neural code. Our main finding is that these objects can be expressed in a “canonical form” that directly translates to a minimal description of the receptive field structure intrinsic to the code. We also find connections to Stanley–Reisner rings, and use ideas similar to those in the theory of monomial ideals to obtain an algorithm for computing the primary decomposition of pseudo-monomial ideals. This allows us to algorithmically extract the canonical form associated to any neural code, providing the groundwork for inferring stimulus space features from neural activity alone.

Appendices

Appendix 1: Proofs

1.1 6.1 Proof of Lemmas 3.1 and 3.2

To prove Lemmas 3.1 and 3.2, we need a version of the Nullstellensatz for finite fields. The original “Hilbert’s Nullstellensatz” applies when k is an algebraically closed field. It states that if f∈k[x ₁,…,x _n ] vanishes on V(J), then $f \in\sqrt{J}$. In other words,

$$I\bigl(V(J)\bigr) = \sqrt{J}. $$

Because we have chosen $k = \mathbb{F}_{2} = \{0,1\}$, we have to be a little careful about the usual ideal-variety correspondence, as there are some subtleties introduced in the case of finite fields. In particular, $J = \sqrt{J}$ in $\mathbb{F}_{2}[x_{1},\ldots,x_{n}]$ does not imply I(V(J))=J.

The following lemma and theorem are well known. Let $\mathbb{F}_{q}$ be a finite field of size q, and $\mathbb {F}_{q}[x_{1},\ldots,x_{n}]$ the n-variate polynomial ring over $\mathbb{F}_{q}$.

Lemma 6.1

For any ideal $J \subseteq\mathbb{F}_{q}[x_{1},\ldots,x_{n}]$, the ideal $J+\langle x_{1}^{q}-x_{1}, \ldots, x_{n}^{q}-x_{n} \rangle$ is a radical ideal.

Theorem 6.2

(Strong Nullstellensatz in Finite Fields)

For an arbitrary finite field $\mathbb{F}_{q}$, let $J \subseteq\mathbb {F}_{q}[x_{1},\ldots,x_{n}]$ be an ideal. Then

$$I\bigl(V(J)\bigr) = J + \bigl\langle x_1^q-x_1, \ldots, x_n^q-x_n \bigr\rangle. $$

1.1.1 6.1.1 Proof of Lemma 3.1

We begin by describing the maximal ideals of $\mathbb{F}_{2}[x_{1},\ldots,x_{n}]$. Recall that

$$m_v \stackrel{\mathrm{def}}{=}I(v) = \bigl\{f \in \mathbb{F}_2[x_1,\ldots ,x_n] \bigm| f(v) = 0 \bigr\} $$

is the maximal ideal of $\mathbb{F}_{2}[x_{1},\ldots,x_{n}]$ consisting of all functions that vanish on $v \in\mathbb{F}_{2}^{n}$. We will use the notation $\bar{m}_{v}$ to denote the quotient of m _v in $R_{\mathcal{C}}$, in cases where $m_{v} \supset I_{\mathcal{C}}$.

Lemma 6.3

$m_{v} = \langle x_{1}-v_{1}, \ldots, x_{n}-v_{n} \rangle\subset\mathbb{F} _{2}[x_{1},\ldots ,x_{n}]$, and is a radical ideal.

Proof

Denote A _v =〈x ₁−v ₁,…,x _n −v _n 〉, and observe that V(A _v )={v}. It follows that I(V(A _v ))=I(v)=m _v . On the other hand, using the Strong Nullstellensatz in Finite Fields we have

$$I\bigl(V(A_v)\bigr) = A_v + \bigl\langle x_1^2-x_1,\ldots,x_n^2-x_n \bigr\rangle= A_v, $$

where the last equality is obtained by observing that, since v _i ∈{0,1} and $x_{i}^{2}-x_{i} = x_{i}(1-x_{i})$, each generator of $\langle x_{1}^{2}-x_{1},\ldots,x_{n}^{2}-x_{n} \rangle$ is already contained in A _v . We conclude that A _v =m _v , and the ideal is radical by Lemma 6.1. □

In the proof of Lemma 3.1, we make use of the following correspondence: for any quotient ring R/I, the maximal ideals of R/I are exactly the quotients $\bar{m} = m/I$, where m is a maximal ideal of R that contains I (Atiyah and Macdonald 1969).

Proof of Lemma 3.1

First, recall that because $R_{\mathcal{C}}$ is a Boolean ring, $\mathrm {Spec}(R_{\mathcal{C}} ) = \mathrm{maxSpec}(R_{\mathcal{C}})$, the set of all maximal ideals of $R_{\mathcal{C}}$. We also know that the maximal ideals of $\mathbb{F}_{2}[x_{1},\ldots,x_{n}]$ are exactly those of the form m _v for $v\in\mathbb{F}_{2}^{n}$. By the correspondence stated above, to show that $\mathrm{maxSpec}(R_{\mathcal{C}}) = \{ \bar{m}_{v} \mid v \in\mathcal {C}\}$ it suffices to show $m_{v} \supset I_{\mathcal{C}}$ if and only if $v\in\mathcal{C}$. To see this, note that for each $v\in\mathcal{C}$, $I_{\mathcal{C}}\subseteq m_{v}$ because, by definition, all elements of $I_{\mathcal{C}}$ are functions that vanish on each $v \in\mathcal{C}$. On the other hand, if $v\notin\mathcal{C}$ then $m_{v} \nsupseteq I_{\mathcal{C}}$; in particular, the characteristic function $\rho_{v} \in I_{\mathcal{C}}$ for $v \notin \mathcal{C}$, but ρ _v ∉m _v because ρ _v (v)=1. Hence, the maximal ideals of $R_{\mathcal{C}}$ are exactly those of the form $\bar{m}_{v}$ for $v\in\mathcal{C}$. □

We have thus verified that the points in $\mathrm{Spec}(R_{\mathcal{C}})$ correspond to codewords in $\mathcal{C}$. This was expected given our original definition of the neural ring, and suggests that the relations on $\mathbb{F} _{2}[x_{1},\ldots,x_{n}]$ imposed by $I_{\mathcal{C}}$ are simply relations ensuring that $V(\bar{m}_{v}) = \emptyset$ for all $v \notin\mathcal{C}$.

1.1.2 6.1.2 Proof of Lemma 3.2

Here we find explicit relations for $I_{\mathcal{C}}$ in the case of an arbitrary neural code. Recall that

$$\rho_v = \prod_{i=1}^n \bigl((x_i-v_i)-1\bigr) = \prod _{\{i\,|\,v_i=1\} }x_i\prod_{\{j\,|\,v_j=0\}}(1-x_j), $$

and that ρ _v (x) can be thought of as a characteristic function for v, since it satisfies ρ _v (v)=1 and ρ _v (x)=0 for any other $x \in\mathbb{F}_{2}^{n}$. This immediately implies that

$$V(J_\mathcal{C}) = V\bigl(\bigl\langle\{ \rho_v \mid v \notin\mathcal{C}\} \bigr\rangle\bigr) = \mathcal{C}. $$

We can now prove Lemma 3.2.

Proof of Lemma 3.2

Observe that $I_{\mathcal{C}}= I(\mathcal{C}) = I(V(J_{\mathcal{C}}))$, since $V(J_{\mathcal{C}}) = \mathcal{C}$. On the other hand, the Strong Nullstellensatz in Finite Fields implies $I(V(J_{\mathcal{C}})) = J_{\mathcal{C}}+ \langle x_{1}^{2}-x_{1},\ldots,x_{n}^{2}-x_{n} \rangle= J_{\mathcal{C}}+ \mathcal{B}$. □

1.2 6.2 Proof of Theorem 4.1

Recall that for a given set of receptive fields $\mathcal{U}= \{ U_{1},\ldots ,U_{n}\}$ in some stimulus space X, the ideal $I_{\mathcal{U}}\subset\mathbb {F}_{2}[x_{1},\ldots ,x_{n}]$ was defined as

$$I_\mathcal{U}\stackrel{\mathrm{def}}{=} \biggl\langle\biggl\{ x_\sigma \prod_{i \in\tau} (1-x_i) \biggm| U_\sigma\subseteq \bigcup_{i \in\tau}U_i \biggr\} \biggr\rangle. $$

The Boolean relations are present in $I_{\mathcal{U}}$ irrespective of $\mathcal{U}$, as it is always true that U _i ⊆U _i and this yields the relation x _i (1−x _i ) for each i. By analogy with our definition of $J_{\mathcal{C}}$, it makes sense to define an ideal $J_{\mathcal{U}}$ which is obtained by stripping away the Boolean relations. This will then be used in the proof of Theorem 4.1.

Note that if σ∩τ≠∅, then for any i∈σ∩τ we have U _σ ⊆U _i ⊆⋃ _j∈τ U _i , and the corresponding relation is a multiple of the Boolean relation x _i (1−x _i ). We can thus restrict attention to relations in $I_{\mathcal{U}}$ that have σ∩τ=∅, so long as we include separately the Boolean relations. These observations are summarized by the following lemma.

Lemma 6.4

$I_{\mathcal{U}}= J_{\mathcal{U}}+ \langle x_{1}^{2}-x_{1},\ldots,x_{n}^{2}-x_{n} \rangle$, where

$$J_\mathcal{U}\stackrel{\mathrm{def}}{=} \biggl\langle\biggl\{ x_\sigma \prod_{i \in\tau} (1-x_i) \biggm| \sigma\cap\tau= \emptyset\;\;\mathrm{ and }\;\; U_\sigma\subseteq \bigcup_{i \in\tau}U_i \biggr\} \biggr\rangle. $$

Proof of Theorem 4.1

We will show that $J_{\mathcal{U}}=J_{\mathcal{C}(\mathcal{U})}$ (and thus that $I_{\mathcal{U}}= I_{\mathcal{C}(\mathcal{U})}$) by showing that each ideal contains the generators of the other.

First, we show that all generating relations of $J_{\mathcal {C}(\mathcal{U})}$ are contained in $J_{\mathcal{U}}$. Recall that the generators of $J_{\mathcal{C}(\mathcal{U})}$ are of the form

$$\rho_v = \prod_{i \in\operatorname{supp}(v)}x_i \prod_{j \notin \operatorname{supp}(v)}(1-x_j) \quad \text{for } v \notin\mathcal{C}(\mathcal{U}). $$

If ρ _v is a generator of $J_{\mathcal{C}(\mathcal{U})}$, then $v \notin\mathcal{C}(\mathcal{U})$ and this implies (by the definition of $\mathcal{C}(\mathcal{U})$) that $U_{\operatorname{supp}(v)} \subseteq\bigcup_{j \notin \operatorname{supp}(v)} U_{j}$. Taking $\sigma= \operatorname{supp}(v)$ and $\tau= [n] \setminus \operatorname{supp}(v)$, we have U _σ ⊆⋃ _j∈τ U _j with σ∩τ=∅. This in turn tells us (by the definition of $J_{\mathcal{U}}$) that x _σ ∏ _j∈τ (1−x _j ) is a generator of $J_{\mathcal{U}}$. Since ρ _v =x _σ ∏ _j∈τ (1−x _j ) for our choice of σ and τ, we conclude that $\rho_{v} \in J_{\mathcal{U}}$. Hence, $J_{\mathcal{C}(\mathcal{U})} \subseteq J_{\mathcal{U}}$.

Next, we show that all generating relations of $J_{\mathcal{U}}$ are contained in $J_{\mathcal{C}(\mathcal{U})}$. If $J_{\mathcal{U}}$ has generator x _σ ∏ _i∈τ (1−x _i ), then U _σ ⊆⋃ _i∈τ U _i and σ∩τ=∅. This in turn implies that ⋂ _i∈σ U _i ∖⋃ _j∈τ U _j =∅, and thus (by the definition of $\mathcal{C}(\mathcal{U})$) we have $v\notin\mathcal {C}(\mathcal{U})$ for any v such that $\operatorname{supp}(v)\supseteq\sigma$ and $\operatorname {supp}(v)\cap\tau=\emptyset$. It follows that $J_{\mathcal{C}(\mathcal{U})}$ contains the relation $x_{\operatorname{supp} (v)}\prod_{j\notin\operatorname{supp}(v)}(1-x_{j})$ for any such v. This includes all relations of the form x _σ ∏ _j∈τ (1−x _j )∏ _k∉σ∪τ P _k , where P _k ∈{x _k ,1−x _k }. Taking f=x _σ ∏ _j∈τ (1−x _j ) in Lemma 6.5 (below), we can conclude that $J_{\mathcal {C}(\mathcal{U})}$ contains x _σ ∏ _j∈τ (1−x _j ). Hence, $J_{\mathcal{U}}\subseteq J_{\mathcal{C}(\mathcal{U})}$. □

Lemma 6.5

For any f∈k[x ₁,…,x _n ] and τ⊆[n], the ideal 〈{f∏ _i∈τ P _i ∣P _i ∈{x _i ,1−x _i }}〉=〈f〉.

Proof

First, denote $I_{f}(\tau)\stackrel{\text{def}}{=}\langle \{f\prod_{i\in\tau} P_{i} \mid P_{i} \in\{x_{i}, 1-x_{i}\} \} \rangle$. We wish to prove that I _f (τ)=〈f〉, for any τ⊆[n]. Clearly, I _f (τ)⊆〈f〉, since every generator of I _f (τ) is a multiple of f. We will prove I _f (τ)⊇〈f〉 by induction on |τ|.

If |τ|=0, then τ=∅ and I _f (τ)=〈f〉. If |τ|=1, so that τ={i} for some i∈[n], then I _f (τ)=〈f(1−x _i ),fx _i 〉. Note that f(1−x _i )+fx _i =f, so f∈I _f (τ), and thus I _f (τ)⊇〈f〉.

Now, assume that for some ℓ≥1 we have I _f (σ)⊇〈f〉 for any σ⊆[n] with |σ|≤ℓ. If ℓ≥n, we are done, so we need only show that if ℓ<n, then I _f (τ)⊇〈f〉 for any τ of size ℓ+1. Consider τ⊆[n] with |τ|=ℓ+1, and let j∈τ be any element. Define τ′=τ∖{j}, and note that |τ′|=ℓ. By our inductive assumption, I _f (τ′)⊇〈f〉. We will show that I _f (τ)⊇I _f (τ′), and hence I _f (τ)⊇〈f〉.

Let g=f∏ _i∈τ′ P _i be any generator of I _f (τ′) and observe that both f(1−x _j )∏ _i∈τ′ P _i and fx _j ∏ _i∈τ′ P _i are both generators of I _f (τ). It follows that their sum, g, is also in I _f (τ), and hence g∈I _f (τ) for any generator g of I _f (τ′). We conclude that I _f (τ)⊇I _f (τ′), as desired. □

1.3 6.3 Proof of Theorem 4.3

We begin by showing that $J_{\mathcal{U}}$, first defined in Lemma 6.4, can be generated using the Type 1, Type 2, and Type 3 relations introduced in Sect. 4.2. From the proof of Theorem 4.1, we know that $J_{\mathcal{U}}= J_{\mathcal{C}(\mathcal{U})}$, so the following lemma in fact shows that $J_{\mathcal{C}(\mathcal{U})}$ is generated by the Type 1, 2, and 3 relations as well.

Lemma 6.6

For $\mathcal{U}= \{U_{1},\ldots, U_{n}\}$ a collection of sets in a stimulus space X,

$$\begin{aligned} J_{\mathcal{U}} =& \biggl\langle\{x_\sigma\mid U_\sigma= \emptyset \}, \biggl\{ \prod_{i \in\tau} (1-x_i) \biggm| X \subseteq\bigcup_{i\in\tau} U_i \biggr\}, \\&\phantom{\biggl\langle} \biggl\{x_\sigma\prod_{i\in\tau} (1-x_i) \biggm| \sigma,\tau\neq \emptyset,\; \sigma\cap\tau=\emptyset,\; U_\sigma\neq\emptyset ,\\&\phantom{\biggl\langle\biggl\{} \bigcup_{i \in\tau} U_i \neq X , \text{ and } U_\sigma\subseteq \bigcup _{i\in\tau}U_i \biggr\} \biggr\rangle. \end{aligned}$$

$J_{\mathcal{U}}$ (equivalently, $J_{\mathcal{C}(\mathcal{U})}$) is thus generated by the Type 1, Type 3 and Type 2 relations, respectively.

Proof

Recall that in Lemma 6.4 we defined $J_{\mathcal{U}}$ as

$$J_\mathcal{U}\stackrel{\mathrm{def}}{=} \biggl\langle\biggl\{ x_\sigma \prod_{i \in\tau} (1-x_i) \biggm| \sigma\cap\tau= \emptyset\ \mathrm{and}\ U_\sigma\subseteq \bigcup_{i \in\tau}U_i \biggr\} \biggr\rangle. $$

Observe that if U _σ =∅, then we can take τ=∅ to obtain the Type 1 relation x _σ , where we have used the fact that ∏ _i∈∅(1−x _i )=1. Any other relation with U _σ =∅ and τ≠∅ would be a multiple of x _σ . We can thus write:

$$\begin{aligned} J_\mathcal{U} =& \biggl\langle\{x_\sigma\mid U_\sigma= \emptyset\},\\&\phantom{\biggl\langle} \biggl\{ x_\sigma \prod_{i \in\tau} (1-x_i) \biggm| \tau\neq\emptyset, \sigma\cap \tau= \emptyset, U_\sigma\neq\emptyset, \ \mathrm{and}\ U_\sigma\subseteq \bigcup_{i \in\tau}U_i \biggr\} \biggr\rangle. \end{aligned}$$

Next, if σ=∅ in the second set of relations above, then we have the relation ∏ _i∈τ (1−x _i ) with U _∅=X⊆⋃ _i∈τ U _i . Splitting off these Type 3 relations, and removing multiples of them that occur if ⋃ _i∈τ U _i =X, we obtain the desired result. □

Next, we show that $J_{\mathcal{U}}$ can be generated by reduced sets of the Type 1, Type 2, and Type 3 relations given above. First, consider the Type 1 relations in Lemma 6.6, and observe that if τ⊆σ, then x _σ is a multiple of x _τ . We can thus reduce the set of Type 1 generators needed by taking only those corresponding to minimal σ with U _σ =∅:

$$\bigl\langle\{x_\sigma\mid U_\sigma= \emptyset\}\bigr\rangle= \bigl\langle\{ x_\sigma\mid\sigma\text{ is minimal w.r.t. } U_\sigma= \emptyset\} \bigr\rangle. $$

Similarly, we find for the Type 3 relations:

$$\biggl\langle \biggl\{\prod_{i \in\tau} (1-x_i) \biggm| X \subseteq \bigcup_{i\in\tau} U_i \biggr\} \biggr\rangle= \biggl\langle \biggl\{\prod _{i \in\tau} (1-x_i) \biggm|\tau\text{ is minimal w.r.t. } X \subseteq\bigcup_{i\in\tau} U_i \biggr\} \biggr\rangle. $$

Finally, we reduce the Type 2 generators. If ρ⊆σ and $x_{\rho}\prod_{i\in\tau} (1-x_{i}) \in J_{\mathcal{U}}$, then we also have $x_{\sigma}\prod_{i\in\tau} (1-x_{i}) \in J_{\mathcal{U}}$. So we can restrict ourselves to only those generators for which σ is minimal with respect to U _σ ⊆⋃ _i∈τ U _i . Similarly, we can reduce to minimal τ such that U _σ ⊆⋃ _i∈τ U _i . In summary:

$$\begin{aligned} & \biggl\langle \biggl\{x_\sigma\prod_{i\in\tau} (1-x_i) \biggm|\sigma ,\tau \neq\emptyset,\; \sigma\cap\tau=\emptyset, \; U_\sigma\neq \emptyset ,\;\bigcup_{i \in\tau} U_i \neq X, \text{ and } U_\sigma\subseteq \bigcup _{i\in\tau}U_i \biggr\} \biggr\rangle\\&\quad = \biggl\langle \biggl\{x_\sigma\prod_{i\in\tau} (1-x_i) \biggm|\sigma ,\tau \neq\emptyset,\; \sigma\cap\tau=\emptyset, \; U_\sigma\neq \emptyset ,\\&\phantom{\quad = \biggl\langle\biggl\{ } \bigcup_{i \in\tau} U_i \neq X, \text{ and } \sigma, \tau\text{ are each minimal } \text{ w.r.t. } U_\sigma\subseteq\bigcup _{i\in\tau}U_i \biggr\} \biggr\rangle. \end{aligned}$$

We can now prove Theorem 4.3.

Proof of Theorem 4.3

Recall that $\mathcal{C}= \mathcal{C}(\mathcal{U})$, and that by the proof of Theorem 4.1 we have $J_{\mathcal{C}(\mathcal{U})} = J_{\mathcal{U}}$. By the reductions given above for the Type 1, 2, and 3 generators, we also know that $J_{\mathcal{U}}$ can be reduced to the form given in the statement of Theorem 4.3. We conclude that $J_{\mathcal{C}}$ can be expressed in the desired form.

To see that $J_{\mathcal{C}}$, as given in the statement of Theorem 4.3, is in canonical form, we must show that the given set of generators is exactly the complete set of minimal pseudo-monomials for $J_{\mathcal{C}}$. First, observe that the generators are all pseudo-monomials. If x _σ is one of the Type 1 relations, and x _σ ∈〈g〉 with 〈x _σ 〉≠〈g〉, then g=∏ _i∈τ x _i for some τ⊆̷σ. Since U _τ ≠∅, however, it follows that $g \notin J_{\mathcal{C}}$ and hence x _σ is a minimal pseudo-monomial of $J_{\mathcal{C}}$. By a similar argument, the Type 2 and Type 3 relations above are also minimal pseudo-monomials in $J_{\mathcal{C}}$.

It remains only to show that there are no additional minimal pseudo-monomials in $J_{\mathcal{C}}$. Suppose f=x _σ ∏ _i∈τ (1−x _i ) is a minimal pseudo-monomial in $J_{\mathcal{C}}$. By Lemma 4.2, U _σ ⊆⋃ _i∈τ U _i and σ∩τ=∅, so f is a generator in the original definition of $J_{\mathcal{U}}$ (Lemma 6.4). Since f is a minimal pseudo-monomial of $J_{\mathcal{C}}$, there does not exist a $g \in J_{\mathcal{C}}$ such that g=x _σ′∏ _i∈τ′(1−x _i ) with either σ′⊆̷σ or τ′⊆̷τ. Therefore, σ and τ are each minimal with respect to U _σ ⊆⋃ _i∈τ U _i . We conclude that f is one of the generators for $J_{\mathcal{C}}$ given in the statement of Theorem 4.3. It is a minimal Type 1 generator if τ=∅, a minimal Type 3 generator if σ=∅, and is otherwise a minimal Type 2 generator. The three sets of minimal generators are disjoint because the Type 1, Type 2, and Type 3 relations are disjoint, provided X≠∅. □

1.4 6.4 Proof of Proposition 4.5

Note that every polynomial obtained by the canonical form algorithm is a pseudo-monomial of $J_{\mathcal{C}}$. This is because the algorithm constructs products of factors of the form x _i or 1−x _i , and then reduces them in such a way that no index is repeated in the final product, and there are no powers of any x _i or 1−x _i factor; we are thus guaranteed to end up with pseudo-monomials. Moreover, since the products each have at least one factor in each prime ideal of the primary decomposition of $J_{\mathcal{C}}$, the pseudo-monomials are all in $J_{\mathcal{C}}$. Proposition 4.5 states that this set of pseudo-monomials is precisely the canonical form $\operatorname{CF}(J_{\mathcal{C}})$.

To prove Proposition 4.5, we will make use of the following technical lemma. Here,z _i ,y _i ∈{x _i ,1−x _i }, and thus any pseudo-monomial in $\mathbb{F} _{2}[x_{1},\ldots,x_{n}]$ is of the form ∏ _j∈σ z _j for some index set σ⊆[n].

Lemma 6.7

If $y_{i_{1}}\cdots y_{i_{m}}\in\langle z_{j_{1}},\ldots, z_{j_{\ell}}\rangle$ where {i _k } and {j _r } are each distinct sets of indices, then $y_{i_{k}}=z_{j_{r}}$ for some k∈[m] and r∈[ℓ].

Proof

Let $f = y_{i_{1}}\cdots y_{i_{m}}$ and $P = \{ z_{j_{1}},\ldots,z_{j_{\ell}} \}$. Since f∈〈P〉, then 〈P〉=〈P,f〉, and so V(〈P〉)=V(〈P,f〉). We need to show that $y_{i_{k}} = z_{j_{r}}$ for some pair of indices i _k ,j _r . Suppose by way of contradiction that there is no i _k ,j _r such that $y_{i_{k}}=z_{j_{r}}$.

Select a∈{0,1}ⁿ as follows: for each j _r ∈{j ₁,…,j _ℓ }, let $a_{j_{r}} = 0$ if $z_{j_{r}} = x_{j_{r}}$, and let $a_{j_{r}} = 1$ if $z_{j_{r}} = 1-x_{j_{r}}$; when evaluating at a, we thus have $z_{j_{r}}(a) = 0$ for all r∈[ℓ]. Next, for each $i_{k}\in\omega\stackrel{\mathrm{def}}{=}\{i_{1},\ldots ,i_{m}\}\backslash\{ j_{1},\ldots,j_{\ell}\}$, let $a_{i_{k}} = 1$ if $y_{i_{k}} = x_{i_{k}}$, and let $a_{i_{k}} = 0$ if $y_{i_{k}} = 1-x_{i_{k}}$, so that $y_{i_{k}}(a)=1$ for all i _k ∈ω. For any remaining indices t, let a _t =1. Because we have assumed that $y_{i_{k}}\neq z_{j_{r}}$ for any i _k ,j _r pair, we have for any i∈{i ₁,…,i _m }∩{j ₁,…,j _ℓ } that y _i (a)=1−z _i (a)=1. It follows that f(a)=1.

Now, note that a∈V(〈P〉) by construction. We must therefore have a∈V(〈P,f〉), and hence f(a)=0, a contradiction. We conclude that there must be some i _k ,j _r with $y_{i_{k}}=z_{j_{r}}$, as desired. □

We can now prove the proposition.

Proof of Proposition 4.5

It suffices to show that after step 4 of the algorithm, the reduced set $\widetilde{\mathcal{M}}(J_{\mathcal{C}})$ consists entirely of pseudo-monomials of $J_{\mathcal{C}}$, and includes all minimal pseudo-monomials of $J_{\mathcal{C}}$. If this is true, then after removing multiples of lower-degree elements in step 5 we are guaranteed to obtain the set of minimal pseudo-monomials, $\operatorname{CF}(J_{\mathcal{C}})$, since it is precisely the nonminimal pseudo-monomials that will be removed in the final step of the algorithm.

Let $J_{\mathcal{C}}= \bigcap_{i=1}^{s} P_{i}$ be the primary decomposition of $J_{\mathcal{C}}$, with each P _i a prime ideal of the form $P_{i} = \langle z_{j_{1}},\ldots,z_{j_{\ell}}\rangle$. Recall that $\mathcal{M}(J_{\mathcal{C}})$, as defined in step 3 of the algorithm, is precisely the set of all polynomials g that are obtained by choosing one linear factor from the generating set of each P _i :

$$\mathcal{M}(J_\mathcal{C}) = \{g = z_{p_1}\cdots z_{p_s} \mid z_{p_i} \text{ is a linear generator of } P_i \}. $$

Furthermore, recall that $\widetilde{\mathcal{M}}(J_{\mathcal{C}})$ is obtained from $\mathcal{M}(J_{\mathcal{C}})$ by the reductions in step 4 of the algorithm. Clearly, all elements of $\widetilde{\mathcal{M}}(J_{\mathcal{C}})$ are pseudo-monomials that are contained in $J_{\mathcal{C}}$.

To show that $\widetilde{\mathcal{M}}(J_{\mathcal{C}})$ contains all minimal pseudo-monomials of $J_{\mathcal{C}}$, we will show that if $f \in J_{\mathcal{C}}$ is a pseudo-monomial, then there exists another pseudo-monomial $h \in \widetilde {\mathcal{M}}(J_{\mathcal{C}})$ (possibly the same as f) such that h|f. To see this, let $f = y_{i_{1}}\cdots y_{i_{m}}$ be a pseudo-monomial of $J_{\mathcal{C}}$. Then, f∈P _i for each i∈[s]. For a given $P_{i} = \langle z_{j_{1}},\ldots,z_{j_{\ell}}\rangle$, by Lemma 6.7 we have $y_{i_{k}} = z_{j_{r}}$ for some k∈[m] and r∈[ℓ]. In other words, each prime ideal P _i has a generating term, call it $z_{p_{i}}$, that appears as one of the linear factors of f. Setting $g = z_{p_{1}}\cdots z_{p_{s}}$, it is clear that $g \in \mathcal{M}(J_{\mathcal{C}})$ and that either g|f, or $z_{p_{i}} = z_{p_{j}}$ for some distinct pair i,j. By removing repeated factors in g one obtains a pseudo-monomial $h \in\widetilde{\mathcal{M}}(J_{\mathcal{C}})$ such that h|g and h|f. If we take f to be a minimal pseudo-monomial, we find $f = h \in\widetilde{\mathcal{M}}(J_{\mathcal{C}})$. □

1.5 6.5 Proof of Lemmas 5.8 and 5.9

Here, we prove Lemmas 5.8 and 5.9, which underlie the primary decomposition algorithm.

Proof of Lemma 5.8

Assume f∈〈J,z〉 is a pseudo-monomial. Then $f = z_{i_{1}}z_{i_{2}}\cdots z_{i_{r}}$, where z _i ∈{x _i ,1−x _i } for each i, and the i _k are distinct. Suppose f∉〈z〉. This implies $z_{i_{k}} \neq z$ for all factors appearing in f. We will show that either f∈J or (1−z)f∈J.

Since J is a pseudo-monomial ideal, we can write

$$J = \bigl\langle z g_1, \ldots, z g_k, (1-z) f_1, \ldots, (1-z) f_l, h_1, \ldots, h_m \bigr\rangle, $$

where the g _j ,f _j , and h _j are pseudo-monomials that contain no z or 1−z term. This means

$$f = z_{i_1}z_{i_2}\cdots z_{i_r} = z \sum _{j=1}^k u_j g_j + (1-z) \sum_{j=1}^l v_j f_j + \sum_{j=1}^m w_j h_j + y z, $$

for polynomials u _j ,v _j ,w _j , and $y \in\mathbb{F}_{2}[x_{1},\ldots,x_{n}]$. Now consider what happens if we set z=0 in f:

$$f|_{z=0} = z_{i_1}z_{i_2}\cdots z_{i_r}|_{z=0} = \sum_{j=1}^l v_j\bigg|_{z=0} f_j + \sum _{j=1}^m w_j\bigg|_{z=0} h_j. $$

Next, observe that after multiplying the above by (1−z) we obtain an element of J:

$$(1-z) f|_{z=0} = (1-z) \sum_{j=1}^l v_j\bigg|_{z=0} f_j + (1-z) \sum _{j=1}^m w_j\bigg|_{z=0} h_j \in J, $$

since (1−z)f _j ∈J for j=1,…,l and h _j ∈J for j=1,…,m. There are two cases:

Case 1::

If 1−z is a factor of f, say $z_{i_{1}} = 1-z$, then $f|_{z=0} = z_{i_{2}}\cdots z_{i_{r}}$ and thus f=(1−z)f| _z=0∈J.

Case 2::

If 1−z is not a factor of f, then f=f| _z=0. Multiplying by 1−z we obtain (1−z)f∈J.

We thus conclude that f∉〈z〉 implies f∈J or (1−z)f∈J. □

Proof of Lemma 5.9

Clearly, 〈J,z _σ 〉⊆⋂ _i∈σ 〈J,z _i 〉. To see the reverse inclusion, consider f∈⋂ _i∈σ 〈J,z _i 〉. We have three cases.

Case 1::

f∈J. Then f∈〈J,z _σ 〉.

Case 2::

f∉J, but f∈〈z _i 〉 for all i∈σ. Then f∈〈z _σ 〉, and hence f∈〈J,z _σ 〉.

Case 3::

f∉J and f∉〈z _i 〉 for all i∈τ⊂σ, but f∈〈z _j 〉 for all j∈σ∖τ. Without loss of generality, we can rearrange indices so that τ={1,…,m} for m≥1. By Lemma 5.8, we have (1−z _i )f∈J for all i∈τ. We can thus write

$$f = (1-z_1)f + z_1(1-z_2)f + \cdots+ z_1\cdots z_{m-1}(1-z_m) f + z_1 \cdots z_m f. $$

Observe that the first m terms are each in J. On the other hand, f∈〈z _j 〉 for each j∈σ∖τ implies that the last term is in 〈z _τ 〉∩〈z _σ∖τ 〉=〈z _σ 〉. Hence, f∈〈J,z _σ 〉.

We may thus conclude that ⋂ _i∈σ 〈J,z _i 〉⊆〈J,z _σ 〉, as desired. □

1.6 6.6 Proof of Theorem 5.4

Recall that $J_{\mathcal{C}}$ is always a proper pseudo-monomial ideal for any nonempty neural code $\mathcal{C}\subseteq\{0,1\}^{n}$. Theorem 5.4 is thus a direct consequence of the following proposition.

Proposition 6.8

Suppose $J \subset\mathbb{F}_{2}[x_{1},\ldots,x_{n}]$ is a proper pseudo-monomial ideal. Then, J has a unique irredundant primary decomposition of the form $J = \bigcap_{a \in\mathcal{A}} {\bf p}_{a}$, where $\{{\bf p}_{a}\}_{a \in\mathcal{A}}$ are the minimal primes over J.

Proof

By Proposition 5.11, we can always (algorithmically) obtain an irredundant set $\mathcal{P}$ of prime ideals such that $J = \bigcap_{I \in\mathcal{P}} I$. Furthermore, each $I \in\mathcal{P}$ has the form $I = \langle z_{i_{1}},\ldots ,z_{i_{k}}\rangle$, where z _i ∈{x _i ,1−x _i } for each i. Clearly, these ideals are all prime ideals of the form ${\bf p}_{a}$ for a∈{0,1,∗}. It remains only to show that this primary decomposition is unique, and that the ideals $\{{\bf p}_{a}\}_{a \in\mathcal{A}}$ are the minimal primes over J. This is a consequence of some well-known facts summarized in Lemmas 6.9 and 6.10, below. First, observe by Lemma 6.9 that J is a radical ideal. Lemma 6.10 then tells us that the decomposition in terms of minimal primes is the unique irredundant primary decomposition for J. □

Lemma 6.9

If J is the intersection of prime ideals, $J=\bigcap_{i=1}^{\ell}\mathbf{p}_{i}$, then J is a radical ideal.

Proof

Suppose p ⁿ∈J. Then p ⁿ∈p _i for all i∈[ℓ], and hence p∈p _i for all i∈[ℓ]. Therefore, p∈J. □

The following fact about the primary decomposition of radical ideals is true over any field, as a consequence of the Lasker–Noether theorems (Cox et al. 1997, pp. 204–209).

Lemma 6.10

If J is a proper radical ideal, then it has a unique irredundant primary decomposition consisting of the minimal prime ideals over J.

Appendix 2: Neural Codes on Three Neurons

See Table 1 and Figs. 6 and 7.

Fig. 6

Receptive field diagrams for the 27 non-∗ codes on three neurons listed in Table 1. Codes that admit no realization as a convex RF code are labeled “non-convex.” The code E2 is the one from Lemma 2.2, while A1 and A12 are permutation-equivalent to the codes in Fig. 3A and C, respectively. Deleting the all-zeros codeword from A6 and A4 yields codes permutation-equivalent to those in Fig. 3B and D, respectively

Fig. 7

Boolean lattice diagrams for the 27 non-∗ codes on three neurons listed in Table 1. Interval decompositions (see Sect. 5.2) for each code are depicted in black, while decompositions of code complements, arising from $\operatorname{CF}(J_{\mathcal{C}})$, are shown in gray. Thin black lines connect elements of the Boolean lattice that are Hamming distance 1 apart. Note that the lattice in A12 is permutation-equivalent to the one depicted in Fig. 5

Table 1 Forty permutation-inequivalent codes, each containing 000, on three neurons