have the smallest possible numbers of neurons.

Let $\mathcal{U} = \{U_i\}_{i=1}^n$ be a collection of sets in a stimulus space $X$. Let $C = C(\mathcal{U})$ denote the corresponding receptive field code, and let $J_C$ denote the neural ideal. Then for any subsets $\sigma_1$, $\sigma_2$, $\tau_1$, $\tau_2 \subseteq [n]$, and $m$ indices $1 \leq i_1 < i_2 <...< i_m \leq n$, with $m \geq 2$, the following hold:

-   When $n=1$, $U_{\sigma_1} \cap (\cap_{i \in \tau_1} U^c_i) = U_{\sigma_2} \cap (\cap_{j \in \tau_2} U^c_j) \Rightarrow x_{\sigma_1} \prod_{i \in \tau_1} (1+x_i) + x_{\sigma_2} \prod_{j \in \tau_2} (1+x_j) \in J_C$.

-   When $n\leq 3$,

    1.  $m$ is even and $U_{i_k} \subseteq \cup_{j \in [m] \setminus  \{k\}} U_{i_j}$ for all $k = 1,...,m \Rightarrow x_{i_1} + ... + x_{i_m} \in J_C$, and

    2.  $m$ is odd, $U_{i_k} \subseteq \cup_{j \in [m] \setminus  \{k\}} U_{i_j}$ for all $k = 1,...,m$, and $\cap_{k=1}^m U_{i_k} = \emptyset \Rightarrow x_{i_1} + ... + x_{i_m} \in J_C$.

-   When $n=1$, $\cup^m_{k=1} U_{i_k} = X \Rightarrow x_{i_1} + ... + x_{i_m} + 1 \in J_C$.

[*Converse of Type 4:*]{} For $n=1$, the possible neural codes are $C_1 = \{\emptyset, 1\}$ and $C_2 = \{1\}$.

In the case of $C_1$, $J_C 