we cannot use the Type 4-6 relations in their current form to conclude anything about the associated neural ideal. It is therefore natural to wonder whether each of these statements can be reversed, and in Section 3 we show that in fact none of them can. However, in Section 4 we will present modified versions of the Type 4-6 relations that are if-and-only-if statements.

Disproving the Converses of the Type 4-6 Relations
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In this section, we show by counterexample that none of the converses of the Type 4-6 relations hold (Theorem 3.3). The converses of the relations are stated below:

Again, each of the converses is false. The receptive fields for all of the counterexample codes are shown in Figure \[fig:Counters\]. In the case of the Type 5 and 6 relations, we will first use the counterexample codes to prove that the converses of these statements do not hold [*even when $J_C$ is replaced by the larger ideal $I(C)$*]{}.

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node\[left\] [$U_{1}$]{}; node\[right\] [$U_{2}$]{}; (-2,-2) rectangle (4,2); (X) at (3.6,1.6) [$X$]{};

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; ; ; ;

node\[left\] [$U_{2}$]{}; node\[right\] [$U_{3}$]{}; node [$U_4$]{}; (-2,-2) rectangle (4,2); ; (X) at (3.6,1.6) [$X$]{}; (Y) at (-.4,-2.5) [$U_1$]{};

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(0,-2) rectangle (2,2); (-2,-2) 