SlowGuess/NeurIPS_2023_OCR huggingface dataset:
impose two linear constraints between the 4 atoms of information having  $S$  as target and the 4 atoms of information having  $Y$  as target (all atoms are:  $I_{\partial}(S; \{X\} \{Y\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{XY\})$ ,  $I_{\partial}(Y; \{X\} \{S\})$ ,  $I_{\partial}(Y; \{S\})$ ,  $I_{\partial}(Y; \{X\})$ ,  $I_{\partial}(Y; \{XS\}))$ ):
$$
I (S; Y) = I _ {\partial} (S; \{X \} \{Y \}) + I _ {\partial} (S; \{Y \}) = I _ {\partial} (Y; \{X \} \{S \}) + I _ {\partial} (Y; \{S \})
$$
$$
I (S; Y | X) = I _ {\partial} (S; \{X Y \}) + I _ {\partial} (S; \{Y \}) = I _ {\partial} (Y; \{X S \}) + I _ {\partial} (Y; \{S \}) \tag {S17}
$$
Combining the two equations in the system of eqs. S17 reveals an equality among the differences in the amount of information carried by pairs of similar atoms (the two redundancies  $I_{\partial}(S; \{X\} \{Y\})$  and  $I_{\partial}(Y; \{X\} \{S\})$ , the two synergies  $I_{\partial}(S; \{XY\})$  and  $I_{\partial}(Y; \{XS\})$ , and the two unique information  $I_{\partial}(Y; \{S\})$  and  $I_{\partial}(S; \{Y\})$ ):
$$
I _ {\partial} (S; \{X \} \{Y \}) - I _ {\partial} (Y; \{X \} \{S \})
$$
$$
= I _ {\partial} (Y; \{S \}) - I _ {\partial} (S; \{Y \}) = I _ {\partial} (S; \{X Y \}) - I _ {\partial} (Y; \{X S \}) \tag {S18}
$$
Therefore, 6 atoms of the 8 atoms belonging to the two decompositions (those appearing in eq S17) are not independent, while  $I_{\partial}(S; \{X\})$  and  $I_{\partial}(Y; \{X\})$  are independent from all other atoms. In Ref. [27] the authors showed that, due to the two constraints of eq. S17, the 8 atoms can be described by 6 finer independent information components (that they called information subatoms). In Ref. [27] they quantify these 6 subatoms as follows: three subatoms are the minimum between pairs of similar atoms belonging to the two decomposition (i.e., the two redundancies  $I_{\partial}(S; \{X\} \{Y\})$  and  $I_{\partial}(Y; \{X\} \{S\})$ , the two synergies  $I_{\partial}(S; \{XY\})$  and  $I_{\partial}(Y; \{XS\})$ ) and the two unique information  $I_{\partial}(S; \{Y\})$  and  $I_{\partial}(Y; \{S\})$ ); one subatom is equal to the difference between the maximum and the minimum in each of the above pairs (which is equal for the three pairs, see eq. S18); two subatoms are equal to the unconstrained atoms not appearing in eq. S17 ( $I_{\partial}(S; \{X\})$ ) and  $I_{\partial}(Y; \{X\})$ ).
A novel perspective on the relationships between the amounts of information carried by specific sets of atoms from different decompositions is to conceptualize the eight atoms as forming an eight-dimensional vector space,  $\mathcal{V}$ . We can represent a generic column vector in  $\mathcal{V}$  as  $\underline{\upsilon}$  and the  $2\times 8$  matrix of constraints imposed by Shannon information quantities relating the atoms of the two decompositions as  $\underline{B}$ . With these definitions, we can express the system of eqs. S17 as a homogeneous linear system:
$$
\underline {{B v}} = \underline {{0}} \tag {S19}
$$
Specifically, the coefficients of  $\underline{B}$  are obtained by taking the difference between the middle- and the right-term in the two eqs. S17. Ordering the dimensions of  $\mathcal{V}$  as  $(I_{\partial}(S; \{X\} \{Y\}), I_{\partial}(S; \{Y\}), I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(S; \{Y\})$ ,  $I_{\partial}(\{Y; \{X\}\})$ ,  $I_{\partial}(\{Y; \{X\}\})$ ,  $I_{\partial}(\{Y; \{X\}\})$  has the following form:
$$
\underline {{\underline {{B}}}} = \left[ \begin{array}{l l l l l l l l} 1 & 1 & 0 & 0 & - 1 & - 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & - 1 & 0 & - 1 \end{array} \right] \tag {S20}
$$
It can easily be verified that, for instance,  $I_{\partial}(S; \{X\} \{Y\}) = I_{\partial}(Y; \{X\} \{S\})$ , is a solution of the homogeneous system in eq. S19 for the matrix  $\underline{\underline{B}}$  defined as in eq. S20. Consider a matrix multiplication between  $\underline{\underline{B}}$  and the vector  $\underline{v}_{SI}$ , whose only non-zero components are  $I_{\partial}(S; \{X\} \{Y\})$  and  $I_{\partial}(Y; \{X\} \{S\})$  (i.e.,  $\underline{v}_{SI} = (I_{\partial}(S; \{X\} \{Y\}), 0, 0, 0, I_{\partial}(Y; \{X\} \{S\}), 0, 0, 0)$ ). This matrix multiplication is equivalent to multiplying the coefficients in columns 1 and 5 of  $\underline{\underline{B}}$  by the two atoms, respectively, and doing the element-wise sum of the two resulting vectors. This sum is zero if and only if  $I_{\partial}(S; \{X\} \{Y\}) = I_{\partial}(Y; \{X\} \{S\})$ . Put simply, columns of  $\underline{\underline{B}}$  with element-wise opposite coefficients correspond to pairs (or triplets) of atoms that form a solution of eq. S19 when they have equal value. As a result, the following are all nontrivial solutions of the homogeneous system in eq. S19, or equivalently, they belong to the null space of  $\underline{\underline{B}}$ :  $I_{\partial}(S; \{X\} \{Y\}) = I_{\partial}(Y; \{X\} \{S\})$ ,  $I_{\partial}(S; \{Y\}) = I_{\partial}(Y; \{S\})$ ,  $I_{\partial}(S; \{XY\}) = I_{\partial}(Y; \{XS\})$ ,  $I_{\partial}(S; \{X\})$ ,  $I_{\partial}(Y; \{X\})$ , and  $I_{\partial}(S; \{X\} \{Y\}) = I_{\partial}(S; \{XY\}) = I_{\partial}(Y; \{S\})$ . These solutions uncover specific relationships between pairs or triplets of atoms across different decompositions. Importantly, considering that the eight atoms are not independent and can be represented by six finer, independent quantities, these solutions lend support to the notion that these finer components of information are shared among the atoms linked by a single solution. Remarkably, the atoms identified as related by a solution precisely correspond to the six subatoms previously defined in Ref [27].
Similar to the case of  $N = 2$  source variables, for  $N = 3$  source variables, there are 36 information atoms (18 per lattice) that belong to two decompositions with different targets. Shannon information
quantities impose constraints relating these 36 atoms, implying the existence of finer information components (or subatoms) that can describe the two decompositions even when there are  $N = 3$  source variables. Our goal here is not to uncover the complete set of components that describe all atoms belonging to the two decompositions. Rather, we aim to demonstrate that a specific algebraic relationship, similar to the ones discussed above, exists between the two atoms present in the FIT definition. To do this, we generalize the homogeneous linear system in eq. S19 to the four-variable case ( $S, X_{past}, Y_{past}, Y_{pres}$ ). In this scenario, the two decompositions that have  $S$  and  $Y_{pres}$  as their respective targets (represented by the two lattices in Fig. S1B) are constrained by the following four Shannon information quantities:
$$
I (S; Y _ {p r e s})
$$
$$
I (S; Y _ {p r e s} | X _ {p a s t})
$$
$$
I (S; Y _ {p r e s} | Y _ {p a s t}) \tag {S21}
$$
$$
I (S; Y _ {p r e s} | Y _ {p a s t}, X _ {p a s t})
$$
Similarly to eq. S17, we can use eq. S7 to express the 4 quantities in eq. S21 as sums of atoms either belonging to the decomposition with  $S$  as target, or the one with  $Y_{pres}$  as target. As an example, in Fig. S1B, we demonstrate that  $I(S;Y_{pres})$  is the sum of atoms belonging to both decompositions, which together consist of 36 atoms (18 per decomposition). In general, each quantity in eq. S21 imposes constraints between two sets of many atoms from the two decompositions.
To numerically study the solutions of S19 for these 36 atoms, we wrote a MATLAB script named FIT_nullB.m. This script computes the four quantities in eq. S21 as the sum of atoms from either the decomposition with  $S$  or  $Y_{pres}$  as target. It then constructs the  $4 \times 36$  matrix  $\underline{B}$  (Fig. S2) in a similar way to how we derived the  $2 \times 8$  matrix in eq. S20 from the eqs. in S17. From Fig. S2, it is clear that in this four-variable case, some atoms, such as  $I_{\partial}(Y_{pres};\{X_{past}\} \{Y_{past}\})$ , are not constrained by  $\underline{B}$  and can vary independently (analogously to how for  $N = 2$  the two terms  $I_{\partial}(S;\{X\})$  and  $I_{\partial}(Y;\{\overline{X}\}))$  were unconstrained). However, there are also pairs of atoms that are not independent when considered individually, but that are specifically related by eq. S19 (i.e. the equality between the two atoms in the pair is a solution of eq. S19). A notable example of these pairwise solutions is made by the pair of atoms appearing in the FIT definition:  $I_{\partial}(S;\{X_{past}\} \{Y_{pres}\}) = I_{\partial}(Y_{pres};\{X_{past}\} \{S\})$ . This relationship can be easily verified from Fig. S2, where the first and the second atom are highlighted in red and white, respectively. Indeed, drawing from the intuition developed in the  $N = 2$  source variables case, these two atoms belong to columns of  $\underline{B}$  with element-wise opposite coefficients. This solution (Fig. S2) reveals a specific pairwise relationship between the two atoms appearing in the FIT definition and supports the existence of a finer component of information shared by these two atoms. It is actually apparent from the plot in Fig. S2 that this is the only pairwise relationship involving any of the two atoms. We quantify this finer component of information by taking the minimum between the two related atoms.
SM1.3.4 Proofs and summary of the main mathematical properties of FIT
Here we prove that FIT defined as in eq. S11 satisfies the two following properties:
1. FIT is simultaneously upper bounded by  $I(S; X_{\text{past}})$ ,  $I(S; Y_{\text{pres}})$ , and  $TE(X \to Y)$ .
2. FIT depends on  $P(S, X_{past}, Y_{pres})$  through all the pairwise marginal distributions  $P(S, X_{past}), P(S, Y_{pres})$ , and  $P(X_{past}, Y_{pres})$ . Thus, FIT can rule out confounding scenarios where  $X$  and  $Y$  independently encode  $S$  with a temporal lag in absence of within-trial correlations between  $X$  and  $Y$  at fixed stimulus.
To prove that FIT is simultaneously upper bounded by  $I(S; X_{past})$ ,  $I(S; Y_{pres})$ , and  $TE(X \to Y)$ , it is sufficient to note that FIT is simultaneously upper bounded by all quantities that set an upper bound to the two atoms appearing in its definition. This can be seen from eqs. S12, S13, and S15, which show:
$$
F I T \leq I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \}) \leq I (S; X _ {p a s t})
$$
$$
F I T \leq I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \}) \leq I (S; Y _ {p r e s}) \tag {S22}
$$
$$
F I T \leq I _ {\partial} (Y _ {p r e s}; \{X _ {p a s t} \} \{S \}) \leq T E (X \rightarrow Y)
$$
A particularly important consequence of the upper bound set by  $TE(X \to Y)$  is that if  $X$  and  $Y$  are independent, then  $FIT = 0$ . Indeed, if  $X$  is independent of  $Y$ , then  $X_{past}$  is independent of
Shannon information constraints
Figure S2: Matrix of constraints imposed by Shannon information-theoretic quantities relating the PID having  $(Y_{pres},X_{past},Y_{past})$  as source variables and  $S$  as target variable to the PID having  $(S,X_{past},Y_{past})$  as source variables and  $Y_{pres}$  as target variable. For brevity, we used the notation  $Y_{t} = Y_{pres},X_{p} = X_{past},Y_{p} = Y_{past}$  and denoted each atom (y axis) directly with the collection it is computed on, with a subscript indicating the target variable of the decomposition (e.g.  $\{X_p\} \{Y_t\} _S = I_\partial (S;\{X_{past}\} \{Y_{pres}\}))$ . For better visibility, we plotted the transpose of the  $4\times 36$  matrix appearing in eq. S19. The red dashed line highlights the first atom appearing in FIT definition  $I_{\partial}(S;\{X_{past}\} \{Y_{pres}\})$ , the white line highlights the second atom in FIT definition  $I_{\partial}(Y_{pres};\{X_{past}\} \{S\})$ . Importantly, only atom highlighted in red has coefficients that are opposite to the ones of the atom highlighted in white.
$(Y_{past}, Y_{pres})$ , and therefore  $I(X_{past}; Y_{past}, Y_{pres}) = 0$ . By applying the information theoretic chain rule [9] to  $I(X_{past}; Y_{past}, Y_{pres})$ , we obtain:
$$
\begin{array}{l} I \left(X _ {p a s t}; Y _ {p a s t}, Y _ {p r e s}\right) = I \left(X _ {p a s t}; Y _ {p r e s} \mid Y _ {p a s t}\right) + I \left(X _ {p a s t}; Y _ {p a s t}\right) \tag {S23} \\ \geq I \left(X _ {p a s t}; Y _ {p r e s} \mid Y _ {p a s t}\right) \geq F I T \\ \end{array}
$$
Proving that, if  $X$  and  $Y$  are independent, then  $FIT = 0$ . Another important point is that none of the 36 atoms belonging to either decomposition with  $S$  or decomposition with  $Y_{pres}$  as target satisfies the first property. Indeed eq. S7 does not establish any relationship between atoms of the decomposition with  $S$  as target and Shannon information between the sources of the decomposition, including  $TE(X \rightarrow Y)$  (see Fig. S1B, left), nor between atoms of the decomposition with  $Y_{pres}$  as
Matrix of constraints between the two decompositions having S and Yt as targets
target and Shannon information between the sources of the decomposition, including  $I(S; X_{past})$  (see Fig. S1B, right). Therefore it is necessary to simultaneously consider atoms belonging to different decompositions to obtain a quantity that satisfies the first property.
To prove that FIT depends on  $P(S, X_{past}, Y_{pres})$  through all the pairwise marginal distributions  $P(S, X_{past}), P(S, Y_{pres})$ , and  $P(X_{past}, Y_{pres})$  we also leveraged on the simultaneous dependence of FIT on  $I_{\partial}(S; \{X_{past}\} \{Y_{pres}\})$  and on  $I_{\partial}(Y_{pres}; \{X_{past}\} \{S\})$ .  $I_{\partial}(S; \{X_{past}\} \{Y_{pres}\})$  and  $I_{\partial}(Y_{pres}; \{X_{past}\} \{S\})$  depend of  $P(S, X_{pres}, Y_{past})$  through the marginals  $P(S, X_{pres})$  and  $P(S, Y_{past})$ , and  $P(Y_{pres}, X_{pres})$  and  $P(S, Y_{past})$ , respectively (see SM1.3.1 and SM1.3.2). Therefore:
$$
F I T = f \left(P \left(S, X _ {p r e s}, Y _ {p a s t}\right)\right) = f \left(P \left(S, X _ {p a s t}\right), P \left(S, Y _ {p r e s}\right), P \left(X _ {p a s t}, Y _ {p r e s}\right)\right) \tag {S24}
$$
This implies that if we partially disrupt the dependency structure of our data and create surrogate data where the individual dependencies of  $X$  and of  $Y$  on  $S$  are preserved (i.e., the pairwise marginals  $P(S, X_{past})$  and  $P(S, Y_{pres})$  do not change) and the within-trial correlations at a fixed stimulus are disrupted (i.e., the conditional distribution  $P(X_{pres}, Y_{past} | S)$  changes), the value of the FIT can differ from its original value. This change occurs because this operation generally disrupts  $P(X_{pres}, Y_{past})$ . Therefore, FIT can rule out confounding scenarios where  $X$  and  $Y$  encode  $S$  independently with a temporal lag, with no information transfer at fixed stimulus value.
SM1.4 The conditional feature specific information cFIT
Here we discuss the definition and the properties of the conditioned version of FIT, termed cFIT.
SM1.4.1 Definition and derivation of cFIT
We defined a conditioned version of FIT, to remove from the feature information transmitted from  $X$  to  $Y$  (that in this Section we term  $FIT_{X}$ ) the information potentially routed through the past activity of a third region  $Z$  ( $Z$  can, in principle, also be the multivariate activity of a set of regions). To do so, we identified subcomponents of the two atoms in the FIT definition that quantified pieces of information that were also shared with the past of  $Z$ , and removed them from FIT.
In this subsection, we will be working with atoms computed on collections belonging to decompositions with  $N = 3$  and  $N = 4$  source variables. To avoid any confusion, we will explicitly denote the number of source variables of each collection in the following discussion. For example, we will use the notation  $\{X_{past}\} \{Y_{pres}\}_{(3)}$  to indicate the collection on which the first atom in the FIT definition is computed. This collection refers to the PID with  $N = 3$  source variables  $(X_{past}, Y_{pres}, Y_{past})$  and target variable  $S$ .
Previous studies showed that, using eq. S7, atoms on the PID with target  $T$  and  $N$  source variables  $\underline{X}_N = (X_1,\dots ,X_N)$  can be written as the sum of finer atoms belonging to the PID with same target and an additional source variable  $\underline{X}_{N + 1} = (X_{1},\ldots ,X_{N},X_{N + 1})$  [7]. Importantly, collections  $\alpha$  present in the PID with  $N$  source variables, denoted as  $\alpha (N)$ , also exist in the PID with  $N + 1$  source variables, denoted as  $\alpha_{(N + 1)}$ , since  $\underline{X}_N\subset \underline{X}_{N + 1}$ . However, the opposite does not necessarily hold.
The atom  $I_{\partial}(T;\alpha_{(N)})$  is the sum of atoms  $I_{\partial}(T;\beta_{(N + 1)})$ , where  $\beta_{(N + 1)}$  simultaneously precedes  $\alpha_{(N + 1)}$  in the PID with  $N + 1$  source variables (as per the ordering relationship of eq. S4, in which precedence includes equality), but does not precede any collections  $\gamma_{(N + 1)}$  such that  $\gamma_{(N)}$  precedes  $\alpha_{(N)}$  [7]. For example, the collection  $\{X_{past}\} \{Y_{pres}\}_{(3)}$ , present in the PID with  $N = 3$  source variables  $(X_{past},Y_{past},Y_{pres})$  and target  $S$ , is preceded only by the collection  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\}_{(3)}$  (i.e., the information that all source variables  $(X_{past},Y_{past},Y_{pres})$  share about  $S$ ) and by itself (see Fig.S1B, left). When adding  $Z_{past}$  to the set of source variables, the collection  $\{X_{past}\} \{Y_{pres}\}_{(4)}$  is preceded by four collections (additionally to itself):  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\}_{(4)}$ ,  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\}_{(4)}$ ,  $\{X_{past}\} \{Y_{pres}\} \{Z_{past}\}_{(4)}$ ,  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\}_{(4)}$ , and  $\{X_{past}\} \{Y_{pres}\} \{Z_{past}Y_{past}\}_{(4)}$  (see Fig.S7A, right). Collection  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\}_{(4)}$ , which was not preceded by any collection apart from itself in the PID with  $N = 3$  variables, is preceded also by  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\} \{Z_{past}\}_{(4)}$ . Therefore:
$$
\begin{array}{l} I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} \{Y _ {p a s t} \} _ {(3)}) \\ = I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} \{Y _ {p a s t} \} _ {(4)}) + I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} \{Y _ {p a s t} \} \{Z _ {p a s t} \} _ {(4)}) \\ \end{array}
$$
which intuitively means that, when considering also the past of a third region  $Z$ , the information that  $(X_{past}, Y_{past}, Y_{pres})$  share about  $S$  breaks down into a component that is also shared with  $Z_{past}$  and a component that is unique with respect to  $Z_{past}$ .
The other two collections  $\{X_{past}\} \{Y_{pres}\} \{Z_{past}\}_{(4)}$  and  $\{X_{past}\} \{Y_{pres}\} \{Z_{past}Y_{past}\}_{(4)}$  precede  $\{X_{past}\} \{Y_{pres}\}_{(4)}$  but do not precede  $\{X_{past}\} \{Y_{pres}\} \{Y_{past}\}_{(4)}$  (Fig.S7A). Therefore:
$$
\begin{array}{l} I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}) = I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} _ {(4)}) \\ + I _ {\partial} \left(S; \{X _ {p a s t} \} \{Y _ {p r e s} \} \{Y _ {p a s t} Z _ {p a s t} \} _ {(4)}\right) + I _ {\partial} \left(S; \{X _ {p a s t} \} \{Y _ {p r e s} \} \{Z _ {p a s t} \}\right) \tag {S26} \\ \end{array}
$$
which shows how the first atom in FIT definition (eq. S11) breaks down into three components in the PID with  $(X_{past},Y_{pres},Y_{past},Z_{past})$  as source variables and  $S$  as target variable (one component that is unique with respect to  $Y_{past}$  but shared with  $Z_{past}$ , one that is unique with respect to both  $Y_{past}$  and  $Z_{past}$  but shared with  $\{Y_{past}Z_{past}\}$ , and one that is also unique with respect to  $\{Y_{past}Z_{past}\}$ ). One of these atoms is the information that  $X_{past},Y_{pres},$  and  $Z_{past}$  share about  $S$ , i.e. the component of the first FIT atom that is also shared with  $Z_{past}$ :  $I_{\partial}(S;\{X_{past}\}\{Y_{pres}\}\{Z_{past}\}_{(4)})$ .
Similarly, the second atom appearing in the FIT definition, is the sum of finer atoms belonging to the PID with  $(X_{past}, S, Y_{past}, Z_{past})$  as source variables and  $Y_{pres}$  as target variable:
$$
\begin{array}{l} I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} _ {(3)}\right) = I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} _ {(4)}\right) + \\ + I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} \{Y _ {p a s t} Z _ {p a s t} \} _ {(4)}\right) + I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} \{Z _ {p a s t} \} _ {(4)}\right) \tag {S27} \\ \end{array}
$$
One of these atoms is the information that  $X_{past}, S$ , and  $Z_{past}$  share about  $Y_{pres}$ , i.e. the component of the second FIT atom that is also shared with  $Z_{past}$ :  $I_{\partial}(Y_{pres}; \{X_{past}\} \{S\} \{Z_{past}\}_{(4)})$ .
To remove from FIT the information that is also shared with  $Z_{past}$  we defined the conditional FIT (cFIT) from  $X$  to  $Y$  conditioned to  $Z$  as:
$$
\begin{array}{l} c F I T _ {X \mid Z} = \min  \left[ I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}), I _ {\partial} (Y _ {p r e s}; \{X _ {p a s t} \} \{S \} _ {(3)}) \right] + \\ \begin{array}{l} \text {S 2 8)} \\ - \min  \left[ I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} \{Z _ {p a s t} \} _ {(4)}), I _ {\partial} (Y _ {p r e s}; \{X _ {p a s t} \} \{S \} \{Z _ {p a s t} \} _ {(4)}) \right] \end{array} \\ \end{array}
$$
Therefore  $cFIT_{X|Z}$  is equal to FIT from  $X$  to  $Y$  (cf. eq. S11) minus a term that is the minimum between two similar information atoms (both quantifying intuitively the feature information about  $S$  that both the past of  $X$  and the past of  $Z$  share with the present of  $Y$ , but is unique with respect to the past of  $Y$ ) on the two PID having  $S$  and having  $Y_{pres}$  as target variables, respectively.
SM1.4.2 Properties of cFIT
In this Section we prove two properties of cFIT, under the assumption that we compute PID atoms using a redundancy measure (such as  $I_{\min}$ ) that is non-negative for each atom. The first property we prove (i) is that  $cFIT_{X|Z}$  is upper bounded by  $FIT_X$  and is lower bounded by the maximum between 0 and  $FIT_X - FIT_Z$  (where we denote as  $FIT_X$  the feature information transmitted from  $X$  to  $Y$  and  $FIT_Z$  the one transmitted from  $Z$  to  $Y$ ). The second property that we prove (ii) is that if  $S \to Z_{past} \to Y_{pres}$  is a Markov chain (i.e.  $P(S;Y_{pres}|Z_{past}) = P(S|Z_{past})P(Y_{pres}|Z_{past})$ ) then  $cFIT_{X|Z} = 0$ . This second property is important because it means that if the present of  $Y$  received all its feature information from the past of a recorded region  $Z$ , then there is no residual FIT through  $X$  once any contribution from  $Z$  is eliminated.
We start by proving property (i). From eq. S28, since we subtract from FIT the minimum between two non-negative quantities, it immediately follows that  $cFIT_{X|Z} \leq FIT_X$ . This proves that  $cFIT_{X|Z}$  is upper bounded by  $FIT_X$ . Then, since from eqs. S26 and S27 we have that
$$
\begin{array}{l} I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}) \geq I _ {\partial} (S; \{X _ {p a s t} \} \{Z _ {p a s t} \} \{S \} _ {(4)}) \\ I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}\right) \geq I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{Z _ {p a s t} \} \{S \} _ {(4)}\right) \tag {S29} \\ \end{array}
$$
from which it follows that: