SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \min  \left[ I _ {\partial} (S; \{X _ {p a s t} \} \{Z _ {p a s t} \} \{Y _ {p r e s} \} _ {(4)}), I _ {\partial} (Y _ {p r e s}; \{X _ {p a s t} \} \{Z _ {p a s t} \} \{S \} _ {(4)}) \right] \\ \leq \min  \left[ I _ {\partial} (S; \{X _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}), I _ {\partial} (Y _ {p r e s}; \{X _ {p a s t} \} \{S \} _ {(3)}) \right] \tag {S30} \\ \end{array}
$$
Eq. S30 shows that the term that we subtract from the right-hand side in eq. S28 is lower or equal to the first one, proving that  $cFIT_{X|Z} \geq 0$ .
Finally, we prove that  $cFIT_{X|Z} \geq FIT_X - FIT_Z$ . We do so by proving that the term we subtract from  $FIT_X$  in the definition of  $cFIT_{X|Z}$  (eq. S28) is smaller than  $FIT_Z$ .  $FIT_Z$  is defined on the two decompositions having  $(Y_{pres}, Y_{past}, Z_{past})$  as sources and  $S$  as target variable, and the one having  $(S, Y_{past}, Z_{past})$  as sources and  $Y_{pres}$  as target variable:
$$
F I T _ {Z} = \min  \left[ I _ {\partial} (S; \{Z _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}), I _ {\partial} \left(Y _ {p r e s}; \{Z _ {p a s t} \} \{S \} _ {(3)}\right) \right] \tag {S31}
$$
Similarly to eqs. S26 and S27, the two atoms in S31 break down into the sum of finer information atoms - when adding variable  $X_{past}$  to the respective sets of source variables (in Fig.S7A we show a graphical depiction of the decomposition of the first atom in  $FIT_Z$  definition, depicted in light blue):
$$
I _ {\partial} (S; \{Z _ {p a s t} \} \{Y _ {p r e s} \} _ {(3)}) = I _ {\partial} (S; \{Z _ {p a s t} \} \{Y _ {p r e s} \} _ {(4)})
$$
$$
+ I _ {\partial} \left(S; \{Z _ {p a s t} \} \{Y _ {p r e s} \} \{Y _ {p a s t} X _ {p a s t} \} _ {(4)}\right) + I _ {\partial} \left(S; \{Z _ {p a s t} \} \{Y _ {p r e s} \} \{X _ {p a s t} \} _ {(4)}\right) \tag {S32}
$$
$$
\begin{array}{l} I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} _ {(3)}\right) = I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} _ {(4)}\right) \\ + I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} \{Y _ {p a s t} Z _ {p a s t} \} _ {(4)}\right) + I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} \{Z _ {p a s t} \} _ {(4)}\right) \tag {S33} \\ \end{array}
$$
From eqs. S32 and S33 it follows that  $I_{\partial}(S; \{Z_{past}\} \{Y_{pres}\}_{(3)}) \geq I_{\partial}(S; \{Z_{past}\} \{X_{past}\} \{Y_{pres}\}_{(4)})$  (i.e. the information the past of  $Z$  and the present of  $Y$  share about  $S$  is larger than the information that they both share also with the past of  $X$  about  $S$ ) and  $I_{\partial}(Y_{pres}; \{Z_{past}\} \{S\} \}_{(3)}) \geq I_{\partial}(Y_{pres}; \{Z_{past}\} \{X_{past}\} \{S\} \}_{(4)})$ . Thus:
$$
F I T _ {Z} \geq \min  \left[ I _ {\partial} (S; \{Z _ {p a s t} \} \{Y _ {p r e s} \} \{X _ {p a s t} \} _ {(4)}), I _ {\partial} \left(Y _ {p r e s}; \{X _ {p a s t} \} \{S \} \{Z _ {p a s t} \} _ {(4)}\right) \right] \tag {S34}
$$
The above proves that  $cFIT_{X|Z} \geq FIT_X - FIT_Z$ . This is important because it assures that the component that we subtract from  $FIT_X$  when removing from it any contribution potentially due to  $Z_{past}$  cannot exceed the feature information transmitted from  $Z$  to  $Y$  (if we only remove the  $FIT_Z$  that is shared with  $FIT_X$ ). To summarize, we proved that  $cFIT_{X|Z} \leq FIT_X$ , that  $cFIT_{X|Z} \geq 0$  and that  $cFIT_{X|Z} \geq FIT_X - FIT_Z$ , meaning that  $cFIT_{X|Z}$  is upper bounded by  $FIT_X$  and is lower bounded by  $max[0, FIT_X - FIT_Z]$ .
We now prove property (ii): if  $S \to Z_{past} \to Y_{pres}$  is a Markov chain (i.e.  $I(S;Y_{pres}|Z_{past}) = 0$ ) then  $cFIT_{X|Z} = 0$ ). If  $S \to Z_{past} \to Y_{pres}$  is a Markov chain, that is  $P(S;Y_{pres}|Z_{past}) = P(S|Z_{past})P(Y_{pres}|Z_{past})$ , then  $I(S;Y_{pres}|Z_{past}) = 0$  [9]. Using the information-theoretic chain rule [9] we can write:
$$
\begin{array}{l} I (S; Y _ {p r e s} | Z _ {p a s t}) = I (S; Y _ {p r e s}, Z _ {p a s t}) - I (S; Z _ {p a s t}) = \\ = I \left(Y _ {p r e s}; S, Z _ {p a s t}\right) - I \left(Y _ {p r e s}; Z _ {p a s t}\right) \tag {S35} \\ \end{array}
$$
Therefore  $I(S;Y_{pres}|Z_{past}) = 0$  implies  $I(S;Y_{pres},Z_{past}) = I(S;Z_{past})$  (meaning that all PID atoms that are a subpart of  $I(S;Y_{pres},Z_{past})$ , but not of  $I(S;Z_{past})$ , are zero) and also  $I(Y_{pres};S,Z_{past}) = I(Y_{pres};Z_{past})$  (meaning that all PID atoms that are a subpart of  $I(Y_{pres};S,Z_{past})$ , but not of  $I(Y_{pres};Z_{past})$ , are zero). In particular, in eqs. S26 all atoms are computed on collections preceding (according to eq. S4) collection  $\{Y_{pres}Z_{past}\}$ , meaning that, due to eq. S7, they are all a subcomponent of  $I(S;Y_{pres},Z_{past})$ . However, among these atoms, only the collection in  $I_{\partial}(S;\{X_{past}\} \{Y_{pres}\} \{Z_{past}\}_{(4)})$  precedes collection  $\{Z_{past}\}$  on this decomposition and, therefore, is a subcomponent of  $I(S;Z_{past})$ . Since in our case  $I(S;Y_{pres},Z_{past}) = I(S;Z_{past})$ , the other two atoms on the right-hand side of S26 are zero. Thus, if  $S \to Z_{past} \to Y_{pres}$  is a Markov chain, the following identity holds for the first atom in FIT definition  $I_{\partial}(S;\{X_{past}\} \{Y_{pres}\}_{(3)}) = I_{\partial}(S;\{X_{past}\} \{Y_{pres}\} \{Z_{past}\}_{(4)})$ . Similarly, in eqs. S27 all atoms are computed on collections preceding collection  $\{SZ_{past}\}$  (meaning that, they are a subcomponent of  $I(Y_{pres};S,Z_{past})$ ). However, among these atoms, only  $I_{\partial}(Y_{pres};\{X_{past}\} \{S\} \{Z_{past}\}_{(4)})$  precedes collection  $\{Z_{past}\}$  on this decomposition and, therefore, is a subcomponent of  $I(Y_{pres};Z_{past})$ . Since in our case  $I(Y_{pres};S,Z_{past}) = I(Y_{pres};Z_{past})$ , the other two atoms on the r.h.s. of S26 are zero. Thus, if  $S \to Z_{past} \to Y_{pres}$  is a Markov chain, the following identity holds for the second atom in FIT definition  $I_{\partial}(Y_{pres};\{X_{past}\} \{S\}_{(3)}) = I_{\partial}(Y_{pres};\{X_{past}\} \{S\} \{Z_{past}\}_{(4)})$ . Altogether, we found that, if  $S \to Z_{past} \to Y_{pres}$  is a Markov chain, the two atoms appearing in FIT definition (eq. S11) are exactly equal to the two atoms between which we minimize to remove the effect of  $Z$  from  $FIT_X$  in eq. S28, proving that in this scenario  $cFIT_X|Z = 0$ .
SM1.5 PID decomposition of DFI
We next use PID to examine a previously introduced measure of the information about a specific stimulus feature  $S$  flowing from  $X$  to  $Y$ , called Directed Feature Information (DFI) [17].
This measure was defined by reasoning to first consider TE between  $X$  and  $Y$  as a measure of the overall information transmitted from  $X$  to  $Y$  and then to subtract out from it the information that is not due to changes in the value of the stimulus feature. The latter was estimated as  $TE(X \to Y|S)$ , the value of TE conditioned on the stimulus feature, that is the expected value of the TE when it is conditioned on the value of a particular stimulus feature. The reasoning of [17] is that the conditioning removes information not related to variations of the stimulus feature, and that thus  $TE(X \to Y|S)$  quantifies the amount of information transferred from  $X$  to  $Y$  that is not related to the variations in the stimulus feature. With this reasoning, the authors of [17] defined the DFI to measure stimulus-feature specific information transfer by subtracting out from the total information their estimate of the one that is not related to variations in stimulus features [17]:
$$
D F I (X \rightarrow Y) = T E (X \rightarrow Y) - T E (X \rightarrow Y | S) \tag {S36}
$$
The authors of Ref [17] showed that DFI is equivalent to the difference between the sum of the information about  $S$  that each of  $X_{past}$  and  $Y_{pres}$  individually carry, minus the information about  $S$  jointly carried by  $X_{past}$  and  $Y_{pres}$ , with all the information quantities conditioned on  $Y_{past}$ :
$$
D F I (X \rightarrow Y) = I (S; X _ {p a s t} | Y _ {p a s t}) + I (S; Y _ {p r e s} | Y _ {p a s t}) - I (S; X _ {p a s t}, Y _ {p r e s} | Y _ {p a s t}) \tag {S37}
$$
The difference between information individually carried and information jointly carried is often referred to as co-information [38; 3]. This measure of co-information has been used in the literature as a measure of the net effect of redundancy and synergy and it indicates prevalent redundancy when positive and prevalent synergy when negative [33; 30]. In this rewriting, DFI has some similarities with FIT, in that it uses a measure of redundancy (although conflating synergy and redundancy) between stimulus information in the past of  $X$  and in the present of  $Y$ , as well as a discounting, by conditioning, of the past activity of  $Y$ .
Previous work on PID has shown that co-information can be expressed as the difference between two non-negative pieces of information which properly quantify synergy and redundancy [38; 3]. Therefore a simple difference between DFI and FIT is that DFI possibly also includes terms of synergy between  $X_{past}$  and  $Y_{pres}$  than should not be included in a definition of transmission of feature information from  $X$  to  $Y$ . Moreover, given that DFI conditions on the past activity of  $Y$  rather requiring uniqueness with respect to the past feature information of  $Y$  (as in FIT), it does not isolate information in the present activity of  $Y$  that has not been present before in  $Y$ . To understand better the consequences of these facts in terms of the difference between DFI and FIT, we reformulated DFI as a sum of the partial information terms from the PID, as follows:
$$
\begin{array}{l} D F I _ {X \rightarrow Y} = I (S; X _ {p a s t} | Y _ {p a s t}) + I (S; Y _ {p r e s} | Y _ {p a s t}) \\ - I (S; Y _ {p r e s}, Y _ {p a s t}, X _ {p a s t}) + I (S; Y _ {p a s t}) \\ = I _ {\partial} \left(S; \left\{X _ {p a s t}, Y _ {p a s t} \right\} \left\{Y _ {p r e s}, Y _ {p a s t} \right\}\right) \tag {S38} \\ + I _ {\partial} (S; \{X _ {p a s t}, Y _ {p a s t} \} \{Y _ {p r e s}, Y _ {p a s t} \} \{X _ {p a s t}, Y _ {p r e s} \}) + \\ + I _ {\partial} \left(S; \left\{X _ {p a s t} \right\} \left\{Y _ {p r e s}, Y _ {p a s t} \right\}\right) + I _ {\partial} \left(S; \left\{Y _ {p r e s} \right\} \left\{X _ {p a s t}, Y _ {p a s t} \right\}\right) \\ + I _ {\partial} \left(S; \left\{X _ {p a s t} \right\} \left\{Y _ {p r e s} \right\}\right) - I _ {\partial} \left(S; \left\{X _ {p a s t}, Y _ {p r e s} \right\}\right) - I _ {\partial} \left(S; \left\{X _ {p a s t}, Y _ {p a s t}, Y _ {p r e s} \right\}\right) \\ \end{array}
$$
Note that in the above expression all terms involving pieces of redundant information are positive and those only involving synergistic information are negative. Thus this decomposition of DFI demonstrates that it is the linear combination of (mostly) redundant information terms appearing with a positive sign and synergistic information terms appearing with a negative sign. This explains why, as a result of not separating redundancy from synergy, DFI can be negative and difficult to interpret as information about a stimulus feature transmitted from  $X$  to  $Y$ .
The fact that DFI can become negative also shows that using  $TE(X \to Y|S)$  to remove from the total transmitted information the one not about the stimulus feature  $S$  (as done in DFI, see [17]) is incorrect. This is because  $TE(X \to Y|S)$  does not really quantify the information from  $X$  to  $Y$  which is not about  $S$ , as conceptualized in Ref [17]. It actually quantifies the information transmitted on average within each feature condition. This can overestimate the information from  $X$
to  $Y$  which is not about  $S$ . In simple terms, when using data from the same feature conditions, some information sent from  $X$  to  $Y$  in this subset of data could be about the specific value of the feature in the considered set of trials. When the strength of communication about  $S$  between  $X$  and  $Y$  varies from one feature value to another, this overestimation may become even more severe, because in this case additional information about  $S$  is encoded synergistically within the network of  $X$  ad  $Y$  by their feature-dependent relationship [30].
SM1.6 Numerical computation of FIT and other information quantities
FIT and all other information theoretic quantities were computed from both simulated and real data by plugging into the corresponding equations the numerical evaluation of the response probabilities from the data. We computed the response probabilities by discretizing neural activity into a number  $R$  of equipopulated bins [22] and then computing empirically the frequency of occurrence of each binned response across all available trials.
In Table S1 we summarize the number of bins we used to discretize neural activity for each figure in the paper. In Section SM2.5 we study the accuracy of the FIT and TE estimates with the number of available trials and we show that the estimates of FIT and TE are accurate and unbiased for the number of bins and number of trials used for all analyses. However, in the code we provide to compute FIT and TE, we also implemented limited-sampling bias correction routines that can be used to obtain more accurate estimates when data are more scarce (see Section SM2.5).
Number of bins	2	3	4
Figures	Fig.3
Fig.S8
Fig.S9
Fig.S11
Fig.S12B
Fig.S13B
Fig.S14B
Fig.S15D-E	Fig.2A,B,E
Fig.S3
Fig.S4
Fig.S5
Fig.S7
Fig.S13A
Fig.S14A
Fig.S15A-B	Fig.2C
Fig.4
Fig.S10
Fig.S12A
Fig.S15C
Table S1: Number of bins used to discretize neural activity for information-theoretic analyses of simulated and real data, for each main text and SM figure
SM1.7 Permutation-based non-parametric null hypotheses for FIT and TE
To test for significance of the information theoretic quantities, we used non-parametric permutation tests, described below.
To test for the significance of mutual information values about the feature of interest, we used established non-parametric procedures [8; 12; 19]. We constructed surrogate datasets in which we destroyed any feature information by randomly permuting across trials the values of  $S$ , and then we recomputed information on the surrogate data to obtain a null-hypothesis distribution of null information values.
To test for the significance of FIT, we developed a permutation test in which we created surrogate data in which we preserve the feature information in the past of  $X$  and the present of  $Y$  while destroying the communication of this information between  $X$  and  $Y$ . We shuffled  $X$  within trials with the same value of the feature  $S$ , destroying any within-trial statistical relationship between the activity of  $X$  and the activity of  $Y$  at fixed values of  $S$ , and recomputed FIT on the surrogate data. This data shuffling preserves the marginal distributions between the feature and the past activity of  $X$  and between the stimulus and the present activity of  $Y$ , thereby preserving the information about the stimulus that each carries. However, it destroys the within-trial statistical relationship at fixed stimulus between  $X$  and  $Y$  that would be present if  $X$  sends stimulus information to  $Y$ . Because FIT depends on  $P(X_{past},Y_{pres})$  (see mathematical proof in Section SM1.3.4), the values of FIT on the permuted data will be smaller than the ones on the original data whenever there is direct within-trial communication of stimulus information between  $X$  and  $Y$ , but will be similar to the value of the original data when
there is no such direct within-trial communication. As shown by numerical simulations (see Section SM2.6) the so generated null hypothesis distribution of FIT values when  $X$  and  $Y$  encode but do not communicate stimulus feature information is more conservative (see Fig. S7C) than the simpler one that would be obtained by a permutation test destroying all information about  $S$  in  $X$  and  $Y$  by randomly permuting  $S$  across all trials, as for the mutual information quantities above. (This permutation test would implement the idea that if no stimulus-feature information is present, it cannot be transmitted). However, in limiting cases in which the stimulus information in the neural data is absent, we found it numerically better to perform this second random permutation of the label of  $S$  across trials (because more possible independent data permutation are available in this second permutation, which therefore may have some advantages in the case of zero or negligible stimulus information, see Fig. S7E). To reduce the probability of false positives in such cases of no information present in the network, we computed and then intersected the two above describe possible permuted distributions by taking the element-wise maximum between the two distributions, and obtained a null distribution for FIT. (In practice, in real data and simulations with stimulus information present, the maximum of the two permuted values coincided in all simulations with the maximum of the first permutation. This is exemplified in Fig. S7C, in which for higher value of the parameter  $W_{ZY}$  some information about the stimulus is created in both  $X$  and  $Y$  in absence of communication between  $X$  and  $Y$ , the null hypothesis for FIT taking the maximum between the first and second permutation has values not only larger than the FIT value measures in the simulation, but also much larger values then the ones based on only shuffling  $S$ . In simulations with null information, the maximum value of the permuted data in each simulation could instead belong to either permutation.)
An identical procedure was applied to test for the significance of DFI. For TE, since by design the measure captures the total amount of information flowing from  $X$  to  $Y$ , we permuted the neural activity of the sender  $X$  across all trials.
Since in all simulations and real data analyses we wanted to test for the significance of the information values averaged either across simulations or participants, we computed the average over simulations and participants of the information values in each realization of the random permutation and we used this distribution of null hypothesis of averaged values for testing the significance of the averaged information value. (To compute the null-hypothesis distribution, we generated 500 different realizations of the permuted average information for each test we conducted.)
In some analyses (e.g. Fig. 3C and 4B,C) we had to identify the cluster of post-stimulus times and transmission delays for which FIT or TE were significantly different from zero (shown, e.g., in Figs. S9G, S10A,B). We individuated these clusters of points in the time-delay space using a cluster-based permutation test [24; 8] using as null hypothesis values those obtained from the permutation test described above. We computed the cluster forming threshold as the 99th percentile of information values in the surrogate data. We created information clusters in the original and shuffled datasets by summing together all adjacent information values above the cluster forming threshold. We then determined a null distribution for information clusters using the maximum cluster value from each shuffled dataset. Finally, we assigned significance to clusters in the original dataset if their value was larger than the 99th percentile of the clusters null distribution  $(\mathrm{p} < 0.01)$ .
SM2 Details of simulations and and additional analyses of simulated data
SM2.1 Simulations of FIT and TE as a function of signal and noise transmission
This section pertains to the description of Fig. 2A-B of the main text.
The goal of the first simulation (whose results are reported in Fig. 2A) was to evaluate the dependence of FIT and TE on stimulus-feature-related and -unrelated transmission. The goal of the second simulation (whose results are reported Fig. 2B) was to test the ability of FIT and TE to localize in time the stimulus-feature-related information transmission. The setting of both simulations was identical and is described in the following.
We simulated 500ms of activity of activity of  $X$  and  $Y$ , in time steps of 10ms. The sending region  $X$  encoded a stimulus  $S$  over time and transmitted stimulus-feature-related and -unrelated activity to the receiver  $Y$  with a given temporal lag  $\delta$ . The stimulus feature  $S$  being encoded by  $X$  and transmitted to  $Y$  was an integer (between 1 and 4) drawn independently and uniformly in each trial (500 trials per stimulus for each of the 50 simulations). The activity of the sender was a two-dimensional
variable with one feature-informative  $X_{stim}$  and one feature-uninformative component  $X_{noise}$ . The stimulus-feature-informative dimension had a temporally-localized feature-dependent bump in the activity (from 200 to 250ms) and multiplicative Gaussian noise
$$
X (t) _ {\text {s t i m}} = S (t) \left(1 + \mathcal {N} \left(0, \sigma_ {\text {s t i m}}\right)\right) \tag {S39}
$$
where  $S(t)$  was a function equal to the value of the stimulus  $s \in [1,4]$  during the time window [200, 250]ms and was zero outside of this window. The presence of noise in  $X(t)_{stim}$  was needed to test for the impact of within-trial encoding of  $S$  in  $X$  on the within-trial encoding of  $S$  in  $Y$ , at fixed values of  $S$  (i.e., when  $X$  encodes  $S$  incorrectly, also  $Y$  encodes  $S$  in a similar way). If  $X(t)_{stim}$  encoded the stimulus perfectly (no noise in  $X(t)_{stim}$ , therefore  $X(t)_{stim} = S$  for  $t \in [200,250]ms$ ), it would be impossible to determine whether  $Y$  is receiving stimulus information from  $X$  or directly from  $S$ . We choose the noise in the stimulus-feature-informative dimension to be multiplicative because it made it a more challenging scenario for FIT. In fact using multiplicative noise  $X$  developed a stimulus-dependent noise in the encoding of  $S$ . The stimulus-dependent noise in the encoding of  $S$  leads to stimulus-dependent within-trial correlations between  $X$  and  $Y$ , which potentially induces synergies in the encoding of  $S$  in  $X$  and  $Y$  [30]. Since FIT computes information transmission by identifying a component of redundant information between the past of  $X$  and the present of  $Y$ , using simulations that have both redundancy between  $X$  and  $Y$  induced by information transmission and synergy between  $X$  and  $Y$  induced by stimulus-dependent amount of noise encoded and transmitted (using this kind of multiplicative noise), makes it potentially harder for a measure of feature-specific information transmission to separate out the redundant information that was transmitted. In fact, we will see that measures that do not separate well redundancy and synergy, such as DFI, will suffer under such conditions (leading to negative values of transmitted information (Fig. S15A), whereas FIT seems to work well even under this condition because it uses PID to only include redundant time-lagged information about  $S$  in  $X$  and  $Y$ , discarding synergy. (However, we found similar results for TE and FIT by replacing the multiplicative noise with an additive noise in Eq. S39 (Fig. S3)). The stimulus-feature-unrelated component was, at any time point, a zero-mean Gaussian noise  $X(t)_{noise} = \mathcal{N}(0,\sigma)$ . The activity of the receiver  $Y$  was the weighted sum of  $X_{stim}$  and  $X_{noise}$  with a delay  $\delta$ , plus a Gaussian noise:  $Y(t) = W_{stim}X_{stim}(t - \delta) + W_{noise}X_{noise}(t - \delta) + \mathcal{N}(0,\sigma)$ . The delay  $\delta$  was chosen randomly in each repetition from a uniform distribution in the range between 40ms and 60ms, in steps of 10ms. Therefore, across repetitions of the simulation,  $Y$  received information from  $X$  only in the time window [240, 310]ms. In all simulations we set a standard deviation  $\sigma = 2$  for the additive Gaussian noise in  $X_{noise}$  and  $Y$ , and a standard deviation  $\sigma_{stim} = \sigma /5 = 0.4$  for the multiplicative Gaussian noise in  $X_{stim}$ .