SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In the first simulation, we computed FIT and TE at the first time instant in which  $Y$  received information from  $X$  ( $t = 200ms + \delta$ ), and at the ground truth delay  $\delta$ , for all combinations of  $W_{stim}$  and  $W_{noise}$  in the range between 0 and 1, in steps of 0.1. In the second simulation we set  $W_{stim} = 0.5$  and  $W_{noise} = 1$  and computed FIT and TE at all time points, in a rage of communication delays between 0 and 100ms, and averaged their values over delays to obtain temporal profiles of transmitted information.
SM2.2 FIT can detect feature specific information flow even with overlapping time courses of stimulus information
This section pertains to the description of Fig. 2C-E of the main text.
One often used method to infer hierarchical flow of information across ares is to consider the timing of neural activation or of stimulus selectivity of activity across brain regions [32]. However, this method is neither necessary nor sufficient to determine real communication. On the one hand time lagged information selectivity between two regions may arise in absence of communication for example if the two regions received a partly shared input signal with a different delay (see also Section SM4.3). On the other hand, as we will exemplify in this section, real features-specific communication between two brain regions could take place even without detectable differences in timing of information across the considered regions. The purpose of this subsection is to illustrate that this can happen and also to show that in such case FIT has power to discriminate between cases in which feature-specific information flow does or does not take place. We will show that this is because FIT can assess that in cases of real communication the format of information encoding is the same in the past activity of the sender and in the present activity of the receiver.
Figure S3: Further tests of FIT on simulated data with additive noise in  $X$ . This figure is similar to Fig. 2 of the main text except that the results are now obtained with additive (rather than multiplicative) noise in  $X$ . (A) FIT and TE as function of stimulus-feature-related ( $W_{stim}$ ) and -unrelated ( $W_{noise}$ ) transmission strength. * indicate significant values ( $p < 0.01$ , permutation test) for the considered parameter set. (B) Dynamics of FIT and TE in a simulation with time-localized stimulus-feature-information transmission. The red area shows the window of stimulus-feature-related information transfer. Yellow dots show time points with significant information ( $p < 0.01$ , permutation test). Results plot mean (lines) and SEM (shaded area) across 50 simulations (2000 trials each).
We simulated a scenario where a sending regions  $X$  encodes and transmits to a receiving region  $Y$  information about an integer stimulus-feature  $S$  (ranging between 1 and 4). Importantly, the past of  $Y$  and the present of  $X$  carry the same amount of feature information of the past of  $X$  and the present of  $Y$ , but encode the information with different formats.
The feature encoding format of each region is determined by the encoding function  $f(S)$  controlling the average response of each region to individual stimulus values  $s = [1,2,3,4]$ . We simulated responses with three different encoding functions:
$$
f _ {1} (S) = 1 + \delta [ 0, 1, 2, 3 ]
$$
$$
f _ {2} (S) = 1 + \delta [ 1, 0, 2, 3 ] \tag {S40}
$$
$$
f _ {3} (S) = 1 + \delta [ 0, 1, 3, 2 ]
$$
where  $\delta$  is a parameter controlling the separation of the average responses to different stimuli, and therefore the amount of feature information carried by each region at a specific time point. We set  $\delta = 1$  in all simulations. The three encoding functions  $f_{1}(S), f_{2}(S), f_{3}(S)$  are depicted in Fig.2C in blue, green and red, respectively. The encoding function determined the feature values that each region preferentially encodes at each specific time point. Specifically, due to the presence of additive Gaussian noise, regions were most informative (according to Eq. S10) about stimulus values for which the response was either minimum (i.e. equal to 1 in eq. S40) or maximum (i.e. equal to  $1 + 3\delta$  in eq. S40). Indeed, activity distributions in response to these stimulus values were less overlapped with activity in response to other stimuli. For example, regions encoding the stimulus as  $f_{1}(S)$  would carry high specific information about stimulus values 1 and 4, and low specific information about stimulus values 2 and 3. On the other hand, regions encoding the stimulus as  $f_{2}(S)$  would carry high specific information about stimulus values 2 and 4, and low specific information about stimulus values 1 and 3. Therefore, since the  $I_{min}$  measure quantifies redundancy as the overlap in the distributions of specific information across individual values of target variable, the responses of two regions  $X$  and  $Y$  would be maximally redundant if they encoded the feature with the same encoding format (e.g.,  $f_{X}(S) = f_{Y}(S) == f_{1}(S)$ ), partially redundant if they both carried high specific information about one stimulus value (e.g.,  $f_{X}(S) == f_{1}(S)$  and  $f_{Y}(S) == f_{2}(S)$ ) or minimally redundant if they carried high specific information about different pairs of stimulus values (e.g.,  $f_{X}(S) == f_{2}(S)$  and  $f_{Y}(S) == f_{3}(S)$ ).
In our simulation,  $X$  and  $Y$  activity at different time points was described by the following set of equations:
$$
\begin{array}{l} X _ {p a s t} = f _ {1} (S) + E _ {1} \\ Y _ {p r e s} = X _ {p a s t} \\ \end{array}
$$
(S41)
$$
Y _ {p a s t} = f _ {2} (S) + E _ {2}
$$
$$
X _ {p r e s} = f _ {3} (S) + E _ {3}
$$
where  $E_1, E_2$ , and  $E_3$  are additive Gaussian noise with standard deviation equal to  $\sigma$ . Importantly, real feature transfer only occurs in the  $X \to Y$  direction, causing the past of the sender and the present of the receiver to encode the feature with the same format. Since  $\sigma$  was equal for all noise terms in eq. S41, both  $X$  and  $Y$  carried the same amount of stimulus-feature information in the past and in the present, removing any contribution to FIT due to time-lagged information levels in the sender and the receiver region. We measured FIT in the two directions ( $X \to Y$  and  $Y \to X$ ) for different levels of noise  $\sigma$ . By changing  $\sigma$ , we controlled the  $SNR = \delta / \sigma$  in both past and present activity of  $X$  and  $Y$ . We repeated the simulation 100 times for each  $SNR$  value ranging between 0.05 and 1 with a precision of 0.05. We measured the FIT significance in the two directions using the permutation test described in section SM1.7.
We found that, because FIT could correctly detect that the format of information representation of  $S$  in the present of  $Y$  was equal to that of the past of  $X$  but different to that of the past of  $Y$  (Fig.2D), and that feature information flowed from  $X$  to  $Y$  (Fig.2E).
SM2.3 Simultaneous transfer of information about more than one feature
We performed a simulated study of how FIT performs when studying neural system that encode and transmit more than one feature (Fig. S4).
We simulated two independent features (e.g. of a sensory stimulus) S1,S2 simultaneously encoded in a brain region  $X$  and transmitted to a brain region  $Y$ . In the simulation,  $S_{1}$  is more strongly and encoded and transmitted than  $S_{2}$ . The equation for simulating the data are as follows:
$$
X = S _ {1} + D S _ {2} + E _ {x} \tag {S42}
$$
where  $S_{1}, S_{2}$  are independent binary variables (values equal to  $\pm 1$ ),  $E_{x}$  is Gaussian noise with standard deviation equal to 1, and  $Y$  equals  $X$  with a time lag, plus independent Gaussian noise with standard deviation equal to 1.
We simulated the system with different values of  $D$  (the strength of encoding and transmission of  $S2$  relative to  $S1$ ). We found (Fig. S4) that FIT identifies correctly that both features are transmitted, and ranks correctly the features about which most information is transmitted. FIT also identified correctly the limiting case ( $D = 1$ ) in which both features are encoded and transmitted with equal strength.
Figure S4: Simulation of a system that encodes independently two features  $S1$  and  $S2$  in the activity of a brain region  $X$  and transmits them to another brain region  $Y$ . We compute FIT about each of the two stimulus features  $S1, S2$  or their combination  $S = (S_1, S_2)$  as a function of the parameter  $D$  describing the strength of encoding and transmission of  $S_2$  with respect to  $S_1$ . We plot mean  $\pm$  SEM over 50 simulations
SM2.4 Simulations of bidirectional transmission between  $X$  and  $Y$
Here we describe the simulations whose results are presented in Fig. S5.
To further investigate the ability of FIT to determine the direction of stimulus-feature information flow, we simulated a scenario with bidirectional (back and forth) communication between  $X$  and  $Y$  with stimulus-feature-related transfer from  $X$  and stimulus-feature-unrelated transfer from  $Y$  to  $X$  (Fig. S5A).
In brief, both  $X$  and  $Y$  received information directly from a feature-information-sending region  $S$ . The region  $X$  received stimulus information from  $S$  early on (between 50 and 90 ms) and  $Y$  received stimulus information from  $S$  at a later time (between 110 and 150 ms).  $X$  sent its entire activity to  $Y$  (therefore communicating its stimulus information when it became available).  $Y$  instead only sent to  $Y$  a part of its activity that did not carry stimulus information. The details of how this was achieved are reported below.
We simulated 180ms of activity of  $X$  and  $Y$ , in steps of 1ms. The stimulus feature  $S$  being encoded and transmitted from a stimulus region  $S$  to  $X$  and  $Y$  was an integer (between 0 and 3) drawn independently and uniformly in each trial. The activity of  $X$  was one-dimensional. The activity of  $Y$  was two-dimensional. Both dimensions of  $Y$  ( $Y_{+}$  and  $Y_{-}$ ) were generated with a Poisson process whose mean was modulated over time by a Gaussian bump (whose amplitude was equal to the stimulus-feature value  $S$ ) in the time window [110, 150]ms, plus an additive Gaussian noise and time-lagged readout of  $X$  activity (with a  $X$  to  $Y$  transmission delay  $\delta_{xy} = 10\mathrm{ms}$ ). Importantly,  $Y_{+}$  encoded the stimulus as a positive Gaussian bump and  $Y_{-}$  encoded the stimulus as a negative Gaussian bump. Therefore, the entire activity of  $Y$ , i.e. the sum of the two components,  $Y_{noise} = Y_{+} + Y_{-}$  carried no information about the stimulus, and the difference of the two components  $Y_{stim} = Y_{+} - Y_{-}$  carried all the stimulus information in  $Y$ .  $X$  was a Poisson process whose mean was positively modulated over time by a Gaussian bump - whose amplitude was modulated by the stimulus - in the time window [50, 90]ms, plus an additive Gaussian noise and time-lagged readout of the entire activity of  $Y$ $Y_{noise}$  (with a  $Y$  to  $X$  transmission delay  $\delta_{yx} = 15\mathrm{ms}$ ).
We measured FIT at each time step of the simulation over a range of communication delays and averaged the resulting information values over delays to obtain temporal profiles of information transmission. We found that FIT correctly captured the flow of information about  $S$  between  $X$  and  $Y$  that we put by design into these simulated data. FIT revealed that there was a significant stimulus-feature-related information transmission from  $X$  to  $Y$ , that was temporally localized in the actual "ground-truth" [60, 100]ms window in which  $Y$  received stimulus information, and no significant stimulus-feature-information transmitted from  $Y$  to  $X$  (Fig. S5B).
We also used these simulations to test the performance of the Directed Feature Information, DFI [17], using the same analysis pipeline described here for FIT. Results are discussed in Section SM4.2.
SM2.5 Limited-sampling bias of FIT and TE
Information-theoretic quantities are known to suffer from a systematic error (called limited sampling bias) when the probabilities used to compute them are estimated from a limited number of experimental trials [26]. While the limited bias of Shannon information quantities such as TE have been studied well and has been shown to be inversely proportional to the number of available trials and directly proportional to the number of bins used to discretize the data [? ], those of FIT remain to be investigated.
Therefore here we simulated a simple scenario to study how FIT and TE scale with the number of trials available in the dataset and the number of bins of the discretized activity. In these simulations  $X$  encoded a binary feature  $S$  with additive Gaussian noise.  $Y$  was equal to  $X$  with a time lag of 1 plus independent Gaussian noise (standard deviation of noise  $= 0.5$ ).
We found (Fig. S6) that FIT behaved much better than TE with the data size and number of bins. The correct value of FIT, which can be estimated from large numbers of trials, was achieved already with smaller number of trials than for TE. We found that accurate calculations of FIT are possible with the number of trials available in empirical datasets (Fig. S6); for comparison FIT calculations with real and simulated data in this paper were done with 2-4 discretization bins, see Table S1). Our understanding is that the better scaling and sampling properties of FIT with respect to Shannon Information quantities arise because FIT considers a PID part of the total information which has
Figure S5: Simulations of performance of FIT and other measures in a case of transfer of stimulus-feature-related and stimulus-unrelated information in different directions. (A) Schematic of the simulation. A stimulus node  $S$  provides partly complementary information about a stimulus feature to  $X$  (in the 50-90ms interval) and to  $Y$  (in the 110-150ms interval).  $X$  transmits stimulus information to  $Y$ .  $Y$  instead has different components of activity and it projects to  $X$  only the component of its activity that is stimulus-unrelated. In other words, we have stimulus-feature-information transfer from  $X$  to  $Y$  and noise transfer from  $Y$  to  $X$ . (B) Results of the analyses of this simulated activity using FIT (left panels) and DFI (right panels). Gray lines plot the value of these quantities Yellow dots plot time points in which the measure was significantly different from null (permutation test of Section SM1.7;  $\mathsf{p} < 0.01$ )
lesser bias compared to other parts of the total information. Given that the PID atoms of FIT do not contain synergistic terms, this is in line with previous work [25] showing that synergistic components of information have much larger limited sampling bias, and that information quantities that do not include synergistic components have much better sampling properties than full multivariate Shannon information quantities. Thus, FIT can be computed from the datasets in which Shannon information measures typically applied to neural data. We applied a widely used bias correction technique, called the Quadratic Extrapolation [37; 26]. This method is based on subtracting the bias estimated from a second-order polynomial fitting of the dependence of the estimated quantity on sub-samples of the available data. We found (Fig. S6) that this bias subtraction technique was helpful in further improving the estimate of information (reducing the limited sampling bias) in cases of very low numbers of trials available. This bias correction technique is made available in the software we provide for both FIT and TE.
SM2.6 Simulation tests of the significance of FIT and cFIT
In this Section we describe the simulations and results presented in Fig. S7.
In this set of simulations, we first evaluated the effectiveness of the permutation-based non-parametric tests for FIT in a difficult scenario in which  $X$  and  $Y$  independently encode stimulus-feature information with a temporal lag, but no actual communication occurs between them. We then evaluated the performance of cFIT in measuring the unique contribution of  $X$  in sending feature information to  $Y$  in presence of an alternative information route through a third node  $Z$  sending information to  $Y$ .
We addressed both questions using a the following simulation setup. We performed a simulation in which two senders  $(X$  and  $Z)$  both transmitted stimulus-feature information to  $Y$  (Fig. S7B).  $X$  encoded the stimulus-feature linearly and  $Z$  non-linearly so that they carried partially different information about  $S$ . This is important because a good measure quantifying the unique contribution of  $X$  (but not  $Z$ ) in the transmission of information to  $Y$  should capture that, even if the total amount of feature information transmitted from  $Z$  to  $Y$  is stronger than the one transmitted from  $X$  to
Figure S6: FIT and TE as a function of the number of simulated trials used to compute them. Vertical lines: number of trials used for the analyses of real and simulated brain data in this paper. "Plugin": plugging empirical probability histograms into FIT, TE eqs. "QE": adding Quadratic Extrapolation bias correction. For all analyses in the rest of the paper we used 2-4 discretization bins (see Table S1). Results in each panel are plotted as mean  $\pm$  SEM over 100 simulations
$Y$ , there can still be a different component of information that is uniquely transmitted by  $X$  and not by  $Z$ . We simulated 500ms of activity, in time steps of 10ms. The encoded and transmitted stimulus feature  $S$  was a stimulus-intensity integer value (0 to 3) drawn independently and uniformly in each trial (500 trials per stimulus). The activity of  $X$  had a temporally-localized square bump between 200 to 250ms whose amplitude depended linearly on  $S$ , with multiplicative Gaussian noise. Activity of  $Z$  had a temporally-localized square bump (from 200 to 250ms) encoded with a different format with respect to  $X$ , and multiplicative Gaussian noise. Specifically,  $X$  encoded the stimulus feature  $S = (0,1,2,3)$  with the encoding function while  $Z$  encoded  $S$  with the encoding function  $S = (1,0,3,2)$ . In this way,  $X$  carries more specific information than  $Z$  (see eq S10) about  $S = 0,3$  and  $Z$  carries more specific information than  $X$  about  $S = 1,2$ , therefore both  $X$  and  $Z$  carry some unique information about  $S$ . Activity of  $Y$  was the weighted sum of  $X$  and  $Z$  with a temporal lag, plus additive Gaussian noise:  $Y(t) = W_{xy}X(t - \delta) + W_{zy}Z(t - \delta) + \mathcal{N}(0,\sigma)$ . The delay  $\delta$  in the transmission of information from  $X$  to  $Y$  was chosen in each repetition of the simulation randomly from a uniform distribution in the range between 40ms and 60ms. We computed FIT and cFIT at the first time instant in which information in  $Y$  was received from  $X$  and  $Z$  ( $t = 200ms + \delta$ ) using to define past activity the ground-truth delay  $\delta$  actually used in that simulation. We set a standard deviation  $\sigma = 2$  for the additive Gaussian noise in  $Y$ , and a standard deviation  $\sigma_{stim} = \sigma /5 = 0.4$  for the multiplicative Gaussian noise in  $X$  and  $Z$ .
SM2.6.1 Tests of significance of FIT accounting for the possible existence of encoded feature information in the absence of transfer of it across regions
We first addressed the first question, that is how to deal with confounding scenarios where  $X$  and  $Y$  independently encode feature information with a temporal lag, but no actual communication occurs between them.
We studied how the FIT from  $X$  to  $Y$  depended on the strength of feature-related transmission from  $Z$  to  $Y$ .  $W_{zy}$  when no stimulus-feature-information was transmitted from  $X$  to  $Y$  ( $W_{xy} = 0$ ). We found that FIT from  $X$  to  $Y$  increased with  $W_{zy}$ , since  $X$  and  $Y$  carried redundant information about  $S$  with a temporal lag. However, FIT was always non-significant (Fig. S7C) using the permutation test described in Section SM1.7, since there were no within-trial correlations between the encoding of  $S$  in  $X$  and the time-lagged encoding of  $S$  in  $Y$ . This proves that, even if  $Z$  was not measured, the permutation test we provided for FIT can correctly rule out confounding scenarios where  $X$  and  $Y$  encode  $S$  with a temporal lag but with no actual communication occurring between  $X$  and  $Y$  (see Section SM1.7 and SM2.6).
SM2.6.2 Simulations testing cFIT in the presence of information transfer through an alternative route involving a third region  $Z$
We next addressed the second question, that is how to evaluate the unique contribution of  $X$  in sending feature information to  $Y$  in presence of an alternative information route through a third node  $Z$  sending information to  $Y$ . We studied how FIT from  $X$  to  $Y$  and cFIT from  $X$  to  $Y$  conditioned on the feature information in  $Z$  depended on the simultaneous transmission of feature information from  $X$  to  $Y$  and from  $Z$  to  $Y$ . To do this, we computed FIT and cFIT for all combinations of  $W_{xy}$  and  $W_{zy}$  in the range between 0 and 1, in steps of 0.1. We found that FIT grew both as a function of  $W_{xy}$  and of  $W_{zy}$  and was significant as soon as some information was transmitted from  $X$  to  $Y$  ( $W_{xy} > 0$ ; Fig. S7D, left). On the contrary, cFIT increased only as a function of  $W_{xy}$  and decreased with  $W_{zy}$ , correctly removing from the FIT from  $X$  to  $Y$  the feature information that was routed to  $Y$  through  $Z$  (Fig. S7D, right). Crucially, cFIT did not simply subtract from FIT through  $X$  the FIT through  $Z$ , but it only removed the amount of feature information that was redundantly transmitted by  $X$  and by  $Z$  to  $Y$ . Indeed, since  $X$  and  $Z$  transmitted partially different feature information to  $Y$ , we have that cFIT was still significant for many combinations of parameters where  $W_{zy} > W_{xy}$  (Fig. S7D, right) and, therefore, FIT through  $X$  was larger than FIT through  $Z$  (not shown).
SM2.7 Simulation studies of how FIT and TE are affected by the mixing of sources
In real electrophysiological recordings, it is possible that that separation and reconstruction of the underlying neural sources is imperfect, due to issues such as for example field spread or common referencing. As a result, electrophysiological recordings from different brain regions may contain, with different weights, a mixture of sources. It has been proposed that such source mixing may affect measures of communication between brain areas [1]. Here, we examine the effect of this source mixing in FIT and TE measures.
We simulated source mixing in different proportions in the sender  $X$  and the receiver  $Y$ . In our simulations we assumed that (as it is expected to be the case in real brain data) the mixing is instantaneous (i.e. sources are mixed with zero lag) and with a proportion of source sharing in  $X, Y$  that is stable across time.
We first simulated a "null model" scenario in which a source  $Z$  (informative about a stimulus feature  $S$ ) is shared between  $X$  and  $Y$  with a different proportion  $A$ :
$$
X = Z (s) + E _ {x} \tag {S43}
$$
$$
Y = A Z (s) + E _ {y}
$$