Attentive Transfer Entropy to Exploit Transient Emergence of Coupling Effect
Authors: Xiaolei Ru
Xin-Ya Zhang
Zijia Liu
Jack Murdoch Moore
Gang Yan
Venue: NeurIPS 2023 spotlight (2023)
Keywords: Directed coupled network reconstruction; Neuronal dynamics; Mutual information estimator; Attention mechanism; Transfer entropy.
TL;DR: Directed coupled network reconstruction based on time series generated by neuronal dynamics.

Abstract: We consider the problem of reconstructing coupled networks (e.g., biological neural networks) connecting large numbers of variables (e.g.,nerve cells), of which state evolution is governed by dissipative dynamics consisting of strong self-drive (dominants the evolution) and weak coupling-drive. The core difficulty is sparseness of coupling effect that emerges (the coupling force is significant) only momentarily and otherwise remains quiescent in time series (e.g., neuronal activity sequence). Here we learn the idea from attention mechanism to guide the classifier to make inference focusing on the critical regions of time series data where coupling effect may manifest. Specifically, attention coefficients are assigned autonomously by artificial neural networks trained to maximise the Attentive Transfer Entropy (ATEn), which is a novel generalization of the iconic transfer entropy metric. Our results show that, without any prior knowledge of dynamics, ATEn explicitly identifies areas where the strength of coupling-drive is distinctly greater than zero. This innovation substantially improves reconstruction performance for both synthetic and real directed coupling networks using data generated by neuronal models widely used in neuroscience.

Attentive Transfer Entropy to Exploit Transient Emergence of Coupling Effect
Xiaolei Ru', Xin-Ya Zhang's Zijia Liu', Jack Murdoch Moore'; Gang Yan
School of Physical Science and Engineering,
National Key Laboratory of Autonomous Intelligent Unmanned Systems,
MOE Frontiers Science Center for Intelligent Autonomous Systems, Tongji University, Shanghai, China
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Abstract
We consider the problem reconstructing coupled networks (e.g., biological neural networks) connecting large numbers of variables (e.g., nerve cells) for which state evolution governed by dissipative dynamics consisting of strong self-drive which dominates the evolution and weak coupling-drive. The core difficulty is sparseness of coupling effect, which emerges with significant coupling force only momentarily and otherwise remains quiescent in time series (e.g., neuronal activity sequence). Here we propose an attention mechanism to guide the classifier to make inference focusing on the critical regions of time series data where coupling effect may manifest. Specifically, attention coefficients are assigned autonomously by artificial neural networks trained to maximise the Attentive Transfer Entropy (ATEn), which is novel generalization of the iconic transfer entropy metric. Our results show that, without any prior knowledge of dynamics, ATEn explicitly identifies areas where the strength of coupling-drive is distinctly greater than zero. This innovation substantially improves reconstruction performance for both synthetic and real directed coupling networks using data generated by neuronal models widely used in neuroscience.
Introduction
In this work, our task infer coupling relationships between observed variables based on time series data and reconstruct the coupled network connecting large numbers of these variables (see Figure [b].. Assume the time series Fi records the state evolution of variable governed by coupled dissipative dynamics, as represented by a general differential equation zhe g Σ B (IT Zzt). where g and are self- and coupling functions respectively. The parent variable influences the evolution behavior of the child variable via the coupling function f. Note that these two functions are hidden and usually unknown for real systems. The asymmetric adjacency matrix B represents the directional coupling relationship between variables. Hence, the goal is to infer matrix B from observed time series Za.
2....,N where N is the number of variables the system. If By 1, the variable j is coupling driver (parent) of variable i, otherwise it is zero
In neural dynamics (e.g., biological neuronal systems observed via neuronal activity sequences), the coupling effect is usually too weak to be detected, making less applicable the classic unsupervised techniques used across multiple research communities to infer coupling relationships [ 2 3 4
5, [6. 7 8 9 [0]. This difficulty manifests in three aspects. First, the dynamics contains self- drive and coupling-drive. The strength of the coupling force f(r,, .) usually many orders of
"Corresponding author
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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activity sequence
Ordered Pair of Sequences
Input
Coupling Attention
Transfer Entropy Estimator
Stage
(a:li
((r arli
MI
MI
Stage 2
Binary Classifier
Infer
ATEn
Max(logit,]...
(a) Reconstruct coupled network
(b) Method
Figure 1: (a) Graphical representation of our task. With the activity sequences recording the state evolution of the variables and small number of known relationships between variables (top panel: solid arrows with and without red crosses denote coupling and non-coupling respectively), our target is infer the remained coupling relationships (bottom panel: dotted arrows) and reconstruct the coupled network; (b) Overall framework of our method. An input sample consists of two activity sequences ordered pair of variables. Stage 1: the neural network Ja assigns attention coefficients fa; ) along sequences to maximize ATEn as Eq. [6] The neural networks fo and fo forming transfer entropy estimator estimate mutual information M7, and M (first and second terms in Eq. 4 Stage 2: Classifier hy infers the probability of coupling between variables.
magnitude smaller than self-drive g(xx)), and the latter dominates the evolution of variable state. Second, the behavior of the coupling-drive can be chaotic, unlike in linear models [1] 12]. The resulting unpredictability and variability of system state means that coupling force can be significant momentarily and otherwise almost vanish, as illustrated in Figure 4 (gray lines). This dilutes the information in time series that can be useful for inferring the coupling relationship. Third, in the heterogeneous networks common in applications, variables, among which difficult to distinguish individual drivers.
variables are hubs coupled with many parent
When coupling effects are weak and sparse, we do not observe clearly the principle of Granger Causality, whereby the parent variable can help to explain the future state change in its child variable 113. Rather, when we train model [5] 8] for prediction task on the neuronal activity sequences, model only exploits the historical information of the child variable itself and that from parent variables is ignored. We posit that coupling-drive makes negligible contribution to state evolution in the majority of samples of time series data. other words, only small fraction of samples is the information of parent variables effective in predicting the evolution of child variables. Taking as an example the gradient algorithm to minimise the regression error over all samples Ex ) the adjustment of model parameters from the tiny samples corresponding to significant coupling force is negligible, but these are the only samples which could induce the model to exploit coupling effects in reducing regression error. Similarly, for transfer entropy [2], which measures the reduction in uncertainty which a potential parent variable provides to potential child variable, there is no significant difference in measured value between ordered pairs of variables with and without coupling relation.
To overcome the difficulty, we introduce coupling attention to identifyy the moments when coupling effect emerges. We design an objective function, Attentive Transfer Entropy (ATEn), comprising a weighted generalisation of transfer entropy. In order maximize ATEn, artificial neural network is trained to autonomously allocate high attention coefficients o, at times where information of parent variables effectively reduces the future uncertainty of child variables, and ignores other positions by setting a, close to zero.
However, coupling attention also detects high transfer entropy regions produced by factors unrelated to coupling-drive e.g., noise in empirical samples, which leads to spurious identification of spurious coupling effects between variables, which disturb the inference. To ameliorate this, consider

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utilising known relationships between variables, small number of which could be inferred. e.g., by manual neuron tracing connectome [4]. We leverage this limited number known relationships for more effective detection of connectivity which advances in technology are making increasingly simple and cheap to observe 15 Specifically, we add binary classification model perform more sophisticated inference under the guidance of coupling attention to focus on these critical regions and recognize different patterns between real and spurious coupling effect. We deal with this coupling relation inference task by way of small sample supervised learning. Although training and test data have a distribution shift in the setting small samples, they arise through an identical underlying generation process. Thus, if the model provides insight into the underlying dynamics the coupling-drinn drive in our task - then the understanding acquired from small samples can be effectively
utilised in the test environment [16 17 18 19]. The role of coupling attention is to help the classification model gain this insight. Our contributions are summarized as follows:
1.
We draw on the coupling attention mechanism to identify the positions time series at which coupling effect emerges and guiding classification model to infer coupling relation by focusing on these critical positions. Without any prior knowledge of dynamics, this mechanism determines the areas where the coupling force substantially different from zero.
2. By formulating Transfer Entropy as the difference between two types mutual information, and based on the dual representation of Kullback-Licbler (KL) divergence, design differentiable metric, Attentive Transfer Entropy, as the objective function of the proposed coupling attention.
3. Our method significantly improves performance synthetic and real directed coupling networks using the data generated by four well-known neural dynamic models, and the number of labels required is very small compared to the size of the coupled networks.
2 Background
2.1 Definition of Transfer Entropy
The transfer entropy, an information-thooreee measure, is able to detect information flow between time series and Y. Transfer Entropy measures the degree of non-symmetric dependence of Y on X, defined 2
(k) xx()) log (( (1++1 1 ,² 21)
TE(X -)---)= Σp (Je+1.
(0))
Vi+1 3/ (8)
where x/ (.., x(-1+1) and v. (Vt, Ut- -+1) and k. are lengths of memory. For an uncoupled system (X and Y independent) that can be approximated by Markov process of order (!))
k, the conditional probability to find Y in state yc+1 at time + satisfies (Yt+1 | vt 24
Me+1 vi ( and so TE(X- + Y) vanishes.
2.2 Mutual Information Neural Estimation
The mutual information is equivalent to the KL divergence between the joint distribution Pxy and the product of the marginal distributions Px @ Py 20 2T). The KL divergence DK admits the neural dual representation 22 23
MI(X,Y) Dez PXxY Py) > sup Exsy [So] log (Erxop [efo]),
(2)
where the supremum is taken over parameter space θ and is the family of functions parameterized by the neural network with parameters 0 E The mutual information neural estimator is strongly consistent and can approximate the actual value with arbitrary accuracy 23

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Figure 2: Visual interpretation of transfer entropy and its attentive version. Transfer Entropy is de- rived the difference of two types of mutual in- formation: MT (x), (blue area) quan- tifies the reduction uncertainty of future state Ut+1 from knowing current states (Vt, F4), and MI (Y,1, Yi) (green area) is same but only Yr is known. Attention coefficients ar (yellow area) assigned to each position of time series by cou- pling attention mechanism to maximize Attentive Transfer Entropy. For brevity, I here.
1
Xr-2
Ness
Xeoz
Ty MIC (l. MICG &0
ATEng-y = Milles (0) 2
3 Method
3.1
Neural Estimator of Transfer Entropy
By the conditional Bayes formula and adding a marginal distribution of Y, we derive the transfer entropy the difference between two types of mutual information. An intuitive description is provided in Figure 2 and the derivation is placed in Appendix A.
TE(X -)
xa(" log ( (K) (1)
De 244
(1++1, V6
Ex (Je+I. ,e,
Σ (w+tin log
(3)
(k) (1)
(K) p U/ )
(Ur) p Ub , Ft
= MM (Yetl, xx!,,xx!))) MI (Xtti)
In these expressions, is sampled from Y randomly and independently of the time step t. The first term MI (Ye+1, (y/(h, xx))) quantifies the reduction in the uncertainty of the future state
Vi+1 from knowing the historical information yr (k) and (1) The second term MI (Y441, (..a.,(h) y,(h)) is
the reduction uncertainty simply from knowing t/t (k) By connecting Eq. and Eq. 2 we define differentiable estimator of transfer entropy as:
P(Y. (xx)x))))))))))) [so] log ( (Frit.i18PPxx0 ( (x1))) [elo]
TENE - Y) = sup F.
sup E P(Y)t))!1)) [sa] - log (F. (FPInnsPPOr!) [efo]).
(5)
Transfer entropy, and even mutual information, is difficult to compute 24 especially for high- dimensional or noisy data. In Appendix B, we offer theoretical proof for the consistency and convergence properties Transfer Entropy Neural Estimation, and examine its bias on a lincar dynamic system where the true values of transfer entropy can be determined analytically.
3.2 Attentive Transfer Entropy
The main difficulty in task is that the coupling effect in certain nonlinear dynamical systems is too weak to be recognized by classic techniques. We discuss the limitation of the iconic transfer entropy in detail that it works well when the three true distributions, i.e., one joint distribution and two conditional distributions in [q can be estimated perfectly. However, sparse coupling effects are easily masked if the estimated probability density deviates even slightly from the real distribution. These momentary sources of evidence of coupling drive are like outliers in the total distribution of time series dominated by self-drive. In order to make the transfer entropy provide clear distinction between coupling and non-coupling pairs, we need to highlight the positions where p yt+1 Vi (k) ..!i)) > ye+11 Vi ) and filter out other times by incorporating a, into Eq.

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while avoiding the problem of distribution approximation. We do so by defnnng ATEn as:
p 31+1 ye (k) (1)
; - Eay ar P POe..ttt.. 31 (4) x²(") log
It
p yh+ (h)
=M (Ye+1,
A4)
MI (Yc+1, x((h) 1 1A).
(6)
In this expression, d, E [0, 1] is the attention coefficient at time step /. and the collection A of attention coefficients the attention series. Comparison of Eq and revells that the transfer entropy can be viewed as a simplified version of ATEn in which attention coefficients are uniformly set to one: Vt, at 1. Because each position has an equal contribution to estimation, the value of transfer entropy is dominated by the majority of positions where coupling effect is negligible, i.e.,
where (v++1 yt Ii = p Ve++1 I vt Similarly to transfer entropy, ATEn is derived as the difference of two mutual informations. but ATEn incorporates the scheme of attention assignment. By connecting Eq. c and Eq. 2 we define differentiable estimator of ATEn as:
1 AT + Y) = sup T,
(k) ((11) Σ at : So (U++1, Di ,
(,, Σ at - e (ef)
log
Σ at fo (M++1.Be (k)) log (1) Σ aj e (o(umiet(t)
sup L
(7)
where i' the total number of time steps and L T max(k: 1). The expectation on the distribution of variables is adapted into the mean over time series.
3.3 Application of Coupling Attention
The overall framework of our model is presented in Figure Lp. In addition two neural networks fe and fo for mutual information estimation, we employ another neural network Ja for coupling attention assignment. Rather than approximating distributions, the neural network go learns to maximize ATEn given by Eq. 7 via gradient descent. However, relying solely on coupling attention mechanism would lead to erroneous identification of spurious coupling effects high transfer entropy regions which can also arise from non-coupling factors, e.g., noise in empirical samples. For more sophisticated inference, we augment our method with a binary classifier h, guided by coupling attention to focus on high transfer entropy regions and recognize different patterns between real and spurious coupling effect. The classifier takes as training samples small number of ordered pairs of variables, among which labels for both coupling and non-coupling relationships are represented.
Then, the training process is divided into two independent stages: coupling attention learning and classification learning. The objectives in the first stage are:
argmax L1 + L2 0
argmax L1 L2
(
a/0
where L1. C2 is the expectation of the first and second sup term of Eq 7 on training set respectively. We update (fo, fø) and 9., alternately. A small learning rate is required to maintain training stability, otherwise the Ia may fall into a trivial solution where attention is almost zero throughout the time series. The objective in the second stage is:
n f- argmin L3,
7/
where L3 is the binary cross entropy and the notation 1) a indicates that coupling attention remains fixed during the second stage of training. The downstream classifier is sensitive to the upstream scheme of attention assignment Implement of our method is presented as Alg. 1 in Appendix C.
When the ATEn reaches stability or convergence, the downstream classifier guided by it usually does not achieve optimal generalization in our experiments. Currently, have not established definitive criterion for determining the best stopping point of the first stage, at which the downstream classifier can achieve optimal generalization. To address this issue, after every fixed number of epochs in the first stage, we retrain a new classifier with a few epochs. When the ATEn converges, we select one with optimal generalization on validation set from those classifiers and further train it until convergence.
5

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Experiment
4.1 Setup
Directed coupling networks. For synthetic networks, we use Scale-Free (SF) graph model [25], which could generate networks with controllable structure characteristics. For real networks, select six neurological connectivity datasets as presented in Table each from a different species: Cat, Macaque, Mouse, C. elegans, Rat and Drosophila.
Dataset
Region
#Nodes
#Edges
Mean degree
Table I: Statistical information of six real networks: dataset name, type of network, number of nodes, number of edges and mean degree (k)) Details are provided in Ap- pendix E.
Cat
Cortex
65
1139
17.5
Macaque
Cortex
242
4090
16.9
Mouse
Cortex
195
214
1.1
C. elegans
Neural
272
4451
16.4
Rat
Cinerea
503
30088
59.8
Drosophila
Medulla
1781
33641
18.9
Dynamic models. We use four dynamic models for neural activity simulation widely used in the field of neuroscience: Hindmarsh-Be Rose (HR), Izhikevich (Izh), Rulkov and FitzHugh-Nagumo (FHN). Dynamic equations are provided in Table and segments of generated time series are represented in Figure
Name
Equations
Table 2: Equations of the four dy- namical models considered. B is the asymmetrical adjacency matrix of the coupling network, record- ing coupling relationships between nodes. Bij
F. 9: ap. bp? Text
HR
dpł 9: [p po) mm]
T= (Vey p.) Σn B1, exp(-A Bxy))))
: 0.04m² 5os + 140 u4 I+
1 if variable i the parent of variable j. other- wise Bij = 0. In these expres- sions, I' describes the coupling- drive, while other terms represent self-drive. The detailed configura- tion of dynamical parameters is pro- vided in Appendix D.
Izh
4. = a(bu; u.). r= o By
F (1 (!)) 8
+ 12 (1z)
Rulkov
Fa(u we)
un;
(u;) Di B (1 exp(A 0.)))
i a
FHN
c(ev u), -=-9. E Big (vz -v)
Baselines. We compare our method with eight baselines. Unsupervised learning: (1) Granger
causality test (Ganger) [1] (2) Transfer Entropy using the Kraskov-estimator [TE_Krakoor [26].
standard method for TE calculation; (3) Convergent cross mapping (CCM) 3i (4) Latent convergent
cross mapping (Latent CCM) [27]: (5) PCMCI 6 and (6) PCMCI* 28 using partial correlation to quantify coupling strength. Unsupervised learning with validation set: (7) Transfer Entropy Neural Estimator (TENE), as in Eq The estimator is trained on the test set unsupervised and terminates upon achieving optimal performance the validation set. Small sample learning: (8) Classifier with traditional attention mechanism 229] which was initially developed for computer vision tasks.
Evaluation metrics. We quantify the performance of methods by evaluating the metries, specifically the area under the receiver operating characteristic curve (AURÔC).
Training details. We employ 4-layer convolutional neural network for model go and ht and 5-layer colnecconeected neural network for model fo and fa. We use the ADAM [30] optimizer with initial learning rate of 10-3 for the classifier ht and 10 for the others. The learning rate decays exponentially by gamma = 0.999 per epoch. The batch size for stage 1 is 32 and for stage 2 is 10. The number of training epochs is 400. We run all experiments this work on local machine with two NVIDIA V V00 32GB GPUs. See codes in Supplementary Materials for more details.

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1.0
1.0
1.0
0.9
0.9
AUROC
0.8
0.8
0.8
0.7
ATEn TENE
0.7
0.7
0.6
0.6
Classiffer
Kraskov
0.6
0.5
0.5
0.5
20
100
180
40
50
0.1
0.2
0.5
(a) Number of Samples
(6) Mean Degree
(4) Coupling Strength
1.0
1.0
0.9
0.9
0.9
AUROC
0.8
0.8
0.7
0.6
0.7
0.5
0.5
0.5
0.05
0.1
0.2
20
30
80
(d) Sampling Period
Measurement Noise(%)
(f)Hidden Variable(%)
Figure 3: We conduct experiments of ablation study in six control enviromments including the number of samples using for supervised training, mean degree of networks (the higher mean degree results in more parents per child variable), coupling strength related to the extent of influence that parent variables have on child variables through the coupling drive, sampling frequency of activity sequence, measurement noise (Gaussian noise is added with mean and standard deviation of certain proportion of the original sequence), and hidden variables (training or validation samples are randomly selected from the remaining observable variables, and test samples are randomly selected from the whole network). In the plots above, the default values of train samples is 20 (10/10 positive/negative samples), mean degree 10, coupling strength 0.2, sampling period 0.1, 10% measurement noise and 0% hidden variables, and each plot considers gradual variation of a single variable. Activity sequences are generated by the HR model on synthetic networks with 500 variables. The size of validation and test set is 100 and 400 respectively, with a uniform distribution of positive and negative samples as in the training set, which are randomly selected from possible ordered pairs within the entire network.
4.2 Ablation study
Our method's innovation can be decomposed into three components: Firstly, trainable transfer entropy calculator (TENE) is proposed: Secondly, TENE is combined with attention to detect region of high transfer entropy; Thirdly, the classifier is guided by coupling attention focus on regions where the coupling effect may emerge. Here, we provide detailed ablation study to determine which component is the key to performance improvement, and show results Figure We observe that Classifier alone (blue lines) exhibits the worse performances than our method (green lines) in most environments. This indicates that the spurious features learned by Classifier, despite quickly reaching low loss on the small training set, are less related to the properties of coupling effect and result in poor generalization capability on test set. When the size of train set increases, the performance of Classifier alone become comparable to that of ATEn (see Figure 34). TENE (cyan lines) exhibits performance marginally superior to transfer entropy estimated using the standard Kraskov method (TE Kraskov; purple lines). The ATEn (green lines) shows substantial performance improvement compared with basclines, which indicates that the classifier with the guidance of coupling attention extract features hidden in activity sequences that closely related to coupling effeet and result in superior generalization. This also demonstrates the importance of identifying and focusing on these critical regions of activity sequences. Moreover, method is robust against changes to experimental conditions other than mean degree and measurement noise, a limitation we diseuss in Sec.
4.3 Insight into the coupling-drive of underlying dynamics
In Figure we demonstrate the ability of coupling attention to catch the transient emergence of coupling effect. The gray lines in Figure child variable over time, and are generated by the coupling term in Table The absolute value of the coupling force rises (the gray lines spike) at occasional moments when the behavior of
Figure represent the change of coupling force from parent to
7

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Power
0
Parent variable
variable
-50
-2
100
200
300
200
300
0
200
400
1.0
1.0
1.0
Coupling-drike drive force Traditional attention Coupling attention
Strength
Emergence of coupling effect
Remain dormant
0.5
0.5
0.5
0.0
0.0
0.0
100
200
300
400
100
200
300
0
200
400
(a) Hindmarsh-Roo Rose
(b) Izhikevich
(c) FitzHugh-Nagumo
Figure 4: Insight into the coupling drive of neural dynamics. Top panel in each subplot: segment of two activity sequences on the ordered pair of variables with a coupling relationship (blue is parent and green is child). Bottom panel in each subplot: the absolute value of coupling force (gray line), the coefficient of traditional attention (light pink line) and coupling attention (orange line) assigned along the sequences. (a) HR: (b) Izh; (c) FHN. Time is index in the time series. In axis labels, "Power" membrane potential, while "Strength" is membrane potential (attention coefficient) for the coupling force (attention). Results are chosen for clarity and are obtained when the training of stage 1 reaches stability.
parent variable substantially influences the evolution of its child variable, and remains almost zero at other times. The orange lines representing the coupling attention keep in step with the gray lines, indicating that the coupling attention mechanism recognizes the effect of coupling force in reducing the uncertainty of the child variable and pays attention to the areas where coupling force is significant. In contrast, the light pink lines in Figure 4 representing the traditional attention [29] remain close to their maximum value, indicating is insensitive to changes in coupling force. This would lead its classifier to extract features from any part of the entire time series, rather than focusing areas where coupling effects may emerge.
4.4 Performance on real networks
We test our method on six empirical neurological connectivity datasets using activity sequences data generated by four neuronal dynamic models. Compared with the baselines, our method usually substantially improves reconstruction performance on real coupled networks, as shown in Table The performance of classical unsupervised methods, for which all positions in the time series are treated equally, is limited by the paucity of distinct coupling effects. In conclusion, our method slightly
Table 3: Performance comparison (model neurons coupled real connectomes). Each point contains the mean and standard deviation of AUROC computed in ten experiments with randomly sampled maainingvalldationnes set (20/100/100) in C. elegans (left) and Drosophila (right) connectomes. Results on other three connectomes are shown in Appendix F.
Dataset
C.elegans
Drosophila
Dynamics
HR
Izh
Rulkov
FHN
HR
Rulkov
FHN
Granger
0.50=0.01
0.64+0.01
0.71+0.01
0.64#0.00
0.6940.00
0.42+0.01
0.74:0.01
0.60+0.01
TE Kraskov
0.63g0.01
0.50s0.01
0.68+0.02
0.54+0.00
0.6010.01
0.4740.02
0.65+0.01
0.58+0.02
CCM
0.79+0.02
0.62+0.02
0.6740.02
0.74#0.01
0.80+0.01
0.42+0.02
0.80s0.01
0.76+0.03
0.7860.01
0.55g0.04
0.53+0.04
0.68:0.01
0.8140.02
0.4740.02
0.76:0.01
0.73x0.01
PCMCI
0.53g0.01
0.53+0.02
0.6640.01
0.5740.02
0,5140.01
0.5140.01
0.51#0.01
0.51+0.01
PCMCI*
0.53g0.01
0.56:0.01
0.69+0.02
0.63+0.00
0.51+0.01
0.51+0.01
0.51=0.01
0.51+0.01
Classifier
0.75+0.06
0.65:0.05
0.7240.06
0.65.0.00
0.8540.05 55
0.65#0.06
0.83:0.09
0.72:0.05
TENE
0.65w0.06
0.43e0.03
0.65+0.08
0.47+0.06
0.5410.07
0.51+0.03
0.54e0.11
0.41:0.06
ATEn
0.94:0.06
0.66s0.12 12
0.90+0.07
0.91+0.06
0.9740.03
0.74+0.09
0.87=0.06
0.83+0.10
8

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increases cost, due to the need for label collection, but obtains a substantial boost in performance compared with those unsupervised methods in this class of coupling network reconstruction tasks. We also observe that the reconstruction performances on identical dynamics across distinct connectomes are significant different (e.g., AUROC of ATEn on Izh across C.elegans Table 3 and Rat in Appendix Table 5), indicating that the underlying network architecture plays an crucial role in shaping the behavior of coupling dynamics.
Limitations
Our methodology has limitations (i.e., cases for which performance improvement is less): 1. Dense networks, where variable is coupled with many driving variables and substantial coupling forces can emerge from distinct parents at overlapping times, making individual drivers harder to distinguish (see green line in Figure B): 2. Intense noise, which makes the coupling attention mechanism falsely identify high transfer entropy regions. The downstream classifier then extracts spurious features, leading to the reduction of its generalization (see Figure 3p); 3. Strongly coupled systems, which are dominated by synchronization phenomena in which the dynamic behaviors of all variables are similar.
In addition, we assume the state evolution of variables in the coupled network is uniformly governed by a group of dynamic equations. However, if the evolution behaviors of variables in the system obey different underlying dynamics, it becomes critical to carcfully select the small training samples, in which the patterns of coupled driving effects are representative. Failure do so may result in the poor generalization of the classifier. That is our next research direction.
Related Work
Coupled Network Reconstruction. Several common methods of causal inference 31 6
are based on conditional independence relations, but differ in the design of condition-selection
strategies or choice of conditional independence test. Granger Causality [1 is extended to nonlinear
dynamics by using neural networks represent nonlinear casual relationships 88 []. Many methods of causal discovery assume that the coupled network is a directed acyclic graph. However, directed cyclic graphs are common in real systems. Conventional frameworks assume separability, i.e., that information about coupling drive are not contained in the parent variable itself. To address the non- non-separability issue, Convergent-cross mapping 3 and its variations [32 27 measure the extent to which the historical record of child can reliably estimate states of the parent in reconstructed state space. However, sparse causal effect neuronal dynamics, particularly in the presence of noise, may lead parent and child time series appear statistically independent, that their contribution state estimation is hard to recognize.
Mutual Information Estimation. Ref. 23 built on dual representation of KL divergence [22 to offer a parametric mutual information neural estimator (MINE) which is linearly scalable in dimension as well as sample size, and is also trainable and strongly consistent. They also discussed another version of MINE based on the f-divergence representation [33] 20). Using the technique of Coitt-Contr Estimation (NCE) [34] based on comparing target and randomly chosen negative samples, Van den Oord information lower bound. An important application of this contrastive learning approach has been
35 proposed InfoNCE loss, minimization of which maximizes a mutual
extracting high-level representations of different data modalities [36] 37 38 39]. In our work, we extend MINE for transfer entropy estimation.
Attention Mechanisms. The attention mechanisms identify key areas in the data by learning a set of weight distributions. Spatial-based based attention [29] 40 41 involves generating attention scores from spatial regions of feature maps, while channel-basee attention 42 43) optimises the representation of each channel. Self-attention
44 45 encodes interactions among all input entities and cross- attention [46 47 48 introduces the interaction of two domains further. However,
recognised that attention mechanisms need to be tailored to the specific problem at hand [49]. In our work, we tailor attention mechanism for coupling relationship inference by accommodating the selection of temporal regions that correspond to the transient emergence of coupling effect in neural activity sequences.

---

7 Discussion
The problem of reconstructing coupling networks from observational data is fundamental in multiple disciplines of science including neuroscience, since it is a prerequisite foundation for the research about structure analysis and behavior control in coupling networks. Especially, several countries have recently launched grand brain projects, and one important goal is to map the connectomes (i.e., directed links between neurons) of different species.
We draw on coupling attention to guide machine learning models to infer coupling relationships while focusing the specific areas where casual effect may emerge. We show that this mechanism identifies weak coupling effects ignored by classical techniques, and helps machine learning models gain insight into the coupling dynamics underlying time series data.
Acknowledgments and Disclosure of Funding
This work is supported by ST12030-Marjor Project 20212D0204500, National Natural Science Foundation of China (Grants No. T2225022, No. 12161141016 No. 12150410309), Shuguang Program of Shanghai Education Development Foundation and Shanghai Municipal Education Com- mission (Grant No. 22SG21), Shanghai Municipal Science and Technology Major Project (Grant No. 202ISHZDZX0100 Shanghai Municipal Commission of Science and Technology Project (Grant No. 19511132101), and the Fundamental Research Funds for the Central Universities.
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13


Reviews:
[{"review_id": "BPbFT7Yh84", "reviewer": "Reviewer_JWtX", "summary": "This paper proposes a novel method to reconstruct coupled networks from observational time series data, where the coupling effects between variables are sparse and weak. The authors propose a new attention mechanism called coupling attention is introduced to identify critical regions in time series where coupling effects momentarily emerge amidst dominant intrinsic dynamics. The authors design a differentiable objective function called Attentive Transfer Entropy (ATEn) to train the coupling attention model in an unsupervised, data-driven manner.\n\n", "questions": "For the Hindmarsh-Rose experiments, why does it appear as though the child variable is leading the parent variable in the time-series in Figure 4?\n\nWhat is the reason for making the attention and classifier convolutional networks while fθ and fφ are fully-connected networks?\n\nFor clarification, Figure 4 TENE refers to using Equation 5?\n\n", "limitations": "No obvious negative societal impacts. Refer to Weaknesses for constructive suggestions for improvement.", "rating": 7, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "ATEn is well motivated and the use of Mutual Information Neural Estimation as a component to both estimate and optimize attention parameters makes sense.\n\nThe concept of using attention to identify sparse coupling effects in time series appears original for this reviewer and the method leverages the use of neural networks well. The method could likely be extended in the future to other neural architectures like transformers, therefore this work is a good initial foray into further work and improvements.\n\nText is well written and clear and the ablation experiments make sense.\n\n", "weaknesses": "There needs to be more clarity in the experimental setup:\n\nSpecifically, the way overfitting is prevented in the proposed ATEn method. Like Mutual Information Neural Estimation method, parameters are optimized in Transfer Entropy, which can result in perfect estimates of the mutual information (due to overfitting). How have the authors addressed this?\n\nWhat are the inputs to fθ and fφ and how are they different from the inputs to the attention and classifier models? Is there a feature space beyond the scalar timeseries of the neurons?\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the referee for stating that ATEn is well motivated and for offering us constructive suggestions for improvement.\n\nTo Weaknesses:\n1. Preventing overfitting.\n\n   The coupling attention model is trained to maximize ATEn on the train samples. When the ATEn reaches stability (or converge to the optimal value), the downstream classifier guided by it usually does not achieve best generalization (performing well on the training set but poorly on the validation set) in our experiment. Currently, we have not established a definitive criterion for determining the optimal stopping point in training the attention model, at which point the downstream classifier guided by attention model can achieve its best generalization. We address this issue as follow: every $n$ training epochs of attention model, we retrain a new classifier with a few epochs and select the one with best generalization on validation set. After finishing the training of attention model, we further train the selected classifier until convergence.\n\n   In our revision, we will clarify this in Sec.4.1, “Setup”.\n\n2. Inputs to $f_θ$ and $f_φ$.\n\n   We derive the transfer entropy to the difference between two mutual information (MI) as in Eq. 5; $f_θ$ estimates the first MI and $f_φ$ estimates the second MI. Taking a simple example, we have two neural activity sequences $X=[ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}]$ and $Y =[ y_{1}, y_{2}, y_{3}, y_{4}, y_{5}]$ with the shape (5,) respectively, where subscripts represent timesteps.\n\n   $f_θ$ and $f_φ$ : Assuming $k=l=1$ in Equation 7, We can construct a pair of samples for $f_θ$ as $[x_{1}, y_{1}, y_{2}]$ and $[x_{1}, y_{1}, y_{\\text{random}}]$ with the shape (3,) (Similarly, $[x_{2}, y_{2}, y_{3}]$ and $[x_{2}, y_{2}, y_{\\text{random}}]$,… ), where $y_{\\text{random}}$ represents it is sampled from sequences $Y$ randomly. We then put these pair of samples into $f_θ$ respectively. $f_θ$ outputs two values with the shape (1,) and is trained to maximize their difference to approximate the mutual information between $Y_{t+1}$ and $(Y_{t}, X_{t})$. We can construct another pair of samples for $f_φ$ as $[y_{1}, y_{2}]$ and $[y_{1}, y_{\\text{random}}]$ with the shape (2,) (Similarly, $[y_{2}, y_{3}]$ and $[y_{2}, y_{\\text{random}}]$,… ). We then put them into $f_φ$, which outputs two values with the shape (1,) and is trained to maximize their difference to approximate the mutual information between $Y_{t+1}$ and $Y_{t}$.\n\n   Attention model and classifier : We stack two sequences vertically and reshape them into an input sample $[[ [x_{1}], [x_{2}],[x_{3}],[x_{4}],[x_{5}] ], [ [y_{1}],[y_{2}],[y_{3}],[y_{4}],[y_{5}] ]]$ with the shape [1,2,5] (like [channel, height, width] for CNN). The attention model is fed by this sample and outputs an attention coefficient vector $[a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ]$ with shape (5,), the same length as the activity sequences. We get a new sample by dot product of original sample and the coefficient vector $[[ a_{1} * [x_{1}], a_{2} * [x_{2}], a_{3} * [x_{3}], a_{4} * [x_{4}], a_{5} * [x_{5}] ]], [ a_{1} *[y_{1}], a_{2} * [y_{2}], a_{3} * [y_{3}], a_{4} * [y_{4}], a_{5} * [y_{5}] ]]$ with the shape [1,2,5] The classifier is fed by this new sample and outputs a logit vector $[logit_{1}, {logit_{2}, logit_{3}, logit_{4}, logit_{5} ]$ with the shape (5,). We take sigmoid (the maximum value in logit vector) as the possibility of a link existing between these two neurons. \n    \n   In the SI of the revision, we will add a detail description about the architectures of models, encompassing explanations of their input and output.\n\n\nTo Questions:\n1. Fig. 4. For the continuously coupled models which we consider, the parent-child relationship does not correspond to a pre-post relationship. We agree that Fig. 4(c) is regrettably confusing, and in our revision we will update it with a pair of variables with more intuitive traces.\n\n2. Connectivity of ANNs. In our response to Weakness 2 we explained these decisions, and we will also clarify these details in our revision.\n\n3.\tTENE does indeed refer to using Equation 5. In our revision, we state in Sec.4.1, “Transfer Entropy Neural Estimator (TENE), as in Eq. 5.” Also, for clarity, in Eq.5 we will change “TE(X→Y)” to “TENE(X→Y)”.\n"}, {"review_id": "MWfQAy3UJ6", "reviewer": "Reviewer_xCSe", "summary": "The authors extend the MiNE method [23], of using a neural network to estimate mutual information, to estimating transfer entropy between two coupled time series. When the coupling is weak on average due to being intermittent, they propose a differentiable attention mechanism that learns to transiently gate the transfer entropy calculation to only these intermittent moments. They test their method to identify coupling in networks created using simple neuron models connected as per connectivity datasets.", "questions": "Already stated in weaknesses.\n\nMinor:\n- L126: ‘adjusting $a_t$ in Eq. 5’ -> ‘adjusting $a_t$ in Eq. 6’ since Eq. 5 does not contain $a_t$.\n", "limitations": "N/A", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "-\tThe authors combine several new techniques to address an important issue of connectivity determination from neural recordings / time series.", "weaknesses": "-\tThe authors straightforwardly extend the theory of the neural estimation of mutual information to their neural estimation of transfer entropy without attention in Appendix A.. However, they do not say anything about their attention based method. Can it be shown that it is also somehow consistent / convergent?\n-\tIt appears to me that all the ‘connectomes’ apart form the C.Elegans one are between macro brain regions. Yet these have been used to connect individual neuron models. Can the authors not use local connectivity of cortical neurons from say the Blue Brain project and deduce this connectivity?\n-\tUltimately the method is only tested on simulated datasets even though the connectivity is based on macro-connectivity of animal brains. Numerous Ca-imaging or spike-train recordings of hundreds of neurons in various small regions of the brain are available. Yet the authors do not seem to have applied their method to these. Thus it calls into question whether their method is suitable for real-world data.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the referee for noting that our paper addresses an important issue.\n\nTo Weakness:\n\n1. Consistency of ATEn. So far, we cannot theoretically demonstrate that ATEn is consistent in some way. At present, we have only experimentally verified that the attention model can converge to stability when the learning rate is small.\n\n2. Neuron-level connectomes. This is a great point! We follow the Reviewer’s suggestion and complement our previous results with a neuron-level connectome from the fruitfly Drosophila medulla (Ref. [1] below). This connectome has 1781 nodes and 33641 edges. We confirm that our method outperforms the baselines using this new connectome, as shown in the following table. \n\n                      HR          Izh        Rulkov         FHN                               \n       Granger     0.69±0.02   0.42±0.01    0.74±0.01    0.60±0.01           \n       TE kraskov  0.60±0.01   0.47±0.02    0.65±0.01    0.58±0.02    \n       CCM         0.80±0.01   0.42±0.02    0.80±0.01    0.76±0.03                   \n       Latent CCM  0.81±0.02   0.47±0.02    0.76±0.01    0.73±0.01              \n       PCMCI       0.51±0.01   0.51±0.01    0.52±0.01    0.51±0.01                                           \n       Classifier  0.85±0.05   0.65±0.06    0.83±0.06    0.72±0.05               \n       TENE        0.54±0.07   0.51±0.03    0.54±0.11    0.41±0.06            \n       ATEn        0.97±0.03   0.74±0.09    0.87±0.09    0.83±0.10\n\n      [1] L. K. Scheffer, et al. A connectome and analysis of the adult Drosophila central brain, eLife, 9: e57443 (2020).\n\n3. Experimental records of neuron activity. Yes, our ultimate goal is applying our method to infer real neuronal connectivity from the experimental recordings of neuronal activities. Before doing that, we wanted to test the method in various data of both neuronal activities and ground-truth connectivity. Unfortunately, neuroscience experimental data currently contain either neuronal activities or connectivity, and it is rare that they have both for a specific biological brain or region. Regarding Ca+imaging activity data, their sampling frequencies are usually low because multilayers of brain slices should be scanned for a specific 3D brain region. Hence, here we use real neuron-level connectome and use instead various neuronal dynamical models to generate activity data to validate our method. We thank the Reviewer for this insightful comment, and will discuss this (and spike-like discrete activity) in the section of limitations and future work in our revision. \n\n\nTo Questions:\n\nThanks. In our revision, we will change “adjusting a_{t} in Eq.5” to “incorporating a_{t} into Eq.5”.\n"}, {"review_id": "4S7BChTUlw", "reviewer": "Reviewer_wNf4", "summary": "This paper proposes \"attentive transfer entropy\" as a way to infer connectivity of real and synthetic coupling networks. The submission suggests extending a 'neural estimator of mutual information' to transfer entropy and adding \"attention\" to focus on moments in time where the influence is larger.\n", "questions": "* Is the estimate of the transfer entropy assumed to be independent of the time-discretization? If yes, please provide evidence for that (e.g. show that changing dt of the numerical integration of the ODE does only have a very minor effect on the estimate of transfer entropy).\n* How does the proposed method coping with confounders (ie. unobserved/unrecorded neurons)?\n* In the single neuron models in the section 4 there is no noise added, so asymptotically, what is the source of entropy in this system? In other words: If the network dynamics is not chaotic, asymptotically the dynamics is expected to relax onto a fixed point or limit cycle. In both cases of fixed points and limit cycle, the true transfer entropy should be zero (observing Y very long is enough to predict it's future even without knowing X) so the full transfer entropy should be zero. What I am confused about is: Is this setup only quantifying some kind of 'transient' transfer entropy? Or is the approach relying on chaotic network dynamics? If yes, how does changing the intensity of network chaos affect the estimate? Or does the method actually implicitly rely on all nodes receiving independent noise? Is the 'numerical noise' enough? I am not expert on transfer entropy and I might be misunderstanding something here, but some clarification seems necessary.\n* Why is noise added to the data sets? (\"Gaussian measurement noise is added with mean zero and standard deviation 1%/10% that of the original time series for synthetic/real networks respectively.\")\n* What are the units of time in figure 3? Units of membrane time constant?\n* How is this method dealing with identifiability issues, e.g. when different network structures can lead to the same dynamics?\n* Could the proposed method be combined with causal interventions? E.g. can it tell me the neuron/synapse it has the biggest uncertainty about, so I could apply activity perturbations/causal interventions to infer if the neuron/synapse is connected or not?\n* Why is the mean connectivity in the mouse data 1.1 according to table 1? It seems that compared to a real brain, possibly not all existing connections are in the data set. How does this change your conclusions? In other words: Real biological cortical neurons have a mean degree more arround 10^3 to 10^4 (see e.g. Potjans, Diesmann 2014), would your method still work for neural networks with realistic degree? ", "limitations": "Limitations (mentioned above) are already addressed in the paper.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* higher performance than other methods this estimator is compared to.\n* fewer data necessary than other methods this estimator is compared to.\n", "weaknesses": "* The introduction of attention appears to be ad hoc, a mathematical/information theoretic justification of the attention mechanisms seems to be missing. \n* The methods seems a direct extension of the Mutual information neural estimator, the only technical innovation seems to be the attention mechanism which seems to lacks theoretic grounding.\n* The whole theory (including the analytical toy example) considers discrete time dynamics but the methods is then applied to inference of couplings in systems of coupled differential equations. Something seems to be missing here to link the two descriptions (continous time and discrete time).\n\nMinor issues:\n* Table 1 is a bit confusing. The second column is \"brain, brain cortex, neural, brain\". This seems rather unspecific. What is the actual brain region in the cat an macaque data? Why don't you use a full connectome, e.g. of a fly (https://www.science.org/doi/10.1126/science.add9330)\n* Weird notation in first equation. Why is t and index, usually ODEs are written as d x(t) /dt =f(x(t)) and not d x_t /dt=f(x_t)\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the referee for noting the strengths of our method: it provides higher performance and requires fewer data.\n\nTo weakness and issues:\n\n1. Discrete and continuous dynamics: In the analytical example and in the writing, for convenience, we used subscripts $t$ and $t+1$ because both experiment and simulation activity time series contain discrete data points. They actually denote $t + \\delta t$ or $t + 2\\delta t$ where $\\delta t$ represents the time interval of each step. We showed the values of simulation time step $\\delta t$ in Supplementary Section D.5.\n \n2. Fruit-fly connectome: The paper about larva fly recently published in \\it{Science} did not directly offer the connectome data and extracting it from their raw data is time consuming. Fortunately, we extract a connectome of the fruitfly Drosophila medulla from another data (Ref. [1] below), which has 1781 nodes and 33641 edges. The new results also confirm that our method outperforms the baselines, as shown in the table below. We will add this into the revision.\n\n                      HR          Izh        Rulkov         FHN                               \n       Granger     0.69±0.02   0.42±0.01    0.74±0.01    0.60±0.01           \n       TE kraskov  0.60±0.01   0.47±0.02    0.65±0.01    0.58±0.02    \n       CCM         0.80±0.01   0.42±0.02    0.80±0.01    0.76±0.03                   \n       Latent CCM  0.81±0.02   0.47±0.02    0.76±0.01    0.73±0.01              \n       PCMCI       0.51±0.01   0.51±0.01    0.52±0.01    0.51±0.01                                           \n       Classifier  0.85±0.05   0.65±0.06    0.83±0.06    0.72±0.05               \n       TENE        0.54±0.07   0.51±0.03    0.54±0.11    0.41±0.06            \n       ATEn        0.97±0.03   0.74±0.09    0.87±0.09    0.83±0.10\n\n   [1] L. K. Scheffer, et al. A connectome and analysis of the adult Drosophila central brain, eLife, 9: e57443 (2020).\n\n3. Thanks. In the first equation, we personally find the notation $\\dot{x}_t=f(x_t)$ more readable and less intimidating than the notation $\\dot{x}(t) =f(x(t))$, especially taking into consideration the next several math equations (otherwise, too many parentheses). We feel that the notation will not induce ambiguity and hope it is okay to continue using our preferred notation.\n\nTo Questions:\n\n1. Impact of sampling frequency. We checked the effect of sampling frequency on the performance based on the estimates of transfer entropy, please see Figure 3(d). For our work, whether the estimates of transfer entropy themselves are precise is less important than whether estimates of transfer entropy between coupled variables are larger than estimates of transfer entropy between uncoupled variables.\n\n2. Hidden nodes. We investigated the effect of hidden nodes on performance of the proposed method, please see Figure 3(f) which showing that the presence of hidden variables does not significantly affect performance. We consider an important reason is our consideration of the specific case of neuron dynamics in which the coupling-drive force is usually weak while self-drive force is strong. Another factor might be the fact that our inference is based on pairwise transfer entropy rather than a conditional transfer entropy which would require time series from all parent variables of each neuron.\n\n3. Chaotic dynamics. So far, we considered chaotic dynamics because this work focuses on neuronal dynamics which are known chaotic. We thank the Reviewer for this insightful comment which is helpful for us to further extend our work to more general dynamical systems. We will acknowledge this by adding to the limitations “Our method may not apply to systems at fixed points or following limit cycles, for which the transfer entropy is zero.”\n\n4. Measurement noise. We tested the effect of measurement noise intensity on the performance of the method in the section of Ablation study, please see Figure 3(e). \n\n5. Units of t in Fig. 4. The units of the x-axis are sampling periods. In the caption to Fig. 4 in our revision, we will clarify this by changing the sentence “Time t is index in the time series” to “Time t is index in the time series and its unit is the sampling period.”.\n\n6. Identifiability issues. Luckily, we would expect cases in which different structures give rise to exactly the same time series to be very rare for the dynamical systems which we consider. \n\n7. The uncertainty is contingent upon the inferred probability of a causal link, $p \\in [0,1]$, which is derived from our methodology. When $p$ approaches 1 or 0, the method exhibits high certainty regarding a connection. Conversely, when $p$ approximates 0.5, the method experiences maximal uncertainty. Nevertheless, we remain uncertain about the optimal locations for intervention experiments as this may rely on the relative value and difficulty associated with establishing or refuting a link. In our revision we will acknowledge this valuable idea by adding to the Conclusion: “In the future we hope to adapt our method to predict which pairs of neurons would be most valuable to have knowledge of whether or not a link exists, so that our method can be applied together with intervention experiments.”\n\n8. Fruitfly and C. elegans neuron-level connectomes have moderate mean degrees. Connectome data of cat, macaque and mouse brains are not neuron-level but macroscopic connections between brain regions. They are all empirical data. The results of performance vs mean degree have been shown in Figure 3(b). Regarding large-degree neuron-level connectomes (i.e. such as dense networks with mean degree 10^3 to 10^4), we already discussed in the section “Limitations”: “Dense networks, where a variable is coupled with many driving variables and substantial coupling forces can emerge from distinct parents at overlapping times, making individual drivers harder to distinguish (see green line in Figure 3b)”."}, {"review_id": "JWdDtfi4dr", "reviewer": "Reviewer_f3Wr", "summary": "This article expands the concept of transfer entropy to include attentive transfer entropy. Implementing attentive transfer entropy on connectomes of realistic biological networks demonstrated that this extension encompasses the transient coupling effect within the connectivity constraints of realistic biological networks. ", "questions": "1) The Classifier outperforms ATEn in mouse connectome across all dynamics. Does this mean ATEn may not work well in very sparse networks? \n2) In many instances, the transfer entropy without attention mechanism performs significantly worse than both classifier and ATEn. Even the conventional attentional mechanism appears to be of great help. Is it possible to compare or discuss in some details the distinction between classifier and ATEn attention mechanisms in real-world networks? \n3) The authors mentioned in a couple places that the train/validate/test split is 20/100/1000. How many samples are there in total? Is it possible to elaborate more training details? \n", "limitations": "This study employed predetermined dynamics, which is hardly realistic for functioning networks in the real world. The sensitivity to the choice of dynamics may hinder this method's potential utility for analyzing experimental results. \n", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "This paper provides convincing empirical evidence that the attentive transfer entropy is particularly useful in understanding coupling effects of realistic biological networks. They achieve this through first conducting ablation study on toy models of simple connectivity (full connectivity?) and comparing the proposed attentive transfer entropy with multiple commonly used baselines. \n\n", "weaknesses": "It is a bit confusing that the classifier with conventional attentional mechanism works almost as well as the ATEn in many scenarios (e.g., the Izh dynamics in C.elegans) and overall in Mouse connectome. In addition, it seems which dynamics being used to simulate the sequences may also influence the performance, but there was no discussion on that. \n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the referee for highlighting how we provided convincing empirical evidence that the attentive transfer entropy is particularly useful in understanding coupling effects of realistic biological networks.\n\nTo Weakness:\n\n1. Similarity of performance of different classifiers. Both the classifier with conventional attentional mechanism and ATEn do not work very well on the Izh dynamics in almost all networks since the strength of the transient couple force is weakest among all the dynamics used in our study. Please see Figure 4 for the gray lines visually depicting a temporal evolution of coupling force over time (Compared with left and right subplot, the spikes of the gray line in middle subplot are narrow and low). Thus, the couple effect of Izh dynamics is relatively hard to detect with either method, leading to similar performance. \n\n2. The mouse connectome is a sparse network with low mean degree 1.1, indicating each child node has only one parent node approximatively. From the experimental results, the classifier indeed performs better than ATEn in mouse connectome. However, performance of classifier decreases quickly when the mean degree become large. Please see green (ATEn) and blue (Classifier) lines in Figure 3(b). ATEn already outperforms the classifier obviously with small supervised training samples when the mean degree of the synthetic network is 10. \n\n3. The dynamics being used to simulate the sequences does indeed influence the performance of the method due to their different strength and frequency of the transient couple force, which ATEn relies upon.\n\nIn our revision we will include a form of the above discussion. \n\n\nTo Questions:\n\n1. Performance of ATEn for sparse networks. The answer is presented in the reply to Weakness 2.\n\n2. Distinction between classifier and ATEn attention mechanism in real-world networks. Thanks for the insightful comment. Biological neuronal networks function in an integrate-and-fire way, i.e., each neuron receives signals from its neighboring neurons, and it will fire only when the cumulative potential exceeds a threshold. Hence, the signal transferring from one neuron to another may not cause subsequential actions, which is the challenge of inferring directed link between neurons. Indeed, as shown in Figure 4 (pink curves), the classifier with conventional attention mechanism cannot capture such subtle fluctuations. Instead, our ATEn (Figure 4, orange curves) can do so. The main distinction is that conventional attention mechanism is to identify the regions of interest in activity series by maximizing cross-entropy while ATEn is to identify the important regions by maximizing transfer entropy. As we know, transfer entropy is a measure of causality between time series, able to capture the information directly transferring from one neuron to another. Such causation implies directionality of information flow, not just the differences between two series. ATEn is motivated by mimicking the transport of neuronal information, which might be the key reason why ATEn outperforms conventional attention mechanism.\n\n3. Elaboration of training details. Taking a simple example, consider a directed network comprising 20 nodes and 100 edges. There are in total 20 * 19 = 380 ordered pairs of nodes, of which 100 have coupled relationships while the remaining 280 have non-coupling relationships. The information of five real networks used in our study is provided in Table 1. The train/validate/test split is 20/100/1000, meaning that we randomly sample 10/50/500 coupled and 10/50/500 non-coupling relationships respectively from all ordered pairs of the network. We will clarify these important details in our revision. \n\n\nTo Limitations:\n\nElectric signals of biological neurons behave in a similar way of the several neuronal dynamical models, yet current experiment techniques usually measure neuronal activities in an indirect way, such as Ca+ signals. This leads to another problem: the measuring process itself is a nonlinear transformation of neuronal activity. Hence, the whole system should be modelled as: $\\dot{x}_t=f(x_t)$ for neuronal dynamics, and $z_t=h(x_t)$ for the measuring process, where $h(*)$ represents a nonlinear function of measuring. To infer the underlying connectome from $z_t$ rather than $x_t$ is an outstanding problem and is also the future work we continue exploring now.\n"}]

Author rebuttal:

Meta-review: The reviewers unanimously voted to accept the paper. The paper presents a method to identify critical regions of time series data where the coupling effect may manifest. To quote reviewer, f3Wr, "This paper provides convincing empirical evidence that the attentive transfer entropy is particularly useful in understanding coupling effects of realistic biological networks. They achieve this through first conducting ablation study on toy models of simple connectivity and comparing the proposed attentive transfer entropy with multiple commonly used baselines". I agree with the reviewer -- the paper is a good contribution and should be presented either as a spotlight or an oral.
Decision: Accept (spotlight)