Author response:
The following is the authors’ response to the original reviews
Summary of our revisions
(1) We have explained the reason why the untrained RNN with readout (value-weight) learning only could not well learn the simple task: it is because we trained the models continuously across trials with random inter-trial intervals rather than separately for each episodic trial and so it was not trivial for the models to recognize that cue presentation in different trials constitutes a same single state since the activities of untrained RNN upon cue presentation should differ from trial to trial (Line 177-185).
(2) We have shown that dimensionality was higher in the value-RNNs than in the untrained RNN (Fig. 2K,6H).
(3) We have shown that even when distractor cue was introduced, the value-RNNs could learn the task (Fig. 10).
(4) We have shown that extended value-RNNs incorporating excitatory and inhibitory units and conforming to the Dale's law could still learn the tasks (Fig. 9,10-right column).
(5) In the original manuscript, the non-negatively constrained value-RNN showed loose alignment of value-weight and random feedback from the beginning but did not show further alignment over trials. We have clarified its reason and found a way, introducing a slight decay (forgetting), to make further alignment occur (Fig. 8E,F).
(6) We have shown that the value-RNNs could learn the tasks with longer cue-reward delay (Fig. 2M,6J) or action selection (Fig. 11), and found cases where random feedback performed worse than symmetric feedback.
(7) We compared our value-RNNs with e-prop (Bellec et al., 2020, Nat Commun). While e-prop incorporates the effects of changes in RNN weights across distant times through "eligibility trace", our value-RNNs do not. The reason why our models can still learn the tasks with cue-reward delay is considered to be because our models use TD error and TD learning itself, even TD(0) without eligibility trace, is a solution for temporal credit assignment. In fact, TD error-based e-prop was also examined, but for that, result with symmetric feedback, but not with random feedback, was shown (their Fig. 4,5) while for another setup of reward-based e-prop without TD error, result with random feedback was shown (their SuppFig. 5). We have noted these in Line 695-711 (and also partly in Line 96-99).
(8) In the original manuscript, we emphasized only the spatial locality (random rather than symmetric feedback) of our learning rule. But we have now also emphasized the temporal locality (online learning) as it is also crucial for bio-plausibility and critically different from the original value-RNN with BPTT. We also changed the title.
(9) We have realized that our estimation of true state values was invalid (as detailed in page 34 of this document). Effects of this error on performance comparisons were small, but we apologize for this error.
Reviewer #1 (Public review):
Summary:
Can a plastic RNN serve as a basis function for learning to estimate value. In previous work this was shown to be the case, with a similar architecture to that proposed here. The learning rule in previous work was back-prop with an objective function that was the TD error function (delta) squared. Such a learning rule is non-local as the changes in weights within the RNN, and from inputs to the RNN depends on the weights from the RNN to the output, which estimates value. This is non-local, and in addition, these weights themselves change over learning. The main idea in this paper is to examine if replacing the values of these non-local changing weights, used for credit assignment, with random fixed weights can still produce similar results to those obtained with complete bp. This random feedback approach is motivated by a similar approach used for deep feed-forward neural networks.
This work shows that this random feedback in credit assignment performs well but is not as well as the precise gradient-based approach. When more constraints due to biological plausibility are imposed performance degrades. These results are not surprising given previous results on random feedback. This work is incomplete because the delay times used were only a few time steps, and it is not clear how well random feedback would operate with longer delays. Additionally, the examples simulated with a single cue and a single reward are overly simplistic and the field should move beyond these exceptionally simple examples.
Strengths:
• The authors show that random feedback can approximate well a model trained with detailed credit assignment.
• The authors simulate several experiments including some with probabilistic reward schedules and show results similar to those obtained with detailed credit assignments as well as in experiments.
• The paper examines the impact of more biologically realistic learning rules and the results are still quite similar to the detailed back-prop model.
Weaknesses:
*please note that we numbered your public review comments and recommendations for the authors as Pub1 and Rec1 etc so that we can refer to them in our replies to other comments.
Pub1. The authors also show that an untrained RNN does not perform as well as the trained RNN. However, they never explain what they mean by an untrained RNN. It should be clearly explained.
These results are actually surprising. An untrained RNN with enough units and sufficiently large variance of recurrent weights can have a high-dimensionality and generate a complete or nearly complete basis, though not orthonormal (e.g: Rajan&Abbott 2006). It should be possible to use such a basis to learn this simple classical conditioning paradigm. It would be useful to measure the dimensionality of network dynamics, in both trained and untrained RNN's.
We have added an explanation of untrained RNN in Line 144-147:
“As a negative control, we also conducted simulations in which these connections were not updated from initial values, referring to as the case with "untrained (fixed) RNN". Notably, the value weights w (i.e., connection weights from the RNN to the striatal value unit) were still trained in the models with untrained RNN.”
We have also analyzed the dimensionality of network dynamic by calculating the contribution ratios of each principal component of the trajectory of RNN activities. It was revealed that the contribution ratios of later principal components were smaller in the cases with untrained RNN than in the cases with trained value RNN. We have added these results in Fig. 2K and Line 210-220 (for our original models without non-negative constraint):
“In order to examine the dimensionality of RNN dynamics, we conducted principal component analysis (PCA) of the time series (for 1000 trials) of RNN activities and calculated the contribution ratios of PCs in the cases of oVRNNbp, oVRNNrf, and untrained RNN with 20 RNN units. Figure 2K shows a log of contribution ratios of 20 PCs in each case. Compared with the case of untrained RNN, in oVRNNbp and oVRNNrf, initial component(s) had smaller contributions (PC1 (t-test p = 0.00018 in oVRNNbp; p = 0.0058 in oVRNNrf) and PC2 (p = 0.080 in oVRNNbp; p = 0.0026 in oVRNNrf)) while later components had larger contributions (PC3~10,15~20 p < 0.041 in oVRNNbp; PC5~20 p < 0.0017 in oVRNNrf) on average, and this is considered to underlie their superior learning performance. We noticed that late components had larger contributions in oVRNNrf than in oVRNNbp, although these two models with 20 RNN units were comparable in terms of cue~reward state values (Fig. 2J-left).”
and Fig. 6H and Line 412-416 (for our extended models with non-negative constraint):
“Figure 6H shows contribution ratios of PCs of the time series of RNN activities in each model with 20 RNN units. Compared with the cases with naive/shuffled untrained RNN, in oVRNNbp-rev and oVRNNrf-bio, later components had relatively high contributions (PC5~20 p < 1.4×10,sup>−6</sup> (t-test vs naive) or < 0.014 (vs shuffled) in oVRNNbp-rev; PC6~20 p < 2.0×10<sup>−7</sup> (vs naive) or PC7~20 p < 5.9×10<sup>−14</sup> (vs shuffled) in oVRNNrf-bio), explaining their superior value-learning performance.”
Regarding the poor performance of the model with untrained RNN, we would like to add a note. It is sure that untrained RNN with sufficient dimensions should be able to well represent just <10 different states, and state values should be able to be well learned through TD learning regardless of whatever representation is used. However, a difficulty (nontriviality) lies in that because we modeled the tasks in a continuous way, rather than in an episodic way, the activity of untrained RNN upon cue presentation should generally differ from trial to trial. Therefore, it was not trivial for RNN to know that cue presentation in different trials, even after random lengths of inter-trial interval, should constitute a same single state. We have added this note in Line 177-185:
“This inferiority of untrained RNN may sound odd because there were only four states from cue to reward while random RNN with enough units is expected to be able to represent many different states (c.f., [49]) and the effectiveness of training of only the readout weights has been shown in reservoir computing studies [50-53]. However, there was a difficulty stemming from the continuous training across trials (rather than episodic training of separate trials): the activity of untrained RNN upon cue presentation generally differed from trial to trial, and so it is non-trivial that cue presentation in different trials should be regarded as the same single state, even if it could eventually be dealt with at the readout level if the number of units increases.”
The original value RNN study (Hennig et al., 2023, PLoS Comput Biol) also modeled tasks in a continuous way (though using backprop-through-time (BPTT) for training) and their model with untrained RNN also showed considerably larger RPE error than the value RNN even when the number of RNN units was 100 (the maximum number plotted in their Fig. 6A).
Pub2. The impact of the article is limited by using a network with discrete time-steps, and only a small number of time steps from stimulus to reward. What is the length of each time step? If it's on the order of the membrane time constant, then a few time steps are only tens of ms. In the classical conditioning experiments typical delays are of the order to hundreds of milliseconds to seconds. Authors should test if random feedback weights work as well for larger time spans. This can be done by simply using a much larger number of time steps.
In the revised manuscript, we examined the cases in which the cue-reward delay (originally 3 time steps) was elongated to 4, 5, or 6 time-steps. Our online value RNN models with random feedback could still achieve better performance (smaller squared value error) than the models with untrained RNN, although the performance degraded as the cue-reward delay increased. We have added these results in Fig. 2M and Line 223-228 (for our original models without non-negative constraint)
“We further examined the cases with longer cue-reward delays. As shown in Fig. 2M, as the delay increased, the mean squared error of state values (at 3000-th trial) increased, but the relative superiority of oVRNNbp and oVRNNrf over the model with untrained RNN remained to hold, except for cases with small number of RNN units (5) and long delay (5 or 6) (p < 0.0025 in Wilcoxon rank sum test for oVRNNbp or oVRNNrf vs untrained for each number of RNN units for each delay).”
and Fig. 6J and Line 422-429 (for our extended models with non-negative constraint):
“Figure 6J shows the cases with longer cue-reward delays, with default or halved learning rates. As the delay increased, the mean squared error of state values (at 3000-th trial) increased, but the relative superiority of oVRNNbp-rev and oVRNNrf-bio over the models with untrained RNN remained to hold, except for a few cases with 5 RNN units (5 delay oVRNNrf-bio vs shuffled with default learning rate, 6 delay oVRNNrf-bio vs naive or shuffled with halved learning rate) (p < 0.047 in Wilcoxon rank sum test for oVRNNbp-rev or oVRNNrf-bio vs naive or shuffled untrained for each number of RNN units for each delay).”
Also, we have added the note about our assumption and consideration on the time-step that we described in our provisional reply in Line 136-142:
“We assumed that a single RNN unit corresponds to a small population of neurons that intrinsically share inputs and outputs, for genetic or developmental reasons, and the activity of each unit represents the (relative) firing rate of the population. Cortical population activity is suggested to be sustained not only by fast synaptic transmission and spiking but also, even predominantly, by slower synaptic neurochemical dynamics [46] such as short-term facilitation, whose time constant can be around 500 milliseconds [47]. Therefore, we assumed that single time-step of our rate-based (rather than spike-based) model corresponds to 500 milliseconds.”
Pub3. In the section with more biologically constrained learning rules, while the output weights are restricted to only be positive (as well as the random feedback weights), the recurrent weights and weights from input to RNN are still bi-polar and can change signs during learning. Why is the constraint imposed only on the output weights? It seems reasonable that the whole setup will fail if the recurrent weights were only positive as in such a case most neurons will have very similar dynamics, and the network dimensionality would be very low. However, it is possible that only negative weights might work. It is unclear to me how to justify that bipolar weights that change sign are appropriate for the recurrent connections and inappropriate for the output connections. On the other hand, an RNN with excitatory and inhibitory neurons in which weight signs do not change could possibly work.
We examined extended models that incorporated inhibitory and excitatory units and followed Dale's law with certain assumptions, and found that these models could still learn the tasks. We have added these results in Fig. 9 and subsection “4.1 Models with excitatory and inhibitory units” and described the details of the extended models in Line 844-862:
Pub4. Like most papers in the field this work assumes a world composed of a single cue. In the real world there many more cues than rewards, some cues are not associated with any rewards, and some are associated with other rewards or even punishments. In the simplest case, it would be useful to show that this network could actually work if there are additional distractor cues that appear at random either before the CS, or between the CS and US. There are good reasons to believe such distractor cues will be fatal for an untrained RNN, but might work with a trained RNN, either using BPPT or random feedback. Although this assumption is a common flaw in most work in the field, we should no longer ignore these slightly more realistic scenarios.
We examined the performance of the models in a task in which distractor cue randomly appeared. As a result, our model with random feedback, as well as the model with backprop, could still learn the state values much better than the models with untrained RNN. We have added these results in Fig. 10 and subsection “4.2 Task with distractor cue”
Reviewer #1 (Recommendations for the authors):
Detailed comments to authors
Rec1. Are the untrained RNNs discussed in methods? It seems quite good in estimating value but has a strong dopamine response at time of reward. Is nothing trained in the untrained RNN or are the W values trained. Untrained RNN are not bad at estimating value, but not as good as the two other options. It would seem reasonable that an untrained RNN (if I understand what it is) will be sufficient for such simple Pavlovian conditioning paradigms. This is provided that the RNN generates a complete, or nearly complete basis. Random RNN's provided that the random weights are chosen properly can indeed generate a nearly complete basis. Once there is a nearly complete temporal basis, it seems that a powerful enough learning rule will be able to learn the very simple Pavlovian conditioning. Since there are only 3 time-steps from cue to reward, an RNN dimensionality of 3 would be sufficient. A failure to get a good approximation can also arise from the failure of the learning algorithm for the output weights (W).
As we mentioned in our reply to your public comment Pub1 (page 3-5), we have added an explanation of "untrained RNN" (in which the value weights were still learnt) (Line 144-147). We also analyzed the dimensionality of network dynamics by calculating the contribution ratios of principal components of the trajectory of RNN activities, showing that the contribution ratios of later principal components were smaller in the cases with untrained RNN than in the cases with trained value RNN (Fig. 2K/Line 210-220, Fig.6H/Line 412-416). Moreover, also as we mentioned in our reply to your public comment Pub1, we have added a note that even learning of a small number of states was not trivially easy because we considered continuous learning across trials rather than episodic learning of separate trials and thus it was not trivial for the model to know that cue presentation in different trials after random lengths of inter-trial interval should still be regarded as a same single state (Line 177-185).
Rec2. For all cases, it will be useful to estimate the dimensionality of the RNN. Is the dimensionality of the untrained RNN smaller than in the trained cases? If this is the case, this might depend on the choice of the initial random (I assume) recurrent connectivity matrix.
As mentioned above, we have analyzed the dimensionality of the network dynamics, and as you said, the dimensionality of the model with untrained RNN (which was indeed the initial random matrix as you said, as we mentioned above) was on average smaller than the trained value RNN models (Fig. 2K/Line 210-220, Fig.6H/Line 412-416).
Rec3. It is surprising that the error starts increasing for more RNN units above ~15. See discussion. This might indicate a failure to adjust the learning parameters of the network rather than a true and interesting finding.
Thank you very much for this insightful comment. In the original manuscript, we set the learning rate to a fixed value (0.1), without normalization by the squared norm of feature vector (as we mentioned in Line 656-7 of the original manuscript) because we thought such a normalization could not be locally (biologically) implemented. However, we have realized that the lack of normalization resulted in excessively large learning rate when the number of RNN units was large and it could cause instability and error increase as you suggested. Therefore, in the revised manuscript, we have implemented a normalization of learning rate (of value weights) that does not require non-local computations, specifically, division by the number of RNN units. As a result, the error now monotonically decreased, as the number of RNN units increased, in the non-negatively constrained models (Fig. 6E-left) and also largely in the unconstrained model with random feedback, although still not in the unconstrained model with backprop or untrained RNN (Fig. 2J-left)
Rec4. Not numbering equations is a problem. For example, the explanations of feedback alignment (lines 194-206) rely on equations in the methods section which are not numbered. This makes it hard to read these explanations. Indeed, it will also be better to include a detailed derivation of the explanation in these lines in a mathematical appendix. Key equations should be numbered.
We have added numbers to key equations in the Methods, and references to the numbers of corresponding equations in the main text. Detailed derivations are included in the Methods.
Rec5. What is shown in Figure 3C? - an equation will help.
We have added an explanation using equations in the main text (Line 256-259).
Rec6. The explanation of why alignment occurs is not satisfactory, but neither is it in previous work on feedforward networks. The least that should be done though
Regarding why alignment occurs, what remained mysterious (to us) was that in the case of nonnegatively constrained model, while the angle between value weight vector (w) and the random feedback vector (c) was relatively close (loosely aligned) from the beginning, it appeared (as mentioned in the manuscript) that there was no further alignment over trials, despite that the same mechanism for feedback alignment that we derived for the model without non-negative constraint was expected to operate also under the non-negative constraint. We have now clarified the reason for this, and found a way, introduction of slight decay (forgetting) of value weights, by which feedback alignment came to occur in the non-negatively constraint model. We have added these in the revised manuscript (Line 463-477):
“As mentioned above, while the angle between w and c was on average smaller than 90° from the beginning, there was no further alignment over trials. This seemed mysterious because the mechanism for feedback alignment that we derived for the models without non-negative constraint was expected to work also for the models with non-negative constraint. As a possible reason for the non-occurrence of feedback alignment, we guessed that one or a few element(s) of w grew prominently during learning, and so w became close to an edge or boundary of the non-negative quadrant and thereby angle between w and other vector became generally large (as illustrated in Fig. 8D). Figure 8Ea shows the mean±SEM of the elements of w ordered from the largest to smallest ones after 1500 trials. As conjectured above, a few elements indeed grew prominently.
We considered that if a slight decay (forgetting) of value weights (c.f., [59-61]) was assumed, such a prominent growth of a few elements of w may be mitigated and alignment of w to c, beyond the initial loose alignment because of the non-negative constraint, may occur. These conjectures were indeed confirmed by simulations (Fig. 8Eb,c and Fig. 8F). The mean squared value error slightly increased when the value-weightdecay was assumed (Fig. 8G), however, presumably reflecting a decrease in developed values and a deterioration of learning because of the decay.”
Rec7. I don't understand the qualitative difference between 4G and 4H. The difference seems to be smaller but there is still an apparent difference. Can this be quantified?
We have added pointers indicating which were compared and statistical significance on Fig. 4D-H, and also Fig. 7 and Fig. 9C.
Rec8. More biologically realistic constraints.
Are the weights allowed to become negative? - No.
Figure 6C - untrained RNN with non-negative x_i. Again - it was not explained what untrained RNN is. However, given my previous assumption, this is probably because the units developed in an untrained RNN is much further from representing a complete basis function. This cannot be done with only positive values. It would be useful to see network dynamics of units for untrained RNN. It might also be useful in all cases to estimate the dimensionality of the RNN. For 3 time-steps, it needs to be at least 3, and for more time steps as in Figure 4, larger.
As we mentioned in our reply to your public comment Pub3 (page 6-8), in the revised manuscript we examined models that incorporated inhibitory and excitatory units and followed Dale's law, which could still learn the tasks (Fig. 9, Line 479-520). We have also analyzed the dimensionality of network dynamics as we mentioned in our replies to your public comment Pub1 and recommendations Rec1 and Rec2.
Rec9. A new type of untrained RNN is introduced (Fig 6D) this is the first time an explanation of of the untrained RNN is given. Indeed, the dimensionality of the second type of untrained RNN should be similar to the bioVRNNrf. The results are still not good.
In the model with the new type of untrained RNN whose elements were shuffled from trained bioVRNNrf, contribution ratios of later principal components of the trajectory of RNN activities (Fig. 6H gray dotted line) were indeed larger than those in the model with native untrained RNN (gray solid line) but still much smaller than those in the trained value RNN models with backprop (red line) or random feedback (blue line). It is considered that in value RNN, RNN connections were trained to realize high-dimensional trajectory, and shuffling did not generally preserve such an ability.
Rec10. The discussion is too long and verbose. This is not a review paper.
We have made the original discussion much more compact (from 1686 words to 940 words). We have added new discussion, in response to the review comments, but the total length remains to be shorter than before (1589 words).
Reviewer #2 (Public review):
Summary:
Tsurumi et al. show that recurrent neural networks can learn state and value representations in simple reinforcement learning tasks when trained with random feedback weights. The traditional method of learning for recurrent network in such tasks (backpropagation through time) requires feedback weights which are a transposed copy of the feed-forward weights, a biologically implausible assumption. This manuscript builds on previous work regarding "random feedback alignment" and "value-RNNs", and extends them to a reinforcement learning context. The authors also demonstrate that certain nonnegative constraints can enforce a "loose alignment" of feedback weights. The author's results suggest that random feedback may be a powerful tool of learning in biological networks, even in reinforcement learning tasks.
Strengths:
The authors describe well the issues regarding biologically plausible learning in recurrent networks and in reinforcement learning tasks. They take care to propose networks which might be implemented in biological systems and compare their proposed learning rules to those already existing in literature. Further, they use small networks on relatively simple tasks, which allows for easier intuition into the learning dynamics.
Weaknesses:
The principles discovered by the authors in these smaller networks are not applied to deeper networks or more complicated tasks, so it remains unclear to what degree these methods can scale up, or can be used more generally.
We have examined extended models that incorporated inhibitory and excitatory units and followed Dale's law with certain assumptions, and found that these models could still learn the tasks. We have added these results in Fig. 9 and subsection “4.1 Models with excitatory and inhibitory units”.
We have also examined the performance of the models in a task in which distractor cue randomly appeared, finding that our models could still learn the state values much better than the models with untrained RNN. We have added these result in Fig. 10 and subsection “4.2 Task with distractor cue”.
Regarding the depth, we continue to think about it but have not yet come up with concrete ideas.
Reviewer #2 (Recommendations for the authors):
(1) I think the work would greatly benefit from more proofreading. There are language errors/oddities throughout the paper, I will list just a few examples from the introduction:
Thank you for pointing this out. We have made revisions throughout the paper.
line 63: "simultaneously learnt in the downstream of RNN". Simultaneously learnt in networks downstream of the RNN? Simulatenously learn in a downstream RNN? The meaning is not clear in the original sentence.
We have revised it to "simultaneously learnt in connections downstream of the RNN" (Line 67-68).
starting in line 65: " A major problem, among others.... value-encoding unit" is a run-on sentence and would more readable if split into multiple sentences.
We have extensively revised this part, which now consists of short sentences (Line 70-75).
line 77: "in supervised learning of feed-forward network" should be either "in supervised learning of a feed-forward network" or "in supervised learning of feed-forward networks".
We have changed "feed-forward network" to "feed-forward networks" (Line 83).
(2) Under what conditions can you use an online learning rule which only considers the influence of the previous timestep? It's not clear to me how your networks solve the temporal credit assignment problem when the cue-reward delay in your tasks is 3-5ish time steps. How far can you stretch this delay before your networks stop learning correctly because of this one-step assumption? Further, how much does feedback alignment constrain your ability to learn long timescales, such as in Murray, J.M. (2019)?
The reason why our models can solve the temporal credit assignment problem at least to a certain extent is considered to be because temporal-difference (TD) learning, which we adopted, itself has a power to resolve temporal credit assignment, as exemplified in that TD(0) algorithms without eligibility trance can still learn the value of distant rewards. We have added a discussion on this in Line 702-705:
“…our models do not have "eligibility trace" (nor memorable/gated unit, different from the original value-RNN [26]), but could still solve temporal credit assignment to a certain extent because TD learning is by itself a solution for it (notably, recent work showed that combination of TD(0) and model-based RL well explained rat's choice and DA patterns [132]).”
We have also examined the cases in which the cue-reward delay (originally 3 time steps) was elongated to 4, 5, or 6 time-steps, and our models with random feedback could still achieve better performance than the models with untrained RNN although the performance degraded as the cue-reward delay increased. We have added these results in Fig. 2M and Line 223-228 (for our original models without non-negative constraint)
“We further examined the cases with longer cue-reward delays. As shown in Fig. 2M, as the delay increased, the mean squared error of state values (at 3000-th trial) increased, but the relative superiority of oVRNNbp and oVRNNrf over the model with untrained RNN remained to hold, except for cases with small number of RNN units (5) and long delay (5 or 6) (p < 0.0025 in Wilcoxon rank sum test for oVRNNbp or oVRNNrf vs untrained for each number of RNN units for each delay).”
and Fig. 6J and Line 422-429 (for our extended models with non-negative constraint):
“Figure 6J shows the cases with longer cue-reward delays, with default or halved learning rates. As the delay increased, the mean squared error of state values (at 3000-th trial) increased, but the relative superiority of oVRNNbp-rev and oVRNNrf-bio over the models with untrained RNN remained to hold, except for a few cases with 5 RNN units (5 delay oVRNNrf-bio vs shuffled with default learning rate, 6 delay oVRNNrf-bio vs naive or shuffled with halved learning rate) (p < 0.047 in Wilcoxon rank sum test for oVRNNbp-rev or oVRNNrf-bio vs naive or shuffled untrained for each number of RNN units for each delay).”
As for the difficulty due to random feedback compared to backprop, there appeared to be little difference in the models without non-negative constraint (Fig. 2M), whereas in the models with nonnegative constraint, when the cue-reward delay was elongated to 6 time-steps, the model with random feedback performed worse than the model with backprop (Fig. 6J bottom-left panel).
(3) Line 150: Were the RNN methods trained with continuation between trials?
Yes, we have added
“The oVRNN models, and the model with untrained RNN, were continuously trained across trials in each task, because we considered that it was ecologically more plausible than episodic training of separate trials.” in Line 147-150. This is considered to make learning of even the simple cue-reward association task nontrivial, as we describe in our reply to your comment 9 below.
(4) Figure 2I, J: indicate the statistical significance of the difference between the three methods for each of these measures.
We have added statistical information for Fig. 2J (Line 198-203):
“As shown in the left panel of Fig. 2J, on average across simulations, oVRNNbp and oVRNNrf exhibited largely comparable performance and always outperformed the untrained RNN (p < 0.00022 in Wilcoxon rank sum test for oVRNNbp or oVRNNrf vs untrained for each number of RNN units), although oVRNNbp somewhat outperformed or underperformed oVRNNrf when the number of RNN units was small (≤10 (p < 0.049)) or large (≥25 (p < 0.045)), respectively.”
and also Fig. 6E (for non-negative models) (Line 385-390):
“As shown in the left panel of Fig. 6E, oVRNNbp-rev and oVRNNrf-bio exhibited largely comparable performance and always outperformed the models with untrained RNN (p < 2.5×10<sup>−12</sup> in Wilcoxon rank sum test for oVRNNbp-rev or oVRNNrf-bio vs naive or shuffled untrained for each number of RNN units), although oVRNNbp-rev somewhat outperformed or underperformed oVRNNrf-bio when the number of RNN units was small (≤10 (p < 0.00029)) or large (≥25 (p < 3.7×10<sup>−6</sup>)), respectively…”
Fig. 2I shows distributions, whose means are plotted in Fig. 2J, and we did not add statistics to Fig. 2I itself.
(5) Line 178: Has learning reached a steady state after 1000 trials for each of these networks? Can you show a plot of error vs. trial number?
We have added a plot of error vs trial number for original models (Fig. 2L, Line 221-223):
“We examined how learning proceeded across trials in the models with 20 RNN units. As shown in Fig. 2L, learning became largely converged by 1000-th trial, although slight improvement continued afterward.”
and non-negatively constrained models (Fig. 6I, Line 417-422):
“Figure 6I shows how learning proceeded across trials in the models with 20 RNN units. While oVRNNbp-rev and oVRNNrf-bio eventually reached a comparable level of errors, oVRNNrf-bio outperformed oVRNNbp-rev in early trials (at 200, 300, 400, or 500 trials; p < 0.049 in Wilcoxon rank sum test for each). This is presumably because the value weights did not develop well in early trials and so the backprop-type feedback, which was the same as the value weights, did not work well, while the non-negative fixed random feedback worked finely from the beginning.”
As shown in these figures, learning became largely steady at 1000 trials, but still slightly continued, and we have added simulations with 3000 trials (Fig. 2M and Fig. 6J).
(6) Line 191: Put these regression values in the figure caption, as well as on the plot in Figure 3B.
We have added the regression values in Fig. 3B and its caption.
(7) Line 199: This idea of being in the same quadrant is interesting, but I think the term "relatively close angle" is too vague. Is there another more quantatative way to describe this what you mean by this?
We have revised this (Line 252-254) to “a vector that is in a relatively close angle with c , or more specifically, is in the same quadrant as (and thus within at maximum 90° from) c (for example, [c<sub>1</sub> c<sub>2</sub> c<sub>3</sub>]<sup>T</sup> and [0.5c<sub>1</sub> 1.2c<sub>2</sub> 0.8c<sub>3</sub>]T) “
(8) Line 275: I'd like to see this measure directly in a plot, along with the statistical significance.
We have added pointers indicating which were compared and statistical significance on Fig. 4D-H, and also Fig. 7 and Fig. 9C.
(9) Line 280: Surely the untrained RNN should be able to solve the task if the reservoir is big enough, no? Maybe much bigger than 50 units, but still.
We think this is not sure. A difficulty lies in that because we modeled the tasks in a continuous way rather than in an episodic way (as we mentioned in our reply to your comment 3), the activity of untrained RNN upon cue presentation should generally differ from trial to trial. Therefore, it was not trivial for RNN to know that cue presentation in different trials, even after random lengths of inter-trial interval, should constitute a same single state. We have added this note in Line 177-185:
“This inferiority of untrained RNN may sound odd because there were only four states from cue to reward while random RNN with enough units is expected to be able to represent many different states (c.f., [49]) and the effectiveness of training of only the readout weights has been shown in reservoir computing studies [50-53]. However, there was a difficulty stemming from the continuous training across trials (rather than episodic training of separate trials): the activity of untrained RNN upon cue presentation generally differed from trial to trial, and so it is non-trivial that cue presentation in different trials should be regarded as the same single state, even if it could eventually be dealt with at the readout level if the number of units increases.”
The original value RNN study (Hennig et al., 2023, PLoS Comput Biol) also modeled tasks in a continuous way (though using BPTT for training) and their model with untrained RNN also showed considerably larger RPE error than the value RNN even when the number of RNN units was 100 (the maximum number plotted in their Fig. 6A).
(10) It's a bit confusing to compare Figure 4C to Figure 4D-H because there are also many features of D-H which do not match those of C (response to cue, response to late reward in task 1). It would make sense to address this in some way. Is there another way to calculate the true values of the states (e.g., maybe you only start from the time of the cue) which better approximates what the networks are doing?
As we mentioned in our replies to your comments 3 and 9, our models with RNN were trained continuously across trials rather than separately for each episodic trial, and whether the models could still learn the state representation is a key issue. Therefore, starting learning from the time of cue would not be an appropriate way to compare the models, and instead we have made statistical comparison regarding key features, specifically, TD-RPEs at early and late rewards, as indicated in Fig. 4D-H.
(11) Line 309: Can you explain why this non-monotic feature exists? Why do you believe it would be more biologically plausible to assume monotonic dependence? It doesn't seem so straightforward to me, I can imagine that competing LTP/LTD mechanisms may produce plasticity which would have a non-monotic dependence on post-synaptic activity.
Thank you for this insightful comment. As you suggested, non-monotonic dependence on the postsynaptic activity (BCM rule) has been proposed for unsupervised learning (cortical self-organization) (Bienenstock et al., 1982 J Neurosci), and there were suggestions that triplet-based STDP could be reduced to a BCM-like rule and additional components (Gjorgjieva et al., 2011 PNAS; Shouval, 2011 PNAS). However, the non-monotonicity appeared in our model, derived from the backprop rule, is maximized at the middle and thus opposite from the BCM rule, which is minimized at the middle (i.e., initially decrease and thereafter increase). Therefore we consider that such an increase-then-decreasetype non-monotonicity would be less plausible than a monotonic increase, which could approximate an extreme case (with a minimum dip) of the BCM rule. We have added a note on this point in Line 355-358:
“…the dependence on the post-synaptic activity was non-monotonic, maximized at the middle of the range of activity. It would be more biologically plausible to assume a monotonic increase (while an opposite shape of nonmonotonicity, once decrease and thereafter increase, called the BCM (Bienenstock-Cooper-Munro) rule has actually been suggested [56-58]).”
(12) Line 363: This is the most exciting part of the paper (for me). I want to learn way more about this! Don't hide this in a few sentences. I want to know all about loose vs. feedback alignment. Show visualizations in 3D space of the idea of loose alignment (starting in the same quadrant), and compare it to how feedback alignment develops (ending in the same quadrant). Does this "loose" alignment idea give us an idea why the random feedback seems to settle at 45 degree angle? it just needs to get the signs right (same quadrant) for each element?
In reply to this encouraging comment, we have made further analyses of the loose alignment. By the term "loose alignment", we meant that the value weight vector w and the feedback vector c are in the same (non-negative) quadrant, as you said. But what remained mysterious (to us) was while the angle between w and c was relatively close (loosely aligned) from the beginning, it appeared (as mentioned in the manuscript) that there was no further alignment over trials (and the angle actually settled at somewhat larger than 45°), despite that the same mechanism for feedback alignment that we derived for the model without non-negative constraint was expected to operate also under the nonnegative constraint. We have now clarified the reason for this, and found a way, introduction of slight decay (forgetting) of value weights, by which feedback alignment came to occur in the non-negatively constraint model. We have added this in Line 463-477:
“As mentioned above, while the angle between w and c was on average smaller than 90° from the beginning, there was no further alignment over trials. This seemed mysterious because the mechanism for feedback alignment that we derived for the models without non-negative constraint was expected to work also for the models with non-negative constraint. As a possible reason for the non-occurrence of feedback alignment, we guessed that one or a few element(s) of w grew prominently during learning, and so w became close to an edge or boundary of the non-negative quadrant and thereby angle between w and other vector became generally large (as illustrated in Fig. 8D). Figure 8Ea shows the mean±SEM of the elements of w ordered from the largest to smallest ones after 1500 trials. As conjectured above, a few elements indeed grew prominently.
We considered that if a slight decay (forgetting) of value weights (c.f., [59-61]) was assumed, such a prominent growth of a few elements of w may be mitigated and alignment of w to c, beyond the initial loose alignment because of the non-negative constraint, may occur. These conjectures were indeed confirmed by simulations (Fig. 8Eb,c and Fig. 8F). The mean squared value error slightly increased when the value-weightdecay was assumed (Fig. 8G), however, presumably reflecting a decrease in developed values and a deterioration of learning because of the decay.”
As for visualization, because the model's dimension was high such as 12, we could not come up with better ways of visualization than the trial versus angle plot (Fig. 3A, 8A,F). Nevertheless, we would expect that the abovementioned additional analyses of loose alignment (with graphs) are useful to understand what are going on.
(13) Line 426: how does this compare to some of the reward modulated hebbian rules proposed in other RNNs? See Hoerzer, G. M., Legenstein, R., & Maass, W. (2014). Put another way, you arrived at this from a top-down approach (gradient descent->BP->approximated by RF->non-negativity constraint>leads to DA dependent modulation of Hebbian plasticity). How might this compare to a bottom up approach (i.e. starting from the principle of Hebbian learning, and adding in reward modulation)
The study of Hoerzer et al. 2014 used a stochastic perturbation, which we did not assume but can potentially be integrated. On the other hand, Hoerzer et al. trained the readout of untrained RNN, whereas we trained both RNN and its readout. We have added discussion to compare our model with Hoerzer et al. and other works that also used perturbation methods, as well as other top-down approximation method, in Line 685-711 (reference 128 is Hoerzer et al. 2014 Cereb Cortex):
“As an alternative to backprop in hierarchical network, aside from feedback alignment [36], Associative Reward-Penalty (A<sub>R-P</sub>) algorithm has been proposed [124-126]. In A<sub>R-P</sub>, the hidden units behave stochastically, allowing the gradient to be estimated via stochastic sampling. Recent work [127] has proposed Phaseless Alignment Learning (PAL), in which high-frequency noise-induced learning of feedback projections proceeds simultaneously with learning of forward projections using the feedback in a lower frequency. Noise-induced learning of the weights on readout neurons from untrained RNN by reward-modulated Hebbian plasticity has also been demonstrated [128]. Such noise- or perturbation-based [40] mechanisms are biologically plausible because neurons and neural networks can exhibit noisy or chaotic behavior [129-131], and might improve the performance of value-RNN if implemented.
Regarding learning of RNN, "e-prop" [35] was proposed as a locally learnable online approximation of BPTT [27], which was used in the original value RNN 26. In e-prop, neuron-specific learning signal is combined with weight-specific locally-updatable "eligibility trace". Reward-based e-prop was also shown to work [35], both in a setup not introducing TD-RPE with symmetric or random feedback (their Supplementary Figure 5) and in another setup introducing TD-RPE with symmetric feedback (their Figure 4 and 5). Compared to these, our models differ in multiple ways.
First, we have shown that alignment to random feedback occurs in the models driven by TD-RPE. Second, our models do not have "eligibility trace" (nor memorable/gated unit, different from the original valueRNN [26]), but could still solve temporal credit assignment to a certain extent because TD learning is by itself a solution for it (notably, recent work showed that combination of TD(0) and model-based RL well explained rat's choice and DA patterns [132]). However, as mentioned before, single time-step in our models was assumed to correspond to hundreds of milliseconds, incorporating slow synaptic dynamics, whereas e-prop is an algorithm for spiking neuron models with a much finer time scale. From this aspect, our models could be seen as a coarsetime-scale approximation of e-prop. On top of these, our results point to a potential computational benefit of biological non-negative constraint, which could effectively limit the parameter space and promote learning.”
Related to your latter point (and also replying to other reviewer's comment), we also examined the cases where the random feedback in our model was replaced with uniform feedback, which corresponds to a simple bottom-up reward-modulated triplet plasticity rule. As a result, the model with uniform feedback showed largely comparable, but somewhat worse, performance than the model with random feedback. We have added the results in Fig. 2J-right and Line 206-209 (for our original models without non-negative constraint):
“The green line in Fig. 2J-right shows the performance of a special case where the random feedback in oVRNNrf was fixed to the direction of (1, 1, ..., 1)<sup>T</sup> (i.e., uniform feedback) with a random coefficient, which was largely comparable to, but somewhat worse than, that for the general oVRNNrf (blue line).”
and Fig. 6E-right and Line 402-407 (for our extended models with non-negative constraint):
“The green and light blue lines in the right panels of Figure 6E and Figure 6F show the results for special cases where the random feedback in oVRNNrf-bio was fixed to the direction of (1, 1, ..., 1) <sup>T</sup> (i.e., uniform feedback) with a random non-negative magnitude (green line) or a fixed magnitude of 0.5 (light blue line). The performance of these special cases, especially the former (with random magnitude) was somewhat worse than that of oVRNNrf-bio, but still better than that of the models with untrained RNN. and also added a biological implication of the results in Line 644-652:
We have shown that oVRNNrf and oVRNNrf-bio could work even when the random feedback was uniform, i.e., fixed to the direction of (1, 1, ..., 1) <sup>T</sup>, although the performance was somewhat worse. This is reasonable because uniform feedback can still encode scalar TD-RPE that drives our models, in contrast to a previous study [45], which considered DA's encoding of vector error and thus regarded uniform feedback as a negative control. If oVRNNrf/oVRNNrf-bio-like mechanism indeed operates in the brain and the feedback is near uniform, alignment of the value weights w to near (1, 1, ..., 1) is expected to occur. This means that states are (learned to be) represented in such a way that simple summation of cortical neuronal activity approximates value, thereby potentially explaining why value is often correlated with regional activation (fMRI BOLD signal) of cortical regions [113].”
Reviewer #3 (Public review):
Summary:
The paper studies learning rules in a simple sigmoidal recurrent neural network setting. The recurrent network has a single layer of 10 to 40 units. It is first confirmed that feedback alignment (FA) can learn a value function in this setting. Then so-called bio-plausible constraints are added: (1) when value weights (readout) is non-negative, (2) when the activity is non-negative (normal sigmoid rather than downscaled between -0.5 and 0.5), (3) when the feedback weights are non-negative, (4) when the learning rule is revised to be monotic: the weights are not downregulated. In the simple task considered all four biological features do not appear to impair totally the learning.
Strengths:
(1) The learning rules are implemented in a low-level fashion of the form: (pre-synaptic-activity) x (post-synaptic-activity) x feedback x RPE. Which is therefore interpretable in terms of measurable quantities in the wet-lab.
(2) I find that non-negative FA (FA with non negative c and w) is the most valuable theoretical insight of this paper: I understand why the alignment between w and c is automatically better at initialization.
(3) The task choice is relevant since it connects with experimental settings of reward conditioning with possible plasticity measurements.
Weaknesses:
(4) The task is rather easy, so it's not clear that it really captures the computational gap that exists with FA (gradient-like learning) and simpler learning rule like a delta rule: RPE x (pre-synpatic) x (postsynaptic). To control if the task is not too trivial, I suggest adding a control where the vector c is constant c_i=1.
We have examined the cases where the feedback was uniform, i.e., in the direction of (1, 1, ..., 1) in both models without and with non-negative constraint. In both models, the models with uniform feedback performed somewhat worse than the original models with random feedback, but still better than the models with untrained RNN. We have added the results in Fig. 2J-right and Line 206-209 (for our original models without non-negative constraint):
“The green line in Fig. 2J-right shows the performance of a special case where the random feedback in oVRNNrf was fixed to the direction of (1, 1, ..., 1) <sup>T</sup> (i.e., uniform feedback) with a random coefficient, which was largely comparable to, but somewhat worse than, that for the general oVRNNrf (blue line).”
and Fig. 6E-right and Line 402-407 (for our extended models with non-negative constraint):
“The green and light blue lines in the right panels of Figure 6E and Figure 6F show the results for special cases where the random feedback in oVRNNrf-bio was fixed to the direction of (1, 1, ..., 1) <sup>T</sup> (i.e., uniform feedback) with a random non-negative magnitude (green line) or a fixed magnitude of 0.5 (light blue line). The performance of these special cases, especially the former (with random magnitude) was somewhat worse than that of oVRNNrf-bio, but still better than that of the models with untrained RNN.”
We have also added a discussion on the biological implication of the model with uniform feedback mentioned in our provisional reply in Line 644-652:
“We have shown that oVRNNrf and oVRNNrf-bio could work even when the random feedback was uniform, i.e., fixed to the direction of (1, 1, ..., 1) <sup>T</sup>, although the performance was somewhat worse. This is reasonable because uniform feedback can still encode scalar TD-RPE that drives our models, in contrast to a previous study [45], which considered DA's encoding of vector error and thus regarded uniform feedback as a negative control. If oVRNNrf/oVRNNrf-bio-like mechanism indeed operates in the brain and the feedback is near uniform, alignment of the value weights w to near (1, 1, ..., 1) is expected to occur. This means that states are (learned to be) represented in such a way that simple summation of cortical neuronal activity approximates value, thereby potentially explaining why value is often correlated with regional activation (fMRI BOLD signal) of cortical regions [113].”
In addition, while preparing the revised manuscript, we found a recent simulation study, which showed that uniform feedback coupled with positive forward weights was effective in supervised learning of one-dimensional output in feed-forward network (Konishi et al., 2023, Front Neurosci).
We have briefly discussed this work in Line 653-655:
“Notably, uniform feedback coupled with positive forward weights was shown to be effective also in supervised learning of one-dimensional output in feed-forward network [114], and we guess that loose alignment may underlie it.”
(5) Related to point 3), the main strength of this paper is to draw potential connection with experimental data. It would be good to highlight more concretely the prediction of the theory for experimental findings. (Ideally, what should be observed with non-negative FA that is not expected with FA or a delta rule (constant global feedback) ?).
We have added a discussion on the prediction of our models, mentioned in our provisional reply, in Line 627-638:
“oVRNNrf predicts that the feedback vector c and the value-weight vector w become gradually aligned, while oVRNNrf-bio predicts that c and w are loosely aligned from the beginning. Element of c could be measured as the magnitude of pyramidal cell's response to DA stimulation. Element of w corresponding to a given pyramidal cell could be measured, if striatal neuron that receives input from that pyramidal cell can be identified (although technically demanding), as the magnitude of response of the striatal neuron to activation of the pyramidal cell. Then, the abovementioned predictions could be tested by (i) identify cortical, striatal, and VTA regions that are connected, (ii) identify pairs of cortical pyramidal cells and striatal neurons that are connected, (iii) measure the responses of identified pyramidal cells to DA stimulation, as well as the responses of identified striatal neurons to activation of the connected pyramidal cells, and (iv) test whether DA→pyramidal responses and pyramidal→striatal responses are associated across pyramidal cells, and whether such associations develop through learning.”
Moreover, we have considered another (technically more doable) prediction of our model, and described it in Line 639-643:
“Testing this prediction, however, would be technically quite demanding, as mentioned above. An alternative way of testing our model is to manipulate the cortical DA feedback and see if it will cause (re-)alignment of value weights (i.e., cortical striatal strengths). Specifically, our model predicts that if DA projection to a particular cortical locus is silenced, effect of the activity of that locus on the value-encoding striatal activity will become diminished.”
(6a) Random feedback with RNN in RL have been studied in the past, so it is maybe worth giving some insights how the results and the analyzes compare to this previous line of work (for instance in this paper [1]). For instance, I am not very surprised that FA also works for value prediction with TD error. It is also expected from the literature that the RL + RNN + FA setting would scale to tasks that are more complex than the conditioning problem proposed here, so is there a more specific take-home message about non-negative FA? or benefits from this simpler toy task? [1] https://www.nature.com/articles/s41467-020-17236-y
As for a specific feature of non-negative models, we did not describe (actually did not well recognize) an intriguing result that the non-negative random feedback model performed generally better than the models without non-negative constraint with either backprop or random feedback (Fig. 2J-left versus Fig. 6E-left (please mind the difference in the vertical scales)). This suggests that the non-negative constraint effectively limited the parameter space and thereby learning became efficient. We have added this result in Line 392-395:
“Remarkably, oVRNNrf-bio generally achieved better performance than both oVRNNbp and oVRNNrf, which did not have the non-negative constraint (Wilcoxon rank sum test, vs oVRNNbp : p < 7.8×10,sup>−6</sup> for 5 or ≥25 RNN units; vs oVRNNrf: p < 0.021 for ≤10 or ≥20 RNN units).”
Also, in the models with non-negative constraint, the model with random feedback learned more rapidly than the model with backprop although they eventually reached a comparable level of errors, at least in the case with 20 RNN units. This is presumably because the value weights did not develop well in early trials and so the backprop-based feedback, which was the same as the value weights, did not work well, while the non-negative fixed random feedback worked finely from the beginning. We have added this result in Fig. 6I and Line 417-422:
“Figure 6I shows how learning proceeded across trials in the models with 20 RNN units. While oVRNNbp-rev and oVRNNrf-bio eventually reached a comparable level of errors, oVRNNrf-bio outperformed oVRNNbp-rev in early trials (at 200, 300, 400, or 500 trials; p < 0.049 in Wilcoxon rank sum test for each). This is presumably because the value weights did not develop well in early trials and so the backprop-type feedback, which was the same as the value weights, did not work well, while the non-negative fixed random feedback worked finely from the beginning.”
We have also added a discussion on how our model can be positioned in relation to other models including the study you mentioned (e-prop by Bellec, ..., Maass, 2020) in subsection “Comparison to other algorithms” of the Discussion):
Regarding the slightly better performance of the non-negative model with random feedback than that of the non-negative model with backprop when the number of RNN units was large (mentioned in our provisional reply), state values in the backprop model appeared underdeveloped than those in the random feedback model. Slightly better performance of random feedback than backprop held also in our extended model incorporating excitatory and inhibitory units (Fig. 9B).
(6b) Related to task complexity, it is not clear to me if non-negative value and feedback weights would generally scale to harder tasks. If the task in so simple that a global RPE signal is sufficient to learn (see 4 and 5), then it could be good to extend the task to find a substantial gap between: global RPE, non-negative FA, FA, BP. For a well chosen task, I expect to see a performance gap between any pair of these four learning rules. In the context of the present paper, this would be particularly interesting to study the failure mode of non-negative FA and the cases where it does perform as well as FA.
In the cue-reward association task with 3 time-steps delay, the non-negative model with random feedback performed largely comparably to the non-negative model with backprop, and this remained to hold in a task where distractor cue, which was not associated with reward, appeared in random timings. We have added the results in Fig. 10 and subsection “4.2 Task with distractor cue”.
We have also examined the cases where the cue-reward delay was elongated. In the case of longer cue-reward delay (6 time-steps), in the models without non-negative constraint, the model with random feedback performed comparably to (and slightly better than when the number of RNN units was large) the model with backprop (Fig. 2M). In contrast, in the models with non-negative constraint, the model with random feedback underperformed the model with backprop (Fig. 6J, left-bottom). This indicates a difference between the effect of non-negative random feedback and the effect of positive+negative random feedback.
We have further examined the performance of the models in terms of action selection, by extending the models to incorporate an actor-critic algorithm. In a task with inter-temporal choice (i.e., immediate small reward vs delayed large reward), the non-negative model with random feedback performed worse than the non-negative model with backprop when the number of RNN units was small. When the number of RNN increased, these models performed more comparably. These results are described in Fig. 11 and subsection “4.3 Incorporation of action selection”.
(7) I find that the writing could be improved, it mostly feels more technical and difficult than it should. Here are some recommendations:
7a) for instance the technical description of the task (CSC) is not fully described and requires background knowledge from other paper which is not desirable.
7b) Also the rationale for the added difficulty with the stochastic reward and new state is not well explained.
7c) In the technical description of the results I find that the text dives into descriptive comments of the figures but high-level take home messages would be helpful to guide the reader. I got a bit lost, although I feel that there is probably a lot of depth in these paragraphs.
As for 7a), 'CSC (complete serial compound)' was actually not the name of the task but the name of the 'punctate' state representation, in which each state (timing from cue) is represented in a punctate manner, i.e., by a one-hot vector such as (1, 0, ..., 0), (0, 1, ..., 0), ..., and (0, 0, ..., 1). As you pointed out, using the name of 'CSC' would make the text appearing more technical than it actually is, and so we have moved the reference to the name of 'CSC' to the Methods (Line 903-907):
“For the agents with punctate state representation, which is also referred to as the complete serial compound (CSC) representation [1, 48, 133], each timing from a cue in the tasks was represented by a 10-dimensional one-hot vector, starting from (1 0 0 ... 0)<sup>T</sup> for the cue state, with the next state (0 1 0 ... 0) <sup>T</sup> and so on.”
and in the Results we have instead added a clearer explanation (Line 163-165):
“First, for comparison, we examined traditional TD-RL agent with punctate state representation (without using the RNN), in which each state (time-step from a cue) was represented in a punctate manner, i.e., by a one-hot vector such as (1, 0, ..., 0), (0, 1, ..., 0), and so on.”
As for 7b), we have added the rationale for our examination of the tasks with probabilistic structures (Line 282-294):
“Previous work [54] examined the response of DA neurons in cue-reward association tasks in which reward timing was probabilistically determined (early in some trials but late in other trials). There were two tasks, which were largely similar but there was a key difference that reward was given in all the trials in one task whereas reward was omitted in some randomly determined trials in another task. Starkweather et al. [54] found that the DA response to later reward was smaller than the response to earlier reward in the former task, presumably reflecting the animal's belief that delayed reward will surely come, but the opposite was the case in the latter task, presumably because the animal suspected that reward was omitted in that trial. Starkweather et al.[54] then showed that such response patterns could be explained if DA encoded TD-RPE under particular state representations that incorporated the probabilistic structures of the task (called the 'belief state'). In that study, such state representations were 'handcrafted' by the authors, but the subsequent work [26] showed that the original value-RNN with backprop (BPTT) could develop similar representations and reproduce the experimentally observed DA patterns.”
As for 7c), we have extensively revised the text of the results, adding high-level explanations while trying to reduce the lengthy low-level descriptions (e.g., Line 172-177 for Fig2E-G).
(8) Related to the writing issue and 5), I wished that "bio-plausibility" was not the only reason to study positive feedback and value weights. Is it possible to develop a bit more specifically what and why this positivity is interesting? Is there an expected finding with non-negative FA both in the model capability? or maybe there is a simpler and crisp take-home message to communicate the experimental predictions to the community would be useful?
There is actually an unexpected finding with non-negative model: the non-negative random feedback model performed generally better than the models without non-negative constraint with either backprop or random feedback (Fig. 2J-left versus Fig. 6E-left), presumably because the nonnegative constraint effectively limited the parameter space and thereby learning became efficient, as we mentioned in our reply to your point 6a above (we did not well recognize this at the time of original submission).
Another potential merit of our present work is the simplicity of the model and the task. This simplicity enabled us to derive an intuitive explanation on why feedback alignment could occur. Such an intuitive explanation was lacking in previous studies while more precise mathematical explanations did exist. Related to the mechanism of feedback alignment, one thing remained mysterious to us at the time of original submission. Specifically, in the non-negatively constraint random feedback model, while the angle between the value weight (w) and the random feedback (c) was relatively close (loosely aligned) from the beginning, it appeared (as mentioned in the manuscript) that there was no further alignment over trials (and the angle actually settled at somewhat larger than 45°), despite that the same mechanism for feedback alignment that we derived for the model without non-negative constraint was expected to operate also under the non-negative constraint. We have now clarified the reason for this, and found a way, introduction of slight decay (forgetting) of value weights, by which feedback alignment came to occur in the non-negatively constraint model. We have added this in Line 463-477:
“As mentioned above, while the angle between w and c was on average smaller than 90° from the beginning, there was no further alignment over trials. This seemed mysterious because the mechanism for feedback alignment that we derived for the models without non-negative constraint was expected to work also for the models with non-negative constraint. As a possible reason for the non-occurrence of feedback alignment, we guessed that one or a few element(s) of w grew prominently during learning, and so w became close to an edge or boundary of the non-negative quadrant and thereby angle between w and other vector became generally large (as illustrated in Fig. 8D). Figure 8Ea shows the mean±SEM of the elements of w ordered from the largest to smallest ones after 1500 trials. As conjectured above, a few elements indeed grew prominently.
We considered that if a slight decay (forgetting) of value weights (c.f., [59-61]) was assumed, such a prominent growth of a few elements of w may be mitigated and alignment of w to c, beyond the initial loose alignment because of the non-negative constraint, may occur. These conjectures were indeed confirmed by simulations (Fig. 8Eb,c and Fig. 8F). The mean squared value error slightly increased when the value-weightdecay was assumed (Fig. 8G), however, presumably reflecting a decrease in developed values and a deterioration of learning because of the decay.”
Correction of an error in the original manuscript
In addition to revising the manuscript according to your comments, we have made a correction on the way of estimating the true state values. Specifically, in the original manuscript, we defined states by relative time-steps from a reward and estimated their values by calculating the sums of discounted future rewards starting from them through simulations. However, we assumed variable inter-trial intervals (ITIs) (4, 5, 6, or 7 time-steps with equal probabilities), and so until receiving cue information, agent should not know when the next reward will come. Therefore, states for the timings up to the cue timing cannot be defined by the upcoming reward, but previously we did so (e.g., state of "one timestep before cue") without taking into account the ITI variability.
We have now corrected this issue, having defined the states of timings with respect to the previous (rather than upcoming) reward. For example, when ITI was 4 time-steps and agent existed in its last time-step, agent will in fact receive a cue at the next time-step, but agent should not know it until actually receiving the cue information and instead should assume that s/he was at the last time-step of ITI (if ITI was 4), last − 1 (if ITI was 5), last − 2 (if ITI was 6), or last − 3 (if ITI was 7) with equal probabilities (in a similar fashion to what we considered when thinking about state definition for the probabilistic tasks). We estimated the true values of states defined in this way through simulations. As a result, the corrected true value of the cue-timing has become slightly smaller than the value described in the original manuscript (reflecting the uncertainty about ITI length), and consequently small positive TD-RPE has now appeared at the cue timing.
Because we measured the performance of the models by squared errors in state values, this correction affected the results reporting the performance. Fortunately, the effects were relatively minor and did not largely alter the results of performance comparisons. However, we sincerely apologize for this error. In the revised manuscript, we have used the corrected true values throughout the manuscript, and we have described the ways of estimating these values in Line 919-976.
