Learning Sequence Attractors in Recurrent Networks with Hidden Neurons

We thank the reviewer for the dedicated proofreading and helpful suggestions on improving our paper. Below, we address each point of yours.

• Comparison of encoding efficiency to other methods/experiments
  In the revised paper, we have compared our approach with networks of continuous neurons trained with backpropagation and modern Hopfield networks. Please see Appendix A and B in the revised paper.

• Experiments for effect of noise level of the first patter on the performance.
  We observed a trade-off between the noise level that can be tolerated and the length of the stored sequence in our experiments. Additionally, we provide the code of our experiments in the supplementary material for further exploration.

• Further, the paper doesn't address how inputs to the network influence the output or performance.
  We agree that the inputs are important. However, in this paper, we focus on spontaneous neural activities and sequence memory for which the inputs are absent. The exploration of inputs to the network is beyond the scope of this paper.

• For Figure 4 and 11 for example it is unclear what is to be gained from seeing the convergence of the algorithm, if it is not compared to other algorithms to see which converges in less epochs for example.
  Figure 4 is to demonstrate that why the algorithm works despite that $\mathbf{P}$ is a random matrix and fixed during learning. Figure 11 is to demonstrate that both errors $\mu_i(t)$ and $\nu_j(t)$ would jointly reduce to zero, rather than conflicting with each other. Additionally, we provide comparison experiments in Figure 12 in Appendix A.

• For Figure 7 and 8 it would be beneficial to see how increasing influences retrieval success as and are changed.
  Due to space limitation, we cannot provide all possible experiments suggested by the reviewer in the paper, as the number of combinations of the hyperparameters is enormous. Nevertheless, we provide the code of our experiments in the supplementary material for further exploration.

• Better describe how the margin of the margin perceptron is related to the robustness here.
  In margin perceptron, the margin is disproportional to the magntitude of the weights, which is related to robustness, as can be been in eq (12) in Proposition 4. We improved the presentation in the revised paper.

• Why does such a $\mathbf{U}^*$ exist?
  The assumption that such a $\mathbf{U}^*$ exists is a condition that the algorithm converges in finite steps. This condition is also in convergence proof of the perceptron learning algorithm.

• After Equation (19): "due to (1)". Should this be "due to (17)"?
  We corrected this typo in the revised paper.

• In Eq. (33) $q$ is appearing without introduction
  $q$ is an index which ranges from $1$ to $p$. It should be clear from the context.

• Clarify step (32) to (33) in the derivation.
  We improved our writing of the derivation in the revised paper.

• In Eq. (39) $\Omega$ has not been introduced.
  $\Omega$ (Big-Omega) is a standard notation in for analyzing complexity of algorithms. It is defined as the asymptotic lower bounds.

• The justification of the relevance of this work to V1 neurons is insufficient.
  The reviewer made good points on our speculation on V1. After reconsideration, we found our speculation on V1 is too premature and therefore removed it in the revised paper.

• What is the expressivity of the Heaviside function for a fixed $M$ and $N$ in terms of $T$?
  In the revised paper, we derived the lower bound of the capacity of our network model. The maximal length of sequences that can be stored is at least $M$ (number of hidden neurons). Please see the updated Proposition 1 in the revised paper.

• On page 8, just above Figure 4, when is $M$ large enough?
  Empirically, we found when $M$ is slightly larger than the sequence length $T$, the network can learn to store the sequence. Interestingly, we also found that larger $M$ leads to faster convergence in learning.

• When is the solution the pseudo-inverse and when the transpose of $P$?
  Both works fine when $M$ is large enough as shown in Figure 4.

• How can high-dimensional probability theory explain this?
  The phenomenon that $\mathbf{x} = \text{sign}(\mathbf{V} \text{sign}(\mathbf{Px}))$ for $\mathbf{V}=\mathbf{P}^\top$ or $\mathbf{V}=\mathbf{P}^+$ only happens when $M$ is sufficiently large. We cannot prove it at the moment but offer our intuition and conjecture for the interested reader to explore further.

We agree with the reviewer that external inputs are crucial in neural information processing. However, it is beyond the scope of this paper. We have acknowledged this limitation in the Conclusion and Discussion section of the revised paper and plan to explore this important topic in the future work.

It seems like the trained neural networks for the comparison are feedforward ones. Why did the authors not choose RNNs, which would also make the comparison more biologically relevant.

Without inputs, RNN is equivalent to a feedforward network when it is unfolded in time. Please see Figure 1 and 3.

Finally, being able to respond to incoming stimuli is an important aspect of biological networks, and it is possible to account for with RNNs, which is missing for the proposed framework.

We focus on spontaneous neural activities and sequence memory for which the inputs are absent. Spontaneous or internally generated activities have been observed in hippocampus [Pastalkova, et al. 2008]. Our comparison work does not consider inputs either [Amari 1972, Bressloff and Taylor 1992, Chaudhry et al., 2023].

This seems like an inefficient way to encode sequences by using almost the number of neurons for the sequence length.

Encoding sequences by using almost the number of neurons for the sequence length also appears in our comparison work [Chaudhry et al., 2023]. In the future, when considering multi-layer hidden neurons, we might be able to derive more efficient capacity bound.

Also, this is considering a single sequence, which limits the applicability of this result.

One can concatenate multiple sequences into a single sequence as in our comparison work [Chaudhry et al., 2023].

Without this neuroscientific justification, the proposed framework is missing a clear contribution for neuroscience.

As discussed in the Abstract and Introduction, it remains largely unclear how the brain learns to store and retrieve sequence memories. Our work provides a possible biologically plausible mechanism in elucidating sequence memory in the brain. We have added this in the revised paper. Thanks for the suggestion.

References

Internally generated cell assembly sequences in the rat hippocampus. Pastalkova, et al. Science, 2008.

Learning patterns and pattern sequences by self-organizing nets of threshold elements. Amari. IEEE Transactions on Computers, 1972.

Perceptron-like learning in time-summating neural networks. Bressloff and Taylor. Journal of Physics A: Mathematical and General, 1992.

Long sequence hopfield memory. Chaudhry et al, NeurIPS, 2023.

We thank the reviewer for the feedback and helpful suggestions on improving our paper. Below, we address each point of yours.

• No, there exist many works looking at non-static attractors.
  What we meant is most previous works on attractor networks of Hopfield type considered only static attractors. Although there are also some other works on sequence attractors as the reviewer pointed out and we cited, the amount of them is smaller compared to those on static attractors. We have clarified this in the revised paper.

• There is a lack of comparison with these prior methods.
  In the revised paper, we have compared our approach with networks of continuous neurons trained with backpropagation and modern Hopfield networks. Please see Appendix A and B in the revised paper.

• Where is the evidence that the examples shown in Figure 2 cannot be generated by a network without hidden neurons?
  "In Figure 2, we show additional examples of sequences that cannot be generated by the network. The sequences are synthetically constructed. We then test if the perceptron learning algorithm can learn the sequences. Since the algorithm converges if the linear separability condition is met [Minsky & Papert, 1969], the divergence of the algorithm implies that the sequences cannot be generated by the networks." This is stated in the text above Figure 2.

• How is your approach conceptually different to hierarchical attractor network architectures?
  We do not claim our approach is conceptually different to hierarchical attractor network architectures such as [Chaudhry et al, NeurIPS 2023] for sequence learning. However, our model and algorithm are more biologically plausible, as stated in Appendix B in the revised paper.

• What is the memory capacity of this network? How does this depend on memory load?
  In the revised paper, we derived the lower bound of the capacity of our network model. The maximal length of sequences that can be stored is at least $M$ (number of hidden neurons). Please see the updated Proposition 1 in the revised paper.

• In the second paragraph of the conclusion, there is some speculation that neurons in V1 with unknown function may be akin to the hidden neurons of this model.
  The reviewer raised good questions for us to reflect on our speculation. After reconsideration, we found our speculation on V1 is too premature and therefore removed it in the revised paper.

We thank the reviewer for the encouragement and helpful suggestions on improving our paper. Below, we address each point of yours.

• A discussion of the capacity of the proposed architecture is missing
  In the revised paper, we derived the lower bound of the capacity of our network model. The maximal length of sequences that can be stored is at least $M$ (number of hidden neurons). Please see the updated Proposition 1 in the revised paper.

• The experiments section also does not have sufficient details about hyper-parameters
  We have tested different hyper-parameters and found the results are consistent. Meanwhile, we provide the code in the supplementary material for reproduction of our experiments.

• How many times is each sequence presented to the model?
  In training, we ran our algorithm for maximal 500 epochs. Each epoch correspond to presenting the sequences to the model once. Empirically, we found the algorithm usually converges within 100 epochs.

• What's the capacity of the model? How many sequences can it store?
  The maximal length of an arbitrary sequence that can be stored in the model is at least $M$ (number of hidden neurons). Please see the updated Proposition 1 in the revised paper. The model can store as many sequences as possible as long as the total length of the sequences is within the model capacity.

• The sign function and the Heaviside function seem to be identical the way it's been defined in the paper.
  Both functions are needed. The sign function is to simplify the math such that one does not have to explicitly convert the values of network states from $\{0,1\}$ to $\{-1,1\}$, as in the classical Hopfield networks. The Heaviside function is needed as $\mu_i(t)$ and $\nu_j(t)$ are errors which need to be non-negative.

• The first sentence of Related work just lists a bunch of papers, which seems redundant, since these papers are explained later anyway.
  We modified the writing as you suggested in the revised paper.

• Would have been useful to have Fig. 2 and Fig. 5 side by side for comparison. Merge the two figures perhaps?
  We considered this suggestion but found merging two figures would break the logic line of our presentation for the first-time readers.

• Bar plots in Fig. 7 and 8 are hard to read. Having concrete values for each bar, and mentioning the specific values of and used would be very useful.
  We considered this suggestion but found adding specific values in the figure would make the figures visually messy.

• I would suggest passing the text through a grammar check.
  We thank the reviewer for the writing suggestions and incorporated them in the revised paper. We will polish our paper further by a large language model later.

We thank the reviewer for the feedback. However, the reviewer seems to misunderstand our paper in several aspects.

1. Fundamentally, we are not trying to solve a "real problem" in the sense of engineering applications, but to answer a biological question: how does the brain learn to store and retrieve sequence memory? Storing the sequences without biologically plausible mechanisms is not the interest of our paper.

2. Given a sequence $\mathbf{x}(1),...,\mathbf{x}(T)$, our goal is not simply storing it as it is, but storing it as an attractor such that given a perturbed state $\hat{\mathbf{x}}(t)$, the network can recover the rest of the stored sequence robustly. Please refer to Figure 9 and 10 and the supplementary material for such a demonstration.

3. While the standard RNN in deep learning literature can certainly store the sequences, it is trained with backpropagation, whose biological plausibility is in question. While in our work, the algorithm has a clear biological interpretation.

4. Our focus is on theoretical analysis. We proved how the algorithm can converge and give rise to sequence attractors. Such results are missing for the standard RNN trained with backpropagation.

Below, we address several of your points.

You’re training a new network for each sequence?

No. For the Moving MNIST experiment, we use a single network for storing 20 sequences.

the brain communicates in spike rates, not just a single spike…

It is questionable that the brain communicates only in spike rates. There is a classic debate between rate coding and temporal coding. As to single spikes, please refer to the following paper for their role in neural coding:

The Role of Spike Timing in the Coding of Stimulus Location in Rat Somatosensory Cortex (Panzeri et al., Neuron 2001)

We conducted experiments on networks of continuous neurons trained with backpropagation, in response to the reviewer's concern. The results are in Appendix A of the revised paper. Our algorithm converges to zero training error much faster empirically.

We use the term "recurrent" to refer to a neuron's output is feedback to itself in a biological sense. As it causes confusion, we would like to consider changing the title to, for example,

Learning Sequence Attractors in Associative Networks with Hidden Neurons

Thank you so much for the suggestion!

Our current work cannot handle multiple sequences when there may be the same sequence element across sequences. The related work in comparison cannot handle such case either. For example,

Learning patterns and pattern sequences by self-organizing nets of threshold elements. Amari. IEEE Transactions on Computers, 1972.

Perceptron-like learning in time-summating neural networks. Bressloff and Taylor. Journal of Physics A: Mathematical and General, 1992.

Long sequence hopfield memory. Chaudhry et al. NeurIPS, 2023.

This can be handled by considering time delay which maps $\mathbf{x}(t), \mathbf{x}(t-1),...,\mathbf{x}(t-\tau)$ to $\mathbf{x}(t+1)$ or introducing stochastic neurons and is left to the future work.