Sequential Memory with Temporal Predictive Coding
Authors: Mufeng Tang
Helen Barron
Rafal Bogacz
Venue: NeurIPS 2023 poster (2023)
Keywords: Predictive coding
sequential memory
hippocampus
TL;DR: We propose a predictive coding-based model for sequential memory in the hippocampus.

Abstract: Forming accurate memory of sequential stimuli is a fundamental function of biological agents. However, the computational mechanism underlying sequential memory in the brain remains unclear. Inspired by neuroscience theories and recent successes in applying predictive coding (PC) to \emph{static} memory tasks, in this work we propose a novel PC-based model for \emph{sequential} memory, called \emph{temporal predictive coding} (tPC). We show that our tPC models can memorize and retrieve sequential inputs accurately with a biologically plausible neural implementation. Importantly, our analytical study reveals that tPC can be viewed as a classical Asymmetric Hopfield Network (AHN) with an implicit statistical whitening process, which leads to more stable performance in sequential memory tasks of structured inputs. Moreover, we find that tPC exhibits properties consistent with behavioral observations and theories in neuroscience, thereby strengthening its biological relevance. Our work establishes a possible computational mechanism underlying sequential memory in the brain that can also be theoretically interpreted using existing memory model frameworks.

Sequential Memory with Temporal Predictive Coding
Mufeng Tang, Helen Barron, Rafal Bogacz MRC Brain Network Dynamics Unit, University of Oxford, UK
{mufeng. tang, helen. barron, rafal. bogacz/(bndu. ox ac.uk
Abstract
Forming accurate memory of sequential stimuli is a fundamental function of biological agents. However, the computational mechanism underlying sequential memory in the brain remains unclear. Inspired by neuroscience theories and recent successes in applying predictive coding (PC) to static memory tasks, in this work we propose a novel PC-based model for sequential memory, called temporal predictive coding (IPC). We show that our tPC models can memorize and retrieve sequential inputs accurately with biologically plausible neural implementation. Importantly, our analytical study reveals that tPC can be viewed a classical Asymmetric Hopfield Network (A (AHN) with an implicit statistical whitening process, which leads more stable performance sequential memory tasks of structured inputs. Moreover, we find that tPC exhibits properties consistent with behavioral observations and theories in neuroscience, thereby strengthening its biological relevance. Our work establishes a possible computational mechanism underlying sequential memory in the brain that can also be theoretically interpreted using existing memory model frameworks.
Introduction
The ability to memorize and recall sequences events with temporal dependencies is crucial for biological memory systems that also underpins many other neural processes (1H3). For example, forming correct memories words requires humans to memorize not only individual letters but also the sequential order following which the letters appear (e.g., "cat" and "act"). However, despite extensive research into models static, temporally unrelated memories from both neuroscience and machine learning [4 TO]
computational modeling of sequential memory is not developed. Existing models for sequential memory are either analytically intractable or have not yet been system- atically evaluated in challenging sequential memory tasks (1H15). which hinders comprehensive understanding of the computational mechanism underlying sequential memory, arguably a more general form of memory in the natural world than static memories.
In this work, we propose a novel approach to modeling sequential memory based on predictive coding (PC) [16. biologically plausible neural network model able to reproduce many phenomena observed in the brain 19], which has also shown to have a close relationship to backpropagation in artificial neural networks (20 21. Using PC to model sequential memory motivated by two key factors: Firstly, neuroscience experiments and theories have suggested that (temporal) predictive processing and memory are two highly related computations in the hippocampus, the brain region crucial for memory [22 23] Secondly, the modeling of static memories (e.g., a single image) using PC has recently demonstrated significant success X TO
raising the question of whether PC can also be employed to model sequential memory. To take into account the temporal dimension in sequential memory, in this work we adopt a temporal extension of the original PC models, which has been employed in filtering problems and in modeling the visual system [26 Here we investigate its performance sequential memory tasks. Our contributions can be summarized as follows:
24 2
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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We propose temporal predictive coding (1PC), family of PC models capable of sequential m tasks that inherit the biologically plausible neural implementation of classical PC models [T]]
We present an analytical result showing that the single-layer tPC can be viewed as the classical Asymmetric Hopfield Network (AHN) performing an implicit statistical whitening step during memory recall, providing possible mechanism of statistical whitening in the brain 30
Experimentally, we show that the whitening step in single-layer tPC models results in more stable performance than the AHN and its modern variants [15] in sequential memory, due to the highly variable and correlated structure of natural sequential inputs;
We show that tPC can successfully reproduce several behavioral observations in humans, including the impact of sequence length in word memories and the primacy/rrcency effect;
Beyond memory, we show that our tPC model can also develop context-dependent representations [1 and generalize learned dynamics to unseen sequences, suggesting a potential connection to cognitive maps in the brain 33 34
2 Background and related work
Predictive Coding Models for Static Memory The original PC model for memory follows a hierarchical and generative structure, where higher layers generate top-down predictions of lower layers' activities and the network's 's learning and inference are driven by the minimization of all the prediction errors [8]. Since the memorized patterns minimize the prediction errors in PC, they can be considered attractors of the energy function defined as the (summed) prediction errors, which is similar to the energy perspective of Hopfield Networks (HNs) [4]. Subsequent research has also shown that PC models for memory can be formulated as recurrent networks to account for the recurrently connected hippocampal network [10] 25), and that continual memorization and recall of stream of patterns can be achieved by combining the hierarchical PC model with conjugate Bayesian updates (24]. Despite these abundant investigations into the memory capability of PC models, they all focused on static memories. Although the BayesPCN model [24] is capable of recalling patterns memorized in an online manner, the memories stored in this model are still statie, order-invariant patterns, rather than sequences of patterns with underlying temporal dependencies.
Predictive Coding with Time PC was extended include temporal predictions by earlier works (see [35) for a review) noticing its relationship to Kalman filters [6). However, the Kalman filtering approach to temporal predictions sacrifices the plausible neural implementation of PC models due to non-local computations, and a recent work proposed an alternative way of formulating PC with temporal predictions that inherits the plausible implementation of static PC, while approximating Kalman filters (37] However, none of these models were examined in memory tasks. Broadly, models based on temporal predictions were proposed in both neuroscience and machine learning 29 (0) However, these models are either trained by the implausible backpropagation or rely on complex architectures achieve temporal predictions. In this work, our (PC model for sequential memory is based on the model in (37), which inherits the simple and biologically plausible neural implementation of classical PC models that requires only local computations and Hebbian plasticity.
Hopfield Networks for Sequential Memory Although there exist other models of sequential memory 13 these models are mostly the conceptual level, used to provide theoretical accounts for physiological observations from the brain, and are thus hard to analyze mathematically. Therefore, here we focus our discussion on the AHN (14) and its modern variants (15), which extended the HN 4 to account for sequential memory, and have a more explicit mathematical formulation. Denoting a sequence of P + 1 patterns x" (p = 1...., P + 1), x" E 1-1, the N x N weight matrix of an AHN is set as follows:
AHN EE (x4)
Notice that it differs from the static HN only by encoding the asymmetric autocovariance rather than the symmetric covariance E x" (xx)T thus the name. The single-shot retrieval, which we define
as R, is then triggered by query, q E {
R AAHN (q) = sgn W AHN 9) = sgn (Ii x (x0))

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The retrieval process in Eq. 2 can be viewed as follows: the query is first compared with each x using a dot product function (x") q that outputs similarity score, then the retrieval is a weighted sum of all x (u = 1, ...., P) using these scores weights. Thus, if is identical to a certain x", the next pattern x will be given the largest weight in the output retrieval. Following the Universal Hopfield Network (UHN) framework [7], we can generalize this process to define retrieval function of a general sequential memory model with realevalued patterns x E IR::
RU (q) = -p x sep (sim(xx .9)
where sim is similarity function such as dot product cosine similarity, and sep is separation function that separates the similarity scores i.e., emphasize large scores and de-emphasize smaller ones [77. When sim is dot product and sep an identity function, we get the retrieval function of the original AHN (for binary patterns an ad-hoc sgu function can be applied to the retrievals). Chaudhry al. [15] have shown that it is possible to extend AHN to a model with polynomial sep function with degree d, and a model with softmax sep function, which we call "Modern Continuous AHN" (MCAHN):
RAHN (q. = Ef x"=1 ((x*)Ta)
(4)
MCCAHN (q, = Ep x softmax (3(x")Tq)
(5)
where 3 is the temperature parameter that controls the separation strength of the MCAHN. Note that these two models can be respectively viewed sequential versions of the Modern Hopfield Network [5] and the Modern Continuous Hopfield Network [6]. which is closely related to the self-attention mechanism in Transformers [4] However, the family of AHNS has not yet been investigated in sequential memory tasks with structured and complex inputs such as natural movies.
Other Models for Sequential Memory Beyond Hopfield Networks, many other computational models have been proposed to study the mechanism underlying sequential memory. Theoretical properties of self-organizing networks in sequential memory were discussed as early as in [42]. In theoretical neuroscience, models by Jensen et al.
43 and Mehta et al. 44 suggested that the hippocampus performs sequential memory via neuron firing chains. Other models have suggested the role of contextual representation in sequential memory [1] 45), with contextual representations successfully reproducing the recency and contiguity effects in free recall (46). Furthermore, Howard et al. (47 proposed that sequential memory is represented in the brain via approximating the inverse Laplacian transform of the current sensory input. However, these models were still at the conceptual level, lacking neural implementations of the computations. Recurrent networks with backpropagation
and large spiking neural networks also demonstrate sequential memory 48 49 model with [48] to validate (PC's alignment with behavior.
We compare our
Our model is also closely related to the concept of cognitive map in the hippocampal formation (50-52), which is often discussed within the context of sequence learning to explain knowledge abstraction and generalization. In this work, we present two preliminary results related cognitive maps, showing that our tPC model can 1) disambiguate aliased observation via latent representations and 2) generalize with simple sequential dynamics as result of performing sequential memory (52) However, as this work centers on memory, we leave cognitive maps for future explorations of TPC.
3 Models
In this section, we introduce the IPC models by describing their computations during memorization and recall respectively, as well as the neural implementations of these computations. We describe the single-layer tPC first, and then move to the 2-layer tPC.
3.1 Single-layer tPC
The most straightforward intuition behind "good" sequential memory models is that they should learn the transition between every pair of consecutive patterns in the sequence, so that accurate recall of the full sequence can be achieved recursively by recalling the next pattern based on the current one. Given sequence of real-valued patterns x" o 1,... P + 1, this intuition can be formalized as the model minimizing the following loss at each time-step:
Memorization
Fu (W)+ = I|x" W Wfxx"-1))
(6)
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which is simply the squared temporal prediction enror. W is the weight parameter of the model, and f(+) is a nonlinear function. Similar to static PC models 8 T0]. we assume that the model has two populations of neurons: value neurons that are loaded with the inputs x at step jr, and error neurons representing the temporal prediction error e := x W fx(xx-1). To memorize the sequence, the weight parameter W is updated at each step following gradient descent:
AW (W)
E" f(x*~1)
and the model can be presented with the sequence for multiple epochs until W converges. Note that the model only has one weight parameter for the whole sequence, rather than weight parameters for a sequence of length 1.
Recall During recall, the weight matrix W is fixed to the learned values, and the value neurons no longer receive the correct patterns x". Instead, while trying to recall the pattern x" based on the query 9, the value neurons are updated to minimize the squared temporal prediction error based on the query q and the learned W:
(x*)
|| W f
where we denote the value neurons' activities during recall as X to differentiate it from the memorized patterns x". The value neurons then perform the following inferential dynamics to minimize the loss Fx"":
k x OF ( e
and the converged x is the final retrieval. Note that the error neurons' activities during recall are defined as e" := x² (W), which is also different from their activities during memorization.
In the case of sequential memory, there are two types of recall. We define the first type as "online" recall, where the query q at each step μ is the ground-truth pattern at the previous step x"-1_ It is called online as these ground-truth queries can be viewed as real-time online feedback during the sequence recall. In this case, the original x" will define a memory attractor as it defines an optimum of the loss, or energy in Eqs. Eqs. [nd The second type is referred to as "offline" recall, where q is the recall from the previous step i.e., x=-1, except at the first step, where ground-truth x is supplied to elicit recall of the whole sequence. This called offline as there is no real-time feedback. In this case, errors from earlier steps may accumulate through time and x" is no longer an ascertained attractor x" - which makes it more challenging and analogous to the replay of memories
unless x"
during sleep 153
Neural implementation A possible neural network implementation of these computations shown in Fig. [T, which is similar to that of static PC models [18] characterized by separate populations of value and error neurons. The difference from static models that the predictions are now from the previous time-step - 1. To achieve this, we assume that the value neurons are connected to the error neurons via two pathways: the direct pathway (the straight arrows between value and error neurons) and the indirect pathway through an additional population of inhibitory interneurons, which provides inhibitory inputs to the error neurons via W. These interneurons naturally introduce a synaptic delay of one time-step, such that when the inputs from step μ reach the error neurons through the indirect pathway, the error neurons are already receiving inputs from step μ via the direct pathway, resulting in the temporal error. Moreover, we assume that memory recall is much faster process than the time-steps μ so that the interneurons can hold a short working memory of q during the (iterative) inferential dynamics in Eq. 9 at step 14, which can be achieved by the mechanisms described in (54]. Notice that in this implementation, the learning rule (Eq. 7 is Hebbian and the inference rule (Eq. is also local, inheriting the plausibility of static PC implementations [8]
3.2 2-layer tPC
Similar to multi-layer static PC models for memory, we can have multiple layers in tPC to model the hierarchical processing of raw sensory inputs by the neocortex, before they enter the memory system 10. We focus on 2-layer tPC model in this work. In this model, we assume set of hidden value neurons z" to model the brain's 's internal neural responses to the sequential sensory inputs x". The hidden neurons make not only hierarchical, top-down predictions of the current activities in the sensory layer like in static PC models 3 but also temporal predictions like in the single-layer IPC. We also assume that the sensory layer x" does not make any temporal predictions in this case. Thus, this 2-layer tPC can be viewed as instantiation of the hidden Markov model (55).

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Figure Neural implementations of the tPC models. A: single-layer tPC. B: 2-layer tPC.
Memorization
During memorization, the 2-layer tPC tries to minimize the sum of squared errors step μ4, with respect to the model parameters and the hidden activities:
Fulz², WH. W 1z W Wnf )1 ||x*
W WWF(z"))
(10)
where WH governs the temporal prediction in the hidden state, Wr is the forward weight for the top-down predictions, and 2n the hidden state inferred the previous time-step. During memorization, the 2-layer tPC follows a similar optimization processing to that of static hierarchical PC 8 first infers the hidden representation of the current sensory input x" by:
2 x - (F, WH, W rr)/2"" = E + f'(z") O We
(11)
where @ denotes the element-wise product between two vectors, and ga and xx are defined as the hidden temporal prediction error 2 W [2(=)) and the top-down error x Wr (2") respectively. After zz converges, WH and W are updated following gradient descent on Fu:
AW H x -0F, W WH, W WF)/OW,, = = f(2
(12)
AWF F (z", WH. H, wr)/OW/ = ExI ((2
which are performed once for every presentation of the full sequence. The converged z" is then used as for the memorization at time-step μ +
Recall After leamnin/meemorization, W H and W are fixed. We also assume that the hidden activities z are unable to store memories, by resetting their values to randomly initialized ones. Thus, the sequential memories can only be recalled through the weights W and W Again, the sensory layer has no access to the correct patterns during recall and thus needs to dynamically change its value to retrieve the memories. The loss thus becomes;
Fu (zz x |z
W (=""("""1) 1 1
| WFf(z*)ll
(13)
where R denotes the activities of value neurons in the sensory layer during recall. Both the hidden and sensory value neurons updated to minimize the loss. The hidden neurons will follow similar dynamics specified in Eq. 11, with the top-down error xx, defined as x f(("), whereas the sensory neurons are updated according to:
L x OF, (z", \0x/ = -.
(14)
and the converged x is the final retrieval. Similar to the single-laye case, if the converged z" is used for the recall at the next step directly, the recall is offline: on the other hand, if we query the model with q = x" i.e., the ground-trutt and use the query infer 2", and then use z" for the recall at the next step, the recall is online.
neural implementation of the computations above is shown in Fig. DB. The hidden layer follows the same mechanism in the single-layer tPC, with interneurons introducing a synaptic delay for temporal errors and short-term memory to perform (Eq. IT.. The connection between the hidden layer and the sensory layer W,r is modeled in the same way as in static PC models, which requires only Hebbian learning and local computations [18. The memorization and recall pscudocode for the tPC models is provided in SM.
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Results
4.1 Theoretical relationship to AHNS
We first develop a theoretical understanding of single-layer tPC by relating it to AHNs:
Property Assume, without loss of generality, sequence of memories x" E ERN (μ 1,... P P+1l 1)
with zero mean. With an identity nonlinear function f(z) = x in Eq. the retrieval of the single- layer IPC with query q defined as RCPC (q), can be written as:
RtPC = E x*+ (Mx") Mq
(15)
where M is an empirical whitening matrix such that:
(Mx" (Mx") T) IN
(16)
where ( is the expectation operation over x!
Proof of this property is provided in the SM. Essentially, this property implies that our single-layer IPC, in its linear form, can be regarded as a special case of the UHN for sequential memories (Eq, 3 with "whitened dot product" similarity function where the two vectors x" and q are first normalized and decorrelated (to have identity covariance Iv) before the dot product. AHNs, on the other hand, calculate the dot product directly. Biologically, this property provides a possible mechanism of statistical whitening in the brain that, unlike earlier models of biological whitening with explicit objectives 32). performs this computation implicitly via the circuit shown in Fig. that minimizes the temporal prediction errors.
4.2 Experimental comparison AHNS
To understand how the whitening step affects the performance in sequential memory tasks, we compare our single-layer (PC with the family of AHNs excperimentally. To ensure consistency to Property [ above, we use identity nonlincarity f(z) for all these experiments. Empirically, we found that using a tanh nonlinearity makes subtle differences irrelevant to our main discussion in this work, and we discuss it in SM.
We first compare tPC with polynomial AHNs (Eq. sequences of uncorre- lated binary patterns, where AHNs are known to work well 44 [5] We plot their sequence capacity
Polynomial AHNS

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Figure 3: A: Recall MSE of MNIST sequences with increasing length; B: Recall MSE of MovingM- NIST sequences of a fixed length 10 but with an increasing number of sequences. Error bars obtained with 5 seeds
P Pmaz against the number of value neurons of the models i.e., the pattern dimension N. Here, Pmoza defined as the maximum length of memorized sequence, for which the probability of incorrectly recalled bits is less than or equal to 0.01. Fig. 2 shows that the capacity of our single-layer tPC is greater than that of the original AHN (d = L) but smaller than that of a quadratic AHN (d = 2). Notice that the single-layer tPC has identity sep function like the original AHN. Therefore the whitening operation has indeed improved the performance of the dot product sim function. Inspired by the decorrelation effect of statistical whitening, we then generated binary patterns with N 100 correlated features, with parameter b controlling the level of correlation (b /correlation)) ) The approach that we followed to generate the correlated patterns is provided in SM. As shown in Fig. 23. as the correlation increases, all AHNS up d suffer from quick decrease of capacity P Pmaaz whereas the capacity of single-layer tPC almost remains constant. This observation is consistent with the theoretical property that the whitening transformation essentially decorrelates features such that patterns with any level of correlation are regarded as uncorrelated in tPC recall (Eq. [5]. This result also explains the comparison in Fig. 2A: although the patterns generated in this panel are theoretically uncorrelated, the small correlation introduced due to experimental randomness will result in the performance gap between AHN and single-layer tPC.
We then investigate the performance of these models with sequences of binarized MNIST images [56] in Fig. 2C. It can be seen that the AHNs with d = 1 and d = 2 quickly fail as the sequence length P reaches 3, whereas the single-layer tPC performs well. These results suggest that our tPC with whitening step is superior to simple AHNs due the inherently correlated structure of natural inputs such as handwritten digits. For all the experiments with binary patterns, we used the online recall mentioned above, and polynomial AHNs already fail in this simpler recall scenario.
MCAHN Due to the quick failure of AHNS with a polynomial sep function, we now compare our single-layer tPC with the MCAHN (Eq. 55. In static memory tasks, it is known that the softmax separation leads to exponentially high capacity, especially when 3 is high 77] (7]. In this work we use 3 for all MCAHNs. We first compare the performances of our single-layer (PC model and an MCAHN on random sequences of MNIST digits with varying lengths. Here we trigger recalls with online queries. Fig. 3 shows that the performance of our single-layer tPC, measured as the mean squared error (MSE) between the recalled sequence and the ground truth, is better than that of the MCAHN, further demonstrating the usefulness of the implicit whitening in our model.
Despite the superior performance of our model in this task, we note that random sequences of MNIST images are not naturally sequential inputs i.e., there are no sequential dynamics underlying them. We thus examine the models on the Moving-MNIST dataset (58). Each video in this dataset consists of 20 frames of 2 MNIST digits moving inside a 64 x 64 patch. Due the fixed sequence length, for experiments with MovingMNIST we vary the total number of sequences to memorize and fix the sequence length to the first 10 frames of the videos. The performance of the models is shown in B.g. On average, the recall MSE of MCAHN has slower increase than that of the single-layer 1PC as the total number of sequences increases. However, the performance of MCAHN has very large variations across all sequence numbers. To probe into this observation, we visually examined 3 examples of the Movin movies recalled by MCAHN and our single-layer tPC in Fig. 4A, when the total number of sequences to memorize is 40. MCAHN produces very sharp recalls for the first 2 example sequences, but totally fails to recall the third one by converging to a different memory sequence after the red triangle in Fig. 4A. On the other hand, the recall by single-layer tPC is less
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sharp but stably produces the correct sequence. This phenomenon can be understood using the UHN framework 77: in Eq. 5 when 3 is large, the softmax separation function used by MCAHN will assign a weight close to to the memory whose preceding frame is most similar to (measure by dot product), and weights close to 0 to all other memories, which results in the sharp recall. In contrast, our single-layer tPC model uses an identity separation function that fails to suppress the incorrect memories, resulting in blurry recalls. Importantly, however, the failure of MCAHN in sequence 3 in Fig. 4A suggests that there are "strong attractors" in the memories with an undesirable advantage in dot product similarity which resulted in the large variance in the numerical results, and is addressed by the whitening step in single-layer tPC as it effectively normalized the patterns before dot product.
The importance of whitening in tPC is further demonstrated in our experiments with random sequences of CIFARIO 10 [59 images and movies from the UCF101 [60] dataset, shown in Figs. 4 and C. When recalling these colored sequences, MCAHN can easily converge to strong attractors preceded by frames with many large pixel values that lead to large dot products e.g., the third image in the CIFARIO example with bright backgrounds. and the penultimate frame in the UCF101 example with a large proportion of sand, which give their subsequent frames large similarity scores. This problem is consistent with earlier findings with static memories [ and circumvented in our single-layer tPC model with the whitening matrix M normalizing the pixel values across the sequence, yielding the correct and more stable memory recalls. However, they are less sharp due to the identity separation function. For all the experiments in Fig. B3 and Fig. A we used offline recall to make the tasks more challenging and more consistent with reality. Results with online recalls are shown in SM.
4.3 Comparison to behavioral data in sequential memory experiments
We further demonstrated the biological relevance of tPC by comparing it to data from behavioral experiments in sequential memory. In Fig 5J,, our 2-layer tPC is compared with Crannell and Parrish's [6]] study on sequence length's 's impact on serial recall of English words. Using one-hot vectors to represent letters (i.e., each "word" in our experiment is an ordered combination of one-hot vectors of "letters", a minimal example of word with letters being: [0, 1, Oj. [1. O, Oj, [0, O, 1)), we demonstrate accuracy the proportion of perfectly recalled sequences across varying lengths. Our model aligns consistently with experimental data in [6]] as well as the recurrent network model by Botvinick and Plaut [48), displaying sigmoidal accuracy drop with increasing sequence length.
Fig 5B introduces qualitative comparison to Henson's 62 experimental data, examining pri- macy/recency effects in serial recall of letters. These effects involve higher accuracy in recalling early (primacy) and late (recency) entries in a sequence, with the recency effect slightly weaker than the primacy effect. Using one-hot vectors and a fixed sequence length 7 (6 positions are shown as the first position is given as the cue to initiate recall our experiment), we visualize recall frequency at different positions across simulated sequences (100 repetitions, multiple seeds for error bars).
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Figure 5: Replicating behavioral data with tPC. A: Experimental data from 61 that studies the impact of sequence length on serial recall of English words and the replications by Botvinick and Plaut 48 and tPC. B: tPC replicates the primacy/recency effects in serial recall experiments [62].
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Figure 6: Context representation and generalization of tPC. A: A simple aliased example with MNIST. B: Numerical investigation into the impact of aliased repeating elements model performance. C: Different latent representations of aliased inputs by the 2-layer tPC. D/E: Recall/generalization of sequences with rotational dynamics. "GT" stands for ground truth and 16 and 1024 are the numbers of training sequences. F: Recall and generalization MSE of seen and unseen rotating MNIST images. Error bars obtained with seeds.
Each bar in Fig 5B indicates the frequency of an entry at particular position being recalled at each position. Our 2-layer tPC reproduces primacy/recency effects, albeit weaker than the model in [62] and previous models [48]. Additionally. the model tends to recall neighboring entries upon errors, echoing Henson's data. We attribute the weaker effects to tPC's memory storage weights, leading to overall improved performance across positions. Details for these experiments are shown in SM.
4.4 tPC develops context-dependert representations
In Fig. 3, we plotted the recall MSE of the 2-layer (PC model in MovingMNIST, which is similar to that of the single-layer tPC. The close performance of these models raises the question of whether and when the hidden layer and hierarchical processing of sequential inputs are necessary. Inspired by carlier neuroscience theories that the hippocampus develops neuron populations signaling the sensory inputs and the context of the inputs separately [I]. we hypothesize that the hidden neurons in our model represent when an element occurs memory sequence i.c., its context. We thus designed sequential memory task with aliased, or repeating inputs at different time-steps 288 63 An example of such an aliased sequence can be seen 6/A where the second and the fourth frames of a short MNIST sequence are exactly the same ("2"). Recalling such a sequence is inherently more challenging as the models have to determine, when queried with the "2" during recall (either online or offline), whether they are at the second or the fourth step give the correct recall at the next step ("1" or "3"). As can be seen in Fig. (A, both MCAHN (which can be regarded as a single-layer network (7) and single-layer tPC fail in this task, recalling average frame of "1" and "3" after the

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aliased steps, whereas the 2-1ayer tPC can recall the correct sequence. We then conducted numerical investigation into this problem. We first plotted the (online) recall MSEs of the single- and 2-layer IPC models in random MNIST sequences (sampled from the training set) of varying lengths, which are shown as the solid lines in F We then randomly replaced 20% (rounded up to the closest integer) of the elements in these sequences with a single digit from the test set of MNIST so that each sequence now has 20% repeating i..., aliased elements, and plotted the recall MSEs as the dotted lines in Fig. BB. The result suggests that sequences with aliased inputs affect the single-layer model much more than it affects the 2-layer one, producing significantly larger MSES than recalls without aliased inputs. It is worth noting that aliased inputs are ubiquitous in natural sequential memories. prrrovii In SM, we provide a natural example from the UCF101 dataset where the 2-layer tPC successfully de-aliased repeating inputs whereas the MCAHN and single-layer tPC failed.
To further understand the mechanism underlying the 2-layer tPC with aliased memories, we used a simpler synthetic sequential memory shown in Fig. GC bottom, where a white har moves first down and then up in a x 5 frame so that the steps 2 and are aliased (28.. We then trained a 2-layer tPC with hidden size 5 to memorize this sequence and queried it offline. The smaller hidden size allows us to plot, when the recall dynamics (Eq. TT have converged, the exact hidden activities in Fig. 6C top, where each vertical line represents the activity of a hidden value neuron 2 As can be seen in the circled time-steps, the 2-layer model represents the aliased inputs differentially in its hidden states, which helps it recall the next frame correctly. This property is consistent with early observations in neuroscience that when memorizing sequences. the hippocampus develops a conjunction of neurons representing individual inputs, well as neurons signaling the temporal context within which an individual appears 3 [T. In our simple 2-layer model, the "sensory" layer represents individual inputs, whereas the hidden layer plays the role of indexing time and context.
4.5
tPC generalizes learned dynamics to unseen sequences
After memorizing a large number of training sequences sharing underlying dynamics, can tPC generalize the dynamies to unseen sequences? In this experiment, we train the 2-layer tPC with sequences of rotating MNIST digits of identical rotating dynamics (i.e., the digits are rotating towards a fixed direction with fixed angle at each time step) and vary the number of training sequences ("training size"). An example of these rotating MNIST digits can be seen in Fig row "ground truth". The model's performance is assessed by its ability to rotate both seen MNIST digits and unseen EMNIST letters. For small training sizes (16), tPC can recall seen rotating digits but struggles with generalizing to unseen letters (Fig 6 and E, second row). Increasing the training size to 1024 improves generalization, evident in clearer rotating sequences (Fig 6 and E, bottom row). Panel F quantitatively confirms this trend: the generalization MSE on unseen EMNIST drops as MNIST training size increases, indicating the model Icarns the underlying dynamics. Interestingly, the recall MSEs for seen MNIST sequences also decrease due to the model extracting rotational dynamics from the larger training set, differing from the behavior observed in random MNIST sequences (A.. Generalization and the capability of developing contextual representations that disambiguate aliased inputs are two critical functions underlying the flexible behavior of animals, thus connecting our tPC to models of cognitive maps in the hippocampus and related brain regions (51)
5 Conclusion
Inspired by experimental and theoretical discoveries in neuroscience, in this work we have proposed temporal predictive coding model for sequential memory tasks. We have shown that our tPC model can memorize and recall sequential inputs such as natural movies, and performs more stably than earlier models based on Asymmetric Hopfield Networks. We have also provided a theoretical understanding of the stable performance of the (PC model, showing that it is achieved by an additional statistical whitening operation that missing in AHNs. Importantly, this whitening step is achieved implicitly by plausible neural circuit performing local error minimization. Moreover, our 2-layer IPC has exhibited representational and behavioral properties consistent with biological observations, including contextual representations and generalization. Overall, our model has not only provided a possible neural mechanism underlying sequential memory the brain but also suggested a close relationship between PC and HN, two influential computational models of biological memories. Future directions include systematic investigations into tPC with more than layers and modelling cognitive maps with tPC.
10

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Acknowledgement
This work has been supported by Medical Research Council UK grant MC CUL0UUUU00000000 00003/1001111 RB, an E.P. Abraham Scholarship in the Chemical, Biological/L.ife and Medical Sciences to MT, and UKRI Future Leaders Fellowship HB (MR/W008939/1), The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work. http://dx.doi.or//100 22000/2/2////ooo22555555

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15


Reviews:
[{"review_id": "dvolzIe6JZ", "reviewer": "Reviewer_vgoV", "summary": "The paper generalizes predictive coding as a method of training neural networks to Hopfield networks, giving a model of temporal predictive coding (tPC).  tPC proves itself able to memorize discrete sequences at a level competitive with Asymmetric Hopfield Networks in experiments, and provides an intriguing hint as to the potential function of statistical whitening in the hippocampus.", "questions": "Can the authors address any of the literature in the learning of cognitive maps in the hippocampus?  Those often provide a connection to discrete event sequences and would give the authors a baseline to compare to beyond just the Hopfield networks they use.", "limitations": "Are the authors aiming their model and their claims at the computational or algorithmic Marr levels of analysis?  Even if they're aiming at the algorithmic level, predictive coding has been proposed to approximate backpropagation in certain limits, and so they could have compared against backprop-trained memory models in which the backprop training was replaced with a sufficient predictive coding architecture.  Likewise, there are plenty of cognitive mapping models at the computational model which are not analytically tractable, but which do admit sampling/Monte Carlo implementations that can attain neural plausibility.  Why the restricted class of comparisons?  Are the authors specifically proposing that the brain performs all necessary computations in closed form?\n\nEDIT: The authors have fully addressed my concern about closed-form computation, which then places their tPC model within the class of things to which I requested comparison.  This is very good!", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The authors provide a strong mathematical grounding and linkage to the predictive coding literature. Temporal predictive coding shows the interesting strength that as the correlation between features in a sequence increases, it does not appear to significantly lose much of its capacity -- no doubt due to the implicit statistical whitening.  They perform an experimental evaluation against sequential versions of the Modern Hopfield Network and the Modern Continuous Hopfield Network, though not against non-Hopfield sequence learning or cognitive mapping models.", "weaknesses": "The authors overclaim about biological/neural \"memories\" from the first sentence of the abstract. This becomes important because lossless sequence memorization is not what the hippocampus does, and if it did, it would be useless.  This leaves the major unaddressed question being: can the paper's tPC model generalize to unseen but reasonably similar sequences?  What features of the sequences can change without disrupting memorization?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive comments on generalization and connection of our model to cognitive maps. Specific responses are provided below and we kindly request that the reviewer consider reevaluating their score in light of our responses:\n\n***\n> “The authors overclaim about biological/neural \"memories\" from the first sentence of the abstract”\n\nWe thank the reviewer for pointing this out. Indeed, the survival of biological agents requires much more beyond sequential memory e.g., generalization. We will change our narrative on the significance of memory in the camera-ready version.\n\n***\n> “can the paper's tPC model generalize to unseen but reasonably similar sequences? What features of the sequences can change without disrupting memorization?”\n\nWe have added experiments to examine the generalization capability of our 2-layer tPC model, and the results are presented in Figure 1 of the PDF file in the global response section. In this experiment, we train the 2-layer tPC with sequences of rotating MNIST digits and vary the number of training sequences (“training size”). An example of these rotating MNIST digits can be seen in Fig 1A, row “ground truth”. The model's performance is assessed by its ability to rotate both seen MNIST digits and unseen EMNIST letters. For small training sizes (16), tPC can recall seen rotating digits but struggles with generalizing to unseen letters (Fig 1A and B). Increasing the training size to 1024 improves generalization, evident in clearer rotating sequences (Fig 1A and B). Panel C quantitatively confirms this trend: the generalization MSE on unseen EMNIST drops as MNIST training size increases, indicating the model learns the underlying dynamics. Interestingly, the recall MSEs for seen MNIST sequences also decrease due to the model extracting rotational dynamics from the larger training set, differing from the behavior observed in random MNIST sequences (Fig 3B in the original paper).\n***\n> “Can the authors address any of the literature in the learning of cognitive maps in the hippocampus? Those often provide a connection to discrete event sequences and would give the authors a baseline to compare to beyond just the Hopfield networks they use.”\n\nPlease see the discussion paragraph in the global response, where we discussed models of cognitive map and many other models related to sequential memory. This paragraph will be added to our Related Works section in the camera-ready version of this paper. In summary, two properties of our model have already shown tPC’s potential connections to cognitive map: 1) generalization 2) latent representation to disambiguate observations (Whittington et al., 2022). However, the focus of this paper is to investigate predictive coding in **sequential memory** whereas cognitive map models concentrate on the hippocampal formation’s role in flexible behavior based on abstracted knowledge. Therefore, we leave the direct comparison between tPC and cognitive map models (like TEM) to future explorations. \n***\n> “Are the authors aiming their model and their claims at the computational or algorithmic Marr levels of analysis? “\n\nAlgorithmic level\n***\n> “and so they could have compared against backprop-trained memory models in which the backprop training was replaced with a sufficient predictive coding architecture”\n\nIt is indeed an interesting direction to explore the relationship between backprop and predictive coding as learning rules in sequential memory, as most discussions on predictive coding as a plausible substitute to backprop focus on supervised learning. However, the focus of this paper is to investigate **whether predictive coding can support sequential memory in the brain and its computational principles** and we do not aim to compare with backprop. We will explore this problem in future works.\n***\n> “Why the restricted class of comparisons?”\n\nWe mainly compare with AHN because of the shared computational principles between tPC and AHN. This is interesting because it connects two classical computational models in neuroscience: predictive coding and Hopfield Nets. However, we do agree with the reviewer that comparing with Hopfield Nets only is restricted. Thus, we added a comparison to a recurrent model by Botvinick and Plaut (2006), shown in Fig3 of the attached PDF file in global responses. Their model has been compared with data collected from benchmark behavioral tasks and thus the comparison to it establishes a connection between tPC and behavioral data. Our result shows that tPC aligns well with the experimental data and can reproduce the effect of sequence length (Fig3A) and the primacy/recency effects (Fig3B).\n***\n> “Are the authors specifically proposing that the brain performs all necessary computations in closed form?”\n\nNo we are not. In fact the computations in tPC are not closed form: Property 1 in our original paper only holds when the temporal prediction is linear. When it is not linear, the retrieval cannot be expressed in closed form and has to be obtained iteratively (See Algorithm2 in SM). We stated this property here to establish a theoretical understanding of tPC and a connection to AHNs.\n***\n> References:\n- James CR Whittington, Timothy H Muller, Shirley Mark, Guifen Chen, Caswell Barry, Neil\nBurgess, and Timothy EJ Behrens. The tolman-eichenbaum machine: unifying space and\nrelational memory through generalization in the hippocampal formation. Cell, 183(5):1249–1263, 2020.\n- James CR Whittington, David McCaffary, Jacob JW Bakermans, and Timothy EJ Behrens. How to build a cognitive map. Nature neuroscience, 25(10):1257–1272, 2022.\n- Matthew M Botvinick and David C Plaut. Short-term memory for serial order: a recurrent neural network model. Psychological review, 113(2):201, 2006.\n"}, {"review_id": "Fh8WsLclxd", "reviewer": "Reviewer_EgZ5", "summary": "The authors propose a temporal predictive coding model that can memorize and recall sequences. The model performs better than a model based on asymmetric Hopfield networks. The authors provide a theoretical evaluation end explain the reasons for better performance. This work is inspired by neuroscience results and the authors argue that it establishes a possible computational mechanism underlying sequential memory in the brain.", "questions": "While the authors mentioned some of the related work, here are a few more papers that build neural models of memory for sequences. \n\nGraves et al. 2014 Neural Turing machines\nVoelker et al. 2019 Legendre memory units: Continuous-time representation in recurrent neural networks\nEliasmith et al. 2013 A large-scale model of the functioning brain\nHoward et al. 2014 A unified mathematical framework for coding time, space, and sequences in the hippocampal region\nWhittington et al. 2020 The Tolman-Eichenbaum machine: unifying space and relational memory through generalization in the hippocampal formation\n\nThe authors could comment on similarities/differences with respect to these approaches or perform a comparison.", "limitations": "The authors provided a sentence about future directions, but not about the limitations.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper proposes a new model for learning sequences using temporal predictive coding. The method is well explained, and results consist of several experiments showing better performance than when using an asymmetric Hopfield network. The authors provide a connection between temporal predictive coding and asymmetric Hopfield network - they identified how temporal predictive coding actually performs the same operation as asymmetric Hopfield network but with an implicit statistical whitening step during memory recall. They showed that when using multi-layer temporal predictive coding, the model develops latent representations of contextual information in sequential memories.", "weaknesses": "The authors mentioned that this work helps establish a possible computational mechanism underlying sequential memory in the brain. In its current form, the paper lacks direct comparison with neural or behavioral data. Behavioral tasks such as free recall could be used to evaluate if the properties of the sequential memory resemble those in the brain.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive comments on connecting our models to behavioral data and additional references. Specific responses are provided below and we kindly request that the reviewer consider reevaluating their score in light of our responses:\n***\n> “In its current form, the paper lacks direct comparison with neural or behavioral data.”\n\nWe have added a comparison to behavioral data and the results can be seen in Fig3 of the attached PDF file in global responses. In Fig3A, our 2-layer tPC is compared with Crannell and Parrish's (1957) study on sequence length's impact on serial recall of English letters/words. Using one-hot vectors to represent letters/words (a minimal example with a sequence 3 letters/words can be: [0,1,0], [1,0,0], [0,0,1]), we demonstrate accuracy as the proportion of perfectly recalled sequences across varying lengths. Our model aligns consistently with experimental data in Crannell and Parrish (1957) as well as the model by Botvinick and Plaut (2006), displaying a sigmoidal accuracy drop with increasing sequence length.\n\nFig3B introduces a qualitative comparison to Henson's (1998) experimental data, examining primacy/recency effects in serial recall. These effects involve higher accuracy in recalling early (primacy) and late (recency) entries in a sequence, with the recency effect slightly weaker than the primacy effect. Using one-hot vectors and fixed sequence length, we visualize recall frequency at different positions across simulated sequences (100 repetitions, multiple seeds for error bars). Each bar in Fig3B indicates the frequency of an entry at a particular position being recalled at each position. Our 2-layer tPC reproduces primacy/recency effects, albeit weaker than Henson (1998) and previous models (Botvinick and Plaut, 2006). Additionally, the model tends to recall neighboring entries upon errors, echoing Henson's data. We attribute the weaker effects to tPC's memory storage in weights, leading to overall improved performance across positions.\n***\n> “While the authors mentioned some of the related work, here are a few more papers that build neural models of memory for sequences.”\n\nWe have added a paragraph to the global responses, where we reviewed the papers that the reviewer pointed out. This paragraph will be added to the Related Works section in the camera-ready version of our paper.\n***\n> \"The authors provided a sentence about future directions, but not about the limitations.\"\n\nThis work is limited to modeling sequential memories, whereas the model can possibly be extended to address other functionalities of the hippocampus, such as generalization, based on our initial experiments. We aim to explore this direction further in future works. \n***\n> References\n\n- Henson, Richard NA. \"Short-term memory for serial order: The start-end model.\" Cognitive psychology 36.2 (1998): 73-137.\n- Crannell, C. W., and J. M. Parrish. \"A comparison of immediate memory span for digits, letters, and words.\" The Journal of Psychology 44.2 (1957): 319-327.\n- Matthew M Botvinick and David C Plaut. Short-term memory for serial order: a recurrent neural network model. Psychological review, 113(2):201, 2006.\n\n"}, {"review_id": "aEADClOq4v", "reviewer": "Reviewer_SUeD", "summary": "The paper presents work on (relatively) biologically-plausible neural networks for remembering sequences of inputs, extending work on temporal predictive coding nets (a simple architecture of a layer of neurons for feature values and a layer for prediction error, with some interneurons) and asymmetric modern hopfield networks.\nAnalysis shows a direct link between temporal Predictive Coding networks and Asymmetric Hopfield Networks, with improved performance vs AHN for correlated patterns reflecting the implicit whitening process built into the tPC net. \nThe multilayer tPC net develops interesting representations of items and context while solving the problem of storing sequences with repeated items.", "questions": "Would an alternative be to explicitly make predictions or learn associations between items that are more than one place apart in the list?\nHow does this model compare to those explicitly involving a sequential contextual signal (e.g. the \"temporal context model\")?", "limitations": "See above for potential limitations/comparisons that could be discussed ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "There are nice clear theoretical results to explain the links between tPC and AHN, and new results showing good performance in challenging sequential memory tasks with complex repeated images.\nThe development of interesting higher order representations of sequential order in this simple-to-analyse system will be of interest to neuroscientists given the development of these representations in the mammalian brain.  ", "weaknesses": "The sequential memory solutions considered here use changes to connection weights to store the sequence, these might be compared with deep networks that are able to reproduce a sequence having been pretrained on similar sequences, but not the one in question (the first example I think is Botvinick & Plaut, Psych Rev, 2006; now transformers). \nIt seems that the only advantage in performance, compared to the AHN, is the whitening process (which could easily be added to AHN), but perhaps this really reflects the fact that this is a biologically plausible implementation of AHN (which introduces whitening as a by product) - perhaps it could be compared to alternative ways of implementing AHN in a biologically plausible way (if they exist)?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive comments on the additional references and possible extensions of our model. Specific responses are given below:\n***\n> “The sequential memory solutions considered here use changes to connection weights to store the sequence, these might be compared with deep networks that are able to reproduce a sequence having been pretrained on similar sequences, but not the one in question (the first example I think is Botvinick & Plaut, Psych Rev, 2006; now transformers)”\n\nWe examined the generalization capability of our 2-layer tPC in an experiment shown in Fig1 of the attached PDF file in global responses. Details of these experiments and results can be found in global response, “Description of Figure 1”.\n\nHowever, we did not compare with Botvinick and Plaut (2006) directly as their sequential memory task is different from ours: their model was trained to memorize **and** recall sequences after a given recall cue. The testing phase is performed on unseen data, to examine whether the model can memorize and recall the new unseen sequences. On the other hand, our model is trained only to memorize, and recall is triggered by presenting to the model the first item in one of the training sequences. This difference in task prevents us from comparing with their model directly.\n\nWe value the reviewer's insight regarding Botvinick and Plaut (2006), which prompted us to incorporate a comparison with experimental data. The results are shown in Fig3 of the attached file to global responses and descriptions of the results can be found in the global response, “Description of Figure 3”.\n***\n> “perhaps it could be compared to alternative ways of implementing AHN in a biologically plausible way (if they exist)”\n\nCurrently, we are unaware of any other biologically plausible implementation of AHN that seamlessly incorporates the whitening operation. A possible approach is to combine the implementation in Chaudhry et al. (2023) with circuits performing statistical whitening such as in Duong et al. (2023). However, this requires further thinking and experiments and we defer it for future explorations.\n***\n> “Would an alternative be to explicitly make predictions or learn associations between items that are more than one place apart in the list? “\n\nWe agree this is an interesting direction to follow, as it may lead to a richer representation of contexts and time in the latent layers. However, this is beyond the scope of the current paper where we aim to explore the computational principle of the single-layer tPC and present preliminary results of the 2-layer tPC. We aim to explore this question in future works.\n***\n> “How does this model compare to those explicitly involving a sequential contextual signal (e.g. the \"temporal context model\")?”\n\nSimilar to the previous point, we aim to explore the contextual signal in the 2-layer tPC in future works. However, we appreciate the additional reference on Temporal Context Model and have added it to our literature review in the global responses.\n***\n> References:\n- Duong, Lyndon R., et al. \"Statistical whitening of neural populations with gain-modulating interneurons.\" arXiv preprint arXiv:2301.11955 (2023).\n- Chaudhry, Hamza Tahir, et al. \"Long Sequence Hopfield Memory.\" arXiv preprint arXiv:2306.04532 (2023).\n- Matthew M Botvinick and David C Plaut. Short-term memory for serial order: a recurrent neural network model. Psychological review, 113(2):201, 2006.\n"}, {"review_id": "isgkvqs2q8", "reviewer": "Reviewer_zZJv", "summary": "This paper proposes to use Predictive Coding Networks for temporal association of sequences.", "questions": "See above.", "limitations": "See above.", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "The paper is well-structured and easy to follow. The motivation is clear: a deep network model with biologically plausible learning algorithms for sequence learning.", "weaknesses": "1. This paper lacks of novelty and is in fact a trivial extension of [1]. In [1], the single-layer and two-layer predictive coding networks are proposed to associate the input and the output. In this paper, the input is replaced with frame x[t] in a sequence and the output is replaced with frame x[t+1].\n\n2. Property 1 in Section 4 is a trivial result for linear regression. It should be noted that Property 1 is not rigorously presented. A condition that the data covariance matrix must be full-ranked should be imposed.\n\n3. Lack of robust retrieval evaluation. In [1] and classic Hopfield networks, the model can recover the stored memories given noisy initialize state. Is the model in this paper robust to noise for sequence storage? Further experimental evaluation is needed.\n\n4. Missing references [2,3]. [2] is the very first work for temporal sequence association. How do the authors compare their work to [3], which is also about predictive coding for sequences?\n\n[1] Associative Memories via Predictive Coding. Tommaso Salvatori, Yuhang Song, Yujian Hong, Simon Frieder, Lei Sha, Zhenghua Xu, Rafal Bogacz, Thomas Lukasiewicz. arXiv, 2021.  \n[2] Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements. S.-I. Amari. IEEE Transactions on Computers, 1972.  \n[3] Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning. William Lotter, Gabriel Kreiman, David Cox. ICLR, 2017.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive comments on our paper. Specific responses are provided below and we kindly request that the reviewer consider reevaluating their score in light of our responses:\n***\n> “This paper lacks of novelty and is in fact a trivial extension of [1]. In [1], the single-layer and two-layer predictive coding networks are proposed to associate the input and the output. In this paper, the input is replaced with frame x[t] in a sequence and the output is replaced with frame x[t+1]”\n\nWe agree with the reviewer that this work extends [1] and this is also what we claimed in Related Works. However, the extension is not trivial as [1] only addresses static memories whereas our model performs sequential memory, arguably a more important type of memory, a point we made at the beginning of the paper. Notably, the seminal introduction of Asymmetric Hopfield Networks by Sompolinsky and Kanter (1986) can be viewed as an analogous extension of the original Hopfield Network towards sequential memory. Such advancements, aiming to address broader biological phenomena, should not be dismissed as trivial. Therefore, we wouldn’t consider our work as trivially extending [1].\n\nThe models in [1] did not directly associate input and output; they learned configurations representing inputs and recalled them when given a corrupted cue. Moreover, they didn't propose single-layer models. Our work differs by introducing temporal predictions for sequential inputs, unlike [1], where predictions were limited to interactions between layers. Thus, our model's innovation extends beyond replacing [1]'s input and output.\n***\n> “Property 1 in Section 4 is a trivial result for linear regression. It should be noted that Property 1 is not rigorously presented. A condition that the data covariance matrix must be full-ranked should be imposed.”\n\nWe thank the reviewer for pointing out that a full-rank covariance matrix condition is needed for Property 1 to hold. We will add it to the paper.\n\nHowever, we respectfully disagree with the reviewer that Property 1 is a trivial result from linear regression. While it does stem from regressing x[t+1] to x[t] within our model, its significance is far from trivial for the following reasons: \n\n1. Property 1 originates from a biologically plausible neural circuit illustrated in Figure 1 of our original paper. This depiction elucidates how the seemingly straightforward linear regression can be executed in a biologically plausible manner within the hippocampus, with local computations and Hebbian plasticity. Consequently, the result holds substantial importance from a biological context. \n\n2. The expression of Eq 15 in Property 1 is in a form akin to Universal Hopfield Network (Millidge et al., 2022), which establishes a vital connection between two influential computational models: predictive coding and Hopfield Network. This linkage between independently proposed models offers a significant contribution to the realm of computational modeling of neural systems. \n\n3. Property 1 offers an insightful interpretation of linear regression coefficients. By decomposing $(X^TX)^{-1}$, it elucidates how linear regression employs a whitened similarity function to compare the input with training data, leveraging this similarity score to weigh target variables and compute the weighted sum of all targets. This interpretation has a noteworthy contribution to the theoretical comprehension of linear regression. \n***\n> “Lack of robust retrieval evaluation. In [1] and classic Hopfield networks, the model can recover the stored memories given noisy initialize state. Is the model in this paper robust to noise for sequence storage?”\n\nWe acknowledge the reviewer's interest in noisy query retrieval and have incorporated experimental outcomes in the attached PDF file in global responses (Fig2). We conducted experiments similar to Fig3A in our original paper, introducing Gaussian noise with varying standard deviation (std) to the recall query. Fig2A illustrates single-layer tPC's performance in retrieving from noisy queries with std 0.1 (top row) and 1.0 (bottom row). As noise levels increase, recall quality diminishes, yet tPC's recall inference progressively refines and denoises retrieved images. In Fig2B, the same evaluation is presented for MCAHN, showcasing consistent retrieval of clear and sharp images. This clarity stems from MCAHN's softmax separation function (Eq 5), absent in tPC (Eq 15). Fig2C provides a quantitative assessment of noise impact on recall MSE for memorized sequence lengths of 16 and 32. Generally, both tPC and MCAHN experience increased recall MSE with higher noise levels. However, while MCAHN performs better than tPC at sequence length 16, the reverse holds true at length 32. This observation, combined with Fig3A in our original paper, affirms sequence length's predominant role in recall MSE under these noise levels. It's noteworthy that MCAHN's recall MSE variance is substantial due to the softmax separation function's potential for convergence to erroneous sequence entries, as explained in our original paper. \n***\n> “Missing references [2,3]. [2] is the very first work for temporal sequence association. How do the authors compare their work to [3], which is also about predictive coding for sequences”\n\nWe have added [2] to our literature review, in global responses. The “predictive coding” used in [3] is in a broader sense: they trained a deep network using backprop, end-to-end, whereas our model is trained using local and Hebbian learning rules derived from local prediction errors. However, we agree that it would be an interesting future direction to investigate whether tPC could achieve comparable results to [3]. We will add this to our Discussion and Related Works section.\n***\n> References\n- Sompolinsky, Haim, and Ido Kanter. Physical review letters. 1986.\n- Millidge, Beren, et al. International Conference on Machine Learning. 2022.\n"}]

Author rebuttal:
**We performed additional experiments as requested by the reviewers and presented the results in the attached PDF file. Since experiments in Fig 1 and 3 are related to the comments from multiple reviewers, we include descriptions of them here for reference:**

> Description of Figure 1:

In this experiment, we train the 2-layer tPC with sequences of rotating MNIST digits and vary the number of training sequences (“training size”). An example of these rotating MNIST digits can be seen in Fig 1A, row “ground truth”. The model's performance is assessed by its ability to rotate both seen MNIST digits and unseen EMNIST letters. For small training sizes (16), tPC can recall seen rotating digits but struggles with generalizing to unseen letters (Fig 1A and B). Increasing the training size to 1024 improves generalization, evident in clearer rotating sequences (Fig 1A and B). Panel C quantitatively confirms this trend: the generalization MSE on unseen EMNIST drops as MNIST training size increases, indicating the model learns the underlying dynamics. Interestingly, the recall MSEs for seen MNIST sequences also decrease due to the model extracting rotational dynamics from the larger training set, differing from the behavior observed in random MNIST sequences (Fig 3B in the original paper).

> Description of Figure 3:

In Fig3A, our 2-layer tPC is compared with Crannell and Parrish's (1957) study on sequence length's impact on serial recall of English letters/words. Using one-hot vectors to represent letters/words (a minimal example with a sequence 3 letters/words can be: [0,1,0], [1,0,0], [0,0,1]), we demonstrate accuracy as the proportion of perfectly recalled sequences across varying lengths. Our model aligns consistently with experimental data as well as the model by Botvinick and Plaut (2006), displaying a sigmoidal accuracy drop with increasing sequence length.

Fig3B introduces a qualitative comparison to Henson's (1998) experimental data, examining primacy/recency effects in serial recall. These effects involve higher accuracy in recalling early (primacy) and late (recency) entries in a sequence, with the recency effect slightly weaker than the primacy effect. Using one-hot vectors and fixed sequence length, we visualize recall frequency at different positions across simulated sequences (100 repetitions, multiple seeds for error bars). Each bar in Fig3B indicates the frequency of an entry at a particular position being recalled at each position. Our 2-layer tPC reproduces primacy/recency effects, albeit weaker than Henson (1998) and previous models (Botvinick and Plaut, 2006). Additionally, the model tends to recall neighboring entries upon errors, echoing Henson's data. We attribute the weaker effects to tPC's memory storage in weights, leading to overall improved performance across positions.

***

**Since all reviewers have pointed us to additional references, we added the following paragraphs of an additional discussion/literature review.**

Beyond Hopfield Networks, many other computational models have been proposed to study the mechanism underlying sequential memory. Theoretical properties of self-organizing networks in sequential memory were discussed as early as in [1]. In theoretical neuroscience, models by Jensen et al. [2] and Mehta et al. [3] suggested that the hippocampus performs sequential memory via neuron firing chains. Other models have suggested the role of contextual representation in sequential memory [4, 5], with contextual representations successfully reproducing the recency and contiguity effects in free recall [6]. Furthermore, Howard et al. [7] proposed that sequential memory is represented in the brain via approximating the inverse Laplacian transform of the current sensory input. However, these models were still at the conceptual level, lacking neural implementations of the computations. Recurrent networks with backpropagation and large spiking neural networks also demonstrate sequential memory [8, 9]. We compare our model with [8] to validate tPC’s alignment with behavior.

Our model is also closely related to the concept of cognitive map in the hippocampal formation
[10 –12], which is often discussed within the context of sequence learning to explain knowledge abstraction and generalization. In this work, we present two preliminary results related to cognitive maps, showing that our tPC model can 1) disambiguate aliased observation via latent representations and 2) generalize with simple sequential dynamics as a result of performing sequential memory [12]. However, as this work centers on memory, we leave cognitive maps for future explorations of tPC.

***

**References:**

[1] S.-I. Amari, IEEE Transactions on computers 100, 1197 (1972).

[2] O. Jensen, M. Idiart, and J. E. Lisman, Learning & Memory 3, 243 (1996).

[3] M. R. Mehta, M. C. Quirk, and M. A. Wilson, Neuron 25, 707 (2000).

[4] G. V. Wallenstein, M. E. Hasselmo, and H. Eichenbaum, Trends in neurosciences 21, 317
(1998).

[5] W. B. Levy, in Psychology of learning and motivation (Elsevier, 1989), vol. 23, pp. 243–305.

[6] M. W. Howard and M. J. Kahana, Journal of mathematical psychology 46, 269 (2002).

[7] M. W. Howard, C. J. MacDonald, Z. Tiganj, K. H. Shankar, Q. Du, M. E. Hasselmo, and H. Eichenbaum, Journal of Neuroscience 34, 4692 (2014).

[8] M. M. Botvinick and D. C. Plaut, Psychological review 113, 201 (2006).

[9] C. Eliasmith, T. C. Stewart, X. Choo, T. Bekolay, T. DeWolf, Y. Tang, and D. Rasmussen, science 338, 1202 (2012).

[10] J. Whittington, T. Muller, S. Mark, C. Barry, and T. Behrens, Advances in neural information processing systems 31 (2018).

[11] J. C. Whittington, T. H. Muller, S. Mark, G. Chen, C. Barry, N. Burgess, and T. E. Behrens, Cell 183, 1249 (2020).

[12] J. C. Whittington, D. McCaffary, J. J. Bakermans, and T. E. Behrens, Nature neuroscience 25, 1257 (2022).


Meta-review: In this theoretical neuroscience study, the authors propose a temporal predictive coding model that can memorize and recall sequences. 3 of the reviewers appreciates the theoretical contributions and comparison to the Asymmetric Hopfield Network.
Decision: Accept (poster)