Recurrent neural network dynamical systems for biological vision
Authors: Wayne WM Soo
Aldo Battista
Puria Radmard
Xiao-Jing Wang
Venue: NeurIPS 2024 spotlight (2024)
Keywords: convolutional neural networks
recurrent neural networks
neuroscience
vision
cognition
perception
Primary area: neuroscience_and_cognitive_science

Abstract: In neuroscience, recurrent neural networks (RNNs) are modeled as continuous-time dynamical systems to more accurately reflect the dynamics inherent in biological circuits. However, convolutional neural networks (CNNs) remain the preferred architecture in vision neuroscience due to their ability to efficiently process visual information, which comes at the cost of the biological realism provided by RNNs. To address this, we introduce a hybrid architecture that integrates the continuous-time recurrent dynamics of RNNs with the spatial processing capabilities of CNNs. Our models preserve the dynamical characteristics typical of RNNs while having comparable performance with their conventional CNN counterparts on benchmarks like ImageNet. Compared to conventional CNNs, our models demonstrate increased robustness to noise due to noise-suppressing mechanisms inherent in recurrent dynamical systems. Analyzing our architecture as a dynamical system is computationally expensive, so we develop a toolkit consisting of iterative methods specifically tailored for convolutional structures. We also train multi-area RNNs using our architecture as the front-end to perform complex cognitive tasks previously impossible to learn or achievable only with oversimplified stimulus representations. In monkey neural recordings, our models capture time-dependent variations in neural activity in higher-order visual areas. Together, these contributions represent a comprehensive foundation to unify the advances of CNNs and dynamical RNNs in vision neuroscience.

Recurrent neural network dynamical systems for biological vision
Wayne W.M. Soo Department of Engineering University of Cambridge
Aldo Battista Center for Neural Science
New York University aldo. battista@nyu. edu
wmws20cam. ac uk
Puria Radmard
Xiao-Jing Wang Center for Neural Science New York University
Department of Engineering University of Cambridge pr4500cam ac. uk
xjwang@nyu. edu
Abstract
In neuroscience, recurrent neural networks (RNNs) are modeled as continuous-time dynamical systems more accurately reflect the dynamics inherent in biological circuits. However, convolutional neural networks (CNNs) remain the preferred architecture in vision neuroscience due to their ability to efficiently process visual information, which comes at the cost of the biological realism provided by RNNs. To address this, we introduce hybrid architecture that integrates the continuous- time recurrent dynamics of RNNs with the spatial processing capabilities of CNNs. Our models preserve the dynamical characteristies typical of RNNs while having comparable performance with their conventional CNN counterparts on bench- marks like ImageNet. Compared conventional CNNs, our models demonstrate increased robustness to noise due to suppesuppressiin mechanisms inherent in recurrent dynamical systems. Analyzing our architecture dynamical system is computationally expensive, so we develop toolkit consisting of iterative methods specifically tailored for convolutional structures. We also train multi-area RNNs us- ing our architecture as the front-end to perform complex cognitive tasks previously impossible to learn or achievable only with oversimplified stimulus representations. In monkey neural recordings, neural activity in higher-order visual areas. Together, these contributions represent comprehensive foundation to unify the advances of CNNs and dynamical RNNs in vision neuroscience.
models capture deeeddepndeent variations in
Introduction
Dynamical systems have long been cornerstone the field neuroscience, tracing their origins back to the modeling of single neurons [ (2) Early biophysical models provided critical insights into the dynamic behavior of neurons. Building on this foundation, neuroscientists progressed to constructing networks of dynamical neurons 2112 The idea was to understand not just how individual neurons
operate but how networks of neurons interact to produce complex behaviors (5/7] [2] and cognitive functions 10) The rise of deep learning and artificial recurrent neural networks (RNNs) marked significant turning point in this endeavor. Neuroscience quickly adopted RNNs due to their ability to model time-dependent processes [1], much like how biological circuits process information over time. these models, each artificial neuron can behave according to some biophysical dynamics, whether through rate coding 113 14
30
spiking mechanisms [9 20 This alignment with
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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biological plausibility made RNNs powerful tool in neuroscience. Today, advances in computational power and learning algorithms have enabled these networks to be trained on diverse and complex
tasks, ranging from motor control [4] 24 to cognitive functions [5 16 18
27 31 Modern
RNNs are capable of learning intricate patterns in data, making them invaluable for modeling a wide array of neural processes. We provide details of key references in Appendix A 733 30 32).
RNNs are not the only models used in neuroscience. Convolutional neural networks (CNNs) have also been widely used, particularly in the domain of biological vision (3456). CNNs excel at processing spatial hierarchies in images, making them ideal for object recognition and scene understanding tasks. While there are recurrent CNNs that integrate temporal dynamics into the spatial processing capabilities of CNNs [36 38 41 49 51 57I. the specific advances of dynamical RNNs in neuroscience have largely remained inapplicable to vision models. To bridge this gap, propose hybrid architecture that integrates the continuous-time recurrent dynamics of RNNs with the image processing capahilities of CNNs, which refer as CordsNet (Convolutional RNN dynamical system). Briefly, our contributions and results are:
Dynamical expressivity analysis. We rigorously compare CordsNet with other recurrent dynamical architectures by training them all multiple cognitive tasks in neuroscience. We find that CordsNet can achieve the same range of dynamical regimes as other architectures.
New training algorithm. Training a continuous-time model is computationally expensive. We propose a computationally cheaper algorithm to efficiently initialize CordsNets and successfully train them on standard image classification benchmarks like ImageNet.
Autonomous and robust inference. We show that our trained models can perform inference autonomously and exhibit robustness to noise compared to static and discrete-time CNNs, which are actually just properties inherent to dynamical systems.
Analytical toolkit. There are many ways to analyze CordsNets, but they are memory intensive. We thus develop a toolkit consisting of iterative and cannnsiionalit-rddetttiii methods needed to analyze the model within reasonable computational limits and show some examples.
Image-computable models. We demonstrate the effectiveness of CordsNet the front-end of an image-computable multi-area model by training such models to perform tasks using the actual stimuli seen by subjects in experiments rather than abstract inputs.
Prediction of neural activity. We find that CordsNets (trained on ImageNet) are able to predict temporal signatures in neural activity of higher-order visual areas (V4 and IT).
Together, our contributions represent a comprehensive and directed effort to bring decades of ad- vancements dynamical systems to vision neuroscience.
Model architecture
We first briefly introduce CNNs and dynamical RNNs in neuroscience. typical convolutional layer in CNN consists of a 2D-convolution, normalization (commonly batch normalization (58)) and a non-linearity (such as ReLU (59), although the order of these operations can change (60). For two consecutive convolutional layers x and Xt, this can be written as:
(a Norm Conv)(xi))
where a is the non-linear activation function. For a single image, the output of convolutional layer is characterized by three dimensions (channels, height, and width), which can be interpreted as the latent state of the CNN (Figure IA, left). On the other hand, a continuous-time RNN dynamical system typically found in neuroscience literature (28 is described by:
dr T The dr== = a (Wier + b + W Winphinp)
(2)
dt
where T represents diagonal matrix of neuron time constants, r represents the latent state of the RNN, W ree the recurrent weight matrix, b represents the vector of neuron biases, and -r the leaky term which mimics the dynamics of biological neurons. hupp is the external input to the network that first undergoes a linear transformation through Winp (Figure middle). Just like in CNNs, a is the non-linear activation function. Note that there are no artificial normalizing operations as they are generally incompatible with continuous-time dynamical systems. The latent state of the network can be simulated forward in time by Euler's method with suitable choice of time discretization.

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A
Convolutional network
Dynamical
network
CordsNet (Hybrid architecture)
continuous-üime
continuous-line
convolution
recurrent
recurrent
normaizzaiion
dynamics
dynamics
non-linearily
convolution
B
Stable regime
Osciliatory regime
Chaolic regime
Examples
Eigenvalues
Examples
Eigenvalues Real
Examples
Eigenvalues
Noural acivity
Real
Neural activity
www
Noural adivity
JUT
MWW
WWWN
Imag
Imag
WWW
C
Perceptual dacision-making
Multi-sensory decision- makinn
Memory-pro delayed response (Ring attractor)
CordsNet
Low-rank RNN
CordsNet
Sperse RNN
PC3
PC3
Oulput
PC2
PC1
PC2
PC1
PC2
PC1
PC2
PC1
Stimulus
Stimulus
PC1
Figure 1: A. Overview of the proposed architecture (right) and its relation to CNNs (left) and RNNs (middle). B. Various dynamical regimes exhibited by candomly-nitiialin neural trajectories between a CordsNet and low-rank RNN trained perceptual decision-making task (left), as well as between a CordsNet and a sparse RNN trained on a multi-sensory decision- making task (right). D. Example of CordsNet trained perform memory pro delayed response task by storing a circular variable in memory using ring attractor.
CordsNets. C. Aligned
Unlike regular RNNs, dynamical RNNs typically evolve (and are therefore backpropagated) over hundreds of time steps 22
In CordsNet, which our proposed hybrid architecture, every convolutional layer is replaced by dynamical RNN (Figure IA, right). The activations x represent firing rates rı. In order to preserve the convolutional structure, the input and recurrent weight matrices (Wing and ree) now convolutions (Convinp and ro)) while hipp and b are reshaped to have convolutional structures. This can be summarized as:
T dr:
(Conv b Conving (himp)
dt
In the rest of this work, we will show that this proposed extension highly non-trivial and brings the best (and worst) features of dynamical RNNs into CNNs.
2.1 Dynamical characteristies
Since convolutions are linear operations, they can expressed as a 2-D weight matrix operating on flattened 1-D vector, just like in equation (2). which we describe in detail in Appendix B. However, the convolutional recurrent weight structure of CordsNet spans only a small subspace of all possible recurrent weight matrices, so there is need to verify if such restriction would limit the range of dynamical properties that our networks can express. We know that (sufficiently large) Gaussian-in -nnttaaiini fully-connected networks using tanh activations exhibit different dynamical regimes based on initialization variance 4 61]. We successfully replicate this in CordsNets to express stable, oscillatory and chaotic behaviors (Figure (T)) under the same conditions. Further analysis of dimensionality and autocorrelations can be found in Appendix B.2.
Multiple variants of dynamical RNNs have been trained cognitive tasks in neuroscience 15 and thus we want to know if CordsNets will produce similar solutions when trained on such tasks despite the restrictions from the convolutional weight structure. For this investigation, choose
28

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the same set of five tasks that was previously adopted for analyzing low-rank dynamical RNNs 662]
consisting of a perceptual decision pocision-making task [6]. a parametric working memory task 644.
multi-sensory decision-making task [65) contextual decision-making task [3] and delayed match- to-sample task [66]. We independently train CordsNets, fully-connected RNNs, low-rank RNNs 62]
and sparsely-connecter RNNs [15] on these tasks across different activation functions, learning rates, network sizes and initializations. We find that trained networks of all architectures produce similar neural trajectories when aligned using canonical correlation analysis (Figure [FF) suggesting that CordsNets are able to utilize the same dynamical motifss as other established architectures in neuroscience to perform cognitive tasks. We also provide a more rigorous and quantitative analysis using recently proposed metrics on representational similarity [67 in Appendix B.3 which further supports this conclusion. In fact, from these metrics, we find that our networks produce more similar solutions to fully-connected RNNs than low-rank or sparsely-connected RNNs.
RNNs can perform tasks requiring long-term dependencies by storing information in memory. A dynamical RNN typically achieves this using attractors in neural activity space [68 69] We show that our networks can represent well-known classes of attractors such as ring (Figure [D), line attractors, and discrete fixed-points (see Appendix B.4). Taken together, our results suggest that the convolutional recurrent weight structure of CordsNet does not constrain its expressiveness as recurrent dynamical system.
3 Training and results
We now focus on the functionality of CordsNets as an image recognition model. We train networks of four different sizes, named CordsNet-RX, where X [2, 4, 6, 8) represents the number of recurrent layers. Exact model specifications can be found in Appendix C.I. where we also review important design choices. We train our models on MNIST 770.. Faahioo-MNIST (7), CIFAR-I0 (72), CIFAR- 100 and ImageNet [73], each with dataset-specifi augmentations [74] as detailed in Appendix C.2. The neuron time constant set to 10 ms (constant for all neurons), and the network is simulated with 2 ms time steps. The loss function that we aim to minimize is computed as:
loss 1ogspace sttpsspss
CEloss (output [170: labels)
(4)
+ 1e-3 * MSEloss (activity [290: ,spontaneous)
We first simulate the networks for 200 (time steps 0 to 100) without any input that they converge to some steady state spontaneous activity (spontaneous). A batch of images is then presented for 200 ms (time steps 100 to 200). During this time, the cross-entropy loss computed for the last 60 ms (time steps 170 to 200) and combined using a log-weighted sum. Finally, the networks are simulated for another 200 ms (time steps 200 to 300), and the mean- equared error between activity in the last 20 ms (time steps 290 to 300) and spontaneous activity is added to the loss. This additional term encourages the networks to return to spontaneous activity after stimulus presentation. This loss function has been carefully designed to produce mono-stable solution so that the network may perform accurate inference indefinitely across time, which we will elaborate on the next section. We also provide a detailed ablation study of every coefficient and every term in Appendix D. Simulating these networks for 300 steps across time computationally expensive. For comparison, we compute the total multiply-accumulae operations (MACs) for 79 well-known CNN models found in torchvision. models library and compare them against their parameter counts (Figure 2C). Our largest model, CordsNet-R8, has approximately the same number of parameters as ResNet-18, the smallest model of the ResNet series. In contrast, our smallest model, CordsNet-R2, requires more MACs to simulate than ViT-H-14 14, the largest vision transformer in the model library currently.
While training the models by computing and minimizing the loss function in equation (4) is inevitable, we can reduce the number of training iterations needed by carefully initializing our models. We do this in three computationally cheaper steps (Figure 2 We first train a feedforward CNN model without the recurrent component and replace it with a one-time convolutional layer. We also include batch normalization here to improve training efficiency. In the second step, we fold the batch normalization (75) into the convolution operation and replace the one-time convolutional layers with dynamical linear RNNs. Batch normalization folding is done in the following way:
BN(W cony
b)
7BN
W COBY r ++
BBN
(b- MEN) + BEN
(5)
VOIN -
VOLN
+E
WIS
bunt

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A
3x3, 64
conv 64
3xx3,
3x3, 128
Svggpao 3x3, ,2
CONV 9x3 84, 72
60nV 363 64
conv 3x3, 128, /2
conv 3x3. 128
classificaion
fattonn
softmax
B
convolution normalization
one-time
log (MACs)
R4 R6 R2000
VIT-H-14
CordsNet
convolution
variants
non-linearity
Vision transformers
ResNet-18
Others DenseNet, EfficienNNet, RegNet,
7
nomallzatiio folding
log, (parametors)
analytical intiialization
D
CordsNet-R2
CordsNet-RA
convolution non-linearity
recurrent
Accuraty (%)
8.3%
19.7%
dynamics
00000
000000000
10
10
CordsNet-R8
CordsNet-R8
parametric annealing
Accuracy (%)
29.1%
......... 41.2%
non-linear
convolution
dynamics
0000000000
0008800000
10
Epochs 10
Epochs
CordsNet
Fully connected RNN
Figure 2: A. Architecture of CordsNet-R4, B. Proposed initialization method. A feedforward CNN is first trained (top). The parameters are then used to initialize and train linear RNNs (middle). Non- linearity is then introduced by annealing (bottom). C. CNN models (green/grey) and CordsNet (purple) plotted against parameter counts. D. Validation accuracy of our models trained ImageNet using the aforementioned three steps (dashed lines), compared to training CordsNet (purple circles) and fully-connected RNNs (green circles) directly.
operations (MACs) of 79
dynamical linear RNN is described by equation (2). except that is simply the identity function. Such network can be analytically solved for every time step
W coes Di + L c (W.. -)(--) W Wap
F
(6)
We derive its steady-state solution, which be approximated and compared to convolutional layer:
=I
W conv) -1(b Winphap) (I+ W cunv
W + (b..)(b.. conv
W inp Ding
(7)
cutput
conv input
The parameters of the linear RNN are W cony and b, which optimize by minimizing the mean- squared error between its steady state Teo and the output of the convolutional layer, After optimization, we train the full linear model (end-to-end) using reduced cost function:
loss reduced logspace 0, steps=30) CEloss (output [70: 100] .labels) (8)
present the image for 200 ms (time steps 100) and compute the weighted classification
loss for the last 60 ms (time steps 70 to 100). We do not need to compute the spontaneous penalty term because linear RNNs are almost guaranteed to be mono-stable during training. Finally, the third step, replace the identity activation function with parametric ReLU non-linearity (76)
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Table 1: Test accuracies of CordsNets obtained using our initialization method and after fine-tuning. compared their equivalent feedforward CNN counterparts. For controls, we trained CordsNets directly for the same time the initialization method (C) took. Fully-connected RNNs were also trained with matched parameter counts (R).
Model
Dataset
Control (R)
Control (C)
Initialization
Fine-tuned
CNN
MNIST
97.45
98.07
97.85
98.48
98.56
F-MNIST
74.17
77.75
85.60
88.12
88.44
R2
CIFAR-
45.68
44.44
62.83
71.83
73,89
CIFAR-100
16.19
10.00
24.65
39.56
40.42
ImageNet
0.22
5.55
8.27
14.23
15.57
MNIST
97.81
99.26
98.82
99.24
99.59
F-MNIST
84.39
87.63
91.64
92.68
93.65
R4
CIFAR-10
48.79
60.64
81.38
88.76
90.29
CIFAR-100 100
17.78
16.93
44.28
60.24
63.95
ImageNet
0.31
11.67
19.67
33.78
36.28
MNIST
98.16
99.32
99.26
99.38
99.76
F-MNIST
86.62
90.82
93.36
94.62
95.32
R6
CIFAR-10
52.68
76.82
88.57
91.32
93.87
CIFAR-100
22.88
42.56
56.98
71.32
75.70
ImageNet
0.22
19.33
29.14
50.14
52.06
MNIST
87.51
99.40
99.37
99.36
99.74
F-MNIST
87.51
91.82
94.26
95.88
96.13
CIFAR-I0
53.74
83.15
91.57
94.56
95.99
CIFAR-100
25.11
51.34
66.89
77.32
78.44
ImageNet
0,26
23.21
41.24
57.90
63.16
with parameter a, where PReLU(x) = max(0, (0, a x min(0, We train the networks starting from and gradually reducing a until a at the end of training. This provides us with an initialization of the original model for further fine-tuning, We additionally perform two control experiments, where we train both fully-connected RNNs (with matched parameter counts) and CordsNets directly without our initialization method. In both controls, the number of training epochs determined by matching the time required for CordsNets be trained directly with the time required for CordsNets to be initialized using our method. Additional details about this time benchmark can be found in Appendix C.3. Test accuracies of experiments can be found Table (for ImageNet, the validation accuracy is shown instead). We draw three main conclusions:
Initialization vs Control (C). Our initialization method consistently produces models with higher test accuracies compared to models that were directly trained on every dataset except for MNIST, highlighting the effectiveness of initialization approach. Figure 2 compares the ImageNet validation accuracy of CordsNets trained directly (purple circles) across all epochs against the validation accuracy of the models initialized using our method (purple dashed lines). MNIST classification is simple and does not require extensive feature extraction; it may therefore benefit from a more straightforward training approach.
Control (R) vs Control (C). When trained for the same number of epochs, CordsNets R6/R8 consistently outperforms their parameter-nn matched fully-connected RNN counterparts on all datasets by significant margin. Fully-comnected RNNs perform remarkably poorly on ImageNet due to their poor scaling with image size (Figure DD, green circles). However, the results become more ambiguous when it comes smaller models on simpler datasets. We attribute this to the flexibility of fully-connected models, which allow them to attain early and fast gains in performance, especially when learning simpler features smaller datasets.
Fine-tuned vs CNN. Our fine-tuned CordsNets are able to attain accuracies that are slightly lower than (but reasonably close to) their feedforward CNN counterparts, suggesting that we have successfully trained continuous-tinn dynamical system with competitive image processing capabilities. Closing the performance gap with CNNs remains goal for future work.

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A
Presented stimulus
feature map
CordsNet festure map
Time
Stimulus (400 ms)
Stimulus (400 ms)
Stimulus (400 ms)
No stimulus
(400 ms)
stimulus (400 ms)
Stimulus (430 ms)
Normalized activations
Stimulus sofimax autput
Normalized activations
Sémulus softmax pulput
1000000oo
0.5 oooooooo
pooooooo
pooo0oooo
CNN
booooood
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error bars x300
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Accuracy
CNN
CordeNet
CNN
CordeNet
0
0.2
MSE
0
0.3
0.1
noise 0.5 0.1
noise 0.5
Figure 3: A. Evolution of single interpretable feature map over time the first layer of feedforward CNN (middle) and CordsNet-R8 (bottom). B. Neural activity and softmax output of a feedforward CNN (green), discrete-tine CNN (red), and CordsNet-R8 (purple) response to various stimuli sequences. C. Mcan-squared deviation from noiseless activations of the output layer across 50000 noisy images. D. ImageNet validation accuracies over noise levels.
4 Model analysis
As a continuous-timg model, the feature maps of CordsNets will change over time. We handpick a single interpretable feature map from CordsNet-Rot that depicts a gradual evolution over time when the input image is changed (Figure 33. To emphasize the rich temporal dynamics that trained CordsNets express, we consider an additional comparison with CORNet-Re -T [38] a class of discrete- time recurrent CNN trained on ImageNet. In addition to architectural differences, the two models are trained differently. CordsNet has been trained to classify images over some time interval (and expected to perform accurately indefinitely across time), while CORNet-RT has been trained to classify for a single time step (and expected to accurate for that particular time step).
We compare the temporal activities of the two models (as well as the feedforward CNN) in two scenarios. In the first scenario, an image is presented for 400 ms and then removed for the next 400 ms (Figure 3B, left). The feedforward CNN, having no concept of time, responds with the same activations across time and drops to zero when there is no input. The activity CORNet-RT varies over time and spikes briefly at the time step, during which it is trained to accurately classify the input image. not able to make an accurate prediction at all other times. In contrast, the activity in CordsNet rises and stays at fixed level for as long as the image is presented and returns to some baseline activity when the image removed. In the second scenario, an image is presented for 400 ms, followed by different image for 400 ms (Figure B3, right). For its particular time step.
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Iteration
Iteration 50
lleration 75
lleration 100
lteration 300
True values
Imaginary
ooob
-1
Real
Real
Real
Real
C
Normalized activations
Top 100 eigenvalues
Irish water spaniel: 94.9%
Curly-oosted 0.0%
Dimension
Fast
Layer
0
Skow
400
Im
Layer 2
Re
Re
Irish water spaniel: 0.0%
Curly- retriever: 99.3%
Dimension 2
400
Im
Im
Layer 3
Slow
Dimension
0
400
water spaniel: 37.9%
Curly-coator retriever: 50.0%
Layer
Dimension
Fast
0
Time (ms)
400
Slow model
Fast model
Dimension
Slow
Figure 4: A. Arnoldi iteration 777 applied to a convolutional recurrent weight matrix. B. Model activations in CerdsNet-R4 trained to classify images in time intervals [140 ms, 200 ms] (purple) and [240 ms, 300 ms] (green), along with their top 100 eigenvalues (right). C. Neural trajectories in the final layer of CordsNet-R8 projected onto two dimensions when presented with 3 different images.
CORNet-RT correctly classifies the first image but cannot correctly classify the second image, even though it can correctly classify the second image if it were presented first. CordsNet can correctly classify both images and again maintains the correct classification throughout the duration of the stimulus. These results showcase the autonomous nature of CordsNet compared to discrete-timg CNNs can perform inference indefinitely in time, self-reset a baseline state, and react flexibly to stimuli changes, all while being governed by a single differential equation.
In the feedforward CNN, when temporally uncorrelated white noise is applied to the input image each time step, model activations deviate from baseline noiseless values, which results in a decrease in classification accuracy (Figure FC-, green). When the same noisy images instead presented to CordsNet, its deviations are attenuated by more than an order of magnitude compared to those of the feedforward CNN. It also robustly maintains its classification accuracy at high noise levels (Figure 3C- D. purple). This is a result of natural noise attenuation from continuous-ting dynamics and filtering by the recurrent weights. We rewrite the dynamies
update equation depict these effects:
Alltering (next stop)
A4
Al
r++1
Ft +
(Wr +hr+
Wing himp + noise)
attenuation
We next present an analysis of our trained models from a dynamical systems point of view. For dynamical RNN, this typically involves performing some form of matrix decomposition on its recurrent weight matrix or applying dimensionality-eduuee CordsNet, both of these approaches are particularly challenging due to the size of the networks. Memory limitations prevent the full recurrent weight matrix from being expanded from its kernel representation. One solution that we found is compute the eigenvalues of the recurrent weight matrix directly from its kernel form using Aroldi iteration [77] (Figure [A)), which we use to uncover an important dynamical motif present in our trained networks. We notice that the dynamical characteristics of our networks are different when we train them to correctly classify images at different times. When trained to classify early, the networks exhibit an oscillatory behavior, but not when trained to classify late (Figure 4B, left). We attribute this to the effect of transient overshooting,
techniques on neural trajectories. For

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A
Numerical quantificaton task
500
Non-match trials only
Noural aclisty
Example inputs
Output
B0g
1002 ms
12:0 mi
Visuel features of varying quanittess
Lever press
1000 Delay period (ms)
Categorical discrimination task
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1000
my
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1000
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Laft
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stimuli moving seccade
at varioss coherence levals
Video stimulus
Sümulus period
1000
Figure 5: A multi-area model consisting of CordsNet-R8 connected to a fully-connected RNN trained on various cognitive tasks in neuroscience. A. Numerical quantification task where monkeys required to remember the number visual features on the screen. B. Discrimination task where monkeys must discern whether image is predominantly monkeys have to saccade to the location previously shown stimulus. D. Evidence integration task where monkeys required to tell if random dots moving left right.
dog. C. "Delayed-respons, task where
where being in the oscillatory regime results in an early overshoot in activity, thereby speeding up propagation the signal. In the linear approximation, according to equation (6), this would manifest if the eigenvalues of (W comv I) of the model trained for faster inference contain larger imaginary components. We confirm this to be the case (Figure 4B, right). is also possible to perform dimensionality reduction within reasonable memory limits, but these methods require some adaptation for convolutional operations. Therefore, we release toolkit consisting of the aforementioned functions that are specifically tailored for convolutional weights. These convenient tools unlock many possible approaches for analyzing CordsNets. For example, we identify a particular dimension in CordsNet-R8 R8 which represents whether the model perceives image Irish water spaniel or curly- curly-coated retriever (Figure 4C).
Applications
CordsNet is an ideal front-end of any RNN dynamical system that requires mmage-computability. To demonstrate this, we connect the final layer CordsNet-R8 with fully-connected RNN (with 512 neurons) and train only the fully-connected RNN on set of cognitive tasks that explicitly requires visual information processing (Figure 5).. We use the actual images the monkeys see the input to our multi-area model. This is done either by obtaining the stimuli set from the authors of the experiments or generated according to the specifications described in the original experimental papers 80 In our trained models, we found interpretable neurons in the fully-connected RNN layer, such as neurons tuned stimulus quantity (Figure 5|A right), cats or dogs (Figure 5 right), spatial positions the screen (Figure 5' right) and motion direction (Figure D. right).
78 79

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A
B
Noural recordings primate system
CordsNet activity
Neural recordings
V1
V2
Normalized acteviy:
300
1
0
Model activations from artifiiaal neural network
Time since stimulus (ms) 300
C
Area V4 (Penultimaee layer)
Area IT (Final layer)
Pearson correaaion
0.13
0.1%
R2
R4
R8
RB
R2
R4
R6
RB
Base score between CordsNet activations and neural recordings Score obtained from temporally shuffled data
Figure 6: A. Framework of Brain-Scort (Vision). B. Normalized CordsNet- activity in the last two layers compared with experimentally-rrcoreei
[82] normalized activity in visual areas V4 and IT. C. Similarity metrics between CordsNet model activations and neural data (dark purple) and temporally-shuffle neural data (light purple). (paired t-test, ** < 10-1, *** < 10-
Brain-Score is benchmarking framework designed to evaluate the performance of artificial neural networks in terms of their ability to model and predict neural and behavioral data from the brain [39] In particular, model activations from CNN layers have been compared with time-averaged neural data from various parts the primate visual system (Figure ( With our continuous-time model, we want to know if our models are able to model temporal signatures in brain activity. To do we omit the averaging step when computing Brain-Scores (see Appendix E for details). Using the same training- teatt test splits, we additionally fit the same model activations with temporally shhuffllly neural data. Due to the lagged response neural data, we have to temporally shift our model activations such that the time in which activity rises in response an input is aligned with neural data. There are occasions where no shifting is necessary, such as in the final two layers of CordsNet-R8 FFigure(63) on V4 and ITT neural activity [82]. All CordsNet models score higher unshuffled data across V4 and IT (paired t-test, all p-values < 10-4, Figure BC), suggesting that the models are capturing temporal structures within the neural data. For the sake of transparency, we also state that we have performed the same test on a different dataset with VI and activities [83 but obtained inconclusive results.
Discussion and conclusion
We have presented CordsNet, hybrid architecture combining the strengths of CNNs and dynamical RNNs to process visual information with continuous-ting dynamics. CordsNet exhibits various dynamical behaviors, such as oscillations and chaos (Figure [BB). In our analysis of its image processing capabilities, we showcase its ability to classify images indefinitely across time (Figure B3 right), well its robustness noise (Figure3P, right). We have also effectively used CordsNet as means to build image-computable dynamical systems capable of performing cognitive tasks (Figure Finally, CordsNet has been successful at modeling the temporal signatures of neural activity in higher- order visual areas like V4 and IT (Figure (C).
Limitations. The main limitation of CordsNet lies in its substantial memory requirements (Figure 2C). Memory-efficient training algorithms have been explored [84]. Another limitation is the difficulty of analyzing CordsNet dynamical system, primarily due to how it is unfeasible to convert the recurrent kernel weights into its corresponding full-size recurrent weight matrix.
Conclusion. CordsNet bridges crucial gap between dynamical systems and vision neuroscience. While most of cognitive neuroscience continues to build upon decades of RNN research, vision neuroscience remains dominated by CNN models due to the inability of RNNs to efficiently process visual information. CordsNet has resolved this limitation by being a dynamical RNN with image processing capabilities.
10

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Author contributions
W.S. designed the architecture. W.S, A.B. and P.R. trained and analyzed the networks. All authors designed the study, took part in discussions, interpreted the results, and wrote the paper.
Acknowledgments and Disclosure of Funding
This work was supported by the NIH grant RO1MH062349, Office of Naval Research grant N00014, James Simons Foundation grant CULM-00003138 and NYU High Performance Computing. A.B. was supported by the Swartz Foundation.
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NeurIPS Paper Checklist
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer:[Yes]
Justification: We have made 6 claims. The claim on dynamical expressitivity analysis can be found in Section 2, the claim on new training algorithm can be found in section 3, the autonomous nature of our model can observed Figure 3, analytical toolkit described in Section 4, the image-computable models can be found in Figure 5, and the neural data analysis can be found in Figure
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17


Reviews:
[{"review_id": "2Z4bDLin1R", "reviewer": "Reviewer_A7p2", "summary": "Inspired by neuroscience, the authors propose to introduce recurrent connections in conventional convolutional neural networks (CNNs). The resulting model shows comparable performance with regular CNNs, but exhibits higher noise robustness. The authors also developed a toolkit to analyze the resulting architecture, using iterative and dimensionality reduction methods. The architecture is used to model complex cognitive tasks, mimicking the higher order visual areas in the monkey.", "questions": "Questions\n\n- Isn't the task in Fig 5D handled by area MT? Have you considered computing the BrainScore for this? In general, can results in Fig 5 be compared to the experimental literature?\n\nComments\n\n- There is an earlier recurrent CNN than the one cited: Ming Liang and Xiaolin Hu. Recurrent convolutional neural network for object recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3367–3375, 2015.", "limitations": "The discussion/conclusion provides adequate assessment.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- This paper seamlessly weaves inspiration from neuroscience, dynamical systems theory, learning algorithms, analysis methods, performance, and modeling and replicating results from neuroscience experiments.\n\n- Introducing continuous dynamics into CNN+RNN seems to exhibit really interesting properties.\n\n- Analysis of the dynamical behavior by finding the eigenvalues directly from the recurrent weight matrices is interesting. This has been used to uncover some really interesting dynamical patterns dependent on the time of classification (fater or slower inference).\n\n- The use of the model in various visual cognitive tasks (in V4 and IT) is also quite impressive.", "weaknesses": "- Too much technical details are hidden in the appendix. Please at least provide a brief sketch in the main text.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We truly appreciate the reviewer for the encouraging and positive review.\n\n**Technical details in the appendix**\n\nThe reviewer has expressed concern that important technical details that are in the appendix should briefly be mentioned in the main text. We agree with this point and refer the reviewer to the general rebuttal for all the changes we plan to make to our submission.\n\n**Random dot motion task and area MT**\n\nThe reviewer has correctly mentioned experimental literature pertaining to evidence of neural activity tuned to random dot direction and coherence in area MT in the dorsal stream. In general, the RNN that we append to the back of our model represents either a decision-making part of the brain, such as the prefrontal cortex, or a motor region such as the frontal eye fields (which is still prefrontal). While MT contains direction-tuned cells, the eventual motor response (in the form of eye saccades) would likely be similarly tuned. However, this is ultimately an (unsubstantiated) interpretation on our end. A more rigorous analysis is required for any conclusions to be drawn.\n\n**Interpretation of results in Figure 5**\n\nThe main goal of the results in Figure 5 is to illustrate the point that we can now build completely continuous-time models that accept real images as an input, rather than abstract one-hot vectors as found in many past works. We do provide some interpretation of the solutions we found, but to draw any conclusions about the brain from this would require a much more in-depth analysis that is currently outside the scope of this work. We do not have the neural data to compute BrainScore, and we foresee that the dataset would not be big enough (small number of coherence levels) for strong statistical significance. \n\n**Missing citation**\n\nWe thank the reviewer for pointing out this missing citation, we will add this citation into our introduction. We will also thoroughly find and include other relevant literature in our final version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EyTXjaZDUQ", "reviewer": "Reviewer_8dkQ", "summary": "In this work, the authors proposed CordsNet (Convolutional RNN dynamical system), which provides incorporates a conventional neuroscientific model of RNN incorporating the convolutional layer.", "questions": "1. Is the analytical formulation of CordNets limited to only linear layer?", "limitations": "The authors discussed the limitations.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The authors analyzed the proposed dynamic convolutional RNN from different aspects. \n\n2. Derived batch normalization for the case of linear dynamic RNN", "weaknesses": "1. The authors failed to compare it with a series of works named Deep Equilibrium Models (DEQs), which model a layer or a block of layers including the non-linearities as a fixed-point problem.  CordsNet is limited to the linear part of the layer while DEQs are not. And DEQs achieved SoTA performance on many large-scale tasks.\n\n2. In the comparison seems Convolutional LSTM/GRU is missing, which is an important family of RNN with Conv layers. Could the authors explain why they are not necessary to be included in the benchmarks?\n\n3. The contribution points are really scattered and don’t feel elaborated enough. Maybe it’s more suitable as a journal paper with more elaborated experimental details for each section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for raising these important points in their review. Our general interpretation of the reviewer's concern is that this work failed to provide what seems to be minimally-required benchmarks, which in turn translates to lack of evidence for any performance improvements. The reviewer is also well-read in the field of convolutional, recurrent and fixed-point architectures, and thus was able to pinpoint key literature that overlap with our proposed work. From this point of view, it can be hard to find significance in this work.  \n\nAs such, we would like to provide a personalized summary of this work that brings the reviewer's aptly-raised concerns into perspective and explains why we believe our work is still a significant contribution despite those concerns. \n\n- **Dynamical systems in neuroscience** study how neural activity evolves over time and how these changes relate to brain functions. They are prevalent in neuroscience, recently in the form of continuous-time RNN dynamical systems with biological constraints such as realistic neuron time constants.\n- **Training with a handicap.** Such dynamical RNNs are harder to train than vanilla RNNs in machine learning, because of the continuous-time property that inflates the number of time steps when coupled with the fast time constants present in biological neurons. Since vanilla RNNs are already ill-suited for image processing, this makes dynamical RNNs even harder to train on ImageNet, and for this reason no such model has been proposed to date. \n- **Motivation and potential.** Yet, there is a strong need for such a model to be built, so that decades of dynamical systems theory can be applied to a model that can actually process natural images. Such a model would be studied by the theoretical neuroscience community to understand continuous-time dynamics underlying biological visual information processing, which we hope will give rise to new ideas and theories to be tested.\n- **Defining expectations.** To that end, our objective is to train the first dynamical system to classify ImageNet at an accuracy that would be deemed as “comparable to CNNs”. We approach this goal with the understanding that the constraints we impose on our model are likely to be detrimental to performance, and we will most likely not be achieving anything SoTA outside the field of neuroscience at this stage; we therefore omit benchmarks typically found in machine learning literature.\n- **Significance of our architecture.** CNNs and convolutional RNNs have already been proposed and extensively analyzed in vision neuroscience as candidate models of the visual system. The logical approach would therefore be to build a continuous-time dynamical extension of these models. Such a model would not only build on the work on convolutional architectures by the vision neuroscience community, but also incorporate the legacy of dynamical systems from the wider neuroscience community. \n- **Evaluating our success.** The best top-1 accuracy achieved by our models on ImageNet (<60%) is modest compared to what vision models today can do. Understandably, as a result of the aforementioned biological constraints, our models fail to outperform basic CNNs with the same convolutional layers (but we are close). But ultimately, we can reasonably say that the models that we trained are indeed performing some meaningful processing of visual information, and that we have successfully built a dynamical system that can classify natural images.\n- **Focusing on the objective.** Finally, we dedicated the majority of the main text, including figures 3,4,5 and 6, to show how our trained model can be studied as a continuous-time dynamical system in various fields of neuroscience, thereby highlighting its potential applications and how it opens up new avenues of research in these areas.\n\nWe hope that this summary is able to help the author gain a different (and hopefully more optimistic) perspective of our work. We now address specific points raised by the reviewer below.\n\n**Issue on DEQs**\n\nWe thank the reviewer for bringing DEQs into the discussion. Due to the character limit, we refer the reviewer to our discussion on DEQs in the general response.\n\n**Issue on linearity**\n\nCordsNet is nonlinear in the exact same way that RNNs or DEQs are nonlinear. The evolution of activity across time in CordsNet is described by equation (1) in the main text, which includes a ReLU activation function. While training the nonlinear model, we split training into several stages (Figure 2B), and one of the stages involves linear models (lines 142-152), which may be the source of this misunderstanding. \n\n**Comparison with convolutional LSTM/GRU**\n\nThere are two key reasons why comparison with gated RNNs is not necessary. \n- Artificial gating mechanisms are not biologically-realistic and do not fall within our objectives.\n- We are trying to model the biological visual system, which is (generally) not the part of the brain that stores memories or makes decisions based on long-term dependencies. Our task is classifying ImageNet, where gating mechanisms may not offer much advantage.\n\n**Issue on scattered points**\n\nWe thank the reviewer for the suggestion. Overall, we feel that this work lies comfortably in the intersection of machine learning and neuroscience, and that there would be an audience for this work in this conference. We also acknowledge the reviewer’s concern that the work presents too many points that have not been sufficiently covered. CordsNet is both a dynamical RNN and a CNN, where each architecture, on their own, has been studied by the neuroscience community for years. As such, the narrative of our paper is focused on introducing the model, how it compares with past RNN or CNN models in neuroscience, and briefly demonstrating the many ways in which new research directions can be explored using this hybrid model. It is also why we developed tools to help analyze our model (lines 214-216).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7THiib3qe9", "reviewer": "Reviewer_7DkU", "summary": "The paper proposes a recurrent convolutional neural network architecture and training algorithm that simulates the biological visual system of mammals. The technical novelty is mostly in the training algorithm which is meant to mimic biological systems, with a first stage of spontaneous activities followed by learning and then again spontaneous activities. Since this process is too computational intensive, the authors introduce approximations, such as initializing from a supervised trained network. The empirical analysis shows that the method is not too far from standard supervised learning while being robust to noise and while exhibiting a behavior that better matches biological systems. Also, the approach is used to predict brain activity data as an application. \n\nDisclaimer: My background is machine learning rather than comp. neuroscience. I am not qualified to assess the impact that this work can have in that community and my assessment is mostly limited to the ML side of this contribution.", "questions": "I am curious whether the model oscillates when there are multiple interpretation of the input. In the simplest setting, the input could be linearly \"mixed-up\". I am also curious whether \"easy\" examples require less iterations to converge.", "limitations": "No concern.", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "+ simple and intuitive architecture\n+ generally well written and fluent paper\n+ the loss definition (in its original formulation of eq. 2) is very interesting and novel\n+ nice analysis of how this work relates and contributes to the field of comp. neuroscience\n+ overall motivation and research topic", "weaknesses": "- important technical details are missing. For instance, in sec. 2 only few lines are used to give a high level description of the architecture. Too much material that should be in the main paper is placed in the appendix (e.g., comparison to supervised CNNs mentioned in the abstract) . Overall the paper is not self-contained and misses critical details.\n- there is a big approximation between the original loss function and what actually is minimized in practice. If the goal is to simulate a biological system, it is unclear to me how the approximation made fit in the context of that goal.\n- overall, claims are not entirely supported. For instance, it is not true that on ImageNet the performance is \"comparable\" to a standard CNN. A difference of 5% is big in that context. Some accomplishments of this work should be toned down a bit.\n- missing references: There is prior work on recurrent CNNs that is not cited. For instance, \nRecurrent Convolutional Neural Network for Object Recognition by Liang et al. CVPR 2015 uses the same architecture as far as I can tell.\n- missing ablations: The authors well ablated the contribution of the recurrent and convolutional part of their model. However, their approach makes lots of design choices which are not very well justified empirically. For instance, what happens if fewer recurrent iterations are used? or what happens if only the top-most fully connected layer are made recurrent? what happens if some terms of the loss are removed? etc.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the helpful suggestions. In response to these points, we have performed additional analyses and made several changes to our submission, as explained below (or referred to general response). \n\n**Important information in appendix**\n\nWe thank the reviewer for raising this important point. We would like to direct the reviewer to our general response, which provides a detailed plan on how we plan to rebalance the results in the main text and the appendix. \n\n**Issue of unsupported claims**\n\nWe agree with the reviewer that the performance of CordsNets fine-tuned on ImageNet is lower than their CNN counterparts with the same convolutional layers, and that this discrepancy may possibly be large enough to challenge our claim. We have made the necessary changes to the submission to account for this, as detailed below. \n\n- Line 163 updated to read, \"We also find that our fine-tuned models perform only slightly poorer than their initial feedforward CNN counterparts, which nonetheless represents a significant improvement over existing continuous-time dynamical models in visual processing.\"\n- Line 387 updated to read, \"Our fine-tuned CordsNets are able to attain accuracies that are only slightly poorer than their feedforward CNN counterparts.\"\n\nWe feel that the term \"comparable\" is modestly subjective. We initially chose this word because of the current inability of continous-time dynamical systems to process visual information, as seen in the third column of Table S5. Dynamical RNNs are barely beating chance level in a classification task with 1000 classes, which is rightfully \"incomparable\" to CNNs. By being in a very broad performance range (50% to 80%), we feel that we may use the term \"comparable\" in this very specific context. However, we will still make the changes above for clarity.\n\nAt the same time, we wish to also review and defend the other claims that we have made about this work, which we hope that the reviewer would agree with.\n\n- **Dynamical expressivity analysis.** We claimed that we have rigorously explored the possible dynamical regimes that our model can express, and also made comparisons to other model architectures. We provided instances of different dynamical regimes in Figure 1B. In addition to CordsNet, we also trained low-rank, dense and sparse RNNs (Section B.3, which is in the appendix in the initial submission) on five different cognitive tasks in neuroscience (Section B.4), for three different network sizes (Section B.5). We then compared the activity trajectories of all these trained models (results from Figures S2 and S3). \n- **New training algorithm.** We have provided full details of our training algorithm in Section 3 in the main text. In addition, we explain certain design choices and perform ablation studies for the loss function in the general response.\n- **Autonomous and robust inference.** We show in Figure 3B that our model is able to reset to baseline levels after stimulus presentation, and can then accurately classify a new input image. This supports our claim on autonomous inference. We also provide evidence that our model is robust to noise, due to the \"evidence integration\" mechanism that is present in dynamical systems. This is explained in lines 190-196, and equation (7).\n- **Analytical toolkit.** We claim to provide a toolkit consisting of Arnoldi and power iteration algorithms and partial SVD specifically for convolutional architectures (i.e. our model). We demonstrate the effectiveness of our implementation of Arnoldi iteration in Figure 4A, and applied partial SVD to a particular scenario in Figure 4C. \n- **Image-computable models.** We have trained CordsNet appended to a fully-connected RNN on four actual tasks in neuroscience literature (Figure 5, left), and provided evidence of our trained models (Figure 5, right). At the same time, we acknowledge that this analysis is brief, which is why we make no predictions or neuroscientific claims based off our results here. An in-depth investigation on this topic would be outside the scope of this work. The intention is to show the potential applications of CordsNet as a multi-area, image-computable model, which we have delivered.\n- **Prediction of neural activity.** We leveraged on a robust and popular benchmark platform known as BrainScore to compute similarity metrics between model activations in CordsNet to actual neural activity recorded in the visual system of the macaque monkey (Figure 6). We show statistical significance in our results.\n\nWe hope that with this overview, the reviewer would agree with the stated accomplishments of this work.\n\n**Missing references**\n\nWe thank the reviewer for the citation. and thoroughly comb through relevant literature in our final version.\n\n**Motivation for loss function and ablation studies**\n\nWe thank the reviewer for this suggestion. We have done an extensive ablation study for our proposed loss function, as detailed in the general response.\n\n**Model behavior subjected to ambiguous input**\n\nThe concept of multistable perception has been extensively studied in neuroscience, with many theories proposed based on dynamical systems. Coincidentally, we have shown preliminary results on this phenomenon in Figure 4C, where the activity just goes to some interpolated activity space between the two interpretations, which is currently still a steady-state response.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cRCOKO1iAT", "reviewer": "Reviewer_7SU8", "summary": "Biological neural networks are continuous-time dynamical systems. However, the previous models that are carefully designed to explain biological networks are discrete-time systems or even non-dynamical ones like convolutional neural networks (CNNs). However, the model that best explains biological vision and achieves high performance on downstream tasks is CNNs. This work fills the gap by creating novel networks that combine CNNs and continuous-time recurrent neural networks called CordsNet. CordsNet is evaluated on a standard ImageNet classification task and also across several standard cognitive neuroscience tasks.", "questions": "1. Citation [36] is wrong. That one is CORnet-S, not CORnet-RT [1].\n\n2. Why are brain scores low compared to the leader board on the same public datasets? Is the score a neural predictivity score defined in the Brain Score paper [2] or a different metric?\n\n3. How are brain-scores of the proposed method compared to CORnet-S and CORnet-RT? How many total parameters of the CordsNet compared to those models?\n\nReference\n\n[1] Kubilius, Jonas, Martin Schrimpf, Aran Nayebi, Daniel Bear, Daniel LK Yamins, and James J. DiCarlo. \"Cornet: Modeling the neural mechanisms of core object recognition.\" BioRxiv (2018): 408385.\n\n[2] Schrimpf, Martin, Jonas Kubilius, Ha Hong, Najib J. Majaj, Rishi Rajalingham, Elias B. Issa, Kohitij Kar et al. \"Brain-score: Which artificial neural network for object recognition is most brain-like?.\" BioRxiv (2018): 407007.", "limitations": "A lot if memory is required during training.", "rating": 9, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "1. Biological realism during inference: CordsNet might be one of the most biomimetic deep learning models, as its dynamics are more realistic than other state-of-the-art models like CORnet-RT and CORnet-S. It is motivated by neural dynamics models.\n\n2. A broad range of cognitive tasks are studied: Beyond the ImageNet classification task, this work investigates the responses of CordsNet on several standard cognitive tasks that neuroscientists use to study humans and animals.\n\n3. Benefit of recurrence dynamics over CNN: CordsNet demonstrates the advantage of classification with noisy images over CNN. This suggests the missing piece to make computer vision models more robust, like humans.\n\n4. Great visualization: The visualization is very insightful and provides intuition on how CordsNet works.", "weaknesses": "1. Other methods like CORnet-RT and CORnet-S are not evaluated on the cognitive tasks that plug RNN. Therefore, we cannot see if CordsNet is better than others on this benchmark. I believe they can perform the same tasks but they may have different characteristics or different representations than CordsNet\n\n2. CordsNet is trained to predict stimuli at a certain duration, but it cannot accurately predict other durations. \n\n3. The task is supervised image classification where the models learn from rich human labels. Moreover, brains perform with minimal or no strong supervision from others. Instead, brains may learn in a self-supervised manner. It is possible to explore self-supervised learning tasks that are more like what brains do.\n\n4. Embodied tasks are not included: To study the benefits and characteristics of neural dynamics, embodied tasks may be more interesting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are heartened by the extremely positive review and high regard the reviewer has for our work. We thank the reviewer for their time and encouragement.\n\n**Comparison of CORNet on tasks in Figure 5**\n\nWe agree with the reviewer that CORNets could potentially arrive at different solutions compared to CordsNets. This would be in line with our efforts to bring analyses that are commonly found in dynamical systems (such as the cognitive tasks in Figure 5, where a dynamical RNN would have to be plugged) to the CNN vision neuroscience community. It is exactly our hope that our analysis of CordsNet as a dynamical system would spark such applications for CORNets and other models in vision neuroscience.\n\n**Performance across time of CordsNet**\n\nWe note that since CordsNet arrives at a steady-state at time of inference, it can indefinitely maintain the accurate classification for as long as the image is still presented, even far beyond the trained duration. In addition, CordsNet is also able to reset back to baseline when an image is removed, and correctly classify another new image when it is presented at a later time. These results can be found in Figure 3.\n\n**Self-supervised learning and embodied tasks**\n\nWe fully agree with the reviewer that the literature of CNNs in neuroscience have progressed far beyond straightforward supervised learning. As a starting point for incorporating dynamical system, we believe that our work will motivate future endeavors on these ideas (including for ourselves).\n\n**Citation error**\n\nWe thank the reviewer for the clarification.\n\n**BrainScore clarification**\n\nIn order to evaluate our models' ability to capture temporal signatures in neural data, we had to extend the BrainScore similarity metric to account for time. \n\nThe original formulation performs a partial-least-squares fit between CNN activations, with shape [number of images, number of nodes] with neural data, with shape [number of images, number of neurons]. This is done with time-averaged neural activity. \n\nHere, we make the most minimal extension, so that we are now fitting CNN activations, with shape [timesteps $\\times$ number of images, number of nodes] with neural data, with shape [timesteps $\\times$ number of images, number of neurons], which means that the time-averaging step is omitted. Without the time averaging step, the BrainScore drops.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fE1WN2mWPG", "reviewer": "Reviewer_UEUJ", "summary": "The paper proposes a hybrid architecture that integrates continuous-time recurrent neural networks (RNNs) with convolutional neural networks (CNNs), named CordsNet, to improve biological realism in vision models. The authors claim that CordsNet matches CNN performance on benchmarks like ImageNet while showing increased robustness to noise. They also present a toolkit for analyzing these models and demonstrate the model's ability to capture time-dependent neural activity.", "questions": "What’s your model’s advantage over a simple sequential design of convolutional preprocessing followed by a dynamical RNN?", "limitations": "1. The use case of the proposed model is limited and the advantage over discrete-time models is not clear. I believe continuous-time vision tasks are needed to fully demonstrate the effectiveness of this method. Tasks such as video prediction, and video feature extraction might be good candidates. Small-scale experiments can be also conducted on augmented image datasets with more temporal information.\n2. The proposed model is novel, but if it provides advantages over existing hybrid models remains questionable. It shares similar design intuitions to convolutional RNN, effectively mimicking the visual preprocessing of visual information followed by temporal processing in a later stage in the brain. Such designs are widely used in the form of CNN + vision Transformer in video processing (XMem, Cheng & Schwing, 2022) and video generation (Video Diffusion Model, Ho et al., 2022) domain where both spatial and temporal processing are conducted concurrently.  However, a comparison with such an existing design is not provided. The provided CNN baseline is relatively weak compared to the above-mentioned recent architecture. \n3. Intuitively the spontaneous activity attraction is functionally similar to weight decay, whether this regularization term provides advantages remains questionable. Ablation studies can be conducted to provide more rationales behind your design choice since multiple design options are present in your model.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The proposed model demonstrates connections to neural behavior and shows effectiveness in dealing with continuous-time modeling. It might inspire future biologically feasible network design.\n2. The sharing of weights for convolutional and recurrent operations is novel since these are typically done sequentially in a backbone.\n3.  The introduction of tools for analyzing convolutional structures in dynamical systems is potentially helpful for future research, but more details are needed.", "weaknesses": "1. The comparative result can be conducted more comprehensively. Some comparative results are not conducted on models with the same temporal processing power (Figure 3 only CNN shown).  Comparison to similar temporal models such as convolutional RNN  and vanilla RNN are lacking. These models can likely exhibit similar behaviors while requiring a less complex training scheme. \n2. The choice of image datasets is questionable. The dataset is in discrete-time space, and continuous-time training might not provide any advantages over traditional CNN. The proposed model enables intra-batch information flow between images, which can provide certain robustness to discrete data. Still, the advantages should be more prominent when images in a batch are naturally sequential and have dense temporal information.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and insightful suggestions. For the sake of consistency across reviews, we would first like to clarify a few definitions used in our response:\n- We think that the reviewer is referring to \"convolutional RNN\" as a CNN connected to an RNN (typically LSTM in literature but could be any type of RNN). This is most likely the widely-adopted definition that the reviewer is using. However, in our response, we will refer to these models as CNN-RNN models.\n- In contrast, we will refer to convolutional RNNs as RNNs whose recurrence is represented by a convolution, just like in this work. This is also consistent with the terminology used by other reviewers.\n\n**Advantages over CNN-RNN**\n\nIn the first point raised in the weaknesses section, as well as in the question section, the reviewer made a sharp observation that simply using CNN-RNN architectures would be sufficient to reproduce the results in Figure 3, and therefore questioned the advantage of our proposed model. We raise several orthogonal points below that we hope would help the reviewer see our work in a better light.\n\n- **Not all results can be reproduced by CNN-RNN.** We agree that the results in Figures 3 and (quite likely but we cannot confirm) Figure 5 can be obtained by the CNN-RNN architecture, but the results in Figures 4B and 6 remain exclusive to our continuous-time model. The phenomenon in Figure 4B is attributed to the temporal lag that arises due to every layer being continuous in time and require time to ramp up. Figure 6 compares neural trajectories across time in CNN layers with neural activity in the macaque ventral stream (their biological visual system), but the CNN in a CNN-RNN architecture is static in time.\n\n- **Applying results from CNN and RNN neuroscience.** CNNs and RNNs have both been proposed as models of the brain (or parts of it). By building a hybrid model, we may integrate results from both fields which may lead to unified results and potentially generate new hypotheses about how the brain works. \n\n- **Redefining expectations for this work.** If the end goal of this work is to find a model that can perform the feats that we show in the paper, then CNN-RNNs would probably have an edge over our model.  But our goal here is to build models of the brain and help improve our understanding of how continuous-time dynamics work in a competent model of vision. This means that we incorporate biological constraints in our models, even if they are detrimental to final performance. Just like how the brain operates in continuous-time, including its visual system, we build a model that also runs continuous in time.\n\n- **CNN-RNNs requiring less complex training.** We absolutely agree that the computational resources required to train a model with a static CNN is considerably less than the resources required to train CordsNet. We like to view this as a positive aspect of our work, rather than a negative one. We are knowingly building a model that is harder to train, with constraints that are likely to impede training and testing performance (short inference time, continuous-time and recurrent properties offering nothing to image classification performance), so that we can obtain a model that can be used to study the dynamics underlying visual processing in a continuous-time dynamical system. In fact, we consider the success of our training, using our proposed algorithm, to be a key result of this work. \n\n**Suitability of ImageNet and other use cases**\n\nThe reviewer has raised concern that ImageNet, being a static dataset that does not change in time, may not be the most suitable task for showcasing how our model works. We agree with this viewpoint and are impressed by their pursuit for engineering optimality. Fundamentally, the theoretical and experimental sides of neuroscience work closely with each other. New experiments are conducted from proposed theories, and in turn new models are built from experimental results. Similarly, this work is designed based on influences from a variety of past works in neuroscience. There has been decades of results on dynamical models of the brain, which are still being worked on today. In recent years, CNNs have gained traction in vision neuroscience, and one of the main driving forces of this is the BrainScore platform [(link here)](https://www.brain-score.org/vision/), where CNNs are trained on ImageNet, and their neural activities are then compared to neural data. This also comes with years of results reported by various neuroscience groups. Our goal here is to bridge the gap between results from dynamical system models and results from CNN models for vision. To that end, training and evaluating our model on ImageNet would best facilitate this.\n\n**Spontaneous activity penalty**\n\nWe thank the reviewer for the suggestion. We have performed ablation studies in the general response that would help clarify your doubts on the impact of this term. \n\nWe would also like to specifically comment on the reviewer's intuition, which is largely in the correct direction. We cannot make strong claims for non-linear models, but in the linear case of our model, it would be the largest eigenvalue of the recurrent weight matrix that would determine whether the model is monostable or not. This is definitely related to the magnitude of the weights to some extent, just as the reviewer has suggested.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their time and effort in reviewing our submission. We will address common issues here.

**Important details in Appendix**

Reviewers A7p2 and 7DkU raised the issue that important results are reported in the appendix, and recommended for their (minimally) brief inclusion in the main text. We agree with this point, and provide a summary of all the changes to our submission below. This summary includes additional changes requested from other reviewers as well.

1. Introduction (additional 3-4 lines)
- mention existing works on recurrent CNNs in machine learning literature
- briefly mention DEQs and their relevance to this work (complete review in the appendix)

2. Model architecture (additional 15-20 lines)
- introduce the model architecture in a more complete way
- include details of the analysis from Appendix B

3. Training and results (additional 5-10 lines)
- move the technical derivations to the appendix (lines 139-156) and briefly describe them in the main text
- include brief descriptions and state results from the ablation studies in the appendix
- move parts of Table S5 into the main text and interpret the results

4. Model analysis (reduced by 15-20 lines)
- remove Figure 3A as Figure 3B illustrates a similar point
- remove explanations about Figure 3A

5. Applications  (reduced by 5-10 lines)
-  remove the methodology for BrainScore in the main text (lines 236-242) and instead elaborate in detail as a section in the appendix

These changes are expected to stay within the 9-page limit. However, if our work is accepted, we would also make full use of the additional page to remedy the information load in the appendix.

**Comparison to DEQs**

Reviewer 8dkQ commented that comparing our model to DEQs is necessary and important. DEQs are inherently similar to RNNs, and therefore CordsNets by transitivity. To address this point, we will mention DEQs in the introduction in the main text, as well as a more in-depth review of the differences between CordsNets and DEQs as a subsection in Appendix A. Fundamentally, DEQs are focused on arriving at a fixed point required for the completion of some task. In contrast, we proposed CordsNet as a model of the biological visual system, and we are concerned with the model activations before, during, and after steady-state inference has been achieved, so that we may compare them with experimental data and generate new hypotheses on how the brain works. 

- CordsNet expresses a range of dynamical behaviors depending on the required task, including oscillatory, chaotic and stable dynamical regimes (Figure 1B). Even within the stable regime, which results in fixed points, they manifest in different patterns, such as point, line and ring attractors (Figures 1C, 1D and S3). Evidence of these temporal behaviors have been found in the brain, which is the focus of this work, even though they may not be ideal for tasks. On the other hand, DEQs prefer fixed points only, in the true spirit of machine learning that seeks optimal performance.

- When DEQs complete a task, the simulation ends. They are run again from scratch for the next task. In contrast, a biological brain runs perpetually, even when there are no tasks. CordsNet follows the same principles, and returns to some baseline activity by itself without external interference after the input image is removed (Figure 3B). Another image can then be presented for future inference.

**Ablation studies**

Reviewers 7DkU and UEUJ have requested for certain ablation studies to be performed in order to justify several design choices in our models. We carefully disect each term in our loss function, as shown in equation (1) of the rebuttal PDF.  Firstly, we note that the goal of our model is to bring together the results from CNN vision neuroscience and RNN cognitive neuroscience communities. To that end, we have to select a suitable objective that is guided by experimental literature (Figure R1 in rebuttal PDF). This influences the inference window in which we aim to minimize the cross-entropy classification loss, and also determines the neuron time constant of 10ms, as well as the way stimuli is presented to the model.

We next look at the effects of introducing the spontaneous penalty term (Figure R2). In order to do so, we train 20 CordsNet-R2s on the CIFAR-10 dataset for 6 different values of penalty coefficients, for a total of 120 models in this analysis. Without the spontaneous penalty term, we find that the solutions can fall in three broad dynamical regimes: unstable, multistable and monostable (Figure R2A). In the unstable regime (top row), neural activity explodes after stimulus presentation, and never recovers, which is undesirable. In the multistable regime (middle row), the model remains stable throughout the first stimulus presentation, but is also unable to return to baseline due to the presence of additional attractors in its activity space. As such, when a second image is presented, it is unable to make the correct classification, despite being stable. The only solution that we want is the monostable case. Figure R2B presents the effects of different penalty coefficients on the types of solutions obtained. We find that our choice of $10^{-3}$ is ideal for only allowing monostable solutions.

We also performed an ablation study on the log-weights that we have introduced to our cross-entropy term. Here, we trained 10 CordsNet-R2s for 5 different log-weighing scales. We find that without this term, there is a small chance that a transient solution is obtained (Figure R3B), where the model does not arrive at a fixed point, but is instead optimized to only classify for the particular inference window. As such, it is unable to exhibit most of the other properties that our models in the main text have. By introducing this weighing scheme, we have eliminated this class of solutions.

Meta-review: There are abundant recurrent connections in biological circuits. Although there have been multiple neural network models that incorporate such recurrent connectivity, non-recurrent networks still constitute the mainstream of applications in computer science and there is ample room for improvement. 
Reviewers were mostly enthusiastic about the current work. 
Many of the concerns could be addressed during the revision before submitting the final version, including:
Citation and discussion of other work with recurrent neural networks, and potentially comparison with other benchmarks (see especially specific suggestions from 8dkQ, UEUj, 7SU8, and 7DkU). 
Reviewer 7DkU rightly noted that it would be important to provide more technical details, but I think that it would be easy for the authors to provide this information before the final version.
The authors provided clear and rigorous responses to the questions and should be able to improve the presentation in the final version. 
I share the reviewers’ enthusiasm for this paper.
Decision: Accept (spotlight)