Measuring and Controlling Solution Degeneracy across Task-Trained Recurrent Neural Networks
Authors: Ann Huang
Satpreet Harcharan Singh
Flavio Martinelli
Kanaka Rajan
Venue: NeurIPS 2025 spotlight ()
Keywords: recurrent neural network
dynamical systems
neuroscience
neural computations
Primary area: neuroscience_and_cognitive_science

Abstract: Task-trained recurrent neural networks (RNNs) are widely used in neuroscience and machine learning to model dynamical computations. To gain mechanistic insight into how neural systems solve tasks, prior work often reverse-engineers individual trained networks. However, different RNNs trained on the same task and achieving similar performance can exhibit strikingly different internal solutions, a phenomenon known as solution degeneracy. Here, we develop a unified framework to systematically quantify and control solution degeneracy across three levels: behavior, neural dynamics, and weight space. We apply this framework to 3,400 RNNs trained on four neuroscience-relevant tasks: flip-flop memory, sine wave generation, delayed discrimination, and path integration, while systematically varying task complexity, learning regime, network size, and regularization. We find that increased task complexity and stronger feature learning reduce degeneracy in neural dynamics but increase it in weight space, with mixed effects on behavior. In contrast, larger networks and structural regularization reduce degeneracy at all three levels. These findings empirically validate the Contravariance Principle and provide practical guidance for researchers seeking to tune the variability of RNN solutions, either to uncover shared neural mechanisms or to model the individual variability observed in biological systems. This work provides a principled framework for quantifying and controlling solution degeneracy in task-trained RNNs, offering new tools for building more interpretable and biologically grounded models of neural computation.

Measuring and Controlling Solution Degeneracy across Bask-Trained Recurrent Neural Networks
Ann Huang .2,3, Satpreet H. Singh? 2:3, Flavio Martinelli Kanaka Rajan
Harvard University
Harvard Medical School
*Kempner Institute 4EPFL
annhuang@g. harvard. edu
Abstract
Task-trained recurrent neural networks (RNNs) are widely used neuroscience and machine learning to model dynamical computations. To gain mechanistic insight into how neural systems solve tasks, prior work often reverse-engineers individual trained networks. However, different RNNs trained on the same task and achieving similar performance can exhibit strikingly different internal solutions, a phenomenon known solution degeneracy. Here, develop unified framework to systematically quantify and control solution degeneracy across three levels: behavior, neural dynamics, and weight space. We apply this framework 3,400 RNNs trained on four neuroscience-relevant tasks: flip-flop memory, sine wave generation, delayed discrimination, and path integration, while systematically varying task complexity, learning regime, network size, and regularization. We find that higher task complexity and stronger feature learning reduce degeneracy neural dynamics but increase it in weight space, with mixed effects on behavior. In contrast, larger networks and structural regularization reduce degeneracy at all three levels. These findings empirically validate the Contravariance Principle and provide practical guidance for researchers seeking to tune variability RNN solutions, either to uncover shared neural mechanisms or to model the individual variability observed in biological systems. This work provides a principled framework for quantifying and controlling solution degeneracy in task-trained RNNs, offering new tools for building more interpretable and biologically grounded models neural computation.
Introduction
Recurrent neural networks (RNNs) are widely used in machine learning and computational neu- roscience to model dynamical processes. They typically trained with standard nonconvex op- timization methods and have proven useful as surrogate models for generating hypotheses about the neural mechanisms underlying task performance [1, 2, 3, 4, 5, 6]. Traditionally, the study of task-trained RNNs has focused on reverse-enge eeneineringgniiinngg a single trained model, implicitly assuming that networks trained on the same task would converge to similar solutions, trained differently. However, recent work has shown that this assumption does not hold universally, and the solution space of task-trained RNNs can be highly degenerate: networks may achieve the same level of training loss, yet differ in cottoo-distribution (OOD) behavior, internal representations, neural dynamies, and connectivity [7, 9, 10, 11, 12, 13J. For instance, [8] found that while trained RNNs may share certain topological features, their representational geometry can vary widely. Similarly, [7] showed that musk-trained networks can develop qualitatively distinct neural dynamics and OOD generalization behaviors.
when initialized or
These findings raise fundamental questions about the solution space of task-trained RNNs: What factors govern the solution degeneracy across independently trained RNNs? When the solution space of task-trained RNNs is highly degenerate, what extent trust conclusions drawn from a single model instance? While feedforward networks have been extensively studied in terms of how
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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weight initialization and stochastic training (e.g., mini- batch gradients) lead to divergent solutions, RNNs still lack systematic and unified understanding of the factors that govern solution degeneracy [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Cao and Yamins [24] proposed the Contravariance Principle, which posits that as the computational objective (i.e., the task) becomes more complex, the solution space should become less dispersed-sinn since fewer models can simultaneously satisfy the stricter constraints imposed by harder tasks. While this principle is intuitive and compelling, it has thus far remained largely theoretical and has not been directly validated through empirical studies.
In this paper, we introduce a unified frame- work for quantifying solution degeneracy three levels: behavior, neural dynamics, and weight space (Figure I). Leveraging this framework, we isolate four key factors that control solution degeneracy: task com- plexity, learning regime, network width, and structural regularization. We apply this framework a large-scale experiment, training 50 independently initialized RNNs tasks.
Factors controlling solution degeneracy
Task complexity
Solution degeneracy
Network
Behavioral degeneracy
Dynamical degeneracy
Learning regime
Weight degeneracy
Structural regularization
on each of four neuroscience-relevee By systematically varying task complex- ity. learning regime, network width, and regularization, map how each factor shapes degeneracy across behavior, dynam- ics, and weights. We find that task complexity increases (whether via more input-output channels, higher memory de- mand, or auxiliary objectives, or as net- works undergo stronger feature learning), their neural dynamics become more con- sistent, while their weight configurations
Figure 1: Key factors shape degeneracy across behav- ior, dynamics, and weights. Schematic of our frame- work for analyzing solution degeneracy in task-trained RNNs. We evaluate how task complexity, learning regime, network size, and structural regularization influ- ence degeneracy at three levels: behavior (network out- puts), neural dynamics (state trajectories). and weight space (connectivity).
grow more variable. In contrast, increasing network size or imposing structural regularization during training reduces variability at both the dynamics and weight levels. At the behavioral level, each of these factors reliably modulates behavioral degeneracy; however, the relationship between behavioral and dynamical degeneracy not always consistent.
Table summarizes how task complexity, learning regime, network size, and regularization affect degeneracy across levels. In both machine learning and neuroscience, the desired level of degeneracy may vary depending on the specific research questions being investigated. This framework offers practical guidance for tailoring training to a given goal, whether encouraging consistency across models [25], or promoting diversity across learned solutions [26, 27, 28].
Our key contributions are follows:
A unified framework for analyzing solution degeneracy in task-trained RNNs across behavior, dynamics, and weights.
A systematic sweep of four factors: task complexity, feature learning, network size, and regular- ization, and a summary of their effects across levels (Table 1), with practical guidance for tuning consistency vs. diversity [25, 26, 27, 28].
A double dissociation: task complexity and feature learning yield contravariant effects on weights vs. dynamics, while network size and regularization yield covariant effects. Here, contravariant means that factor decreases degeneracy at one level (e.g., dynamics) while increasing it at another (c.g., weights), whereas covariant means both levels change in the same direction.
2 Methods
2.1 Model architecture and training procedure
We use discrete-time nonlinear vanilla recurrent neural networks (RNNs), defined by the update rule: h, = tanh (W W W W2x;= + b) where h, E RD the hidden state, x, E IR is the input, WA E gaxa and W p are the recurrent and input weight matrices, and b E IR" is bias

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vector. A learned linear readout is applied to the hidden state to produce the model's output at each time step. Networks are trained with Backpropagation Through Time (BPTT) [29], which unrolls the RNN over time to compute gradients at each step. All networks are trained using supervised learning with the Adam optimizer without weight decay. Learning rates are tuned per task (Appendix B). For each task, we train 50 RNNs with 128 hidden units. Weights are initialized from the uniform distribution 2/ (-1/Vn, 1/)n) and hidden states are initialized to be zeros.
In all experiments, we train networks until them reach a near-asympiotic, task-specific mean-squred error (MSE) threshold on the training set (see Appendix B), after which we allow a patience period of 3 epochs and stop training to measure degeneracy. This early-stopping criterion ensures that networks trained on the same task achieve comparable final losses before any degeneracy analysis.
2.2 Task suite for diagnosing solution degeneracy
We selected a diverse set of four tasks designed to elicit distinct neural dynamics commonly studied in neuroscience. The N-Bit Flip-Flop task captures pattern recognition and memory retrieval processes, analogous to Hopfield-type attractor networks that store discrete binary patterns and retrieve them from partial cues [30, 31]. The Delayed Diserimination task models working memory maintenance in classic delayed-response paradigms [32, 33j. The Sine Wave Generation task represents pattern generation, analogous to Central Pattern Generators (CPGs) that produce self-sustaining rhythmic outputs underlying motor control [34], as well as oscillatory activity observed in motor cortex during movement [35]. Finally, the Path Integration task inspired by hippocampal and entorhinal circuits that build a cognitive map of the environment to track position by integrating self-motion cues [36]. These tasks have also been used in prior benchmark suites for neuroscience-relevane RNN training [37, 38, 8], underscoring their broad relevance for studying diverse neural computations. Below, we briefly describe the task structure and the typical dynamics required to solve each one.
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Figure 2: Our task suite spans memory, integration, pattern generation, and decision-making. Task schematics and representative network trajectories projected onto the top principal components are shown in (A)-B). The four tasks are: N-Bit Flip-Flop: The network must remember the last nonzero input on each of N independent channels. Delayed Discrimination: The network compares the magnitude of two pulses, separated by variable delay, and outputs their sign difference. Sine Wave Generation: A static input specifies a target frequency, and the network generates the corresponding sine wave over time. Path Integration: The network integrates velocity inputs to track position in a bounded 2D or 3D arena (schematic shows 2D case).
N-Bit Flip-Flop Task Each RNN receives N independent input channels taking values in (-1, which switch with probability Pswitch The network has N output channels that must retain the most recent nonzero input on their respective channels. The network dynamics form 2 fixed points, corresponding to all binary combinations of +1| The output range of this task is [-1, 1] and we apply an carly-stopping training MSE threshold at 0.001.
Delayed Discrimination Task The network receives two pulses of amplitudes S1, /2 [2, 10), separated by a variable delay E (5, 20) time steps, and must output sign(ʃ2 -f). In the N-channel variant, comparisons made independently across channels. The network forms task-relevant fixed
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points to retain the amplitude of fi during the delay period. The output range of this task is [-1,1] 1] and we apply an early-stopping training MSE threshold at 0.01.
Sine Wave Generation The network receives a static input specifying a target frequency f E [1,30) and must generate the corresponding sine wave sin(2 ft) over time. We define Nirey target frequencies, evenly spaced within the range [1, 1,3, and use them during training. In the N-channel variant, each input channel specifies frequency, and the corresponding output channel generates a sine wave at that frequency. For each frequency, the network dynamics form and traverse a limit cycle that produces the corresponding sine wave. The output range of this task is [-1, 1] and we apply an early-stopping training MSE threshold at 0.05.
Path Integration Task Starting from a random position in 2D, the network receives angular direction 0 and speed v at each time step and updates its position estimate. In the 3D variant, the network takes as input azimuth 0, elevation Φ, and speed v, and outputs updated (ar, y, z) position. The network performs path integration by accumulating velocity vectors based on the input directions and speeds. After training, the network forms map of the environment in its internal state space. The output range of this task is [-5, 5] and we apply an early-stopping training MSE threshold at 0.05.
In our task suite, trained RNNs develop distinct stable dynamical objects: fixed-point (N-Bit Flip Flop. Delayed Discrimination), limit cycle (Sine Wave Generation), and attractor manifold (Path Integration). In Appendix E, extend our task suite include next-step prediction task on the Lorenz 96 chaotic attractors [39], where networks exhibit chaotic dynamical regime.
2.3
Multi-level framework for quantifying degeneracy
2.3.1 Behavioral degeneracy
We define novel metric for behavioral degeneracy as the variability in network responses to out-of- distribution (OOD) inputs. We quantify OOD performance as the mean squared error of all converged networks that achieved near-asymptotic training loss under temporal generalization condition. For the Delayed Discrimination task, we doubled the delay period. For all other tasks, we doubled the length of the entire trial to assess generalization under extended temporal contexts. Behavioral degen-
eracy is defined as standard deviation of the OOD losses: GOOD = N EN (coOD- IOOD)
where ZoOD is the mean OOD loss. While we focus primarily on the temporal generalization condition for behavioral degeneracy since it directly probes RNNs' sequence processing capacities and their ability to generalize across extended temporal horizons, the same metric can be readily applied to other OOD conditions, such as input noise or external perturbations. In the rest of the paper, we use the term behavioral degeneracy [temporal generalization) to explicitly indicate the OOD condition being tested.
2.3.2 Dynamical degeneracy
We use Dynamical Similarity Analysis (DSA) [40] to compare the neural dynamics of networks through pairwise analyses. While previous comparison methods mostly focus on geometry of the data [41, 42, 43, 44], RNNs implement computations through time-varying trajectories rather than static representations, and two RNNs exhibiting similar representational geometry can implement distinct dynamical computations, and vise versa. DSA compares the topological structure of the neural dynamics and has been shown to be more robust to noise and better at identifying behaviorally relevant differences than geometry-base
comparison method [45]. For a pair of networks X and Y, DSA projects their time series of activities to a higher-dimensionall space and identifies linear dynamic operator for each system via next-step prediction. The DSA distance between two systems then computed by minimizing the Frobenius norm between the operators, up to an orthogonal transformation (rotation and reflection):
dusa (Au; min
||2 CA,C-'lle
CEO(M)
where O(n) is the orthogonal group. We define dynamical degeneracy as the average DSA distance across all network pairs. Additional details on the DSA metric are provided in Appendix F. We note that scale of the DSA distance used to quantify dynamical degeneracy can depend on the choice of DSA hyperparameters. To ensure fair comparison across conditions, we keep all DSA

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hyperparametters fixed for RNNs trained on the same task. To assess if the neural dynamics across different trained networks are statistically different, we also establish null distribution by comparing neural trajectories sampled from the same underlying network, see Appendix F.3 for details.
We focus comparing neural dynamics because RNNs implement computations through time- evolving trajectories rather than static input representations. In addition. we assess representational degeneracy using Singular Vector Canonical Correlation Analysis (SVCCA) [41]. As shown in Appendix G, the four factors that influence dynamical degeneracy do not impose the same constraints on representational degeneracy.
2.3.3 Weight degeneracy
We quantify weight-level degeneracy via permutation-invariaatt invariant version of the Frobenius norm, defined as:
deIf 11, W2)
min -W-PP PEP(n)
where Wi and W2 are the recurrent weight matrices for a pair of RNNs, P(n) is the of permutation matrices of size n x and || -| denotes the Frobenius distance. See Appendix F.2 for additional details. For comparing dpiF computed on networks of different sizes, we normalize the above norm by the number of parameters in the weight matrix.
Results
3.1 Task complexity modulates degeneracy across levels
To investigate how task complexity influences dynamical degeneracy, we varied the number of independent input-output channels. This increased the representational load by forcing networks to solve multiple input-output mappings simultaneously. To visualize how neural dynamics vary across networks, applied woo-dimensional Multidimensional Scaling (MDS) to their pairwise distances. As task complexity increased, network dynamics became more similar, forming tighter clusters in the MDS space (Figure 3A). This contravariant relationship between task complexity and dynamical degeneracy was consistent across all tasks (Figure 3B). Higher task demands constrain the space of viable dynamical solutions, leading to greater consistency across independently trained networks.
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Figure 3: Higher task complexity reduces dynamical and behavioral degeneracy, but increases weight degeneracy. (A) Two-dimensional MDS embedding of network dynamics shows that inde- pendently trained networks converge more similar trajectories as task complexity increases. (B) Dynamical, (C) weight, and (D) behavioral degeneracy [temporal generalization) across networks as function of task complexity. Shaded area indicates +1 standard error.
At the behavioral level, networks trained on more complex tasks consistently showed lower variability in their responses to OOD test inputs (Figure 3D) in the temporal generalization condition. This finding suggests that increased task complexity, by reducing dynamical degeneracy. also leads to more consistent and less degenerate behavior on the temporal generalization condition across networks. Together, the results at the behavioral and dynamical levels support the Contravariance Principle, which posits an inverse relationship between task complexity and the dispersion of network solutions [24].
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At the weight level. we found that pairwise distances between converged RNNs' weight matrices increased consistently with task complexity (Figure 3C). This likely reflects increased dispersion of local minima in weight space for harder tasks. This interpretation is consistent with prior work on mode averaging and loss landscape geometry in feedforward networks, showing that harder tasks tend to yield increasingly isolated minima, separated by steeper barriers [46, 47, 48, 49, 50, 51, 52]. complementary perspective comes from [53] who introduced the intrinsic dimension as the lowest- dimensional weight subspace that still contains solution, which can serve
proxy for task complexity. As task complexity increases, the intrinsic dimension of the weight space expands and each solution occupies thinner slice of a higher-dimentt digger-dimeesionall space, leading to minima that lie further apart. In Section 3.2, we propose an additional mechanism: an interaction between task complexity and the network's learning regime that further amplifies weight-space degeneracy.
3.1.1 Additional axes of task complexity
In earlier experiments, we controlled task complexity by varying the number of independent input-output channels, effectively dupli- cating the task across di- mensions. Here. we explore two alternative approaches: increasing the task's mem-
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Changing memory de- mand. Of the four tasks, only Delayed Discrimi- nation requires extended m as its performance depends on maintaining the first stimulus across a variable delay. See Appendix D for quantification of each task's memory demand. We increased the memory load in Delayed Diserimination by lengthening the delay period. This manipulation reduced degeneracy at the dynamical and behavioral levels but increased it at the weight level, mirroring the effect of increasing task dimensionality (Figure 4A).
Figure 4: changes task complexity, which in turn modulates degeneracy. In the Delayed Discrimination task, both manipulations reduce dynamical and behavioral degeneracy [temporal generalization] while increasing weight degeneracy. The auxiliary loss also induces additional line attractors in the network's dynamics, as shown in (C).
Increasing memory demand adding auxiliary loss
Adding auxiliary loss. We next examined how adding an auxiliary loss affects solution degeneracy in the Delayed Discrimination task. Specifically, the network outputs both the sign and the magnitude of the difference between two stimulus values (f2 - fi), using separate output channels for each. This manipulation added second output channel and increased memory demand by requiring the network to track the magnitude of the difference between incoming stimuli. Consistent with our hypothesis, this manipulation reduced dynamical and behavioral degeneracy [temporal generalization] while increasing weight degeneracy (Figure 4B). Crucially, the auxiliary loss induced additional line attractors in the network dynamics, further structuring internal trajectories and aligning neural responses across networks (Figure 4C). While the auxiliary loss increases both output dimensionality and temporal memory demand, we interpret its effect holistically as a structured increase in task complexity.
3.2 Feature learning
3.2.1
Task complexity scales feature learning
In deep learning theory, neural networks can either solve tasks using their random features at initialization, or adapt their weights and internal features to capture task specific structure [54, 55, 56, 57]. These are referred to as the lazy learning regime, where weights and internal features remain largely unchanged during training, and the rich learning, or feature learning regime, where networks reshape their hidden representations and weights to capture task-specific structure [54, 58, 59, 55). As the complexity a task grows, the initial random features no longer suffice to solve it, pushing the network beyond the lazy regime and into feature learning, where weights and internal representations

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adapt more substantially. [60, 61]. If more complex task variants, like those in Section 3.1, truly induce greater feature learning, then networks should adapt more from their initializations and traverse greater distance in the weight space, resulting in more dispersed final weights.
We therefore hypothesize that the increased weight degener- acy observed in harder tasks reflects stronger feature learn- ing within the network. To test this idea, we measured feature learning strength in networks trained on different task variants using two complementary met-
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rics [62, 58]: Weight-change norm: WW Woll where larger values indicate stronger feature learning. Kernel align- ment (KA): The geometry of learning under gradient descent can be described by the neu- ral tangent kernel (NTK), which captures how weight updates af- feet the network outputs. The NTK is defined by K Vwğ" Vwū where ý denotes the network output. KAA. measures the direc- TT(K(ax
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Figure 5: More complex tasks drive stronger feature learning in RNNs. Increased input-output dimensionality leads to higher weight-change norms ([AW][F) and lower kernel alignment (KA). Error bars indicate +1 standard error.
tional change of the NTK before and after training:
Lower K km]
KA indicates greater NTK rotation and thus stronger feature learning.
We find that more complex tasks consistently drive stronger feature learning and greater dispersion in weight space, as reflected by increasing weight-change norm and decreasing kernel alignment across all tasks (Figure 5).
3.2.2 Controlling feature learning reshapes degeneracy across levels
Our earlier results show that harder tasks induce stronger feature learning, which in turn shapes the dispersion of solutions in the weight space. To test whether feature learning causally affects degeneracy, we used principled network parameterization known as maximum update parameteri- zation (μPP), which allows stable feature learning across network widths, even in the infinite- width limit [57, 54, 56, 55]. In this setup, a single hyperparameter (x) controls the strength of feature learning: higher 7 values induce richer feature-learning regime. Under this parameterization, the network update rule, initialization, and learning rate are scaled with respect to network width N. For the Adam optimizer, the output scaled f(() 7 (he)donn(((())... The hidden state update is scaled as hi(t + 1) - h(tt) recurrent weights and á is the tanh nonlinearity. The learning rate scales as 17 = MMo- A detailed explanation of μPP and its relationship to the standard parameterization is in Appendix K and L. For each task, we trained networks with multiple y values and confirmed that larger y consistently induces stronger feature learning, as evidenced by increased weight-change norm and decreased kernel alignment (Appendix M).
(-h(t) + Jo(h(t)) + Uz(()), where Jij ~ (N, .) are the
We observed that stronger feature learning reduced degeneracy at the dynamical level but increased it the weight level. We see that when 7 is high, networks tend to learn similar task-specific features and converge to consistent dynamics and behavior. In contrast, lazy networks (with small 7) rely on their initial random features, leading more divergent solutions across eveds-even though their weights move less overall (Figure 6). This finding aligns with prior work in feedforward networks, where feature learning was shown to reduce the variance of the neural tangent kernel across converged models [60]. At the behavioral level, however, increasing feature-learming strength leads networks to overfit the training distribution (Appendix J.2). We hypothesize that stronger feature learning exacerbates overfitting, increasing both average OOD loss and the variability of OOD behavior across models (Figure 6) [63, 64, 65, 66). Although stronger feature learning increases behavioral degeneracy [temporal generalization), this may partially reflect overfitting to the training distribution,
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Figure 6: Stronger feature learning reduces dynamical degeneracy but increases weight and behavioral degeneracy. Panels show degeneracy at the dynamical, weight, and behavioral levels (top to bottom). Shaded area indicates =1 standard error.
an effect we highlight in Appendix J.2. Clarifying the mechanistic link between dynamical and behavioral degeneracy [temporal generalization] remains an important direction for future work. In Appendix I, we demonstrate that the observed effects of feature learning on degeneracy both interpolates smoothly within the range of y values and extrapolates beyond the range reported in Figure 6.
3.3 Larger networks yield more consistent solutions across levels
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Figure 7: Larger networks reduce degeneracy across weight, dynamics, and behavior. After controlling for feature learning strength (? held constant across network widths), wider RNNs yield more consistent solutions across all three levels of analysis. Panels show degeneracy at the dynamical, weight, and behavioral levels (top to bottom). Shaded area indicates + standard error.
Prior work in machine learning and optimization shows that over-parameeeriizaion improves conver- gence by helping gradient methods escape saddle points [67, 68, 69, 70, 71, 72, 16]. We therefore hypothesized that larger RNNs would converge to more consistent solutions across seeds. How- ever, increasing width also tends to push models towards the lazy regime, where feature learning suppressed [73, 59, 54, 55, 56]. To disentangle these competing effects, we again use the uP parameterization, which holds feature learning strength constant (via fixed 7) while scaling width. Although larger networks may yield more consistent solutions via aelf-averaging, this outeome not guaranteed without controlling for feature learning, In standard RNNs. increasing width often

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Table 1: Summary of how each factor affects solution degeneracy. Arrows indicate the direction of change for each level as the factor increases. Contravariant factors shift dynamic and weight degeneracy in opposite direction; covariant factors shift them in the same directions.
Factor
Dynamics
Weights
Behavior
Higher Task complexity (contravariant)
More Feature learning (contravariant)
Larger Network size (covariant) Regularization (covariant)
induces lazier dynamics, which can paradoxically increase dynamical degeneracy rather than reduce it. The pP setup enables to isolate the size effect cleanly.
Across all tasks, larger networks consistently exhibit lower degeneracy at the weight, dynamical, and behavioral levels, producing more consistent solutions across random seeds (Figure 7). Our dense sweep over 12 intermediate network sizes from 32 to 512 on the 3-Bits Flip Flop task Appendix I further confirms the observed effect of network width on degeneracy. This pattern aligns with findings in vision and language models, where wider networks converge to more similar internal representations [74, 75, 41, 76, 77, 65]. In recurrent networks, only a few studies have investigated effect using representation-base these findings by (1) focusing neural computations across time (i.e., neural dynamics) rather than static representations, and (2) demonstrating convergence-withe withsselle across weight, dynamical, and behavioral levels in RNNs.
this "convergence-witl-scaee
metrics [74, 78]. Our results extend
3.4 Structural regularization reduces solution degeneracy
Low-rank and sparsity constraints are widely used structural regularizers in neuroscience- inspired modeling and efficient machine learn- ing [4, 79, 80, 81, 82]. A low-rank penalty com- presses the weight matrices into a few dominant modes, while an L penalty drives many param- eters to zero and induces sparsity. musk-irrleevat features are pruned, nudging in- dependently initialized networks toward more consistent solutions on the same task. To test this idea, we augmented the task loss with either nuclear-norm penalty on the recurrent weights
Low-Rank Ragularization
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both cases,
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f = Lusk + Arank Ei oi where a; are the singular values of the recurrent matrix, l sparsity penalty: L = Lrask + Xe, E. /tyl.
25
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1--5
5e
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5e
We focused the Delayed Discrimination task to control for baseline difficulty, and observe that both regularizers consistently reduced de- generacy across all levels. Similar effects hold in other tasks (Appendix O, Figure 8) and in- termediate regularization strengths (Appendix
Regularizstion Strength
Figure Low-rank and sparsity regularization reduce solution degeneracy across all levels. On the Delayed Discrimination task, both regularizers lower degeneracy in dynamics, weights, and be- havior. Shaded area indicates +:1 standard error.
4 Discussion
In this work, we introduced unified framework for quantifying solution degeneracy in recurrent neural networks (RNNs) at three complementary levels: behavior, neural dynamics, and weights. We systematically varied four factors within our generalizable framework: (i) task com- plexity (via input-output dimensionality, memory demand, or auxiliary loss), (ii) feature learning strength, (iii) network size, and (iv) structural regularization. We then evaluated their effects on solution degeneracy across diverse set of neuroscience--ellvvat tasks.

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Two consistent patterns emerged from this analysis. First, increasing task complexity or boosting feature learning produced a contravariant effect: dynamical degeneracy decreased while weight degeneracy increased. Second, increasing network size or applying structural regularization reduced degeneracy at both the weight and dynamical levels-that is, a covariant effect. Here, covariant and contravariant refer to the relationship between weight and dynamic degeneracy, not whether degeneracy increases or decreases overall. For example, task complexity and feature learning reduce dynamical degeneracy but increase weight degeneracy, whereas network size and regularization reduce both.
We also observed that the relationship between dynamical and behavioral degeneracy depends on the varying factor. For instance, stronger feature learning leads more consistent neural dynamics on the training task but greater variability in OOD generalization This suggests that tightly constrained dynamics on the training set do not guarantee more consistent behavior on OOD inputs. This highlights the need for further empirical and theoretical work on how generalization depends on the internal structure of task-trained networks [83, 84, 85]. This divergence highlights a key open question: how much of behavioral consistency generalizes beyond maining-aligned dynamics, and what task or network factors drive this decoupling?
These knobs allow researchers to tune the level of degeneracy in task-trained RNNs to suit specific research questions or application needs. For example, researchers may want to suppress degeneracy to study common mechanisms underlying a neural computation. Conversely, to probe individual differences, they can increase degeneracy expose solution diversity across independently trained networks [86, 87, 88, 89, 90]. Our framework also supports ensemble-base modeling of brain data. By comparing dynamical and behavioral degeneracy across trained networks, it may be possible to match inter-individe variability in models to that observed in animals, helping capture the full distribution of task-solving strategies [91, 92, 93, 94].
Although our analyses use artificial networks, several of the mechanisms we uncover may translate di- rectly experimental neuroscience. For example, introducing an auxiliary sub-task during behavioral shaping, which mirrors our auxiliary-Ioss manipulation, could constrain the solution space animals explore. thereby reducing behavioral degeneracy [95]. Finally, our contrasting findings motivate theoretical analysis, e.g., using linear RNNs to understand why some factors induce contravariant versus covariant relationships across behavioral, dynamical, and weight-level degeneracy.
In summary, our work takes a first step toward addressing this classic puzzle in task-driven modeling: What factors shape the variability across independently trained networks? We present unified framework for quantifying solution degeneracy in task-trained RNNs, identify the key factors that shape the solution landscape, and provide practical guidance for controlling degeneracy to match specific research goals in neuroscience and machine learning.
Limitations and future directions. This work considers networks equivalent if they achieve similar training loss. Future work could extend the framework to tasks with multiple qualitatively distinct solutions, to examine whether specific factors bias the distribution of networks across those solutions. Another open question is the observed decoupling between dynamical and behavioral degeneracy: how much of behavioral consistency generalizes beyond training-alignae dynamics, and what task or network factors drive this divergence.
5 Acknowledgments
We acknowledge funding from NIH (RFIDA056403, UOINS James S. McDonnell Founda- tion (220020466), Simons Foundation (Pilot aateenioon-00003322-00 McKnight Endowment Fund, CIFAR Azrieli Global Scholar Program, NSF (2046583), Harvard Medical School Neurobiology Lefler Small Grant Award, Harvard Medical School Dean's Innovation Award, Alice and Joseph Brooks Fund Postdoctoral Fellowship, and Kempner Graduate Fellowship. This work has been made possible in part by a gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner Institute for the Study of Natural and Artificial Intelligence at Harvard University.
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[103] Kenneth D Miller and Francesco Fumarola. Mathematical equivalence of two common forms firing rate models of neural networks. Neural computation, 24(1):25-31, 2012.
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Appendix
A Task details
A.1 N-Bit Flip Flop
Task Parameter
Value
Probability of flip
0.3
Number time steps
100
A.2 Delayed Discrimination
Task Parameter
Value
Number of time steps
60
Max delay
20
Lowest stimulus value
2
Highest stimulus value
10
A.3 Sine Wave Generation
Task Parameter
Value
Number of time steps
100
Time step size
0.01
Lowest frequency
1
Highest frequency
30
Number of frequencies
100
A.4 Path Integration
Task Parameter
Value
Number of time steps
100
Maximum speed (Umax)
0.4
Direction increment std (0ma @std)
T/ /10
Speed increment std
0.1
Noise std
0.0001
Mean stop duration
30
Mean go duration
50
Environment size (per side)
10
18

---

B Training details
B.1
N-Bit Flip Flop
Training Hyperparameter
Value
Optimizer
Adam
Learning rate
0.001
Learning rate scheduler
None
Max epochs
300
Steps per epoch
128
Batch size
256
Early stopping threshold
0.001
Patience
3
Time constant (nP)
B.2 Delayed Discrimination
Training Hyperparameter
Value
Optimizer
Adam
Learning rate
0.001
Learning rate scheduler
CosineAnnnellng WermRestarts
Max epochs
500
Steps per epoch
128
Batch size
256
Early stopping threshold
0.01
Patience
3
Time constant (pP)
0.1
B.3 Sine Wave Generation
Training Hyperparameter
Value
Optimizer
Adam
Learning rate
0.0005
Learning rate scheduler
None
Max epochs
500
Steps per epoch
128
Batch size
32
Early stopping threshold
0.05
Patience
3
Time constant (μP)
19

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B.4 Path Integration
Training Hyperparameter
Value
Optimizer
Adam
Learning rate
0.001
Learning rate scheduler
ReduceLROnnPlteau
Learning rate decay factor
0.5
Learning rate decay patience Max epochs
40
1000
Steps per epoch
128
Batch size
64
Early stopping threshold
0.05
Patience
3
Time constant (μP)
0.1
C Task performance of trained networks
In all experiments, train networks until them reach near-asymptotic, task-specific mean- squred error (MSE) threshold (0.001 for N-BFF, 0.01 Delayed Discrimination, and 0.05 for Sine-Wave Generation and Path Integration), after which we allow patience period of 3 epochs and stop training to measure degeneracy. This carly-stopping criterion ensures that networks trained on the same task/condition achieve comparable final losses before any degeneracy analysis.
To quantify the residual variation, we report the coefficient of variation (CV) of the final training loss across seeds for each condition, expressed as % of the mean. Header labels match the x x-axis levels used in the main-text figures. Final losses cluster tightly near small values of the loss threshold, so even a double-digit CV translates very small absolute variation. For example, a 10% CV at an MSE 0.001 implies an s.d. of 10-4; at 0.01 it's 10-3, Additionally, the networks converged well a global scale. Across our experiments, the mean MSE after training is under 2% of the mean MSE at initialization, indicating that training has converged well. Individual values: 0.059% (N-BFF), 1.6% (Delayed Discrimination), 0.32% (Sine-Wave Generation), 0.94% (Path Integration). CV can look large when the mean is tiny (the denominator is small). For example, a 16% CV on Sine-Wave& Generation task corresponds to 0.05% of the initialization loss, which is consistent with minor differences due to the stochastic gradients rather than under-training
These variability values also monotonic in any factor and sometimes move opposite the degeneracy trends, arguing against loss-dispersion confound to solution degeneracy.
Table 2: Coefficient of variation (CV) of the final training loss across 50 networks for each task complexity level.
Task Complexity
Level
Level 2
Level 3
Level 4
N-BFF
6.30%
4.60%
9,30%
3.50%
Delayed Discrim.
15.90%
8.40%
9.50%
Sine Wave Gen.
9.94%
9.20%
8.70%
Path Integr.
9.16%
2.85%
Table 3: Coefficient of variation (CV) of the final training loss across 50 networks for each feature learning strength (7)
Feature Learning Strength
7/2
7Y4
N-BFF
9.70%
9.10%
13.40%
11.70%
Delayed Discrim.
8.70%
12.60%
11.70%
12.30%
Sine Wave Gen.
3.50%
3.90%
10.90%
11.70%
Path Integr.
5.40%
5.20%
6.20%
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Table 4: Coefficient of variation (CV) of the final training loss across 50 networks for each network width.
Network Width
64 units
128 units
256 units
N-BFF
3.80%
4.20%
3.50%
Delayed Diserim.
3.30%
3.00%
3.20%
Sine Wave Gen.
17.80%
16.60%
16.40%
Path Integr. 5.10%
5.40%
5.90%
Table 5: Coefficient of variation (CV) of the final training loss across 50 networks for each L1 regularization strength.
L1 Regularization
A1
A2
Às
A4
N-BFF
2. 10%
6.90%
1.10%
Delayed Discrim.
15.90%
14.50%
16.70%
14.90%
Sine Wave Gen.
10.40%
11.10%
11.10%
Path Integr.
9.00%
7.10%
3.00%
Table 6: Coefficient of variation (CV) of the final training loss across 50 networks for each rank regularization strength.
Rank Regularization
A
Aa
A
X4
N-BFF
2.10%
7.20%
4.30%
Delayed Discrim.
15.90%
16.90%
13.60%
12. 10%
Sine Wave Gen.
13.90%
14.30%
15.90%
Path Integr.
7.70%
7.90%
6.70%
D Memory demand of each task
In this section, we quantify each task's memory demand by measuring how far back in time its inputs influence the output. Specifically, for each candidate history length h, we build feature vectors
(1) [zt-h+1:
233 yo
E
IR du touss
and train two-layer MIP to predict the subsequent target 9t++1 We then evaluate the held-out mean-squared error MSE(h), averaged over multiple random initializations. We identify the smallest history length h² at which the error curve plateaus or has a minimum, and take h" as the task's intrinsic memory demand.
From the results, we can see that the N-Bits Flip-Flop task requires only one time-step of mem- ory-exactly what's needed recall the most recent nonzero input each channel. The Sine Wave Generation task demands two time-stepp steps, reflecting the need to track both phase and direction of change. Path Integration likewise only needs one time-step, since the current position plus instanta- neous velocity and heading suffice to predict the next position. Delayed Discrimination is the only memory-intensive task: our method estimates a memory demand 25 time-steps, which happens to be the time interval between the offset of the first stimulus and the onset of the response period, during which the network needs to first keep track of the amplitude of the first stimulus and then its decision.
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Hewe
03:
641-
0.00
Lench
Figure 9: Memory demand of each task. The held-out mean-squared error MSE(h) of two-layer MLP predictor is plotted against history length h. The intrinsic memory demand h", defined by the plateau minimum of each curve, I for the N-Bits Flip-Flop and Path Integration tasks, for Sine Wave Generation, and 25 for Delayed Disrinination-mmtciing the inter-stimulus delay interval in that task.
E Solution degeneracy in chaotic RNNs
Our original task suite comprises catroosciencemmttivae tasks that produce stable-attractors: fixed- point (N-Bit Flip Flop, Delayed Discrimination), limit cycle (Sine Wave Generation), and attractor manifold (Path Integration). To further demonstrate that the observed effect of the four factors on degeneracy extend to RNNs with chaotic activity, here add chaotic attractor task and verified that the effects all four factors on dynamical and weight degeneracy are consistent with Table 1.
Lorenz 96 Attractor Dataset We simulated trajectories from the Lorenz 96 dynamical system [39], defined by
dx; -1 24 -2--2)22--1111 Iz F. 1
dt =
with eyclic boundary conditions x-1 -N
IN, NN+1 F1. The external forcing parameter was set F 8.0, a standard choice that induces chaotic dynamies.
To generate the dataset, we numerically integrated the system for N = 16, 24, and 32 dimensions. Each simulation used time step of A/ 0.01 and produced 15000 time points after discarding initial transient of 1000 steps to remove non-stationary behavior, Initial conditions were sampled as small random perturbations around the fixed point x;
z (0) = F 0.11 N
For each condition, trained 50 RNNs next-step prediction until the networks achieve asymptotic MSE loss at 0.0005. After training, the average Lyapunov exponent RNNs trained on the Lorenz 96 attractor with 16 dimensions is 12.58 + 0.74, indicating chaotic neural dynamics.
RNN Hidden State
Lorenz 96 Attractor
PC3
PC3
PC2
PC2
PC1
Figure 10: RNN recurrent activities and Lorenz 96 attractor (N 16) trajectories projected onto their respective top principle components.
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Task Complexity
Feature Learning Strength
Width
Regulerization Strength
2.89
0:2
Dynamical
Dergeneracy
0.50
02753
3.85-
0.22
0.80-
252
0.9003
0.55
0.20
24
32
5
10
20
128
256
512
Degeneras eereey
1.55
X10
200-
X90
120
Weigght
200-
100-
5
10
20
120
200
512
5e-5
Tedl
Se-4
attacior dimensionally
Network widhh
Rank regularzztion strength
Figure 11: Varying the four factors on Lorenz 96 next-step prediction task changes solution degeneracy across the dynamical and weight level a way that consistent with Table 1.
F
Additional details on the degeneracy metries
F1 Dynamical Degeneracy
Briefly, DSA proceeds follows: Given two RNNs with hidden states h1 (t R" and h2 E R", we first generate a delay-embedded matrix, H1 and H2 of the hidden states in their original state space. Next, for each delay-em embedded matrix, we Dynamic Mode Decomposition (DMD) [96] to extract linear forward operators A₁ and A₂ of the two systems' dynamics. Finally, a Procrustes distance between the matrices A and A is used to quantify the dissimilarity between the two dynamical systems and provide an overall DSA score, defined as:
dprocrustes (A,, A2)
min ||A, QA 1| QEO(m)
where Q a rotation matrix from the orthogonal group O(n) and || || the Frobenius norm. This metric quantifies how dissimilar the dynamics of the two RNNs are after accounting for orthogonal transformations. We quantify Dynamical Degeneracy across many RNNs as the average pairwise distance between pairs of RNN neural-dynamics (hidden-state trajectories).
After training, we extract each network's hidden-state activations for every trial in the training set, yielding tensor of shape (trials x time steps x neurons). We collapse the first two dimensions and yield matrix of size (trials x time steps) X neurons. We then apply PCA to retain the components that explain 99% of the variance to remove noisy and low-variance dimensions of the hidden state trajectories. Next, we perform grid search over candidate delay lags, with a minimum lag of and maximum lag of 30, selecting the lag that minimizes the reconstruction error of DSA on the dimensionality reduced trajectories. Finally, we fit DSA with full rank and the optimal lag to these PCA-projected trajectories and compute the pairwise DSA distances between all networks.
F.2
Weight degeneracy
We computed the pairwise distance between the recurrent matrices from different networks using Two-sided Permutation with One Transformation [97, 98] function from the Procrustes Python package [99].
E.3 Establishing null distribution for dynamical and weight degeneracy
The DSA scores that we used to define the dynamical degeneracy are inherently context-dependent. Specifically, the absolute scale of DSA distances can vary with hyperparameters, particularly the delay embedding dimension and the rank used in DSA, because the underlying Procrustes analysis between two dynamics matrices relies the Frobenius norm, which in turn depends the dimension of the dynamic operator being compared. Following the procedure described in the original DSA paper, we fixed these hyperparameters across all groups within each task to ensure fair comparison.
To further validate the interpretation of DSA values, we computed null distributions of the DSA scores, i.c the distribution of DSA scores when sampled neural activities come from identical networks. For
23

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each of the 50 networks analyzed in Figure 3B, we randomly split the sampled neural trajectories from the same network into two subsets and computed DSA distances between them. This procedure yields a distribution of DSA scores expected from identical dynamical systems, which serves as a reference noise floor. The 95% confidence intervals (CIs) for these null distributions are reported below (header labels such as "Level 1" correspond to the task-complexity levels shown in the main-text figures). These CIs are, on average, an order of magnitude smaller than the computed dynamical degeneracy, indicating that the observed differences between networks trained from different initializations are statistically significant.
Table Establishing a null distribution for dynamical degeneracy: 95% confidence intervals of null DSA scores computed by comparing trajectories from the same network. CIs are on average an order of magnitude smaller than across network distances.
Task Complexity
Level 1
Level
Level 3
Level 4
N-BFF
[0.011, 0.013]
[0.009, 0.016]
10.008, 0.013]
10.006, 0.0091
Delayed Discrimination
[0.039. 0.064]
[0.014, 0.076]
[0.025, 0.032]
Sine Wave Generation
[0.057, 0.10
[0.054, 0.081]
[0.048, 0.073]
Path Integration
[0.023, 0.037]
[0.010, 0.018]
For the PIF distance used define weight degeneracy, we similarly established noise floor by randomly permuting each trained network's recurrent weight matrix and computing the distance between the permuted and original matrices. The PIF metric reliably recovers PIF distance of under this null setting, confirming its robustness to noise and the meaningfulness of the reported cross-network PIF differences.
G Representational degeneracy
We further quantified solution degeneracy at the representational level-that is, the variability in each network's internal feature space when presented with the same input usinn Singular Vector Canonical Correlation Analysis (SVCCA). SVCCA works by first applying singular value decomposition (SVD) to each network's activation matrix, isolating the principal components that capture most of its variance, and then performing canonical correlation analysis (CCA) to find the maximally correlated directions between the two reduced subspaces. The resulting canonical correlations therefore measure how similarly two networks represent the same inputs: high average correlations imply low representational degeneracy (i.e., shared feature subspaces), whereas lower correlations reveal greater divergence in what the models learn. We define the representational degeneracy (labeled as the SVCCA distance below) as
drepe (Az;
SVCCA (VCCA(A,;
We found that we vary the four factors that robustly control the dynamical degeneracy across task-trained RNNs, the representational-leeel degeneracy isn' It necessarily constrained by those same factors in the same way. In RNNs, task-relevant computations are implemented at the level of network's dynamics instead of static representations, and RNNs that implement similar temporal dynamics can have disparate representaional geometry. Therefore, it is expected that task complexity, learning regime, and network size change the task-relevant computations learned by the networks by affecting their neural dynamics instead of representations. DSA captures the dynamical aspect of the neural computation by fitting forward operator matrix A that maps the network's 's activity at one time step to the next, therefore directly capturing the temporal evolution of neural activities. By contrast, SVCCA aligns the principal subspaces of activation vectors at each time point but treats those vectors as independent samples-it never examines how one state evolves into the next. As a result, SVCCA measures only static representational similarity and cannot account for the temporal dependencies that underlie RNN computations. Nonetheless, we expect SVCCA might be more helpful in measuring the solution degeneracy in feedforward networks.
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Task Complexity
Delayed Discrimination
Parh Integration
8V000 055316
SUOOA daterra
SVOCA disane
9.977
0.9
0.00
0.78
08-
charnels
Feature Learning
Delayd Discrimination
Sinewave
Path Integration
S0OKA araecc
SYCDA desance
6.52
0,4000
0.50
0.4875
C.51
0.4550
SUCOA
0.72
0.50
Network Size
SRFF
Daiaydd Diserimination
Sinewave
Integration
SWCLK corance
0.78
SYCCAdianno oipanco
0.480
SVCCA
0.74
SVCCK
0.50
0.500
0.488
196
Network witth
Network
Network with
Network with
Low-rank Regularization
Delayed
Integration
0.745
SYCCA dtrance
0
SVCCA desanoe
a 0.980
SySGA derance
0.740
O.8
0
0.ST
0.735
0.500
1 Regulatizaion
strength
Reguluization strengt
Sparsity (L1) Regularization
3BFF
Delayed Diectimination
Sineweve
Path Inlegration
SWOSA dttinee
0.800
SWCGA ditaroaa
0.52
SVCCA datanee
SWOCA dateen
3.58
0.50
0.700
0.78
5.500
3.54
Se"
Reguletiationn shrngh
ahanghh
Representational degeneracy, as measured by the average SVCCA distance between networks, does not necessarily change uniformly vary task complexity, feature learning strength, network size, and regularization strength.
Figure 12:
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H Task complexity effect on degeneracy in Gated RNNs
To examine whether the observed trends in dynamical and weight degeneracy generalize beyond vanilla RNNs, we conducted additional experiments using gated recurrent units (GRUs). Note that prior work suggests that architectural choices influence the geometry but not the topology of neural dynamics, which is primarily shaped by task structure [100]. Meanwhile, the Dynamical Similarity Analysis (DSA) metric we employ to quantify dynamical degeneracy is designed to precisely capture the topological organization of neural dynamics while remaining invariant to geometric transformations [40].
As a preliminary test, we trained GRUs on the Sine Wave Generation task while systematically varying task complexity by changing the number of input-output channels. Consistent with our findings in vanilla RNNs, increasing task complexity led to decrease in dynamical degeneracy and rise in weight degeneracy.
Dynamical Degeneracy
Dynamical Degeneracy
Weight Degeneracy
Weight Degeneracy
0.25
0.004
0.20
0.002
2
Number of Channels
Number of Channels
Figure 13: Increasing task complexity in the Sine Wave Generation task produces the same effect on dynamical and weight degeneracy in both vanilla RNNs and GRUs.
I Dense sweep on feature learning, network width, and regularization strength
is important to know whether the degeneracy trends generalize to intermediate values and beyond the ranges reported in the main paper. To test this, used 3-BFF example and dense sweep both interpolating within and extrapolating beyond the ranges shown in Figs. and 6-8. We demonstrate that cross dynamical and weight levels, the degeneracy trends remain consistent and interpolate smoothly across these intermediate values.
Feature Learning Strength
Network Width
Rank Regularization Strength
Dynamical Degeneracy
Dynanical Degeneracy
Dynamical Degenoracy
0.072
0.07
3070
0 065
0088
0.4
85
1.2
16
20
24
2.8
32
3.5
4.0
120
700
2300
2000
100
$12"
Network
Rank regularization
Weight degeneracy
0.1350
Weight degeneracy
0.25
Weight regenerdyg
10
0.1325
0.90
0.20
0.1300
0.85
0.1275
8,00
0.1250
111
0.
2.0
244
28
32-
36-
400
512
766
80-5
16-5
Network width
Rank regularization
Figure 14: Feature learning, network width, and regularization strength's 's effect on degeneracy over
denser sweep of conditions on the 3-Bits Flip Flop task.
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Detailed characterization of OOD generalization performance
In addition to showing the behavioral degeneracy in the main text, here we provide a more detailed characterization the OOD behavior of networks by showing the mean versus standard deviation, and the distribution of the OOD losses.
J.1 Changing task complexity
fe-3
3BFF
Se-3
Delayed Discrim.
Sine Wave Gen.
1e-1
Iniegr.
chammels
24-
22-
chamss
20-
1.-
chonnic
chamnels
2.6
3.0
3.2
155
0.
01
1o-4
Ter2
1e-1
18-
21-
10-
14-
12
Frequeryy
Frapaency
Recquncy
0.5
1.0
000
GOD
0C0 Losa
000
Loss
Figure 15: Detailed characterization the OOD performance of networks while changing task complexity.
J.2
Changing feature learning strength
14-
16-4
1--3 Delayed Discrim.
Sine Wave Gen.
Path Integr.
G-5
16-
12-
12-
10-
32-
03-
03-
2.5-
2.5
0.9
0.4
2.0
27
10-1
12-4
12-4
10-
10-
Frepurey
Trequesy
15-
Frequency
12-
1.5 OC0 Loss
LOD
2.0
COD Loos
2.5
3.0
Figure 16: Detailed characterization of the OOD performance of networks while changing feature learning strength. Across Delayed Discrimination, Sine Wave Generation, and Path Integration tasks, networks trained with larger 7 - and thus undergoing stronger feature learning exhibit higher mean OOD generalization loss together with higher variability, potentially reflecting overfitting to the training task.
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J.3 Changing network size
3BFF
Delayed Discrim.
1e-2
Path Integr.
unt
unis
128
120 unia
256
25d
9.00-
52-
9 8.25-
unit
120 unis
2.-
250 unis
1.8
2.0
7:50
155
1.0
1.5
2.55
2.70
18-
18-
12-
10-
6.0
0.0
2.0 0.00
2.0
2.5
3.0
3.5
X190
Loos
000
Figure 17: Detailed characterization of the OOD performance of networks while changing network size.
J.4
Changing regularization strength
J.4.1 Low-rank regularization
3-BFF.
S--3
Delayed Discrim.
1e-2
Sine Wave Gen.
Path Integr.
2.0
Ie
325
55-
1.6
Se
Se
5.0-
2
2.75
4.5
2:5-
4.-
1.0
1.
1u-3
1--2
32-
18-
14-
10
Frequens
20-
24
Frequenoy
10
12-
Frequency
0.0
1.0
000 Lane
0C0 Loza
000 Loss
x10
Figure 18: Detailed characterization of the OOD performance of networks while changing low-rank regularization strength.
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J.4.2 Sparsity (L1) regularization
3BFF
Delayed Discrim.
Ie-2
Sine Wave Gen
---1
Path Integr.
20-
2 Le
Ze
Se
12-
Ie
1.5
Sp-t
24-
10-
1.0-
2.3-
0.8-
0.5-
0.-
2:1
0.0
0.5
3
3.0
3.2
3.4
3.6
3.5
1e-2
16-1
32-
10
28-
20-
Inequency
15-
16-
12-
D- x10"
00
02
0.6
1.0
000
OCD
x100
OCD
Figure 19: Detailed characterization of the OOD performance of networks while changing sparsity (L.1) regularization strength.
K
A short introduction to Maximal Update Parameterization (nP)
Under the NTK parametrization, as the network width goes to infinity, the network operates in the lazy regime, where its functional evolution is well-approximated by first-order Taylor expansion around the initial parameters [73?, 54, 55J. In this limit feature learning is suppressed and training dynamics governed by the fixed Neural Tangent Kernel (NTK).
To preserve non-trivial feature learning at large width, the Maximal Update Parametrization (μP) rescales both the weight initialisation and the learning rate. jP keeps three quantities ridth-invariant at every layer-(i) (i) the norm/variance of activations (ii) the norm/variance of the gradients, and (iii) the parameter updates applied by the optimizer [101, 102, 56, 57].
For recurrent neural networks, under Stochastic Gradient Descent (SGD), the network output, initial- ization, and learning rates are scaled as
1
NoNV W dh).
(1)
0,h- = - h + VN
J (h),
Jo ~ N(0,1)
(2)
TSCD no ro N.
(3)
Under Adam optimizer, the network output, initialization, and learning rates are scaled as
f NoNN V (h).
(4)
0,h = h + N (d),, J(0,N(
(5)
AAaam = 170 "/--
L Theoretical relationship between parameterizations
We compare two RNN formalisms used in different parts of the main manuscript: a standard discrete- time RNN trained with fixed learning rate and conventional initialization, and JPP-style RNN trained with leaky integrator dynamics and width-aware scaling.
In the standard discrete-time RNN, the hidden activations are updated as h(t + 1) o (Wnh((t) War(t)),
29

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In pPP RNNs, the hidden activations are updated as
1it + 1))+ = ((-h(() + 1/ N (0(0((0))+x(()))
When h(t 1) -(t) -h(t)+ + 1 N
Jo(h(t))+ + Uz(())
h(t+1)= = - JO(h(t)) U
Aside from the overall scaling factor, the difference between the two parameterizations lies in the placement of the non- non-lincarity:
Standard RNN: é is applied post-activation, i.c. after the recurrent and input terms linearly combined,
μP RNN: applied pre-activation; i.e. before the recurrent weight matrix, so the hidden state is first non- non-linearized and then linearly combined
Miller and Fumarola [103] demonstrated that two classes of continuous- time firing-rate models which differ in their placement of the non-linearity are mathematically equivalent under change of variables:
du v-model T = - I(t)+ + +W)w
dt
dr T ---++f(Wr)
r-model:
with equivalence holding under the transformation vtt) = Wr(t) + I(t) and ī(t) I(t) dl assuming matched initial conditions.
Briefly, they show that Wr + T evolves according to the e-equation follows:
v()) Wr(1) + I(11
du (Wr(1)+10) + +(()) dr dt
di di
dt +
dt
(-r + f(Wr+ 11) + dl
W
dP
du
Wr + W WWrr + + dT dt
dt
-(V 1) + W f(v) + 1p
U + r dI
dt + WWI))
du dt
+ I(t) Wf(1)
This mapping applies directly to RNNs viewed continuous- time dynamical systems and helps relate v-type yP-sttyl RNNs to standard discrete-time RNNs. It suggests that the μP RNN (in v-type form) and the standard RNN (in r-type form) can be treated as different parameterizations of the same underlying dynamical system when:
Initialization scales matched
The learning rate is scaled appropriately with y
30

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Ouiput weight norms are adjusted according to width
In summary, while theoretical equivalence exists, it contingent on consistent scaling across all components of the model. In this manuscript, we use the standard discrete-time RNNs due to its practical relevance for task-driven modeling community, while switching p&P to isolate the effect of feature learning and network size. Additionally, we confirm that the feature learning and network size effects on degeneracy hold qualitatively the same in standard discrete-ting time RNNs, unless where altering network width induces unstable and lazier learning in larger networks (Figure P and Q).
M Verifying larger 7 reliably induces stronger feature learning in μ.P
In puP parameterization, the parameter y interpolates between lazy training and rich, feature-learning dynamics, without itself being the absolute magnitude of feature learning. Here, we assess feature- learning strength in RNNs under varying y using two complementary metrics:
Weight-change norm which measures the magnitude of weight change throughout training. A larger weight change norm indicates that the network undergoes richer learning or more feature learning.
WT T Woll.
N
where N is the number of parameters in the weight matrices being compared.
Kernel alignment (KA), which measures the directional change of the neural tangent kernel (NTK) before and after training. A lower KA score corresponds to a larger NTK rotation and thus stronger feature learning.
T((K5) K(O) |
KA(KO), KO)
K
V W Vwū.
|
We demonstrate that higher y indeed amplifies feature learning inside the network.
M.1
N-BFF
Per-seed weight change
Mean SE
0.106
I| Woll
0.106
0.104
0.102-
0.103
0.5
0.5
2
Per-seed kerne alignment
Mean SE
0.35
0.30
KA(Ki) K(0)
0.30
0.28
0.26
0.25-
0.24
0.20
0.22
0.20
0.15
0.5
0.5
2
Figure 20: Weight change norm and kernel alignment for networks trained the 3-Bits Flip Flop task as we vary 7. On the left panels, we show the per-seed metrics where connected dots of the same color are networks of identical initialization trained with different 7. On the right panels, we show the mean and standard error of the metrics across 50 networks. For larger 7, the weights move further from their initializations shown by the larger weight change norm, and their NTK evolves more distinct from the network's 's NTK at initialization as shown by the reduced KA. Both indicate stronger feature learning for networks trained under larger 7
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M.2 Delayed Discrimination
Per-seed weight change
Moan SE
0.35
Well
0.30
0.30
0.25
I[W,
0.20
0.20
Per-seed kernel alignment
Mean
KA(Km, k(0))
0.6-
0.4
0.3-
0.2
0.2
Figure 21: Stronger feature learning for networks trained under larger on the Delayed Discrimination task.
M.3 Sine Wave Generation
Per-seed waight change
Mean + SE
0.114
0.115
0.112
I|W, Woll
0.110
0.110
106-
0.104
0.1
0.5
0.1
0.5
Per-seed kemel alignment
0.10
KA(K K10))
0.08
0.06
0.08
0.04
0.02
0.00
0.1
0.5
0.1
0.5
Figure Stronger feature learning for networks trained under larger 7 on the Sine Wave Generation task.
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M.4 Path Integration
Per-seed weight change
0.240
0.20
0.235
IlW W
0.24
0.230
0.225
0.220
0,215
Per-seed kemel alignment
Mean SE
0.74
0.58
KA(kin K(o)
0.6-
0.5-
0.58
0.54
0.3
0.52
0.-
Figure 23: Stronger feature learning for networks trained under larger y the Path Integration task.
N
Verifying μP reliably controls for feature learning across network width
Here, only use Kernel Alignment assess the feature learning strength in the networks since the unnormalized weight-change norm - -Wllp scales directly with matrix size (therefore network size) and there exists no obvious way normalize across different dimensions. In our earlier analysís where we compared weight-change norms at varying network size remained fixed, so those Frobenius-nome norm measures were directly comparable. We found that, for all tasks except Delayed Discrimination, the change in mean KA across different network sizes remains extremely small (less than 0.1), which demonstrates that pP parameterization with the same y has effectively controlled for feature learning strength across network sizes. On Delayed Discrimination, the networks undergo slightly lazier learning for larger network sizes. Nevertheless, we still include Delayed Diserimination in our analyses of solution degeneracy ensure our conclusions remain robust even when j.P can't I perfectly equalize feature-learning strength across widths. As shown the main paper, lazier learning regime generally increases dynamical degeneracy; yet, larger networks which exhibit lazier learning in the N-B task actually display lower dynamical degeneracy. This reversed trend confirms that the changes in solution degeneracy arise from network size itself, not from residual variation in feature learning strength.
N.1 N-BFF
Per-seed kernel alignment
Mean SE
0.30
0.24
KA(Km, k)))
0.23
0.25
0.22
0.20-
0.21
0.20
0.15
0.18
64
128
255
64
128
256
Network width
Network witth
Figure 24: Kernel alignment (KA) for different network width on the 3 Bits Flip-Flop task. (Lower KA implies more feature learning.)
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N.2 Delayed Discrimination
alignment
0.50
0.46
0.4
0.40
KAlkin
0.35
02-
0.30
64
256
84
120
268
Networs width
Network with
Figure 25: Kernel alignment for different network width on the Delayed Discrimination task.
N.3 Sine Wave Generation
Per-seed kernel alignment
Mean SE
KA(k- Kion
0.08
0.02
0.00
Figure 26: Kernel alignment for different network width the Sine Wave Generation task.
N.4 Path Integration
Per-seed kernel alignment
Mean a SE
0.80
0.8
K4(K
0.76
0.6
0.74
0.51
64
128
256
84
120
2b6
Network width
Network wicth
Figure 27: Kernel alignment for different network width on the Path Integration task.
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Regularization's effect on degeneracy for all tasks
In addition to showing regularization's is effect degeneracy in Delayed Discrimination task in the main paper, here we show that heavier low-rank regularization and sparsity regularization also reliably reduce solution degeneracy across neural dynamics, weights, and OOD behavior in the other three tasks.
0.1 Low-rank regularization
3-BFF
Sine Wave Gen.
Path Integr.
Dynamical Degeneracy
0.2350
0.1445
0.2345
0.1440
0.095
1e-4
1e-3
1e-3
Degeneracy
92
Weight
1.10
9.0-
0.8
1.09
Behavioral
Degeneracy
2.0
3
5-
2-
5e
le"
5e
1e-6
1e
1e"
Ragularization Strength
Regularization Strength
Regularization Strength
Figure 28: Low-rank regularization reduces degeneracy across neural dynamics, weight, and OOD behavior on the N-BFF, Sinewave Generation, and Path Integration task.
0.2 Sparsity regularization
3-BFF
Sine Wave Gen.
Path Integr.
Dynamical
Degeneracy
0.10
0.2345
0.144
0.08
0.2340
0.143
Degeneracy
1.1
Weight
7.5
0.8
1.0-
25
1e-4
1e-2
18-1
Behavioral
Degeneracy
1.5
1.0
3-
0.5
1ea
5e 6
5et
1e"
le
lea
le
Regularization Strength
Regularization Strength
Regularization Strength
Figure 29: Sparsity regularization reduces degeneracy across neural dynamics, weight, and OOD behavior on the N-BFF, Sinewave Generation, and Path Integration task.
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P Test feature learning effect on degeneracy in standard parameterization
While μP lets systematically vary feature-learning strength to study its impact on solution degeneracy, we confirm that the same qualitative pattern appears in standard discrete-time RNNs: stronger feature learning lowers dynamical degeneracy and raises weight degeneracy (Figure 30).
To manipulate feature-learning strength in these ordinary RNNs we applied the --trick-scaling the network's 's outputs by raand multiplied the learning rate by the same factor. With width fixed, these two operations replicate the effective changes induced by j.P. Figure shows that this combination reliably tunes feature-learning strength. Besides weight-change norm and kernel alignment, we also report representation alignment (RA), giving more fine-grained view how much the learned features deviate from their initialization [62]. Representation alignment is the directional change of network's 's representational dissimilarity matrix before after training, and defined by
RA(R()) R = TH (2) R(O) | |
R H,
lower RA means more change in the network's representation of inputs before and after training. and indicates stronger feature learning.
N-BFF
Delayed Discrim.
Sine Wave Gen.
Path Integr.
Degeneracy Dynamical
Degeneracy
0.245
0.11
0.235
0.148
0.001375
Weight
0.00100
0.001350
0.00995
0.001325
0.0011
0.5
V
V
Figure 30: Stronger feature learning reliably decreases dynamical degeneracy while increasing weight degeneracy in standard discrete-time RNNs.
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NBFF
Delayed Discrim.
weight change
Weight change
Wwint SE
Well
well
1.5-
5.50-
0.5
08-1
Pen-sced kernel alignment
241
Komel aligmmen mean SE
Por sood komel a aigmment
Komel alignment moan a
KAIK .e0)
04-
54
02
0.10
0.-
22
0.5
For soed repe. aligmmen
Repr. aligmment mean +
aligmmnn
Repr allgnment
RAIRI A18
0.-
58-
RAUgin Rilt)
06-
0.--
26-
04
0.1-
0.6
Sine Wave Gen.
Path Integr.
Weight change moon
Perisodd weight change
Weight change
weight change
0000/5-
-Wall
1:20
0.00150-
0.00125-
0.1
0.5
0.5 Kemal aignoent mean ST
Per-osed iermel algement
Pepsasnc kurnel sigrmani
Kamall wigrmant maan 5E
KAMK? ggbng
0.5-
200-
KAyK
025
0.2-
2.25-
0.1
0.
0.1
0.7
Per owed rept algrmant
Repe. algemar muan SE
Par-seed rupe. algoment
Rape. algement mean SE
RAIRM g00))
1.51
1.0
3.80
0.00
n
0.2
Figure 31: Larger y reliably induces stronger feature learning in standard discrete-time RNNs.
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Q
Test network size effect on degeneracy in standard parameterization
When we vary network width, both the standard parameterization and pP parameterization display the same overall pattern: larger networks exhibit lower dynamical and weight degeneracy. An exception arises the 3BFF task, where feature learning becomes unstable and collapses in the wider models. In that setting instead see higher dynamical degeneracy, which suspect because the feature learning effect (lazier learning leads to higher dynamical degeneracy) dominates the network size effect.
N-BFF
Delayed Discrim.
Sine Wave Gen.
Path Integr.
Waight Dynamical
Degeneracy Degeneracy
0.215
0.1505
0.235
0.15
0.210
0.230
0.1500
0.225
0.003
0.003
0.0050
0.002
0.002
0.002
0.001
0.001
0.001
128
Network Size
Network Size
Network Size
Network Size
Figure 32: Larger network sizes lead to lower dynamical and weight degeneracy, except in the case where feature learning is unstable across width (in N-BFF).
NBFF
Delayed Discrim.
seed kamel
SE
Per kereel
Kernel
0.4-
0.6
0.6-
KAY gelt)
03-
0.4
0.4-
0.-
0.2
0.2
0.1
0.-
0.0-
128
04
150
256
250
125
Por-soed
Por-sood alignment
alignment SE
0.5-
RAge Re0a
06-
0.6-
0.8
6.6
05-
0.5-
0.2
0.2
Network
Network Size
Network Size
Network
Sine Wave Gen.
Path Integr.
alignment
alignment mean
Per-owad kernel alignment
Kamai aligrment meen SE
0.2
KAUK Rent
0.4-
0.1-
02
256
1/8
250
Per-oeed repr. aliggment
Repe. al alggment meon SE
Pur-seed rapr. alignmant
Repr. alignment mean' SE
1.06
100
10-
RA
1.00
1.00
030
0.00
0.98
04
158
206
125
250
Network Size
Network. Size
Figure 33: When changing network width in standard discrete-time RNNs, feature learning strength remains stable across width except in N-BFF, where notably lazier learning happens in the widest network.
38

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R Disclosure of compute resources
In this study, we conducted 50 independent training runs on each of four tasks, systematically sweeping four factors that modulate solution degeneracy- task complexity (15 experiments), learning regime (15 experiments), network size (12 experiments), and regularization strength (26 experiments), resulting in a total of 3400 networks. Each experiment was allocated 5 NVIDIA V V100/A 100 GPUs, 32 CPU cores, 256 GB RAM, and 4-hour wall-clock limit, for total compute cost of approximately 68 000 GPU-hours.
39

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Reviews:
[{"review_id": "VomLMycDyP", "reviewer": "Reviewer_pYdB", "summary": "This paper investigates solution degeneracy in task-trained RNNs, which occurs when different networks achieve similar task performance (as measured by their training loss) but vary widely in weights, neural dynamics, and behavior.  The authors identify 4 crucial factors (task complexity, learning regime, network size, regularization) that control solution degeneracy in various task-trained RNNs, providing empirical support for the “contravariance principle.” They propose novel metrics to quantify solution degeneracy and show that task complexity and learning regime have contravariant effects (i.e., affect weights and dynamics in opposite ways), while network size and regularization have covariant effects.", "questions": "1. Definition of degeneracy: Has the definition of behavioral degeneracy you use in Section 2.3.1 appeared in prior literature? If not, please state that it is novel; if it has, cite the appropriate references.\n2. Task motivation: Could you elaborate on why these four tasks were chosen? Do they correspond to known computational functions of brain areas, or are they commonly used RNN benchmarks in neuroscience or machine learning?\n3. Path integration claim: On line 127, you say that “After training, the network forms a Euclidean map of the environment in its internal state space.” Can you back this up? Many studies involving RNNs trained on the task of path integration find toroidal neural manifolds in the internal space, similar to grid cells in the mammalian entorhinal cortex. These are non-Euclidean. \n4. Clarify the $\\phi$ model: In Section 3.3, you introduce a hidden state update involving $\\phi$ without context. Is this a new model? How does it tie into your prior experiments?\n5. Initialization and $h_0$: How is the initial hidden state $h_0$ set in your experiments?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strengths:**\n\n- Provides empirical validation for the Contravariance Principle.\n- Identifies four key factors controlling degeneracy: task complexity, learning regime, network size, and regularization.\n- Uses a solid set of degeneracy metrics.\n- Well-written and thoughtfully designed set of experiments.\n\n**Weaknesses:**\n\n- Terms like *“learning regime”* and *“feature learning”* are used interchangeably. The paper relies on the Rich vs Lazy regime literature, but this may not be familiar to all readers (especially in neuroscience), and could benefit from clearer terminology introduced earlier.\n- In Figure 2, subfigure 2B is not clearly described (presumably PCA of neural activity). It should be clarified in the caption.\n- In Section 3.3, $\\phi$ is not defined, and a new update rule is introduced without context—unclear how this relates to prior experiments.\n- Line 267–268: Inconsistent reasoning—on one hand, the text says that high γ leads to consistent behavior; on the other, it says stronger feature learning leads to less consistent behavior.\n- Minor typo on line 160: “compleity”.\n- Some sections could benefit from better citations.", "was_revised": true, "final_justification": "This is a technically solid paper and the authors have adequately addressed my remaining questions and concerns in their rebuttal. Therefore, I believe it should be accepted, and I am maintaining my score of 5.", "rebuttal": "Thank you for your thoughtful and supportive review! We are glad that you find the paper well-written, and like our empirical validation of the Contravariance Principle, and our systematic analysis of four factors shaping solution degeneracy with solid metrics. Here we address the points you raised in the Questions and Weaknesses section.\n\n### Questions\n\n> Q1: Has the definition of behavioral degeneracy you use in Section 2.3.1 appeared in prior literature?\n\nTo the best of our knowledge, our definition of behavioral degeneracy has not appeared in prior literature. Previous work, such as [1], has reported that image classification models trained from random initializations can have varied performance on out-of-distribution data. However, no explicit definition of behavioral variability or behavioral degeneracy was made there. \n\n> Q2: Could you elaborate on why these four tasks were chosen?\n\nIndeed we picked these tasks by surveying prior work in computational neuroscience and neural computation. Below we list several papers that motivated our choices:\n* N-BFF: pattern recognition / memory retrieval; Hopfield‑type\nattractor networks store binary patterns and retrieve them from partial cues [2,3].\n* Delayed Discrimination: working memory maintenance; dorsolateral prefrontal cortex neurons show direction‑ or stimulus‑specific persistent “delay‑period” firing activities that bridges the sample and test period in delayed‐response / discrimination tasks [4,5].\n* Sine Wave Generation: Central Pattern Generators (CPGs) produce self‑sustaining sinusoidal motor commands [6]; rhythmic and oscillatory activity in the motor cortex during movement [7].\n* Path Integration: Hippocampal place cells and entorhinal grid cells integrate self‑motion to update an internal position estimate when external cues are absent [8].\n\nFurthermore, previous work that developed benchmark neuroscience-relevant task suites have used similar tasks including the N-BFF, Sine Wave Generation, and Delayed Discrimination tasks [9-11].\n\n> Q3: Networks form a map of the environment on Path Integration\n\nWe appreciate that the reviewer pointed out that it may not be an Euclidean map. We apologize for any confusion and will remove that from our revised manuscript.\n\n> Q4: Clarify the $\\phi$ model\n\nTo test feature learning's effect on solution degeneracy, we used a principled network parameterization called muP (maximal update parameterization) to systematically control for feature learning strength. muP enables stable feature learning strength across network widths, even in the infinite-width limit [12]. Line 259-261 in Section 3.3 describes the update rule, and scaling of initialization and learning rate under the muP parameterization. Here, $\\phi$ is the tanh nonlinearity, consistent with our discrete-time RNN model introduced in section 2.1. The time constant is 1. In this manuscript, we use the standard discrete-time RNNs due to its practical relevance for task-driven modeling community, while switching to μP to isolate the effect of feature learning and network size in Section 3.3 and 3.4, respectively. In Appendix H, we show a theoretical equivalence between the muP parameterization and the discrete-time RNN setting under proper scaling of the network initialization and update. Moreover, we confirm that the feature learning and network size effects on degeneracy hold qualitatively the same in standard discrete-time RNNs, unless where altering network width induces unstable and lazier learning strength in larger networks in Appendix L-M. \n\n> Q5: How is the initial hidden state $h_0$ set in your experiments?\n\nWe initialize our hidden state to be all zeros.\n\n&nbsp;  \n\n### Weaknesses\n> The paper relies on the Rich vs Lazy regime literature, but this may not be familiar to all readers (especially in neuroscience), and could benefit from clearer terminology introduced earlier.\n\nThank you for your suggestion! We will expand our current introduction on Rich vs Lazy regime in section 3.2., to incorporate more detailed explanation of the terminology, and also discuss their relevance for the neuroscience community [13-15]. \n\n\n> In Figure 2, subfigure 2B is not clearly described (presumably PCA of neural activity)\n\nThanks for catching this! Figure 2B indeed plots the PCA of neural activity in the top principle components' state space. We will clarify this point in the caption of figure 2B.\n\n> Line 267-268: Inconsistent reasoning—on one hand, the text says that high γ leads to consistent behavior; on the other, it says stronger feature learning leads to less consistent behavior.\n\nWe clarify that higher $\\gamma$, which corresponds to stronger feature learning, leads to more consistent neural dynamics (lower dynamical degeneracy) but less consistent weight structure (higher weight degeneracy).\n\n> Minor typo on line 160: “compleity”\n\nThanks for noticing it. We will fix the typo. \n\n&nbsp;  \nMany thanks again for your careful reading and constructive suggestions! Hope our replies above clarify your questions, and we appreciate the comments that helped us improve the presentation of the paper, which will be incorporated into the revised version of the paper.\n\n&nbsp;  \n\n\n### References\n[1] A. D’Amour et al., “Underspecification presents challenges for credibility in modern machine learning,” JMLR, 2022.\n\n[2] J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA, vol. 79, no. 8, pp. 2554–2558, 1982.\n\n[3]  I. Jarne, “Exploring flip flop memories and beyond: training recurrent neural networks with key insights,” Frontiers in Systems Neuroscience, 2024.\n\n[4] S. Funahashi, C. J. Bruce, and P. S. Goldman‑Rakic, “Mnemonic coding of visual space in the monkey’s dorsolateral prefrontal cortex,” J. Neurophysiol., vol. 61, no. 2, pp. 331–349, 1989.\n\n[5] P. S. Goldman‑Rakic, “Cellular basis of working memory,” Neuron, vol. 14, no. 3, pp. 477–485, 1995.\n\n[6] E. Marder and D. Bucher, “Central pattern generators and the control of rhythmic movement,” Curr. Biol., vol. 11, no. 23, pp. R986–R996, 2001.\n\n[7] M. M. Churchland et al., “Neural population dynamics during reaching,” Nature, vol. 487, no. 7405, pp. 51–56, 2012.\n\n[8] B. L. McNaughton, F. P. Battaglia, O. Jensen, E. I. Moser, and M.-B. Moser, “Path integration and the neural basis of the ‘cognitive map’,” Nat. Rev. Neurosci., vol. 7, pp. 663–678, 2006.\n\n[9] G. R. Yang, M. R. Joglekar, H. F. Song, W. T. Newsome, and X.‑J. Wang, “Task representations in neural networks trained to perform many cognitive tasks,” Nat. Neurosci., 2019. \n\n[10] M. Khona, S. Chandra, J. J. Ma, and I. R. Fiete, “Winning the Lottery With Neural Connectivity Constraints: Faster Learning Across Cognitive Tasks With Spatially Constrained Sparse RNNs,” Neural Comput., vol. 35, no. 11, 2023, doi: 10.1162/neco_a_01613.\n\n[11] N. Maheswaranathan et al., “Universality and individuality in neural dynamics across recurrent networks,” NeurIPS, 2019.\n\n\n[12] G. Yang and E. J. Hu, “Feature Learning in Infinite-Width Neural Networks,” in Proc. ICML, 2021.\n\n[13] Y. H. Liu, A. Baratin, J. Cornford, S. Mihalas, E. Shea-Brown, and G. Lajoie, “How connectivity structure shapes rich and lazy learning in neural circuits,” ICLR, 2024.\n\n[14] T. Flesch, K. Juechems, T. Dumbalska, A. Saxe, and C. Summerfield, “Orthogonal representations for robust context-dependent task performance in brains and neural networks,” Neuron, vol. 110, no. 7, pp. 1258–1270, 2022.\n\n[15] M. Farrell, S. Recanatesi, and E. Shea-Brown, “From lazy to rich to exclusive task representations in neural networks and neural codes,” Curr. Opin. Neurobiol., vol. 83, p. 102780, 2023."}, {"review_id": "sLwR0ZrqEc", "reviewer": "Reviewer_VYSY", "summary": "This paper studies solution degeneracy in vanilla recurrent neural networks (RNNs) across three levels: behavior, neural dynamics, and weight space. The paper runs a systematic empirical analysis by training ~50 networks on each of four neuroscience inspired tasks, varying: task complexity, feature learning, network size, and regularization. Previous work has mostly studied trained networks after fixing these variables, so this systematic study is quite timely.\n\nThe paper finds interesting relationships between the behavioral, dynamical, and weight space similarities of trained networks as a function of the task complexity, network size, and regularization. In particular, it finds that for some features (such as task complexity), the dynamical degeneracy and the weight degeneracy move in opposite directions (\"contravariant\") while for other features (such as regularization), they move in the same direction (\"covariant\").\n\nOverall, the paper provides a clear summary of the effect of key architectural choices on RNN solution degeneracy, which both can serve as a practical guide for neuroscientists using RNNs as well as opening the door for future studies to better explore the solution space of RNNs on nonlinear tasks.", "questions": "1. Do you think the results would hold for gated RNNs?\n2. Do you normalize the weight-change norm for the number of steps (T) required for convergence? If not, does incorporating some normalization there affect the results (I'm not sure what the right normalization is, maybe divide by $\\sqrt(T)$?)\n3. What about $\\ell_2$ regularization? I'm assuming it has a similar effect as $\\ell_1$ and low-rank penalties. I think it would be helpful to add $\\ell_2$ regularization experiments in section 3.5 for completeness.\n4. In figure 2B, for the sine-wave generation task, are the colored dots fixed points or samples from trajectories? I would assume the latter. If so, it might be helpful to show the fixed points as points and the trajectories as lines, to be consistent with the other panels.", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n- The paper studies a critical and (as pointed out in the introduction) relatively understudied problem of solution degeneracy in RNNs.\n- The paper is really well written, with clear motivation and thorough descriptions of the experiments and results.\n- The inclusion of both code and the trained model weights will be a valuable resource for the community.\n- The study examines networks trained across four disparate (but all strongly motivated in neuroscience) tasks. The consistency of the results across these tasks really strengthens the likely generalizability of the findings.\n\n# Weaknesses\n- Narrow definition of behavioral degeneracy: The behavioral degeneracy is defined as out-of-distribution generalization to longer time sequences (temporal generalization). However, this represents only a narrow slice of possible generalization we might expect networks to exhibit. Calling this category \"behavioral degeneracy\" slightly overstates the claims, in my opinion. I think being explicit about this and referring to this category as \"temporal generalization\" instead of \"behavioral degeneracy\" would be a more accurate description. (Or, run additional experiments to examine other types of generalization: such as generalization to noise, perturbations, or other types of task variability).\n- Limitation to vanilla RNNs. The study exclusively studies vanilla RNNs with tanh nonlinearities, which lack the gating mechanisms of LSTMs or GRUs. It is unclear whether the findings would generalize to gated networks.\n- Potential confound in the weight change norm: When quantifying the weight-change norm (Section 3.2.1), do you control for the number of training steps required for each task? I would imagine that due to the stochastic nature of SGD, the change in weight norm trivially grows with the number of steps (e.g. as discussed in https://arxiv.org/pdf/1806.08805). So if the number of steps varies across tasks, the weight change norm will also, in a way that has nothing to do with weight degeneracy.", "was_revised": true, "final_justification": "After reading the other reviews and the author rebuttal, I am keeping my score at 5, but I still think this is a solid paper and am in favor of acceptance.", "rebuttal": "Thank you for the thoughtful, constructive review and suggestions! We are glad you find the study of solution degeneracy critical and timely. Below we address the comments you raised in Questions and Weaknesses part of the review.\n\n### Questions\n\n> [Q1, W2]: Do you think the results would hold for gated RNNs?\n\nWe agree that testing if results generalize to gated RNNs is an important question! We do think the trend in dynamical degeneracy will hold with gated RNNs, as pointed out by [1], architectural choices affect the geometry but not topology of neural dynamics, which is shaped by the task structure. At the same time, the DSA metric we used to quantify dynamical degeneracy compares the topological structure of neural dynamics while being invariant to changes in geometry. As a preliminary investigation into the results’ generalizability with respect to architectural choices, we have trained GRUs to perform the Sine Wave Generation task, while changing task complexity by varying number of input-output channels. As task complexity increases, the dynamical degeneracy drops while weight degeneracy rises, consistent with what we observed for vanilla RNNs. We will add this result to the supplementary material in the revised version of the paper.\n\n| Task Complexity |1 channels | 2 channels | 3 channels |\n|---|---|---|---|\n| Dynamic | 2.61e-01 ± 2.09e-04 | 2.07e-01 ± 2.09e-04 | 1.86e-01 ± 7.89e-05 |\n| Weight | 1.25e-03 ± 1.16e-05 | 2.82e-03 ± 3.17e-05 | 4.43e-03 ± 2.71e-05 |\n\n> [Q2, W3]: Do you normalize the weight-change norm for the number of steps (T) required for convergence?\n\n* We plotted the weight change norm of our networks after every gradient step, and found it plateaued early in training, and the $||\\Delta W||$ between consecutive gradient steps fell below $10^{-7}$ per weight towards end of training, which can be considered neglectable compared to the weight change norm on the order of $10^{-3}$ per weight. We therefore do not normalize the weight change norm by T when measuring feature learning strength, since longer training brings minimally effect on the weight change norm.\n\n* We would also like to point out that in Section 3.2.1, we used weight change norm $||W_T-W_0||$ as a measure for feature learning strength [2-3]. The weight change norm computes the Frobenius distance between the trained weights and their initialization for each network (within network comparison, trained vs initialized weights). This is distinct from the weight degeneracy, measured as the mean permutation-invariant Frobenius distance across 50 networks at the end of training (across-network comparison). A higher weight change norm from initialization indicates stronger feature learning, but may not necessarily lead to higher weight degeneracy, since the latter depends on the loss landscape geometry. For example, all networks travel far from their initialization, but are clustered close together in the loss landscape after training will have high weight change norm, but low weight degeneracy.\n\n* The reviewer also hinted at an excellent point: the permutation-invariant Frobenius distance used to quantify weight degeneracy can change with respect to the magnitude of the weight matrices being compared. Here, we might be tempted to normalize the weight matrices by the number of training steps, or directly by their magnitude to bring them to the same scale then compute PIF distance. However, with the presence of tanh nonlinearity, scaling the weight magnitude changes the input-output mapping of the networks (e.g. for a network with smaller weight norms, upscaling it may bring the hidden activations to saturation in tanh nonlinearity). In other words, normalizing the weights by their magnitude / training steps distorted the computation they perform. We therefore do not normalize of the weight matrices before computing the weight degeneracy either, since the weight norm is entangled with its computation. \n\n> Q3: What about $l_2$ regularization? I'm assuming it has a similar effect as $l_1$ and low-rank penalties. I think it would be helpful to add regularization experiments in section 3.5 for completeness.\n\nYes! We do confirm that L2 penalty induces qualitatively same effect on degeneracy as L1 and rank regularization. Here we provide data on the 3BFF task, and will add L2 effect for all tasks to the supplementary material in the camera ready version.\n\n| Regularization Strength | $\\lambda$=0 | $\\lambda$=1e-5 | $\\lambda$=5e-5 | $\\lambda$=1e-4 |\n|---|---|---|---|---|\n| Dynamic | 0.122 ± 1.58e-04 | 0.120 ± 1.52e-04 | 0.063 ± 1.20e-04 | 0.050 ± 8.00e-05 |\n| Weight | 9.49e-04 ± 8.66e-07 | 9.25e-04 ± 8.46e-07 | 5.49e-04 ± 1.80e-06 | 3.49e-04 ± 1.43e-06 |\n| Behavior | 1.14e-03 | 8.64e-04 | 9.89e-05 | 4.23e-05 |\n\n\n> Q4: In figure 2B, for the sine-wave generation task, are the colored dots fixed points or samples from trajectories?\n\nWe thank the reviewer for noticing this, the colored dots are indeed trajectories of neural activities. We will change the visualization to show the fixed points as points and the trajectories as lines in the revised version of the paper to enhance consistency across figures.\n\n&nbsp;  \n\n\n### Weaknesses\n\n> W1: The behavioral degeneracy is defined as out-of-distribution generalization to longer time sequences (temporal generalization). However, this represents only a narrow slice of possible generalization we might expect networks to exhibit. \n\nWe will add temporal generalization in brackets everywhere to clarify the specific form of behavioral degeneracy that we are assessing in our study. Your suggestion to  explore other types of generalization such as response to noise, to input / activation perturbations etc. is an excellent suggestion for future work!\n\n&nbsp;  \n\n### References\n[1] N. Maheswaranathan et al., “Universality and individuality in neural dynamics across recurrent networks,” NeurIPS, 2019.\n\n[2] Y. H. Liu, A. Baratin, J. Cornford, S. Mihalas, E. Shea-Brown, and G. Lajoie, “How connectivity structure shapes rich and lazy learning in neural circuits,” ICLR, 2024.\n\n[3] T. George, G. Lajoie, and A. Baratin, “Lazy vs. hasty: Linearization in deep networks impacts learning schedule based on example difficulty,” TMLR, 2022."}, {"review_id": "wTxOkrOGEV", "reviewer": "Reviewer_vngU", "summary": "This paper proposes a unified framework to measure degeneracy in RNNs across three dimensions: behavioral, weight, and dynamic degeneracy. It quantifies the degree of degeneracy as a function of various factors, including task complexity, network size, learned features, and regularization.", "questions": "- Can the authors demonstrate that all models of interest converge to similar losses (e.g., within 2% of the average)? How robust are the results if this threshold is varied (e.g., within 5% of the average)?\n\n- The paper defines network size in terms of width, but depth also contributes to model capacity. Do the relationships reported in Table 1 hold for networks of varying depth?\n\nWe may consider increasing the score if these questions are addressed.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n- The authors propose three metrics to quantify degeneracy across behavioral, weight, and dynamic dimensions.\n- The paper analyzes multiple factors and establishes qualitative relationships between these factors and different types of degeneracy.\n\nWeaknesses:\n- The theoretical (even qualitative) rationale behind the observed relationships is limited.\n- The degeneracy measurements assume that different networks achieve comparable training losses, yet no explicit justification for this assumption is provided.", "was_revised": true, "final_justification": "This work investigates the degeneracy of RNNs across different dimensions and common factors shared across tasks. The rebuttal and discussion have effectively addressed my concerns regarding training convergence, the impact of network depth, and the variability across trained models. I recommend accepting this work and raise my score to 4.", "rebuttal": "We thank the reviewer for the thoughtful questions! \n\n> [Q1, W2]: Can the authors demonstrate that all models of interest converge to similar losses? How robust are the results if this threshold is varied?\n\n* In all experiments, we train networks until them reach a **near‑asymptotic, task‑specific mean-squred error (MSE) threshold** (0.001 for N-BFF, 0.01 for Delayed Discrimination, and 0.05 for Sine‑Wave Generation and Path Integration), after which we allow a patience period of 3 epochs and stop training to measure degeneracy. This early‑stopping criterion ensures that networks trained on the same task/condition achieve comparable final losses before any degeneracy analysis. While we applied this protocol throughout, we recognize it was not stated explicitly in the paper; in the revision we will describe it explicitly in Section 2.1 Model architecture and training procedure section.\n\n* To quantify the residual variation, we report the coefficient of variation (CV) of the final training loss across seeds for each condition, expressed as % of the mean. Header labels match the x‑axis levels used in the main‑text figures. Final losses cluster tightly near small values of the loss threshold, so even a double-digit CV translates to very small absolute variation. For example, a 10% CV at an MSE of 0.001 implies an s.d. of $1 \\times 10^{-4}$; at 0.01 it’s $1 \\times 10^{-3}$.\nWe further note that these variability values are not monotonic in any factor and sometimes move opposite to the degeneracy trends, arguing against a loss‑dispersion confound. \n\n| Task Complexity |Level 1| Level 2 | Level 3| Level 4|\n|---|---:|---:|---:|---:|\n|N‑BFF|6.30%|4.60%|9.30%|3.50%|\n|Delayed Discrim.|15.90%|8.40%|9.50%||\n|Sine Wave Gen.|9.94%|9.20%|8.70%||\n|Path Integr.|9.16%|2.85%|||\n\n| Feature Learning Strength |$\\gamma_1$|$\\gamma_2$|$\\gamma_3$|$\\gamma_4$|\n|---|---:|---:|---:|---:|\n|N‑BFF|9.70%|9.10%|13.40%|11.70%|\n|Delayed Discrim.|8.70%|12.60%|11.70%|12.30%|\n|Sine Wave Gen.|3.50%|3.90%|10.90%|11.70%|\n|Path Integr.|5.40%|5.20%|6.20%||\n\n| Network Width| 64 units|128 units|256 units|\n|---|---:|---:|---:|\n|N‑BFF|3.80%|4.20%|3.50%|\n|Delayed Discrim.|3.30%|3.00%|3.20%|\n|Sine Wave Gen.|17.80%|16.60%|16.40%|\n|Path Integr.|5.10%|5.40%|5.90%|\n\n| L1 Regularization |$\\lambda_1$|$\\lambda_2$|$\\lambda_3$|$\\lambda_4$|\n|---|---:|---:|---:|---:|\n|N‑BFF|2.10%|6.90%|1.10%||\n|Delayed Discrim.|15.90%|14.50%|16.70%|14.90%|\n|Sine Wave Gen.|10.40%|11.10%|11.10%||\n|Path Integr.|9.00%|7.10%|3.00%||\n\n\n| Rank Regularization |$\\lambda_1$|$\\lambda_2$|$\\lambda_3$|$\\lambda_4$|\n|---|---:|---:|---:|---:|\n|N‑BFF|2.10%|7.20%|4.30%||\n|Delayed Discrim.|15.90%|16.90%|13.60%|12.10%|\n|Sine Wave Gen.|13.90%|14.30%|15.90%||\n|Path Integr.|7.70%|7.90%|6.70%||\n\n* **Degeneracy trends for a looser stopping criterion on N-BFF, consistent with Table 1.** We verified that using a slightly looser stopping criterion for N‑BFF (MSE = 0.005 instead of 0.001) leaves the degeneracy trend for all four factors unchanged, consistent with Table 1. We also computed the CV of final training losses and confirmed that the higher threshold yields slightly greater across‑seed variability, yet the degeneracy trends are robust to both the magnitude and variability of the final loss. Due to the character limit, we show degeneracy trend at this higher loss threshold for varying feature learning strength, network size, and rank regularization an examples. We will include the full analysis for all four tasks in the supplementary material of the revised manuscript.\n\n| Feature Learning Strength| $\\gamma=0.5$ | $\\gamma=1$ | $\\gamma=2$ | $\\gamma=3$ |\n|---|---|---|---|---|\n| Dynamical Degeneracy | 5.51e-02 ± 6.08e-05 | 5.27e-02 ± 6.98e-05 | 4.73e-02 ± 7.92e-05 | 4.54e-02 ± 8.83e-05 |\n| Weight Degeneracy | 1.25e-01 ± 1.00e-04 | 1.26e-01 ± 9.52e-05 | 1.28e-01 ± 9.78e-05 | 1.31e-01 ± 1.01e-04 |\n\n| Network Width | Width=64 | Width=128 | Width=256 |\n|---|---:|---:|---:|\n| Dynamical Degeneracy | 4.29e-02 ± 8.41e-05 | 4.11e-02 ± 8.04e-05 | 3.94e-02 ± 8.48e-05 |\n| Weight Degeneracy    | 1.81e-01 ± 2.74e-04 | 1.27e-01 ± 1.08e-04 | 8.90e-02 ± 3.79e-05 |\n\n| Rank Regularization |$\\lambda=0$ | $\\lambda=1e-06$ | $\\lambda=5e-06$ |\n|---|---|---|---|\n| Dynamical Degeneracy | 1.22e-01 ± 1.58e-04 | 1.21e-01 ± 1.54e-04 | 1.15e-01 ± 1.78e-04 |  |\n| Weight Degeneracy | 9.49e-04 ± 8.66e-07 | 9.42e-04 ± 8.50e-07 | 8.78e-04 ± 8.08e-07 |  |\n\n\n* We also remark that precisely matching final losses and their variability across runs is inherently noisy; moreover, post‑hoc filtering to include only runs within a tight loss band can introduce selection bias (e.g., favoring certain optimization paths or initializations). Our method motivates future theoretical analysis where the variability of the loss can be controlled exactly, to provide a theoretical relationship between loss dispersion and solution degeneracy. We thank you for your insightful question again!\n\n\n&nbsp;  \n\n> Q2: The paper defines network size in terms of width, but depth also contributes to model capacity. Do the relationships reported in Table 1 hold for networks of varying depth?\n\n\nWe thank the reviewer for raising the important point of examining the network depth effect on solution degeneracy. Below we position the effect of depth on degeneracy within the scope of our contributions, clarify our design choice, and report the requested experiment.\n\n* Our study is motivated by task‑driven modeling in neuroscience, where researchers reverse‑engineer RNNs trained on neuroscience/cognitive tasks to understand neural computation. In this literature, recurrent models are predominantly **single‑layer (depth‑1), vanilla RNNs** [5, 18, 57, 72, 76, 79]; increasing depths would imply a very specific **biological** architectural bias that has not been explored much, to the best of our knowledge. More biologically grounded alternatives are multi‑region RNNs, which can be viewed as a single‑layer RNN with structured/sparse connectivity. While we focus on RNNs due to their relevance for modeling neural computations in our paper, depth is of fundamental interest to **deep feedforward networks** (e.g. CNNs). In feedforward networks, computations are carried out in terms of a limited number of transformations of a model’s representations over it's layers, instead of dynamics. Therefore, alternative metrics should be used to measure degeneracy (representational instead of dynamical degeneracy). Lots of work has been conducted to study the representations across depth in feedforward networks [1-3]. To the best of our knowledge the specific relation between solution degeneracy and model depth has not been studied yet, and could be a promising future direction!\n\n* **Controlling feature learning to isolate the effect of depth in deep RNNs requires a substantial new theory**. To isolates the network width effect on solution degeneracy, we use μP to equalize feature‑learning strength across width, but a depth‑consistent parameterization that achieves the same control for deep RNNs is not available and, to our knowledge, remains an open problem. To the best of our knowledge, only one recent paper [4] attempts a scaling limit for infinite depth, and it does so for feedforward residual networks with additional constraints (including effectively width=1), explicitly noting the difficulty of obtaining a depth‑consistent scaling for general width and nonlinear settings. Given these constraints, we made the conscious choice to keep depth fixed and vary width. We will update the paper to explicitly refer to width when we discuss model size.\n\n* We ran the requested experiment, and found **that dynamical and weight degeneracy are in general lower for deeper RNNs, although not perfectly monotonic.** We varied the number of recurrent layers from 1 to 4, while keeping the width for each layer fixed at 128 units, and examined dynamical and weight degeneracy on the Sine Wave Generation task. Overall, degeneracy decreases for deeper networks, consistent with Table 1. However, the trend is not perfectly monotonic. Upon closer examination of the feature learning strength of networks with different depth, we found that the feature learning strength is indeed unstable across depth, which could confound the change in degeneracy.\n\n| Model Depth |1 | 2 | 3 | 4 |\n|---|---|---|---|---|\n| Dynamical Degeneracy | 2.33e-01 ± 1.22e-04 | 2.35e-01 ± 2.20e-04 | 2.32e-01 ± 5.72e-05 | 2.30e-01 ± 5.66e-05 |\n| Weight Degeneracy | 1.10e-03 ± 2.17e-06 | 1.06e-03 ± 2.12e-06 | 1.06e-03 ± 1.76e-06 | 1.06e-03 ± 3.00e-06 |\n\n&nbsp;  \n\n> W1: The theoretical rationale behind the observed relationships is limited.\n\nWe agree that a theoretical account of the relationships between the four factors and degeneracy would be valuable. Our contribution is a systematic empirical framework for quantifying solution degeneracy across behavior, dynamics, and weight space. We establish robust empirical relationships between solution degeneracy and task complexity, learning regime, network width, and regularization. The consistency of results across tasks strongly motivates future theoretical work.\n\n\nWe hope our responses address your questions and concerns. If any further details or clarifications would be helpful, please let us know!\n\n&nbsp;  \n### References\n[1] M. Raghu, J. Gilmer, J. Yosinski, and J. Sohl-Dickstein, “SVCCA: Singular Vector Canonical Correlation Analysis for deep learning dynamics,” NeurIPS, 2017.\n\n[2] F. Chen, D. Kunin, A. Yamamura, and S. Ganguli, “Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks,” NeurIPS, 2023.\n\n[3] F. Cagnetta, L. Petrini, U. M. Tomasini, A. Favero, and M. Wyart, “How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model,” Phys. Rev. X, vol. 14, no. 3, p. 031001, 2024. doi: 10.1103/PhysRevX.14.031001.\n\n[4] G. Yang, D. Yu, C. Zhu, and S. Hayou, “Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks,” ICLR, 2024."}, {"review_id": "risg6oDcfH", "reviewer": "Reviewer_fDhN", "summary": "This paper investigates the solution degeneracy of recurrent neural networks (RNNs) trained on a suite of tasks, evaluating solution diversity across behavioral, dynamical, and weight-space levels. The central claim is that task complexity predictably modulates degeneracy: increasing complexity leads to more similar neural dynamics across training runs, but greater divergence in weight space. The authors use recently proposed tools such as Dynamical Similarity Analysis (DSA) and introduce the Permutation-Invariant Frobenius distance (PIF) to quantify similarity between trained RNNs. The paper proposes methods to quantify and control degeneracy (e.g., via task complexity, auxiliary loss functions, regularization).", "questions": "•\tThe paper needs stronger empirical or theoretical validation for DSA and PIF as measures of degeneracy. What threshold should count as significant difference in DSA or PIF? Why is a DSA of ~0.12 meaningful if previous work considered that range as \"invariant\"?\n\n•\tCan the authors provide flow fields, phase portraits, or other dynamical visualizations to show the difference in dynamics?\n\n•\tHow do the measures and trends extend to chaotic or high-dimensional RNNs? For instance, are the same trends visible with chaotic attractors?\n\n•\tIn Figure 3, are network sizes held constant across varying complexities? Could increasing complexity simply lead to networks using more of their capacity, thereby reducing variability?\n\n•\tThe paper does not report task performance. How well do these RNNs actually solve the tasks? Are higher-degeneracy networks lower-performing?\n\n•\tThe paper refers to dispersion of local minima but does not show empirical support. Could the observed weight variability be due to wide basins in a large solution space?\n\n•\tAre DSA comparisons based on autonomous (free-running) dynamics or input-driven trajectories? This distinction is crucial for interpreting the meaning of “similar dynamics.”", "limitations": "While the idea of the paper is interesting and novel, some limitations remain:\n\n•\tLack of analysis for chaotic or more realistic dynamical systems.\n\n•\tAbsence of flow field visualizations to confirm qualitative differences in neural dynamics.\n\n•\tLack of explicit model performance reporting.\n\n•\tLimited number of datapoints (e.g., 2–4 variants per task) which weakens broad generalization claims.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n•\tThe paper is clearly written, with well-organized figures and a clean narrative structure\n\n•\tThe focus on degeneracy and its controllability is relevant for both interpretability and robustness in ML models\n\nWeaknesses:\n\n\n•\tThe validation of the proposed complexity and degeneracy measures is limited. For example, DSA scores in the paper in part (e.g. Fig. 3B and Fig. 6) show values on the order of 0.12—similar to what's considered \"invariant\" in previous DSA work (e.g., Ostrow et al. 2023, Fig. 4,[1]). Without a stronger argument for statistical significance, these differences appear too small to support strong claims\n\n•\tSeveral key claims (e.g., the significance of degeneracy differences, the effect of “dispersion of local minima”) are made without sufficient visual or quantitative evidence. The paper would benefit from more direct visualizations of flow fields or attractors, especially for chaotic or high-dimensional tasks, to support the dynamical claims\n\n•\tThe scale of experiments is modest—typically 2–4 data points per task variant—which limits the strength of the generalization claims (e.g. Figs. 3,6,7). Tasks are also relatively low-dimensional and may not reflect the complexity of real-world RNN applications\n\n\n\n\nReferences\n[1] Ostrow, M., Eisen, A., Kozachkov, L., & Fiete, I. (2023). Beyond geometry: Comparing the temporal structure of computation in neural circuits with dynamical similarity analysis. Advances in Neural Information Processing Systems, 36, 33824-33837.", "was_revised": true, "final_justification": "All of my concerns have been addressed by the authors, and they have provided additional results that directly respond to the major issues I raised.\nThese new results substantially strengthen the paper.\nProvided that the described changes and additions are incorporated into the final version, I recommend accepting the paper.\nBased on the other reviews, it appears that all (or most) remaining concerns have also been addressed.", "rebuttal": "Thank you for your thoughtful and constructive questions! We are glad you find this work well-written and relevant for both interpretability and robustness in ML models research. \n\n> Q1: What counts as a meaningful difference in DSA and PIF?\n\n* **DSA scores are context dependent**.\nWe would like to point out that DSA does not provide a context-invariant score but rather the scale of DSA distance can depend on hyperparameters (particularly the delay and rank of DSA) since the Procrustes analysis over two dynamics matrices which relies on the *Frobenius* norm is not scale-invariant.\nTo select these hyperparameters, we followed the original DSA paper's process. We carefully keep these hyperparameters the same for all groups within each task for a fair comparison. In figures 3,4,6-8, we report the average pairwise DSA distance with std. error bands that clearly indicate when mean pairwise DSA differences between groups are significant.\n\n* **Calculating null distributions for DSA scores**:  For each of our 50 networks (in Fig 3B), we split the sampled neural trajectories from each network into 2 subsets, and computed DSA distances between them. This produced a \"null distribution\" of DSA scores that come from comparing identical dynamical systems, for which we report the 95% CI below (Header labels, e.g. \"Level 1\" for task complexity match the x‑axis levels used in the main‑text figures). The CIs below are on average an order of magnitude smaller than the DSA distance across networks. Therefore, it is valid to consider networks trained from different initialization as having statistically significantly different neural dynamics.\n\n| Task Complexity| Level 1 | Level 2 | Level 3 | Level 4 |\n|---|---|---|---|---|\n| N-BFF | [0.011, 0.013] | [0.009, 0.016] | [0.008, 0.013] | [0.006, 0.009] |\n| Delayed Discrim. | [0.039, 0.064] | [0.014, 0.076] | [0.025, 0.032] | | |\n| Sine Wave Gen | [0.057, 0.102] | [0.054, 0.081] | [0.048, 0.073] | | |\n| Path Integr. | [0.023, 0.037] | [0.010, 0.018] | | | | |\n\n* For PIF distance, we similarly establish a noise floor by randomly permuting the trained networks' recurrent matrix and computing the distance between the permuted weight matrix and the original one. The PIF metric reliably recovers a PIF distance of 0.\n\n&nbsp;  \n\n> Q2: Absence of flow field visualizations to confirm qualitative differences in neural dynamics.\n\nThank you for the suggestion! Unfortunately, the change in the rebuttal rules does not allow us to upload figures, but we do find cases where the dynamics look qualitatively different and will include these visualizations in the revised supplemental materials. That said, we believe visualization is not always the most reliable way to detect differences in neural dynamics, because:.\n\n* **Dynamical differences may lie in high-dimensional space and be\nvisually subtle.** As suggested by [56], networks trained on the same task can develop the same dynamical objects and transitions between them. We therefore aim to go beyond looking at the presence of dynamical objects (fixed points, limit cycles, etc.) and their transitions, and instead capture neural dynamics via a high‑dimensional linear operator with DSA, directly quantifying differences in the learned dynamics operators. Moreover, we found it hard to compare dynamics in visualizations beyond 3D, and the dimensionality of the neural activity of our tasks often exceeded that, as reported below for Q3.\n\n* **Visible differences can reflect geometry, not topology.** Flow fields and phase portraits primarily reflect the geometry of neural activity, while we define “similar” neural dynamics as topologically conjugate neural dynamics. Moreover, even if two networks have topologically equivalent dynamics, they can be related by a rotation, making their flow fields look very different (see, for example, Figure 6 in the appendix of [67]).\n\n* **We do observe qualitative differences in some cases.** In two‑channel Delayed Discrimination, networks develop two sets of fixed points (one per channel) representing the previous stimulus during the delay period. The PCA plots reveal either factorized, disentangled versus entangled fixed points for different channels in the state space. Linking this entanglement to behavioral/OOD generalization would be a promising next step to connect dynamical and behavioral degeneracy.\n\n&nbsp;  \n\n> Q3: How do the measures and trends extend to chaotic or high-dimensional RNNs? For instance, are the same trends visible with chaotic attractors?\n\nWe remark that our original task suite comprises neuroscience‑motivated tasks that can produce high‑dimensional neural dynamics as the number of input–output channels increase. We see that the average number of principal components explaining 95% variance are: N‑BFF (3/8/16/32 channels: 5, 10.32, 14.82, 29.2), Delayed Discrimination (2/3/4 channels: 21.5, 42.08, 42.1), Sine‑Wave Generation (2/3/4 channels: 17.38, 22.9, 29), Path Integration (2/3 channels: 5.86, 7.26).\n\nWe now **added a chaotic attractor task and verified that the effects of all four factors on dynamical and weight degeneracy are consistent with Table 1**. We will add full details  to the revised supplementary materials. Meanwhile, we summarize our results below:\n\nWe implemented the Lorenz 96 model [1] with 16, 24, and 32 dimensions and trained 50 RNNs on next-step prediction until the networks achieve a near asymptotic MSE loss at 0.0005. After training, the average Lyapunov exponent of RNNs trained on the Lorenz 96 attractor with 16 dimensions is $12.58 \\pm 0.74$, indicating chaotic neural dynamics.\n\n| Task Complexity |N=16 | N=24 | N=32 |\n|---|---|---|---|\n| Dynamical Degeneracy | 2.58e-01 ± 7.02e-05 | 2.57e-01 ± 6.97e-05 | 2.52e-01 ± 6.03e-05 |\n| Weight Degeneracy | 9.68e-04 ± 9.39e-07 | 1.02e-03 ± 1.14e-06 | 1.14e-03 ± 1.76e-06 |\n\n| Feature Learning Strength |$\\gamma=4$ | $\\gamma=5$ | $\\gamma=10$ | $\\gamma=20$ |\n|---|---|---|---|---|\n| Dynamical Degeneracy | 2.79e-01 ± 1.68e-04 | 2.77e-01 ± 1.74e-04 | 2.75e-01 ± 1.49e-04 | 2.70e-01 ± 2.16e-04 |\n| Weight Degeneracy | 1.35e-01 ± 1.08e-04 | 1.42e-01 ± 1.04e-04 | 1.76e-01 ± 1.24e-04 | 2.25e-01 ± 2.13e-04 |\n\n| Network Width |128 units | 256 units | 512 units |\n|---|---|---|---|\n| Dynamical Degeneracy | 2.80e-01 ± 1.39e-04 | 2.66e-01 ± 1.09e-04 | 2.55e-01 ± 1.04e-04 |\n| Weight Degeneracy | 1.23e-01 ± 8.82e-05 | 8.79e-02 ± 3.30e-05 | 6.23e-02 ± 1.21e-05 |\n\n| Rank Regularization |$\\lambda=0$ | $\\lambda=5e-5$ | $\\lambda=1e-4$ | $\\lambda=5e-4$ |\n|---|---|---|---|---|\n| Dynamical Degeneracy | 2.58e-01 ± 7.02e-05 | 2.10e-01 ± 1.12e-04 | 2.02e-01 ± 9.60e-05 | 1.98e-01 ± 1.35e-04 |\n| Weight Degeneracy |  9.68e-04 ± 9.39e-07 | 7.27e-05 ± 4.71e-07 | 1.51e-05 ± 3.71e-07 | 1.74e-06 ± 6.91e-09 |\n\nOur original tasks already span fixed‑point, limit‑cycle, and attractor manifold regime; the new chaotic regime extends this coverage.\n\n\n&nbsp;  \n\n> Q4: In Figure 3, are network sizes held constant across varying complexities?\n\nYes, the network sizes are held constant (128 units) across varying complexities, and indeed increasing task complexity forces equally-sized networks to utilize higher capacity . It is natural to assume that this would reduce the degeneracy in their solutions. This is argued by Cao & Yamins (2021) proposing the Contravariance Principle, but not rigorously tested with experiments. Our findings suggest that when networks are forced to operate at higher capacity, only behavioural and dynamical degeneracy decreases, while weight degeneracy increases. Hence, we reveal a more complex, unintuitive relationship between task complexity and degeneracy of solution. And we believe that the computational neuroscience community (and anyone trying to reverse engineer RNNs) should be aware of this.\n\n> Q5: How well do these RNNs actually solve the tasks? \n\nWe applied an early stopping to all networks trained on the same task, regardless of the task complexity. The early stopping threshold is a mean-squared error (MSE) of 0.001 for N-BFF, 0.01 for delayed discrimination, and 0.05 for Sine Wave Generation and Path Integration. The dynamic range for Path Integration is [-5, 5] and [-1, 1] for the other tasks. We confirmed that all networks used for analysis in the paper reached this threshold, and groups of networks trained on the same task all achieve **comparable loss** regardless of their levels of degeneracy. We will explicitly report the model performance in the revised paper, and add figures to the supplementary material showing example trials that juxtapose the ground truth and predicted signals.\n\n> Q6: The paper refers to dispersion of local minima but does not show empirical support.\n\nAt line 177, we present this as speculation (not a main claim). We apologize for the confusion and will move it to the Discussion section. It is also possible that the increased weight degeneracy is caused by wider basins in a large solution space. Clarifying the cause of increased weight degeneracy for harder tasks, or providing a mechanistic link between loss‑landscape geometry and weight degeneracy, remains an exciting direction for future work, but is beyond the scope of the current paper.\n\n> Q7: Are DSA comparisons based on autonomous (free-running) dynamics or input-driven trajectories? \n\nThank you for raising this important point! In the original DSA paper, the authors demonstrate the use of DSA using either autonomous systems or systems with controlled, low-dimensional inputs. We believe that we are performing our experiments in exactly the same regime as the original paper, since our tasks involve either autonomous dynamics or use simple controlled inputs. We also take care to ensure that the inputs being used to compare networks are exactly matched.\n\n&nbsp;  \n### References\n[1] E. N. Lorenz, “Predictability: a problem partly solved,” in *ECMWF Seminar on Predictability*, 4–8 Sept. 1995. Reading, U.K.: European Centre for Medium-Range Weather Forecasts, 1996."}]

Author rebuttal:

Meta-review: This paper investigates solution degeneracy in task-trained recurrent neural networks (RNNs), a well-documented phenomenon in which different networks achieve similar task performance using different network solutions (i.e., different weights and/or dynamics). This paper identifies 4 crucial factors governing this degeneracy and propose new metrics to quantify it. The problem is timely and important, and is likely to be of interest to a broad audience at NeurIPS. The reviewers were unanimous in their assessment that the paper makes a worthwhile contribution to the literature and should be accepted.  Congratulations! Please be sure to address all reviewer comments and criticisms in the final manuscript.
Decision: Accept (spotlight)