Source-linked AI summary
Universality and individuality in neural dynamics across large populations of recurrent networks
Niru Maheswaranathan, Alex H. Williams, Matthew D. Golub, Surya Ganguli, David Sussillo
TL;DR
The paper asks which aspects of RNN dynamics depend on architectural choices and which are universal, a foundational issue for using task-based models to interpret brain dynamics. It studies large populations of RNNs across architectures and tasks, finding architecture-sensitive representational geometry but broadly universal computational dynamical structure.
Problem
It asks which RNN dynamical properties vary across architectural choices and which are universal, an issue important for neuroscientific modeling.
Method
The authors train and tune large populations of Vanilla, UGRNN, LSTM, and GRU networks across canonical memory, pattern-generation, and integration tasks, then analyze representations, fixed-point topology, and linearized dynamics.
Results
Representational geometry varies significantly with architecture and activation function, whereas fixed-point topologies and linearized dynamics converge to qualitatively similar solutions across architectures.
Takeaways & Limitations
SVCCA and CKA may reflect modeling choices more than task performance, while fixed-point topology and linearized dynamics may provide more universal indicators of computational structure.
Takeaways & Limitations
The fixed-point analysis requires compared networks to have the same number of fixed points, motivating methods robust to differing numbers.
Abstract
from arXiv · showhide
Task-based modeling with recurrent neural networks (RNNs) has emerged as a popular way to infer the computational function of different brain regions. These models are quantitatively assessed by comparing the low-dimensional neural representations of the model with the brain, for example using canonical correlation analysis (CCA). However, the nature of the detailed neurobiological inferences one can draw from such efforts remains elusive. For example, to what extent does training neural networks to solve common tasks uniquely determine the network dynamics, independent of modeling architectural choices? Or alternatively, are the learned dynamics highly sensitive to different model choices? Knowing the answer to these questions has strong implications for whether and how we should use task-based RNN modeling to understand brain dynamics. To address these foundational questions, we study populations of thousands of networks, with commonly used RNN architectures, trained to solve neuroscientifically motivated tasks and characterize their nonlinear dynamics. We find the geometry of the RNN representations can be highly sensitive to different network architectures, yielding a cautionary tale for measures of similarity that rely representational geometry, such as CCA. Moreover, we find that while the geometry of neural dynamics can vary greatly across architectures, the underlying computational scaffold---the topological structure of fixed points, transitions between them, limit cycles, and linearized dynamics---often appears universal across all architectures.
1 Introduction
The paper asks which RNN dynamical properties are architecture-dependent and which are universal across models trained on the same task. It studies large network populations to compare representational geometry with dynamical structure.
- Motivation: Infinitely many model networks and expanding choices of architectures, initialization, regularization, and automated design complicate scientific interpretation of task-trained RNNs.These modeling degrees of freedom motivate separating architectural individuality from task-related invariants.
- Research question: The central question is whether RNN dynamics vary across architectural choices or contain properties that remain universal across architectures.Universal properties could recur across artificial networks and brain circuits solving the same task.
- Approach: The study examines thousands of RNNs trained on neuroscientifically motivated tasks involving discrete memory, oscillatory dynamics, and integration.The paper frames this as an extensive numerical study of individuality and universality in nonlinear RNN dynamics.
- Main findings: Representational geometry varies considerably across architectures, whereas fixed-point topology, limit cycles, transitions, and linearized dynamics show greater universality.This contrast motivates caution when using geometry-based similarity measures to infer shared computation.
- Significance: The paper positions universality and individuality in RNN dynamics as both theoretically important and relevant to neuroscientific modeling.It connects the analysis to broader ideas of universality classes in dynamical systems and statistical mechanics.
2 Methods
The authors train broad populations of RNNs on canonical neuroscience tasks, then compare representational geometry and dynamical-system structure across models. Their analyses include SVCCA, CKA, fixed points, transitions, and local linearizations.
- Model architectures: The four architectures are Vanilla RNN, UGRNN, GRU, and LSTM, with relu or tanh non-gate activations and sigmoid gating units.The RNN state is recursively updated from the previous state and input, with predictions produced by a linear readout.
- Model training: The study trains 2,880 networks per task across 96 configurations spanning four architectures, two activations, three sizes, four regularization strengths, and 30 seeds.Configurations also receive separate searches over gradient clipping and learning-rate schedule parameters.
- Tasks: The networks are evaluated on canonical tasks covering discrete memory, frequency-cued sine-wave generation, and context-dependent integration.These tasks probe memory maintenance, oscillatory pattern generation, and selective integration of noisy streams.
- Representational geometry: SVCCA and CKA compare whether two networks’ responses to the same inputs have aligned representational geometry after handling low-variance directions.SVCCA retains approximately 95% of variance before CCA, while CKA weights components by singular value.
- Dynamical analysis: Dynamical-systems analysis finds fixed points under static task conditions and studies transitions, limit cycles, and linearized behavior around those points.The Jacobian and its eigenvector decomposition characterize local dynamics near each fixed point.
- Visualization: MDS projects pairwise network-similarity matrices into two dimensions to visualize clustering across configurations for each task.The same procedure is applied to geometry and dynamical similarity analyses.
3 Results
Across tasks, RNNs trained with different architectures showed architecture-sensitive representational geometry but broadly similar computational dynamics. Fixed-point topology and linearized dynamics exposed shared computational scaffolds that SVCCA-based geometry sometimes obscured.
- 3 Results: The study trained large RNN populations on discrete memory, sine-wave generation, and context-dependent integration tasks.Architectures included vanilla, UGRNN, LSTM, and GRU variants with different activations and modeling choices.
- 3 Results: Across tasks, representational geometry was sensitive to architecture, while fixed-point topology and linearized dynamics were qualitatively similar across architectures.The authors frame this contrast as a caution against treating SVCCA or CKA similarity as a direct indicator of functional similarity.
- 3 Results: Three-dimensional trajectories for the 3-bit memory task roughly traced cube corners across architectures, yet SVCCA revealed systematic representational differences.The task encoded all 2^3 possible outputs as stable fixed points, with saddle points routing activity toward appropriate memories.
- 3 Results: Sine-wave networks showed architecture- and activation-related SVCCA differences, whereas fixed-point topology did not systematically cluster by either factor.Each command frequency produced one input-dependent fixed point, and a mildly unstable oscillatory eigenmode aligned generally well with the commanded frequency.
- 3 Results: Context-dependent integrators exhibited geometry clustering by architecture and activation, while fixed-point topology clustered weakly by architecture but more clearly by activation.All architectures approximated line attractors; gated networks used more eigenvalues near 1 than vanilla RNNs for integration.
4 Related Work
Prior work established universal properties and dynamical-systems analyses for neural networks, but systematic comparisons across common RNN architectures remained limited. This paper addresses that gap while also illustrating risks in interpreting representation-based similarity.
- 4 Related Work: Earlier studies examined universal properties in deep networks, including representational geometry in optimally trained deep linear networks and capacity changes with depth and width.Related RNN work also established universal approximation and compared per-parameter capacity across architectures.
- 4 Related Work: Figure 4 demonstrates that SVCCA can place trained and untrained networks of one nonlinearity closer than trained networks using different nonlinearities.This motivates caution when using representational similarity to infer functional similarity.
- 4 Related Work: Neural-network research has long applied nonlinear dynamical-systems methods to optimized networks and compared artificial dynamics with biological circuits.These efforts provide the broader context for analyzing fixed points, attractors, and transitions in task-trained RNNs.
- 4 Related Work: Previous RNN studies had not systematically surveyed dynamical variability across commonly used architectures such as LSTMs and GRUs.They also had not quantified how dynamical solutions vary with architecture and nonlinearity.
5 Discussion
The study finds both individuality and universality in recurrent-network dynamics: representational geometry varies across architectures, while several computational structures and strategies are shared. Fixed-point analysis is constrained by requiring compared networks to have the same number of fixed points.
- Representational geometry varies significantly with architecture and activation function, whereas fixed-point topologies vary much less across networks.The study also identifies common linear strategies, including essentially linear oscillations for the sine-wave task and linear integration for the CDI task.
- Linear analyses reveal common computational strategies across architectures but also small systematic differences clustered by architecture.For example, the frequency of the oscillatory mode can differ from the input frequency.
- The degree to which higher-order terms contribute to the total solution appears to be another aspect of individuality.
- The fixed-point analysis requires compared networks to have the same number of fixed points.Most trained networks in the three studied tasks did have the same number, but future work should handle differing counts more robustly.
- The study aims to establish a foundation for methods comparing RNN dynamics and future connections between biological circuits and artificial networks.
A RNN Architectures
The paper examines four RNN architectures spanning different levels of complexity, reflecting distinct historical preferences in computational neuroscience and machine learning.
- Four RNN architectures were examined to represent varying degrees of complexity and sophistication.
- Vanilla RNNs have historically been favored by computational neuroscientists, whereas LSTM and GRU networks have been favored by machine learning practitioners for performance advantages.
- Neuroscientists are increasingly using gated RNNs to study more complex phenomena.
Vanilla RNN
The vanilla RNN uses a nonlinear recurrent update, while the provided gated formulation combines the prior hidden state with a candidate state through a gate.
- The gated formulation interpolates between the previous hidden state and a candidate state using a gate.
- The candidate state is computed from the previous hidden state and current input through a nonlinear transformation with a bias.
B Non-normal linear dynamical systems analysis
The non-normal linear-dynamics analysis represents the Jacobian through left and right eigenvectors and a diagonal matrix of complex eigenvalues, without requiring orthogonal eigenvectors.
- The analysis uses an eigenvector decomposition for non-normal matrices while suppressing dependence on the fixed point and input for clarity.
- The columns of R are right eigenvectors of Jrec, the rows of L are left eigenvectors, and L equals R^-1.
- Λ is diagonal and contains complex-valued eigenvalues sorted by magnitude, with no requirement that R^T R = I.This permits potentially sophisticated locally linear dynamics.
C Network performance
All retained networks achieve low error on their respective tasks, with final performance summarized across networks using mean squared error.
- All networks retained for analysis achieve low error on their respective tasks.
- Figure 5 summarizes the distribution of final performance across networks as mean squared error.
- The performance visualization covers all networks used for the paper’s analyses.
D SVCCA
SVCCA compares state-vector representations from two RNNs by reducing each representation with singular value decomposition and then applying canonical correlation analysis.
- SVCCA takes two matrices containing the state-vector representations of two RNNs over P test inputs.
- The two representation matrices may have different neuron counts, denoted N1 and N2.
- SVCCA truncates each singular-value decomposition to the top R singular vectors before applying canonical correlation analysis.
E Centered kernel alignment (CKA)
CKA is compared with SVCCA as a representation-similarity measure across the flip flop, sinewave, and context-dependent integration tasks.
- CKA measures similarity between representations and is invariant to orthogonal transformation and isotropic scaling.
- Unlike SVCCA, CKA is not invariant to invertible linear transformations.
- Both SVCCA and CKA show representation differences that cluster by RNN architecture.
- The figures compare SVCCA and CKA for the flip flop, sinewave, and context-dependent integration tasks.