State Space Models Naturally Produce Time Cell and Oscillatory Behaviors and Scale to Abstract Cognitive Functions

arXiv:2507.13638 · q-bio.NC, cs.LG · Submitted 2025-07-18 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: I'm Ines.

Marcus: And I'm Marcus, here with Yuki.

Ines: Today's paper: "State Space Models Naturally Produce Time Cell and Oscillatory Behaviors and Scale to Abstract Cognitive Functions".

Marcus: The gist The proposed model suggests that State Space Models can naturally produce time cells and oscillatory behaviors, which are essential for cognitive functions,

Ines: First, who's behind it and why it matters.

Title and authors: Ines: The paper we're looking at is titled "State Space Models Naturally Produce Time Cell and Oscillatory Behaviors and Scale to Abstract Cognitive Functions." And the authors are Sen Lu, Xiaoyu Zhang, Mingtao Hu, Eric Yeu-Jer Lee, Soohyeon Kim, and Wei D. Lu.

Marcus: It’s a pretty ambitious title. It suggests two big things: first, that State Space Models can naturally create time cells and oscillations without explicit training second, that this framework scales up to abstract cognitive functions like counting or event recognition.

Yuki: I'm curious if the paper actually delivers on both fronts, because linking simple timing to complex counting tasks is a significant leap for us as population geneticists trying to understand behavioral evolution.

Ines: Well, what they show is that the model, through its pre-configured structural basis, can naturally produce those sequences. They test this by showing an S5-based agent trained on a temporal discrimination task exhibiting emergent time cell sequences during delay periods.

Marcus: And they didn't just stop at timing; they showed it works for more complex things, like a lap-counting task where the network achieved ninety-seven point zero percent accuracy against random chance.

Yuki: Ninety-seven percent accuracy on a lap count is substantial, but I need to know what that means biologically in terms of how this structure supports counting versus just simple timing sequences we discussed earlier.

Ines: The analysis of the hidden state dynamics showed that the model naturally achieved this lap-counting task through reward without needing explicit instructions regarding temporal coding.

Marcus: That's a key part—the reward drove the structure to handle the abstract variable, not an instruction to count in terms of time intervals.

Yuki: So it suggests that abstract counting isn't just a fancy way of measuring time cells; it’s something this underlying dynamic structure can do on its own when given the right setup.

Ines: The framework is also shown to be scalable beyond simple interval timing, which means it can handle more complex cognitive functions if you build the right architecture on top of those SSM dynamics.

Marcus: And they also looked at how learning interacts with these pre-configured structures, which leads into some interesting ideas about adaptation.

The paper's summary: Yuki: So to summarize what they found in "State Space Models Naturally Produce Time Cell and Oscillatory Behaviors and Scale to Abstract Cognitive Functions," it seems the main point is that State Space Models provide a potential biological model because their differential equations align with biophysical processes.

Ines: Right. They adopt a diagonalized version of the HiPPO framework for initialization, which means they start with these optimal pre-configured oscillatory states acting as a structural basis for temporal cognition.

Marcus: The summary emphasizes that these pre-configured oscillatory states act as a structural basis, and this leads to the hypothesis that learning acts primarily as a selection mechanism that fine-tunes these modes rather than building them de novo.

Yuki: So they're saying the inherent structure is biological, and learning is just refining it, which aligns with ideas about innate structures being modulated by experience.

Ines: They show that time cells emerge without explicit learning, and they also classify hidden state units into time cells and ramping cells based on their trial-averaged activity.

Marcus: The classification part is interesting because it helps us characterize the temporal coding properties of those hidden state units, separating them from other types of dynamics.

Yuki: And they found that ramping cells are defined by a monotonic increase or decrease in firing rates over a timed interval, while time cells have a higher temporal information based on the Skaggs et al. formula.

Ines: That information measure is what identifies the candidates for time cells, and they are considered candidates if their temporal information is higher than the 99th percentile.

The paper's improvements: Marcus: Now let's talk about the improvements they suggest, which are pretty interesting because they address how this structure can be used more flexibly in practice.

Ines: One improvement is developing deep learning architectures initialized with High-order Polynomial Projection Operators, or HiPPO, to inherently capture those optimal pre-configured oscillatory states.

Marcus: So the idea is that the initialization itself ensures you start with these structures already built in, making learning just a selection mechanism for them.

Yuki: That sounds like a way to enforce biological plausibility from the very beginning of the model development, ensuring we don't accidentally create something purely synthetic.

Ines: Another point is applying the State Space Model framework to create computationally tractable systems capable of producing emergent behaviors such as event counting.

Marcus: This addresses one of the big hurdles—making these models practical enough to actually run on real data, moving them beyond just theoretical constructs.

Yuki: If we can get this framework to handle event counting, it could offer a scalable framework for understanding neural activities far beyond simple interval timing tasks that we've discussed.

Ines: They also suggest implementing SSMs with frozen internal dynamics to enable intrinsic scaling, which allows agents to adapt without needing full re-training.

Marcus: That’s a practical way to handle adaptation. If you only retrain the linear readout layer while keeping the internal dynamics, like Lambda and B, frozen, performance fully restores itself.

Conclusion: Ines: So to wrap up on "State Space Models Naturally Produce Time Cell and Oscillatory Behaviors and Scale to Abstract Cognitive Functions," the paper concludes that the simple principle of rotational dynamics may be an essential computation for time perception.

Marcus: It really boils down to this: rotational dynamics might be an essential computation for time perception in a biological sense. They are suggesting that this structural idea is fundamental to how we perceive and encode time.

Yuki: I think it means we should look at the broader implications for how we model development across species, seeing if these intrinsic dynamics are conserved even as we move away from simple timing tasks toward more abstract counting.

Ines: It’s a big piece of work because it offers a framework that bridges the gap between millisecond time-scale neuronal behaviors and long time-scale macroscale functions.

Marcus: It provides a way to understand how pre-configured structures can guide learning in complex systems, whether we're talking about biological neural circuits or sophisticated AI architectures.

Yuki: We'll be keeping an eye on this, especially as they look into validating this framework using real spiking data to see if these structural ideas hold up in reality.

Ines: And that’s where the next step is—validating this framework with real spiking data to truly test whether these hypothesized structures exist in living systems.

Sen Lu, Xiaoyu Zhang, Mingtao Hu, Eric Yeu-Jer Lee, Soohyeon Kim, Wei D. Lu

University of Michigan

q-bio.NC, cs.LG

Submitted: 2025-07-18

Updated: 2026-10-14

Comments: Sen Lu and Xiaoyu Zhang contributed equally. Wei D. Lu is the corresponding author. 5 figures are included in 15 pages

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: The gist The proposed model suggests that State Space Models can naturally produce time cells and oscillatory behaviors, which are essential for cognitive functions, by leveraging pre-configured

Key concepts

State Space Models (SSMs): SSMs are mathematical models where the current state of a system depends on its past history and external inputs. The paper uses them because their differential equations mimic how physical systems, like neurons, evolve over time. This structure is hypothesized to be biologically plausible.
Rotational Dynamics/Basis: This refers to using complex-valued coefficients in the SSM initialization, specifically a diagonalized version of the HiPPO framework. This pre-configured rotational structure acts as a structural basis that naturally organizes temporal information, suggesting it is an essential computation for time perception.
Time Cells and Ramping Cells: These are specific types of neural activity patterns identified in the model. Time cells show selective activation at distinct moments, while ramping cells exhibit monotonic changes in activity over time. Classifying these helps characterize how the hidden state units encode temporal information.
Scalability to Abstract Cognition: The framework is shown to go beyond simple timing tasks. It successfully performed a lap-counting task by naturally achieving high accuracy, demonstrating that this SSM approach can model complex, abstract cognitive functions without needing explicit instructions for temporal coding.

Terminology

Reinforcement Learning — 1× in this paper · explained in 4 episodes

Summary

The gist The proposed model suggests that State Space Models can naturally produce time cells and oscillatory behaviors, which are essential for cognitive functions, by leveraging pre-configured structural basis functions and rotational dynamics.

Theoretical Framework

The paper proposes that State Space Models (SSMs) offer a potential biological model because the differential equations governing elements in an SSM are conceptually consistent with the dynamics of biophysical processes. The mathematical formulation of a linear time-invariant SSM is given by equations (1) and (2), where the hidden state evolves based on its history and external inputs The authors adopt a diagonalized version of the HiPPO framework for initialization, denoted as M with complex-valued coefficients. This mathematical formulation leads to the main hypothesis that these optimal pre-configured oscillatory states act as a structural basis for temporal cognition.

Emergent Temporal Dynamics

The model demonstrates that time cells and other oscillations can naturally emerge without explicit learning. The results showed that an S5-based agent trained on a temporal discrimination task exhibited emergent time cell sequences during delay periods. Different units became selectively active at different moments, effectively cascading the entire temporal interval. The internal state dynamics remain largely stable during training when initialized according to HiPPO framework.

Role of Learning and Initialization

The study distinguishes between pre-configured structures and learning by comparing networks initialized with HiPPO versus randomly initialized ones. The HiPPO-initialized network exhibits sequentially organized ‘time cell’ patterns even before training, which is preserved and refined throughout the learning process without specifically training for it. In contrast, a randomly initialized network only produced unstructured, high-entropy dynamics with no discernible temporal order after extensive training. The structural hypothesis is validated by parameter-freezing experiments where the HiPPO-initialized network maintained accuracy when internal parameters were frozen.

Scalability to Abstract Cognition

The framework is shown to be scalable beyond simple interval timing to abstract cognitive functions. The model successfully performed a lap-counting task where the network achieved 97.0% accuracy against random chance. The analysis of hidden state dynamics showed that the model naturally achieved this through reward without explicit instructions regarding temporal coding. Furthermore, a linear classifier could solve the lap-counting task with 100% accuracy on unseen test sets by partitioning the state space into distinct regions corresponding to each lap count.

Biological Plausibility and Future Directions

The rotational dynamics inherent in the HiPPO framework are suggested to be physically instantiated implementations of basis functions, aligning with biological requirements. The use of complex-valued states aligns with the irreducible two-dimensional requirement for oscillation found in simplified biophysical models. Future work should investigate whether evolving a population of SSMs over generations causes weight matrices to spontaneously converge toward HiPPO-like structures. Finally, it will be critical to validate this framework using real spiking data.

Classification of Functional Cell Types

To characterize the temporal coding properties of hidden state units, units were classified into time cells and ramping cells based on their trial-averaged activity. Ramping Cells were classified if their temporal tuning curve showed a monotonic increase or decrease after fitting a linear regression. Time Cells were identified by calculating the temporal information of the tuning curve based on the Skaggs et al. formula, and they were considered candidates if their temporal information was higher than the 99th percentile

Actor-Critic RL Framework

The framework employs an actor-critic architecture with an SSM memory module where the SSM processes the raw input signal stream to generate a state representation. The loss function for training is defined as L = Lpi + LV, where Lpi = ∑− log pi ∗ (Rt − V̂)(T−1) t=0 and LV = ∑l1(V̂,Rt)(T−1) t=0. During training, the actor updates its policy in the direction suggested by the critic using temporal-difference error based on discounted cumulative reward. The agent was trained to minimize this loss function L = Lpi + LV.

Extended Data

The experiments utilized specific hyperparameters summarized in Extended Table 1. Key settings included training episodes of 250000, a batch size of 1 trial at a time, and the use of the Adam optimizer. The SSM discretization used was Zero-Order Hold (ZOH) with an S5 discretization. The surrogate function employed was a Sigmoid-shaped surrogate with its smooth, bellshaped profile providing a continuous gradient.

Conclusion

The study concludes that the simple principle of rotational dynamics may be an essential computation for time perception.

Improvements for AI systems

textbf Intellectual Property for Temporal Encoding via HiPPO Initialization: Develop deep learning architectures initialized with High-order Polynomial Projection Operators (HiPPO) to inherently capture optimal pre-configured oscillatory states, ensuring that learning acts primarily as a selection mechanism that fine-tunes these pre-configured oscillatory modes, rather than constructing temporal codes de novo.
textbf Scalable Cognitive Function Modeling: Apply the State Space Model (SSM) framework to create computationally tractable systems capable of producing emergent behaviors such as event counting, offering a scalable framework for understanding neural activities beyond simple interval timing.
textbf Adaptive Temporal Re-mapping: Implement SSMs with frozen internal dynamics to enable intrinsic scaling, allowing agents to adapt without re-training by leveraging the ability of the model to process extended delays, as demonstrated by the finding that retraining only the linear readout layer (while keeping internal dynamics Λ and B frozen) fully restores the performance.
textbf Abstract Event Recognition: Design systems using SSMs initialized with HiPPO to solve complex tasks like lap-counting, allowing them to naturally 'unroll' repetitive sensory inputs into a structured trajectory and use linear classifiers on hidden states to achieve 100% accuracy on the unseen test set for abstract variables like count.