Can state space models explain time cells and abstract cognitive functions?

A rotational basis, not a trained time code

The paper asks whether a single mathematical principle can generate several temporal response profiles seen in neural recordings: time cells that fire at a specific moment in a delay, ramping cells whose activity rises or falls monotonically, and persistent oscillations. Its answer is that these can all be projections of one underlying process, a set of complex-valued states that rotate at different frequencies.

The framework is a state space model, or SSM, a sequence model that carries information forward in a hidden state vector rather than through recurrent connections. In this implementation the state update during a delay reduces to independent rotations, one per state element, so each unit behaves like a latent oscillator. The authors argue that time cells and ramping cells need not be separate biological entities: a rotating state appears as a time cell if its trajectory crosses the positive real axis during the delay, and as a ramp if it travels from the positive toward the negative real axis.

The key qualifier is stated by the authors themselves. The framework offers mathematical sufficiency and computational plausibility for generating profiles that resemble these cells; it is not a proof that biological evolution produced an equivalent basis. Direct comparisons with neural data or biologically constrained circuit models remain future work.

HiPPO initialization supplies the temporal basis

The structure comes from HiPPO initialization, a mathematical construction that projects a continuous input history onto a set of optimal basis functions. In the model, this initialization acts as the proxy for a pre-configured circuit: it gives the network a spectrum of rotational frequencies before any task training.

The paper tests this by comparing HiPPO-initialized SSMs with randomly initialized ones. Both learn the task with comparable accuracy, but their internal solutions differ. The HiPPO network shows sequentially organized time-cell patterns even before training, and this structure is preserved and refined during learning rather than constructed from scratch. The authors describe learning as selecting and adapting an existing temporal basis for the specific task.

The supplementary initialization ablation adds a boundary. Random real-diagonal and random complex initializations reached base accuracies of 86.8% and 86.9%, close to the HiPPO-based 88.1%, yet their temporal coverage at initialization was only 1.0 and 2.1 of 30 bins, versus 13.7 of 30 for HiPPO. Accuracy on this task therefore does not by itself demonstrate a structured temporal code.

A harder timing task separates the architectures

The main behavioral test is a three-stimuli temporal discrimination task. An S5-based agent, trained with actor-critic deep reinforcement learning, must identify the longest of three stimuli presented sequentially, with two delay periods between them. The authors chose three stimuli because preliminary tests showed that the standard two-stimuli version was too easy: both LSTM and SSM architectures quickly reached ceiling, making their capabilities hard to distinguish.

The SSM layer communicates with downstream value and policy networks through spiking signals, generated when the real part of a hidden state crosses a threshold, with a sigmoid surrogate gradient used for backpropagation. After training, the agent identified the longest sequence with an average accuracy of 88.1 ± 1.9% across 10 independently trained agents.

The architecture comparison is reported in the supplement. Under a matched parameter budget, an LSTM with 32 hidden units averaged 76.3%, an LSTM with 50 hidden units averaged 78.2%, the HiPPO SSM averaged 87.9%, and a randomly initialized SSM averaged 86.6%. The authors read this as an architectural advantage for precise temporal representation, while noting that the random-initialization result also suggests the task may be solvable without HiPPO's long-term history compression.

Time cells and ramps coexist in one population

Sorting hidden units by the time of their peak activation during the delays reveals a sequential cascade: different units become active at different moments, covering the interval in the way time-cell sequences are described in hippocampus and prefrontal cortex recordings. The heatmaps also show high-frequency oscillatory patterns in the remaining units.

Across the two delay periods and 5000 trial episodes, 21.0% of the population met the criteria for time cells and 61.8% showed monotonically increasing or decreasing activity classed as ramping cells. Classification required tuning curves from even- and odd-numbered trials to correlate above the 99th percentile of 100 shuffled comparisons, with a separate temporal-information criterion for time-cell candidacy.

The frequency organization is not random. Because rotational frequency is inversely proportional to unit index in the HiPPO spectrum, low-index units retain high-frequency oscillations while higher-index units rotate more slowly, producing the smoother time-cell and ramping profiles. A lesion study supports the functional weight of this arrangement: removing the first ~20 low-index, high-frequency units left accuracy largely intact, whereas removing high-index, low-frequency units caused an immediate and steep drop.

Retiming works only when the basis stays intact

The paper tests whether the model can adapt to different durations without relearning its internal dynamics. In the retiming experiment, HiPPO-initialized and randomly initialized models are compared on zero-shot transfer to extended delays and on recovery after retraining only the readout while state-transition and input-projection parameters stay frozen.

A related control shows that freezing the internal dynamics essentially freezes model performance, and retraining only the readout layer cannot recover it. The authors use this to argue that the state-transition structure, not just the readout, carries the temporal computation.

This is also where the LSTM comparison becomes interpretable. The authors conclude that LSTMs can produce oscillator-based, time-tiling representations, but through learning rather than as an intrinsic property of the architecture. The distinction the paper draws is between a model that must build a temporal basis and one that starts with it.

Repetition becomes a linearly decodable count

The scaling test moves from timing to an abstract cognitive function: event counting. In a lap-counting task, each lap has a random duration from 10 to 60 steps and 0 to 2 random pauses of 0 to 25 steps, so absolute elapsed time cannot identify the lap. The trained network reached a normalized edit-distance score of 0.999, where 1.0 means predicted event timings exactly match the ground truth.

The hidden-state analysis shows the same sequential cascade re-emerging within each of four laps. Because the temporal pattern resets each lap, the paper asks how count information persists. A linear support vector classifier trained on a 50/50 split decoded lap identity from hidden-state activity at a fixed timestep, achieving near-perfect performance on unseen test data.

The authors interpret this as the recurrent dynamics unrolling repeated inputs into distinct regions of state space, so that a downstream readout can extract event count through simple linear projections. The claim is about linear separability of the count variable in this model, not about how biological circuits represent number.

A sufficient model, not a demonstrated mechanism

The paper's own limitations set the boundary of its claim. The tasks may not fully exploit the model's capacity, and the fact that randomly initialized networks also performed well suggests the chosen tasks might be solvable with naive designs rather than requiring HiPPO's long-term compression. Performance varied across seeds, so the reported numbers may be below the best achievable under this framework.

The biological mapping is also explicitly partial. HiPPO is a mathematically optimal solution for online function approximation, but the authors state there is no guarantee the brain uses this specific orthogonal basis rather than another pre-configured dynamic. The model abstracts away hippocampal microarchitecture and cell-type heterogeneity, and it relies on artificially encoded inputs rather than real spiking data.

The most useful reading is therefore a comparison rather than a verdict. Within this paper, the decisive contrast is between a model whose temporal basis exists before training and one that must learn it: HiPPO initialization produced structured temporal coverage of 13.7 of 30 bins at the outset, while random initializations covered 1.0 to 2.1, even though final task accuracy was similar. That gap, not accuracy alone, is the paper's central evidence for a pre-configured rotational basis.

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