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State space models (SSMs) are an emerging alternative to Transformers. SSMs offer linear-time inference and parallel training, but most implementations make the parallel scan efficient by making the recurrence diagonal: each channel of the hidden state is multiplied by an input-dependent real or complex number. These numbers commute, so the state records which inputs occurred but not their order. We formalize this with representation theory, which explains and fixes the state-tracking failures previously reported for SSMs. The state update defines a linear representation of the monoid of input sequences, and we show that an SSM tracks a group at every sequence length exactly when that group is a quotient of the group generated by its transition matrices. Two consequences follow. Diagonal real- and complex-valued SSMs can track only commutative groups, while the transition matrices of any SSM that can track a group must split into independent blocks over the real, complex, and quaternion numbers. We then introduce HyperSSM, which adds quaternion channels while keeping linear-time inference and parallel training. Trained only on final answers, HyperSSM solves the group tasks of recent state-tracking benchmarks exactly with at most five state numbers. This includes $S_5$ with 32 generators, where previous SSMs are at chance, and HyperSSM generalizes to sequences 25 times longer than those seen in training. Because the theory says what the transitions must encode, the learned weights of the linear recurrence can be interpreted directly. We demonstrate this by recovering the calibration of a real gyroscope, the rotations of imperfect qubit gates and the geometry of protein backbones.

Biological neural circuits are both modular and recurrent, with dense communities sparsely linked to one another. Spiking neural networks (SNNs) for reinforcement learning are usually feed-forward, and it is unclear how to obtain functional modules in a recurrent SNN or what they buy. In this work, we introduce a class of spiking networks, called Community Recurrent SNNs (CoRe SNNs): recurrent networks partitioned into communities with fixed roles. A controller community receives the stimulus, each action has its own community, and latent communities carry internal state. Spiking activity is then a walk between communities, and the action is the community the walk reaches. All learning is local and three-factor, and the recurrent communities are closer to cortical microcircuits than a feed-forward actor. We compare three regimes for obtaining the communities: dynamic modularity, in which a new learning rule we introduce, Map-STDP, encourages modular structure during learning; partial modularity, in which a stochastic block model (SBM) initialization enforces it partly; and full modularity, in which sealing enforces it fully. On CartPole, across three spiking neuron models, Map-STDP improves learning over the same network without it matches full modularity (sealed communities) on two of the three models, and makes the observed spike walk more compressible. Replacing the feed-forward actor of a TD-STDP actor-critic with recurrent action communities improves on the feed-forward network on CartPole and LunarLander, and matches it on Acrobot. How strongly to enforce modularity depends on the task: full modularity helps most on Acrobot, while on LunarLander the dynamic, partial and full regimes perform alike. Overall, these recurrent modular networks can match or even improve upon feed-forward approaches, with improvements in bio-plausibility and energy efficiency.