Dynamical Systems

Uddipan Banik derives a dynamical account of dark-matter halo density profiles using the Boltzmann–Poisson equations for an expanding Universe populated by subhalos of many masses. A self-similar solution links density slope to halo mass growth; including gravitational scattering and subhalo capture gives gamma = 6/(s + 2). Initial collapse with s = 2 produces a prompt cusp of slope 3/2. Hierarchical growth connects the slope to the primordial power spectrum and recovers the Syer–White relation during matter domination. Stable clustering selects an inner Navarro–Frenk–White cusp of slope one, while stalled accretion approaches the outer slope of three. The framework connects these profiles to successive stages of halo assembly.

One of the many ocean swells in the history of computational approaches to mind and brain is the rise and fall of cognitive architectures. Although supplanted by neural networks as the dominant modelling modality, cognitive architectures possess many attributes sorely coveted by the computational modeller: large, architectural-level scales, unified explanations of disparate cognitive functions in single models, and rich, hierarchical-modular decompositional structure conducive to building and modifying large computational systems. In contrast, we find in the neural network literature a proliferation of small-scale, localised models of single tasks, isolated functions, or dissected brain areas. What would it take to unify all this local knowledge into an architectural neural model? The key property that ensures (de-)compositional structure is functional encapsulation – the preservation of component behaviour in disparate contexts. Here, by analysing the behaviour of attractor networks under composition, we demonstrate that neural networks lack functional encapsulation – a property we call ‘structural flatness’. The core conclusion drawn from the analysis is this: the notion of localised, encapsulated function, essential to the more ‘cartographic’ approaches to the brain, is conceptually consistent only when applied to single networks as wholes, and not to their parts. Following from this, if the brain possesses localised, encapsulated cognitive functions, then neural networks are not the proper formalism to realise this on a grand scale; if the brain does not possess encapsulated functionality, then a critical evaluation must be made of the conceptual utility of functional localisation within computational neuroscience, and how it shapes our empirical and theoretical efforts.

Taillte May investigates how unstable black-hole quasinormal-mode spectra affect observed gravitational-wave signals. Small changes in a gravitational potential can substantially shift the frequencies normally used to test astrophysical black holes. A simple time-domain model reveals a delay before those shifts appear, tied to the echo travel time between the perturbation and the main potential. In the perturbative regime, the frequency change after one echo agrees with the conventional frequency-domain prediction. Beyond that regime, the intermediate waveform cannot be represented by a single constant frequency shift.

Develop data-driven theories of how neural circuit structure shapes computation in the Allen Institute Neural Dynamics accelerator. The scientist will combine large-scale connectivity atlases, simultaneous recordings from thousands of neurons across brain regions, and targeted perturbations during behavior. Work includes mechanistic models fitted to connectivity and activity, coarse-grained descriptions of collective dynamics, analysis of which models remain compatible with data, and design of experiments that distinguish between them. The role involves close collaboration with anatomical, electrophysiological and imaging teams, and communicating research through publications and conferences. Based in Seattle, Washington, with most hours on site. The stated annual salary range is $87,450–$108,250. Relocation assistance is offered; work visa sponsorship may be available. Remote work, when permitted, must take place in Washington State.

How do humans stabilize an inverted pendulum, and why does a balanced pole eventually fall? Drawing on 25 years of fingertip pole-balancing research, this talk examines neural correction delays: longer poles move more slowly relative to the nervous system’s response time. Delay-differential models can stabilize the upright position, yet skilled people still experience falls. The proposed explanation is microchaos arising from interactions among delay, sensory dead zones and frequency-dependent force encoding. A region of transient falling solutions lies next to stable microchaotic dynamics. Such microchaos is absent in virtual frontal-plane balancing tasks, while models of standing postural sway lack the corresponding transient regime. The comparison suggests that human falls, unlike pole falls, are more plausibly associated with medical events or slips and trips.

Navid Azizan presents methods that make neural models obey physical, safety and operational constraints at deployment. Hard-constrained neural networks, or HardNets, enforce input-dependent constraints by construction while preserving universal approximation within the feasible function class. Applications include models of chaotic dynamics with bounded trajectories, energy-constrained operator learning, safe reinforcement learning and control with formal guarantees. The talk then turns to enforcing constraints during sampling from pretrained diffusion and flow-matching models. Formulating generation as trajectory optimization allows receding-horizon control to guide outputs toward feasibility without retraining or imposing excessive restrictions on the sampling process. Examples from fluid dynamics, robot planning and control, PDE control and language-guided image editing illustrate how constraints can define admissible behavior while retaining expressive learning and generation.