Topology of Human and Synthetic Minds

Topology and Neuroscience

Topology is, in roughest terms, the mathematics of shape, connectivity, and what remains invariant under continuous deformations of objects, e.g. stretching, rotating, bending but not e.g. cutting. It also includes studying the presence of holes and loops in an object.

Why do we care about topology in neuroscience? Because the brain is not just a collection of parts, but a highly interconnected system, and many of its functionally relevant properties are about how nodes are connected, how cycles and feedback loops exist, how modules nest within one another.

At the molecular level, topology already plays a role (e.g. in protein folding, in the topology of RNA loops, in the topology of chromatin). But in the brain, we may ask: how do circuits “wire up” in a topologically efficient way? What patterns exists in the collective activity of neurons? How might certain patterns of connectivity lead to patterns in neural activity, support computation, memory, or robustness? And crucially, how does topology evolve when disease or development intervene?

Topological signatures sometimes change in diseased brains – loops may break, modules may fragment, connectivity shells may reorganize. Topological data analysis can explain a scaffold for cortical networks, and alterations in this scaffold can signal disease states or a breakdown of integrative processes.

So topology offers a lens: not just what connects to what, but how connectivity is qualitatively organized. And that matters for understanding resilience, plasticity, and failure modes in the brain.

Some neuroscientific questions that topology might help answer include:

• In neurodegenerative disease, can we detect “holes” or loss of cycles before overt functional degeneration occurs?
• How do learning and plasticity rewire connectivity in topological space (e.g. adding loops, merging modules)?
• Can we design artificial networks (or neuromorphic substrates) with desirable topologies mimicking human brains, but optimized for energy, fault tolerance, etc.?

Topology and Intelligence

In human intelligence, one can imagine that thought processes, memory states, associative structures, and perceptual networks all inhabit high-dimensional spaces with topological constraints. Topological data analysis can help us uncover “holes” in concept spaces (i.e. missing connections), or detect loops of recall, or measure connectivity across semantic modules. As a metaphor: ideas might live on a “manifold” and topology may help us track how they deform and connect over time.

In synthetic intelligence, especially in deep learning and neural networks, topology may help in multiple ways:

• During training, the weight landscape (loss surface) is high-dimensional and its topology may influence optimization behaviours (local minima, saddle points). Understanding that shape could help design better optimizers.
• In latent spaces (e.g. in autoencoders or generative models), topological invariants might reflect the learned manifold of inputs. We can probe whether a model has captured the correct homology.
• In robustness and adversarial examples, topology might reveal hidden vulnerabilities—small perturbations that break connectivity in latent space.
• In neuromorphic hardware, embedding connectivity in topologically efficient ways (e.g. wiring loops, modular connectivity with few “holes”) could yield power efficiency and fault tolerance. 

The Workshop: Reflections from King’s

When we and Federico (Turkheimer) first floated the idea of a two-day workshop on “Topology of Human and Synthetic Minds,” in September 2025, we were hopeful but also wary: would topics align, would people talk past each other? The actual event proved far richer. In the hum of conversation, one could sense the excitement of crossing disciplinary boundaries.

The format was simple: a series of short ‘crosswalk’ talks, each speaker presenting their preferred topological tool or neural/AI challenge. They sparked lively discussions ranging from the topology of latent spaces in neural population models, as well as in generative models, to the contrast between network and topological descriptions—what do we lose when we reduce neglect higher-order information?—to the challenge of detecting the emergence of new topological structures during learning and plasticity.

Throughout, three threads received particular attention:

1. Bridging scales: one group proposed tracking topology in hybrid models that map molecular / small-network processes upward to meso-scale circuits, thereby preserving topological signatures across scales.

1. From invariants to dynamics: another group asked: can we not just compute static Betti numbers or homology classes, but define topological flows—how topology itself evolves over time—and use those as dynamical signatures of cognition or adaptation?

1. Mathematically robust tools: how can one perform proper topological analyses of data without getting into statistical loop holes?

By day two, new sparks were flying. A neuroscientist and an AI researcher realized they might collaborate: one will feed topological summaries of EEG dynamics into a generative AI to “predict” forthcoming brain states; another pair began sketching a joint grant proposal on “topological fingerprinting of cognitive styles.” We left convinced that this workshop had catalysed genuine connections.