Brains as graphs
In graph theory, a graph G = (V, E) is a set of vertices V and a set of edges E connecting pairs of them. In a connectome, vertices are neurons (or, at larger scales, brain regions) and edges are synapses (or bundles of fibres). Edges can carry a direction, since signals flow from one neuron to the next, and a weight, such as how many synapses link two cells.
G = (V, E), E ⊆ V × V, w: E → ℝ (connection strength)
Milestones in mapping real nervous systems
- C. elegans (1986). John White and colleagues mapped the nervous system of a tiny roundworm: 302 neurons. It was the first complete connectome, reconstructed by hand from electron microscope images.
- The adult fruit fly (2024). The FlyWire consortium published the first complete wiring diagram of an adult brain in Nature: about 139,000 neurons and roughly 54.5 million synapses, proofread by researchers and citizen scientists.
- Mouse visual cortex (2025). The MICrONS project mapped a cubic millimetre of mouse cortex, containing around 200,000 cells including roughly 120,000 neurons, with more than 523 million synapses. Crucially, it combined this wiring with recordings of what tens of thousands of those neurons did while the mouse watched videos.
- The human brain. The human brain has around 86 billion neurons and an estimated hundred trillion or so synapses. A full synapse-level human connectome is far beyond current technology, so human studies map connections between regions using MRI.
What graph theory reveals
- Degree: how many connections a vertex has. Some neurons and regions are far more connected than others.
- Small-world structure: brains combine tight local clustering with short paths between distant areas, a pattern Watts and Strogatz described in 1998. It allows both specialised processing and fast integration.
- Hubs and the rich club: highly connected hub regions connect densely to each other, forming a "rich club" that acts as the brain's communication backbone.
- Modules: groups of vertices that connect more within the group than outside it, matching functional systems like vision or movement.
Structure is not the whole story
A wiring diagram tells you which neurons can talk to each other, not what they say. The same graph can produce very different activity depending on connection strengths, neuromodulators and the body's state. That is why datasets pairing structure with activity, like MICrONS, are so valuable: they let us test whether a theory explains both the graph and what flows over it.
Our name comes from this notation. We model minds as graphs whose activity evolves over time, xt+1 = f(W xt, st, ht), where W is the weighted connectome. Our research asks which structural and dynamical properties of such graphs, like integration and recurrence, matter for consciousness.
Quick answers
What is a connectome?
A connectome is a complete map of the connections in a nervous system, usually represented as a graph where neurons or regions are vertices and synapses or fibre tracts are edges.
Has the human brain been fully mapped?
No. Complete synapse-level connectomes exist for C. elegans and the adult fruit fly, and for a cubic millimetre of mouse cortex. The human brain, with around 86 billion neurons, is mapped only at the level of regions.
Further reading
- White, J. G. et al. (1986). The structure of the nervous system of the nematode Caenorhabditis elegans. Phil. Trans. R. Soc. B, 314.
- Dorkenwald, S. et al. (2024). Neuronal wiring diagram of an adult brain. Nature, 634.
- The MICrONS Consortium (2025). Functional connectomics spanning multiple areas of mouse visual cortex. Nature.
- Watts, D. & Strogatz, S. (1998). Collective dynamics of 'small-world' networks. Nature, 393.
- van den Heuvel, M. & Sporns, O. (2011). Rich-club organization of the human connectome. Journal of Neuroscience, 31(44).
- Sporns, O., Tononi, G. & Kötter, R. (2005). The human connectome: a structural description of the human brain. PLOS Computational Biology.