distance dependent synaptic connectivity
1946 – Wiener & Rosenblueth’s excitable‐medium “ring” model.
They were studying cardiac tissue, not cortex, but it’s the earliest mathematical network that lets coupling strength fall off with physical distance around a loop. The paper treats cells as identical excitable units and makes propagation depend on neighbor spacing.
1956 – Beurle’s “grass-fire” cortical net.
Beurle put neuron-like elements on a 2-D grid, let synaptic influence decay with lattice distance, and watched travelling waves and reverberating patterns.
1972-73 – Wilson & Cowan’s neural-field equations.
Their second paper adds an explicit kernel w(|x-x’|) so the input to any point is a weighted integral over nearby points. Modern neural-field theory starts here.
1975-77 – Amari’s lateral-inhibition fields.
Amari generalised the kernel idea, proved stability results, and popularised distance-dependent excitation vs inhibition for pattern formation.
Since then, countless variants (Ermentrout, Bressloff, Kilpatrick, etc.) have used distance-tuned kernels for waves, bumps, grid-cell rings, you name it.
| Model family | Typical connectivity assumption | Reality check |
|---|---|---|
| Point-neuron “vanilla” RNNs (e.g. Sompolinsky chaotic nets, most machine-learning RNNs) | IID random weights, no spatial term | Ignores distance completely because math is easier and GPUs don’t care about microns. |
| Cognitive-task recurrent rate/spiking models | Often still random or low-rank + random; some add functional structure (low-rank modes) but not real geometry | Distance left out unless the task itself is spatial. |
| Large-scale cortical simulations (Blue Brain, Potjans-Diesmann, NEST “hex-grid” demos) | Frequently use an exponential or Gaussian fall-off for local axons; long-range patchy projections modeled separately | Distance kept, because they’re trying to match anatomy. |
| Neural-field / bump / wave models | Distance-dependent kernel is the whole point | Always included; kernel shape is a free parameter. |
A 2022 review of connectivity modelling notes that most population or abstract network studies still default to uniform random connectivity unless spatial layout is central to the question. Even in modern papers that do add geometry, inhibition is usually made broader than excitation, but exact decay constants are rarely tuned from data.