"More Is Different'' in Neural Circuits: Algebraic Emergence of Effective Theories in Canonical Recurrent Motifs of Biological Neuronal Networks
Nima Dehghani
q-bio.NC, cs.FL, cs.NE, nlin.CD, physics.bio-ph
2026-08-31
Modeled as finite maps, every frozen DN or WTA generator is aperiodic, yet WTA-to-DN composition yields a two-cycle that moves both coordinates.
Two canonical cortical motifs are usually sold as functions. Divisive normalization (DN) rescales a population by a pooled signal. Winner-take-all (WTA) selects one pattern by recurrent excitation plus shared inhibition. One changes scale, the other changes identity. Each story is clean in isolation.
Biology composes them. A normalized field is the arena for selection, and the selected winner can rewrite the gain that follows. Trajectory plots show one response. They do not list which state maps become legal under arbitrary finite input sequences. This paper writes the motifs as finite transformation systems and asks Anderson's "More is Different" at circuit scale: can updates that are dissipative one drive at a time generate locally reversible structure once they are composed.
Each motif is a triple (X, Σ, δ): finite states, a finite input alphabet, and a deterministic update. Input symbol a induces a map fa. The transition monoid M is the closure of those maps under composition: every transformation a finite word can produce. A map is aperiodic when its functional graph has no cycle longer than one, the algebraic name for a frozen-drive sink.
Three distinctions do the work. Generator-explicit: a primitive map already contains a cycle. Composition-generated: every generator is aperiodic, yet some monoid element is not. Genuinely composite: the cycle moves the DN and WTA coordinates together, so it cannot be charged to either factor. Holonomy, the constructive form of a Krohn–Rhodes cascade, then grades image sets and attaches permutation groups acting on tiles. The software is SgpDec.
The negative control is three-state DN. The intermediate-drive map f1=(2,1,0) is an involution; the 13-element monoid has units isomorphic to Z2. Four-state DN raises the semi-saturation constant to σ=2 so all four generators are aperiodic. WTA is two excitatory populations plus an inhibitory gate, eight binary configurations, parameters G=1, S=L=B=1/2, W=1, R=0. All four frozen-drive generators are sinks. The two motifs then share a 32-state product, compared as an independent product, DN→WTA cascade, WTA→DN cascade, and synchronous versus asynchronous recurrence.
Four-state DN generates 24 monoid elements, 3 of them non-aperiodic. WTA generates 326 elements, 55 idempotents, 49 non-aperiodic, and a trivial group of units. The shortest witness is a two-symbol word: balanced drive then drive to population 1. It funnels every state into {4,5} and transposes that image. Decoded, the winner identity is fixed; only the inhibitory gate toggles. The symmetric word yields {2,3} for population 2.
| System | States | M | Non-ap. | Shortest witness | ||
| DN, four-state | 4 | 24 | 3 | DN-local | ||
| WTA | 8 | 326 | 49 | WTA-local | ||
| Independent product | 32 | 576 | 88 | WTA-local | ||
| DN→WTA | 32 | 3149 | 401 | WTA-local | ||
| WTA→DN | 32 | 3084 | 361 | genuinely composite | ||
| Synchronous recurrent | 32 | 9 | 6 | already in the generator | ||
| Asynchronous recurrent | 32 | 8 | 0 | none |
Of 34 group-carrying image sets in the uncoupled product, 27 move only WTA, 7 only DN, and none both. DN→WTA inflates the algebra to 3149. The shortest witness stays competition-local, yet 15 of 46 windows are composite, and this is the only cascade with a Z4, cycling four joint configurations. Reverse the interface and the shortest witness itself becomes composite: normalization hops between its two lowest states while the winner's gate opens and closes. The pair C={(D:0,W:4),(D:1,W:5)} carries Z2 on two singleton tiles. Composite windows dominate, 35 of 57.
An exhaustive interface sweep pins the effect to coupling. All 4^8=65536 winner-to-normalization maps have aperiodic generators and trivial units, and none of the generated monoids is aperiodic. A composite cycle appears for 65278 maps, 0.9961 of the coupled ones. The 258 exceptions have an exact characterization: the image sits in the two intermediate drive symbols, plus the two constant maps to the extremes. Direction controls legibility. Sweeping the 256 normalization-to-competition maps, 224 of 252 coupled maps contain a composite cycle, but only 30 of 252 make it the shortest witness.
Synchronous recurrence collapses the algebra to 9 elements, 6 non-aperiodic, holonomy Z3. State-dependent interfaces overwrite the external symbol, so the four generators coincide and the cycle is already in that one map. Across all 50331648 interface-schedule pairs, composition never produces non-aperiodicity from an aperiodic recurrent update. At the biological interfaces both asynchronous schemes generate 8-element aperiodic monoids.
If motifs are building blocks, composing them is programming: choose primitives and interfaces so the generated algebra lands on the intended leaf. The transition monoid is the contract a brick presents to later construction. Side-by-side placement inherits; winner-conditioned gain makes the composite cycle the shortest witness. For people compiling onto recurrent substrates, identical wiring with a different interface can separate an aperiodic monoid from one that carries a composite group component.
This is a diagnostic, not a training method. It turns "composition creates new computation" into a checkable algebraic claim rather than another simulated trace.
The models are small finite-state abstractions built so the monoid can be enumerated, not biophysical replicas. Detected non-aperiodicity is sensitive to discretization, interface, and schedule. The authors scan N in {3,4,5,6}, but continuous, stochastic, high-dimensional cortex sits outside the audit.
The cascade of the headline WTA→DN system was not constructed. Its coordinate structure admits 5.78×10^21 tuples, and a single cascade element exhausts 14 GB. Skeleton depth 22, 92 components, and five Z2 groups are exact; existence of the cascade follows from the holonomy theorem, not a machine-checked round-trip. Seven smaller systems did get elementwise verification.
Local group structure is not global reversibility. Units of the WTA and composite monoids are trivial. Reversible action lives on low-rank images after irreversible collapse. The paper also does not show that these finite-state cycles survive as limit cycles or slow manifolds in a continuous rate model.