"More Is Different'' in Neural Circuits: Algebraic Emergence of Effective Theories in Canonical Recurrent Motifs of Biological Neuronal Networks

Nima Dehghani

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"More Is Different'' in Neural Circuits: Algebraic Emergence of Effective Theories in Canonical Recurrent Motifs of Biological Neuronal Networks

Nima Dehghani

arXiv:2608.30231v1Aug 31, 2026q-bio.NCcs.FLcs.NEnlin.CDphysics.bio-ph

TL;DR

The paper asks whether canonical neural motifs retain algebraic simplicity under composition and models their input-conditioned updates as finite transformation systems. It analyzes transition monoids and holonomy decompositions to distinguish generator-level structure from composition-generated structure. The central result is that aperiodic primitives can produce certified composite group structure, although the main cascade’s full decomposition is not machine-verified.

Problem
The paper examines whether canonical divisive-normalization and winner-take-all motifs generate qualitatively new algebraic structure under composition, beyond their individual update rules.
Method
The paper represents motifs and couplings as finite transformation systems, enumerates transition monoids, and analyzes their Krohn–Rhodes holonomy decompositions.
Results
Aperiodic primitive updates generate non-aperiodic monoids, including a genuinely composite WTA-to-DN cycle coupling normalization state with the winner’s inhibitory gate.
Takeaways & Limitations
Composing neural motifs can create computational structure belonging to neither motif separately, making interfaces part of the algebraic repertoire available for computation.
Takeaways & Limitations
For the WTA-to-DN cascade, the holonomy cascade was not constructed because its coordinate structure has 5.78 × 10^21 tuples and exceeds available memory.

Abstract

from arXiv · show

Canonical neural circuit motifs are usually described functionally: divisive normalization rescales population activity by a pooled signal, and winner-take-all competition selects one pattern through recurrent excitation and shared inhibition. We represent them, and their compositions, algebraically as finite transformation systems and analyze the transition monoids generated by their input-conditioned updates, distinguishing structure already present in a generator from structure that appears only through composition, and, on a joint state space, structure inherited from one factor from structure that lives on a joint configuration. Individually aperiodic updates can generate non-aperiodic monoids. In the WTA, every frozen-drive generator collapses to fixed points, yet short input sequences create local cycles of winner-dependent inhibitory gating: globally dissipative dynamics with a reversible action. The strongest result arises in WTA-to-DN composition. The composed monoid then contains a genuinely composite local cycle in which normalization state and the winner's gating state change together, although every primitive generator is aperiodic. Holonomy analysis certifies this as a group component of the Krohn-Rhodes cascade rather than an incidental cycle, and finds most group-carrying image sets on joint configurations, whereas the uncoupled product has none. An exhaustive interface sweep shows that the composite cycle is a property of the coupling rather than of a chosen map. If motifs are building blocks of neural computation, composing them is a form of programming: one chooses primitives and interfaces so that the generated algebra has the intended repertoire. The transition monoid is that repertoire - what a primitive presents to any later construction. Recurrent circuits are compositional transformation systems; their algebra constrains what they can be programmed to compute.

I. INTRODUCTION

The paper asks whether canonical neural motifs remain algebraically simple under composition and formalizes their computational repertoires as transition monoids. It finds that composition can generate certified local group structure absent from individual aperiodic updates.

Motivation: Biological circuits constrain permissible state transitions, motivating an algebraic account of the computations generated when motif transitions are composed.The paper treats recurrent circuits as transformation systems whose algebra contributes to what they compute.
Framework: Transition monoids contain every transformation obtainable from finite sequences of input-conditioned updates, while Krohn–Rhodes holonomy identifies their group components.Aperiodic components represent irreversible transformations; group components certify locally reversible structure.
Winner-take-all: Each frozen-drive WTA generator collapses every orbit to a fixed point, but switching among such updates produces non-aperiodic monoid elements.The WTA monoid has 326 elements, including 49 non-aperiodic transformations whose local reversibility occurs on invariant images.
Compositions: On the joint state space, the WTA-to-DN cascade yields a shortest witness that changes DN and WTA coordinates together, creating a genuinely composite cycle despite aperiodic primitive generators.This distinguishes coupling-generated structure from structure inherited by independent juxtaposition.
Divisive normalization: Three-state DN contains generator-explicit Z2 structure, so the four-state model is used to test whether composition alone creates non-aperiodic behavior.The three-state monoid has 13 elements and an intermediate-drive generator that is its own inverse.
Divisive normalization: In four-state DN, all input-conditioned maps are aperiodic, yet their 24-element monoid contains three non-aperiodic elements and a three-symbol transposition witness.Holonomy confirms one Z2 component acting on two singleton tiles.

B. Coupling gain control to selection creates a cycle neither motif contains

Coupling gain control to winner-take-all selection can create reversible structure that belongs to the joint circuit rather than either motif alone. In the WTA-to-DN direction, the shortest non-aperiodic witness changes both normalization and inhibitory-gating states, and holonomy certifies the pair as a group component.

WTA baseline: The WTA’s internal witness cycles arise from short two-symbol words that toggle the inhibitory gate after a winner has been selected.The word W00W01 produces the symmetric E2 cycle {2,3}, while the corresponding E1 cycle is {4,5}.
Uncoupled product: The uncoupled product inherits WTA or DN cycles without creating any group action that changes both coordinates.Among 34 group-carrying image sets, 27 move only competition and 7 only normalization; none moves both.
DN-to-WTA: DN-to-WTA expands the monoid to 3149 elements, but its shortest non-aperiodic witness remains competition-local with normalization fixed.Composite group structure exists deeper in the algebra: 15 of 46 group-carrying image sets are composite, including a Z4 component.
WTA-to-DN: WTA-to-DN yields 3084 elements and 361 non-aperiodic ones, with a shortest two-cycle changing normalization and the winner’s inhibitory gate together.All four composite generators remain irreversible and aperiodic, while the winner’s identity stays constant as its gating state and applied gain alternate.
Holonomy certification: Holonomy identifies the WTA-to-DN pair as a Z2 group component whose action exchanges two composite states rather than a coarser partition.One exchanging element changes both coordinates, and 35 of 57 group-carrying image sets are composite, compared with none in the uncoupled product.
Interpretation: The composite witness is an emergent property of coupling interfaces, while coupling direction controls whether that structure is the shortest and most accessible witness.The paper contrasts buried composite structure under DN-to-WTA with an immediately visible composite witness under WTA-to-DN.

C. Composite structure is robust across interfaces but shaped by coupling and schedule

Exhaustive interface and schedule analyses show that composite structure is generated by coupling, especially in the WTA-to-DN direction, while update schedules reshape or suppress its visibility. Holonomy decomposition confirms that these cycles constitute joint group structure rather than incidental local behavior.

Holonomy evidence: The uncoupled product has 0 composite windows of 34, compared with 15 of 46 for DN→WTA and 35 of 57 for WTA→DN.These holonomy windows are image sets whose permutation groups move both normalization and competition coordinates.
Interface robustness: 65,278 of 65,536 WTA-to-DN interfaces generate a composite cycle, including 65,278 of 65,532 coupled maps.All interfaces retain four aperiodic generators and a trivial group of units, yet none generates an aperiodic monoid.
Interface robustness: Composite cycles are absent precisely when coupled interfaces never deliver either extreme normalization drive, plus the two constant extreme maps.This exact characterization identifies 258 exceptions, including 254 coupled interfaces.
Coupling direction: Composite structure is common in both coupling directions, but the shortest witness is composite for 30 of 252 normalization-to-competition interfaces versus 224 of 252 containing any composite cycle.In the reverse direction, every injective interface contains a composite cycle, but none makes it the shortest witness.
Schedule dependence: Synchronous recurrent coupling writes a three-cycle into the generator, whereas strict sequential updating substantially suppresses composite cycles.At the biological interfaces, synchronous recurrence yields nine-element monoids with six non-aperiodic elements, including a three-cycle spanning both coordinates.
Schedule dependence: Asynchrony lowers the generic composite-cycle rate from 0.514 to 0.374 rather than eliminating it, while complete elimination at biological interfaces is operating-point specific.The biological asynchronous variants are aperiodic, but the broader sweep shows that asynchronous updating does not generally abolish composite structure.

B. The biological meaning of the WTA local cycle

The WTA’s reversible structure is a local cycle in winner-dependent inhibitory gating, not global alternation between competing populations. In WTA-to-DN composition, normalization and gating change together, producing a genuinely composite cycle certified by holonomy analysis.

WTA local cycle: The WTA local cycle holds winner identity fixed while toggling the inhibitory or gating state.The selected population alternates between winner-active and winner-gated states rather than exchanging winners.
WTA local cycle: The WTA monoid has local Z2-like structure despite a trivial global group of units.Holonomy identifies two Z2 components acting on image sets with singleton tiles, supporting a two-cycle on individual states.
Winner-dependent normalization: The WTA-to-DN witness changes both normalization and WTA coordinates: (D:0,W:4) ↔ (D:1,W:5).Biologically, low normalization with an ungated winner alternates with altered normalization and a gated winner.
Winner-dependent normalization: Winner-dependent gain control creates a local reversible component not confined to normalization or competition alone.Selection changes the gain-control context, yielding an effective computational degree of freedom on the joint state space.
Joint configuration structure: 35 of 57 group-carrying image sets are composite, compared with 12 competition-local and 10 normalization-local sets.The uncoupled product has 34 group-carrying image sets, none of them composite.
Comparative compositions: The independent product and DN-to-WTA cascade contain cycles whose shortest witnesses remain WTA-local, unlike the WTA-to-DN cascade.These controls show that non-aperiodicity alone does not establish genuinely composite structure.
Update schedule: Update schedule determines whether recurrent compositions preserve or generate local cycles.Asynchronous DN-first and WTA-first variants are aperiodic under default interfaces, while the synchronous recurrent system carries a Z3 component.

F. Relation to Krohn–Rhodes theory

The paper uses Krohn–Rhodes and holonomy analysis to distinguish primitive, composition-generated, inherited, and genuinely joint algebraic structure in neural-circuit transformations. Its finite models support local group-like structure while remaining controlled probes rather than biophysical replicas.

Local versus global structure: Local group components can occur on lower-rank images even when the global unit group is trivial.Thus globally dissipative circuits may contain locally reversible components after irreversible collapse onto an effective subspace.
Algebraic distinctions: Krohn–Rhodes analysis distinguishes generator-explicit structure from composition-generated structure and locates group action on image sets and tiles.The framework separates non-aperiodic elements, local permutation groups, and group divisors in the cascade decomposition.
Compositional interpretation: The monoid perspective treats recurrent motifs as transformation systems whose compositional algebra links biological operations with effective theories of computation.The relevant repertoire is generated by composing input-conditioned updates rather than examining only a single trajectory or rule.
Interpretive scope: The models preserve the qualitative logic of gain control, competition, inhibition, selection, and feedback without claiming to replicate cortical circuits.They are controlled finite-state probes for exact algebraic diagnosis.
Interfaces and programming: Coupling structure and interface choices determine whether algebra is inherited, composition-generated, generator-explicit, or aperiodic.The paper therefore treats motif composition as a programming problem in which primitives and interfaces constrain the available computational repertoire.
Interfaces and programming: The paper’s formal target is an exact transition-monoid account of what a motif or motif coupling presents to later composition.This account is positioned as a step from connectome-level structure toward a formal characterization of computational repertoire.

L. Conclusion

The conclusion recasts divisive normalization and winner-take-all competition as finite transformation generators whose compositions can create local reversible structure absent from individual updates. It presents this algebra as a design contract for programming recurrent neural systems.

Conclusion: Canonical motifs become generators of transformation monoids, and their monoids can contain local group structure absent from every primitive update.The holonomy decomposition confirms nontrivial Z2 components in the four-state normalization, WTA, and WTA-to-DN systems despite aperiodic generators.
Conclusion: The WTA-to-DN system contains a composite local cycle coupling normalization state to the winner’s inhibitory gate.This witness assigns computational structure to the composition rather than to either motif separately.
Programming recurrent systems: The finite transformation-system formalism provides a programming layer in which primitives and interfaces are chosen for the algebraic repertoire they generate.The algebraic contract complements specifying what the recurrent system should compute.
Programming recurrent systems: In recurrent constructions, the interface becomes part of the update rule, so the composite algebra cannot be read from any single component.Composition is therefore more than juxtaposition of independent circuit modules.
Conclusion: A transition monoid is the exact algebra of transformations obtainable from finite sequences of input-conditioned updates.It therefore describes the repertoire a recurrent motif presents to subsequent constructions.

B. Divisive-normalization finite-state systems

The paper constructs finite-state divisive-normalization systems to separate structure already present in one update from structure emerging through composition. A three-state baseline contains an explicit toggle, whereas a damped four-state system supplies an aperiodic baseline for testing emergence.

Model construction: Divisive normalization is modeled as recurrent state-dependent gain control in which drive is divided by semi-saturation plus pooled activity.The discrete systems represent macroscopic activation regimes of a continuously motivated normalization process.
Three-state baseline: The three-state baseline uses generators f0 = (0,0,0), f1 = (2,1,0), and f2 = (2,2,1).These maps define the finite input-conditioned transformations used in the baseline audit.
Three-state baseline: The intermediate-drive map f1 contains a period-two toggle between states 0 and 2, making its reversible structure generator-explicit.Because the cycle is already present in one primitive rule, this baseline fails the strict composition-generated criterion.
Four-state audit: The four-state system uses N = 4, α = 2, σ = 2, and β = 1 with nearest-integer quantization and clipping to {0,1,2,3}.Increasing baseline leak relative to the three-state model is intended to eliminate hard-coded oscillation.
Four-state audit: Every four-state generator is designed to be a purely dissipative sink that relaxes to a fixed point under constant drive.This makes the system an aperiodic baseline for testing whether reversible structure can emerge from irreversible components.
Robustness check: A parameter and quantizer sweep tests whether composition-generated cycles persist across normalization discretizations.The sweep varies σ, β, α, and floor, nearest-integer, or ceiling quantization while enumerating the resulting monoids.

C. Winner-take-all finite-state system

The WTA model is a binary two-population recurrent circuit with shared inhibitory gating, represented as a finite transformation system over eight configurations. Each frozen-drive generator collapses to a fixed point, enabling tests of whether changing inputs create non-aperiodic structure.

Model construction: WTA is modeled with two excitatory populations and a shared inhibitory pool, with all three variables binarized.The binary projection isolates the structural invariant of strong recurrent competition: suppressed or saturated population states.
State representation: The eight configurations are encoded by q = 4x1 + 2x2 + y, preserving the binary ordering of (x1,x2,y).This encoding lets each input-conditioned update be written as an 8-tuple.
Update rule: The WTA update uses Heaviside thresholding to map each configuration deterministically to the next state under a fixed input.The threshold dynamics isolate deterministic mappings between attractor basins.
Frozen-drive dynamics: Under drive 10, fW_10 has the unique fixed point 5 = (1,0,1), where x1 wins and inhibition is engaged.Configurations initially favoring x2 are eventually redirected to this winner state.
Frozen-drive dynamics: All four fixed-drive WTA generators are aperiodic, with no cycle longer than one.Repeated application therefore produces dissipative collapse under each frozen input.
Monoid closure: The generated WTA monoid contains every transformation produced by finite sequences of the input-conditioned generators.This closure tests whether switching among fixed-point rules can generate non-aperiodic structure.

D. DN–WTA composite state space

The composite DN–WTA system is represented on a Cartesian product state space, encoding normalization context and competitive identity as joint configurations. An inverse decoding separates these coordinates to distinguish inherited WTA cycles from genuinely composite cycles.

The composite system uses a Cartesian product state space combining normalization and WTA states.This represents every joint combination of normalization context and competitive identity.
Product states are encoded as idx(d,w) = 8d + w, allowing each transformation to be represented as a length-32 tuple.Finite closure can therefore analyze the joint architecture with the same exact procedure used for isolated motifs.
Inverse decoding tests whether a cycle changes only w or changes d and w together, with simultaneous coordinate changes certifying a genuinely composite cycle.A cycle confined to w is inherited from the WTA sub-manifold, whereas joint coordinate changes traverse the coupled phase space.
A single unified external drive aligns inputs to both modules, so newly generated cycles reflect internal recurrent interactions rather than mismatched drives.The DN and WTA input decodings map the shared external alphabet into their respective state updates.

E. DN–WTA composition schemes

Four composition schemes test how uncoupled, feedforward, and feedback interfaces alter the transition algebra of DN–WTA systems. The WTA-to-DN interface explicitly couples inhibitory gating to normalization and is designed to produce structural entanglement.

Four biologically interpretable composition schemes were examined, including an independent product and directional DN-to-WTA and WTA-to-DN cascades.The independent product serves as a null model in which both modules receive the same input without synaptic cross-talk.
In the DN-to-WTA cascade, normalization updates first and determines the subsequent WTA drive through a state-dependent interface.This models normalized upstream representations preparing a field for downstream selection.
The default DN-to-WTA interface maps DN states 0, 1, 2, and 3 to WTA drive conditions 00, 10, 01, and 11, respectively.The competitive drive is therefore a strict function of the normalized state.
In the WTA-to-DN cascade, WTA updates first and its state determines the subsequent DN input through a state-dependent feedback interface.This models selected populations actively recruiting inhibition to reshape the local gain field.
The WTA-to-DN interface maps gated or coactive states to strong pooled normalization and single ungated winners to winner-specific normalization conditions.Because the inhibitory gate is linked to DN state, the interface structurally entangles selection and gain control and generates the paper’s composite Z2 cycle.

4. Recurrently coupled DN–WTA system

The recurrently coupled system lets each motif determine the other’s effective input, with synchronous and asynchronous schedules treated as distinct transformation systems. Exact finite closure then tests fixed points, reversibility, and cycles generated by input sequences.

In the recurrent system, current WTA and DN states determine the next normalization condition and competitive drive, respectively.The synchronous update creates a recurrent normalization-selection loop.
Because recurrent interfaces override the external symbol, all four input-indexed maps coincide and the recurrent monoid is generated by one transformation.The same holds for the asynchronous variants.
Different update schedules were examined because operation order can produce different transition monoids in discrete automata.The schedule controls the topology of the generated phase space and helps separate schedule effects from circuit structure.
Finite closure computes each transition monoid exactly by exhaustively composing generators rather than simulating selected trajectories.The resulting monoid contains every state transformation obtainable from finite input sequences.
Idempotents identify absorbing fixed-point behavior, while global units identify transformations reversible across the entire state space.A trivial unit group indicates globally dissipative and irreversible dynamics.
Non-aperiodic monoid elements indicate local cycles that can emerge from sequential composition even when every primitive generator is dissipative.This is the primary test for conservative cyclic structure arising from irreversible rules.

G. Witness-word search and cycle classification

Witness-word search finds short input sequences producing nontrivial cycles, then decodes those cycles in Cartesian coordinates to classify them as DN-local, WTA-local, or genuinely composite. Holonomy decomposition validates the algebraic structure, while the WTA-to-DN cascade has a major computational boundary.

Breadth-first search finds the shortest generator word whose transformation contains a nontrivial functional-graph cycle.The witness word gives the deterministic stimulus sequence that pushes the system beyond fixed-point behavior.
For DN–WTA systems, witness cycles are decoded into composite coordinates to track normalization context and competitive identity simultaneously.The decoding projects an abstract loop back onto the physical Cartesian phase space.
Cycles are classified as DN-local, WTA-local, or genuinely composite according to which coordinates change along the cycle.A genuinely composite cycle changes both coordinates rather than remaining within one sub-network.
The reported decomposition uses Monoid(G), retaining the identity, and distinguishes representative group components while accounting for conjugate subduction classes.This preserves zero-time evolution and avoids omitting identity-dependent monoid structure.
Holonomy decomposition distinguishes image sets, tiles, permutator groups, and holonomy groups when analyzing group components.Holonomy acts on coarse-grained tiles, whereas the permutator group acts on exact states.
The decomposition was checked by elementwise round trips, with the DN-to-WTA cascade verified for 125 distinct elements without mismatch.Monoid computations also agreed between GAP and the independent Python audit for all nine systems.
The WTA-to-DN cascade was not fully constructed because its cascade state space contains 5.78 × 10^21 tuples and a single cascade element requires 14 GB.Its skeleton, depth, components, and group components are exact, but the cascade itself is not machine-verified.

I. Interface and schedule sweeps

The study exhaustively varies DN–WTA interfaces, update schedules, and controls to distinguish structure inherited from isolated motifs from structure generated by meaningful composition. These sweeps establish whether observed cycles depend on coupling direction, exact maps, discretization, or temporal symmetry.

Interface enumeration: 65,536 winner-to-normalization maps and 256 normalization-to-competition maps were enumerated, covering the specified interface spaces exhaustively.Joint surjectivity ensures that every winner-to-normalization map corresponds to a distinct generator tuple.
Exact monoid audit: Breadth-first search on each exact Cayley graph identified shortest non-aperiodic witnesses without a word-length cap.The audit recorded monoid size, idempotents, non-aperiodic elements, group-of-units order, and witness classification.
Biological stratification: Interfaces were stratified by whether they encode no WTA information, gating, winner identity, or both, linking the combinatorial sweep to biological coupling hypotheses.Winner-only maps match the biologically motivated winner-pooling interface.
Controls: The independent DN–WTA product serves as a no-cross-talk null model, separating inherited non-aperiodicity from structure created by interaction.Any non-aperiodic structure in the product is attributable to the component systems rather than normalization–selection coupling.
Controls: The controls distinguish hard-coded generator cycles, inherited motif structure, and genuine emergence from normalization–selection composition.The study also varies discretization, interface maps, and synchronous versus asynchronous schedules to test robustness and temporal-symmetry effects.

Appendix A: A Physical Primer on Algebraic Decomposition for Dynamical Systems

The appendix translates finite-state algebra into dynamical-systems language and explains how Krohn–Rhodes decomposition certifies reversible structure within dissipative neural dynamics. It also documents the computational boundary reached by the WTA-to-DN cascade.

Algebraic interpretation: Krohn–Rhodes decomposition factors finite-state dynamics into dissipative reset-like components and reversible group components.Group components certify structurally necessary local reversibility rather than merely observed periodic trajectories.
Algebraic interpretation: A monoid is a composition-closed transformation set containing the identity, which represents zero-time evolution in a continuously existing biological network.The identity allows stable fixed points to be represented while time passes.
Algebraic interpretation: Image sets represent restricted attractor regions, permutator groups exact-state shuffling, and holonomy groups the shuffling of coarse-grained phase-space tiles.Matching permutator and holonomy groups indicates precise cycles, whereas a mismatch permits macroscopic oscillation with microscopic variation.
Verification: The decomposition is checked by reconstructing each physical transition from its holonomy coordinates and verifying exact equality with the original transition.This equivalence is presented as evidence that the cascade preserves the network’s dynamical information.
Computational boundary: The WTA-to-DN cascade has 5.78×10^21 coordinate tuples, exhausting available machine memory and preventing explicit cascade reconstruction.The appendix attributes this expansion to representing recurrent feedback within a mandated one-way hierarchy.
Baseline and normalization: The three-state baseline already contains an explicit reversible period-two generator, so it cannot serve as a clean test of emergent composite cycles.Its intermediate-drive map satisfies f1^2 = id and generates a monoid with a Z2 group of units.
Baseline and normalization: The four-state DN generators are individually aperiodic, yet their monoid contains a local two-cycle reached after fluctuating drive inputs.The generated monoid has 24 elements, eight idempotents, trivial group of units, and three non-aperiodic elements.

2. Winner-take-all

The WTA generators are individually aperiodic but can produce local reversible action after input sequences collapse the system into a small attractor image. This cyclic structure is inhibitory-gating dynamics rather than winner switching.

State representation: WTA states encode two competing excitatory populations and a shared inhibitory interneuron as q = 4x1 + 2x2 + y.The encoding maps the circuit’s physical binary variables to eight discrete states.
WTA dynamics: Every frozen-drive WTA generator is aperiodic, but the two-symbol witness h = fW10 ◦ fW00 produces a Z2 action on the image {4,5}.The square h^2 is idempotent, while h exchanges states 4 and 5 on its image.
WTA dynamics: The WTA cycle preserves winner identity while the inhibitory variable y alternates between silent and active states.The active competitor remains x1 = 1 throughout, so the local rhythm is generated by inhibitory gating rather than competition between winners.
Composite state space: Composite states use idx(d,w) = 8d + w, allowing normalization and WTA configurations to be represented on one joint state space.This encoding supports direct identification of cycles that change both motif coordinates.
Composite dynamics: The key composite cycle changes normalization and inhibition together: low normalization with silent inhibition alternates with elevated normalization and active inhibition.Excitation remains active while the normalization context and inhibitory gate change simultaneously, producing a composite rhythmic pattern.
Rank distributions: Non-aperiodic elements concentrate at low ranks, where irreversible collapse funnels the system into narrow attractor basins before local cycling appears.The rank distribution reports 1,548 elements with 159 non-aperiodic elements at rank 2 and 1,427 with 179 at rank 3.
Rank distributions: Synchronous cascades concentrate cycles at ranks 2 and 3, whereas asynchronous recurrent monoids have eight elements and no non-aperiodic elements.The caption attributes the loss of cycles under asynchronous updating to the removal of instantaneous temporal symmetry.