Every route counts, but some count more: a max-plus view of the C. elegans brain

1 Introduction

• Electrical, synaptic, monoaminergic, neuropeptidergic, and hormonal signaling differ in onset and persistence, motivating a functional connectome whose layer contributions vary over time.The functional connectome is described as effective directed interactions under particular conditions; the signal-propagation atlas captures responses mediated by engaged routes.Source: §1 · pp. 2–3
• Persistent behavioral states and evolving population activity motivate examining whether recurrent connectome organization helps support or reconfigure internal states.The introduction presents this as a hypothesis and a way to identify potentially relevant circuits and communication mechanisms.Source: §1 · p. 3
• Existing connectome analyses often omit direction, weights, or separate communication mechanisms, while additive spectral methods combine walks rather than identifying the specific dominant feedback routes.The introduction frames these routes as needed to understand how recurrent organization changes.Source: §1 · p. 3
• Max-plus algebra accumulates weights along connections and selects the strongest alternative route, making dominant paths and recurrent loops explicit in a lattice-ordered state space.Unlike Euclidean representations based on superposition and distance, the proposed order-based description compares activity patterns by inclusion.Source: §1 · p. 4
• A prior threshold-based max-plus study simulated activity on one wiring diagram, whereas this paper analyzes each layer's connectivity spectrum; its underlying theory is developed in a companion paper.The paper presents the method at the level needed to interpret its biological application.Source: §1 · pp. 4–5
• The proposed model composes changing communication layers with max-plus algebra, while five available networks are analyzed separately because their measurements cannot yet be combined into one operator.The analyses identify recurrent domains, dominant feedback circuits, and connected upstream and downstream neuronal populations.Source: §1 · p. 5

2.1 The model

• Max-plus dynamics select the strongest accumulated incoming route, while changing layer engagement can merge recurrent groups and shift the dominant cycles and their neuronal domains.Electrical coupling is treated as a stable scaffold; faster synaptic and more sustained neuromodulatory signals can reorganize critical cycles on different timescales.Source: §2.1 · pp. 6–7

2.2 Empirical analysis

• Because the five network descriptions use different biological quantities, conditions, and neuronal coverage, the study analyzes them independently rather than combining their weights.The empirical networks cover gap junctions, chemical synapses, monoaminergic signaling, neuropeptidergic signaling, and experimentally measured functional propagation; hormonal signaling remains theoretical because no neuron-to-neuron map is available.Source: §2.2 · pp. 7–8

2.2.1 Gap-junction connectome

• The gap-junction network places 296 of 302 neurons in one component, whose dominant critical class is the ALA–CANL/CANR pairings with eigenvalue λ = 401.A second mode, λ = 1, belongs to the isolated reciprocal IL2DL–URADL component; the anatomical edge weights do not measure physiological coupling strength.Source: §2.2.1 · pp. 8–10

2.2.2 Synaptic connectome

• The synaptic network has 11 SCCs, including a 275-neuron somatic core and an 18-neuron pharyngeal core, linked by the single M1→RIPL chemical bridge.The pharyngeal core also projects to terminal MC motor neurons; the two posterior touch neurons provide independent inputs to the somatic core. This topology recovers the bow-tie organization and somatic–pharyngeal division.Source: §2.2.2 · pp. 10–11
• The synaptic spectrum reduces the somatic and pharyngeal recurrent regimes to the RIAR–SMDVL feedback loop at λ = 37 and the M4 self-loop at λ = 15.The corresponding eigendirections extend these critical classes into upstream and downstream domains, including the hierarchy between the pharyngeal and stronger somatic regime.Source: §2.2.2 · pp. 12–14

2.2.3 Monoaminergic connectome

• The monoaminergic network is highly divergent: its 17-neuron recurrent SCC268 projects to 267 downstream SCCs and contains nearly all canonical monoamine-releasing neurons.Two dopaminergic neurons, ADEL and ADER, are the only upstream singleton SCCs feeding the core; the left and right eigendirections contain 19 and 284 finite neurons, respectively [4].Source: §2.2.3 · pp. 14–15
• The 17-neuron recurrent monoaminergic core contains a maximum-mean NSM–RIM cycle (λ = 2.5), isolating an experimentally characterized regulator of feeding and behavioral state.NSM-to-RIM edges have weight 2 and RIM-to-NSM edges weight 3; the critical class comprises NSML, NSMR, RIML, and RIMR. The broader organization distinguishes restricted upstream input, a recurrent core, and widespread downstream broadcasting.Source: §2.2.3 · pp. 15–16

2.2.4 Neuropeptidergic connectome

• The neuropeptidergic network has a 286-neuron recurrent component, but its maximum-mean recurrence is the PQR autocrine self-loop, supported by 14 ligand–receptor relations.Twelve singleton SCCs upstream include IL1, CEP, I6, and M4 neurons; four downstream singletons are RIA and RME. The 298-neuron left and 290-neuron right eigendirections differ only at this directional periphery; PQR’s self-interaction is notably diverse.Source: §2.2.4 · pp. 16–17

2.2.5 Functional signal-propagation network

• In the 188-neuron functional network, the maximum-mean cycle is the reciprocal RIH–SIBVR response (λ = 0.921569), embedded in a 147-neuron recurrent component.RIH activation produces a SIBVR response of 0.973749, and SIBVR stimulation produces a RIH response of 0.869389; λ is their cycle’s mean post-stimulus calcium-response amplitude. RIH’s serotonin regulation and SIBVR’s motor role link modulatory and motor domains, but the atlas does not establish the pathway.Source: §2.2.5 · pp. 18–19

2.3.1 Linear spectral analysis

• Linear spectral analysis matches coarse network organization but misses secondary recurrent circuits and gives less stable, biologically interpretable descriptions than max-plus analysis.For example, M4’s λ = 15 self-loop has no leading linear-mode counterpart, PQR ranks 24th in in-centrality despite its dominant autocrine loop, and RIH–SIBVR appears only in the fourth linear mode. In four directed layers, eigenvector-basis condition numbers exceed 10^18, while more than 100 eigenvalues exceed 10% of the spectral radius in the chemical-synapse and functional networks.Source: §2.3.1 · p. 20

2.3.2 Biological meaning of the spectral quantities

• Max-plus quantities correspond to identifiable routes and remain in the data’s units, whereas linear spectral radii aggregate walks and need not be attained by any circuit.Examples include 37 as the mean of the 43 and 31 reconstructed contacts in the RIAR–SMDVL loop, 14 supported relations in the PQR loop, and 0.92 mean calcium response in the RIH–SIBVR cycle. Max-plus describes domains through order and inclusion, supporting an ordered, lattice-structured rather than Euclidean representation of network-supported states.Source: §2.3.2 · pp. 20–21

2.3.3 Community detection and rich clubs

• Compared with community and rich-club analyses, max-plus identifies which circuits recur, how they are directionally embedded, and which connections dominate within them.It places ADEL and ADER upstream of the monoaminergic core, adds RIC and I5 to that core, and identifies the NSM–RIM recurrence; comparisons to the baseline are qualitative because it used earlier network data. Conventional methods still describe modules, hubs, and global spectral properties, while max-plus distinguishes coexisting circuits and their upstream and downstream domains.Source: §2.3.3 · p. 22

Time-dependent model

• The time-dependent model combines electrical, synaptic, monoaminergic and neuropeptidergic signaling layers, whose changing relative contributions can alter dominant routes and feedback cycles without changing structural connectivity.The full mathematical development and lattice hierarchy are given in a companion paper.Source: §3 · p. 23

Spectral method

• Generalized Rmax,+ spectral analysis links critical cycles to upstream and downstream neuronal domains while retaining the neurons and connections involved in recurrent interactions.The framework combines strongly connected components, critical graphs and fundamental left and right eigendirections, including for networks with distinct recurrent components connected asymmetrically.Source: §3 · p. 23

Empirical contribution

• Across different network architectures, max-plus analysis recovers established C. elegans organization, including the RIAR–SMDVL cycle, monoaminergic broadcasting, NSM–RIM interactions and the PQR self-loop.The synaptic decomposition also reflects the separation between somatic and pharyngeal systems.Source: §3 · p. 24
• The ALA–CANL/R electrical interactions, corresponding ALA–CAN synaptic connections and functional-network RIH–SIBVR cycle are less-explored associations proposed as targets for physiological investigation.Suggested studies include targeted perturbations and brain-wide recordings.Source: §3 · p. 24

Limitations

• Anatomical and molecular layer weights are proxies for physiological strength and omit excitation-versus-inhibition distinctions; testing predicted circuit reorganization requires a common scale and whole-brain recordings across behavioral states.The neuropeptidergic analysis uses a conservative short-range model, and no neuron-to-neuron hormonal map is available.Source: §3 · p. 24

Interpretation

• The framework reads a connectome as dominant feedback circuits embedded among neurons that can drive or be reached from them, and proposes an order-based rather than Euclidean description of internal states.The paper argues that this approach preserves the biological meaning of measurements and identified neurons and connections.Source: §3 · pp. 24–25

4.1 Connectivity datasets

• Five separate adult-hermaphrodite network descriptions cover electrical, chemical synaptic, monoaminergic, neuropeptidergic and experimentally measured functional interactions.They differ in directionality, density, spatial range, experimental origin and edge-weight interpretation.Source: §4.1 · p. 25

4.1.1 Electrical gap-junction connectome

• The electrical connectome retains 302 neurons, excludes non-neuronal cells and neuromuscular contacts, and represents gap-junction coupling as symmetric connections.Weights come from the corrected adjacency matrix and reflect serial electron-microscopy sections over which contacts are observed.Source: §4.1.1 · p. 25

4.1.2 Chemical synaptic connectome

• The chemical synaptic network is a directed, weighted graph over 302 neurons, with edges oriented from presynaptic to postsynaptic cells.Preprocessing accounts for the source-row, target-column orientation of the corrected Cook et al. matrices [3, 55].Source: §4.1.2 · p. 26

4.1.3 Monoaminergic connectome

• The monoaminergic network represents potential extrasynaptic routes from monoamine-producing neurons to compatible receptor-expressing targets.Its inferred weights count supported monoamine–receptor relationships and combine transmitter identities, receptor expression and connectome-aligned neuronal identities [4].Source: §4.1.3 · p. 26

4.1.4 Neuropeptidergic connectome

• The neuropeptidergic network infers directed ligand–receptor-compatible routes from peptide and receptor expression and experimentally characterized compatibility.The analysis selects the spatially conservative short-range model; mid- and long-range versions allow progressively more distant interactions.Source: §4.1.4 · pp. 26–27

4.1.5 Functional signal-propagation connectome

• The functional signal-propagation network maps statistically supported responses after optogenetic stimulation, using calcium changes measured over 30 seconds and averaged across trials and animals.It covers 188 neurons represented in the perturbation atlas; absent edges can mean unmeasured pairs or measured pairs without retained responses.Source: §4.1.5 · p. 27

4.2 Max-plus algebra

• Max-plus algebra replaces ordinary addition with maximization and multiplication with addition, so competing routes are represented by the strongest accumulated contribution.The presentation reports max-plus results; the order-dual min-plus algebra is described as suited to inverse questions.Source: §4.2 · pp. 27–28, Operations and graph interpretation · p. 28

Operations and graph interpretation.

• In the matrix representation, absent edges are ⊥, finite entries encode retained connections or responses, and powers and the Kleene star collect best walks of fixed or arbitrary length.Each neuron receives the largest weighted contribution among its predecessors.Source: Operations and graph interpretation · p. 28

Data availability.

• The five datasets are available from WormWiring, the C. elegans Connectome Toolbox, Ripoll-Sánchez et al. and Randi et al.; processed matrices are included in the code repository.The listed sources include the corrected July 2020 WormWiring release and February 2024 neuropeptidergic update.Source: Data availability · p. 29

Code availability.

• The analysis code is available on GitHub and accepts user-provided connectivity matrices for use with other weighted neuronal networks.The passage provides the repository address.Source: Code availability · p. 29

Competing interests.

• The authors declare no competing interests.Source: Competing interests · p. 30

• The guide uses nodes for neurons and weighted directed edges for interactions, with Aij oriented from neuron j to neuron i.Source: Appendix A · p. 34
• An entry Aij encodes the direction of an interaction from its column-indexed neuron j to its row-indexed neuron i.Source: Appendix A · p. 34
• The guide frames the structures as network representations whose nodes are neurons and whose edges carry interaction weights.Source: Appendix A · p. 34

What does max-plus analysis measure?

• Max-plus algebra replaces ordinary addition with maximization and ordinary multiplication with addition, so route weights accumulate while alternative routes of equal length are compared by their largest total.For recurrent routes, mean weight per connection allows comparison across routes of different lengths.Source: What does max-plus analysis measure? · p. 34
• A two-edge cycle with weights 6 and 4 has mean weight 5, equal to a self-connection of weight 5.Mean weight accounts for recurrent routes of different lengths.Source: What does max-plus analysis measure? · p. 34
• A large weight may represent extensive anatomical contact, supported ligand–receptor relationships, or a large measured response.Dominance is relative to the weights and path-composition rule of the network under study.Source: What does max-plus analysis measure? · p. 35

Strongly connected components: where can recurrence occur?

• An SCC is a maximal group in which every neuron can reach every other through directed paths, which may pass through intermediate neurons.Direct connections between every pair are not required.Source: Strongly connected components: where… · p. 35
• An SCC with multiple neurons necessarily contains recurrent routes, while a singleton supports recurrence only if it has a self-connection.Other singletons may lie along incoming or outgoing routes or be disconnected from the rest of the network.Source: Strongly connected components: where… · p. 35
• Biologically, SCCs identify neuronal domains where signals can circulate and return, and component size describes the extent of mutual reachability..Source: Strongly connected components: where… · p. 35

Condensation graph: how are recurrent regions arranged?

• The condensation graph replaces each SCC with a single node and represents a directed edge when a connection links neurons in the corresponding direction.A represented edge requires at least one such connection.Source: Condensation graph: how are recurrent… · p. 35
• The acyclic graph shows components supplying routes into recurrent regions, components reached from them, and connections between distinct recurrent regions.A cycle between components would make them mutually reachable and therefore part of the same SCC.Source: Condensation graph: how are recurrent… · p. 35
• “Upstream” components have paths leading toward a region of interest, whereas “downstream” components are reachable from it..Source: Condensation graph: how are recurrent… · p. 35

Critical cycles and classes: which recurrent circuits dominate?

• A critical cycle is a closed directed route with the greatest mean edge weight within a recurrent component.It identifies a feedback circuit attaining the maximum average weight per connection.Source: Critical cycles and classes: which… · p. 35
• When several cycles attain the same maximum, their union forms the critical graph, and its SCCs are called critical classes.A class may contain several equally dominant cycles connected through shared critical structure.Source: Critical cycles and classes: which… · p. 35
• Critical classes distinguish the specific circuits dominating weighted organization from the full recurrent region, which may be much larger.A large recurrent component can have a critical class involving only one or a few neurons.Source: Critical cycles and classes: which… · p. 35

Eigenvalue: what does the dominant cycle weight mean?

• The max-plus eigenvalue λ of a strongly connected recurrent component equals its maximum cycle mean.In the example, a critical cycle with weights 6 and 4 gives the eigenvalue.Source: Eigenvalue: what does the dominant… · p. 36
• The eigenvalue measures mean anatomical weight, supported signaling relationships, or response weight, depending on the dataset.For molecular signaling networks, it is the mean number of supported signaling relationships.Source: Eigenvalue: what does the dominant… · p. 36
• Networks with several recurrent components can have several spectral values, interpreted alongside directed connections and separately across datasets with different units.Source: Eigenvalue: what does the dominant… · p. 36

Right eigenvectors: which neurons can the critical circuit reach?

• A right eigenvector describes weighted organization extending downstream from a critical circuit along outgoing paths.Source: Right eigenvectors: which neurons can… · p. 36
• The reachable neurons include those in the circuit’s own recurrent component and those beyond it.Source: Right eigenvectors: which neurons can… · p. 36
• Biologically, the vector describes the neuronal population toward which the circuit has potential routes of influence, with coordinates weighted relative to its associated spectral value.Source: Right eigenvectors: which neurons can… · p. 36

Left eigenvectors: which neurons can reach the critical circuit?

• A left eigenvector relates a critical circuit to neurons with directed paths leading toward it, identifying potential routes for supplying signals.Neurons in the same recurrent component may belong to both upstream and downstream domains; differences between the domains reveal directional organization.Source: Left eigenvectors: which neurons can… · p. 36

How should finite and infinite coordinates be read?

• Only differences between finite eigenvector coordinates are meaningful, because adding the same constant to all finite coordinates leaves the eigenvector unchanged.The finite domain is the set of neurons with finite coordinates.Source: How should finite and infinite… · p. 36
• Infinite coordinates encode network structure rather than activity: ⊥ marks unreachable neurons, while ⊤ marks reachability into another recurrent component whose loops are stronger.In the chemical network, the pharyngeal circuit (λ = 15) reaches the somatic component (λ = 37), so accumulated weights measured against the pharyngeal loop grow without bound.Source: How should finite and infinite… · pp. 36–37

Reading the structures together

• Together, SCCs, condensation graphs, critical cycles, eigenvalues, and left and right eigenvectors connect recurrent interactions to the organization of the whole network.Source: Reading the structures together · p. 37