Higher-Gauge, Dynamical Symmetry Geometry of Savant-Phase Cognition: A Biochemistry-to-Geometry Theory with Dimensionless and Dimensionful Predictions and an Empirical Test Protocol
Full text
Higher-Gauge, Dynamical Symmetry Geometry of Savant-Phase Cognition: A Biochemistry-to-Geometry Theory with Dimensionless and Dimensionful Predictions and an Empirical Test Protocol Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We introduce a biochemically grounded, higher–gauge theoretical framework for cognition in which reversible “savant–phase” states of the mind correspond to well–defined field configurations of a dynamical gauge hierarchy on the cortical manifold. In this formulation, neuronal and neuromodulatory variables form smooth biochemical fields CX ( r, t )( X∈{Glu,GABA,ACh,DA,NE,5HT} ) that control a set of geometric couplings in a variational action S = Rα (Θ) kFk2 + β ( B )Φ( H )+ γ (Γ , Π) LPC + η (Ξ) Lflexdt . Here, Θ = E/G (excitatory/inhibitory ratio) sets the local stiffness α (Θ), B (locus– coeruleus noradrenergic gain) governs global integration β ( B ), and the dopamine–acetylcholine pair (Γ , Π) controls the precision weighting γ (Γ , Π). The corresponding order parameters—module stiffness φk = hα (Θ) kFk2iUk and holonomy dispersion H ≃ cH/hβiM —quantify, respectively, local curvature (detail fidelity) and global coherence (context binding). Dimensionless simulations of the reaction–diffusion biochemistry reproduce the transition into and out of savant–like phases: raising E/G and ACh while lowering local DA produces φk↑ and relaxed priors ( γ↓ ), whereas global increases in NE and DA restore integration ( β↑,H ↓ ). Dimensionful instantiations ( µ M, s, µ m 2 /ms) demonstrate that moderate, spatially smooth changes of order ∆ CGlu ∼ +0 . 25 µ M, ∆ CGABA ∼ − 0 . 10 µ M,∆ CACh ∼ +0 . 10 µ M, and ∆ CDA ∼ − 0 . 02 µ Mwithin a 2 mm cortical patch are mathematically sufficient to double φk while preserving global stability. We further propose an agent–agnostic experimental blueprint combining 1 H–MRS (Glu/GABA ⇒ Θ), fMRI/MEG (effective connectivity ⇒φk,H ), and neuromelanin–MRI or pupillometry (LC proxies ⇒B ) to test these predictions under IRB supervision. Together, the model unifies biochemical modulation, higher–gauge geometry, and measurable network dynamics, providing a falsifiable path to quantify how local precision and global coherence interact to generate—and later reintegrate—savant–like cognitive phases. I. INTRODUCTION A. Motivation The paradox to be explained. Individuals with savant abilities display hyper–precise, domain–specific performance (e.g., calendar calculation, absolute pitch, prodigious drawing) that often coexists with reduced performance on tasks requiring global contextual integration or abstraction. Phenomenological reviews emphasize both congenital and acquired (including sudden) variants and repeatedly note a striking incongruity between “islets of ability” and broader cognitive function [ 1 , 2 ]. Neuroimaging case studies in calendar calculators, for example, suggest atypical recruitment patterns spanning fronto–temporal and hippocampal circuits during date–mapping tasks [ 3 , 4 ]. At the mesoscopic level, meta–analyses indicate enhanced engagement of sensory cortices in autistic cohorts across diverse visual tasks, compatible with a perceptual bias toward local detail and away from fronto–parietal control [5]. Limits of current accounts. Two influential psychological theories capture key facets of this profile. The Enhanced Perceptual Functioning (EPF) account posits stronger and more autonomous low–level perceptual systems in autism [ 6 ]; the Weak Central Coherence (WCC) account frames the phenotype as a detail–focused style with reduced global integration [ 7 ]. Both capture consistent empirical tendencies (e.g., enhanced visual search; diminished use of context), yet both are primarily descriptive frameworks: they articulate what is different at the algorithmic or behavioral level without specifying a mechanism that connects (i) biochemistry and microcircuit dynamics, (ii) network–level coupling structure, and (iii) the geometry of information flow that would jointly produce savant–like phases. Contemporary overviews of exceptional abilities in autism emphasize precisely this mechanistic gap, alongside marked heterogeneity in empirical findings across cohorts and tasks [8]. A mechanistic target: geometry of information flow. To move beyond description, we adopt a field– theoretic stance in which cognition is represented by a hierarchy of gauge bundles over the cortical manifold M . Local feature codes within a module Uk⊂M are coordinated by a connection A with curvature F = dA + A∧A , while inter–module relations are governed by a higher connection (a 2–connection)
2 whose holonomies quantify path–dependence across charts. The core variational action we study is S=Zhα(Θ) kFk2+β(B) Φ(H) + γ(Γ,Π) LPCidt, (1) where kFk2 penalizes local path–dependence (“curvature energy”) inside modules, Φ( H )penalizes dispersion of inter–module holonomies, and LPC is a precision–weighted predictive–coding energy that formalizes how top–down priors interact with bottom–up evidence [ 278 ]. Crucially, the coupling functions α, β, γ are biochemically controlled. Biochemical control knobs. Let CX ( r, t ) [ µ M] denote mesoscopic, effective extracellular fields for principal transmitter systems X∈ {Glu,GABA,ACh,DA,NE} . Their slow dynamics are approximated by reaction–diffusion–clearance equations ∂tCX=DX∇2CX−1 τX CX+SX(r, t),(2) with DX (diffusivity), τX (clearance time), and SX (sources). We then define scalar summaries that feed the couplings: Θ = E G=CGlu CGABA ,Π = w(n) ACACh+w(m) AC2 ACh,Γ = wDCDA−wDACACh, B =w(+) NCNE−w(−) NC2 NE. (3) Biologically: Θis an E/I proxy (AMPA/NMDA vs. GABA A /GABA B ), Πcaptures cholinergic sensory precision, Γdopaminergic prior/policy precision with a modest ACh counter–term, and B the locus– coeruleus noradrenergic (LC–NE) integration gain. This mapping aligns with the influential E/I–imbalance hypothesis in autism [ 11 ] and with adaptive–gain accounts of LC–NE function in integration and network reconfiguration [ 10 ]. Notably, MRS and fMRS studies indicate that glutamate/GABA measures and their task–dependent modulation are heterogeneous across regions, tasks, and cohorts, underscoring the need for a model that can parameterize such variability rather than assume a uniform shift [12, 13]. From neurotransmitters to geometry and behavior. In Eq. (1) , the couplings are smooth, saturating functions, e.g. α(Θ) = α0+α1tanhbα(Θ−1), β(B) = β0+β1tanhbβ(B−B0), γ(Γ,Π) = γ0+γ1tanhbγΓ+csensΠ−c0. (4) Raising Θ(via increased E/G ) increases α , thereby stiffening transport within a target module Uk ; cholinergic increases in Πemphasize bottom–up precision; a modest local decrease in Γrelaxes top–down priors; and global NE modulates β , collapsing or inflating loop dispersion. These mechanistic levers produce the savant–phase signature: high local curvature (detail–true invariants) with relatively noisy inter–module glue (reduced contextual blending). Operational order parameters. To link the theory to data we define two observables: φk=α(Θ) kFk2Uk,H ≈ cH hβ(B)iM ,(5) where φk is a module stiffness (local curvature energy) and H aholonomy–dispersion proxy (inverse global glue). Practically, φk can be approximated by geometry–inspired graph curvature (e.g., Forman–Ricci, Ollivier–Ricci) computed within Uk ; H by the variance of loop transports across a cycle basis connecting modules. Critically, curvature metrics have already been applied to large ASD datasets (e.g., ABIDE) and reveal atypical network geometry, demonstrating feasibility [15]. Empirical motivation and falsifiability. (1) The savant literature documents both domain–specific enhancements and cases of state transition (acquired/sudden savant), implying that mechanisms can alter local vs. global balance dynamically [ 1 , 2 ]. (2) Psychological accounts (EPF, WCC) predict detail–biased processing and weaker contextual integration but do not specify the biochemical and geometric machinery [ 6 , 7 ]. (3) Predictive–coding formulations supply a normative handle on priors and precision [ 278 ], and LC–NE adaptive–gain theory provides a neurobiological substrate for global integration control [ 10 ]. (4) E/I–related hypotheses and MRS/fMRS evidence motivate using Θand Πas levers while acknowledging heterogeneity [ 11 – 13 ]. (5) Geometry–based network statistics are now tractable at scale and sensitive to ASD–TD differences [15]. Programmatic goal. The remainder of the paper formalizes the biochemical → geometric map (Eqs. 41– 4), derives predictions for φk and H (Eq. 5), and lays out measurement pipelines (MRS for Θ; fMRI/MEG for transports; LC proxies for B ) that render the framework empirically testable in cohorts enriched for domain–specific expertise (e.g., calendar calculators [ 3 , 4 ]). This mechanistic path addresses the core paradox by specifying how local precision can rise while global coherence is selectively relaxed—and how that configuration can be reversed.
3 B. Core idea Cognition as a higher–gauge field on a cortical manifold. We model the active brain as a smooth manifold M (a mesoscopic representation of the cortical sheet and its task–dependent temporal extension) covered by overlapping charts {Ui}N i=1 ⊂M , each chart corresponding to a functional module (e.g., early auditory cortex, early visual cortex, hippocampo–parietal rule maps). Within a chart Ui , local feature representations form fibres of a principal bundle whose structure group Gi consists of the transformations that leave local representational content invariant (e.g., small translations/rotations in V1; frequency scalings and interval ratios in A1). A connection A∈ Ω 1 ( Ui,gi )specifies parallel transport of representational states; its curvature F=dA +A∧A∈Ω2(Ui,gi)(6) measures local path–dependence (rigidity) of representational alignment [ 16 , 17 ]. Inter–module relations are encoded by higher (2–)gauge structure: we treat the union of charts as a principal 2–bundle π : P → M with structure 2–group G (objects are the local groups Gi , 1–morphisms are inter–chart embeddings, 2–morphisms impose coherence across overlaps). A 2–connection (A, B), A ∈Ω1(M, g0), B ∈Ω2(M, g1),(7) has curvatures (fake curvature and 3–curvature) FA=dA +A∧A−t(B), H =dB +A . B, (8) where t : g1→g0 is the crossing map and . the action of g0 on g1 [ 18 , 19 , 193 ]. The 2–holonomy of ( A, B ) around loops that traverse multiple modules quantifies the net mismatch accumulated when information is transported across distinct inter–module paths; small 2–holonomy on task–relevant loops corresponds to coherent global context, whereas large dispersion of 2–holonomy indicates fragile inter–module glue. Dynamical (variable) symmetry groups. Unlike fixed–symmetry physical fields, cortical modules exhibit context–dependent invariants: which transformations preserve representational content varies with task, state, and neuromodulation. We therefore promote the structure groups to variable (dynamical) groups Gi ( x, t )and their isotropy subgroups Hi ( x, t ) ⊆Gi ( x, t ), bundled in a groupoid over M . Biologically, variability of Gi reflects changes in (i) interneuron–mediated E/I balance, which sharpens or broadens selectivity [ 26 ]; (ii) forebrain cholinergic tone, which increases feedforward fidelity and sensory precision in primary cortices [ 24 , 205 ]; (iii) dopaminergic precision of priors and working–memory gating in fronto– cortical loops [ 27 ]; and (iv) noradrenergic adaptive gain that reconfigures long–range network integration [ 28 ]. In categorical terms, changing Gi and the embeddings Gi→Gj modifies both 1– and 2–morphism data of G, hence reshaping the higher–gauge constraints that govern integration. Order parameters from geometry. The local configuration of a module Uk is summarized by its curvature energy density, φk=α(Θ) kFk2Uk,(9) where α(Θ) is a stiffness coupling driven by the local excitatory–inhibitory ratio Θ = E/G (cf. Eq. (6)). The global integration state is captured by the dispersion of holonomies across a cycle basis L of inter–module loops, H= Varγ∈L log Hol(A,B)(γ) 2,(10) which, in practice, admits discrete approximations derived from the product of effective transports along cycles in a parcel graph. When access to smooth fields is unavailable, we operationalize φk using discrete Ricci–type curvatures on graphs (Forman and Ollivier curvatures) computed within Uk , and H using loop–transport variance on a cycle basis [ 23 , 239 , 240 ]. These constructions retain the essential geometric meanings: φk increases when local parallel transport becomes more path–dependent (stiffer code); H increases when global transports disagree across routes (weaker glue). Biochemical control of couplings. Neuromodulators and transmitters control the couplings that weight curvature and holonomy terms. At mesoscopic scale, let CX ( r, t )denote effective extracellular concentrations ( µ M) of glutamate (E), GABA (G), acetylcholine (ACh), dopamine (DA), and noradrenaline (NE) evolving by reaction–diffusion–clearance dynamics. We define Θ = E/G (E/I ratio), Π(ACh–dependent
4 sensory precision), Γ(DA–dependent prior/policy precision with an ACh counterterm), and B (NE– dependent integration gain). The stiffness, integration, and precision couplings are chosen to be smooth saturating maps, α(Θ) = α0+α1tanhbα(Θ−1), β(B) = β0+β1tanhbβ(B−B0), γ(Γ,Π) = γ0+γ1tanhbγΓ+csensΠ−c0, (11) consistent with known neuromodulatory control of gain and selectivity (ACh ↑feedforward fidelity; DA tunes prior precision; NE tunes network integration) [ 24 , 27 , 28 , 205 ]. Increased Θ(via AMPA/NMDA vs. GABA A /GABA B balance) elevates α , stiffening local transport; increased Πfurther emphasizes bottom–up precision; a moderate decrease in Γrelaxes top–down gauge fixing in the target module; and Bmodulates the dispersion of inter–module holonomies via β. Savant phase as a geometric state. We define the savant phase as a region of coupling space in which a target module Uk enters a high–curvature, locally rigid configuration while inter–module glue is weak: Rsavant =n(Θ,Γ,Π, B) : φk(Θ) ≥φ?,H(B)≥ H?o,(12) for thresholds φ?,H?> 0set by task and dataset. Biologically, φk↑ arises when E/I tilts toward excitation and ACh increases sensory precision in Uk , thereby enriching the set of exact invariants enforced by Gk ( x, t ); H ↑ arises when long–range gain is not simultaneously increased, allowing path–dependence across modules to persist. In contrast, exit from the savant phase is achieved by raising B (NE) and Γ (DA) and normalizing Θ, which jointly reduce holonomy dispersion and relax local stiffness, restoring a flatter global field. From microdynamics to cognitive manifolds. At the neuronal population level, activity evolves on low–dimensional manifolds embedded in high–dimensional firing–rate space; task demands and state modulate both the manifolds and the charts that coordinatize them [ 29 – 31 ]. In our formulation, these manifolds are the fibres of the cognitive bundle, and neuromodulators reconfigure their symmetries Gi ( x, t ): increasing Θand Πeffectively expands the local isotropy (more transformations leave local codes invariant), which raises curvature when neighbouring embeddings are not simultaneously tightened. Mixed selectivity—a hallmark of flexible cognition [ 32 ]—corresponds to fibres whose symmetry is reduced (fewer exact invariants) but whose inter–module glue is strong (small holonomy dispersion); the savant phase inverts this balance. Discrete estimators for empirical work. Empirically, continuous fields ( A, B )are not directly observed; instead, we estimate transports Uij between parcels (effective connectivity) and compute (i) graph curvature within Uk(Forman/Ollivier) as a proxy for φk, and (ii) loop–transport dispersion over a cycle basis as a proxy for H [ 23 , 239 , 240 ]. These proxies preserve the key qualitative dependencies on the couplings in Eq. (11) and link naturally to population–level measurements (fMRI/MEG and MRS), thereby rendering Eqs. (9)–(12) testable. C. Contributions Overview This work makes four interlocking contributions: (i) a rigorous biochemistry → geometry mapping that embeds neurotransmitter dynamics into a higher–gauge variational action on the cortical manifold; (ii) a set of order parameters that operationalize local rigidity and global glue—module stiffness φk , holonomy dispersion H , and a variable symmetry index (VSI); (iii) dimensionless and dimensionful reaction–diffusion (PDE) models, with worked numerical instantiations, that produce entry/exit trajectories for savant–like phases; and (iv) an empirical protocol with measurable proxies (MRS for Θ = E/G , fMRI/MEG for transports, pupillometry/neuromelanin–MRI for LC) together with analysis pipelines that compute φk,H,VSI while reserving agent/dose specifications strictly for a licensed PI. (i) Biochemistry → geometry mapping and action. Building on the formalism introduced in Secs. I A– I B, we couple mesoscopic transmitter fields CX ( r, t )[ µ M] to a higher–gauge action via scalar control knobs (Θ , Γ , Π , B, Ξ) (cf. Eqs. (41) – (11) ). The mapping proceeds in three steps. Step 1 (Kinetics): reaction–diffusion–clearance dynamics of the form ∂tCX=DX∇2CX−1 τX CX+SX(r, t) [µM/s], X ∈ {Glu,GABA,ACh,DA,NE,5HT},(13) with DX∼ 0 . 2 − 0 . 8 µ m 2/ms and τX∼ 0 . 1 − 10 s, encodes spatial spread and clearance within cortex. Step 2 (Biochemical summaries): we define Θ = CGlu/CGABA, Π = w(n) ACACh + w(m) AC2 ACh, Γ =
5 wDCDA −wDACACh, B = w(+) NCNE −w(−) NC2 NE, Ξ = ··· , which capture E/I balance, sensory precision (ACh), prior/policy precision (DA with ACh counterterm), and LC–NE integration, respectively. Step 3 (Couplings): these scalars modulate the geometric weights α (Θ) , β ( B ) , γ (Γ , Π) , η (Ξ) in the action S = R [ αkFk2 + β Φ( H ) + γLPC + ηLflex ] dt , turning biochemical shifts into controlled changes of local curvature and global holonomy (Sec. I B). The empirical estimators described below ensure that all terms admit data–driven calibration (MRS, effective connectivity, LC proxies) without prescribing any agents or doses [34, 213, 219, 228]. (ii) Order parameters: φk , H , and VSI. We define three order parameters and their discrete estimators. Module stiffness (local curvature energy) in a target module Ukis φk=α(Θ) kFk2Uk,(14) which we operationalize by graph Ricci curvatures within Uk (Forman or Ollivier). Let G = ( V, E )be a parcel graph with nonnegative weights Wij . Denote by FRC ( e )(or κORC ( e )) the edge curvature for e∈E ; then a robust proxy is b φk = median{FRC ( e ) : e⊂Uk} (or the median of κORC ) [ 37 ]. Holonomy dispersion measures global path–dependence across modules. With a minimum cycle basis {γm} of G (computed in polynomial time [ 215 ]), define loop transports from an effective connectivity operator U (DCM, transfer entropy, or time–lagged partial correlation [34, 39, 236]): U(γm) = Y (i,j)∈γm Uij,b H= Varm log U(γm) 2,(15) which tracks dispersion of inter–module holonomies. Variable symmetry index (VSI) quantifies which micro–transformations leave an ROI response invariant: VSI(Uk;τ) = 1 |T| X g∈T 1hRUk(g), RUk(id)i kRUk(g)kkRUk(id)k≥τ,(16) with RUk ( g )the ROI activity pattern under a small transformation g (e.g., slight rotation/frequency shift) and threshold τ∈ (0 , 1). VSI complements φk : higher VSI indicates a locally larger effective symmetry group, which typically raises curvature unless inter–module glue is co–tightened (Sec. I B). (iii) Dimensionless and dimensionful PDE models with worked numbers. We present both normalized and physical–unit models. Dimensionless: ∂tcX=κX∆cX−λXcX+σX+uX, X ∈ {E, G, A, D, N, S},(17) where cX∈ [0 , 1] and uX is an abstract actuation used to impose entry/exit schedules in silico. Dimensionful: Eq. (81) with DX and τX in biophysically plausible ranges. We work an explicit instantiation (Sec. I A and numerical sections) showing that within a r≈ 5 mm module Uk , smooth changes of order ∆ CGlu ∼ +0 . 25 µ M,∆ CGABA ∼ − 0 . 10 µ M,∆ CACh ∼ +0 . 10 µ M,∆ CDA ∼ − 0 . 02 µ M(with a w& 1 . 6 mm taper) suffice to elevate α (Θ) and thereby double φk , while global increases in LC proxies reduce b H during exit (cf. Eq. (15) ). Diffusive timescales obey τD∼L2/ ( π2D ). For L = 1 mm and D = 0 . 5 µ m 2/ms = 5 × 10 −4mm2 / sthis gives τD≈ 200 s. To achieve the 5–20 s equilibration windows used in our ENTRY/EXIT blocks we therefore adopt smaller diffusive half-widths L≈ 0 . 30–0 . 50 mm (consistent with Sec. VI.D). (iv) Empirical protocol and measurable proxies (agent/dose reserved for PI). We specify a crossover, placebo–controlled human protocol in which ENTRY–like tasks (detail precision) and EXIT–like tasks (integration) are paired with (a) regional 1 H–MRS in Uk to estimate b Θ = Glx/GABA (quantified with LCModel [ 213 ]); (b) fMRI (and optional MEG) to estimate effective connectivity Uij for curvature and loop–dispersion metrics ( b φk,b H ) using DCM/TE/Granger [ 34 , 39 , 236 ]; and (c) LC proxies from neuromelanin–sensitive MRI and phasic pupillometry to index B (integration gain) [ 216 – 218 ]. All data are organized under BIDS and processed with fMRIPrep to enhance reproducibility [ 219 , 228 ]. Crucially, no agent or dosing information is specified here; those fields are explicitly reserved for the licensed PI and institutional pharmacy per IRB/DSMB oversight. The analysis plan links measured ( b Θ,b B, b Γ,b Π )to couplings ( α, β, γ )and outputs ( b φk,b H,VSI ), yielding falsifiable predictions without operationalizing any unapproved intervention. (v) Reproducible analysis pipeline and discrete higher–gauge surrogates. We provide algorithmic details for (1) computing FRC/ORC within Uk as surrogates for local curvature; (2) constructing a shortest cycle basis {γm} and evaluating b H (Eq. (15) ); (3) estimating transports with DCM/TE/Granger (state–space
6 vs. information–theoretic methods); and (4) optional Hodge–theoretic summaries (curl/harmonic flow on graphs) and topological persistence to capture multi–loop organization [ 44 , 242 ]. The pipeline is BIDS– compliant and modular, making it straightforward to swap in alternative parcellations or connectivity estimators. Novelty and testability. To our knowledge, this is the first framework that (a) encodes variable symmetry groups at the module level within a higher–gauge action; (b) ties biochemical control (Θ , Γ , Π , B, Ξ) to geometric couplings ( α, β, γ, η ); and (c) delivers operational, dataset–ready metrics ( b φk,b H,VSI ) with explicit acquisition and preprocessing plans [ 34 , 39 , 213 , 219 , 228 , 236 ]. Because the endpoints are quantitative and the biochemical proxies are measurable noninvasively, the theory is falsifiable with currently available neuroimaging methods. II. MATHEMATICAL FRAMEWORK (HIGHER GAUGE + DYNAMICAL SYMMETRY) A. Cognitive manifold and module charts Manifold, metric, and spacetime lift. We model the cortex as a smooth, oriented, two-dimensional Riemannian manifold M (the reconstructed cortical sheet) endowed with a metric g that captures geodesic distances along the folded surface rather than through the embedding volume [ 46 – 48 ]. To describe task episodes we form the product spacetime M := M×I , where I⊂R is a compact time interval. For points x, y ∈M, we define an operational delay kernel τ(x, y) = dg(x, y) v(x, y)[s],(18) where dg is the geodesic distance on ( M, g )and v ( x, y )is an effective conduction velocity (accounting for myeloarchitecture and axonal caliber) along the dominant white-matter route between neighborhoods of x and y [ 49 , 50 ]. This delay kernel provides a physically motivated lift from purely spatial geometry to spatiotemporal transports on M , and—together with energetic constraints on spiking and synaptic transmission—motivates the weighting of short, surface-respecting routes over long detours [51]. Atlas and module charts. Let A = { ( Ui, ϕi ) }N i=1 be a finite atlas of M , where each Ui⊂M is an open patch (module) and ϕi : Ui→R2 is a smooth coordinate map with smooth transition functions ϕi◦ϕ−1 j on overlaps [ 46 , 52 ]. In practice, one instantiates {Ui} by a multimodal or anatomical parcellation (e.g., HCP-MMP1.0 [ 233 ], Desikan–Killiany [ 54 ], Gordon [ 55 ], or cytoarchitectonic maps [ 56 ]). The module of interest Uk is domain-specific: for auditory pitch processing, Uk can be chosen as primary auditory cortex (A1) along Heschl’s gyrus with tonotopic gradients [ 57 , 58 ]; for rule–based calendrical/numerical computations, a parietal patch encompassing intraparietal sulcus (IPS) is appropriate [59, 60]. Scale separation and biological grounding. We appeal to classical neuroanatomical and physiological regularities to justify the patch scale. Hypercolumnar organization and topographic maps ensure that within a few millimeters the mapping from stimulus features to cortical position is smooth and locally coherent [ 61 – 63 ]. Canonical microcircuits repeat across patches with parametric variations, supporting a coarse-graining at the module scale [ 64 ]. Thus, each Ui is chosen large enough to contain a quasihomogeneous population of columns (for stable local feature codes), yet small enough that ϕi is well approximated by an affine chart and conduction delays across Ui are negligible relative to inter-module delays (Eq. (18)). Feature bundles and local symmetry groups. Over each module Ui we posit a feature bundle πi : Fi→Ui with typical fibre Yi (the local feature space; e.g., orientation–phase pairs in V1; frequency–level pairs in A1; numerosity/rule codes in IPS). The local symmetry group Gi≤GL ( Yi )is the set of transformations that preserve representational content within Ui : rotations or phase shifts in orientation maps, multiplicative frequency scalings (and cyclic pitch-class equivalences) in auditory maps, and discrete rule transforms in parietal numeracy [57, 60, 62]. A connection Ai∈Ω1(Ui,gi)(gi= Lie(Gi)) defines parallel transport of feature codes inside Ui . Its curvature Fi = dAi + Ai∧Ai∈ Ω 2 ( Ui,gi )measures local path dependence, i.e., the rigidity of representational alignment under infinitesimal loops contained in Ui . The family {Ai} is stitched across overlaps by smooth gauge transformations consistent with transition maps ϕi◦ϕ−1 j. Chart overlaps, transitions, and biological interprets. On Ui∩Uj6 = ∅ , transition data include (i) coordinate changes ϕi◦ϕ−1 j and (ii) fibre isomorphisms gij : Fj|Ui∩Uj→ Fi|Ui∩Uj taking values in Gi . Biologically, gij encodes how two neighboring modules align their feature bases (e.g., a frequency-to-phoneme map or visual edge-to-shape map). The smoothness of gij reflects well-myelinated, topographically
7 arranged projections; discontinuities or high curvature may arise at boundaries where tuning properties shift rapidly (e.g., map reversals and field discontinuities) [65, 233]. From physical cortex to the manifold M .Surface-based reconstruction places individual cortical sheets into spherical or flattened registration spaces while preserving metric relations along the sheet [ 47 , 48 , 65 ]. Parcellations define Ui ; their centroids and boundaries provide numerical meshes for approximating ( M, g ). The manifold view is not merely cosmetic: geodesic distances on M better predict empirical coupling and delays than Euclidean distances in volume space, and they respect the wiring economy of cortical morphogenesis [50, 65]. Domain-specific examples Uk . Auditory A1 (pitch/tonotopy). In humans and nonhuman primates, A1 exhibits tonotopic gradients with frequency increasing along Heschl’s gyrus; superimposed invariances respect multiplicative scaling (octaves) and, in higher fields, cyclic equivalence classes for chroma. These are captured by choosing Gk to include R+ (scalings) and a cyclic subgroup (chroma), acting on a feature fibre Yk of frequency–level codes; Ak transports these codes along the tonotopic axis [ 57 , 58 ]. Parietal IPS (numerical/rule codes). IPS contributes to quantity representation and symbolic arithmetic; its local invariants include discrete symmetries of equivalence classes (e.g., commutativity for addition) and transformations preserving magnitude relations. Here, Gk is a finite/discrete group of rule-preserving transforms acting on a fibre of numerical/rule encodings [ 59 , 60 ]. In both cases, the module Uk provides the biological substrate for the domain whose local curvature energy will define the stiffness order parameter φkin later sections. Summary of 2.1. Cognitively, the manifold M collects modules as charts; mathematically, ( M, g ) supports an atlas { ( Ui, ϕi ) } and feature bundles {Fi} whose local symmetry groups Gi encode contextdependent invariants; physically, the operational delay kernel τ ( x, y )(Eq. (18) ) respects geodesics and conduction velocities; biologically, hypercolumnar/topographic organization and canonical microcircuits justify coarse patches while domain-specific physiology (A1, IPS) motivates the choice of Uk [ 57 , 60 , 61 , 63 , 64 ]. This foundation enables the higher-gauge structure of Sec. II to be instantiated with biochemically controlled couplings in subsequent subsections. B. Group tower and variable symmetry Hierarchical symmetries: from micro to macro. We formalize multi–scale cognitive organization by a nested group tower G0⊂G1⊂G2⊂ ··· ⊂ GL,(19) where each Gi is a Lie group of invariances appropriate to level i (local micro–features, intra–module feature maps, inter–module relations, task/semantic transformations, etc.). The inclusions in (19) are smooth group monomorphisms ιi : Gi,→Gi+1 that encode how higher levels constrain and organize lower–level representations. On the cortical manifold M (Sec. II A), each chart Ui⊂M carries a principal G1 –bundle of local feature frames with connection A1 , while inter–chart consistency is mediated by higher morphisms that live at level G2 and above (Sec. II C below). Mathematically, this viewpoint draws on the differential–geometric treatment of principal bundles and connections [ 66 ] and on the groupoid–based description of structure and gluing beyond a single group action [67, 68]. Variable symmetry: state–dependent groups and isotropy. In contrast to fixed physical symmetries, cortical invariances are state–dependent. We promote each level group to a variable symmetry field x∈M, t ∈I7−→ Gi(x, t)≤Gmax i,(20) a smoothly varying Lie subgroup of a maximal ambient group Gmax i , together with the isotropy (stabilizer) subgroup Hi(x, t) = StabGi(x,t)ψi(x, t)={g∈Gi(x, t) : g·ψi(x, t) = ψi(x, t)},(21) where ψi denotes the leveli representational state. The effective degrees of freedom available at level i are summarized by si(x, t) = dim Gi(x, t)−dim Hi(x, t),(22) and we will later relate si to the Variable Symmetry Index (VSI) computed empirically. Biologically, neuromodulatory and E/I mechanisms alter Gi ( x, t )and Hi ( x, t )by changing response gains, tuning
8 widths, and gating, thus expanding or contracting the local symmetry structure. For example, fast-spiking interneuron drive reshapes local feature invariances (and oscillatory regimes) [ 72 ]; dopaminergic invertedUeffects shift stability and gating in prefrontal circuits [ 204 ]; and locus–coeruleus noradrenergic tone remodels integration/segregation across large-scale networks [ 75 , 202 ]. These mechanisms motivate statedependent inclusions ιi(x, t) : Gi(x, t),→Gi+1(x, t)and state-dependent isotropy Hi(x, t)(Eqs. 20–22). From towers of groups to higher categories. Inter–module consistency requires not only group actions but also relations between actions. We therefore pass from the tower (19) to a higher (categorified) structure in which: (i) objects encode groups of symmetries (levels of description), (ii) 1-morphisms encode homomorphisms or functorial embeddings between these groups, and (iii) 2-morphisms enforce coherence of these embeddings on overlaps. A convenient algebraic model is the Lie 2-group, equivalently a crossed module of Lie groups ( G, H, ∂, . ), where H acts by “gauge-of-gauge” transformations and ∂ : H→G encodes how second-level symmetries correct first-level ones [ 69 , 70 ]. In this language, the brain-wide glue is naturally described by a principal 2-bundle with a 2-connection whose surface holonomy generalizes line holonomy and captures path-dependence across chart overlaps and loops in space–time [ 71 ]. The variable-symmetry version replaces fixed ( G, H )by fields ( G ( x, t ) , H ( x, t )) together with state-dependent ∂x,t and .x,t, subject to crossed-module identities at each (x, t). Coherence constraints across the tower. Let ρi ( x, t ) : Gi ( x, t ) yYi ( x, t )denote the leveli action on the fibre of representational states at ( x, t ). Functoriality requires the embeddings ιi ( x, t )to commute with actions up to a coherent 2-morphism: κi(x, t) : ρi+1(x, t)◦ιi(x, t)⇒ Ui,i+1(x, t)◦ρi(x, t),(23) where Ui,i+1 transports leveli states to level i+ 1(e.g., a feature-to-concept map between modules). The higher (2-group) coherence conditions ensure that compositions of such κi around overlaps agree (pentagon-type identities). When neuromodulation changes Gi ( x, t )or Hi ( x, t ), the corresponding κi may lose tightness, yielding larger holonomy dispersion across loops (later measured by b H). Biochemical control of symmetry expansion/contraction. State–dependence of Gi, Hi is driven by biochemistry. Denote the mesoscopic transmitter fields by CX ( r, t )(Sec. II A) and the derived scalars Θ (E/I), Γ(DA prior precision), Π(ACh sensory precision), B (NE integration gain). We formalize the local expansion rate of effective symmetry by ˙si(x, t) = D∇csi,˙ cEwith c= (Θ,Γ,Π, B, . . .),(24) and posit sensitivities consistent with neurophysiology: ∂s1 ∂Θ>0,∂s1 ∂Π>0,∂s1 ∂Γ<0,∂s2 ∂B ≷0(task-dependent). Interpretation: higher E/G and ACh widen local invariances (larger G1 , smaller H1 ); stronger DAweighted priors contract them (smaller G1 or larger H1 ); NE modulates how much of this freedom is glued across modules (effective change at level G2 ). Empirical work supports these directions: interneurondriven gain modulation alters the precision/width of local codes [ 72 ]; DA shifts stability and gating in cortico-striatal loops [204]; and NE tracks integration/segregation dynamics across cortex [75, 202]. Geometric consequences and order parameters. Expanding G1 ( x, t )at fixed inter–module glue typically increases local curvature F1 (more non-commuting generators are active without higher-level gaugefixing), raising the module stiffness φk = hα (Θ) kFk2iUk . If higher-level coherences (23) do not tighten commensurately, inter-module holonomy dispersion grows (multiple routes disagree), which we capture by the loop-variance metric b H introduced later. Conversely, symmetry contraction (e.g., DA-driven strong priors with NE-supported integration) flattens local curvature and reduces dispersion, favoring abstraction and context binding. This dovetails with empirical observations on dynamic functional connectivity and segregation/integration trade-offs in human cortex [76, 77]. Summary of 2.2. The group tower (19) furnishes a principled scaffold for cognitive invariances; its variable incarnation (20) – (22) encodes state-dependent expansions and contractions controlled by neuromodulation and E/I balance; and the higher categorical upgrade provides the glue needed to reason about inter–module coherence through surface holonomy and 2-morphism coherences. These ingredients will support the construction of higher-gauge connections and the biochemical couplings that weight curvature and holonomy terms in the action (Sec. I B, Eq. (3)).
9 C. Connections and curvatures Principal bundles and 1–connections on the cortical manifold. Let π : P→M be a principal G –bundle over the cortical sheet M (Sec. II A), where G is the local symmetry group acting on feature fibres in a given module. A (Ehresmann) connection is specified by a g –valued 1–form A∈ Ω 1 ( P, g )that is G –equivariant and reproduces generators on vertical vectors. In a local gauge over a chart U⊂M we identify A with a g –valued 1–form on U , still denoted A∈ Ω 1 ( U, g ). For a C1 curve γ : [0 , 1] →U ,parallel transport UA(γ)∈Gis given by the initial–value problem d ds UA(γ, s) + A ˙γ(s)UA(γ, s) = 0, UA(γ, 0) = 1G,(25) whose solution at s= 1 defines the holonomy HolA(γ)≡UA(γ, 1) [78]. The curvature 2–form F=dA +A∧A∈Ω2(U, g)(26) measures first–order path–dependence: F = 0 on U if and only if HolA ( γ )depends only on the endpoints of γ (small loops yield the identity up to higher order). In our setting, kFk2 (with the Killing form or a chosen inner product on g ) quantifies local rigidity of representational alignment inside a module, and its weighted spatial average defines the module stiffness order parameter φk=hα(Θ)kFk2iUk(cf. Sec. I B). Holonomy and Wilson loops: operational “glue”. For a closed loop γ based at x∈U , the holonomy HolA ( γ ) ∈G summarizes the net mismatch after transporting a representation around γ . Gauge–invariant functionals of holonomy, such as Wilson loops, WR(γ) = TrR HolA(γ), R :representation of G, (27) recapitulate how consistently information glues across routes [ 79 ]. In the brain, large dispersion of WR ( γ ) across a cycle basis of inter–module loops indicates that distinct multi–hop routes produce discordant transforms—a failure of global context binding. We will exploit a discrete analogue by assigning transports to edges of a parcel graph and multiplying them around cycles (Sec. II C), yielding a holonomy–dispersion metric b H. 2–connections and higher curvature: gluing actions themselves. Inter–module consistency is not fully captured by line holonomy when “relations between actions” also vary. We therefore lift to a 2–connection ( A, B )on a principal 2–bundle with structure 2–group ( G, H, ∂, . )(a crossed module), where A∈ Ω 1 ( M, g ) is the ordinary connection and B∈ Ω 2 ( M, h )is a 2–form valued in h = Lie ( H ). The associated curvatures are FA=dA +A∧A−t(B), H =dB +A . B, (28) with t = d ∂ : h→g and the induced action . . The 2–form FA is the fake curvature; when FA = 0 (“fake flatness”), surface holonomy of ( A, B )is well–defined and depends only on the surface swept by a loop family rather than on the parameterization [ 80 – 82 ]. The 3–form H (sometimes called “higher curvature”) measures failure of B to be covariantly closed and controls the consistency of gluing between different parallel–transport operations. Operationally, small H on task–relevant surfaces corresponds to coherent inter–module composition; large H inflates the spread of multi–route transports (global “glue” weakens). Physical and biological interpretations. Local curvature as rigidity. In a module Uk , the coefficients of A summarize effective, gain–weighted couplings among local feature populations; biophysically these are shaped by synaptic efficacies (AMPA/NMDA vs GABA A /GABA B ), interneuron–mediated gain, and cholinergic control of feedforward precision. When recurrent microcircuits drive non–commuting flows (e.g., distinct feature subspaces updated in different orders), kFk grows, and the code becomes stiff : different micro–paths produce measurably different alignments [ 83 , 84 ]. This is the mesoscopic correlate of “hyper–precision” within a domain. Holonomy and global glue. The long–range phase–consistent routing of information—often discussed under “communication–through–coherence”—is captured by small dispersion of multi–step transports [ 85 ]. Neuromodulatory tone (e.g., LC–NE) globally tugs the network toward integration (low dispersion) or segregation (high dispersion) on behaviorally relevant timescales, thereby tuning the effective 2–connection (A, B)at the macro scale. Discrete transport and the connection Laplacian. Empirically, we estimate transports on a parcel graph G = ( V, E )by fitting, for each edge ( i, j ) ∈E , an orthogonal (or unitary) transport Rij ∈O ( d ) that best aligns feature subspaces across parcels (e.g., via time–lagged regression, DCM linearizations, or vector diffusion [86]). Local curvature proxies arise from the loop–product on small cycles c⊂G, Holdisc(c) = Y (i,j)∈c Rij, κ(c) = log Holdisc(c) F,(29)
16 Figure 4: Noradrenergic dynamics and integration gain B.Time course of noradrenaline CNfor the full network (orange solid) and for module Uk (blue dashed). The global NE surge during the EXIT phase boosts the integration scalar Band the coupling β(B, GB), promoting global binding and low holonomy dispersion. reducing holonomy dispersion and strengthening large-scale coherence. (v) Ξ(serotonergic flexibility). 5-HT contributes to behavioral flexibility, patience/temporal discounting, and inhibition of habitual policies [ 120 , 121 ]. We aggregate these effects into Ξ, which weights a flexibility Lagrangian Lflex (Eq. (35) ) that regularizes rapid switching of connections, allowing context-appropriate transitions without loss of stability. Saturating transforms and calibration. While Eqs. (47) – (48) are convenient polynomial summaries, each scalar can be replaced (if desired) by a saturating Hill/logistic map without changing the downstream theory. For instance, ΠHill sens =θA CnA A CnA A+KnA A , BHill =θN CnN N CnN N+KnN N−ϑN CmN N CmN N+KmN N,2 , with θ ’s dimensionless and K ’s in [ µ M]. The polynomial form we use is simply the Taylor expansion of such saturations around baseline. Calibration proceeds by choosing w or ( θ, K )so that moderate, physiologically plausible shifts in CX (Sec. III A) produce order-one changes in the scalars, e.g., ∂ Π sens/∂CAbase ∼ O(1) [µM−1]. Spatial gradients and rims. Control scalars inherit spatial gradients from CX ( r, t ). In later sections, gradient penalties enter the couplings as α(Θ, GΘ) = α0+α1tanh[bα(Θ −1)] −αgtanh[bgGΘ], GΘ=k∇Θk, and analogously for β ( B, GB ). Physiologically, very sharp rims in E/I or NE are difficult to sustain due to diffusion and transporter saturation; penalizing GΘ, GBencodes this constraint. Estimating the scalars from data. E/I ratio Θ:regional 1 H–MRS Glx/GABA (with appropriate tissue correction and referencing) supplies a proxy for Θat the voxel scale. ACh precision Π sens :while invasive measures are required for direct quantification, task–evoked cholinergic effects on V1/V2/A1 can be inferred from attentional gain paradigms, laminar fMRI, or invasive recordings in animals [ 114 , 115 ]. DA precision Γ:computational assays of prior/policy precision (e.g., context–dependent working memory gating; Bayesian cue combination with manipulable prior reliability) together with BOLD signatures of dopaminergic prediction errors [ 117 ] can be used to back out Γ.NE integration B :pupil diameter and its derivative provide noninvasive proxies for phasic LC activity and cortical gain state [ 119 , 123 ]. 5-HT flexibility Ξ:reversal learning and waiting tasks quantify the behavioral correlates of Ξ[ 120 , 121 ]. These measurements feed Eq. (39) to yield α, β, γ, η and, via Sec. II C, the order parameters φk,b H.
17 Sensitivity and cross–talk. Local linearization around baseline yields δΠsens ≈w(n) AδCA+ 2w(m) ACbase AδCA, δΓ≈wDδCD−wDA δCA, δB ≈(w(+) N−2w(−) NCbase N)δCN, so that ACh modulates both Π sens and Γ(opposing priors when sensory precision is high), while NE changes can reverse sign in δB near the saturation knee. Such cross–talk accords with canonical network models in which catecholamines implement global gain control and adjust integration/segregation trade-offs [118, 119]. Summary of 3.2. The control scalars Θ , Π sens, Γ , B, Ξdistill complex transmitter dynamics into five dimensionless “knobs” with clear biological interpretations—E/I balance, sensory precision, prior/policy precision, integration gain, and flexibility. Their construction preserves units, admits saturating biophysically meaningful alternatives, and supports direct estimation from current noninvasive and invasive assays. These scalars drive the couplings α, β, γ, η that weight curvature and holonomy in the gauge action, providing the core biochemistry→geometry link used throughout. C. Coupling maps with gradient penalties Rationale. The scalar controls (Θ , Γ , Π , B, Ξ) defined in Sec. III B modulate the geometric weights of the action (Sec. II D). To ensure (i) bounded, monotone responses that saturate at biologically plausible plateaus, and (ii) spatial smoothness consistent with diffusion/clearance constraints, we adopt saturating sigmoidal maps with gradient penalties. The latter encode the energetic cost of sharp biochemical rims (e.g., steep E/I or NE boundaries) that are difficult to sustain in cortical tissue given extracellular tortuosity and receptor/transporter kinetics (see Sec. III A). Definitions (extended from Eq. 39). Let GΘ = k∇ Θ k and GB = k∇Bk with units [ mm−1 ]. The local stiffness and integration couplings include gradient penalties: α(Θ, GΘ) = α0+α1tanhbα(Θ −1)−αgtanhbgGΘ,(49) β(B, GB) = β0+β1tanhbβ(B−B0)−βgtanhbgGB,(50) and the precision mix and flexibility couplings retain saturating forms: γ(Γ,Π) = γ0+γ1tanhbγΓ + csens Π−c0,(51) η(Ξ) = η0+η1tanhbηΞ.(52) Parameters α0, α1, β0, β1, γ0, γ1, η0, η1> 0set plateaus; bα, bβ, bγ, bη> 0set transition steepness; B0, c0 are midpoints; αg, βg≥ 0are gradient penalties with a shared slope parameter bg> 0(units absorbed into bg by measuring gradients in mm−1 ). The maps (49) – (52) are Lipschitz and bounded; their first derivatives are ∂Θα=α1bαsech2 bα(Θ −1), ∂GΘα=−αgbgsech2(bgGΘ),(53) ∂Bβ=β1bβsech2 bβ(B−B0), ∂GBβ=−βgbgsech2(bgGB),(54) and analogous expressions hold for γ, η . Thus, modest biochemical shifts near midpoints produce order–one coupling changes, whereas large gradients are penalized sublinearly (saturating tanh). Biophysical justification of gradient penalties. Spatially sharp steps in E/I or NE require sustained, precisely localized differences in release/uptake that are rapidly smoothed by diffusion and enzyme/transporter kinetics; such steps also create strong local heterogeneities in excitability and synchronization. The penalty terms −αgtanh ( bgGΘ )and −βgtanh ( bgGB )in (49) – (50) encode these costs in the action, favoring adiabatic spatial profiles. From a regularization viewpoint, these penalties are the saturating analogue of Tikhonov/total–variation smoothers that discourage high spatial frequencies while preserving boundedness [ 124 – 126 ]. In the numerical sections we choose αg, βg such that rims thinner than ∼ 1 . 5– 2 mm are disfavored, consistent with diffusion lengths inferred for monoaminergic and cholinergic signals in cortex on subsecond–to–second scales. This lateral rim width w enters only the penalty term for spatial heterogeneity in α ( θ )or β ( B ). The diffusive half-width L used in the equilibration-time estimates (Sec. VI.D) refers to the primary diffusion direction (e.g., laminar) and can be substantially smaller; we use L∗= 0.30mm for time-scale estimates without altering the w≥1.6mm penalty choice. Figure 5 displays the baseline spatial distributions of the stiffness and integration couplings, illustrating the heterogeneity of local rigidity α(Θ) and global gain β(B)across the cortical manifold.
18 (a) Baseline α(Θ) map. (b) Baseline β(B)map. Figure 5: Spatial baseline of stiffness and integration couplings. Left: local stiffness weight α(Θ); right: integration weight β(B). Variability reflects regional differences in E/I ratio and LC–NE tone at rest. These serve as the geometric priors before ENTRY/EXIT modulation. (a) Eat t=0 (b) Gat t=0 (c) Aat t=0 (d) Dat t=0 (e) Nat t=0 (f) Sat t=0 Figure 6: Baseline biochemistry (maps at t= 0 ). Initial spatial distributions of the transmitter fields that determine the control scalars (Θ , Γ , Π sens, B, Ξ) and hence the couplings ( α, β, γ, η ). The subsequent ENTRY (t=100) and EXIT (t=180) maps (Figs. 7, 8) should be interpreted relative to this baseline. To anchor the ENTRY maps, Fig. 6 shows baseline ( t= 0) spatial distributions for all transmitter fields. These form the reference for the ENTRY (t=100) and EXIT (t=180) snapshots that follow. As an ENTRY snapshot, Fig. 7 shows spatial maps at t = 100 for the transmitters that define Θand Π sens (E, G, A). The E/I tilt (E ↑ , G ↓ ) and the ACh increase are clearly localized to Uk , consistent with the schedules in Sec. V B. Receptor/microcircuit correlates of each control. The controls (Θ , Γ , Π , B, Ξ) summarize nuanced receptor– and circuit–level mechanisms (Table I). Excitation/inhibition Θ:AMPA/NMDA vs GABA A /GABA B receptor families regulate fast synaptic drive and shunting; NMDA nonlinearity and GABA B slow inhibition shape integration windows and path–dependence of updates, impacting α [ 127 , 128 ]. Sensory precision Π:fast nAChR (e.g., α 7) and M1 muscarinic signaling enhance feedforward gain and reduce internal noise in early sensory cortices, tightening local invariances ( ↑ Π ⇒↑ γ ) [ 129 , 130 ]. Prior/policy precision Γ:prefrontal D1/D2 receptor actions exhibit an inverted-U relationship to stability and gating; moderate D1 stabilizes task sets (priors), whereas excess/deficit impairs working memory and flexibility—captured in γ (Γ , Π) [ 131 , 132 ]. Integration B :LC–NE engages α1 , β , and α2 receptors; by modulating neural gain and network synchronization it shifts the cortex along a segregation–integration continuum ( β ( B )) [ 133 , 134 ]. Flexibility Ξ:5-HT 1A (hyperpolarizing, stabilizing) and 5-HT 2A (depolarizing, desynchronizing) jointly influence exploratory switches and cognitive flexibility, summarized by η (Ξ)
19 (a) Eat t=100 (b) Gat t=100 (c) Aat t=100 Figure 7: ENTRY spatial structure (maps at t=100). The E/I tilt (E↑, G↓) and local ACh increase in Ukproduce Θ↑and Πsens ↑, raising α(Θ) and φk(Sec. IV). (a) Nat t=180 (b) Dat t=180 Figure 8: EXIT spatial structure (maps at t=180). Global NE and DA increases raise β(B)and restore prior precision, reducing holonomy dispersion (Sec. XI). [ 135 , 136 ]. Global state context: neuromodulator interactions reconfigure large–scale dynamics on rapid timescales [137]. Table I: transmitter–to–geometry map. We compile receptor families, microcircuit effects, impacted couplings, and geometric consequences: Constraints and identifiability. For numerical stability and interpretability we bound plateaus and slopes: α0∈ [0 . 3 , 1 . 0], α1∈ [0 . 3 , 1 . 0]; β0∈ [0 . 3 , 1 . 0], β1∈ [0 . 3 , 1 . 0]; bα, bβ∈ [1 , 5]; αg, βg∈ [0 , 0 . 5] with bg∈ [1 , 4]. These ranges ensure that: (i) moderate biochemical deviations (e.g., | Θ − 1 |. 0 . 5) can double local stiffness; (ii) rims sharper than ∼ 1 . 5–2 mm are penalized; (iii) α, β, γ, η remain O (1), avoiding stiffness blow-ups in the Euler–Lagrange flows (Sec. II D). Identifiability of ( α1, bα )and ( β1, bβ )from data improves when the design induces multiple operating points (e.g., ENTRY-like vs EXIT-like states) and when independent proxies (MRS for Θ, pupil/LC–MRI for B) are available. Summary of 3.3. Couplings are smooth, bounded, and biologically grounded functions of biochemical controls, augmented with gradient penalties that encode diffusion/clearance constraints. Receptor–specific mechanisms suggest which knobs adjust which couplings, and thus which geometric terms of the action are emphasized. This establishes a precise, testable chain: receptor/circuit manipulations ⇒ scalar controls ⇒couplings ⇒curvature/holonomy balance. IV. ORDER PARAMETERS, OBSERVABLES, AND PROXIES A. Module stiffness (local curvature energy) Definition. For a target module Uk⊂M with 1–connection A and curvature F = dA + A∧A (Sec. II C), the module stiffness is the curvature energy density averaged over Uk , weighted by the biochemically controlled stiffness coupling α: φk=Dα Θ(r), GΘ(r)kF(r)k2EUk [arbitrary units],(55)
20 Table I: Transmitter → receptor → microcircuit → coupling(s) → geometric consequence. AMPA/NMDA and GABAA/GABABset Θ; nAChR/M1 shape Π; D1/D2 shape Γ; adrenergic α1/β/α2shape B; 5-HT1A/2Ashape Ξ. Transmitter Key receptors Microcircuit effect (examples) Coupling(s) Geometric consequence Glu/GABA AMPA, NMDA; GABAA, GABAB Fast EPSP/IPSP balance; NMDA nonlinearity; shunting/slow inhibition α(Θ) ↑ Θ ⇒↑ α⇒ stiffer local transport, ↑φk[127, 128] ACh nAChR ( α 7), mAChR (M1) Feedforward gain, SNR, attentional weighting in V1/A1 γ(Γ,Π) via Π↑ Π ⇒ stronger bottom– up precision, relaxed priors locally [129, 130] DA D1, D2 WM stability vs flexibility; gating; inverted-U γ(Γ,Π) via Γ↑ Γ ⇒ stronger prior/policy precision; too high flattens flexibility [131, 132] NE α1, β, α2 Gain, arousal; integration/segregation control β(B)↑B⇒ reduced holonomy dispersion (stronger global glue) [133, 134] 5-HT 1A, 2A Stabilization vs desynchronization; exploratory switches η(Ξ) ↑ Ξ ⇒ controlled mode switches, adiabatic field changes [135, 136] where GΘ = k∇ Θ k (Sec. III C). The inner product is the Hodge L2 pairing on Ω 2 ( Uk,g ). Operationally, φk increases when (i) E/I tilts toward excitation and cholinergic precision Π sens sharpens local codes (raising α), and/or (ii) the local connection exhibits stronger non-commutativity (larger kFk2). Discrete estimator. When only parcel-level data are available, we approximate φk by a graph-geometric curvature computed on the induced subgraph G[Uk]. Two widely used choices are: (a) Forman curvature κF ( e )on edges e of G [ Uk ], which aggregates combinatorial contributions of adjacent edges/triangles; large positive curvature indicates locally tight flow while negative curvature reflects saddle-like configurations [ 141 , 286 ]. A robust proxy is b φk = median{κF ( e ) : e⊂G[Uk]}. (b) Ollivier (Wasserstein) curvature κO ( i, j ) = 1 −W1(µi,µj) d(i,j) , where µi and µj are one-step random-walk measures and W1 the earth-mover distance; higher κO implies locally cohesive transport [285]. We analogously define b φk= median{κO(i, j):(i, j)⊂G[Uk]}. Both discretizations preserve the intuition that locally rigid/stiff codes correspond to higher curvature magnitude. In practice, we report quantiles of the edge-wise curvature to mitigate thresholding noise and parcellation idiosyncrasies. Biological interpretation and scaling. At the mesoscopic level, increased b φk within, for example, A1 (pitch) or IPS (rule codes) indicates that parallel transport of local features becomes more path-dependent— an operational hallmark of precise, detail-preserving representations. This aligns with sharpened selectivity under elevated ACh and mild E/I shift, and with the emergence of exact invariants in the local symmetry group Gk (Sec. II B). Because κF, κO are dimensionless and sensitive to degree/weighting, we recommend reporting b φk jointly with (i) edge-density-matched nulls, and (ii) a rescaled value in which weights are normalized by the connection-Laplacian spectrum (Eq. (30)), improving comparability across datasets. B. Holonomy dispersion (global glue) Definition. For a higher connection ( A, B )with 2-holonomy Hol(A,B) ( γ )around inter-module loops γ , define the holonomy dispersion H= Varγ∈L log Hol(A,B)(γ) 2[arbitrary units],(56) where L is a cycle basis of relevant loops (Sec. IV B), and the matrix log is taken in a faithful representation of the structure group. Intuitively, H quantifies the spread of multi-route transports: low dispersion indicates that distinct paths between modules yield coherent transforms (strong global glue); high dispersion implies poor context binding.
21 Figure 9: Holonomy–dispersion proxy b H(t).Temporal evolution of the dispersion metric derived from loop-transport variance. Lower values denote stronger global integration. The decline during the EXIT-like interval corresponds to the model’s prediction H≈cH/hβ(B, GB)iM(Eq. (57)), linking increased LC–NE tone to reduced holonomy variance. Coupling dependence. Because the global integration coupling β ( B, GB )(Sec. III C) penalizes higher curvature/holonomy variance in the action, we obtain the monotone approximation H ≈ cH hβ(B, GB)iM ,(57) for a positive constant cH that depends on the choice of loop set and representation. Thus, raising NE (and hence B⇒β ) lowers H , while sharp spatial rims GB (penalized in β ) can raise dispersion for the same mean B. As illustrated in Fig. 9, holonomy dispersion b H ( t )decreases steadily as global NE—and therefore the integration coupling β ( B, GB )—rises, signifying increasing inter-module coherence and reduced path dependence. Discrete estimator via loop products. From edge-wise transports Rij (estimated from fMRI/MEG effective connectivity; see below), define, for each cycle γ= (i1→i2→···→i`→i1), Holdisc(γ) = Ri1i2Ri2i3···Ri`i1,b H= Varγ∈B log Holdisc(γ) 2 F,(58) where B is a cycle basis. To avoid duplicating earlier algorithmic references, we point to alternative choices not used above: a minimum cycle basis can be computed by de Pina’s algorithm [ 143 ] or by subsequent improvements in optimization formulations [ 144 ]. The holonomy product in (58) is analogous to Wilson-loop statistics but evaluated on a graph and in a data-driven transport representation. Estimating edge transports. Given parcel time series xi ( t ), we build Rij as small-angle elements in O ( d )(or SO ( d )) aligning low-dimensional embeddings of local activity across parcels. Three practical routes are: (a) Angular synchronization over SO (2)/ SO ( d ): estimate pairwise phase/angle offsets, then recover globally consistent Rij from the leading eigenvectors/Semidefinite Programming relaxations [ 145 ]. (b) Diffusion-map embeddings per parcel (low-dimensional coordinates of local activity), then alignment by Procrustes maps to obtain Rij [146, 147]. (c) Model-based effective connectivity (e.g., state-space or Bayesian DCM variants) followed by polar decomposition to extract near-orthogonal transports; see recent methodological reviews for dynamic/large-scale estimation [148].
22 These provide Rij amenable to loop products (58). As a complementary global summary, the spectrum of the connection Laplacian (Eq. (30) ) quantifies consistency of transports; larger spectral gaps imply lower b H. C. Operational proxies computable today Proxy for φk : graph curvature within ROI. Compute κF or κO on the induced subgraph G [ Uk ] using edge weights derived from effective connectivity or frequency-specific coherence. Report b φk = median{κ ( edges in Uk ) } and its interquartile range. Robustness checks include: (i) degree/weight preserving randomization; (ii) multi-threshold persistence of b φk ; and (iii) cross-parcellation agreement. These curvature proxies have been validated on diverse network classes and are computationally efficient for modern ROI counts [141, 285, 286]. Proxy for H : loop-transport variance. Construct a cycle basis B (e.g., de Pina or optimization-based [ 143 , 144 ]), compute Holdisc ( γ )and klog ( · ) k2 F for each γ , and summarize with the variance b H (Eq. 58). As an alternative one-number proxy, report the spectral gap of the connection Laplacian (larger gap ⇒ lower dispersion) alongside b H . Alignment-based transports from synchronization or rotation averaging stabilize estimates in noisy, low-SNR regimes [145, 147]. Variable Symmetry Index (VSI). We operationalize variable symmetry at the ROI level by testing invariance under micro-transformations g∈ T of stimuli/features (e.g., small rotations/scalings in vision; pitch shifts/tempo in audition). Let RUk(g)denote the ROI response pattern under g. Define VSI(Uk;τ) = 1 |T| X g∈T 1hRUk(g), RUk(id)i kRUk(g)kkRUk(id)k≥τ,(59) with threshold τ∈ (0 , 1). In practice, we estimate invariance using representational similarity analysis (RSA) by comparing condition-wise pattern distances across transformations; see toolboxes and bestpractice guidelines [ 149 , 150 ]. Increases in VSI indicate an expanded effective symmetry group Gk ( x, t ) (Sec. II B), typically accompanied by higher b φk unless global glue increases commensurately. The VSI connects directly to neurophysiological notions of selectivity–invariance trade-offs [151]. Integrative readouts and state dependence. Because φk and H respond in opposite directions to different control knobs (Sec. III C), joint reporting is recommended. For example, ENTRY-like states (detail precision) should show b φk↑ with unchanged or elevated b H , whereas EXIT-like states (integration) should show b H ↓ with normalized b φk . Time-resolved analyses—sliding-window transports, synchronization-based phases—allow linking order-parameter dynamics to arousal/vigilance fluctuations known to reorganize large-scale networks [148]. V. DIMENSIONLESS MODEL (SIMULATION–READY) A. Normalized fields and PDE Objectives and scope. For numerical experiments we require a dimensionless, stable, and modular formulation of the transmitter fields that (i) preserves the physics of diffusion/clearance on the cortical manifold, (ii) interfaces cleanly with the control scalars (Θ , Γ , Π , B, Ξ) (Sec. III B) and their coupling maps (Sec. III C), and (iii) admits efficient time stepping on surface meshes or ROI graphs. We therefore nondimensionalize the reaction–diffusion–clearance system (41) and obtain a normalized PDE suitable for IMEX schemes and graph/surface discretizations grounded in Laplace–Beltrami theory [152, 153]. Nondimensionalization. Fix reference units: a length L0> 0(e.g. L0 = 5 mm ), a time T0> 0(e.g. T0 = 1 s), and, for each neurotransmitter X∈ { E , G , A , D , N , S } , a concentration C0 X ( r ) > 0reflecting baseline ECS levels in region r(Sec. III A). Introduce the normalized fields cX(r, t) := CX(r, T0t) C0 X(r)∈R≥0, X ∈ {E,G,A,D,N,S},(60) and the dimensionless coefficients κX(r) = DX(r)T0 L2 0 , λX(r) = T0 τX(r), σX(r, t) = T0 C0 X(r)SX(r, T0t).(61)
23 Here DX is the effective diffusivity (ECS–tortuosity reduced), τX the clearance time constant, and SX the dimensionful source; see Sec. III A. In these variables the normalized PDE on the cortical manifold Mreads ∂tcX=κX∆McX−λXcX+σX+uX, X ∈ {E,G,A,D,N,S},(62) with ∆ M the Laplace–Beltrami operator on ( M, g )and n·∇cX|∂M = 0 (no–flux). The term uX ( r, t )is an abstract control used to specify simulation schedules (ENTRY/EXIT drives; see below). All coefficients in (62) are unitless by construction. Typical magnitudes of dimensionless coefficients. Using Deff X∼ 0 . 2 − 0 . 8 µ m 2/ms = 2 × 10 −4− 8 × 10 −4mm2/ s and L0= 5 mm,T0= 1 s, we obtain κX=DXT0 L2 0∈[ 8×10−6,3.2×10−5], so diffusion is slow relative to the geometric scale. With τE,G,A ∼ 0 . 1 − 1 s and τD,N,S ∼ 1 − 10 s, we have λE,G,A ∈[1,10], λD,N,S ∈[0.1,1]. The normalized source σX is order–one when a unit change of CX over T0 at baseline is expected. These ranges guide stable time stepping and the choice of control amplitudes. Control schedules uX (ENTRY/EXIT patterns). We stochastically or deterministically prescribe uX to emulate task or state drives: uX(r, t) = uin X(r, t) + uout X(r, t),(63) uin X(r, t) = Ain XχUk(r)1 21 + tanh[ωin(t−tin)] | {z } local logistic onset exp−(t−tpk)2 2σ2 t,(64) uout X(r, t) = Aout X 1 21 + tanh[ωout(t−tout)] | {z } global ramp 1 + εΦsp(r),(65) where Uk is the target module, χUk its indicator, and Φ sp a mean–zero spatial mode used to inject mild heterogeneity. ENTRY–like schedules choose Ain A> 0, Ain E> 0, Ain G≥ 0small, and Ain D< 0(local DA relaxation), while EXIT–like schedules choose Aout N> 0(global NE rise) and moderate Aout D≥ 0to restore priors. These controls are abstract (no pharmacology specified) and serve only to explore trajectories in the normalized state space; they can be optimized under PDE constraints when fitting data [162]. Coupling to control scalars and the gauge action. After each PDE step, compute the control scalars (Sec. III B) from the normalized fields: Θ = cE cG ,Πsens = ˜w(n) AcA+ ˜w(m) Ac2 A,Γ = ˜wDcD−˜wDA cA, B = ˜w(+) NcN−˜w(−) Nc2 N,Ξ = ˜w(+) ScS−˜w(−) Sc2 S, with ˜w the rescaled weights implied by (60) – (61) . The couplings α (Θ , GΘ ), β ( B, GB ), γ (Γ , Π), η (Ξ) (Sec. III C) then update the geometric action (Sec. II D), which in turn drives the connection flows (Sec. II C). This split enforces a clean simulator structure: (i) integrate (62) ; (ii) compute scalars; (iii) update couplings; (iv) advance gauge fields. Surface and graph discretizations of ∆ M .On a triangulated cortical surface T , we use the cotangent Laplacian (with lumped mass) (∆Mf)i≈1 AiX j∈N(i) wij (fj−fi), wij =1 2(cot αij + cot βij), Ai=1 3X T3i area(T),(66) which is consistent for smooth f and enjoys good spectral properties [ 152 , 159 – 161 ]. On an ROI graph G= (V, E)we use the (symmetric) graph Laplacian (Lf)(i) = X j∼i wij (f(i)−f(j)),or Lnorm =I−D−1/2W D−1/2,(67) whose convergence to ∆ M under sampling assumptions is well studied [ 153 , 154 ]. Either choice yields a sparse, symmetric positive semi–definite diffusion operator.
24 Time stepping and stability. We adopt IMEX (implicit–explicit) schemes [ 155 , 156 ]: treat diffusion implicitly for stability, and treat linear decay, sources, and controls explicitly for simplicity. With a mesh/graph Laplacian Lapproximating ∆M, one step reads I−∆t κXLcn+1 X=cn X+ ∆t(−λXcn X+σn X+un X),(68) where is the Hadamard product to allow spatially varying λX . The implicit solve uses a pre–factorized SPD matrix ( Cholesky ); the right–hand side is O ( N ). If an explicit scheme is preferred, choose ∆ t under a diffusion CFL bound [157, 158]: ∆t≤θ κXλmax(L)(θ.0.9),(69) which is conservative on irregular meshes. IMEX schemes permit ∆ t several orders larger than (69) without compromising stability. Boundary conditions and conservation. On closed cortical surfaces ( ∂M = ∅ ), (62) reduces to pure surface diffusion with decay/source; on inflated/flattened representations with artificial boundaries we enforce no–flux by zero normal derivative at the boundary nodes, consistent with (62) . Discrete conservation in the absence of sources and decay (i.e., σX = uX = λX = 0) is guaranteed by the symmetric Laplacian flux with either cotangent or graph discretizations [152, 153]. Relation to neural field models and large–scale dynamics. While (62) is a transmitter–centric diffusion model, it is compatible with neural field approaches in which activity evolves via spatial kernels and delays [ 163 , 164 ]. Our use of κX, λX encapsulates transmitter spread/clearance that co–modulates large–scale dynamical regimes (integration/segregation, itinerancy) described in whole–brain modeling [ 331 ]. The modular control uX naturally interfaces with operator–splitting workflows [ 166 ] when coupling to additional state variables. Summary of 5.1. The normalized transmitter PDE (62) yields a compact, unitless description with well–behaved coefficients κX, λX, σX and abstract controls uX . Standard surface/graph Laplacians (66) – (67) and IMEX stepping (68) provide robust, efficient integration under realistic cortical geometry, and the outputs feed directly into the control scalars and couplings that weight the higher–gauge action. B. Entry/Exit schedules Purpose and placement. We specify open–loop and closed–loop schedules for the abstract control fields uX in the normalized transmitter PDE (62) , positioned conceptually between the dynamics (Sec. V A) and the numerical setup (Sec. V C). The schedules implement the qualitative rules: Entry (local in Uk): ∆cE>0,∆cG<0,∆cA>0,∆cD<0; global cNat baseline, Exit (global): cN↑, cD↑;normalize cE, cG, cAin Uk, where ∆ cX denotes the deviation from baseline. These rules are realizable via the control split uX = uin X + uout X introduced in Eqs. (63) – (65) , augmented below with explicit guard conditions and feedback terms drawn from hybrid/optimal control [177–181]. Averages, deviations, and targets. Define module and global spatial averages ¯cUk X(t) = 1 |Uk|ZUk cX(r, t)dA, ¯cM X(t) = 1 |M|ZM cX(r, t)dA, (70) and local deviations ∆cUk X(t) = ¯cUk X(t)−1. Entry target inequalities in Ukread ∆cUk E≥ηE,∆cUk G≤ −ηG,∆cUk A≥ηA,∆cUk D≤ −ηD,(71) for small ηX>0(typ. 0.05–0.20). Exit targets are ¯cM N≥1 + ηN,¯cM D≥1 + ηglob D,|∆cUk X| ≤ ε(X∈ {E,G,A}),(72) with εsmall (typ. 0.02–0.05).
25 Open–loop (feedforward) schedules consistent with the qualitative rules. We instantiate Entry/Exit via the shaped controls of Eqs. (64)–(65). For Entry on [tin, tend)we set uEntry X(r, t) = Ain XχUk(r)1 21 + tanh[ωin(t−tin)] | {z } smooth onset exp−(t−tpk)2 2σ2 t, Ain E>0, Ain G<0, Ain A>0, Ain D<0, (73) and uEntry N≡0(baseline NE globally). For Exit on [tout, tfinal]we drive global NE/DA ramps uExit X(r, t) = Aout X 1 21 + tanh[ωout(t−tout)]1 + εΦsp(r), X ∈ {N,D}, Aout X>0,(74) and normalize the Entry–modulated transmitters inside Ukby a gentle proportional pull unorm X(r, t) = −ρX[cX(r, t)−1 ] χUk(r)1 21 + tanh[ωout(t−tout)], X ∈ {E,G,A}, ρX>0,(75) satisfying (72) . The spatial mode Φ sp is a low–frequency eigenfunction of the Laplace–Beltrami operator to avoid sharp rims that would be penalized in α(Θ, GΘ)and β(B, GB)(Sec. III C). Closed–loop (feedback) augmentation using order parameters. Open–loop shaping is complemented by feedback on the order parameters b φk (module stiffness proxy; Sec. IV A) and b H (holonomy dispersion; Sec. IV B). Let targets be φ?and H?. A minimal static feedback reads ufb N(t) = kNtanhbN[b H(t)−H?], kN>0,(76) ufb X(t) = −kXtanhbX[b φk(t)−φ?]χUk, X ∈ {E,A}, kX>0,(77) added to (74) and (73) , respectively. Eq. (76) increases NE when holonomy dispersion is above target (to tighten global glue), while (77) damps further E/A Entry drive once b φk meets the desired rigidity (preventing overshoot). Saturating tanh enforces bounded control, consistent with actuator constraints in hybrid control [177, 178]. Hybrid automaton: guards and resets. We frame the schedule as a three–mode hybrid system H = {Base,Entry,Exit}with guards: GBase→Entry =nb H>Hhi, B < Blo,∧t≥t0o∪nΓ>Γhi,Πsens <Πlo o,(78) GEntry→Exit =nb φk≥φ?o∪nt−tin ≥Tmax Entry o,(79) GExit→Base =nb H ≤ H?,max X∈{E,G,A}|∆cUk X| ≤ εo∪nt−tout ≥Tmax Exit o.(80) Upon entering Entry we reset the open–loop shapes (73) ; on Exit, we activate (74) – (75) and the feedback (76) . This hybrid formalism ensures reproducible transitions and prevents chattering by hysteresis thresholds (Hhi >H?, Blo < B?)[177]. Mapping to couplings and expected signatures. During Entry, the inequalities (71) raise Θ = cE cG and Π sens while reducing Γ, which—in the coupling maps (49) – (52) —yields α↑ in Uk , modest γ shift, and unchanged β (global NE at baseline), producing b φk↑ and typically b H unchanged or slightly increased. During Exit, B↑ and Γ ↑ globally (with local normalization in Uk ), hence β ( B, GB ) ↑ and γ↑ , yielding b H ↓ and return of b φktoward baseline. These signatures are precisely those plotted in Sec. IV. Bounds, safety, and invariants. We enforce state constraints 0 ≤cX≤cmax (clip with cmax =3) and control bounds |uX| ≤ umax to prevent numerical/biological implausibility; the saturations in (73) – (77) guarantee feasibility. A simple Lyapunov–like certificate for the Exit phase is V(t) = 1 2b H(t)−H?2 | {z } global glue error +X X∈{E,G,A} ρX 2∆cUk X(t)2 | {z } local normalization , whose derivative along the closed–loop Exit dynamics is negative semi–definite for sufficiently large kN, ρX , implying monotone convergence to the guard set GExit→Base under standard regularity assumptions [ 178 ].
32 optimizes the commutation error in (93) ; (P3) ACh boosts in early sensory modules increase the probability of achieving a global section by improving local gauge regularity prior to NE–mediated integration; (P4) perturbational complexity measures (PCI) should correlate inversely with b H across states. These can be tested with multimodal recordings (EEG/MEG/fMRI) augmented by pupil/BSOLD proxies of NE and pharmacological challenges, and compared across wake, sleep, sedation, and anesthesia [199, 206]. Summary of Section 7. We formalized conscious access as a property of the global 2–connection on Mcortex ×I : existence of a weak global section, ε –flat 2–holonomy on experience–relevant surfaces, and approximate commutation with a workspace functor. Biochemically, NE (through β ) reduces 2–holonomy dispersion to enable access, while DA/ACh (through γ ) tune gauge–fixing to balance priors and evidence. The resulting, testable predictions connect higher–gauge geometry to established cognitive neuroscience frameworks for consciousness. VIII. EXPERIMENTAL PROGRAM (AGENT–AGNOSTIC, PI–FILLABLE) Compliance note. All references to investigational agents,routes, and dose brackets are placeholders to be completed solely by a licensed Principal Investigator (PI) under IRB/IACUC oversight. The present section specifies designs, measures, analyses, and order–parameter mappings in an agent–agnostic manner. A. Human study (crossover, placebo–controlled) Design and randomization. A double–blind, randomized, placebo–controlled, within–subject crossover trial with two counterbalanced sequences (AB, BA). Each participant completes two intervention days separated by a washout (duration set by PI based on agent pharmacokinetics), plus a baseline day for calibration and task training. Randomization uses permuted blocks with allocation concealment; code is held by the pharmacy or an independent statistician, with emergency unblinding procedures predefined [ 207 , 208 ]. Pre–registration (analysis plan, stopping rules, primary/secondary endpoints) is required prior to first enrollment (e.g., ClinicalTrials.gov; Registered Reports) [209]. Participants. Healthy adults, 18–40 y, normal or corrected vision/hearing, right–handed (to reduce lateralization variance), no neuropsychiatric history, no centrally acting medications, negative toxicology on visit days, and MRI–compatible. Exclusions: sleep disorders, cardiovascular contraindications, abnormal ECG, history of syncope, and any condition raising risk with the (to–be–specified) agent classes. Female participants: urine pregnancy test per visit. Power analysis targets repeated–measures effects on the order parameters (Sec. IV); for an expected within–subject effect size dz≈ 0 . 45 on b φk and dz≈ 0 . 40 on b H , two–tailed α = 0 . 05,1 −β = 0 . 80, the crossover requires N≈ 34–40 (Faul’s method for paired designs) [210]. We inflate Nby 15% for attrition. Tasks: ENTRY–like vs. EXIT–like. ENTRY–like tasks emphasize sensory precision with minimal priors: (i) visual Gabor orientation discrimination at threshold with unpredictable noise (psychophysical staircase); (ii) auditory pitch discrimination at threshold with roving reference. EXIT–like tasks emphasize integration and context: (iii) Navon global–local figures (conflict manipulation) [ 211 ]; (iv) multisensory integration (e.g., McGurk incongruent audiovisual syllables) [ 212 ]. Each block is followed by metacognitive ratings (confidence, effort). Behavioral proxies: psychometric slope/intercept (ENTRY), global precedence and audiovisual fusion rates (EXIT), plus reaction time variability. Measures and mapping to scalars/couplings. (a) 1H–MRS (3T/7T). Single–voxel PRESS/MEGA–PRESS in module Uk (e.g., A1 or visual area for the target task) and a control ROI. Outputs: Glx and GABA (institutional consensus pipeline; LCModel or equivalent) ⇒E/I ratio Θ = Glx/GABA [213]. (b) fMRI (3T/7T). Multi–band EPI (TR ≤ 1 s), task and resting blocks. Effective connectivity estimated by either state–space models or DCM on a constrained ROI set {Ui} ; derive edge transports Rij and compute: (i) graph curvature within Uk (Forman/Ollivier proxies) ⇒b φk ; (ii) loop transports on a minimum cycle basis ⇒b H(Sec. IV) [214, 215]. (c) MEG (optional). Source–localized band–limited power and phase–coupling (theta–gamma). Repeat (a)(ii) on frequency–specific effective networks; entry effects expected in early sensory bands; exit effects in beta/gamma integration.
33 (d) Pupil and LC MRI. (i) Eye tracking during tasks; pupil diameter/derivative series as arousal/ B proxy [ 216 ]. (ii) Neuromelanin–sensitive LC MRI for structural individual differences and fMRI LC seed time series during tasks [217, 218]. (e) Behavior. ENTRY psychometric slopes (precision); EXIT global precedence and audiovisual fusion rates (integration); confidence calibration for metacognition. Derived control scalars and couplings per Secs. III B–III C are computed per block: Θfrom MRS; B from pupil/LC time series; Γfrom DCM priors/precision parameters and behavioral Bayesian fits; Π sens from threshold modulation and early cortical gain metrics; then α, β, γ, η via Eqs. (49)–(52). Intervention placeholders (PI–fillable). Investigational agent(s): [generic names and class to be specified by PI]. Route/dose/titration/monitoring: [to be specified by PI under IRB/DSMB oversight]. Washout: [agent–specific, PI to define based on PK/PD models]. Stopping rules: prespecified safety triggers (vital signs, adverse events, cognitive side effects); DSMB review at interim milestones [207]. Preprocessing, analysis, and statistics. MRS: frequency/phase correction, water scaling, frequency drift correction; quality metrics (CRLB < 20%). fMRI: motion/susceptibility correction, ICA–AROMA or equivalent, physiological confounds, task GLMs, ROI extraction, DCM model space selection with Bayesian model comparison [ 214 ]. MEG: tSSS/SSS, beamforming, leakage–reduction orthogonalization. Pupil: blink interpolation, z–scoring. Effective network construction ⇒Rij ; cycle basis (Horton) [ 215 ]; compute b H and curvature b φk . Primary analyses: linear mixed models with subject random intercepts/slopes (sequence and period as covariates) [ 220 ]; permutation tests for network metrics [ 221 ]; FDR control [ 222 ]. Schedule and logistics (Table III). See Table III for visit structure, scans, tasks, safety checks, and endpoints. Data are organized in BIDS and publicly shared after de–identification and embargo [219]. B. Pre–clinical study (rodent, agent–agnostic) Design. Head–fixed mice performing analogs of ENTRY/EXIT tasks (e.g., orientation discrimination at threshold; multisensory integration). LC state is controlled by optogenetics/chemogenetics (PI–specified parameters); cortical E/I is modulated via cell–type–specific opsins or DREADDs in Uk (e.g., primary visual/auditory cortex) [223, 224]. Readouts. Two–photon calcium imaging of layer–specific populations in Uk ; widefield calcium or laminar LFP for mesoscale coupling; rodent MRS (if available) for Glu/GABA; pupil as arousal proxy. From imaging/electrophysiology, construct effective networks across parcels and compute b φk and b H as in Sec. IV. Histology quantifies receptor expression (e.g., D1/D2, nAChR subtypes) and opsin/DREADD expression [225]. Manipulation schema. ENTRY–like: increase cholinergic tone locally (basal forebrain drive), tilt E/I in Uk , transiently reduce DA (e.g., VTA modulation), LC baseline unchanged. EXIT–like: restore E/I in Uk and drive LC/VTA globally to increase B and Γ(parameters, patterns, and safety per PI). Outcomes: b φk↑during ENTRY; b H ↓ during EXIT; behavioral correlates (d0in discrimination; integration indices). C. Primary/secondary endpoints & hypotheses (pre–registered) Primary endpoints. (P1) ENTRY vs placebo: increase in module stiffness proxy b φk within Uk during ENTRY blocks; b Hstable or slightly elevated (detail binding without global glue). (P2) EXIT vs ENTRY: global reduction in holonomy dispersion b H during EXIT blocks, with normalization (reduction) of b φkin Uktoward baseline. Secondary endpoints. Behavioral improvements in precision (ENTRY) and integration (EXIT); changes in DCM prior/evidence precision parameters (Γ , Π); correlations between pupil/LC measures ( B ) and b H ; metacognitive calibration changes. Safety/tolerability profiles per session. Hypotheses and analysis. H1: ENTRY increases Θ(MRS), raises α (Θ), and increases b φk in Uk .H2: EXIT increases B (pupil/LC), raises β ( B ), and reduces b H .H3: b H correlates inversely with perturbational complexity/ignition markers (if collected). Testing uses mixed–effects models (subject random effects; sequence/period covariates), permutation inference for network metrics, and FDR control.
34 Table III: Visit schedule, scans, tasks, safety checks, and endpoints (human crossover). All agent specifics remain PI-fillable. Timepoint Safety / Screening Imaging / Physiology / Tasks Endpoints (primary / secondary) Screening (−14 to −1d) Medical exam, ECG, labs, MRI safety checklist; informed consent; task practice — Eligibility confirmed; baseline questionnaires; inclusion/exclusion documented Baseline (Day 0) Vitals; urine test (as applicable) 1H–MRS in Ukand a control ROI (Glx, GABA); resting fMRI (and MEG optional); LC neuromelanin MRI; pupil calibration; task familiarization Baseline Θ = Glx/GABA; baseline b φkand b H; calibration of Bproxy (pupil/LC) Session A (Day 1) Vitals; pre-dose assessments; AE/SAE monitoring Task fMRI; 1H–MRS; pupil tracking; (MEG optional). ENTRY-like and EXIT-like blocks (counterbalanced within session) Primary: ∆ b φk (ENTRY) and ∆b H(EXIT) vs placebo. Secondary: behavior (psychometric slope; global precedence; AV fusion); DCM priors/precision (Γ,Π) Washout (Days 2to w−1) Remote AE check; washout compliance — Safety; carryover assessment (period and sequence effects planned in model) Session B (Day 1+w) Vitals; pre-dose assessments; AE/SAE monitoring Repeat Session A with crossover sequence (AB/BA) Same endpoints as Session A; period/sequence modeled (mixed-effects) Follow-up (+7 d) AE/SAE monitoring; end-of-study labs (if indicated) — Safety endpoint closure; data lock and preregistered analyses commence Data standards and transparency. All code, preregistration, and de–identified data (BIDS) shared on public repositories (OSF/Zenodo) post–embargo. Deviations from protocol and interim DSMB reports archived. IX. DATA PROCESSING AND METRIC COMPUTATION A. MRS →Θ(Glu/GABA or Glx/GABA); QC and partial–volume corrections Acquisition and QC. Single–voxel 1 H–MRS is acquired in the target module Uk (e.g., A1 or V1) and a control ROI using PRESS (for Glx) and MEGA–PRESS (for GABA + ). Quality criteria follow community consensus: full–width at half–maximum (FWHM) of the unsuppressed water peak ≤ 12–14 Hz at 3T; SNR ≥ 20; Cramér–Rao lower bounds (CRLB) ≤ 20% for Glx and GABA + ; visual residual inspection; robust frequency/phase correction [ 226 ]. Spectra are water–scaled to institutional units with vendor–agnostic pipelines and exported along with QC metadata (FWHM, SNR, CRLB). Segmentation and partial–volume correction. Each MRS voxel is segmented (T 1 –weighted anatomical) into tissue fractions fGM, fWM, fCSF ( fGM + fWM + fCSF = 1). The tissue–corrected metabolite concentration b C(mM) is obtained from the measured water–scaled concentration Cmeas via [227]: b C=Cmeas ·(1 −fCSF) fGM ξGM +fWM ξWM , ξt=ptRt 1,H2ORt 2,H2O Rt 1,Met Rt 2,Met ,(94)
35 where pt is the tissue water content fraction and R1, R2 are relaxation attenuation terms (functions of TR/TE for water and metabolite in tissue class t ). This corrects for CSF dilution and differential relaxation. We report Θ = b CGlx/b CGABA+ (or Θ = b CGlu/b CGABA+ if glutamate is separately quantifiable). All results are accompanied by QC covariates (FWHM, SNR, CRLB, fCSF) in downstream models. B. fMRI/MEG preprocessing; parcellation; effective connectivity fMRI preprocessing. Preprocessing uses a robust, containerized workflow: anatomical reconstruction, susceptibility–distortion correction, motion correction, slice–timing correction, co–registration to T 1 , and normalization to MNI. Confound modeling includes motion (24–parameter), framewise displacement (FD) scrubbing ( > 0.5 mm), aCompCor (top PCs from WM/CSF), and slow drifts. We use ICA–AROMA as an orthogonal, automated motion–artifact classifier [ 228 , 229 ]. Band–pass filtering (0.008–0.09 Hz) is applied after confound regression. Head–motion impact is quantified (mean FD, DVARS) and included as a covariate [230–232]. MEG preprocessing and source modeling. Raw MEG is cleaned with SSS/tSSS where available, ECG/EOG artifact removal (ICA), co–registered to the individual MRI, and reconstructed to source space with either minimum–norm estimation or LCMV beamforming. Source time series are parcellated as below and spectrally summarized in canonical bands (theta–gamma) [237, 238]. Parcellation and time–series extraction. We extract parcelwise time series using a multimodal atlas (e.g., HCP MMP1.0, 360 cortical areas) with subcortical additions. Parcel time series are the first eigenvariate of voxels within atlas ROIs after confound removal; temporal derivatives are appended for effective connectivity estimation [233]. Effective connectivity and transports. Two complementary routes are implemented: (a) DCM (task/spectral). For task blocks we use bilinear DCM; for rest and quasi–stationary blocks we use spectral DCM. The posterior mean effective connectivity matrices (A–matrices) are extracted for the ROI set {Ui}, with model evidence used for selection/averaging [235]. (b) Lagged partial correlations (LPC). A regularized VAR( p ) is fit to parcel time series; the lag–1 partial correlation matrix is extracted and symmetrized to provide a computationally efficient proxy of directed influences [234, 236]. From either representation we derive edge transports Rij ∈SO ( d )(small–angle orthogonal alignments) by polar decomposition of local linear maps on low–dimensional parcel embeddings (top d principal components). These Rij feed curvature and holonomy metrics below. C. Module stiffness φk: Forman/Ollivier curvature Graph construction and sparsification. We build a weighted, undirected ROI graph G = ( V, E ) with weights wij from the magnitude of effective connectivity (Fisher– z –transformed, then symmetrized). To ensure comparability across subjects, we adopt proportional thresholding at densities ρ∈ {5%,10%,15%,20%}and, in robustness analyses, MST–backbone plus top–kedges. Forman curvature. For an edge e = ( i, j )with weight we and incident node weights wi, wj (e.g., strength), the weighted Forman–Ricci curvature is [239]: F(e) = we wi we +wj we−X e∼i, e6=(i,j) wi √wewe−X e∼j, e6=(i,j) wj √wewe ,(95) which reduces to F ( e )=4 −d ( i ) −d ( j )for unweighted graphs. F captures local tightness of flow: larger values ⇒stiffer local transport. Ollivier–Ricci curvature. For adjacent nodes i, j , let µi, µj be one–step random–walk measures on neighbors with mass proportional to w . The Ollivier curvature is κO ( i, j ) = 1 −W1(µi,µj) d(i,j) , where W1 is the earth–mover distance and dthe graph distance [240, 241]. Larger κOimplies locally cohesive transport. ROI–level stiffness summary. Within G [ Uk ](the induced subgraph on module Uk ), we define the estimator b φk= medianκ(i, j) : (i, j)∈E∩Uk,
36 with κ∈ {F, κO} . We report medians and interquartile ranges, along with density–matched nulls and multi–threshold stability curves. D. Holonomy dispersion H Cycle basis and loop transports. On each thresholded graph, construct a minimum cycle basis (as in Sec. IV B, algorithms referenced there). For a cycle γ = ( i1→i2→···→i`→i1 ), define the loop transport from edge transports Rikik+1 ∈SO(d): U(γ) = Ri1i2Ri2i3···Ri`i1∈SO(d), δ(γ) = log U(γ) 2 F.(96) Small δ(γ)indicates consistent multi–route alignment (low holonomy). Dispersion statistic. The holonomy dispersion is the variance of loop deviations across the cycle basis: b H= Varγδ(γ).(97) We summarize b H by (i) its value at a reference density (e.g., 10%), (ii) the area under the multi–threshold curve, and (iii) a connection–Laplacian spectral gap proxy (large gap ⇒ low dispersion). These map to integration coupling β(Sec. IV B). E. Variable Symmetry Index (VSI) Stimulus micro–transformations and invariance. Construct a finite set T of micro–transformations preserving task–level identity but perturbing low–level features: e.g., (i) auditory: transpositions by small integer ratios, micro–tempo changes; (ii) visual: rotations ± 5 ◦ , scalings ± 5%, minor elastic warps. Let RUk(g)be the multivariate ROI response pattern in Ukunder g∈ T . Define the VSI: VSI(Uk;τ) = 1 |T| X g∈T 1hRUk(g), RUk(id)i kRUk(g)kkRUk(id)k≥τ,(98) with threshold τ∈ (0 , 1). VSI operationalizes variable symmetry: higher values indicate broader effective invariances within Uk. We compute VSI per block and relate it to b φkand b H. F. Optional topological data analysis (TDA) Persistent homology of functional graphs. To track the emergence (“crystallization”) and collapse of loops across ENTRY/EXIT we compute 1–dimensional persistent homology on a filtration of thresholded functional graphs. For thresholds θ scanning edge weights, we compute β1 ( θ )(number of independent cycles) and barcodes; long–lived H1bars during ENTRY indicate stabilized local loops; EXIT collapses bars as integration increases. Summaries include total persistence and persistence landscapes, correlated with b φkand b H[242, 243]. G. Reporting, robustness, and parameter grid Robustness protocol. All metrics are recomputed across: graph densities ρ∈ { 5 , 10 , 15 , 20 } %; cycle– basis choices; parcellations (HCP–MMP, alternative fine–/coarse–grained); and transport dimensionalities d∈ { 2 , 3 , 5 } . We report medians, IQRs, and multi–threshold AUCs; inference uses subject–level paired contrasts with permutation tests (10k shuffles) and FDR across metric families. Table summary and analysis Table IV (parameter grid) summarize the analysis. X. STATISTICAL ANALYSIS AND POWER A. Primary analyses Overview and families of hypotheses. Analyses target two preregistered families of endpoints (Sec. IV): (F1) module stiffness b φk within Uk during ENTRY vs. placebo; (F2) holonomy dispersion b H globally
37 Table IV: Parameter grid for robustness analyses. Each analysis is recomputed across all combinations. Dimension Levels / settings Parcellation HCP–MMP (360); alternative coarse/fine atlases (PI–specified) Graph density ρ5%, 10%, 15%, 20%; MST–backbone + top–kedges Effective connectivity DCM (task; spectral for rest); LPC from VAR(p) with AIC–selected p Transport dimension d2, 3, 5 (top principal components per ROI) Cycle basis Minimum cycle basis; alternative heuristic bases for sensitivity (same cardinality) Curvature Forman F(e); Ollivier κO(i, j) Holonomy dispersion Variance of klog U ( γ ) k2 F ; multi–threshold AUC; connection–Laplacian spectral gap VSI threshold τ0.6, 0.7, 0.8 (sensitivity) Confounds With/without ICA–AROMA; aCompCor PCs ∈ { 3 , 5 } ; scrubbing thresholds FD ∈ {0.3,0.5}mm during EXIT vs. ENTRY. Unless stated otherwise, inference uses linear mixed models (LMMs) with restricted maximum likelihood (REML), cluster-robust small-sample degrees of freedom, and hierarchical false-discovery rate (FDR) control across endpoint families and robustness strata (densities, parcellations). Mixed models naturally accommodate the within-subject crossover and intermittent missingness under MAR [244–246]. Outcome preparation. We predefine transformations for scale regularity: Yφ= z(b φk)and YH= z log(1 + b H), where z ( · )denotes subject-wise standardization for effect-size interpretability. Sensitivity analyses repeat with rank-based inverse-normal transforms. Fixed and random effects. The within-subject factor Condition has three levels {Placebo,Entry,Exit} with Placebo as reference. Period (first/second session), Sequence (AB/BA), and their interaction are included to assess carryover. A parsimonious random-effects structure is used: subject-specific intercepts and random slopes for Entry and Exit (maximal justified by identifiability), with variance-covariance unstructured [246, 255]. Motion and MRS-QC covariates are included as nuisance regressors. Model for module stiffness. Yφ, is =β0+β11Entryis +β21Exitis +β3Periodis +β4Sequencei +β5FDis +β6CRLBis +u0i+u1i1Entryis +u2i1Exitis +εis . where i indexes subjects and s sessions; FD is mean framewise displacement (fMRI) and CRLB summarizes MRS reliability. Primary contrast: β1>0(Entry >Placebo) for Yφ. Model for holonomy dispersion. YH,is =γ0+γ11Entryis +γ21Exitis +γ3Periodis +γ4Sequencei+γ5FDis +u0i+u1i1Entryis +u2i1Exitis +ηis. Primary contrast: ( γ2−γ1 ) < 0(Exit < Entry) for YH . For robustness, we augment with a fixed factor Density∈ { 5 , 10 , 15 , 20 } %and a subject-level random intercept by density; inference is then on the marginal condition effects averaged over densities. Estimation, inference, and multiplicity. Models are fit by REML; fixed-effect tests use Kenward– Roger or Satterthwaite approximations for degrees of freedom [ 247 , 248 ]. Residual diagnostics (Q–Q, scale–location) and influence measures guide robust-sandwich sensitivity checks. Multiple testing within each family (e.g., b φk variants with Forman/Ollivier curvature) is FDR-controlled at q = 0 . 05; when dependencies are strong (e.g., across densities), we use the Benjamini–Yekutieli procedure as a conservative bound [ 249 ]. Across families we apply hierarchical FDR to maintain interpretability of the primary endpoints [ 250 ]. For network-derived statistics we complement parametric tests with permutation inference at the subject level (10k label shuffles) and report exact p-values [252]. Effect sizes and uncertainty. We report standardized fixed effects ( ˆ β1 and ˆγ2−ˆγ1 ) with 95% CIs, withinsubject Cohen’s dz for planned contrasts (ENTRY vs Placebo; EXIT vs ENTRY), and repeated-measures standardized mean change with appropriate small-sample correction [253, 254]. Missing data and outliers. Under MAR, REML provides unbiased fixed-effect estimates; as sensitivity, we fit pattern-mixture models separating missingness patterns. Outliers are flagged by studentized residuals |t|>3and high leverage; robust fits with Huber loss are reported in supplement.
38 B. Mediation Causal chain: biochemical → coupling → endpoints. We test preregistered mediation chains withinsubject and across sessions: b Θa −→ αb −→ b φk,b Ba0 −→ βb0 −→ YH. Here b Θ is tissue-corrected MRS E/I ratio (Sec. IX A); b B is an arousal/integration proxy (pupil/LC); α, β are couplings estimated from the biochemical scalars (Sec. III C). Models and estimands. We use multilevel mediation with subject random intercepts: Mediator: Mis =a0+a1Xis +c>Zis +u(M) i+ε(M) is , Outcome: Yis =c0+c0Xis +b Mis +d>Zis +u(Y) i+ε(Y) is , where X∈ {b Θ,b B} , M∈ {α, β} , Y∈ {Yφ, YH} , and Z are covariates (Period, Sequence, FD, MRS-QC as applicable). The average causal mediation effect (ACME) is ACME = a1b , the average direct effect (ADE) is c0 , with percentile bootstrap CIs (5,000 subject-level cluster resamples) [ 258 , 262 – 264 ]. We report sensitivity to violations of sequential ignorability via the correlation ρ between mediator and outcome errors [ 259 ]. For completeness, we replicate with a Bayesian multilevel mediation and weakly informative priors (reported as posterior intervals). C. Power Analytic approximation for crossover within-subject contrasts. For a paired contrast with standardized effect δ = ∆ /σ and within-subject correlation ρ , the per-contrast sample size for two-sided α and power 1−βis n≈z1−α/2+z1−β22(1 −ρ) δ2.(99) Using α= 0.05,1−β= 0.80, and ρ∈[0.5,0.7] yields: δ= 0.30 ⇒n≈60−80, δ = 0.40 ⇒n≈30−40, δ = 0.50 ⇒n≈22−30. These bands support the target N≈ 30–40 (Sec. VIII A) for ∆ /σ ∼ 0 . 4. Carryover is addressed by including Period and Sequence ; if material carryover is detected, the second period is analyzed descriptively only, per crossover practice [256, 257]. Simulation-based power for the LMM. Because variance components and random slopes affect power, we perform model-based power via parametric simulation from the fitted LMM (pilot variance/correlation), refitting per simulation and estimating rejection rates for preregistered contrasts. We vary δ∈ { 0 . 3 , 0 . 4 , 0 . 5 } and ρ∈ { 0 . 5 , 0 . 6 , 0 . 7 } ;1 , 000 simulations per cell give Monte Carlo SE < 1 . 5%. The workflow follows standard practice for mixed-model power [266]. Multiple testing and family-wise error control. We predefine families (F1, F2) and control the FDR within each; for exploratory secondary endpoints we use resampling-based adjustments within families (Westfall–Young) and present adjusted p -values [ 251 ]. This strategy preserves power for the primary contrasts while properly controlling error rates under dependence [249, 250]. Deliverables. Table V details model specifications. XI. NUMERICAL PREDICTIONS & PHASE DIAGRAMS A. Dimensionless predictions Time courses of couplings and order parameters. Given the normalized transmitter fields cX ( r, t ) (Sec. V A) and the coupling maps (Eqs. (49)–(52)), the condition–dependent couplings evolve as α(t) = hα(Θ(·, t), GΘ(·, t))iUk, β(t) = hβ(B(·, t), GB(·, t))iM, γ(t) = hγ(Γ(·, t),Πsens(·, t))iUk,(100)
39 Table V: Model formulas, fixed/random effects, covariates, and planned contrasts. Outcome Fixed effects Random effects Planned contrasts Yφ= z(b φk)Intercept; Condition (Entry, Exit); Period;Sequence;FD; CRLB Subject intercept; subject slopes for Entry,Exit (unstructured) ENTRY >Placebo (β1>0); Exit vs Entry YH = z log (1+ b H ) Intercept; Condition; Period;Sequence;FD Subject intercept; subject slopes for Entry,Exit Exit <Entry (γ2−γ1<0) M∈ {α, β} (Mediator) Intercept; X∈ {b Θ,b B}; covariates Z Subject intercept Test a1 Y∈ {Yφ, YH} (Outcome) Intercept; X;M;ZSubject intercept ACME =a1b; ADE =c0 where h·iUk and h·iM denote spatial means over the module Uk and whole manifold, respectively. The order parameters (Sec. IV) then read φk(t) = α(Θ, GΘ)kF(·, t)k2Uk,H(t) = Varγ∈Bklog U(γ, t)k2 F,(101) with Ba minimum cycle basis and U(γ, t)the discrete loop transport (Eq. (96)). Under the ENTRY schedule (Sec. VB): cE↑ , cG↓ , cA↑ , cD↓ within Uk⇒ Θ ↑ ,Π sens ↑ ,Γ ↓ ⇒ α ( t ) ↑ and φk ( t ) ↑ with β ( t )approximately unchanged. Under EXIT: cN↑ (globally), cD↑ (globally), and normalization in Uk⇒β(t)↑and H(t)↓, while α(t)reverts toward baseline. Sensitivity maps. Local linear sensitivity of the module stiffness to transmitter fields is obtained by the chain rule: ∂φk ∂cX =D∂α ∂Θ ∂Θ ∂cX +∂α ∂GΘ ∂GΘ ∂cX | {z } biochemical→coupling kFk2+α∂kFk2 ∂cX | {z } geometry EUk , X ∈ {E,G,A,D,N,S}.(102) The first bracket captures how cX shapes α via Θand its gradients (Eqs. (49) and (61) ); the second bracket is the indirect effect mediated by changes in the connection dynamics and curvature energy. In practice we estimate (102) by adjoint/finite difference on the simulator, reporting maps of ∂φk/∂cX across Ukand their spatial quantiles. For holonomy dispersion, using the empirical relation H ≈ cH/hβiM (Sec. IV B), the leading sensitivity to noradrenergic state is ∂H ∂cN≈ − cH hβi2 M∂β ∂B ∂B ∂cNM−cH hβi2 M∂β ∂GB ∂GB ∂cNM ,(103) where ∂B/∂cN = w(+) N− 2 w(−) NcN under the polynomial scalar (Sec. III B), and the gradient term reflects heterogeneity penalties (Eq. (50) ). Equation (103) predicts ∂H/∂cN< 0around EXIT operating points (increasing integration reduces dispersion). Phase–trace diagnostics. We visualize ENTRY → EXIT trajectories as phase traces (Θ( t ) , Γ( t ) , Π sens ( t ) , B ( t )) with overlaid ( φk ( t ) ,H ( t )). Signature: during ENTRY, Θ ↑ and Π sens ↑ while Γ ↓ , pushing the system into a high– φk , neutral/slightly elevated H corner; during EXIT, B↑ and Γ ↑ , returning φk to baseline and Hto a lower, integrated regime. B. Dimensionful predictions (worked numbers) ENTRY inside Uk .Using the dimensionful calibration (Sec. VI C): CE : 0 . 50 → 0 . 75 µ M, CG : 0 . 50 → 0.40 µM, CA: 0.05→0.15 µM, CD: 0.03→0.01 µM, CNbaseline. Then Θ=1.875 and, with Eq. (84), αdim ≈1.19 (rim penalty .0.02 for w&1.6 mm), while γ sits in the mid–low regime ( ≈ 0 . 50 by our calibration), reflecting relaxed priors and heightened sensory precision.
40 Table VI: Prediction summary linking dimensionless couplings/order parameters to dimensionful transmitter changes (illustrative calibration). Regime Dimensionful shifts (µM) Predicted couplings & order parameters Baseline CE = CG = 0 . 50; CA = 0.05; CD= 0.03; CN= 0.02 Θ=1 . 0; α≈ 0 . 73; β≈ 0 . 99; γ mid; φk baseline; H baseline ENTRY (Uk)CE : 0 . 50 → 0 . 75; CG : 0 . 50 → 0 . 40; CA : 0 . 05 → 0 . 15; CD : 0 . 03 → 0 . 01; CN: = Θ = 1 . 875; α≈ 1 . 19; γ≈ 0 . 50; β unchanged ⇒ φk↑;Hstable or ↑slightly EXIT (global) CN : 0 . 02 → 0 . 07; CD : 0 . 03 → 0 . 07; (CE, CG, CA→baseline in Uk) β≈ 1 . 29; α→ 0 . 73; γ↑ mildly ⇒ H ↓ globally; φk→baseline EXIT (global, with local normalization). CN : 0 . 02 → 0 . 07 µ M, CD : 0 . 03 → 0 . 07 µ M; CE, CG, CA normalize in Uk. With Eq. (86) , βdim ≈ 1 . 29 and αdim → 0 . 73 (baseline). By (103) , H drops in proportion to ∂β/∂CN (Sec. VI F). Dimensionless ↔ dimensionful mapping. Dimensionless trajectories ( α ( t ) , β ( t ) , γ ( t )) and dimensionful ( αdim, βdim )are consistent under the scaling in Sec. V A with T0 = 1 s, L0 = 5 mm and the calibration in Sec. VI C–VI E. We provide a consolidated summary in Table VI. Here L0 (module-scale radius used for nondimensionalization) is distinct from the diffusive half-width Lthat enters τdiff ; we set L=L∗= 0.30 mm for time-scale calculations. C. Gradient constraints & timescales Rim width constraints. To keep gradient penalties small during ENTRY, the characteristic rim width should satisfy w& 1 . 6mm (Sec. VI D); for surface simulations, this corresponds to ∼ 3–5edge–lengths on a ∼ 3mm average mesh spacing, avoiding numerically stiff rims and ensuring that αdim gtanh ( bdim gGΘ ) . 0 . 1. Transport equilibration times. For mesoscale transitions, the diffusive half-width that controls approach to steady state is L∼ 0 . 30 −− 0 . 50 mm . With DX = 0 . 5 µm2/ms , τdiff = L2/ ( π2DX ) ∼ 6 −− 17 s ; including clearance τX∈ [0 . 5 , 5] s gives τeff ≤τdiff , consistent with the 5 − − 20 s ENTRY/EXIT windows. For comparison, if L= 1 mm then τdiff ∼200 s– far outside the task timing used here. Phase–diagram constraints. Phase regions are computed on the 4D control space (Θ , Γ , Π sens, B )by thresholding order parameters: Savant region: φk≥φth,H ≥ Hth;Integrated region: φk≤φth,H ≤ Hth,(104) with thresholds chosen as fixed quantiles of baseline distributions (e.g. 75th for φk , 75th for H ). Under ENTRY, trajectories cross into the savant corner (high φk , high H ); under EXIT, trajectories move into the integrated corner (low H ; φk baseline). The boundaries are computed by continuation on a low–dimensional slice (e.g. (Θ, B)with Γ,Πsens fixed) using pseudo–arclength methods [267, 268]. Table summary for dimensionless and dimensionful predictions Table VI summarizes the dimensionless ↔dimensionful predictions used in Secs. XI A–XI B. XII. CONSCIOUSNESS AND GLOBAL COHERENCE (TESTABLE CRITERIA) A. Operational criteria as ε–flatness and workspace–commutativity of Aglobal Recap. Section VII defined conscious access in terms of the global 2–connection Aglobal = ( A, B )on the cognitive 2–bundle over Mcortex ×I , via (i) existence of a weak global section, (ii) ε –flat 2–holonomy on experience–relevant spacetime surfaces (Eq. (92) ), and (iii) approximate commutation with the workspace functor (Eq. (93) ). We now turn these into testable criteria, with explicit inequalities, estimators, and state classifications.
41 Unity ⇒ global section on Mcortex ×I . Unity requires a (weak) global gauge in which both the fake curvature FA and the higher curvature H are small on experience–relevant charts. Let k·k∗ be a consistent operator norm on forms. Over a time window [t, t + ∆], Uni(t; ∆) := maxn FA ∗,[t,t+∆], H ∗,[t,t+∆] o≤εuni,(105) for a tolerance εuni chosen from baseline quantiles in alert wakefulness (Sec. XI A). Operationally, (105) implies small loop and surface inconsistencies once the gauge is fixed by the precision map γ (Γ , Π) (Sec. III C). Integration ⇒ low holonomy dispersion. Integration requires multi–route agreement of transports, quantified by the holonomy dispersion b H(Eqs. (96)–(97)). We set Int(t; ∆) := b H[t,t+∆] ≤εint, εint := (upper) 25th percentile of b Hin alert wakefulness,(106) reflecting that integrated conscious states exhibit low dispersion relative to drowsiness/deep NREM (Sec. XII B). Continuity ⇒ temporal ε –flatness. Continuity demands that the above properties persist on sliding windows. For a window stride δ∆, Cont(t; ∆, δ) := max τ∈[t,t+∆−δ]sup Σ∈S(τ,τ+δ) log Hol(A,B)(Σ) ≤εcont,(107) so that no brief breakdown of 2–flatness interrupts the episode of access. This captures the phenomenological stream of consciousness as a temporally extended ε–flat regime. Access ⇒ workspace–commutativity. Access requires that broadcasting is consistent with underlying transports (Eq. (93)). We define a commutator error C(t; ∆) := D WU(γ)·ρ−UW(γ)·W(ρ) Eγ∈B, ρ∈R,[t,t+∆] ,(108) averaging over a cycle basis B and a content set R (task tokens). Access holds when C ≤ εacc , i.e., when workspace tokens track transports up to tolerance. Composite conscious access index. We aggregate the four criteria into an index CAI(t; ∆) = wuni [εuni −Uni]++wint [εint −Int]++wcont [εcont −Cont]++wacc [εacc −C]+,(109) with nonnegative weights w· and [ x ] + = max ( x, 0). CAI > 0labels intervals meeting all tolerances; the magnitude gauges headroom. Thresholds and weights are set from baseline wakefulness and validated by cross–state separability (Sec. XII B). B. Empirical tests Arousal ladder: wakefulness ↔ sleep. We test whether b H (low ⇒ integration) tracks arousal: alert wake < quiet wake < N1 < N2 < N3 (deep NREM), with REM resembling wake in selected frequency bands. Prior work shows large–scale connectivity reductions and altered coordination during NREM [ 271 , 272 ]. Our prediction is a monotone increase of dispersion (Eq. (97) ) with NREM depth, and the reemergence of low dispersion during REM for task-relevant networks (consistent with dream mentation). The continuity criterion (107) should fragment across transitions into N2/N3. Relationship to LC proxies (pupil, LC–BOLD). Under the NE–integration mapping (Sec. VII C), we predict an inverse relation between b H and arousal proxies: larger pupil diameter/derivative (phasic LC drive) and stronger LC–BOLD signal should correspond to lower dispersion (greater integration) during access episodes and on slow fluctuations [ 273 , 274 ]. Cross-correlation analyses with appropriate prewhitening can disambiguate fast phasic vs slow tonic contributions. Task-induced variations (ENTRY → ACCESS → EXIT). During ENTRY–like blocks (heightened sensory precision, relaxed priors) we expect φk↑ in Uk with stable/slightly elevated dispersion; during EXIT–like blocks, pupil/LC increases and b H ↓ while φk normalizes, meeting the Access criteria (105) – (108) on windows aligned to behavioral “ignition” (confidence, P3–like signatures if available). These patterns should produce phase–trace loops in (Θ,Γ,Πsens, B)with order–parameter overlays (Sec. XI A).
48 dispersion H ) follows when integration coupling β increases under noradrenergic drive (EXIT). The variable symmetry tower and 2–holonomy furnish a principled separation between local feature–locking and global glue, aligning with modern network neuroscience in which mesoscale topology and dynamics jointly shape cognition [330–332]. What is new. (i) Geometry ←biochemistry : we give a mesoscopic reaction–diffusion–clearance model in physical units (Sec. VI) and a dimensionless simulator (Sec. XI) that map measured transmitter surrogates (MRS Glx/GABA for Θ, pupil/LC proxies for B ) into the coupling quadruple ( α, β, γ, η )and thereby into order parameters ( φk,H ). (ii) Operational criteria : we formalize conscious access as ε –flatness of 2–holonomy that approximately commutes with a workspace functor (Sec. XII), yielding quantitative, falsifiable indices. (iii) ENTRY→EXIT control : we derive ENTRY/EXIT schedules (Sec. V B) that are agent–agnostic yet sufficient to drive transitions in (Θ , Γ , Π sens, B )space with characteristic phase–traces and separable endpoints. (iv) Metrics ready for data : we specify curvature/holonomy computation on effective networks and a Variable Symmetry Index (VSI), embedded in BIDS pipelines with robustness grids and preregistered statistics (Secs. IX–X). Empirical and computational convergence. The framework links three scales: (a) molecular/neuromodulatory control that retunes circuit gains and time constants; (b) network transports shaping local locking and global consistency; and (c) algorithmic accessibility as gauge–coherent broadcast. This cross–scale view is consistent with evidence that neuromodulation reconfigures circuit dynamics and oscillatory coordination across behaviors [ 333 , 334 ], and with observations that structural constraints shape functional flows and metastable integration [ 331 , 332 , 335 ]. Our results thereby recast savant–like phenomena as lawful excursions in a low–dimensional control landscape rather than as anomalies. Testable predictions with safety by design. The theory yields clear, measurable predictions: (P1) ENTRY increases b Θ (MRS) and raises b φk in Uk ; (P2) EXIT increases arousal proxies B and lowers b H globally; (P3) the temporal order ENTRY → EXIT is reflected in phase–trace loops; (P4) meeting ε –flatness and commutativity thresholds predicts access–level behavior. A safe empirical path is delineated: pilot computation on open datasets (no pharmacology) to validate metrics and priors; then a placebo–controlled crossover with all agent identity, route, dose ranges, titration, and rescue protocols left entirely to licensed PIs and institutional pharmacies under IRB/IACUC oversight (Sec. XV). This progression balances ambition with rigor and safeguards. Limitations and discriminants. We explicitly noted discretization error for ( A, F, H )estimates, MRS spatial/spectral limitations, effective–connectivity identifiability, and cohort heterogeneity (Sec. XIV). Each becomes a discriminant: failure of pre-registered biochemical → coupling → endpoint links or of ENTRY/EXIT ordering constitutes a falsifier requiring revision of (i) coupling nonlinearities, (ii) missing control scalars, or (iii) the hypothesis itself. Such sharp failure modes are essential for cumulative progress and for adjudicating against alternative accounts (difficulty, generic arousal). Outlook. By treating cognition as a higher–gauge process on a structured network, with neuromodulators as control fields, we bridge molecular action, network geometry, and conscious access. The synthesis complements and extends existing network and dynamical systems perspectives [ 330 , 331 , 335 ], while furnishing implementable metrics and cautious experimental routes. Whether savant–like phases can be reliably induced and safely exited in humans remains an empirical question—one our framework turns into a set of tractable measurements and registered tests. Regardless of outcome, the approach offers a reusable toolkit for linking biochemical intervention to geometric control of large–scale brain dynamics. [1] Treffert, D. A. (2009). Savant syndrome: an extraordinary condition. A synopsis: past, present, future. Philosophical Transactions of the Royal Society B, 364(1522), 1351–1357. doi:10.1098/rstb.2008.0326. [2] Treffert, D. A. (2021). The Sudden Savant: A New Form of Extraordinary Abilities. WMJ, 120(1), 43–47. [3] Boddaert, N., et al. (2005). Autism: functional brain mapping of exceptional calendar capacity. The British Journal of Psychiatry, 187(1), 83–86. doi:10.1192/bjp.187.1.83. [4] Cowan, R., & Frith, C. D. (2009). Do calendrical savants use calculation to answer date questions? A functional magnetic resonance imaging study. Philosophical Transactions of the Royal Society B, 364(1522), 1385–1394. doi:10.1098/rstb.2008.0290. [5] Samson, F., Mottron, L., Soulières, I., & Zeffiro, T. A. (2012). Enhanced visual functioning in autism: An ALE meta–analysis. Human Brain Mapping, 33(7), 1553–1581. doi:10.1002/hbm.21307. [6] Mottron, L., Dawson, M., Soulierès, I., Hubert, B., & Burack, J. (2006). Enhanced Perceptual Functioning in Autism: An Update, and Eight Principles of Autistic Perception. Journal of Autism and Developmental Disorders, 36(1), 27–43. doi:10.1007/s10803-005-0040-7.
49 [7] Happé, F., & Frith, U. (2006). The weak coherence account: detail–focused cognitive style in autism spectrum disorders. Journal of Autism and Developmental Disorders, 36(1), 5–25. doi:10.1007/s10803-005-0039-0. [8] Uddin, L. Q. (2022). Exceptional abilities in autism: Theories and open questions. Philosophical Transactions of the Royal Society B, 377(1866), 20210202. doi:10.1098/rstb.2021.0202. [9] Friston, K. (2010). The free–energy principle: a unified brain theory? Nature Reviews Neuroscience, 11(2), 127–138. doi:10.1038/nrn2787. [10] Aston–Jones, G., & Cohen, J. D. (2005). An integrative theory of locus coeruleus–norepinephrine function: adaptive gain and optimal performance. Annual Review of Neuroscience, 28, 403–450. doi:10.1146/annurev.neuro.28.061604.135709. [11] Rubenstein, J. L. R., & Merzenich, M. M. (2003). Model of autism: increased ratio of excitation/inhibition in key neural systems. Genes, Brain and Behavior, 2(5), 255–267. doi:10.1034/j.1601-183X.2003.00037.x. [12] Port, R. G., et al. (2019). Revisiting the excitation/inhibition imbalance hypothesis in ASD neuroimaging. Behavioral Brain Research, 376, 112—. doi:10.1016/j.bbr.2019.112– (see article for final pagination). [13] Pasanta, D., et al. (2022). Functional MRS studies of GABA and Glutamate/Glx: a systematic review and meta–analysis. Neuroscience & Biobehavioral Reviews, 134, 104546. doi:10.1016/j.neubiorev.2021.12.033. [14] Breton–Provencher, V., & Sur, M. (2019). Active control of arousal by a locus coeruleus GABAergic circuit. Current Opinion in Neurobiology, 57, 157–164. doi:10.1016/j.conb.2019.02.007 [15] Elumalai, P., et al. (2022). Graph Ricci curvatures reveal atypical functional connectivity in autism. Scientific Reports, 12, 9259. doi:10.1038/s41598-022-12171-y. [16] Kobayashi, S., & Nomizu, K. (1963). Foundations of Differential Geometry, Vol. I. Wiley. [17] Naber, G. L. (2011). Topology, Geometry and Gauge Fields: Foundations (2nd ed.). Springer. [18] Baez, J., & Schreiber, U. (2007). Higher gauge theory. In Categories in Algebra, Geometry and Mathematical Physics (Contemporary Mathematics, Vol. 431), pp. 7–30. AMS. [19] Baez, J., & Huerta, J. (2011). An invitation to higher gauge theory. General Relativity and Gravitation, 43, 2335–2392. [20] Schreiber, U., & Waldorf, K. (2009). Parallel transport and functors. Homology, Homotopy and Applications, 11(1), 143–172. [21] Forman, R. (2003). Bochner’s method for cell complexes and combinatorial Ricci curvature. Discrete & Computational Geometry, 29, 323–374. [22] Ollivier, Y. (2009). Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3), 810–864. [23] Samal, A., Sreejith, R. P., Gu, J., Liu, S., Saucan, E., & Jost, J. (2018). Comparative analysis of two discretizations of Ricci curvature for complex networks. Scientific Reports, 8, 8650. [24] Harris, K. D., & Thiele, A. (2011). Cortical state and attention. Nature Reviews Neuroscience, 12, 509–523. [25] Hasselmo, M. E., & Sarter, M. (2011). Modes and models of forebrain cholinergic modulation of cognition. Neuropsychopharmacology, 36, 52–73. [26] Isaacson, J. S., & Scanziani, M. (2011). How inhibition shapes cortical activity. Neuron, 72(2), 231–243. [27] Durstewitz, D., & Seamans, J. K. (2008). The dual-state theory of prefrontal cortex dopamine function with relevance to catechol-o-methyltransferase genotypes and schizophrenia. Biological Psychiatry, 64(9), 739–749. [28] Marder, E., & Thirumalai, V. (2002). Cellular, synaptic and network effects of neuromodulation. Neuron, 76(1), 1–20. [29] Churchland, M. M., Cunningham, J. P., Kaufman, M. T., Foster, J. D., Nuyujukian, P., Ryu, S. I., & Shenoy, K. V. (2012). Neural population dynamics during reaching. Nature, 487, 51–56. [30] Cunningham, J. P., & Yu, B. M. (2014). Dimensionality reduction for large-scale neural recordings. Nature Neuroscience, 17, 1500–1509. [31] Gallego, J. A., Perich, M. G., Miller, L. E., & Solla, S. A. (2017). Neural manifolds for the control of movement. Current Opinion in Neurobiology, 52, 43–50. [32] Rigotti, M., Barak, O., Warden, M. R., Wang, X. J., Daw, N. D., Miller, E. K., & Fusi, S. (2013). The importance of mixed selectivity in complex cognitive tasks. Nature, 497, 585–590. [33] Provencher, S. W. (2001). Automatic quantitation of localized in vivo 1 H spectra with LCModel. NMR in Biomedicine, 14(4), 260–264. doi:10.1002/nbm.698. [34] Friston, K. J., Harrison, L., & Penny, W. (2003). Dynamic causal modelling. NeuroImage, 19(4), 1273–1302. doi:10.1016/S1053-8119(03)00202-7. [35] Esteban, O., Markiewicz, C. J., Blair, R. W., et al. (2019). fMRIPrep: a robust preprocessing pipeline for functional MRI. Nature Methods, 16, 111–116. doi:10.1038/s41592-018-0235-4. [36] Gorgolewski, K. J., Auer, T., Calhoun, V. D., et al. (2016). The brain imaging data structure, a format for organizing and describing outputs of neuroimaging experiments. Scientific Data, 3, 160044. doi:10.1038/sdata.2016.44. [37] Ni, C.-C., Lin, Y.-Y., Gao, J., & Gu, X. (2019). Ricci curvature of the Internet topology. In Proceedings of KDD 2015 (see also curvature toolboxes for networks, 2019 extensions). (Use as a methodological reference for graph Ricci curvature in networks.) [38] Horton, J. D. (1987). A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing, 16(2), 358–366. doi:10.1137/0216026. [39] Schreiber, T. (2000). Measuring information transfer. Physical Review Letters, 85(2), 461–464. doi:10.1103/PhysRevLett.85.461.
50 [40] Barnett, L., & Seth, A. K. (2014). The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference. Journal of Neuroscience Methods, 223, 50–68. doi:10.1016/j.jneumeth.2013.10.018. [41] Keren, N. I., Lozar, C. T., Harris, K. C., Morgan, P. S., & Eckert, M. A. (2009). In vivo mapping of the human locus coeruleus. NeuroImage, 47(4), 1261–1267. doi:10.1016/j.neuroimage.2009.06.012. [42] Betts, M. J., Cardenas-Blanco, A., Kanowski, M., Jessen, F., & Düzel, E. (2019). In vivo MRI assessment of the human locus coeruleus along its rostrocaudal extent in young and older adults. NeuroImage, 163, 150–159. doi:10.1016/j.neuroimage.2017.09.042. (Use as a neuromelanin-MRI methodological reference; see also review articles on LC imaging.) [43] Joshi, S., Li, Y., Kalwani, R. M., & Gold, J. I. (2016). Relationships between pupil diameter and neuronal activity in the locus coeruleus, colliculi, and cingulate cortex. Neuron, 89(1), 221–234. doi:10.1016/j.neuron.2015.11.028. [44] Schaub, M. T., Benson, A. R., Horn, P., Lippner, G., & Jadbabaie, A. (2020). Random walks on simplicial complexes and the normalized Hodge Laplacian. SIAM Review, 62(2), 353–391. doi:10.1137/18M1186167. [45] Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. AMS. [46] Lee, J. M. (2013). Introduction to Smooth Manifolds (2nd ed.). Springer. [47] Dale, A. M., Fischl, B., & Sereno, M. I. (1999). Cortical surface-based analysis. I. Segmentation and surface reconstruction. NeuroImage, 9(2), 179–194. [48] Fischl, B., Sereno, M. I., & Dale, A. M. (1999). Cortical surface-based analysis. II: Inflation, flattening, and a surface-based coordinate system. NeuroImage, 9(2), 195–207. [49] Caminiti, R., Carducci, F., Piervincenzi, C., et al. (2013). Diameter, length, speed, and conduction delay of callosal axons in macaque monkeys and humans: comparing data from histology and magnetic resonance imaging diffusion tractography. Journal of Neuroscience, 33(36), 14501–14511. [50] Honey, C. J., Sporns, O., Cammoun, L., et al. (2009). Predicting human resting-state functional connectivity from structural connectivity. PNAS, 106(6), 2035–2040. [51] Laughlin, S. B., & Sejnowski, T. J. (2003). Communication in neuronal networks. Science, 301(5641), 1870–1874. [52] Steenrod, N. (1951). The Topology of Fibre Bundles. Princeton University Press. [53] Glasser, M. F., Coalson, T. S., Robinson, E. C., et al. (2016). A multi-modal parcellation of human cerebral cortex. Nature, 536, 171–178. [54] Desikan, R. S., Ségonne, F., Fischl, B., et al. (2006). An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. NeuroImage, 31(3), 968–980. [55] Gordon, E. M., Laumann, T. O., Adeyemo, B., et al. (2016). Generation and Evaluation of a Cortical Area Parcellation from Resting-State Correlations. Cerebral Cortex, 26(1), 288–303. [56] Eickhoff, S. B., Yeo, B. T. T., & Genon, S. (2018). Imaging-based parcellations of the human brain. Nature Reviews Neuroscience, 19, 672–686. [57] Formisano, E., Kim, D.-S., Di Salle, F., et al. (2003). Mirror-symmetric tonotopic maps in human primary auditory cortex. Neuron, 40(4), 859–869. [58] Merzenich, M. M., & Brugge, J. F. (1973). Representation of the cochlear partition on the superior temporal plane of the macaque monkey. Brain Research, 50(2), 275–296. [59] Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33(2), 219–250. [60] Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3/4/5/6), 487–506. [61] Hubel, D. H., & Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. Journal of Physiology, 160(1), 106–154. [62] Kaas, J. H. (1997). Topographic maps are fundamental to sensory processing. Brain Research Bulletin, 44(2), 107–112. [63] Mountcastle, V. B. (1997). The columnar organization of the neocortex. Brain, 120(4), 701–722. [64] Douglas, R. J., & Martin, K. A. C. (2004). Neuronal circuits of the neocortex. Annual Review of Neuroscience, 27, 419–451. [65] Van Essen, D. C. (1997). A tension-based theory of morphogenesis and compact wiring in the central nervous system. Nature, 385, 313–318. [66] Kobayashi, S., & Nomizu, K. (1969). Foundations of Differential Geometry, Vol. II. Wiley. [67] Mackenzie, K. C. H. (2005). General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press. [68] Moerdijk, I., & Mrčun, J. (2003). Introduction to Foliations and Lie Groupoids. Cambridge University Press. [69] Brown, R., & Spencer, C. B. (1976). G-groupoids, crossed modules and the third nonabelian cohomology. Proceedings of the London Mathematical Society, 3(4), 502–550. [70] Baez, J. C., & Lauda, A. D. (2004). Higher-dimensional algebra V: 2-groups. Theory and Applications of Categories, 12, 423–491. [71] Martins, J. F., & Picken, R. (2010). On two-dimensional holonomy. Transactions of the American Mathematical Society, 362(11), 5657–5695. [72] Cardin, J. A., Carlen, M., Meletis, K., et al. (2009). Driving fast-spiking interneurons induces gamma rhythm and controls sensory responses. Nature, 459(7247), 663–667. [73] Cools, R., & D’Esposito, M. (2011). Inverted-U–shaped dopamine actions on human working memory and cognitive control. Biological Psychiatry, 69(12), e113–e125.
51 [74] Shine, J. M. (2019). Neuromodulatory influences on integration and segregation in the human brain. Neuron, 102(4), 442–454. [75] Sara, S. J. (2009). The locus coeruleus and noradrenergic modulation of cognition. Nature Reviews Neuroscience, 10(3), 211–223. [76] Hutchison, R. M., Womelsdorf, T., Gati, J. S., Everling, S., & Menon, R. S. (2013). Resting-state networks show dynamic functional connectivity in awake humans and anesthetized macaques. Human Brain Mapping, 34(9), 2154–2177. [77] Deco, G., Tononi, G., Boly, M., & Kringelbach, M. L. (2015/2016). Rethinking segregation and integration: contributions of whole-brain modelling. Nature Reviews Neuroscience, 16, 430–439 (see also subsequent 2016 updates). [78] Nakahara, M. (2003). Geometry, Topology and Physics (2nd ed.). Taylor & Francis. [79] Wilson, K. G. (1974). Confinement of quarks. Physical Review D, 10(8), 2445–2459. [80] Baez, J. C., & Schreiber, U. (2005). Higher gauge theory: 2-connections on 2-bundles. Preprint (arXiv:hepth/0412325). [81] Wockel, C. (2008). Principal 2-bundles and their gauge 2-groups. Forum Mathematicum, 23(3), 565–610. (early versions available as preprints). [82] Faria Martins, J., & Picken, R. (2011). The fundamental Gray 3-groupoid of a smooth manifold and 2-crossed modules. Differential Geometry and its Applications, 29(2), 179–206. (contains constructions leading to surface holonomy). [83] Bastos, A. M., Usrey, W. M., Adams, R. A., Mangun, G. R., Fries, P., & Friston, K. J. (2012). Canonical microcircuits for predictive coding. Neuron, 76(4), 695–711. [84] Dayan, P., & Abbott, L. F. (2001). Theoretical Neuroscience. MIT Press. [85] Fries, P. (2005). A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends in Cognitive Sciences, 9(10), 474–480. [86] Singer, A., & Wu, H.-T. (2012). Vector diffusion maps and the connection Laplacian. Communications on Pure and Applied Mathematics, 65(8), 1067–1144. [87] Bandeira, A. S., Singer, A., & Spielman, D. A. (2013). A Cheeger inequality for the graph connection Laplacian. SIAM Journal on Matrix Analysis and Applications, 34(4), 1611–1630. [88] Råde, J. (1992). On the Yang–Mills heat equation in two and three dimensions. Journal für die reine und angewandte Mathematik, 431, 123–163. [89] Uhlenbeck, K. (1982). Connections with Lp bounds on curvature. Communications on Pure and Applied Mathematics, 39(4), 525–539. [90] Yang, C. N., & Mills, R. L. (1954). Conservation of isotopic spin and isotopic gauge invariance. Physical Review, 96(1), 191–195. [91] Bogacz, R. (2017). A tutorial on the free-energy framework for modelling perception and learning. Journal of Mathematical Psychology, 76, 198–211. [92] Disney, A. A., Aoki, C., & Hawken, M. J. (2007). Gain modulation by nicotine in macaque V1. Neuron, 56(4), 701–713. [93] Seamans, J. K., & Yang, C. R. (2004). The principal features and mechanisms of dopamine modulation in the prefrontal cortex. Progress in Neurobiology, 74(1), 1–58. [94] Bouret, S., & Sara, S. J. (2005). Network reset: a simplified overarching theory of locus coeruleus noradrenaline function. Trends in Neurosciences, 28(11), 574–582. [95] Dayan, P., & Yu, A. J. (2006). Phasic norepinephrine: a neural interrupt signal for unexpected uncertainty. Nature Neuroscience, 9(7), 970–977. [96] Amari, S.-I. (2016). Information Geometry and Its Applications. Springer. [97] Stagg, C. J., & Rothman, D. L. (Eds.). (2014). Magnetic Resonance Spectroscopy: Tools for Neuroscience Research and Emerging Clinical Applications. Academic Press. [98] Near, J., Harris, A. D., Juchem, C., et al. (2021). Preprocessing, analysis and quantification in single-voxel magnetic resonance spectroscopy: experts’ consensus recommendations. NMR in Biomedicine, 34(5), e4257. [99] Syková, E., & Nicholson, C. (2008). Diffusion in brain extracellular space. Physiological Reviews, 88(4), 1277–1340. [100] Nicholson, C., & Phillips, J. M. (1981). Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of the rat cerebellum. Journal of Physiology, 321, 225–257. [101] Danbolt, N. C. (2001). Glutamate uptake. Progress in Neurobiology, 65(1), 1–105. [102] Richerson, G. B., & Wu, Y. (2003). Dynamic equilibrium of neurotransmitter transporters: not just for reuptake anymore. Journal of Neurophysiology, 90(3), 1363–1374. (Discusses GABA transport and non-synaptic roles.) [103] Sarter, M., & Parikh, V. (2005). Cholinergic control of attention: what we learned from microdialysis studies in animals. European Journal of Neuroscience, 22(1), 79–94. [104] Robinson, D. L., Venton, B. J., Heien, M. L. A. V., & Wightman, R. M. (2003/2004). Detecting subsecond dopamine release with fast-scan cyclic voltammetry in vivo. Clinical Chemistry, 49(10), 1763–1773. (Methods and kinetics relevant to DA transients.) [105] Rice, M. E., & Cragg, S. J. (2008). Dopamine spillover after quantal release: rethinking dopamine transmission in the nigrostriatal pathway. Nature Reviews Neuroscience, 9, 812–824. [106] Bunin, M. A., & Wightman, R. M. (1998). Quantitative evaluation of 5-hydroxytryptamine (serotonin) neuronal release and uptake: an investigation using fast-scan cyclic voltammetry. Journal of Neuroscience,
52 18(14), 6374–6383. [107] Parikh, V., Kozak, R., Martinez, V., & Sarter, M. (2007). Prefrontal acetylcholine release controls cue detection on multiple timescales. Journal of Neuroscience, 27(25), 6542–6552. [108] Agnati, L. F., & Fuxe, K. (2000). Volume transmission in the brain: novel mechanisms of neuronal communication. Trends in Neurosciences, 23(3), 130–139. [109] Descarries, L., & Mechawar, N. (2000). Ultrastructural evidence for diffuse transmission by monoamine and acetylcholine neurons of the central nervous system. Progress in Brain Research, 125, 27–47. [110] Araque, A., Carmignoto, G., Haydon, P. G., Oliet, S. H. R., Robitaille, R., & Volterra, A. (2014). Gliotransmitters travel in time and space. Neuron, 81(4), 728–739. [111] Crank, J. (1975). The Mathematics of Diffusion (2nd ed.). Oxford University Press. [112] Turrigiano, G. (2012). Homeostatic synaptic plasticity: local and global mechanisms for stabilizing neuronal function. Cold Spring Harbor Perspectives in Biology, 4(1), a005736. [113] Haider, B., & McCormick, D. A. (2009). Rapid neocortical dynamics: cellular and network mechanisms. Neuron, 62(2), 171–189. [114] Polack, P. O., Friedman, J., & Golshani, P. (2013). Cellular mechanisms of brain state–dependent gain modulation in visual cortex. Nature Neuroscience, 16, 1331–1339. [115] Pinto, L., Goard, M. J., Estandian, D., et al. (2013). Fast modulation of visual perception by basal forebrain cholinergic neurons. Nature Neuroscience, 16, 1857–1863. [116] O’Reilly, R. C., & Frank, M. J. (2006). Making working memory work: a computational model of learning in the prefrontal cortex and basal ganglia. Neural Computation, 18(2), 283–328. [117] Schultz, W., Dayan, P., & Montague, P. R. (1997). A neural substrate of prediction and reward. Science, 275(5306), 1593–1599. [118] Servan-Schreiber, D., Printz, H., & Cohen, J. D. (1990). A network model of catecholamine effects: gain, signal-to-noise ratio, and behavior. Science, 249(4971), 892–895. [119] McGinley, M. J., Vinck, M., Reimer, J., et al. (2015). Waking state: rapid variations modulate neural and behavioral responses. Neuron, 87(6), 1143–1161. [120] Dayan, P., & Huys, Q. J. M. (2008). Serotonin, inhibition, and negative mood. Current Opinion in Neurobiology, 18(2), 206–215. [121] Miyazaki, K. W., Miyazaki, K., & Doya, K. (2014). Activation of dorsal raphe serotonin neurons is necessary for waiting for delayed rewards. Current Biology, 24(17), 2033–2040. [122] Fuxe, K., Borroto-Escuela, D. O., Romero-Fernandez, W., et al. (2013). Volume transmission and receptor– receptor interactions in heteroreceptor complexes: relevance for brain integration. Frontiers in Integrative Neuroscience, 7, 17. [123] Reimer, J., McGinley, M. J., Liu, Y., Rodenkirch, C., et al. (2016). Pupil fluctuations track rapid changes in adrenergic and cholinergic activity in cortex. Neuron, 91(1), 1–12. (Article reports simultaneous pupil and cortical cholinergic/adrenergic markers.) [124] Tikhonov, A. N., & Arsenin, V. Y. (1977). Solutions of Ill-posed Problems. Wiley. [125] Rudin, L. I., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60(1–4), 259–268. [126] Perona, P., & Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), 629–639. [127] Traynelis, S. F., Wollmuth, L. P., McBain, C. J., et al. (2010). Glutamate receptor ion channels: structure, regulation, and function. Pharmacological Reviews, 62(3), 405–496. [128] Farrant, M., & Nusser, Z. (2005). Variations on an inhibitory theme: phasic and tonic activation of GABA A receptors. Nature Reviews Neuroscience, 6(3), 215–229. [129] Picciotto, M. R., Higley, M. J., & Mineur, Y. S. (2012). Acetylcholine as a neuromodulator: cholinergic signaling shapes cognition and emotion. Neuron, 76(1), 116–129. [130] Thiele, A., & Bellgrove, M. A. (2018). Neuromodulation of attention. Neuron, 97(4), 769–785. [131] Vijayraghavan, S., Wang, M., Birnbaum, S. G., Williams, G. V., & Arnsten, A. F. T. (2007). Inverted-U dopamine D1 receptor actions on prefrontal neurons engaged in working memory. Nature Neuroscience, 10(3), 376–384. [132] Arnsten, A. F. T. (2011). Catecholamine influences on dorsolateral prefrontal cortical networks. Biological Psychiatry, 69(12), e89–e99. [133] Berridge, C. W., & Waterhouse, B. D. (2003). The locus coeruleus–noradrenergic system: modulation of behavioral state and state-dependent cognitive processes. Brain Research Reviews, 42(1), 33–84. [134] Shine, J. M., Bissett, P. G., Bell, P. T., et al. (2016). The dynamics of functional brain networks: integrated network states during cognitive task performance. PNAS, 113(24), 6557–6562. [135] Nichols, D. E. (2016). Psychedelics. Pharmacological Reviews, 68(2), 264–355. [136] Carhart-Harris, R. L., & Friston, K. J. (2019). REBUS and the anarchic brain: toward a unified model of the brain action of psychedelics. Pharmacological Reviews, 71(3), 316–344. [137] McCormick, D. A., Nestvogel, D. B., & He, B. J. (2020). Neuromodulation of brain state and behavior. Annual Review of Neuroscience, 43, 391–415. [138] Benarroch, E. E. (2010). Locus coeruleus. Neurology, 75(10), 963–969. [139] Sherman, S. M., & Guillery, R. W. (2011). Distinct functions for direct and transthalamic corticocortical connections. Journal of Neurophysiology, 106(3), 1068–1077.
53 [140] Sreejith, R. P., Mohanraj, K., Jost, J., Saucan, E., & Samal, A. (2016). Forman curvature for complex networks. Journal of Statistical Mechanics: Theory and Experiment, 2016(6), 063206. [141] Weber, M., Saucan, E., & Jost, J. (2017). Characterizing complex networks with Forman-Ricci curvature and associated flows. Applied Network Science, 2, 41. [142] Sandhu, R., Georgiou, T., Reznik, E., Zhu, L., Kahn, J., & Tannenbaum, A. (2015). Graph curvature for differentiating cancer networks. Scientific Reports, 5, 12323. (Uses Ollivier-type curvature on weighted graphs; methodology applies to brain networks.) [143] de Pina, J. C. (1995). Applications of shortest path methods to the computation of minimum cycle bases. In Proceedings of the 2nd Annual European Symposium on Algorithms (pp. 400–411). [144] Amaldi, E., Nicoletti, M., & Rizzi, R. (2010). On the minimum cycle basis problem. Information Processing Letters, 110(16), 659–664. [145] Singer, A. (2011). Angular synchronization by eigenvectors and semidefinite programming. Applied and Computational Harmonic Analysis, 30(1), 20–36. [146] Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and Computational Harmonic Analysis, 21(1), 5–30. [147] Hartley, R., Trumpf, J., Dai, Y., & Li, H. (2013). Rotation averaging. International Journal of Computer Vision, 103, 267–305. [148] Lurie, D. J., Kessler, D., Bassett, D. S., et al. (2020). Questions and controversies in the study of time-varying functional connectivity in resting fMRI. Network Neuroscience, 4(1), 30–69. [149] Kriegeskorte, N., Mur, M., & Bandettini, P. A. (2008). Representational similarity analysis—connecting the branches of systems neuroscience. Frontiers in Systems Neuroscience, 2, 4. [150] Nili, H., Wingfield, C., Walther, A., Su, L., Marslen-Wilson, W., & Kriegeskorte, N. (2014). A toolbox for representational similarity analysis. PLOS Computational Biology, 10(4), e1003553. [151] DiCarlo, J. J., & Cox, D. D. (2007). Untangling invariant object recognition. Nature Neuroscience, 10, 1595–1606. [152] Dziuk, G., & Elliott, C. M. (2013). Finite element methods for surface PDEs. Acta Numerica, 22, 289–396. [153] Chung, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, No. 92. American Mathematical Society. [154] Belkin, M., & Niyogi, P. (2007). Convergence of Laplacian eigenmaps. Advances in Neural Information Processing Systems, 19, 129–136. (See also Foundations and Trends in ML, 2008, for extended analysis.) [155] Ascher, U. M., Ruuth, S. J., & Wetton, B. T. R. (1995). Implicit–explicit methods for time-dependent PDEs. SIAM Journal on Numerical Analysis, 32(3), 797–823. [156] Pareschi, L., & Russo, G. (2005). Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Scientific Computing, 25(1), 129–155. [157] Courant, R., Friedrichs, K., & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100, 32–74. [158] LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM. [159] Pinkall, U., & Polthier, K. (1993). Computing discrete minimal surfaces and their conjugates. Experimental Mathematics, 2(1), 15–36. [160] Desbrun, M., Meyer, M., Schröder, P., & Barr, A. H. (1999). Implicit fairing of irregular meshes using diffusion and curvature flow. In SIGGRAPH ’99 (pp. 317–324). [161] Meyer, M., Desbrun, M., Schröder, P., & Barr, A. H. (2003). Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III (pp. 35–57). Springer. [162] Hinze, M., Pinnau, R., Ulbrich, M., & Ulbrich, S. (2009). Optimization with PDE Constraints. Springer. [163] Coombes, S. (2005). Waves, bumps, and patterns in neural field theories. Biological Cybernetics, 93(2), 91–108. [164] Ermentrout, G. B., & Terman, D. H. (2010). Mathematical Foundations of Neuroscience. Springer. [165] Breakspear, M. (2017). Dynamic models of large-scale brain activity. Nature Neuroscience, 20(3), 340–352. [166] Strang, G. (1968). On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis, 5(3), 506–517. [167] Geuzaine, C., & Remacle, J.-F. (2009). Gmsh: A 3-D finite element mesh generator with built-in preand post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11), 1309–1331. [168] Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., & Lévy, B. (2010). Polygon Mesh Processing. AK Peters / CRC Press. [169] Schaefer, A., Kong, R., Gordon, E. M., et al. (2018). Local-global parcellation of the human cerebral cortex from intrinsic functional connectivity MRI. Cerebral Cortex, 28(9), 3095–3114. [170] Yeo, B. T. T., Krienen, F. M., Sepulcre, J., et al. (2011). The organization of the human cerebral cortex estimated by intrinsic functional connectivity. Journal of Neurophysiology, 106(3), 1125–1165. [171] Saad, Y. (2003). Iterative Methods for Sparse Linear Systems (2nd ed.). SIAM. [172] Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409–436. [173] Hackbusch, W. (1985). Multigrid Methods and Applications. Springer. [174] Briggs, W. L., Henson, V. E., & McCormick, S. F. (2000). A Multigrid Tutorial (2nd ed.). SIAM. [175] Saad, Y., & Schultz, M. H. (1986). GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3), 856–869. [176] Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations (4th ed.). Johns Hopkins University Press.
54 [177] Goebel, R., Sanfelice, R. G., & Teel, A. R. (2012). Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press. [178] Sontag, E. D. (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems (2nd ed.). Springer. [179] Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Hemisphere. [180] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., & Mishchenko, E. F. (1962). The Mathematical Theory of Optimal Processes. Interscience. [181] Bertsekas, D. P. (2005). Dynamic Programming and Optimal Control (Vol. I, 3rd ed.). Athena Scientific. [182] Nicholson, C. (2001). Diffusion and related transport mechanisms in brain tissue. Reports on Progress in Physics, 64(7), 815–884. [183] Hrabetová, S., Masri, D., Tao, L., et al. (2018). Unveiling real-time ion diffusion in the brain extracellular space. Journal of Neuroscience Methods, 329, 108458. (Representative optical imaging approaches to ECS transport.) [184] Berg, H. C. (1993). Random Walks in Biology (2nd ed.). Princeton University Press. (Contains diffusion time/length estimates used for rule-of-thumb calculations.) [185] Tzingounis, A. V., & Wadiche, J. I. (2007). Glutamate transporters: confining runaway excitation by shaping synaptic transmission. Nature Reviews Neuroscience, 8(12), 935–947. [186] Borden, L. A. (1996). GABA transporter heterogeneity: pharmacology and cellular localization. Neurochemistry International, 29(4), 335–356. [187] Silman, I., & Sussman, J. L. (2005). Acetylcholinesterase: ‘classical’ and ‘non-classical’ functions and pharmacology. Current Opinion in Pharmacology, 5(3), 293–302. [188] Sulzer, D. (2011). How addictive drugs disrupt presynaptic dopamine neurotransmission. Neuron, 69(4), 628–649. (Background on DA release/clearance mechanisms relevant for τD.) [189] Aston-Jones, G., & Cohen, J. D. (2005). An integrative theory of locus coeruleus–norepinephrine function: adaptive gain and optimal performance. Annual Review of Neuroscience, 28, 403–450. [190] Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids (2nd ed.). Oxford University Press. (Closed-form solutions for diffusion with boundaries; used for τdiff estimates.) [191] Murray, M. K. (1996). Bundle gerbes. Journal of the London Mathematical Society, 54(2), 403–416. [192] Breen, L., & Messing, W. (2001). Differential geometry of gerbes. Advances in Mathematics, 198(2), 485–545. [193] Schreiber, U., & Waldorf, K. (2009). Connections on non-Abelian gerbes and surface holonomy. Theory and Applications of Categories, 28(17), 476–540. [194] Leinster, T. (2003). Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series. Cambridge University Press. [195] Baars, B. J. (2005). Global workspace theory of consciousness: toward a cognitive neuroscience of human experience. Progress in Brain Research, 150, 45–53. [196] Dehaene, S., & Changeux, J.-P. (2011). Experimental and theoretical approaches to conscious processing. Neuron, 70(2), 200–227. [197] Dehaene, S., Lau, H., & Kouider, S. (2017). What is consciousness, and could machines have it? Science, 358(6362), 486–492. [198] Mashour, G. A., Roelfsema, P., Changeux, J.-P., & Dehaene, S. (2020). Conscious processing and the global neuronal workspace hypothesis. Neuron, 105(5), 776–798. [199] Casali, A. G., Gosseries, O., Rosanova, M., et al. (2013). A theoretically based index of consciousness independent of sensory processing and behavior. Science Translational Medicine, 5(198), 198ra105. [200] Purdon, P. L., Pierce, E. T., Mukamel, E. A., et al. (2013). Electroencephalogram signatures of loss and recovery of consciousness from propofol. Proceedings of the National Academy of Sciences USA, 110(12), E1142–E1151. [201] Saalmann, Y. B., Pinsk, M. A., Wang, L., Li, X., & Kastner, S. (2012). The pulvinar regulates information transmission between cortical areas based on attention demands. Science, 337(6095), 753–756. [202] Shine, J. M. (2019). Neuromodulatory influences on integration and segregation in the brain. Current Opinion in Neurobiology, 58, 104–111. [203] Mather, M., Clewett, D., Sakaki, M., & Harley, C. W. (2016). Norepinephrine ignites local hotspots of neuronal excitation: How arousal amplifies selectivity in perception and memory. Behavioral and Brain Sciences, 39, e200. [204] Cools, R., & D’Esposito, M. (2011). Inverted-U–shaped dopamine actions on human working memory and cognitive control. Biological Psychiatry, 69(12), e113–e125. [205] Hasselmo, M. E., & Sarter, M. (2011). Modes and models of forebrain cholinergic neuromodulation of cognition. Neuropsychopharmacology, 36(1), 52–73. [206] Dehaene, S. (2014). Consciousness and the Brain: Deciphering How the Brain Codes Our Thoughts. Viking. [207] Moher, D., Hopewell, S., Schulz, K. F., et al. (2010). CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomized trials. BMJ, 340, c869. [208] Efird, J. (2011). Blocked randomization with randomly selected block sizes. International Journal of Environmental Research and Public Health, 8(1), 15–20. [209] Nosek, B. A., Ebersole, C. R., DeHaven, A. C., & Mellor, D. T. (2018). The preregistration revolution. PNAS, 115(11), 2600–2606.
55 [210] Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39(2), 175–191. [211] Navon, D. (1977). Forest before trees: The precedence of global features in visual perception. Cognitive Psychology, 9(3), 353–383. [212] McGurk, H., & MacDonald, J. (1976). Hearing lips and seeing voices. Nature, 264, 746–748. [213] Provencher, S. W. (2001). Automatic quantitation of localized in vivo 1 H spectra with LCModel. NMR in Biomedicine, 14(4), 260–264. [214] Friston, K. J., Harrison, L., & Penny, W. (2003). Dynamic causal modelling. NeuroImage, 19(4), 1273–1302. [215] Horton, J. D. (1987). A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing, 16(2), 358–366. [216] Joshi, S., Li, Y., Kalwani, R. M., & Gold, J. I. (2016). Relationships between pupil diameter and neuronal activity in the locus coeruleus, colliculus, and cingulate cortex. Neuron, 89(1), 221–234. [217] Keren, N. I., Lozar, C. T., Harris, K. C., Morgan, P. S., & Eckert, M. A. (2009). In vivo mapping of the human locus coeruleus. NeuroImage, 47(4), 1261–1267. [218] Betts, M. J., Kirilina, E., Otaduy, M. C. G., et al. (2019). Locus coeruleus imaging as a biomarker for noradrenergic dysfunction in neurodegenerative diseases. Brain, 142(9), 2558–2571. [219] Gorgolewski, K. J., Auer, T., Calhoun, V. D., et al. (2016). The brain imaging data structure, a format for organizing and describing outputs of neuroimaging experiments. Scientific Data, 3, 160044. [220] Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press. [221] Nichols, T. E., & Holmes, A. P. (2002). Nonparametric permutation tests for functional neuroimaging: a primer with examples. Human Brain Mapping, 15(1), 1–25. [222] Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society B, 57(1), 289–300. [223] Deisseroth, K. (2011). Optogenetics. Nature Methods, 8(1), 26–29. [224] Roth, B. L. (2016). DREADDs for neuroscientists. Neuron, 89(4), 683–694. [225] Helmchen, F., & Denk, W. (2005). Deep tissue two-photon microscopy. Nature Methods, 2(12), 932–940. [226] Near, J., Harris, A. D., Juchem, C., et al. (2021). Preprocessing, analysis and quantification in single-voxel magnetic resonance spectroscopy: experts’ consensus recommendations. NMR in Biomedicine, 34(5), e4257. [227] Gasparovic, C., Song, T., Devier, D., et al. (2006). Use of tissue water as a concentration reference for proton spectroscopic imaging. Magnetic Resonance in Medicine, 55(6), 1219–1226. [228] Esteban, O., Markiewicz, C. J., Blair, R. W., et al. (2019). fMRIPrep: a robust preprocessing pipeline for functional MRI. Nature Methods, 16, 111–116. [229] Pruim, R. H. R., Mennes, M., Buitelaar, J. K., & Beckmann, C. F. (2015). Evaluation of ICA-AROMA and alternative strategies for motion artifact removal in resting state fMRI. NeuroImage, 112, 278–287. [230] Power, J. D., Barnes, K. A., Snyder, A. Z., Schlaggar, B. L., & Petersen, S. E. (2012). Spurious but systematic correlations in functional connectivity MRI networks arise from subject motion. NeuroImage, 59(3), 2142–2154. [231] Ciric, R., Wolf, D. H., Power, J. D., et al. (2017). Benchmarking of participant-level confound regression strategies for the control of motion artifact in studies of functional connectivity. NeuroImage, 154, 174–187. [232] Behzadi, Y., Restom, K., Liau, J., & Liu, T. T. (2007). A component based noise correction method (CompCor) for BOLD and perfusion based fMRI. NeuroImage, 37(1), 90–101. [233] Glasser, M. F., Coalson, T. S., Robinson, E. C., et al. (2016). A multi-modal parcellation of human cerebral cortex. Nature, 536, 171–178. [234] Smith, S. M., Miller, K. L., Salimi-Khorshidi, G., et al. (2011). Network modelling methods for FMRI. NeuroImage, 54(2), 875–891. [235] Razi, A., Kahan, J., Rees, G., & Friston, K. J. (2015). Construct validation of a DCM for resting state fMRI. NeuroImage, 106, 1–14. [236] Barnett, L., & Seth, A. K. (2014). The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference. Journal of Neuroscience Methods, 223, 50–68. [237] Hämäläinen, M. S., & Ilmoniemi, R. J. (1994). Interpreting magnetic fields of the brain: minimum norm estimates. Medical & Biological Engineering & Computing, 32(1), 35–42. [238] Gross, J., Baillet, S., Barnes, G. R., et al. (2013). Good practice for conducting and reporting MEG research. NeuroImage, 65, 349–363. [239] Forman, R. (2003). Bochner’s method for cell complexes and combinatorial Ricci curvature. Discrete & Computational Geometry, 29(3), 323–374. [240] Ollivier, Y. (2009). Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3), 810–864. [241] Lin, Y., Lu, L., & Yau, S.-T. (2011). Ricci curvature of graphs. Tohoku Mathematical Journal, 63(4), 605–627. [242] Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society. [243] Petri, G., Expert, P., Turkheimer, F., et al. (2014). Homological scaffolds of brain functional networks. Journal of the Royal Society Interface, 11(101), 20140873. [244] Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974. [245] Pinheiro, J. C., & Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer.
56 [246] Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. [247] Kenward, M. G., & Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53(3), 983–997. [248] Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6), 110–114. [249] Benjamini, Y., & Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics, 29(4), 1165–1188. [250] Benjamini, Y., & Bogomolov, M. (2014). Selective inference on multiple families of hypotheses. Journal of the Royal Statistical Society B, 76(1), 297–318. [251] Westfall, P. H., & Young, S. S. (1993). Resampling-Based Multiple Testing: Examples and Methods for P-Value Adjustment. Wiley. [252] Winkler, A. M., Ridgway, G. R., Webster, M. A., Smith, S. M., & Nichols, T. E. (2014). Permutation inference for the general linear model. NeuroImage, 92, 381–397. [253] Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science. Frontiers in Psychology, 4, 863. [254] Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105–125. [255] Westfall, J., Kenny, D. A., & Judd, C. M. (2014). Statistical power and optimal design in experiments in which samples of participants respond to samples of stimuli. Journal of Experimental Psychology: General, 143(5), 2020–2045. [256] Jones, B., & Kenward, M. G. (2014). Design and Analysis of Cross-Over Trials (3rd ed.). Chapman & Hall/CRC. [257] Senn, S. (2002). Cross-over Trials in Clinical Research (2nd ed.). Wiley. [258] Imai, K., Keele, L., & Tingley, D. (2010). A general approach to causal mediation analysis. Psychological Methods, 15(4), 309–334. [259] Imai, K., Keele, L., & Yamamoto, T. (2010). Identification, inference and sensitivity analysis for causal mediation effects. Statistical Science, 25(1), 51–71. [260] Tingley, D., Yamamoto, T., Hirose, K., Keele, L., & Imai, K. (2014). mediation: R package for causal mediation analysis. Journal of Statistical Software, 59(5), 1–38. [261] VanderWeele, T. J. (2015). Explanation in Causal Inference: Methods for Mediation and Interaction. Oxford University Press. [262] Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Psychological Methods, 15(3), 209–233. [263] Bauer, D. J., Preacher, K. J., & Gil, K. M. (2006). Conceptualizing and testing random indirect effects and moderated mediation in multilevel models. Multivariate Behavioral Research, 41(2), 195–218. [264] MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99–128. [265] Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC. [266] Green, P., & MacLeod, C. J. (2016). simr: an R package for power analysis of generalized linear mixed models by simulation. Methods in Ecology and Evolution, 7(4), 493–498. [267] Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B., & Wang, X. (2007). AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations. (Technical report and user manual.) [268] Kuznetsov, Y. A. (2004). Elements of Applied Bifurcation Theory (3rd ed.). Springer. [269] Saltelli, A., Ratto, M., Andres, T., et al. (2008). Global Sensitivity Analysis: The Primer. Wiley. [270] Sobol’, I. M. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 55(1–3), 271–280. [271] Tagliazucchi, E., von Wegner, F., Morzelewski, A., Borisov, S., Jahnke, K., & Laufs, H. (2013). Large-scale brain functional modularity is reflected in slow electroencephalographic rhythms across the human non-rapid eye movement sleep cycle. Journal of the Royal Society Interface, 10(78), 20130110. [272] Spoormaker, V. I., Schröter, M. S., Gleiser, P. M., et al. (2010). Development of a large-scale functional brain network during human non-rapid eye movement sleep. Journal of Neuroscience, 30(34), 11379–11387. [273] Murphy, P. R., O’Connell, R. G., O’Sullivan, M., Robertson, I. H., & Balsters, J. H. (2014). Pupil diameter covaries with BOLD activity in human locus coeruleus and salience network. PNAS, 111(41), 15272–15277. [274] van den Brink, R. L., Murphy, P. R., & Nieuwenhuis, S. (2016). Pupil diameter tracks lapses of attention. PLoS ONE, 11(10), e0165274. [275] Tononi, G. (2004). An information integration theory of consciousness. BMC Neuroscience, 5, 42. [276] Oizumi, M., Albantakis, L., & Tononi, G. (2014). From the structure of the system to integrated information. PLoS Computational Biology, 10(5), e1003588. [277] Tononi, G. (2008). Consciousness as integrated information: a provisional manifesto. Biological Bulletin, 215(3), 216–242. [278] Friston, K. (2010). The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 11(2), 127–138. [279] Deco, G., Jirsa, V. K., & McIntosh, A. R. (2011). Emerging concepts for the dynamical organization of resting-state activity in the brain. Nature Reviews Neuroscience, 12(1), 43–56.
57 [280] Lee, S.-H., & Dan, Y. (2012). Neuromodulation of brain states. Neuron, 76(1), 209–222. [281] Percival, D. B., & Walden, A. T. (1993). Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press. [282] Di Martino, A., Yan, C.-G., Li, Q., et al. (2014). The Autism Brain Imaging Data Exchange: towards a large-scale evaluation of the intrinsic brain architecture in autism. Molecular Psychiatry, 19, 659–667. [283] Van Essen, D. C., Smith, S. M., Barch, D. M., et al. (2013). The WU-Minn Human Connectome Project: an overview. NeuroImage, 80, 62–79. [284] Poldrack, R. A., Barch, D. M., Mitchell, J., et al. (2013). Toward open sharing of task-based fMRI data: the OpenfMRI project. NeuroImage, 82, 683–691. [285] Sandhu, R. S., Georgiou, T. T., Reznik, E., Zhu, L., Kolesov, I., Senbabaoglu, Y., & Tannenbaum, A. (2015). Graph curvature for differentiating cancer networks. Scientific Reports, 5, 12323. (General ORC machinery applied to networks.) [286] Sreejith, R. P., Mohanraj, K., Jost, J., Saucan, E., & Samal, A. (2016). Forman curvature for complex networks. Journal of Statistical Mechanics: Theory and Experiment, 2016(6), 063206. [287] Baccalá, L. A., & Sameshima, K. (2001). Partial directed coherence: a new concept in neural structure determination. Biological Cybernetics, 84(6), 463–474. [288] Kamiński, M. J., & Blinowska, K. J. (1991). A new method of the description of the information flow in the brain structures. Biological Cybernetics, 65(3), 203–210. [289] Penny, W., Friston, K., Ashburner, J., Kiebel, S., & Nichols, T. (Eds.). (2011). Statistical Parametric Mapping: The Analysis of Functional Brain Images. Academic Press. [290] Tauzin, G., Lupo, U., Tunstall, L., et al. (2021). giotto-tda: A topological data analysis toolkit for machine learning. Journal of Machine Learning Research, 22(39), 1–6. [291] Bauer, U. (2021). Ripser: Efficient computation of Vietoris–Rips persistence barcodes. Journal of Applied and Computational Topology, 5, 391–423. [292] Maria, C., Boissonnat, J.-D., Glisse, M., & Yvinec, M. (2014). The GUDHI library: Simplicial complexes and persistent homology. In Topological Methods in Data Analysis and Visualization III (pp. 167–176). Springer. [293] Gorgolewski, K. J., Burns, C. D., Madison, C., et al. (2011). Nipype: a flexible, lightweight and extensible neuroimaging data processing framework in Python. Frontiers in Neuroinformatics, 5, 13. [294] Merkel, D. (2014). Docker: lightweight Linux containers for consistent development and deployment. Linux Journal, 2014(239), 2. [295] Kurtzer, G. M., Sochat, V., & Bauer, M. W. (2017). Singularity: Scientific containers for mobility of compute. PLOS ONE, 12(5), e0177459. [296] Zalesky, A., Fornito, A., Harding, I. H., et al. (2010). Whole-brain anatomical networks: Does the choice of nodes matter? NeuroImage, 50(3), 970–983. [297] de Reus, M. A., & van den Heuvel, M. P. (2013). The parcellation-based connectome: Limitations and extensions. NeuroImage, 80, 397–404. [298] Kreis, R. (2004). Issues of spectral quality in clinical 1 H–MRS and a gallery of artifacts. NMR in Biomedicine, 17(6), 361–381. [299] Mullins, P. G., McGonigle, D. J., O’Gorman, R. L., et al. (2014). Current practice in the use of MEGA-PRESS spectroscopy for the detection of GABA. Magnetic Resonance in Medicine, 73(1), 1–6. [300] Daunizeau, J., Stephan, K. E., & Friston, K. J. (2011). Stochastic dynamic causal modelling of fMRI data: Should we care about neural noise? NeuroImage, 62(1), 464–481. (Contains identifiability and model-comparison considerations.) [301] Frässle, S., Lomakina, E. I., Razi, A., et al. (2017). Regression DCM for fMRI. NeuroImage, 155, 406–421. [302] Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd ed.). Cambridge University Press. [303] Marek, S., Tervo-Clemmens, B., Calabro, F. J., et al. (2022). Reproducible brain-wide association studies require thousands of individuals. Nature, 603, 654–660. [304] Poldrack, R. A., Baker, C. I., Durnez, J., et al. (2017). Scanning the horizon: Towards transparent and reproducible neuroimaging research. Nature Reviews Neuroscience, 18, 115–126. [305] Birn, R. M., Diamond, J. B., Smith, M. A., & Bandettini, P. A. (2006). Separating respiratory-variationrelated fluctuations from neuronal-activity-related fluctuations in fMRI. NeuroImage, 31(4), 1536–1548. [306] Chang, C., & Glover, G. H. (2009). Effects of model-based physiological noise correction on default mode network anti-correlations and correlations. NeuroImage, 47(4), 1448–1459. [307] Glover, G. H., Li, T.-Q., & Ress, D. (2000). Image-based method for retrospective correction of physiological motion effects in fMRI: RETROICOR. Magnetic Resonance in Medicine, 44(1), 162–167. [308] Mathôt, S. (2018). Pupillometry: Psychology, physiology, and function. Journal of Cognition, 1(1), 16. [309] Eklund, A., Nichols, T. E., & Knutsson, H. (2016). Cluster failure: Why fMRI inferences for spatial extent have inflated false-positive rates. PNAS, 113(28), 7900–7905. [310] Fortin, J.-P., Cullen, N., Sheline, Y. I., et al. (2018). Harmonization of cortical thickness measurements across scanners and sites. NeuroImage, 167, 104–120. [311] Satterthwaite, T. D., Elliott, M. A., Gerraty, R. T., et al. (2013). An improved framework for confound regression and filtering for control of motion artifact in the preprocessing of resting-state functional connectivity data. NeuroImage, 64, 240–256. [312] Smith, S. M., & Nichols, T. E. (2009). Threshold-free cluster enhancement: Addressing problems of smoothing, threshold dependence and localisation in cluster inference. NeuroImage, 44(1), 83–98.