Branched Optimal Transport for Stimulus to Reaction Brain Mapping

Branc hed Optimal T ransp ort for Stim ulus to Reaction Brain Mapping Cristian Mendico 1 1 Institut de Mathématique de Bourgogne - UMR 5584 CNRS, Université Bourgogne Europe , cristian.mendico@u-b ourgogne.fr Marc h 23, 2026 Abstract A cen tral problem in systems neuroscience is to determine how an external stimulation is propagated through the brain so as to pro duce a reaction. Curren t deterministic and sto c hastic con trol mo dels quantify transition costs b et ween brain states on a prescrib ed netw ork, but do not treat the transp ort net w ork itself as an unknown. Here we prop ose a v ariational framew ork in which the inferred ob ject is a graph/current connecting a stimulation source measure to a reaction target measure. The mo del is posed as an anisotropic branched optimal transp ort problem, where conca vit y of the flux cost promotes aggregation and branc hing. The supp ort of an optimal current defines a stimulus-to-reaction routing architecture, interpreted as a brain reaction map. W e prov e existence of minimizers in discrete and con tin uous form ulations and introduce a hybrid sto chastic extension combining ramified transp ort with a path-space Kullbac k–Leibler con trol cost on the induced graph dynamics. This approach pro vides a mathematical mec hanism for inferring propagation architectures rather than con trolling tra jectories on fixed substrates. Key words: Branched Optimal T ransp ort ; Mathematical Biology ; V ariational mo dels. MSC 2020: 35Q49 - 35Q92 - 58E25. Con ten ts 1 In tro duction 2 2 Notations 4 3 Estimation of µ + stim , µ − react , and c ( x, τ ) from data 5 3.1 General principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Estimation from fMRI and EEG/MEG data . . . . . . . . . . . . . . . . . . . . . 6 3.3 Estimation of the anatomical cost density . . . . . . . . . . . . . . . . . . . . . . 6 4 Discrete graph mo del 7 4.1 W eigh ted directed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 Finite-library existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5 Con tinuous current form ulation 9 5.1 Rectifiable transp ort currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Con tinuous branc hed transp ort functional . . . . . . . . . . . . . . . . . . . . . . 10 1 6 Hybrid sto c hastic extension 14 6.1 Graph-induced linear sto chastic dynamics . . . . . . . . . . . . . . . . . . . . . . 14 6.2 P ath- space relativ e en trop y and dynamic cost . . . . . . . . . . . . . . . . . . . . 16 6.3 Hybrid graph-selection functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7 In terpretation of the minimizer 18 8 Conclusion 19 1 In tro duction A fundamental problem in systems neuroscience is to understand how an externally applied stim ulus is propagated through the brain so as to pro duce a reaction. Dep ending on the exp erimen tal setting, the reaction ma y b e motor, p erceptual, autonomic, or cognitive; in all cases, ho wev er, one is faced with the same conceptual question: thr ough which neur al r outing ar chite ctur e is the stimulus c onveye d to the r e action-pr o ducing c onfigur ation? While mo dern neuroimaging and connectomics provide increasingly detailed information on structural and functional connectivit y , they do not b y themselv es identify a mathematically canonical map of pr op agation from stimulation to resp onse. The aim of this pap er is to prop ose such a map as the solution of a v ariational problem. A large b o dy of recen t work addresses related questions through control-theoretic mo dels of brain dynamics. In these approac hes, one t ypically fixes an ambien t dynamical system, deterministic or sto chastic, on a prescrib ed state space or netw ork, and studies the cost of driving the system from an initial state to a target state by means of admissible con trol inputs [ 14 , 17 , 16 ]. This p ersp ective has b een highly successful in quantifying controllabilit y prop erties of structural brain netw orks, in measuring the difficulty of task-dep enden t state transitions, and in incorp orating sto c hasticit y through entropic optimal control and Sc hrö dinger bridge form ulations. Math ematically , the optimization v ariable in those mo dels is a control pro cess, or equiv alently a controlled tra jectory or controlled path-space law. The underlying netw ork—or state-space geometry—is fixed in adv ance. More broadly , the present work fits within a long tradition of mathematical biology and mathematical neuroscience devoted to linking neural structure, collectiv e dynamics, and large- scale communication through explicit mo dels. At mesoscopic scales, neural mass and neural field mo dels ha v e long provided a mathematical language for describing spatially extended cortical activit y , including trav eling wa ves, lo calized bumps, oscillations, and pattern formation [ 5 , 4 , 24 ]. Their sto chastic counterparts ha ve also b een placed on a rigorous analytical fo oting, with results on w ell-p osedness, gradient-flo w structure, inv arian t measures, and delay ed or diffusive extensions that are directly relev ant for coarse-grained descriptions of brain activity [ 8 , 20 , 26 ], see also [ 11 ]. A t larger scales, whole-brain mo dels com bine structural connectivity with lo cal p opulation dynamics and noise in order to study how anatom y constrains functional co ordination, effectiv e connectivit y , and task-dep enden t reconfiguration [ 13 , 6 , 21 ]. Recent w ork has further emphasized that realistic macroscale mo deling of the brain requires integrating netw ork top ology with lo cal biological annotations and with quantitativ e measures of structure–function coupling [ 2 , 9 ]. Our approac h is complemen tary to this literature. Rather than starting from a fixed connectome and studying the resulting dynamics, we take the source–target pair and the anatomical transp ort cost as primary inputs, and seek the routing architecture itself as the solution of a v ariational problem. In many situations, the primary quan tit y of interest is not the energetic or informational cost of realizing a transition on a known substrate, but rather the substr ate itself . If one knows where the stimulus is injected, and one has a mathematical represen tation of the reaction-pro ducing configuration, can one infer a preferred transp ort arc hitecture connecting the tw o? Differently , can one formulate a mo del in whic h the output of 2 the optimization problem is not a con trolled path, but a gr aph —or, in the contin uous setting, a r e ctifiable curr ent —describing the routing of the stim ulus through the brain? This is the p ersp ectiv e adopted here. The key observ ation is that such a question is naturally geometric rather than purely dynamical. If information or activ ation propagates jointly through nearb y neural path wa ys b efore branc hing to w ard different downstream regions, then the corresp onding cost should not dep end only on endp oint displacements or on state tra jectories, but on the actual network supp orting the flux . This leads directly to the mathematical paradigm of br anche d optimal tr ansp ort . In branched transp ort, one minimizes a transportation cost that is conca ve with resp ect to the transp orted flux. Concavit y fa v ors aggregation: transp orting mass together for part of the route is strictly cheaper than transp orting it separately all the wa y . As a consequence, minimizers are not diffuse flo ws but branching structures. This ph enomenon was first formalized in the discrete Gilb ert mo del, where one minimizes ov er weigh ted directed graphs satisfying Kirc hhoff balance laws [ 12 ]. It w as later extended by Xia to a contin uous Eulerian formulation in terms of divergence-constrained rectifiable vector measures, or equiv alently one-dimensional curren ts [ 27 ]. These mo dels, together with their Lagrangian traffic-plan counterparts, n o w form a w ell-established mathematical theory of ramified transp ort [ 3 , 25 ]. Our prop osal is to adapt this framew ork to stim ulus-to-reaction brain mapping. W e represent the external stimulation by a source measure µ + stim ∈ M + (Ω) , and the reaction-pro ducing neural configuration by a target measure µ − react ∈ M + (Ω) , with equal total mass, on a compact domain Ω ⊂ R d represen ting the relev ant anatomical or mesoscopic neural space. The unknown is a transp ort current v = τ θ H 1 ⌞ M , where M is a countably 1 -rectifiable set, τ is an orientation field, and θ is a scalar multiplicit y enco ding transp orted flux. The current is constrained by the balance la w div v = µ + stim − µ − react , and is selected by minimizing an anisotropic branc hed transp ort energy of the form M α,c ( v ) = Z M c ( x, τ ( x )) θ ( x ) α d H 1 ( x ) , 0 < α < 1 . Here the exp onent α enco des the ramification incen tiv e, while the co efficient c : Ω × S d − 1 → (0 , ∞ ) w eigh ts transp ortation according to lo cal anatomical or functional features, suc h as tissue geometry , preferred orien tation, conductivit y , or tractographic information. The supp ort of an optimal current is then interpreted as the br ain r e action map : the preferred routing arc hitecture that con v eys the stimulation tow ard the reaction. This p oin t of view differs conceptually from standard con trol formulations in one essential resp ect. In deterministic or sto c hastic netw ork con trol, the geometry is prescrib ed and the optimization determines how b est to mo ve on that geometry . In the present framew ork, the geometry itself is inferred fr om the source–target pair by v ariational selection. The curren t or graph is the primary optimization v ariable; controls and tra jectories, when they app ear, are secondary ob jects built on top of that geometry . This distinction is not merely formal. It reflects 3 a different mo deling ob jectiv e: not the cost of using a known neural substrate, but the inference of an effective routing architecture from stimulation and resp onse data. A t the same time, the branc hed transport picture do es not aim to replace dynamical con trol mo dels. Rather, it provides a geometric backbone onto which dynamical mo dels may subsequen tly b e imp osed. This motiv ates the h ybrid extension dev elop ed later in the pap er. Once an admissible graph/curren t has b een selected, one ma y associate with it a sto chastic neural dynamics and then quantify the discrepancy b etw een uncontrolled and controlled evolutions on that graph through a path-space Kullback–Leibler cost. More precisely , if ( G, w ) denotes an admissible weigh ted graph and P G,w, 0 and P G,w,u denote the uncontrolled and con trolled path-space la ws induced b y a sto c hastic dynamics on G , one ma y define a secondary dynamic term J dyn ( G, w ) := inf u KL  P G,w,u    P G,w, 0  . In this wa y , a graph ma y b e p enalized not only by its ramified geometric transp ort cost, but also by the minimum informational effort required to realize a compatible controlled sto c hastic ev olution on it. The resulting mo del preserves the graph/current as the main unkno wn while incorp orating a dynamic regularization principle inspired by Schrödinger bridge theory [ 17 , 16 ]. F rom a mathematical viewp oint, the pap er develops this idea at t wo levels. First, w e form ulate discrete and contin uous branched transp ort mo dels adapted to the neuroscience setting and prov e existence of minimizers under natural assumptions. Second, w e prop ose a sto chastic extension in which the geometric optimizer is coupled to a path-space con trol cost. F rom a mo deling viewp oint, the framework offers a candidate v ariational definition of a brain reaction map, com bining the geometric efficiency of ramified transp ort with the flexibility of sto chastic con trol. W e b elieve that this dual p ersp ective may provide a useful bridge b etw een geometric measure theory , optimal transp ort, and systems neuroscience. Numerical simulations asso ciated with the mo del we are prop osing in this w ork will app ear in Multimo dal br anche d tr ansp ort infers anatomic al ly aligne d br ain r e action maps b y the author. Structure of the pap er. Section 2 fixes the notation and the admissible data. In Section 3 w e explain ho w these quan tities may b e mo deled and estimated in realistic settings. Section 4 in tro duces the discrete graph formulation and pro ves its basic w ell-p osedness prop erties. Section 5 dev elops the contin uous Xia-type curren t form ulation. Section 6 presents a sto chastic extension inspired b y Sc hrö dinger bridge control. Finally , in Section 7 w e discuss the neuroscien tific in terpretation of the minimizer and its connection with data-driv en estimation. 2 Notations Let Ω ⊂ R d b e a compact connected domain represen ting the brain, with d = 3 in a volumetric mo del or d = 2 in a cortical-surface mo del. W e denote by P (Ω) th e space of Borel probability measures on Ω and by M (Ω; R d ) the space of finite R d -v alued Radon measures on Ω . Definition 2.1 (Stimulus and reaction measures) . A stimulus sour c e me asur e is a nonnegative finite Borel measure µ + stim ∈ M (Ω) , µ + stim (Ω) = m > 0 , represen tin g the entry distribution of the external stimulation. A r e action tar get me asur e is a nonnegativ e finite Borel measure µ − react ∈ M (Ω) , µ − react (Ω) = m, represen ting the neural distribution asso ciated with the pro duction of the reaction. The pair ( µ + stim , µ − react ) is called b alanc e d if µ + stim (Ω) = µ − react (Ω) . 4 The balanced condition is the natural conserv ation la w for the transp ort problem. The common mass m may b e normalized to 1 without loss of generality . In the brain, the transp ort cost should not dep end only on Euclidean length. One should also p enalize propagation through regions or directions that are anatomically implausible or metab olically exp ensive. Definition 2.2 (Anatomical cost densit y) . An anatomic al c ost density is a con tinuous function c : Ω × S d − 1 → (0 , ∞ ) suc h that there exist constants 0 < c ∗ ≤ c ∗ < ∞ with c ∗ ≤ c ( x, τ ) ≤ c ∗ ∀ ( x, τ ) ∈ Ω × S d − 1 . The costs c ( x, τ ) ma y encode structural connectivit y , tractograph y constraints, tissue anisotrop y , or metab olic p enalties. Throughout the pap er we fix α ∈ (0 , 1) . The concavit y of w 7→ w α is what drives the branching effect: grouping tw o flows b efore they split is cheaper than transp orting them indep enden tly o ver the same distance. 3 Estimation of µ + stim , µ − react , and c ( x, τ ) from data This section explains how the abstract ingredients of the mo del ma y b e estimated from empirical neuroimaging and tractograph y data. Our goal is not to prescrib e a unique prepro cessing pip eline, but rather to indicate a mathematically consisten t interface b etw een the v ariational form ulation and realistic data analysis. 3.1 General principle The source measure µ + stim is mean t to represen t the neural en try distribution of the external stim ulation, while the target measure µ − react represen ts the spatial distribution of activity asso ciated with the pro duction of the reaction. In practice, b oth measures can b e estimated either at the level of regions of interest (R OI-lev el atomic measures) or at the vo xel/source-space lev el (absolutely con tin uous or k ernel-smo othed measures). A common template is the following. Let ( R i ) N i =1 b e a brain partition in to R OIs with represen tativ e p oin ts x i ∈ Ω , and let a stim i , a react i ≥ 0 b e nonnegativ e activit y scores extracted from early and late temp oral windows, resp ectiv ely . Then one sets µ + stim = m N X i =1 a stim i P N j =1 a stim j δ x i , µ − react = m N X i =1 a react i P N j =1 a react j δ x i , pro vided the denominators are nonzero. If a con tinuous represen tation is preferred, one replaces eac h Dirac mass δ x i b y a smo oth kernel φ i ≥ 0 supp orted in R i and normalized by R Ω φ i ( x ) dx = 1 . The anatomical cost density c ( x, τ ) should enco de the lo cal ease or difficulty of information propagation. In applications, it may b e estimated from diffusion MRI, tractography , structural connectivit y , or other anatomical priors. This reduction of neuroimaging data to source and target measures is consistent with the standard in terface u sed in large-scale brain mo deling, where parcellated regional activity and structural priors pro vide the natural bridge b etw een empirical measurements and mesoscale mathematical models [ 13 , 22 ]. In the same spirit, recen t work on biologically annotated connectomes and macroscale structure–function coupling emphasizes that regional summaries should b e interpreted jointly with anatomical and dynamical constrain ts, rather than as purely descriptiv e observ ables [ 2 , 9 ]. 5 3.2 Estimation from fMRI and EEG/MEG data F or fMRI data, let y i ( t ) denote the prepro cessed BOLD signal in R OI R i , and let e y i ( t ) = y i ( t ) − ¯ y base i b e the baseline-corrected signal. In task-based fMRI, task-evok ed resp onses are routinely summarized through BOLD con trasts and generalized-linear-mo del-based statistics, whic h pro vide a natural source of nonnegative regional scores for the construction of µ + stim and µ − react [ 10 , 7 ]. More generally , task-based fMRI is explicitly designed to map the neural resp onse to cognitiv e, p erceptual, or motor manipulations through BOLD fluctuations, making early and late activ ation windows a natural pro xy for stim ulus-entry and reaction-related activity , resp ectiv ely [ 7 ]. Early stim ulus-related scores may then b e defined b y a stim i = Z t 1 t 0  e y i ( t )  + dt, or, more generally , b y a p ositive task-evok ed contrast such as a GLM co efficient or a t -statistic. Lik ewise, late reaction-related scores may b e defined by a react i = Z t 3 t 2  e y i ( t )  + dt. These quan tities are then inserted into the general normalization form ula ab o v e. F or EEG or MEG data, after source lo calization one obtains regional source time series u i ( t ) . In this case, the same construction can b e applied with a stim i = Z t 1 t 0 | u i ( t ) | dt, a react i = Z t 3 t 2 | u i ( t ) | dt, or with signed v ariants if one wishes to distinguish excitatory increases from other effects. F or EEG and MEG, source reconstruction pro vides spatially resolved time series asso ciated with cortical or sub cortical generators, thereb y offering a natural route to define regional activity scores at muc h finer temp oral resolution than fMRI [ 18 , 1 ]. This high temp oral resolution is particularly adv an tageous when one aims to distinguish early stimulus-driv en comp onents from later reaction-related comp onen ts, which is precisely the temp oral separation required by the presen t source–target construction [ 18 ]. Remark 3.1. If deactiv ations are considered relev ant, one may either treat p ositive and negativ e parts separately or in tro duce multi-source/m ulti-target signed decomp ositions. In the presen t pap er we restrict attention to positive measures, whic h is the natural setting for branc hed transp ort. 3.3 Estimation of the anatomical cost densit y W e now indicate sev eral p ossible constructions of the anatomical cost density c ( x, τ ) . The use of diffusion MRI and tractograph y to construct anatomical transp ort costs is consisten t with a broad connectomic literature in whic h white-matter architecture is used to estimate structural connectivity and to constrain large-scale communication mo dels [ 15 , 29 , 28 ]. In particular, b oth tract-sp ecific and connectome-based analyses routinely extract quantitativ e information from tractography that can b e incorp orated into spatially heterogeneous or direction- dep enden t w eigh ts, exactly of the kind enco ded here b y c ( x, τ ) [ 29 , 28 ]. More generally , recent w ork on biologically annotated connectomes and structure–function coupling suggests that macroscale brain mo dels should in tegrate wiring information with lo cal biological or geometric 6 annotations, whic h further supp orts the use of anisotropic and biologically informed transp ort costs in the present framework [ 2 , 9 ]. A simple isotropic heterogeneous mo del is obtained from a smo oth white-matter score w : Ω → [0 , 1] by setting c ( x, τ ) = c min + ( c max − c min )(1 − w ( x )) , 0 < c min < c max , so that propagation through white matter is c heap er than propagation through less fa v orable regions. A direction-dependent mo del can b e deriv ed from a con tinuous field of symmetric n onnegativ e matrices D ( x ) ∈ R d × d , for instance estimated from diffusion MRI or tractograph y , by defining c ( x, τ ) = q τ ⊤  D ( x ) + εI  − 1 τ , ε > 0 . This makes transp ort c heap er along directions aligned with strong local diffusion or fib er orien tation. A more flexible mixed mo del is c ( x, τ ) = a ( x ) q τ ⊤ A ( x ) τ + b ( x ) , where a, b : Ω → (0 , ∞ ) are con tin uous and b ounded ab o v e and b elo w by p ositive constants, and A ( x ) is a uniformly p ositive definite symmetric matrix field. Finally , if one has a contin uous directional plausibility score p : Ω × S d − 1 → [0 , 1] , one ma y set c ( x, τ ) = c ∗ + ( c ∗ − c ∗ )  1 − p ( x, τ )  , so that more plausible propagation directions carry lo wer transp ort cost. Remark 3.2. The regularization term εI in tensor-based mo dels guarantees uniform p ositive definiteness and therefore ensures the contin uity and p ositive upp er/low er b ounds required in the definition of anatomical cost density . 4 Discrete graph mo del 4.1 W eigh ted directed graphs W e first form ulate a discrete mo del, since it directly returns a graph and is particularly conv enient for computational implementations. Definition 4.1 (Embedded weigh ted directed graph) . An emb e dde d weighte d dir e cte d gr aph in Ω is a quadruple G = ( V , E , w ) with the following comp onents: ( i ) V = { v 1 , . . . , v N } ⊂ Ω is a finite set of vertices; ( ii ) E is a finite set of orien ted edges; ( iii ) for eac h e ∈ E , the edge is represented by an injectiv e Lipsc hitz path γ e : [0 , ℓ e ] → Ω , with γ e (0) = v e − and γ e ( ℓ e ) = v e + , where e − , e + ∈ { 1 , . . . , N } denote the tail and head of the edge; 7 ( iv ) w = ( w e ) e ∈ E with w e ≥ 0 is the family of transp orted fluxes. V ertices may represent observed ROIs, laten t rela y regions, or free Steiner-type branching p oin ts. Definition 4.2. F or each edge e ∈ E , define the anatomical-geometric cost β e := Z ℓ e 0 c  γ e ( s ) , ˙ γ e ( s ) | ˙ γ e ( s ) |  ds, whenev er ˙ γ e ( s )  = 0 almost everywhere. Let b : V → R b e a prescrib ed vertex supply/demand function satisfying X v ∈ V b ( v ) = 0 . Heuristically , we interpret: 1. b ( v ) > 0 : stim ulus source at v ; 2. b ( v ) < 0 : reaction sink at v ; 3. b ( v ) = 0 : pure relay/branc hing vertex. Definition 4.3. A weigh ted directed graph ( G, w ) is K ir chhoff-admissible with resp ect to b if for ev ery v ertex v ∈ V , X e ∈ Out( v ) w e − X e ∈ In( v ) w e = b ( v ) . This is the discrete mass-conserv ation law. Definition 4.4. Given α ∈ (0 , 1) , the discr ete br anche d br ain tr ansp ort ener gy of an admissible w eighted graph ( G, w ) is E α ( G, w ) := X e ∈ E β e w α e . The problem of interest is therefore: min {E α ( G, w ) : ( G, w ) ∈ G adm } , (1) where G adm is a prescrib ed admissible family of em b edded weigh ted directed graphs satisfying the Kirc hhoff constrain ts and the source/sink sp ecification induced b y ( µ + stim , µ − react ) . Remark 4.5. The supp ort of a minimizer ( G ∗ , w ∗ ) is interpreted as the br ain r e action map . The edge multiplicities w ∗ e quan tify the amount of routed information (or effective signal load), while the branching structure indicates shared pathw ays and integrativ e hubs. 4.2 Finite-library existence theorem The broadest v ersion of the discrete problem inv olves optimization ov er b oth graph top ology and geometry . W e first consider the imp ortan t case in which one starts from a finite candidate library of p ossible edges. Assumption 4.6 (Finite admissible library) . Ther e is a finite dir e cte d gr aph G 0 = ( V 0 , E 0 ) emb e dde d in Ω , and G adm c onsists of al l weighte d dir e cte d sub gr aphs of G 0 satisfying the K ir chhoff b alanc e law for a pr escrib e d b : V 0 → R . 8 Theorem 4.7. Under the finite-libr ary assumption, the optimization pr oblem ( 1 ) admits at le ast one minimizer. Pr o of. Since the candidate edge set E 0 is finite, an admissible weigh ted subgraph is completely determined b y a nonnegative vector w = ( w e ) e ∈ E 0 ∈ [0 , ∞ ) | E 0 | , with the conv en tion that edges with w e = 0 are inactive. The Kirc hhoff constrain ts are linear: Aw = b, for a suitable incidence matrix A . Hence the admissible set A := { w ∈ [0 , ∞ ) | E 0 | : Aw = b } is closed and conv ex. Because the total outgoing mass from source v ertices is fixed, every admissible w has uniformly b ounded ℓ 1 norm. Indeed, summing the p ositive supplies gives X e ∈ E 0 w e ≤ C ( b, G 0 ) , for some constant dep ending only on the graph incidence structure and the source/sink v ector b . Hence A is b ounded. Since it is also closed in a finite-dimensional space, it is compact. The ob jective w 7→ X e ∈ E 0 β e w α e is contin uous on [0 , ∞ ) | E 0 | . Therefore, by the W eierstrass theorem, it attains its minimum on A . Prop osition 4.8. A ssume that β e > 0 for every e dge e . Then every minimizer of ( 1 ) c an b e chosen cycle-fr e e. Pr o of. Supp ose that an admissible minimizer contains a directed cycle C carrying strictly positive flo w. Let ε := min e ∈ C w e > 0 . Reduce the flow by ε on every edge of the cycle, leaving all other edge w eights unc hanged. The Kirc hhoff constrain ts remain satisfied, since every vertex on the cycle loses the same incoming and outgoing amount. Because t 7→ t α is strictly increasing on (0 , ∞ ) and each β e > 0 , the total energy strictly decreases: X e ∈ C β e  ( w e − ε ) α − w α e  < 0 . This con tradicts minimalit y . Hence a minimizer ma y alw ays b e c hosen without cycles. Remark 4.9. Prop osition 4.8 sho ws that the optimal graph has a tree-like flav or, as exp ected in branched transp ortation. In the present application, such a cycle-free minimizer is naturally in terpreted as a preferred routing tree for the stim ulus-to-reaction propagation. 5 Con tin uous curren t form ulation The discrete mo del is often the most in tuitiv e. How ever, when one wishes to optimize not only w eights but also the geometric placemen t of branches and paths, it is more natural to pass to a con tinuous Eulerian form ulation. 9 5.1 Rectifiable transp ort curren ts Definition 5.1. A vector measure v ∈ M (Ω; R d ) is called r e ctifiable if there exist: ( i ) a coun tably 1 -rectifiable set M ⊂ Ω , ( ii ) a measurable unit tangent field τ : M → S d − 1 , ( iii ) a m ultiplicit y function θ : M → [0 , ∞ ) integrable with resp ect to H 1 ⌞ M , suc h that v = τ θ H 1 ⌞ M . Here M is the unkno wn transp ort netw ork, τ its orientation, and θ the flow m ultiplicity . Definition 5.2. Giv en balanced measures ( µ + stim , µ − react ) , an admissible curren t is a rectifiable v ector measure v satisfying div v = µ + stim − µ − react in the sense of distributions on Ω . Definition 5.3. Let Ω ⊂ R d b e compact. Giv en balanced finite nonnegative measures µ + stim , µ − react ∈ M (Ω) , µ + stim (Ω) = µ − react (Ω) =: m, w e denote b y A ( µ + stim , µ − react ) the class of acyclic rectifiable vector measures v = τ θ H 1 ⌞ M ∈ M (Ω; R d ) suc h that div v = µ + stim − µ − react . 5.2 Con tinuous branc hed transp ort functional F or a rectifiable curren t v = τ θ H 1 ⌞ M , define M α,c ( v ) := Z M c ( x, τ ( x )) θ ( x ) α d H 1 ( x ) , and set M α,c ( v ) = + ∞ if v is not rectifiable. The con tin uous brain mapping problem is: min n M α,c ( v ) : v ∈ M (Ω; R d ) , div v = µ + stim − µ − react o . (2) Remark 5.4. The unkno wn support M ∗ = supp ( v ∗ ) of a minimizer v ∗ is the con tinuous analogue of the optimal transp ort graph. It is the inferred brain routing arc hitecture. The m ultiplicity θ ∗ quan tifies the amoun t of information routed through each p oint of the netw ork. Lemma 5.5. L et α ∈ (1 − 1 d , 1) . A ssume that Ω ⊂ R d is c omp act and that c : Ω × S d − 1 → (0 , ∞ ) is c ontinuous and satisfies 0 < c ∗ ≤ c ( x, τ ) ≤ c ∗ < ∞ ∀ ( x, τ ) ∈ Ω × S d − 1 . L et ( v n ) n ⊂ A ( µ + stim , µ − react ) b e a se quenc e such that sup n M α,c ( v n ) < ∞ 10 and v n ∗ ⇀ v we akly-* in M (Ω; R d ) . Then v ∈ A ( µ + stim , µ − react ) and M α,c ( v ) ≤ lim inf n →∞ M α,c ( v n ) . Pr o of. The argument relies on the standard Eulerian–Lagrangian corresp ondence for branched transp ort, compactness of traffic plans, and low er semicontin uit y of the corresp onding Lagrangian H -mass. Since v n ∗ ⇀ v in M (Ω; R d ) and div v n = µ + stim − µ − react ∀ n ∈ N , the div ergence constrain t passes to the limit. Indeed, for every φ ∈ C 1 (Ω) , Z Ω ∇ φ · dv = lim n →∞ Z Ω ∇ φ · dv n = − lim n →∞ Z Ω φ d ( µ + stim − µ − react ) = − Z Ω φ d ( µ + stim − µ − react ) , hence div v = µ + stim − µ − react . Moreo v er, since c ≥ c ∗ , the sequence has uniformly b ounded standard branched transp ort mass: c ∗ Z M n θ α n d H 1 ≤ Z M n c ( x, τ n ( x )) θ α n d H 1 = M α,c ( v n ) , and therefore sup n Z M n θ α n d H 1 < ∞ . A t this p oin t we pass from the Eulerian formulation to the Lagrangian one. Since eac h v n b elongs to A ( µ + stim , µ − react ) , it is in particular acyclic. By the standard corresp ondence b et w een acyclic transp ort currents and go o d irrigation plans, there exists for every n an irrigation plan η n whose asso ciated current is exactly v n and such that the corresp onding energy formula holds: M α,c ( v n ) = I α,c ( η n ) . Here I α,c ( η ) := Z Γ Z + ∞ 0 c  γ ( t ) , ˙ γ ( t ) | ˙ γ ( t ) |  | ˙ γ ( t ) | Θ η ( γ ( t )) α − 1 dt dη ( γ ) , where Γ denotes the space of 1-Lipschitz even tually constant curves in Ω , and Θ η ( x ) is the traffic m ultiplicit y at x . The equiv alence b et ween the Eulerian and Lagrangian formulations of branc hed transp ort, together with the corresp onding energy iden tity , is standard; see [ 3 , 23 ]. Let m := µ + stim (Ω) = µ − react (Ω) . Since the multiplicit y of a traffic plan is b ounded ab o v e b y the total transp orted mass, one has 0 ≤ Θ η n ( x ) ≤ m ∀ x ∈ Ω . Because α − 1 < 0 , it follo ws that Θ η n ( x ) α − 1 ≥ m α − 1 . 11 Using also the low er b ound c ≥ c ∗ , w e obtain I α,c ( η n ) ≥ c ∗ m α − 1 Z Γ Z + ∞ 0 | ˙ γ ( t ) | dt dη n ( γ ) = c ∗ m α − 1 Z Γ ℓ ( γ ) dη n ( γ ) , where ℓ ( γ ) := Z + ∞ 0 | ˙ γ ( t ) | dt. Hence sup n Z Γ ℓ ( γ ) dη n ( γ ) < ∞ . By the compactness theorem for traffic plans with uniformly b ounded av erage length (see [ 3 , Chapter 3]), after extraction of a subsequence not relab elled, there exists a traffic plan η suc h that η n ⇀ η narro wly in P (Γ) . Since th e endp oint ev aluation maps are contin uous on the relev ant compact subsets of Γ , the initial and terminal marginals are preserved in the limit, so that η is an admissible irrigation plan from µ + stim to µ − react . These facts are standard in the theory of traffic plans; see [ 3 ]. Let v η denote the current induced b y η . By contin uit y of the sup erp osition map from traffic plans to vector-v alued curren ts under narrow conv ergence and uniform length control, one has v n ∗ ⇀ v η in M (Ω; R d ) . By uniqueness of the weak-* limit, it follo ws that v = v η . W e now use lo w er semicon tin uit y on the Lagrangian side. The functional I α,c is a weigh ted Lagrangian H -mass with H ( r ) = r α . The low er semicontin uit y of the Lagrangian H -mass is pro v ed in [ 19 ]; in the present setting the same argument applies to the contin uous b ounded anisotropic w eigh t ( x, τ ) 7− → c ( x, τ ) , and therefore yields I α,c ( η ) ≤ lim inf n →∞ I α,c ( η n ) . Com bining this with the energy identit y gives M α,c ( v ) = M α,c ( v η ) = I α,c ( η ) ≤ lim inf n →∞ I α,c ( η n ) = lim inf n →∞ M α,c ( v n ) . Finally , since I α,c ( η ) < ∞ , the asso ciated current v η is rectifiable; see again [ 3 , 23 ]. Therefore v = v η ∈ A ( µ + stim , µ − react ) , and the pro of is complete. W e now state a standard well-posedness result in the spirit of branc hed transp ort theory . Theorem 5.6. L et α ∈ (1 − 1 d , 1) . A ssume that: (i) Ω ⊂ R d is c omp act and c onne cte d; (ii) µ + stim and µ − react ar e b alanc e d finite nonne gative me asur es on Ω ; 12 (iii) c : Ω × S d − 1 → (0 , ∞ ) is c ontinuous and ther e exist c onstants 0 < c ∗ ≤ c ∗ < ∞ such that c ∗ ≤ c ( x, τ ) ≤ c ∗ ∀ ( x, τ ) ∈ Ω × S d − 1 . Then the minimization pr oblem inf n M α,c ( v ) : v ∈ A ( µ + stim , µ − react ) o admits at le ast one minimizer. Pr o of. W e first sho w that the admissible class is nonempt y and that the infim um is finite. Since α > 1 − 1 d and the measures µ + stim and µ − react are balanced on the compact set Ω , the classical irrigability theorem yields the existence of an admissible irrigation plan with finite α -energy . Passing to the asso ciated Eulerian current, one obtains some v 0 ∈ A ( µ + stim , µ − react ) suc h that M α,c ( v 0 ) ≤ c ∗ M α ( v 0 ) < ∞ . Hence the infimum is finite. Let ( v n ) n ⊂ A ( µ + stim , µ − react ) b e a minimizing sequence: M α,c ( v n ) − → inf n M α,c ( v ) : v ∈ A ( µ + stim , µ − react ) o . In particular, there exists C > 0 such that M α,c ( v n ) ≤ C ∀ n ∈ N . Since the energy density is strictly p ositive, any cyclic comp onent can b e remo v ed without c hanging the div ergence and while decreasing the energy . Therefore, without loss of generality , w e may assume that ev ery v n is acyclic. By the corresp ondence with go o d irrigation plans and the compactness theorem for traffic plans with uniformly b ounded av erage length, after extraction of a subsequence not relab elled there exists v ∈ M (Ω; R d ) suc h that v n ∗ ⇀ v in M (Ω; R d ) . W e may now apply Lemma 5.5 . It follo ws that v ∈ A ( µ + stim , µ − react ) and M α,c ( v ) ≤ lim inf n →∞ M α,c ( v n ) . Since ( v n ) n is minimizing, the right-hand side is exactly the infim um of the problem. Hence M α,c ( v ) = inf n M α,c ( w ) : w ∈ A ( µ + stim , µ − react ) o , whic h pro ves that v is a minimizer. 13 6 Hybrid sto c hastic extension The branc hed transp ort mo del in tro duced ab o v e is purely geometric: it selects a routing arc hitecture b y minimizing a ramified transp ort cost b etw een a stim ulation source measure and a reaction target measure. In many situations, how ever, one also wishes to enco de whether a given routing architecture is dynamically compatible with observ ed brain-state transitions. This motiv ates a hybrid extension in which eac h admissible weigh ted graph induces a sto c hastic neural dynamics, and the graph is p enalized not only by its transp ort energy , but also b y the minim um con trol effort required to realize prescrib ed initial and terminal distributions under that dynamics. 6.1 Graph-induced linear sto c hastic dynamics W e work in the finite-library discrete setting introduced in Section 4 . Let ( G, w ) ∈ G adm b e an admissible weigh ted graph with vertex set V = { v 1 , . . . , v n } . W e identify each vertex with one neural state v ariable, so that the neural activit y is describ ed b y a v ector X t = ( X 1 t , . . . , X n t ) ∈ R n . Definition 6.1. Giv en an admissible w eighted graph ( G, w ) , define the directed w eigh ted adjacency matrix W G,w = ( W ij ) 1 ≤ i,j ≤ n b y W ij := X e ∈ E e − = i, e + = j w e , that is, W ij is the total flux carried by edges directed from v i to v j . Since the sto chastic d ynamics b elow is meant to represent effective propagation rather than purely directed mass balance, we introduce the symmetrized w eighted adjacency S G,w := 1 2  W G,w + W ⊤ G,w  . W e then define the w eigh ted degree matrix D G,w := diag   n X j =1 ( S G,w ) 1 j , . . . , n X j =1 ( S G,w ) nj   , and the asso ciated weigh ted graph Laplacian L G,w := D G,w − S G,w . The matrix L G,w is symmetric and p ositive semidefinite. It is therefore natural to use it as the basic graph-induced diffusion op erator in a linear neural dynamics. Definition 6.2. Fix constants κ > 0 , β > 0 , σ 0 > 0 , σ 1 ≥ 0 . F or each admissible w eighted graph ( G, w ) , define A G,w := − κI n − β L G,w , (3) and C G,w := σ 0 I n + σ 1 D 1 / 2 G,w . (4) 14 The in terpretation is straigh tforward: the Laplacian term describes graph-mediated spreading of activity , the scalar term − κI n mo dels intrinsic relaxation tow ard baseline, and the matrix C G,w allo ws the amplitude of sto chastic fluctuations to dep end on the lo cal weigh ted degree while preserving uniform nondegeneracy through σ 0 I n . W e then consider the con trolled linear sto chastic differential equation dX t = A G,w X t dt + B stim a t dt + u t dt + C G,w dW t , t ∈ [0 , T ] , (5) where: (i) X t ∈ R n is the neural state vector; (ii) a t is a prescrib ed exogenous stimulus profile; (iii) B stim ∈ R n × m stim injects the stimulation into designated entry regions; (iv) u t is an adapted control pro cess with v alues in R n ; (v) W t is a standard n -dimensional Brownian motion. The uncon trolled pro cess corresp onds to u t ≡ 0 . Prop osition 6.3. F or every admissible weighte d gr aph ( G, w ) , the matrix A G,w define d in ( 3 ) is symmetric ne gative definite. In p articular, al l its eigenvalues ar e strictly ne gative, and the unc ontr ol le d line ar system is exp onential ly stable. Pr o of. Since L G,w is symmetric p ositive semidefinite, all its eigenv alues are nonnegative. Hence the eigen v alues of A G,w = − κI n − β L G,w are of the form − κ − β λ, λ ≥ 0 . Since κ > 0 and β > 0 , ev ery such eigenv alue is strictly negative. Prop osition 6.4. F or every admissible weighte d gr aph ( G, w ) , the matrix C G,w define d in ( 4 ) is symmetric p ositive definite. Mor e over, C G,w C ⊤ G,w ≥ σ 2 0 I n for al l ( G, w ) ∈ G adm . Pr o of. Since D G,w is diagonal with nonnegativ e entries, D 1 / 2 G,w is well defined and p ositive semidefinite. Therefore C G,w = σ 0 I n + σ 1 D 1 / 2 G,w is symmetric and all its eigen v alues are b ounded b elow b y σ 0 > 0 . Hence C G,w is p ositive definite. The matrix inequality follows immediately . Remark 6.5. The maps ( G, w ) 7− → W G,w , ( G, w ) 7− → S G,w , ( G, w ) 7− → D G,w , ( G, w ) 7− → L G,w are con tinuous in the finite-dimensional edge-weigh t parameterization of the admissible graph library . Consequently , ( G, w ) 7− → A G,w , ( G, w ) 7− → C G,w are con tin uous as well. 15 6.2 P ath-space relative en tropy and dynamic cost Fix T > 0 and tw o Gaussian marginals X 0 ∼ N ( m 0 , Σ 0 ) , X T ∼ N ( m T , Σ T ) , with Σ 0 , Σ T symmetric p ositive definite. F or eac h admissible w eighted graph ( G, w ) , let P G,w,u denote the law on path space Ω T := C ([ 0 , T ]; R n ) induced b y ( 5 ), and let P G,w, 0 denote the corresp onding uncontrolled law. Definition 6.6. The graph-dep endent dynamic cost is defined by J dyn ( G, w ) := inf u KL  P G,w,u    P G,w, 0  , where the infim um is tak en ov er all adapted controls u suc h that the corresp onding con trolled la w P G,w,u satisfies the prescrib ed initial and terminal Gaussian marginal constrain ts. Th us, J dyn ( G, w ) is a graph-dep enden t Sc hrö dinger bridge cost: it measures the minimum path-space relativ e entrop y needed to deform the uncontrolled diffusion induced by the graph in to a con trolled pro cess realizing the required endp oint distributions. Prop osition 6.7. F or every admissible weighte d gr aph ( G, w ) , KL  P G,w,u    P G,w, 0  = Z Ω T log d P G,w,u d P G,w, 0 ! d P G,w,u . Mor e over, under the standar d assumptions of Girsanov’s the or em, KL  P G,w,u    P G,w, 0  = 1 2 E G,w,u " Z T 0 u ⊤ t  C G,w C ⊤ G,w  − 1 u t dt # . Pr o of. The first iden tity is the definition of relativ e en tropy . Since the con trolled and uncon trolled pro cesses differ only b y the drift term u t , and since C G,w C ⊤ G,w is in vertible by Prop osition 6.4 , Girsano v’s theorem yields the Radon–Nik o dym deriv ativ e of P G,w,u with resp ect to P G,w, 0 . T aking the logarithm and integrating with resp ect to the controlled la w giv es the stated quadratic form ula. Remark 6.8. Prop osition 6.7 sho ws that J dyn ( G, w ) is a graph-dep endent quadratic sto c hastic con trol cost. The hybrid mo del therefore supplemen ts the ramified transp ort energy with a minim um-energy con trol p enalt y on the graph-induced diffusion. Because the con trol en ters additively in all state comp onents and the noise is uniformly nondegenerate, the endp oin t-constrained linear-Gaussian bridge problem is feasible under the presen t assumptions. Prop osition 6.9. F or every admissible weighte d gr aph ( G, w ) ∈ G adm , the minimization pr oblem in Definition 6.6 is fe asible, the value J dyn ( G, w ) is finite, and the infimum is attaine d. Pr o of. By Prop osition 6.3 , the uncontrolled drift is exp onentially stable, and b y Prop osition 6.4 the diffusion is uniformly nondegenerate. Since the control enters through the full state v ector, standard linear-Gaussian steering arguments imply that one can realize the prescrib ed initial and terminal Gaussian marginals ov er the finite time horizon [0 , T ] . Hence the admissible class in Definition 6.6 is nonempty . 16 By Prop osition 6.7 , the dynamic cost is the v alue of a strictly con vex quadratic control problem of the form inf u 1 2 E G,w,u " Z T 0 u ⊤ t  C G,w C ⊤ G,w  − 1 u t dt # under endp oint marginal constraints. Since the in tegrand is nonnegative and co ercive, and the admissible class is nonempty , the direct metho d in the corresp onding Hilb ert control space yields existence of an optimal adapted control. In particular, the v alue is finite and attained. 6.3 Hybrid graph-selection functional W e now combine the geometric branc hed transp ort term with the graph-dep endent dynamic cost. Definition 6.10 (Hybrid functional) . Let λ ≥ 0 . The h ybrid branc hed-sto c hastic functional is F λ ( G, w ) := E α ( G, w ) + λ J dyn ( G, w ) , ( G, w ) ∈ G adm . The asso ciated hybrid optimization problem is min n F λ ( G, w ) : ( G, w ) ∈ G adm o . (6) Remark 6.11. The parameter λ balances: (i) a purely geometric/anatomical routing criterion, enco ded b y the branched transp ort energy E α , and (ii) a dynamical compatibility criterion, enco ded b y the minimum path-space control cost J dyn . F or λ = 0 , one reco vers the purely geometric mo del. F or λ > 0 , graphs that are dynamically harder to supp ort b ecome less fav orable even if they are geometrically efficient. T o obtain existence of minimizers for ( 6 ) , one needs a stability prop ert y of the graph-dep enden t bridge cost. Assumption 6.12. The map ( G, w ) 7− → J dyn ( G, w ) is lower semic ontinuous on G adm with r esp e ct to the finite-dimensional e dge-weight p ar ameteriza- tion of the admissible gr aph libr ary. Remark 6.13. Under the canonical c hoice A G,w = − κI n − β L G,w , C G,w = σ 0 I n + σ 1 D 1 / 2 G,w , con tinuit y of ( G, w ) 7→ A G,w and ( G, w ) 7→ C G,w is immediate. It is therefore natural to exp ect lo w er semicontin uity of J dyn from the stabilit y theory of linear-Gaussian Sc hrö dinger bridge problems. W e record it here as an assumption in order to keep the fo cus on the v ariational graph-selection problem. Theorem 6.14. A ssume the finite-libr ary setting of The or em 4.7 , and assume mor e over A s- sumption 6.12 . Then the hybrid pr oblem min n F λ ( G, w ) : ( G, w ) ∈ G adm o admits at le ast one minimizer. 17 Pr o of. By the finite-library assumption, the admissible set G adm is compact in the finite- dimensional edge-weigh t parameterization; this was established in the pro of of Theorem 4.7 . The geometric energy ( G, w ) 7− → E α ( G, w ) is con tin uous on G adm , while the dynamic term ( G, w ) 7− → J dyn ( G, w ) is finite by Prop osition 6.9 and low er semicontin uous by Assumption 6.12 . Therefore ( G, w ) 7− → F λ ( G, w ) = E α ( G, w ) + λ J dyn ( G, w ) is low er semicontin uous on a compact set. By the direct metho d of the calculus of v ariations, it attains its minimum on G adm . 7 In terpretation of the minimizer W e now discuss the neuroscientific meaning of the optimal solution selected by the v ariational problem. The central mo deling principle of the pap er is that the minimizer is not merely an auxiliary mathematical ob ject, but rather the inferred routing architecture through which the stim ulation is propagated tow ard the reaction-pro ducing configuration. Definition 7.1. Let v ∗ b e a minimizer of the contin uous problem ( 2 ), and write v ∗ = τ ∗ θ ∗ H 1 ⌞ M ∗ for its rectifiable representation. Let ( G ∗ , w ∗ ) b e a minimizer of the discrete problem ( 1 ) . The br ain r e action map asso ciated with the stimulus–reaction pair ( µ + stim , µ − react ) is defined as: (i) the supp ort M ∗ = supp( v ∗ ) in the contin uous setting; (ii) the em b edded weigh ted graph ( G ∗ , w ∗ ) in the discrete setting. This definition reflects the guiding idea of the pap er: the graph/current is the primary unkno wn, and its supp ort is interpreted as the preferred stimulus-to-reaction routing arc hitecture. In particular, the minimizer should not b e viewed merely as a transp ort ob ject connecting t wo measures, but as a candidate mesoscale map of neural propagation. F rom this p ersp ectiv e, the optimal structure carries several distinct lev els of in terpretation. Branc hing p oints or v ertices of high degree indicate regions where information is aggregated, redistributed, or split, and may therefore b e regarded as integrativ e hubs. Edges with large w eigh ts corresp ond to routes that carry a substantial fraction of the transp orted flux and can b e interpreted as principal propagation pathw ays. T erminal branches lo cated near the supp ort of µ + stim iden tify lik ely entry routes of the stimulation into the effective transp ort architecture, whereas terminal branches lo cated near the supp ort of µ − react iden tify the regions and pathw ays most directly inv olved in the pro duction of the reaction. Finally , the total v alue of the ob jective functional pro vides a quantitativ e measure of the ov erall difficulty of pro ducing the reaction from the stimulation under the anatomical and geometric constrain ts enco ded b y the mo del. The brain reaction map should b e in terpreted as an effective routing architecture selected b y a v ariational principle. In this sense, it is conceptually closer to a mesoscale or functional transp ort backbone than to a direct structural connectome. Its role is to identify whic h pathw a ys are join tly preferred by the source–target configuration and b y the transp ort cost. The distinction b etw een the discrete and contin uous f orm ulations is also meaningful from an in terpretative viewp oin t. In the discrete mo del, the minimizer is immediately represented as a graph ov er a prescrib ed family of candidate regions and routes, which is particularly conv enien t 18 for computational implemen tations and R OI-lev el analyses. In the contin uous formulation, by con trast, the minimizer is a rectifiable current whose supp ort is free to select its o wn geometric routing structure. The con tin uous mo del is therefore b etter suited to situations in which one do es not wish to prescribe the branching geometry in adv ance, but instead aims to infer it directly from the source–target pair. A practical implemen tation of the framew ork requires estimating the pair ( µ + stim , µ − react ) together with the anatomical cost density c ( x, τ ) from data. As discussed in Section 3 , the source measure µ + stim ma y b e estimated from early p ost-stimulus activity lo calized in sensory or en try regions, while the target measure µ − react ma y b e estimated from later activit y asso ciated with the realization of the reaction, for instance in premotor, motor, asso ciative, or sub cortical regions. The cost densit y c ( x, τ ) may b e estimated from diffusion MRI, tractography , structural connectivit y , or suitable hand-crafted priors that enco de lo cal ease or difficult y of propagation. In the hybrid mo del introduced in Section 6 , the in terpretation b ecomes richer. In that setting, the minimizer is selected not only by geometric transp ort efficiency , but also by its compatibilit y with a sto chastic neural dynamics induced b y the graph. The matrices A G,w and C G,w enco de, resp ectively , effectiv e propagation and noise structure on the optimal weigh ted graph. The additional dynamic term J dyn ( G, w ) then measures the minimum path-space control cost required to realize prescrib ed endp oin t distributions on that graph. Consequen tly , the hybrid minimizer may b e interpreted as a routing arc hitecture that is sim ultaneously geometrically economical and dynamically plausible. T aken together, these considerations suggest that the prop osed framew ork should b e viewed as a v ariational to ol for inferring effectiv e stim u lus-to-reaction propagation maps from source– target information and anatomical priors. Its main output is not a tra jectory , but a transp ort arc hitecture. This is precisely the sense in whic h the minimizer provides a mathematically defined candidate for a brain reaction map. 8 Conclusion W e hav e prop os ed a v ariational framew ork for stimulus-to-reaction brain mapping in whic h the unkno wn is an optimal transp ort graph or current rather than a control acting on a fixed substrate. The supp ort of the minimizer is in terpreted as a brain reaction map, namely an effective routing architecture through whic h stimulation is propagated tow ard the reaction-pro ducing configuration. The mo del is built on anisotropic branc hed optimal transp ort, b oth in a discrete graph form ulation and in a con tin uous current formulation. In each case, conca vity of the flux cost promotes aggregation and branc hing, yielding routing structures that naturally enco de shared propagation path wa ys. W e established basic w ell-p osedness results in the discrete setting and pro ved existence of minimizers in the contin uous setting. W e also introduced a hybrid sto chastic extension in whic h each admissible graph induces a linear sto chastic neural dynamics, and an additional path-space Kullback–Leibler term measures the minimum control effort required to realize prescrib ed endp oint distributions on that graph. The main conceptual contribution of the pap er is therefore a shift from control of tra jectories on fixed net w orks to v ariational inference of propagation arc hitectures. This makes the framew ork particularly app ealing when the primary quantit y of in terest is not only the cost of a state transition, but the effective routing bac kb one supp orting it. F uture w ork should address finer structural prop erties of minimizers, stability with resp ect to p erturbations of the data, and quan titativ e comparisons with empirical structural and functional neuroimaging datasets. 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