TITLE:
Modeling Dopaminergic Pathways in Schizophrenia with Directed Graphs and Linear Control Theory
AUTHORS:
Samet Güçkiran, Mehmet Enver Analan, Şerife Büyükköse
KEYWORDS:
Schizophrenia, Dopaminergic Pathways, Directed Graph, State-Space Model, Network Controllability, Vandermonde Matrix
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.16 No.2,
August
21,
2026
ABSTRACT: Schizophrenia is considered not as a disorder isolated to a specific region of the brain, but rather as a disconnection syndrome in which interactions and communication between complex brain networks are disrupted. The dopamine hypothesis, which is among the most widely accepted neurobiological approaches for explaining the core mechanism of this disease, is based on a pathological imbalance between cortical and subcortical dopaminergic systems. According to the classical dopamine hypothesis, increased subcortical and striatal dopaminergic function is associated with positive symptoms, while decreased or dysregulated prefrontal dopaminergic function is linked to negative and cognitive symptoms. To fully comprehend these descriptive biochemical findings in neuroscience and to quantitatively model circuit-level disruptions, new mathematical frameworks are needed. In this study, the most extensively researched mesolimbic and mesocortical pathways in the pathophysiology of schizophrenia are examined; the nigrostriatal and other dopaminergic pathways are excluded from the scope of the study. The aim of the study is to transform the classic dopaminergic pathway hypotheses related to schizophrenia into a theoretical mathematical model within the frameworks of directed graphs, adjacency matrices, and linear control systems. As in network neuroscience approaches, specific anatomical regions of the brain affected by dopamine are considered as the “nodes” of the graph, while the directed communication and dopamine flow between these regions are regarded as the “edges” of the graph. While the adjacency matrix represents the presence and direction of anatomical connections, the magnitude and temporal changes of dopaminergic effects are modeled via a weighted state-transition matrix. In this way, the anatomical and dynamic properties of the dopaminergic pathways associated with schizophrenia are expressed as a theoretical network model that can be analyzed with graph theory, matrix analysis, and controllability criteria. For the proposed five-node linear model, the controllability matrix is reduced to Vandermonde form, and it is shown that complete controllability of the system is equivalent to all weights from the VTA to the target regions being nonzero and the self-dynamic coefficients of the target regions being pairwise distinct.