Mathematical and Dynamic Modeling of the Anatomical Localization of the Insula in the Brain

Abstract

The insula, a deeply situated cortical structure beneath the Sylvian sulcus, plays a critical role in sensory integration, emotion regulation, and cognitive control in the brain. Although several studies have described its anatomical and functional characteristics, mathematical models that quantitatively represent the insula's complex structure and connectivity are lacking. This study aimed to develop a mathematical model to represent the anatomical localization and functional organization of the insula, drawing on current neuroimaging findings and established anatomical data. A three-dimensional (3D) ellipsoid model was constructed to mathematically represent the anatomical boundaries of the insula using Montreal Neurological Institute (MNI) coordinate data. This geometric model adapts the ellipsoid equation to reflect the spatial configuration of the insula and is primarily based on cytoarchitectonic mapping and anatomical literature. Relevant findings from prior imaging research, particularly those reporting microstructural variations across insular subdivisions, were reviewed and conceptually integrated to guide the model's structural assumptions and interpretation of potential applications. The ellipsoid-based 3D model accurately represented the anatomical dimensions and spatial localization of the right insula, centered at the MNI coordinates (40, 5, 5 mm), and matched well with the known volumetric data. Functional regions (face, hand, and foot) were successfully plotted within the model, and statistical analysis confirmed significant differences along the anteroposterior and superoinferior axes (p < 0.01 and p < 0.05, respectively). Dynamic simulations revealed oscillatory patterns of excitatory and inhibitory neural activity, consistent with established insular neurophysiology. Additionally, connectivity modeling demonstrated strong bidirectional interactions between the insula and key regions, such as the prefrontal cortex and anterior cingulate cortex (ACC), reflecting its integrative role in brain networks. This study presents a scientifically validated mathematical model that captures the anatomical structure, functional subdivisions, and dynamic connectivity patterns of the insula. By integrating anatomical data with computational simulations, this model provides a foundation for future research in neuroimaging, functional mapping, and clinical applications involving insula-related disorders.

Keywords: Anatomical localization; Functional connectivity; Insula; MNI coordinates; Mathematical modeling.

Fig. 1

3D reconstruction of insula. Anterior (a) and lateral (b) views are presented. The frontal, parietal, and temporal lobes cover the insula, which forms the floor of the lateral sulcus. The insula becomes visible when the frontal, parietal, and temporal opercula are removed

Fig. 2

3D diagram illustrating the MNI coordinate system and the central anatomical locations of the right and left insula. The red, green, and blue arrows represent the x (mediolateral), y (anteroposterior), and z (superoinferior) axes, respectively. The magenta and cyan spheres indicate the locations and spatial dimensions of the right and left insula, centered at (40, 5, 5) mm and (–40, 5, 5) mm, respectively

Fig. 3

Mathematical model of the right insula. A 3D plot of the right insula modeled as an ellipsoid in the MNI coordinate space. This visualization shows the spatial extent and location of the insula along the mediolateral (X), anteroposterior (Y), and superoinferior (Z) axes of the brain

Fig. 4

3D Reconstruction of Insular Functional Subdivisions (a) This 3D plot visualizes the anatomical localization of three functional subdivisions of the left insular cortex: face (red), hand (green), and foot (blue), in MNI space. The X-axis represents mediolateral positioning, the Y-axis represents anteroposterior alignment, and the Z-axis represents the superoinferior direction. The coordinates used are based on established MNI-based functional mapping data. (b) This 3D diagram illustrates the standard anatomical reference axes in the MNI space used in mathematical modeling. The x-axis (red) represents the mediolateral direction, with positive values indicating lateral movement and negative values indicating the medial position. The y-axis (green) denotes the anteroposterior direction, with positive values toward the anterior and negative values toward the posterior of the brain. The z-axis (blue) reflects the superoinferior direction, where positive values indicate superior positions and negative values indicate inferior positions. These axes show the anatomical coordinates in the brain, including the localization of the insula and its functional subdivisions (face, hand, and foot)

Fig. 5

Morphometric measurements of the insula (mm). (a) The shortest horizontal distances from the insula to the anatomical midline of the brain were measured. The anatomical midline represents the central sagittal plane that divides the brain into the right and left hemispheres. (b) 3D anatomical illustration of the right (red) and left (blue) insula. Each side was plotted symmetrically around the midline, providing a precise spatial representation of both insulae locations according to the morphometric measurements

Fig. 6

Heat map illustrating the scientific relevance of key parameters involved in the Coordinate System and Mathematical Localization of the insula. The parameters assessed included MNI coordinate accuracy, reliability of coordinate conversions, precision of geometric modeling, microstructural representation, accuracy of functional localization, spatial variability, volumetric consistency, and accuracy of the ellipsoid model. Relevance scores (0–10) highlight the relative importance of each parameter representing the anatomical and functional roles of the insula for model validation and accuracy

Fig. 7

Dynamic simulation of excitatory (x) and inhibitory (y) neuronal activity in the insular cortex (a) The simulation was conducted over a time interval of 0 to 20 s, with both excitatory and inhibitory activity levels ranging from − 1.1 to 1.1 units. The plot demonstrates oscillatory dynamics driven by the interactions between excitatory and inhibitory inputs, reflecting the temporal patterns of neuronal activation in the insula. These dynamics are relevant to functions such as cognitive control, emotional regulation and sensory integration. (b) 3D phase plot of the dynamic model of the insular cortex, illustrating the relationship between excitatory activity (x), inhibitory activity (y), and the rate of change in excitatory activity (dx/dt) over time. The trajectory represents the dynamic interaction between excitatory and inhibitory neuronal activities, generating a cyclic or oscillatory pattern. Such oscillations reflect fundamental neurophysiological processes within the insula, which are implicated in functional domains such as pain perception, emotional regulation, and interoceptive awareness. This visualization highlights how excitatory responses are modulated by inhibitory input, providing insights into the temporal evolution of insular network activity under varying physiological conditions

Fig. 8

Connectivity model of the insula and associated brain regions. Network-level interactions, including the insula, prefrontal cortex, amygdala, anterior cingulate cortex (ACC), and thalamus, were represented. Directed edges indicate the direction of functional connectivity, whereas numerical labels on each edge represent the connection strength wij between regions i and j. The connectivity matrix W underlying this model integrates data from neuroimaging and anatomical studies, highlighting both ascending sensory pathways (thalamus → insula) and descending modulatory pathways (insula → brainstem via the ACC and amygdala)

Fig. 9

Correlation matrix depicting the validation and accuracy of the mathematical and dynamic modeling of the insula. The correlations shown in the matrix underline how effectively different aspects, such as anatomical accuracy, microstructural variations, and functional interactions, are integrated into mathematical modeling. The high correlation between functional connectivity and microstructural differences highlights the biological realism of the mathematical formulations. Conversely, the correlation between mathematical modeling accuracy and neuroimaging validation emphasizes the validation of mathematical predictions, indicating that the proposed geometric and dynamic model reliably represents the actual neuroanatomy and neurophysiology of the insula. These correlations provide a scientific justification for employing mathematical models as effective tools for understanding complex neural structures, such as the insula