Fractional Calculus Operators - Bloch-Torrey Partial Differential Equation - Artificial Neural Networks-Computational Complexity Modeling of the Micro-Macrostructural Brain Tissues with Diffusion MRI Signal Processing and Neuronal Multicomponents

Fractional calculus and fractional-order calculus are arranged in lineage as regards the mathematical models with complexity-theoretical tenets capable of capturing the subtle molecular dynamics by the integration of power-law convolution kernels into time- and space-related derivatives emerging in equations concerning the Magnetic Resonance Imaging (MRI) phenomena to which the fractional models of diffusion and relaxation are applied. Endowed with an intricate level of complexity and a unique physical and structural scaffolding at molecular and cellular levels with numerous synapses forming elaborate neural networks which entail in-depth probing and computing of patterns and signatures in individual cells and neurons, human brain as a heterogeneous medium is constituted of tissues with cells of different sizes and shapes, distributed across an extra-cellular space. Characterization of the unique brain cells is sought after to unravel the connections between different cells and tissues for accurate, reliable, robust and optimal models and computing. Accordingly, Diffusion Magnetic Resonance Imaging (DMRI), as a noninvasive and experimental imaging technique with clinical and research applications, provides a measure related to the diffusion characteristics of water in biological tissues, particularly in the brain tissues. Compatible with these aspects and beyond the diffusion coefficients’ measurement, DMRI technique aims to exceed the spatial resolution of the MRI images and draw inferences from the microstructural properties of the related medium. Thus, novel tools become essential for the description of the biological (organelles, membranes, macromolecules and so on) and neurological (axons, dendrites, neurons and so forth) tissues’ complexity. Mathematical model-based computational analyses with multifaceted methods to extract information from the DMRI with SpinDoctor into neuronal dynamics can provide quantitative parametric instruments in order to reflect the tissue properties focusing on the precise link between the tissue microstructure and signals acquired by employing advanced medical imaging technologies. Coalesced with accurate neuron geometry models as well as numerical DMRI simulations, a novel extended and multifaceted predictive mathematical model based on SpinDoctor and Bloch–Torrey partial differential equation (BTPDE) with the Caputo fractional-order derivative (FOD) with three-parameter [Formula: see text] Mittag-Leffler function (MLF) has been proposed and developed in our study by extending for the application on Brain Neuron Spin Unit dataset with the relevant multi-stage application-related steps. The feedforward neural networks (FFNNs) with BFGS Quasi-Newton equation, as one of the artificial neural network (ANN) algorithms, are applied on BTPDE with Caputo fractional-order derivative for the neurons and their algorithmic complexity is computed by building a BTPDE with Caputo FOD Neuron model based on different fractional orders. The fractional-order degree of the proposed and developed model is applied in relation to their corresponding complexity degrees. Consequently, experimentation and observations from the simulation-driven FFNN (with BFGS Quasi-Newton equation) learning scheme applied to the Bloch–Torrey PDE–Caputo FOD with MLF Neuron model (named as FFNN–BTPDE–CFODMLF Neuron model) proposed in this study, are made. Thus, by investigating whether the mathematical models based on the accurate neuron geometry models obtained can be optimized by comparing the errors in order to define the order parameter and identify to what optimal extent the errors are in relation to the prediction results with a particular focus on the neuron model, we have been able to estimate and predict brain microstructure through DMRI, accentuating mathematical and medical contributions based on the exploitation and corroboration of powerful modeling as well as computational capabilities.