This project aims to simulate and analyze population dynamics in evolutionary systems, using stochastic differential equations and probabilistic rules for crossbreeding and benefits. The study employs advanced computational techniques to model the evolution of populations over an extended period. Specifically, we explore a scenario with ten distinct groups, each starting with an equal population and evolving under defined genetic and cooperative behaviors. The simulation spans 1000 years, incorporating individual lifespans, stochastic updates, and probabilistic interactions. To ensure robustness and reliability, the study includes detailed statistical analysis, encompassing descriptive statistics, confidence intervals, and ANOVA tests. We further visualize the temporal behavior of populations and subpopulations, providing insights into the evolutionary dynamics and stability of the system. The results demonstrate the effectiveness of balanced cooperation and genetic diversity in maintaining stable population distributions. The project showcases the integration of mathematical modeling, statistical analysis, and computational simulations, contributing valuable insights into evolutionary biology and complex systems. It highlights the importance of interdisciplinary approaches in understanding and predicting population behaviors, with implications for both theoretical research and practical applications in fields such as ecology, genetics, probability and social sciences.

Psychiatric disorders, characterized by complex and fluctuating patterns of thoughts, emotions, and behaviors, can be effectively modeled and visualized using mathematical wave equations. This study explores the representation of three major psychiatric conditions— Psychosis, Bipolar Disorder and Obsessive Compulsive Disorder (OCD)—through the application of differential equations, damping factors, and noise perturbations. Psychosis is modeled using Lissajous figures with high-frequency components, damping, and noise, symbolizing the irregular and erratic thought patterns characteristic of this condition. The inclusion of random perturbations and varying frequencies illustrates the chaotic and unpredictable nature of psychotic episodes, while damping effects reflect the temporary reductions in symptom intensity. For bipolar disorder, a system of partial differential equations simulates the cyclical nature of manic and depressive episodes. The model captures high-energy manic phases with periodic disturbances and the subsequent transition to lower-energy depressive states through damping effects. This representation highlights the oscillatory behavior and recurring episodes inherent to bipolar disorder. OCD is represented with Circumferential Complex Equations in Mult directions. The resulting graphical representations provide intuitive visual insights into the dynamics of these psychiatric disorders. By leveraging wave equations and mathematical modeling, this approach offers a novel perspective on understanding and analyzing the complex behaviors associated with OCD, bipolar disorder and psychosis. This method not only aids in the conceptualization of psychiatric phenomena but also underscores the potential of mathematical tools in the study of mental health conditions.

Autism spectrum disorder (ASD) is characterized by diverse patterns of social interaction, communication, and behavior. This study explores the application of clustering and stochastic migration models to visualize and understand the social and neural connectivity patterns associated with autism. The clustering model, with its strong internal connections and limited external connections, mirrors the tight-knit social groups and communication barriers often observed in individuals with autism. In contrast, the stochastic model, characterized by more dispersed connections and greater adaptability, represents the variability and flexibility seen in social interactions across typical individuals. By comparing these models, we can gain deeper insights into the unique challenges and strengths of individuals with autism, highlighting the importance of personalized interventions. Additionally, these models provide a visual framework for understanding the differences in brain connectivity patterns observed in autism, with implications for both behavioral and neural correlates. This study underscores the value of computational models in advancing our understanding of autism and guiding effective support strategies.

In this study, we explore the impact of quantum effects on synaptic activity and their implications for the deterministic model of neural processes. By simulating synaptic events in a large-scale neural network, we demonstrate how small probabilistic quantum events can accumulate and introduce variability in neural behavior. Our model incorporates Dirac delta functions to simulate sudden changes in synaptic probabilities, creating a non-linear, seesaw effect in the cumulative synaptic activity curve. The results reveal that even minimal quantum effects, when scaled to the vast number of synapses in the human brain, can lead to significant dissociations in neural activity. These findings challenge the traditional deterministic view, suggesting that neural processes—and by extension, decision-making and movements—may inherently incorporate elements of randomness. This probabilistic perspective aligns with theories of free will and underscores the need for further empirical research to validate the role of quantum mechanics in neuroscience. Our study provides a theoretical framework for understanding the complex interplay between quantum effects and neural dynamics, opening new avenues for exploring the nature of consciousness and the fundamental principles governing reality.

Multiple Sclerosis (MS) is a chronic autoimmune disorder characterized by the demyelination of neural fibers in the central nervous system. Understanding the impact of demyelination on neural network efficiency is crucial for elucidating the pathophysiological mechanisms underlying MS. This study aims to model the effects of demyelination on neural network efficiency, specifically focusing on how increased resistance and path lengths due to demyelination correlate with the clinical symptoms of MS. We employed a graph-theoretical approach to simulate neural networks, introducing varying degrees of demyelination. The demyelination was represented by increasing the resistance (weights) of selected edges within the network, affecting both random and clustered distributions of network connections. The primary metric analyzed was the average shortest path length, indicative of network efficiency. The models demonstrated a significant increase in the average shortest path length in demyelinated networks compared to intact ones. Specifically, networks with clustered demyelination exhibited localized inefficiencies, correlating with the focal nature of MS lesions. The increase in path length was associated with reduced signal transmission efficiency, providing insights into a range of MS symptoms including motor and sensory impairments, cognitive dysfunction, visual disturbances, and fatigue. The graph-theoretical model of neural network demyelination offers a novel perspective on how structural changes in the brain's connectivity due to MS can lead to diverse clinical manifestations. While the model simplifies the complexity of neural networks, it underscores the potential of computational approaches in understanding and predicting the course of neurological disorders like MS.

This study explores the intriguing phenomenon of action potentials reverberating and propagating in high-resistance tissues, such as gliosis, leading to the emergence of short circuits within the brain. Gliosis, characterized by increased tissue resistance, presents a unique environment for electrical signals. My research investigates the possible mechanisms behind these reverberations and their role in the creation of self-empowering short circuits that can potentially affect the entire brain's functionality. Understanding these processes may have significant implications for the treatment and management of neurological disorders associated with gliosis and abnormal electrical activity in the brain.

Glioblastoma is a highly aggressive brain tumor characterized by rapid growth, extensive infiltration, and poor prognosis. Understanding the complex interplay of biological processes driving glioblastoma progression is crucial for developing effective therapeutic strategies. In this study, we present a comprehensive mathematical model that incorporates key aspects of glioblastoma growth and dissemination, including tumor cell diffusion, angiogenesis, nutrient availability, molecular signaling pathways, and blood-brain barrier (BBB) interactions. The model is based on a system of coupled partial differential equations (PDEs) and ordinary differential equations (ODEs) that capture the spatiotemporal dynamics of tumor cell density, nutrient concentration, signaling molecule concentrations, and BBB integrity. We employ numerical simulation techniques, such as the finite difference method and the Runge-Kutta method, to solve the model equations and visualize the evolution of tumor growth and its associated biological processes. The model parameters are estimated using experimental data from the literature, and the model is validated against clinical observations. Furthermore, we perform a comprehensive sensitivity analysis to identify the key parameters that significantly influence tumor growth, diffusion, and dissemination. The sensitivity analysis reveals the relative importance of each parameter in driving glioblastoma progression and highlights potential targets for therapeutic interventions. Our computational framework provides a powerful tool for understanding the complex dynamics of glioblastoma growth and offers valuable insights into the underlying biological mechanisms. The model can be used to test hypotheses, predict tumor response to various treatment strategies, and guide the design of targeted therapies. The findings from this study contribute to the ongoing efforts in developing personalized treatment approaches for glioblastoma patients and improving their clinical outcomes.

This article presents a novel computational approach to visualizing memristor dynamics within a simulated neural network. Memristors, known for their ability to emulate synaptic plasticity due to their variable resistance characteristics, are key components in neuromorphic computing and hold potential for advancing our understanding of neural processes. Our study introduces a sophisticated memristor model that incorporates non-linear resistance changes, simulating the complex behavior of synaptic connections in a neural network.The neural network, consisting of multiple interconnected neurons with memristor-based synapses, is subjected to a series of electrical stimuli. Each memristor's resistance is modulated in response to the applied voltage, mimicking the synaptic weight adjustments that occur during learning and memory formation in biological neural networks. To effectively illustrate these changes, we employ a high-contrast color mapping scheme, where the varying resistance of each memristor is represented by distinct colors, providing a clear and intuitive visualization of synaptic modifications over time.Our simulation runs through multiple iterations, demonstrating how synaptic weights evolve in response to different input patterns. The use of an extended voltage range and increased scaling factors ensures pronounced changes in memristance, enhancing the visibility of synaptic adaptations. The resulting visualizations offer a compelling representation of how memristors can mimic the dynamic nature of biological synapses, contributing to the field of neuromorphic engineering and deepening our comprehension of neural mechanisms underlying learning and memory.This work not only showcases the potential of memristors in simulating neural behavior but also provides a valuable educational tool for illustrating complex concepts in neuroscience and neuromorphic computing. The insights gained from this study pave the way for further exploration into the development of advanced neural network models and the design of memristor-based computing systems.

This study presents a computational simulation to explore the distinctive patterns of neural connectivity in Autism Spectrum Disorder (ASD), focusing on the concepts of spatial and topological proximities. ASD is characterized by unique cognitive and behavioral profiles, often attributed to atypical neural development and connectivity. Our simulation employs a two-pronged approach: first, by modeling spatial proximity through the generation of clustered 3D node positions, we mimic the observed over-connectivity in localized brain regions in ASD. Second, we construct a topological network to represent the complex interplay of under-connectivity and over-connectivity in functional brain networks, a hallmark of ASD's neural landscape. The spatial model demonstrates pronounced clustering, reflecting the structural brain differences and atypical growth patterns reported in neuroimaging studies of individuals with ASD. In contrast, the topological model reveals a network with a mix of sparse and dense connections, simulating the diverse and often contradictory findings of functional connectivity in ASD, such as reduced long-range connections and increased local connections. Our simulation provides a visual and conceptual framework for understanding the altered neural connectivity in ASD. It underscores the importance of considering both physical (spatial) and functional (topological) aspects of brain connectivity to grasp the full extent of neurodevelopmental deviations in ASD. This approach not only aids in elucidating the neural underpinnings of ASD's core symptoms but also offers a foundation for developing targeted interventions. Future research, integrating more complex and individual-specific data, could further refine this model, enhancing our understanding of ASD's neural dynamics and contributing to personalized therapeutic strategies.

This article explores the intricate dynamics of complexity and emergence through the lens of a graphical model representing hippocampal functions. Utilizing a series of overlapping curves that intersect at varying degrees and frequencies, we create a visual metaphor for the sophisticated processes to store information observed in the hippocampus. This model not only depicts the spatial and frequency-related interactions of these curves but also illustrates the emergent patterns that arise from these interactions. These patterns exemplify the principles of complexity and emergence, where the collective behavior of simple elements results in intricate and unexpected outcomes. This graphical representation serves as a bridge between the abstract mathematical concepts of complexity theory and the tangible biological processes observed in neuroscience. By drawing parallels to the hippocampus, known for its role in memory formation and spatial navigation, the article sheds light on how complex systems, such as neural networks, exhibit properties that are more than just the sum of their parts. The model thus offers a unique perspective on understanding complex systems, not only in neuroscience but in broader scientific inquiries where complexity and emergent phenomena are pivotal.

This article explores the intricate journey of neural progenitor cells from the ventricular zones of the embryonic brain, focusing on their pivotal role in the formation of rich club networks, which are integral to the brain's functional architecture. Initially, the ventricular zones, particularly the cerebral ventricular zone, serve as the primary sites for the proliferation and differentiation of neural progenitors, giving rise to various types of neurons and glial cells. We delve into the sequential migration patterns of these cells, emphasizing how early-migrating neurons, destined to form deeper cortical layers, establish extensive and robust connections, potentially evolving into rich club nodes. These nodes are characterized by their high-degree connectivity and serve as crucial hubs for integrating and disseminating information across disparate brain regions.The article further discusses the implications of this developmental pathway in the context of brain network efficiency, resilience, and cognitive function. We highlight how disruptions in early cell migration can lead to anomalies in rich club formation, drawing connections to various neurodevelopmental disorders such as autism, ADHD and schizophrenia. Additionally, the significance of rich club nodes in maintaining efficient information transfer and network resilience is elucidated, underscoring their role in supporting complex cognitive processes and overall brain dynamics.In summary, this article offers a comprehensive overview of the developmental trajectory from the ventricular zones to the establishment of rich club networks, providing insights into the fundamental processes that shape the brain's intricate connectivity and functional capabilities. Ric