Mathematical Models for Brain functions: Insights into Neurological Disorders

Abstract

Understanding how neurons process and transmit signals is crucial for modeling brain function and diagnosing neurological disorders. This study explores the Leaky Integrate-and-Fire (LIF) model—a simplified yet powerful mathematical representation of spiking neurons—as a framework to simulate neural dynamics under different input conditions. The LIF model is formulated using a first-order differential equation that captures the membrane potential's evolution in response to external currents and inherent leakage. Two simulation scenarios are analyzed: one with constant input current and another with step-varying current. Results demonstrate how the neuron exhibits regular spiking for sufficient stimulation and remains sub-threshold otherwise, highlighting threshold-dependent behavior. These dynamics are linked to real-world phenomena such as sensory gating, delayed neural activation, and seizure-like hyperactivity. The model offers valuable insights into cognitive disorders like ADHD and epilepsy and can serve as a computational basis for developing biologically inspired neural circuits or neuromorphic systems. Overall, the LIF model provides an accessible yet biologically relevant tool for investigating neural behavior and its disruptions in pathological conditions.