The integrate-and-fire neuron model is a widely used framework for analyzing neural systems, focusing on the membrane potential in response to synaptic inputs and injected currents. An action potential (spike) is generated when the membrane potential reaches a threshold, but the detailed changes in voltage and conductances are not modeled. Synaptic inputs are treated as stochastic, following a temporally homogeneous Poisson process. The review examines methods for both current and conductance synapses using the diffusion approximation, where individual contributions to the postsynaptic potential are small. Key mathematical techniques include stochastic differential equations and the Fokker–Planck equation, which describe the time distribution of output spikes. The model has become a canonical tool for understanding spiking neurons due to its mathematical tractability and ability to capture essential neural processing features. Variations of the model, its relationship with the Hodgkin–Huxley model, and comparisons with electrophysiological data are discussed. The review also highlights the model's contributions to understanding the irregularity of cortical spiking and neural gain modulation.The integrate-and-fire neuron model is a widely used framework for analyzing neural systems, focusing on the membrane potential in response to synaptic inputs and injected currents. An action potential (spike) is generated when the membrane potential reaches a threshold, but the detailed changes in voltage and conductances are not modeled. Synaptic inputs are treated as stochastic, following a temporally homogeneous Poisson process. The review examines methods for both current and conductance synapses using the diffusion approximation, where individual contributions to the postsynaptic potential are small. Key mathematical techniques include stochastic differential equations and the Fokker–Planck equation, which describe the time distribution of output spikes. The model has become a canonical tool for understanding spiking neurons due to its mathematical tractability and ability to capture essential neural processing features. Variations of the model, its relationship with the Hodgkin–Huxley model, and comparisons with electrophysiological data are discussed. The review also highlights the model's contributions to understanding the irregularity of cortical spiking and neural gain modulation.