Abstract
Neural networks produce electrical activity that is generated by the biophysical properties of the constituent neurons and synapses. Individual neurons produce electrical signals through processes that are highly nonlinear and communicate these signals to one another through synaptic interactions, resulting in emergent network outputs. The output of neural networks underlies behaviours in all higher animals. Mathematical equations can be used to describe the electrical activity of neurons and neural networks and the underlying biophysical properties. These equations give rise to computational models of neurons and networks that can be analysed using mathematical techniques or numerically simulated with computers. This article briefly reviews the current mathematical and computational techniques involved in modelling neurons and neural networks.
Key Concepts:

An action potential is a brief nonlinear rise and fall of the membrane voltage of a cell and is the primary signal used for neural communication.

Excitability is the ability of neurons and other cell types to produce action potentials when the transmembrane voltage crosses a threshold.

The Hodgkin–Huxley model is a mathematical description of how action potentials are generated in neurons and propagate along their axons.

The integrate‐and‐fire neuron is a simplified mathematical model of excitability in neurons, is useful for the ability to do mathematical analysis and is used primarily in network models.

Bifurcation is a mathematical term for a change in the qualitative structure of a dynamical system when a parameter value is changed.

Neural oscillations are repetitive or rhythmic changes in the voltage activity of a neuron or a network of neurons. Neural oscillations may arise in individual neurons or through network synchrony.

Bursting is the ability of some neurons and networks to produce periodic spiking activity followed by an interval of no activity.

A half‐centre oscillator is a network of two neurons that probursting activity out of phase with one another and is a key subnetwork of central pattern generators.
Keywords: bifurcation; phase plane; synchrony; Hodgkin–Huxley model; compartmental modelling; cable equation; balanced networks