  Rhythmic electrical activity in the brain emerges from regular non-trivial
interactions between millions of neurons. Neurons are intricate cellular
structures that transmit excitatory (or inhibitory) signals to other neurons,
often non-locally, depending on the graded input from other neurons. Often this
requires extensive detail to model mathematically, which poses several issues
in modelling large systems beyond clusters of neurons, such as the whole brain.
Approaching large populations of neurons with interconnected constituent
single-neuron models results in an accumulation of exponentially many
complexities, rendering a realistic simulation that does not permit
mathematical tractability and obfuscates the primary interactions required for
emergent electrodynamical patterns in brain rhythms. A statistical mechanics
approach with non-local interactions may circumvent these issues while
maintaining mathematically tractability. Neural field theory is a
population-level approach to modelling large sections of neural tissue based on
these principles. Herein we provide a review of key stages of the history and
development of neural field theory and contemporary uses of this branch of
mathematical neuroscience. We elucidate a mathematical framework in which
neural field models can be derived, highlighting the many significant inherited
assumptions that exist in the current literature, so that their validity may be
considered in light of further developments in both mathematical and
experimental neuroscience.
