He attended Aireborough Grammar School and, after spending a year teaching in Uganda, Broomhead moved to Merton College, Oxford, where he read chemistry for his first degree.

With Greg King he developed techniques to determine whether an experimental time series had been generated by a deterministic chaotic system by combining the pure mathematical results on topological embedding due to Takens with the engineering method of singular value decompositions.

In 1989 he was awarded the John Benjamin Memorial Prize for work with David Lowe and Andrew Webb that exploited an analogy between neural networks and interpolation using the newly developed radial basis functions from numerical analysis.

Later, as a member of the Manchester Centre for Integrative Systems Biology, he worked with Douglas Kell on large-scale models of metabolism, and with Mike White on the dynamics of intracellular signalling cascades.

He also developed a deep interest in hybrid systems and asynchronous processes, founding the Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA).