  Neuromorphic computing is a non-von Neumann computing paradigm that performs
computation by emulating the human brain. Neuromorphic systems are extremely
energy-efficient and known to consume thousands of times less power than CPUs
and GPUs. They have the potential to drive critical use cases such as
autonomous vehicles, edge computing and internet of things in the future. For
this reason, they are sought to be an indispensable part of the future
computing landscape. Neuromorphic systems are mainly used for spike-based
machine learning applications, although there are some non-machine learning
applications in graph theory, differential equations, and spike-based
simulations. These applications suggest that neuromorphic computing might be
capable of general-purpose computing. However, general-purpose computability of
neuromorphic computing has not been established yet. In this work, we prove
that neuromorphic computing is Turing-complete and therefore capable of
general-purpose computing. Specifically, we present a model of neuromorphic
computing, with just two neuron parameters (threshold and leak), and two
synaptic parameters (weight and delay). We devise neuromorphic circuits for
computing all the {\mu}-recursive functions (i.e., constant, successor and
projection functions) and all the {\mu}-recursive operators (i.e., composition,
primitive recursion and minimization operators). Given that the {\mu}-recursive
functions and operators are precisely the ones that can be computed using a
Turing machine, this work establishes the Turing-completeness of neuromorphic
computing.
