Simulations of the electrical activity of networks of morphologically-detailed neuron models allow for a better understanding of the brain. Short time to solution is critical in order to study long biological processes such as synaptic plasticity and learning. State-of-the-art approaches follow the Bulk Synchronous Parallel execution model based on the Message Passing Interface runtime system. The efficient simulation of networks of simple neuron models is a well researched problem. However, the efficient large-scale simulation of detailed neuron models --- including topological information and a wide range of complex biological mechanisms --- remains an open problem, mostly due to the high complexity of the model, the high heterogeneity of specifications in modern compute architectures, and the limitations of existing runtime systems. In this thesis, we explore novel methods for the asynchronous multi-scale simulation of detailed neuron models on distributed parallel compute networks. In the first study, we introduce the concept of distributed asynchronous branch-parallelism of neurons. We present a method that extracts variable dependencies across numerical resolution of neurons topological trees and the underlying algebraic solver. We demonstrate a method for the decomposition of the neuron simulation problem into interconnected resolutions of neuron subtrees. Results demonstrate a significant reduction in runtime on heterogeneous distributed, multi-core, Single Instruction Multiple Data (SIMD) computing environments. In the second study, we complement the previous effort with graph-parallelism retrieved from mathematical dependencies across the systems of Ordinary Differential Equations (ODEs) that model the activity of neurons. We describe a method for the automatic extraction of read-after-write variable dependencies, concurrent variable updates, and independent ODEs. The automation of the methods expose a computation graph of biological mechanisms interdependency, and an embarrassingly-parallel SIMD execution of mechanism instances, leading to an acceleration of the simulation. The third part of this thesis pioneers the fully-asynchronous parallel execution model, and the "exhaustive yet not speculative" stepping protocol. We apply it to our use case, and demonstrate that by removing collective synchronization of neurons in time, by utilising a memory-linear neuron representation, and by advancing neurons based on their synaptic connectivity, a substantial runtime speed-up is achievable through cache efficiency. The fourth and last part advances the aforementioned fully-asynchronous execution model with variable-order variable-timestep interpolation methods. We demonstrate low dependency of runtime and step count on volatility of solution, and a high dependency on synaptic events. We introduce a variable-step execution model that allows for non-speculative asynchronous stepping of neurons on a distributed compute network. We demonstrate higher numerical accuracy, less interpolation steps, and shorter time to solution, compared to state-of-the-art implicit fixed-step resolution. This thesis demonstrates that utilising asynchronous runtime systems is a requirement to succeed in efficiently simulating complex neuronal activity at large scale. Most contributions presented follow from first principles in numerical methods and computer science, thus providing insights for applications on a wide range of scientific domains.