Reduced neuron models are essential tools in computational neuroscience to aid understanding from the single cell to network level. In this thesis I use these models to address two key challenges: introducing experimentally verifi�ed heterogeneity into neocortical network models, and furthering understanding of post-spike refractory mechanisms. Neocortical network models are increasingly including cell class diversity. However, within these classes significant heterogeneity is displayed, an aspect often neglected in modelling studies due to the lack of empirical constraints on the variance and covariance of neuronal parameters. To address this I quantified the response of pyramidal cells in neocortical layers 2/3-5 to square-pulse and naturalistic current stimuli. I used standard and dynamic I-V protocols to measure electrophysiological parameters, a byproduct of which is the straightforward extraction of reduced neuron models. I examined the between- and within-class heterogeneity, culminating in an algorithm to generate populations of exponential integrate-and-�re (EIF) neurons adhering to the empirical marginal distributions and covariance structure. This provides a novel tool for investigating heterogeneity in neocortical network models. Spike threshold is dynamic and, on spike initiation, displays a jump and subsequent exponential decay back to baseline. I examine extensions to the EIF model that include these dynamics, fi�nding that a simple renewal process model well captures the cell's response. It has been previously noted that a two-variable EIF model describing the voltage and threshold dynamics can be reduced to a single-variable system when the membrane and threshold time constants are similar. I examine the response properties of networks of these models by taking a perturbative approach to solving the corresponding Fokker-Planck equation, �finding the results in agreement with simulations over the physiological range of the membrane to threshold time constant ratio. Finally, I found that the observed threshold dynamics are not fully described by the inclusion of slow sodium-channel inactivation.