The mathematical modelling of behaviour at the single neuron level is an important part of computational neuroscience. Conductance-based models remain dominant because of their biophysical interpretation. Nevertheless, their high dimensionality and number of parameters complicate mathematical analysis and numerical simulation. Integrate-and-fire models have been successful at the accurate prediction of spike times at a reduced computational cost. However, this success comes at the cost of a loss of interpretability and connection to the biophysics, while lacking the ability to capture complex subthreshold dynamics.

This thesis attempts to narrow the gap between conductance-based and integrate-and-fire modelling. Our starting point is the Multi-Quadratic Integrate-and-Fire (MQIF) model, that was recently developed to model excitability on multiple timescales. We show that, in contrast to other integrate-and-fire models, the MQIF model allows realistic modelling of the modulation of excitability type and bursting, by capturing the balance between restorativity and regenerativity in each timescale.

We propose a generalisation of the MQIF model that enables more quantitative modelling of neural behaviours. Two different methods are introduced to identify this multi-scale integrate-and-fire model from either conductance-based models or electrophysiological experiments. The model retains the low dimensionality of the MQIF model by grouping the contributions to the behaviour on each timescale. Furthermore, it is possible to find the parameters for the MQIF model by approximating the model around its organising centre, that is, the transcritical singularity.

The proposed methods allow the study of neural dynamics and modulation in an efficient and interpretable manner. This is shown for modulation both via a change of intrinsic conductances and via a change of mean applied current. The latter example is used in a simulation of an excitatory-inhibitory (E-I) network. The results show the potential of the application of the integrate-and-fire model in network studies.