Dynamical Systems in Neuroscience:

Chapter 8Simple ModelsThe advantage of using conductance-based models, such as the I Na +I K -model, is thateach variable and parameter has a well-defined biophysical meaning. In particular, theycould be measured experimentally. The drawback is that the measurement proceduresmay not be accurate, that the parameters are usually measured in different neurons,averaged, and then fine-tuned (a fancy word meaning “to make arbitrary choices”).As a result, the model does not have the same behavior as one sees in experiments.And even if it “looks” OK, there is no guarantee that the model is accurate from thedynamical systems point of view, i.e., that it exhibits the same kind of bifurcations asthe type of neuron under consideration.Sometimes we do not need or cannot afford to have a biophysically detailed conductancebasedmodel. Instead, we want a simple model that faithfully reproduces all the neurocomputationalfeatures of the neuron. In this chapter we review salient features ofcortical, thalamic, and other neurons, and we present simple models that capture theessence of their behavior from the dynamical systems point of view.8.1 Simplest ModelsLet us start with reviewing the simplest possible models of neurons. As one canguess from their names, the integrate-and-fire and resonate-and-fire neurons capturethe essence of integrators and resonators. The models are similar in many respects:both are described by linear differential equations, both have a hard firing thresholdand a reset, both have a unique stable equilibrium at rest. The only difference is thatthe equilibrium is a node in the integrate-and-fire case, but it is a focus in the resonateand-firecase. One can model the former using only one equation, and the latter usingonly two equations, though multi-dimensional extensions are straightforward. Bothmodels are useful from the analytical point of view, i.e., to prove theorems.Many scientists, including the author of this book, refer to these neural modelsas being “spiking models”’. The models have a threshold, but they lack any spikegenerationmechanism, i.e., they cannot produce a brief regenerative depolarizationof membrane potential corresponding to the spike up-stroke. Therefore, they are not279