Ever since the pioneering work of Hodgkin and Huxley, biological neuron models have consisted of ODEs representing the evolution of the transmembrane voltage and the dynamics of ionic conductances. It is only recently that maps have begun to receive attention as valid phenomenological neuron models. They can not only be computationally advantageous substitutes of ODEs, but, since they accommodate chaotic dynamics in a natural way, they may reproduce rich collective behaviors with a minimum of analytical complexity.
I'll review a family of map-based neuron models that have appeared scattered in the scientific literature in recent years, providing a unified perspective of them. Phase plane analysis shows that there exists in this family a trade-off between the sensitivity of the neuron to steady external stimulation and its resonance properties, which may be tuned by the neutral or asymptotic character of the slow variable. I present implications of the results for the suprathreshold behavior of the neurons in different regimes of interest.
The results establish a consistent link between single-neuron parameters and network dynamics, which I explore further. I analyze spatiotemporal behaviors in networks of bursters with regular topologies, and in small networks with arbitrary balanced inhibitory connections. In the latter case two kinds of patterns are found depending on the symmetry of the network: slow cyclic patterns riding on subthreshold oscillations where almost all neurons contribute bursts in a sparse manner, and fast patterns of bursts in which only one of two mutually exclusive groups of neurons take part. I show how these patterns can be predicted from network topology using the technique of master stability functions.