Map-based models of spiking neurons

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.