This review gives a short historical account of the excitable maps approach for modeling neurons and neuronal networks. Some early models, due to Pasemann (1993), Chialvo (1995) and Kinouchi and Tragtenberg (1996), are compared with more recent proposals by Rulkov (2002) and Izhikevich (2003).We also review map-based schemes for electrical and chemical synapses and some recent findings as critical avalanches in map-based neural networks.We conclude with suggestions for further work in this area like more efficient maps, compartmental modeling and close dynamical comparison with conductance-based models.

Activity in the brain propagates as waves of firing neurons, namely avalanches. These waves’ size and duration distributions have been experimentally shown to display a stable power-law profile, long-range correlations and 1/f b power spectrum in vivo and in vitro. We study an avalanching biologically motivated model of mammals visual cortex and find an extended critical-like region – a Griffiths phase – characterized by divergent susceptibility and zero order parameter. This phase lies close to the expected experimental value of the excitatory postsynaptic potential in the cortex suggesting that critical be-havior may be found in the visual system. Avalanches are not perfectly power-law distributed, but it is possible to collapse the distributions and define a cutoff avalanche size that diverges as the network size is increased inside the critical region. The avalanches present long-range correlations and 1/f b power spectrum, matching experiments. The phase transition is analytically determined by a mean-field approximation.

We introduce a simple generalization of graded response formal neurons which presents very complex behavior. Phase diagrams in full parameter space are given, showing regions with fixed points, periodic, quasiperiodic and chaotic behavior. These diagrams also represent the possible time series learnable by the simplest feed-forward network, a two input single-layer perceptron. This simple formal neuron (‘dynamical perceptron’) behaves as an excitable ele ment with characteristics very similar to those appearing in more complicated neuron models like FitzHugh-Nagumo and Hodgkin-Huxley systems: natural threshold for action potentials, dampened subthreshold oscillations, rebound response, repetitive firing under constant input, nerve blocking effect etc. We also introduce an ‘adaptive dynamical perceptron’ as a simple model of a bursting neuron of Rose-Hindmarsh type. We show that networks of such elements are interesting models which lie at the interface of neural networks, coupled map lattices, excitable media and self-organized criticality studies.

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