In formal spiking neuron models, spikes are fully characterized by their firing time t(f) defined by a threshold criterion. Integrate-and-fire and Spike Response Model are typical examples of spiking neuron models. Leaky integrate-and-fire point neurons with current input can be mapped exactly to the Spike Response Model. Even multi-compartment integrate-and-fire models can be mapped to the Spike Response Model, if indirect effects due to previous output spikes are neglected. An integrate-and-fire model with spike-time dependent parameters, which is a generalization of the leaky integrate-and-fire model, can be seen as a special case of the Spike Response Model. The nonlinear integrate-and-fire model, i.e., a model where parameters are voltage dependent is a different generalization. The quadratic integrate-and-fire model is particularly interesting since it is a generic example for a type I neuron model.
Detailed conductance based neuron models can be approximately mapped to formal spiking neuron models. With the help of formal spiking neuron models, problems of pulse coding can be studied in a transparent graphical manner. The Spike Response Model, defined in this chapter, will be reconsidered in part II where systems of spiking neurons are analyzed.
Formal neuron models where spikes are triggered by a threshold process have been popular in the sixties (Stein, 1967b,1965; Weiss, 1966; Geisler and Goldberg, 1966), but the ideas can be traced back much further (Hill, 1936; Lapicque, 1907). It has been recognized early that these models lend themselves for hardware implementations (French and Stein, 1970) and mathematical analysis (Stein, 1967a,1965), and can be fitted to experimental data (Brillinger, 1988,1992). Recent developments in computation and coding with formal spiking neurons has been reviewed in the book `Pulsed Neural Networks' edited by Maass and Bishop (1998).
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