Point Neuron Models

There have been models of neurons for a long time.
The oldest that I'm aware of is the Integrate and Fire model from Lapicque in 1907.
It wasn't a computer model, and McCullouch and Pitts came up with another version of the model in 1943. Computers were pretty simple then, and I think they came up with the model independently.
The idea is that a neuron collects activation (and inhibition) from neurons that connect to it, when those neurons fire (emitting a spike).
If the activation goes over a threshold, the neuron itself emits a spike continuing the proces. That's the integrate and fire model.
What happens to the activation when a neuron doesn't fire. (It loses all its activation when it fires.) You can timestep it, and you can either lose it all each time step, or keep it all.
LIF: a more accurate model has the activation leaking away when a neuron doesn't fire. That's the leaky integrate and fire.
Adaptation: if neurons are given a constant amount of input (with say an electrode) they often fire regularly, but then more slowly. They get tired, or adapt.
The boltzmann machine is another model that is more probabilistic but still integrates and fires.
Izhikevich neurons are also called reverberate and fire neurons. These can account for a range of behaviour, like bursty neurons.
These are all point models because they model the neuron with some simple equations.
In each of these models, the unit is represented by one or a few variables (e.g. activation and fatigue). They're updated in discrete steps, or via events continously.