Neurons communicate by means of short electrical impulses. We say that neurons fire, spike, or discharge. We speak of spikes, impulses, discharges or action potentials. What concepts do these metaphors convey?
“Spike” seems to refer to the shape of the action potential when looked at on a voltage trace, ie, there is an abrupt rise and fall of the potential. The action potential is quite a different notion: it is a particular form of potential that allows some form of action. That is, the terms action potential convey the notion that those potentials, unlike other ones (subthreshold potentials), have an effect on other neurons. This is a notion that is strikingly not representational.
Firing, impulse and discharge add another aspect: an action potential releases energy. The metaphor is rather accurate as energy is stored in electrochemical gradients across the membrane and the opening of ionic channels releases some of that energy. Firing also conveys a notion of movement: the energy is targeted to some particular place, the axonal terminals. The metaphor is only partially accurate, because when firing a gun, energy is only released at firing time and then the bullet moves to its target. But in neurons, propagation is active and energy is released all along the axon. Thus a better metaphor would be that the neuron ignites, where the axon progressively burns. On the other hand, in myelinated axons, energy is released at discrete locations (Ranvier nodes), so the neuron could be seen as firing in sequence towards the next node: between two nodes, there is a movement that does not use additional energy, as in firing a bullet (dominoes could also be an adequate metaphor). So perhaps a mylienated axon fires (repeatedly), but an unmyelinated axon ignites!
“Discharge” is an interesting term because it relates to a former theory of action potential. The metaphor suggests that the membrane is an electrically charged capacitor, and it gets discharged during the action potential. This seems to correspond to Bernstein's theory (beginning of the twentieth century), according to which the negative resting potential is due to a gradient of potassium concentration across the membrane and the action potential corresponds to a non-selective increase in membrane permeability, resulting in a decrease of the membrane potential (in absolute value). But in 1939, Hodkgin and Huxley made the first intracellar recording of an action potential in an animal and they found out that the membrane potential did not go to 0 mV but actually exceeded it quite substantially. So the discharge metaphor entails a particular model, but one that is now outmoded.
Finally, related to the concept of firing is the notion of threshold. When the membrane potential reaches a threshold, the neuron fires. A threshold is a spatial delimitation between two rooms, or between the outside and the inside of a house. It conveys the notion of a qualitative change. Before threshold, you are outside; after the threshold, you are inside. So the threshold metaphor entails the all-or-none law of neural activity: there is a spike or there is no spike.
In the integration metaphor, inputs and outputs are seen as objects (things that can be manipulated). Specifically, neural output (membrane potential) is a container (inputs are integrated into the membrane potential). In contrast, in the firing metaphor (and related metaphors), neural outputs are seen not as objects but as discrete, timed actions on other neurons (the action potential), which release energy. Thus the integration metaphor and the firing metaphor convey somewhat different views on neural function. Perhaps speculatively, I would suggest that the disonance between these two metaphors is the deep source of the firing rate vs. spike timing debate. In the integration metaphor, the neuron is a container and what matters for a container is what it contains, i.e. the number of inputs. When exactly those inputs come into the container is relatively unimportant. The integration metaphor conveys a representational view of the brain and is consistent with the rate-based view. In the firing metaphor, what is emphasized is the fact that neurons spend energy to act on each other. Actions are events, and therefore time is possibly important. This view is not representational but rather interactional or dynamicist.
An important question is how empirically accurate these metaphors are, especially when some are inconsistent. I have discussed this question indirectly in my series on the firing rate vs. spike timing debate. I will simply point out that the firing metaphor is fairly accurate, as briefly discussed above, possibly if firing is replaced by ignition. There is a release of energy that propagates and acts on other neurons, which occurs discretely when some condition is met. The integration metaphor, on the other hand, is rather loose. It cannot be accurate without substantial qualifications. The main effect of a presynaptic spike is generally short-lived, so an input could be said to be integrated, but only with the qualification that it gets out quickly. The effect of several input spikes on the membrane potential also depends on their relative time of arrival, and this fact does not fit the container metaphor very well.