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This is what I know so far:
Neurons are nodes with a number called the threshold.
Neurons are connected to other neurons through directed axons.
Axons take the signals produced by neurons and deposit them on other neurons, these signals can be "negative" or "positive".
Once all the signals coming into a particular neuron are added up, if the sum is greater than the threshold, that neuron fires.
My question is: are these all the rules? And do all neurons have the same rules?

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My question is: are these all the rules?

No.

Some things you left out:

Plasticity (change)

  • Very-short-term synaptic "plasticity" (changes in synaptic strength); at least a few different forms of this (post-tetanic potentiation, short term synaptic depression, synaptic facilitation).

  • Very-short-term intrinsic "plasticity" (changes in spiking behavior): spiking accommodation, etc.

  • Medium to long-term synaptic plasticity; several forms of this: classical Long Term Potentiation or Depression, Spike Timing Dependent Potentiation, synaptic underpinnings of habituation and sensitization in, e.g., Aplysia, synaptic scaling.

  • Medium to long-term plasticity of intrinsic properties, like threshold, input resistance, firing properties.

  • Metaplasticity (changing the changeability)

Neuromodulation

  • Co-transmission of classical neurotransmitters with a neuropeptides
  • Neuromodulation of ionic conductances directly, postsynaptically
  • Neuromodulation of ionic conductances directly, presynaptically
  • Neuromodulation via metabotropic receptors and signaling to the nucleus
  • Why there are dozens of neuromodulators just for controlling a crab's digestion. (ref)

Odds n' Ends

  • Electrical "synapses" ("gap junctions")
  • Refractory periods preventing firing even if above threshold
  • Extrasynaptic and presynaptic receptors
  • Retrograde transmission
  • Axonal failures and branch points
  • Non-traditional innervation (dendro-axonic, dendro-dendritic, autapses, etc.)
  • Ephatic coupling (ref) (through local electrical fields)
  • Wiggling dendritic spines
  • Glial function as a modifier of neuronal function.

And I'm sure I'm leaving out another 20 or so interesting ones. Arguably, everything I've listed here modifies or has causal relevance to the simple set of rules you provided above. And of course, I'm still leaving out an enormous amount of detail of neuroscience (cell types, architecture/circuit diagrams, molecular cascades, cytoskeletal aspects, development, etc.) that have relevance to "the rules" as well.

And do all neurons have the same rules?

No. Any of the above may vary depending on cell type, part of the nervous system, individual differences, development, species, sex, time of day or other cycles, environmental factors, and possibly other factors.

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Those are very simplified rules which are decent enough for simple neural networks but real neurons in the brain are a lot more complicated.

See this paper for some ways that an axon can play a role beyond the simplified rules you mentioned.

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The rules that govern the behavior of neurons are described by the Nobel prize winning work of Hodgkin and Huxley. The model has seen a huge number of revisions and extension since its first publication, but remains the foundation by which we understand how neurons work. At its heart are a set of nonlinear differential equations that describe how ions flow across the cell membrane.

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There's also Dale's law: in general, a given neuron releases only a single type of neurotransmitter. While that is not really true for neuromodulatory neurons, it is broadly true of neurons that receive glutamate and GABA, the primary excitatory and inhibitory neurotransmitters.

tl;dr Neurons are either inhibitory at all their output synapses or excitatory at all their output synapses but not both.

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    $\begingroup$ We might want to retire "Dale's Law". Henry Dale apparently never stated it, and it's not much of a law if there are exceptions. I am curious to what degree classical neurotransmitter (GABA/glutamate) cotransmission occurs, but here are two example papers: ncbi.nlm.nih.gov/pmc/articles/PMC2832847 and nature.com/neuro/journal/v17/n11/full/nn.3823.html $\endgroup$
    – Chelonian
    Commented Mar 6, 2016 at 22:35
  • $\begingroup$ the easy response to those papers is that things are funky in the midbrain. $\endgroup$
    – honi
    Commented Mar 7, 2016 at 0:04
  • $\begingroup$ but seriously, its a pretty good heuristic. it certainly makes conceptualizing network function much much easier for students. it is also a heuristic that allows for a massive amount of the research that goes on in systems neuroscience. $\endgroup$
    – honi
    Commented Mar 7, 2016 at 0:05
  • $\begingroup$ after all, not all neurons fire action potentials, and not all neurons have separation between axon and dendrite, and neurons don't really have a consistent threshold, and neurotransmission can occur outside of synapses, etc etc etc. but it's still important to teach those rules of thumb because otherwise we can't teach anything. $\endgroup$
    – honi
    Commented Mar 7, 2016 at 0:11

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