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Let's say you want to study the evolution of "intelligence". And we are going to define intelligence as the ability to learn new tasks, identify unique objects, etc.
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Let's say an animal is confronted with food the animal can see, but can't simply reach. Some birds can use a stick to get at bugs they can't reach. Dogs and cats can solve a puzzle get at food. Some apes can build towers or open jars. Octopi can also open jars and learn.
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In the video below an octopus actually reaches behind it's prey and taps it. Scaring the poor shrimp into it's mouth! That's quite clever. And the octopus remembers it. And, apparently, UC Berkeley things it is oddly sexy (listen o the 1970s disco-porn music that accompanies the video).
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Here is why an octupus is both different and important on the scheme of things. And octopus has no vertebra or spine. It is an invertebrate and therefore has branched off from vertebrates (like humans) very far back in history.
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So, all of our studies on brain activity has been on vertebrates.
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Recently a group of scientists have mapped the genome of the octopus. They found some interesting things. First there are two lines that are identified with intelligence that are found in abundance in the octopus and vertebrates (like humans), but are not found in other invertebrates. I am going to summarize because if you care to know more you should check out the details yourself (like in Christian Science Monitor or Nature).
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The first is a genome system that allows tissues to rapidly modify proteins to change their function. This allows "learning". It is found in Octopi, but not other invertebrates - and also found in vertebrates.
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Then there are the "protocadherins" which manage creation and develop of neurons. This is important because neurons can tell the difference between each other, which allows (say) your fingers to work. We humans know from neurons where our arm ends and our fingers start because we can tell the difference between the two. This may seem obvious, but it isn't. Use an example of yourself. Have someone watch you from behind and try to twitch only the left side of your lower back only.
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Anyway, octopi have something like half a billion of these neurons - which is twice as many as say a mouse. But they are spread out a lot more than in vertebrates. For us, the vast vast majority or our neurons are in our brain, followed closely by the spinal cord. Messages are sent via the spinal cord to our brain (see video below).
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But the octopus doesn't have a spine (hence the term invertebrate). So only about 1/3 of its neurons are in the brain, and the rest spread out through the body and eight arms. The arms can "feel" and "taste" independently only giving the heads up to the brain when something is yummy.
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This yields some amazing results of what the arms can do. But limited communication paths (i.e. no spine) means that for many of the very complex functions, the octopus can only control the arm when it can see it.
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Weird huh?