This is a hard question. I’ll do my best to answer it but as many people have already pointed out, we don’t have a complete answer yet, and you probably already know a lot of the things I’m about to say. Let me also preface this by saying that most of the answer I’m about to give has been observed either in cells in a dish or in non-human animals, but almost none of it has been seen in awake, behaving humans (yet).
The best short answer to your question is paraphrased from Donald Hebb: neurons that fire together, wire together. Your representation of a cuttlefish started out as a bunch of neurons that got activated when you thought about cuttlefish, and now those neurons have stronger, more sensitive connections to one another. Together, they are responsible for your experience of knowing about cuttlefish.
Or, another way of thinking about it the way you can tell if a neuron has information in it or not is if it’s alive. A neuron that doesn’t give or receive information will eventually die off.
To go more in depth, we can look at this on multiple levels. The smallest level that’s useful to us is the molecular level. When one neuron sends a message to another neuron through an action potential, the receiving neuron responds by letting calcium ions flow into the cell through AMPA receptors and, if the depolarization is strong enough, NMDA receptors. Once a critical concentration of calcium is reached, it activates calcium-dependent protein kinases, including CaMKII (calcium-calmodulin dependent protein kinases II) and PKC (protein kinase C).
The process I just described is called early LTP. So at this stage in learning, the cells involved are already qualitatively distinguishable from other cells: they have high concentrations of calcium, and active forms of CaMKII and PKC.
But these aren’t permanent changes. For learning to be sustained, the cell has to go through gene transcription and protein synthesis.
The signaling cascade described above (calcium—>PKC + CaMKII) activates a series of second messengers, including MAPK/ERK, which @gasman mentioned, and CREB (cyclic AMP response element binding protein) which @mattbrowne mentioned. Another important molecule that’s constitutively active and necessary for this process is the enzyme PKMzeta.
Together, these messengers initiate gene transcription and subsequent protein synthesis so that the cell can start undergoing structural changes. So that more or less wraps up the molecular perspective, and we can move up to the cellular level.
There are several changes the cell undergoes to reflect its new learning. More AMPA receptors are moved from storage pools up to the cell membrane, new dendritic spines start to grow, and the spines that exist change their shape to become more available to outside signals. Here’s another image of the effects of experience-dependent plasticity on dendritic spines.
So on this level, you could say that a neuron that has this new information in it is distinguishable from one that doesn’t in terms of its number of membrane-bound AMPA receptors, and the number and shape of its dendritic spines.
Now let’s take a big jump up from the molecular/cellular level up to the systems/behavioral/cognitive level. One way you can see if a cell has information stored is to see what it responds to. We can measure the electrical activity of a cell directly by implanting electrodes in or near them, but doing this is in humans is generally frowned upon. So, instead we have to measure the activity of whole networks using methods like EEG, PET, and my tool of choice, functional MRI.
Using these techniques, you can see which parts of the brain become active when you do certain tasks. Visual cortex in the occipital lobe responds strongly whenever you look at anything; the amygdala becomes active in response to emotionally arousing situations, the medial temporal lobe is engaged in memory tasks. Using these techniques, we differentiate cells that “know” a specific kind of information by measuring the magnitude of voltage changes in a given region (in the case of EEG) or the amount of energy being expended measured as glucose consumption (PET) or blood flow (fMRI). The cells involved in a certain kind of information storage become active when they engage in a relevant task, and in a roundabout way we can use these techniques to find out which ones do what.
And that’s the long answer.