Models

In a previous post, I wrote about financial models. The point is that a scientific model generally simplifies. At the same time simplification gives models their power, one must also take care to assess whether adapting or transplanting the model to new fields is valid. Hence some disconnects between economic models and the financial tools based off these models.

Here’s another illustration. I was talking with my friend about his thesis. R. is interested in building a model of the olfactory bulb. This structure is interesting; it is well defined anatomically into three layers. The top layer contains neuropil structures called glomeruli. Glomeruli contain the axon projections from the primary sensory neurons and dendritic branches of the neurons in the bulb. Both these “main” neurons and so-called interneurons form  connections within this layer. Since this is where raw signals from the nose arrive, it is called the input layer. Together, these cells form a network and reshapes the responses into new neural activity patterns, relayed to deeper olfactory processing areas of the brain.

The middle layer contains the cell bodies of the olfactory bulb output neurons. As mentioned, these cells, called mitral or tufted cells (usually termed M/T cells), send a main dendrite to the glomerulus. Each cell also sends secondary dendrites laterally, within the middle layer. The third layer, the granule cell layer, contains interneurons that form connections between the laterally spread dendrites in the middle layer. This forms a second point within the olfactory bulb where the raw input from the nose can be reshaped, repatterned, and repackaged for subsequent processing.

OK: my friend spoke of his troubles. He needed to convert the sensory neuron activity (from the nose), which differ for different smells. The features that are important seem to be when the activity begins (onset latency), how long it lasts for (duration), and how intense (basically how often the neuron “fires” an action potential.) There are some other subtleties, naturally. Each smell evokes activities in a great many olfactory neurons, some of which respond with a different set of characteristics. The idea is to build the model so that the responses from bulb output neurons can be calculated, given the set of parameters (i.e. the input activity patterns).  Ultimately, these input neural patterns can be related to the actual behavior that helped shape them (such as the sniffing that an animal might engage in as they hone in on some odorous.)

His trouble came with integrating the Hodgkin-Huxley model of the action potential (this is basically derived from physical/thermodynami first principles), determining how this model would generate action potential “spikes” in a way that mimics what the olfactory bulb neurons would do, given the pattern of input activity and the 2 layers of interneuronal influence within the bulb. It seemed like a set of nested differential equations – that is, the action potentials varied over time, with the degree of influence from the various interneurons also changing in time. That’s a real cluster-eff.

I thought I had a brilliant idea (and I still think it’s nice.) I suggested that he can simply build a phase space to describe all the possible arrangements of his input patterns. Each point in this abstract descriptive space can be correlated to a set of output profiles (i.e. how the bulb neurons eventually respond.) He can, in the end, identify the bulb response most likely to result from a given set of input patterns.

The problem is that this is a descriptive model. The Hodgkin-Huxley model would have the advantage of being an actual, theoretical model. Once this is in place, they can literally predict, down to the number of spikes and when they fire, the output of the olfactory bulb.

So yes, that, in a nutshell, is the difference between data-mining versus something derived from first principles. While one might be able to infer the same conclusions from a descriptive model, the theoretical model might be easier to work with when extending it slightly further than what had been observed by scientists. As Justin Fox warns, such extensions can be perilous if one does not take care to worry about validity.