Complex Action Potentials

I was inspired to make this for a friend who graduated years ago. They recorded action potentials from a neuron in the Thalamus, and I was very hyped about what the Hilbert Transform could do for pulling information from acoustic signals after reading one of my favorite papers of all time, so I applied it to everything for fun. Doing so gives you 2 things here: the original, real-valued train of action potentials (voltage), and the imaginary part. The real and imaginary parts together make a complex signal, oscillating as they move through time, making a beautiful 3D spiral.

I wanted to make them something they cold hold in their hand. From one angle, you would see the action potentials (voltage), but shift your perspective and you see the imaginary component of the action potentials, revealing something more. I would include some inspirational words about science and math revealing the unknown, or viewing a problem from multiple perspectives. Artistic and geeky combined.

The Process

The goal was to make a rigid structure that could stand on its own, about the size of a paperweight. Probably plastic.

The project hit several snags from the beginning. Learning AutoCAD enough to make what I wanted wasn't bad, but importing custom numerical coordinates without distortion was the first challenge. The resolution I needed to keep the structure smooth was too many datapoints for AutoCAD (or my cheap laptop) to handle. Eventually I found a workaround. Turning those coordinates into a 3D object was a nightmare though -- avoid it intersecting with itself, but keep it smooth. Remove and correct distortions again. Then the 3D printing. I've printed small high-res objects with SLS before, but the material and dimensions I wanted just didn't work. A friend with a resin printer tried, but it required too much intricate support structure and I didn't want to clip and sand for hours and risk breaking the structure half way through.

Neuro Rabbit Hole

I tried to understand if the imaginary part meant anything -- biological or not. When you record from neurons or model them with ordinary differential equations HH style you learn that the 1st derivative of the membrane voltage is proportional to the total membrane current.

At first glance the spiral looks like the neurophysiologist's version of the phase-plane, but that comes from voltage and its 1st derivative; the imaginary part looks similar to dV/dt, but they're different. I thought it might apply to something more nuanced like total conductance through the membrane, but it doesn't seem to be.

Wikipedia says that the Hilbert transform is "given by the Cauchy principal value of the convolution with the function f(t) = 1/(pi*t)", but I don't think that translates to anything neurophysiological for the imaginary part. Another mathematical description involves a ±90° phase shift of all the frequency components (nice visuals here), which makes sense in signal processing world but doesn't seem particularly interesting in the context of neurophysiology.

The Result

I bit off more than I could chew making this thing, especially when that chewing was just for fun when I had the time and energy. Frankly I like doing that...learning new skills to make something with a combination of googling, stumbling through, and asking people who know more than me. That's how I do most of these things. Some succeed, some fail.

I ended up with a few slinkies, and eventually just sent my friend a digital version with the inspirational message. Not nearly as cool.

If the imaginary signal is related to something interesting in neurons, I don't know what it is. I suspect it's meaningless, physiologically speaking, but it looks cool. This project was more art anyway.

Near the end of the process, I realized what I wanted -- small, smooth, and rigid -- was probably only possible with injection molding. This would have required a custom cast, which is only cost-effective if you make them in bulk.

If you know anyone who wants 50,000 of these let me know. We can split the cost.