Computational Scalability in Nature

Discovery through Interdisciplinary Approaches

Did you ever try to measure a wall, but the tape measure wasn’t long enough? Naturally, what you did is put a finger at the end of the tape measure, continue measuring from there, and then add the 2 values. A simple, direct method. What works at home often isn’t scalable though: imagine measuring your neighborhood or even the Great Wall of China with the same method. It would just take too long. Manufacturing a slightly longer tape measure would help and is in principle possible, but its maintenance would be a nightmare, for one because every element to repair is unique.

Although other technologies exist, how might one measure something huge using simple instruments, while maintaining high resolution at the same time? How do you determine the position in a scalable way, using easily replaceable modules? Unexpectedly, nature has a solution; One of those that seems obvious in retrospect, because we have tools that work that way.

Biology — Precision Systems

Biology is well known for its resilience and self-repairing capabilities. It has however a really bad reputation of poor precision, noisiness, even randomness! What else would you expect from systems that live in a dense environment that tends to break down towards entropy?

What is remarkable, though, is that despite these characteristics, nature is capable of incredible precision. Sea Turtles that travel the oceans for months to come back to the same exact beach from where they left, the precise patterns of beehives, out ability to recognize people, the insanely accurate timing of a master pianist playing Chopin!

That’s a little more than the Great Wall of China …each way (non-Leatherback photo by L. Roos).

To consistently achieve this kind of precision at a behavioral level, the underlying systems must themselves be able to measure precisely. This can be achieved directly or through a statistical approach, in which many imprecise sources are combined to reduce common noise. What is interesting from an engineering standpoint is that one can be confident that the underlying principles are highly energy efficient, because that is one of evolution's major drivers.

Uncovering these principles is a difficult task because biology is a mess. Living systems are dynamic and complex, and predicting their behavior from the properties of individual parts may be hard or right out misleading, as a reductionist approach might send years of research down blind alleys (e.g., systems in isolation may behave differently than when embedded with others). A better option is to study systems simultaneously, possibly in the natural environment (as it oughta be done in true Systems Biology) or as close as possible. Under this approach, engineering techniques provide powerful tools because —among other advantages — they allow abstracting whole systems to mathematical terms, and therefore they allow to combine different levels (e.g., organisms and cells) under common denominators. For instance, if you start by observing an animal’s behavior, you can make hypotheses and predictions about the capabilities of the underlying systems. Testing those directly in the freely behaving animal or person, and then abstracting the results to a common level allows to potentially identify a meaningful correlation.

Position measurement, for instance, is clearly needed to navigate the world. Although much is known about animal navigation, the mechanisms remain largely a mystery. A well-studied system that shows a correlation with navigation are the so-called Grid Cells. These provide a computational rationale to our quest of measuring the Great Wall.

Neuroscience — Our very own Matrix

There are many fascinating and interesting aspects about this structure in the mammalian hippocampus. Grid cells were given that name because they activate at regular spatial intervals: if you make a sign of where the animal or person is when a grid cell fires, you’ll get a neat dotted map.

A representation of physical movement in the brain.

What struck me in particular, is their ability to measure at different scales. Depending on the cell you’re measuring on, the spacing between activation dots is different. Having been obsessed with spatial frequencies during my PhD, this made me immediately think about multiple sampling bases. What are multiple sampling bases good for? Calculating position. As shown in a recent paper, the grid cells in the rat’s hippocampus can encode its absolute position. Like an internal GPS.

Now, nobody knows yet how the grid cell system is fed, we just know that by keeping track of your movements at regular intervals, you can know your absolute position. What might constitute such a periodical signal? My guess is a that a simple footstep would be sufficient: like a pedometer, each time you put your foot down, Grid Cells are activated at the appropriate level, which is determined by estimations of the biomechanics. The relative accuracy of a step is the main factor making it less accurate than artificial measurement systems. Using an accurate step would improve the whole system, and make it usable for industrial applications. But that’s nothing new.

Engineering — Linear Encoders

A type of sensor called absolute linear encoder works on the same computational principle. Steps are measured at different intervals and combined into the absolute distance. The signals are either optical, magnetic, inductive, capacitive or Eddy current. In the optical linear encoders, the steps can be simply a regular alternation of black and white on a strip. A strip with a single constant interval provides relative distance. For instance, the mechanical computer mouse used a single on-off alternation arranged on a disc.

Mouse internals with 2 relative rotational encoders, their code, and a Gray disc for absolute positioning.

The number of different intervals determines the length that you can measure. So, what do you do if you want to measure double as much? You just double the system and add a strip with a new interval. Everything else stays the same. Double the length, just one more variable to measure, not double as many. That’s scalable, exponentially scalable.

This means that to measure where one person (meter accuracy) is on the Great Wall of China, you’d only need 25 patterns. A very manageable control system. But get this: to measure each millimeter directly on the 8’851.8 km (5,500.3 miles) of the Great Wall of China, you’d need only 35 systems. Micrometer? 45 systems only.

(Linear) mind blown.

What about repairs? You just need to look at the patterns on the left and right, and color accordingly; the inventory for spare parts would be very small. There would be other aspects of scalability to consider, but in terms of computation, the principle of multi-frequency sampling for position encoding is simple and very scalable. No wonder nature evolved this solution, even in the sloppiness of nature. But that’s nothing new (what, again?).

Mathematics — fighting variability with abstraction

I still remember, many years ago as a Robotics undergrad, the sparkle in the eyes of a friend of mine at EPFL: “No matter what wave, it provides you with the frequency components!” I sparkled back: All of them? —That’s also very scalable. Yes, and the Fourier Transform can also be inverted, which means that a bunch of different frequencies can be combined into a single wave.

If the wave is a single position, a Dirac, the Fourier Transform shows that the spike is equivalent to the combination of all frequencies. If there's noise in the position measurement, distributed as a Gaussian (similar to Grid Cells), its Fourier transform is approximatively a set of frequencies with the same coefficients. In other words, if the input to a population-based positioning system is not very precise, the output can be calculated using a set of waves of different frequencies.

Approximation of the Fourier Transform on a Dirac and a Gaussian bell (axes omitted for visualisation simplicity).

Does this mean that the Grid Cells calculate Fourier Series? Maybe. Very likely not in a rigorous fashion. But it’s certainly a different lens with which to interpret and analyze the system. We can look at the properties of, in this case, the Fourier Transform and test them on the system, even at different levels of abstraction. For instance, the frequency spectrum of the sensory sampling would determine the resolution of the readout, which can have a measurable effect on the behavior of the organism.

The best part about thinking in these interdisciplinary terms (we strolled on the Great Wall through Biology, Neuroscience, Engineering, and Math) is that you can apply these principles to other organisms and other applications beyond positioning. Probably also other structures of the brain use it. For instance the eyes. Wait, why would the eye use a position encoder? That’s another story.

~ Nicola Rohrseitz, PhD