Humans have always sought to make life easier. It is almost what defines us as a species.
We have always sought ways to avoid mundane tasks, from making a pot stand so it doesn’t have to be held over fire, to having an automatic out-of-office reply on your email.
Automation
Since the industrial revolution, systems have become more complex, and we have seen more interest in the automatic control of those systems, one of which is the automatic steering of ships.
One problem with steering large ships is the effect of large wind forces. These cannot be easily corrected when observing directional error alone, and since large ships have a lot of inertia, it takes a lot of effort to get them to change direction or slow down.
Controlling automatic ship steering
In 1911, Elmer Sperry designed a system using calculus to better control ship steering.
His system considered:
- Proportional error: how far off is my current direction from the direction I want to be in?
- Integral error: what is the sum-total error over a given amount of time? And...
- Derivative error: how quickly is the rate of error changing?
Proportional, integral, and derivative error
Derivative error is helpful because it would be easy to overcompensate for the first two errors
Proportional error is self-explanatory; there is always some error from a true bearing that must be constantly corrected. Calculating integral error helps when a ship experiences strong winds because even if the proportional error is zero, the wind will still push the ship off course.
Derivative error is helpful because it would be easy to overcompensate for the first two errors so that the ship overshoots the desired bearing sending it off course still. Sperry’s original design used a gyroscope. It was adopted by the US Navy and utilised in WW1.
Hyper-direct pathway
Moving away from ships and onto biological nervous systems, the nerves for directly controlling movement are in the spinal cord. However, when you decide to catch a ball, there is a process in the brain that decides the action should take place and passes this information to the spinal cord. The spinal cord itself is unable to make conscious decisions.
One pathway, known as the hyper-direct pathway, has been identified as important when different species, including humans, encounter a physical go/stop situation. For example, running for a base in rounders or baseball.
MIT research
The distance to the location was randomised to prevent learning via timing
Researchers from MIT reported some interesting findings in Cell Reports, in July 2022. They set up an experiment where mice navigated a virtual reality corridor via a self-powered treadmill.
When the mice stopped at a precise location for a set amount of time, they received a reward. The distance to the location was randomised to prevent learning via timing so the mice had to rely only on what they saw.
Findings
The researchers determined that mouse brains required some error correction to successfully stop at the landmark in time and collect their reward. So, for example, if the mouse does not factor in how far away it is from the landmark, it will not increase its speed to get there quickly enough.
It needs to keep providing a stimulus for its legs to keep running. If the mouse is running at full tilt, it must compensate for the time it's braking before it overshoots the landmark and misses its reward. It must inhibit leg movement to stop. They discovered the mouse’s brain compares its internal movement plan with its current moving state.
Error signal
From the mouse’s point of view, a stop signal is generated, which we refer to as an error signal
When these two states become significantly different, an ‘error’ signal is produced that is interpreted as a stop signal. Or, from the mouse’s point of view, a stop signal is generated, which we refer to as an error signal.
This presents the same dilemma as with steering ships. If the mouse simply reacts to receiving a single error signal, it will begin stopping too soon and fall short of the landmark.
Integration
The mouse needs to sum-total the error signal over time, otherwise known as integration in mathematics, to ensure it is time to begin stopping. However, the visual system of the mouse is too slow to integrate quickly enough.
Solely relying on this means the mouse will be too delayed, overshoot the landmark, and miss the reward.
Optimal movement
Researchers discovered the mouse compares the difference between the stimulus and inhibitory signals at any point in time just like taking a derivative in mathematics.
It allows the mouse to stop soon enough to hit the landmark without overshooting. The mouse brain is using calculus, both derivatives, and integrals, to create optimal movement.
PID controller
Modern electronic PID controllers are widely found in devices that require feedback to know if they are doing the right thing
Back to ships, and in 1922 a Russian-American engineer named Nicolas Minorsky generalised the mathematics behind Sperry’s navigation system. What emerged was the modern PID (Proportional Integral Derivative) controller, an indispensable device in modern control systems, typically implemented on a microchip.
Modern electronic PID controllers are widely found in devices that require feedback to know if they are doing the right thing. For example, the temperature control of an oven, cruise control of a car, reagent mixing in a chemical plant, or climate control in a building.
Corrective adjustments
There is irony in how Minorsky came to develop this area of control theory. He noticed the helmsman of naval ships provided corrective adjustments to a ship’s direction based upon:
- Current error - “I am 4°off-course, therefore correct by 4°”
- Previous error - “How far was I off-course over the last 10 seconds?” and
- Rate of error - “How quickly am I approaching the direction I want?”
The helmsman was using calculus to optimally navigate the ship just like our mouse trying to get its reward.
Role of engineering
Engineering seeks to reduce complex systems into constituent parts to make them more manageably understandable and to gain deeper insight into the system. But the human brain is the most complex system we have encountered in the universe. If we could reduce it all too basic calculus, the brain would not still present so much mystery.
For all that engineering has solved, it still doesn’t explain the conscious experience. However, it is interesting to know that biology has also removed some of the mundane tasks in life, in the same way humans have done with machines (PID controllers included).
