This past weekend I was wondering about our ability to estimate time, how the hell can we do that? Do we have an internal clock somewhere? How does it tick?

We all have somehow a notion of time. Good musicians are able to keep a perfect tempo, respect the music pauses and respect the notes length. Dancers synchronize their body movements with the rhythm of the music. The most evident examples of our internal clock are related with music but a notion of time might be necessary for all sort of things. When a volleyball player receives a pass he needs to decide the perfect timing for him to jump and hit the ball. He might have a very good idea of what are the dynamics of a falling object (he has seen it many times) and together with an internal clock he would be able to predict the ball’s trajectory and decide the best instant for him to jump and meet the ball. There are , however, other options to achieve this that don’t require the use of an internal clock. For example, the volleyball player could analyze at each instant the ball’s state history and decide if there would be a better instant to jump in the future or not based on past experience, he would then jump when his prediction said that there wouldn't be a better instant for him to do it. This last method seems to be very inefficient method in terms of memory and ability to generalize. It’s a very hard problem to be able to learn a value for each state in the huge state space, composed by the history of the ball state, and it seems even harder to reuse this learned knowledge in a similar situation but with slightly different conditions (e.g. playing with wind). Having an internal clock would certainly simplify the learning and generalization of the task so we will assume for now that we do have this internal clock. This assumption raises another number of questions: do we only have one internal clock or do we have many and we apply them to different tasks? how precise is this clock? is the clock delayed or in advance by a fixed amount of time? is the rate of time different? what is the behaviour of the error of this clock?

What made me think of this problem

Recently I made a simple program that tracked the position of a ball that was moving with constant angular speed over a circle. To do this I used a Particle Filter, i.e. I spawned a bunch of particles that represent hypothesis of the true state of the ball and I simulate their movement based on the dynamics of the ball (constant angular speed). I also wanted to represent the uncertainty that we have about the ball position if the ball gets invisible for a certain period. In the limit if the ball gets invisible for a long time we would no longer have any clue of where in the circle the ball might be, i.e. the ball is equally likely to be anywhere on the circle. There is a particular kind of imprecise internal clock that would converge to this solution. If an internal clock increases its time with a random walk the concentration of particles along the trajectory will converge to an uniform distribution evolving much like the concentration of ink particles in a glass of water $\frac{dc}{dt}=D \frac{d^2c}{dx^2}$, where $c$ is the concentration of particles and $D$ is the diffusion constant. When the concentration of ink particles is uniform on the glass of water ($\frac{d^2}{cdx^2}=0$) the concentration will remain constant along time. This random walk of the ink particles (also called Brownian Motion) is due to the thermal energy and it leads to an increase in entropy of the physical system with a speed dependent on the temperature of the system.

It would make sense for the particles in the Particle Filter to also move with brownian motion when deprived of new measurements and being sampled by an imperfect system, the brownian motion would increase the information entropy as fast as the Diffusion constant that in this case would represent how imperfect is the sampling system. There is no relation between a Particle Filter and what happens in the human mind but either way this made me wonder about the nature of our internal clock so I decided to do a simple experiment.

A subject is asked to press a key every second. Every time he presses a key a slider tells him how much was his error: if he’s going too fast or too slow. After the mean of the absolute error within 10 keypresses is less than 0.1 seconds the subject has essentially learned the correct rhythm and the slider will now always indicate that the error is zero and we start collecting data until we have n keypresses.

I did this experiment over the weekend, using one subject (myself). I did 15 trials and in each trial I made 15 keypresses.

Each keypress should be done after 1 second from the previous keypress, the following plot shows the distribution of the error between keypresses:

The plot shows that the mean is slightly negative which means that I am biased for a shorter second than the actual value (I hope that doesn't mean I am stressed person!!), however, the bias of ~0.05 seconds doesn't seem too bad for me, I'm curious about what would be the performance of a drum player. The distribution looks gaussian-like which is the distribution that a random walk would obtain.

Does my sense of rhythm decays after losing the slider information? The following plot tries to answer that:

It seems that my bias and standard deviation doesn't change much over time.

A particle, placed at the origin, that goes on a biased random walk (e.g. $xt=xt-1+\mathcal{N}(\mu, \sigma)$) has a mean position of $<xt>=t$ and a mean squared position of $<x2>= \sigma^2t+\mu t^2$. Let’s see if the error of my clock behaves like this! We get $\mu$ and $\sigma$ from the distribution plotted in the histogram and we used it to make the predictions that are plotted in the following graphs (note: now the error plot is no longer between keypresses but the cumulation of error until a specific keypress).

The mean error and the mean squared error evolution is similar to the one predicted which strengthens the theory of a random walk.

Final Considerations

The experience that I conducted is so simple that I feel compelled to program it on a website and try to collect hundreds of samples from it. There are other experiments to be made and questions to be answered! It seems that I was able to synchronize my finger's clock with a certain rhythm. Can I use now another finger with the same rhythm and without having to train it again? If so then the clock is more general than just a finger's clock!! Which clock is responsible for this movement? Is it muscular memory? To which extend and to which tasks can this trained clock be applicable?


This was my first blog post! I would love to get some feedback (harsh critics are allowed too!)  and create some discussion :D

Thank you and stay tuned for the Doubts I Have!