Simple pattern formation with cellular automata

A cellular automaton is a dynamical system where space, time and dynamic variable are all discrete. The system is thus composed of a lattice of cells (discrete space), each described by a state (discrete dynamic variable) which evolve into the next time step (discrete time) according to a dynamic rule.
\begin{equation}
x_i^{t+1} = f(x_i^t, \Omega_i^t, \xi)
\end{equation}
This rule generally depends on the state of the target cell $x_i^t$, the state of its neighbors $\Omega_i^t$, and a number of auxiliary external variables $\xi$. Since all these inputs are discrete, we can enumerate them and then define the dynamic rule by a transition table. The transition table maps each possible input to the next state for the cell. As an example consider the elementary 1D cellular automaton. In this case the neighborhood consists of only the 2 nearest neighbors $\Omega_i^t = \{x_{i-1}^t, x_{i+1}^t\}$ and no external variables.

In general, there are two types of neighborhoods, commonly classified as Moore or Von Neumann. A Moore neighborhood of radius $r$ corresponds to all cells within a hypercube of size $r$ centered at the current cell. In 2D we can write it as $\Omega_{ij}^t = \{x^t_{kl}:|i-k|\leq r \wedge |j-l|\leq r\}\setminus x^t_{ij}$. The Von Neumann neighborhood is more restrictive: only cells within a manhattan distance of $r$ belong to the neighborhood. In 2D we write $\Omega_{ij}^t = \{x^t_{kl}:|i-l|+|j-k| \leq r\}\setminus x^t_{ij}$.

Finally it is worth elucidating the concept of totalistic automata. In high dimensional spaces, the number of possible configurations of the neighborhood $\Omega$ can be quite large. As a simplification, we may consider instead as an input to the transition table the sum of all neighbors in a specific state $N_k = \sum_{x \in \Omega}\delta(x = k)$. If there are only 2 states, we need only consider $N_1$, since $N_0 = r – N_1$. For an arbitrary number $m$ of states, we will obviously need to consider $m-1$ such inputs to fully characterize the neighborhood. Even then, each input $N_k$ can take $r+1$ different values, which might be too much. In such cases we may consider only the case when $N_k$ is above some threshold. Then we can define as an input the boolean variable

\begin{equation}
P_{k,T}=\begin{cases}
1& \text{if $N_k \geq T$},\\
0& \text{if $N_k < T$}.
\end{cases}
\end{equation}

In the simulation you can find here, I considered a cellular automaton with the following properties: number of states $m=2$; moore neighborhood with radius $r=1$; lattice size $L_x \times L_y$; and 3 inputs for the transition table:

  • Current state $x_{ij}^t$
  • Neighborhood state $P_{1,T}$ with $T$ unspecified
  • One external input $\xi$\begin{equation}
    \xi_{ij}=\begin{cases}
    1& \text{if $i \geq L_x/2$},\\
    0& \text{if $i < L_x/2$}.
    \end{cases}
    \end{equation}
  • Initial condition $x_{ij} = 0 \; \forall_{ij}$

For these conditions a deterministic simulation of these conditions yields only a few steady states: homogeneous 1 or 0, half the lattice 1 and the other 0, and oscillation between a combination of the previous.

One possibility would be to add noise to the cellular automaton in order to provide more interesting dynamics. There are two ways to add noise to a cellular automaton:

The most straightforward way is to perform the following procedure at each time step:

  • Apply the deterministic dynamics to the whole lattice
  • For each lattice site $ij$, invert the state $x_{ij}$ with probability $p$

This procedure only works of course for $m=2$. In the case of more states there is no obvious way to generalize the procedure and we need to use a proper monte carlo method to get the dynamics.

A second way is to implement a probabilistic cellular automaton. In this case the transition table is generalized to a markov matrix: each input is now mapped not to a specific state but rather to a set of probabilities for a transition to each state ($m$ probabilities). Naturally for each input these sum to one. In this case we have $m$ times more parameters than before.

Preprint review: Parameter Space Compression Underlies Emergent Theories and Predictive Models

So here’s a preprint I found really interesting [arxiv:1303.6738]. I’ll try to give a quick overview of the story in my own words.

The main concept used in the paper is the Fisher Information, which is no more than a measure of the curvature in the space of probability distributions. It is easy to intuitively understand what it is in the 1D case. Suppose you have a probability distribution for some random variable $x$ parametrized by $\theta$: $P(x|\theta)$. If you change $\theta$ by an infinitesimal amount, how will the probability distribution itself change? Will it be vastly different or almost the same? We can quantify that change by averaging the square of the relative changes of the probabilities of all the points: $$\mathcal{I}=E\left[\left(\frac{dP(x)}{d\theta}\frac{1}{P(x)}\right)^2\right]_x$$

Another way to look at it is as quantifying the “resolution” with which we can detect the parameter $\theta$: when the FI is high, we can distinguish between 2 parameters with close values more easily than for low FI, which corresponds to a higher resolution in parameter space. But why can I make this statement? After all the FI quantifies the difference between the probability distributions, not the parameters which specify them. The reason the “resolution” picture makes sense is thanks to the Cramér-Rao bound: $$var(\hat{\theta})=\frac{1}{\mathcal{I}}$$

To understand the bound we must make the following definitions: a hat over a parameter denotes the estimator of that parameter (an estimator is just a function that takes a set of realizations (or “measurements”) of the random variable we are looking at and spits out a number, which we hope will be close to the true value of the estimator) and the variance of an estimator is just the variance of the result of applying the estimator to many independent sets of measurements. With that in mind, the statement is that the variance of an estimator of our parameter is bounded by the inverse of the FI (note: we assume the estimator to be “unbiased“). Because FI for a set of N independent samples scales as N, for a very large sample size the variance of the estimator tends to zero, meaning we always get the same, “correct value”, as we’d intuitively expect.

Now bear in mind this is a very mathematical construct, hence all the quotation marks. The intuition I described above is only valid when the model is simple enough that we can afford to use the concept of unbiased estimators (which imply discarding all prior information we might have) and can assume that the underlying parameter is indeed a single number. Im many interesting cases we cannot assume this, but rather that it is a random variable itself (either due to nature, or due to intermediate processes we neglect to add to our model). I digress. The FI stands on its own as a useful tool in a myriad of applications, often related to quantifying the “resolution” of a system in the sense I tried to convey above.

In the paper we are looking at, the authors consider dynamic systems at a microscopic level with many degrees of freedom. In these systems you can consider the attributes of each particle a parameter, and code the many possible state of the system at some point in time by a probability distribution, which of course depends on this very large set of parameters. In this case the FI for any parameter is likely very small, as if you were to make a small change in one of these parameters the system would evolve in a very similar way leaving the probability distributions relatively unchanged, in the FI sense. The interesting step is now to find the eigenvalues of the FI matrix. This projects the parameters onto a new space where the directions correspond to some natural observables of the system.

Now, if we coarsen the system by allowing a long time to pass (i.e. a diffusion process) or by looking at it from a macroscopic scale (i.e. coarse graining an ising model) it turns out a few of these directions have a very large weight and the rest have comparatively low weight. The authors argue that these directions, when cast as observables, correspond to the macroscopic parameters of the system. Going back to the picture of the FI as resolution, these few observables will be the ones which we will be able to easily distinguish, while all the others will get lost in the noise. This is an appealing statement because it agrees with what we already know from statistical physics: we can accurately model systems at the macroscopic scale even if we have no hope to know what is going on at the microscopic level. Now we can see this idea emerge naturally from probability theory.

Another point they make is that this procedure works for both diffusive type processes, where we attribute this scale separation due to the fact that fluctuations are only relevant at the micro scale but not at the macro; and for processes with phase transitions where fluctuations are relevant at all scales at the critical point (cf. renormalization group). Under this framework there is a single explanation for why universal behavior is so prevalent in physics which I think is pretty cool.

Modeling excitable media with cellular automata

While researching for my seminar I came across a class of cellular automata which models spiral waves in excitable media. Because these models are so simple I had some fun implementing them in processing. Processing is great because you can use the javascript version to embed the visualization in a webpage directly and everyone can play with it. Here are some of the models I played around with:

  • Spiral, a model developed by Gerhard and Schuster to simulate a chemical reaction.
  • CCA, a simple cyclic cellular automaton.
  • StochCCA, where I added stochasticity to the previous model. This is useful to assign more weight to the 4 neighbors of the Von Neumann neighborhood than to the remaining 4 which complete the Moore neighborhood. This appears to make space “more isotropic” and makes the waves actually circular.