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.

Simulating tissues with pressure

One small project I did was to code up a simulation of a growing tissue which feels pressure and where each cell has a dynamic state which depends on its neighbors and the pressure it feels. The idea is to reproduce some essential properties of morphogenesis. You can look at the code here. I am going to talk about the most interesting parts of the code.

I initialize the cells in an ordered lattice, with random perturbations in their positions, except those which are in the borders (bottom, left, right). Those are static and do not evolve in the simulation like the others. They are there just to represent the pressure from the rest of the body (huge) on the simulated tissue (tiny). This is not a very realistic assumption because many developmental systems have a size of the order of the body size, but we have to start somewhere!

All cells are connected with springs, which simulate adhesive and pressure forces in the tissue. If left alone, the system relaxes into a hexagonal configuration, since this minimises the spring potential energy. I integrate the harmonic oscillator equations using a fourth order Adams Moulton algorithm.

Now, it is important to realize that there are two time scales in the system: the pressure equilibration and cell lifetimes. We can assume the mechanical pressure equilibrates very fast, while cell divisions take their time. So what we do is run the oscillator system until equlibrium for each time step of the cellular state evolution, which we will talk about later.

Springs connect each cell. The larger colored circle is of the same size as the rest length of the springs. Thus, overlapping circles mean the spring wants to extend, while spaces mean the spring wants to contract. The color denotes the automaton state of the cell.
Springs connect each cell. The larger colored circle is of the same size as the rest length of the springs. Thus, overlapping circles mean the spring wants to extend, while spaces mean the spring wants to contract. The colors are explained below.

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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.

Information processing systems post-mortem

Slides from the talk

Yesterday I gave an informal talk about information processing systems and lessons learned from the fields of AI and biology. This was a mix of introductory information theory and some philosophical ramblings.

While creating this talk I took the time to review several concepts from machine learning and AI. In Jaynes’ book about probability theory, bayesian inference is presented as a completely general system for logic under uncertainty. The gist of the argument is that an inference system which obeys certain internal consistency requirements must use probability theory as a formal framework. A hypothetical information processing system should obey such consistency requirements when assigning levels of plausibility to all pieces of information, which means its workings should be built upon probability theory. As a bonus, all the theory is developed, so we need only apply it!

To implement such a system we make a connection with biology. I started by arguing that an organism which wants to maximise its long term population growth must be efficient at decoding environmental inputs and responding to them. Thus if we define long term viability of an organism implementing a given information processing system as a finess function, we can obtain good implementations of our system by maximising such a function.

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