A statistical mechanics approach to cultural evolution of structured behavior in non-human primates: From disorder to tetris-like structures

Data concerning [7] have been deposited as an open-source repository in the Dryad database (doi:10.5061/dryad.0f1m0). The experiment described in [7] is divided into 6 transmission chains. A transmission chain is simply a predefined random order of the 12 monkeys. For this experiment, a "time step" or evolutionary step is defined by one monkey interacting with a grid of squares. For each transmission chain, the group of monkeys successively interact towards the emergence of structured behavior (for details of the participants and the experimental protocol, see [7]). At each evolutionary step, a different monkey was exposed to a grid of 16 squares, 12 white and 4 red. This defined a trial. The same participant received a first block of 50 transmission trials.

The procedure for the between-individuals transmission relies on the fact that the actual pattern of squares touched while attempting to reproduce these 50 transmission trials and became the set of target patterns (after randomly reordering) shown to the next individual in that chain. From this, the experimental data of [7] is defined by 6 transmission chains, in which 50 trials are exposed to the group of 12 monkeys. All our calculations with the experimental data of [7] are stored in https://github.com/javiervz/baboons.

Each trial consisted of the interaction between a monkey and a grid made by 16 squares, 12 white and 4 red. Here, each trial $T = (t_{ij})$ is understood as a $4\times 4$ binary matrix where $t_{ij}=1$ if the cell i, j is activated, and 0 otherwise. The (von Neumann) neighborhood of a cell i, j is the set $V_{ij} = \{(i,j+1),(i,j-1),(i+1,j),(i-1,j)\}$ (for details, see fig. 2).

Zoom In Zoom Out Reset image size

To shed light on the formation of tetris-like shapes, as an evidence of the emergence of structured behavior, we used a number of measures pointing out to the same phenomenon: the participants of the experiments described by [7] prefer the activation of closer cells over evolutionary steps. We focus on the description of the tetris-like shapes emerging from activated cells approaching to each other, by means of 1) the connectivity of the grid cells and 2) the distance to the nearest activated cell; and the simplification in cognition, by means of 3) energy-like measure.

We provide two simple measures to quantify the way in which squares become closer to each other. First, given a trail $T = (t_{uv})$ (at each neighborhood V_u ) such that the cell u is activated, the function c_u is defined, which is 1 in the case that one neighbor in V_u is activated, and 0 otherwise. Averaging this quantity over all cells defines the connectivity of the configuration at that time step,

$$\begin{equation} C = \frac{1}{4} \sum_{u} \frac{1}{|V_u|} \sum_{v \in V_u} c_v. \end{equation} \tag{ 1 }$$

From this definition, it is natural to think that the evolutionary steps from random cell grids to the formation of tetrominos can be characterized by an increase of connectivity.

Second, at each neighborhood V_u the function d_u is defined, which measures the distance between u and the nearest activated cell. The distance between two cells is $|i_u - i_v| + |j_u - j_v|$, where i_u , i_v are the respective row numbers, and j_u , j_v are the respective column numbers. Summing this quantity over all cells defines the distance to the nearest activated cell of the configuration at that time step,

$$\begin{equation} D = \sum_{u} \frac{1}{|V_u|} \sum_{v \in V_u} d_v. \end{equation} \tag{ 2 }$$

Random configurations of activated cells could lead to small distances to the nearest activated cells. By contrary, the formation of tetromino structures could led to high distances to the nearest activated cells.

An energy-like measure is proposed to describe the cultural evolution of structured behavior as the tendency to put cells with the same activation state (activated or deactivated) together. In simple terms, each trial matrix is simply understood as a $4\times 4$ binary matrix, formed by activated and deactivated cells. To explicitly describe the amount of local agreement between cells, a function, called the "energy", is defined (for a similar function, see [17]). This energy-based approach arises from a physical interpretation. The energy measures the amount of local instability of the configuration. Large values of energy imply the evolution until ordered low-energy configurations are reached.

At each neighborhood V_u , the function $\delta_u$ is defined, which is 1 in the case that $x_u = x_v$ (agreement between the vertices u and v), and 0 otherwise (disagreement). Thus, it measures the amount of local agreement of the neighborhood $\sum_{v \in V_u} \delta(x_u,x_v)$. Summing this quantity over all cells defines the total energy of the configuration at that time,

$$\begin{equation} E = -\frac{1}{16} \sum_{u} \frac{1}{|V_u|} \sum_{v \in V_u} \delta_v. \end{equation} \tag{ 3 }$$

As a first simple numerical simulation, the average connectivity of grid cells over transmission chains was analyzed in fig. 3 (top). As a general observation, a smooth increase in connectivity C vs. time can be noticed. Two domains can be observed. First, C increases abruptly for t < 7, indicating that for this domain the participants prefer to touch closer cells. Second, a stable phase is attained for t > 7. This indicates that from that time step activated cells remain close to each other over transmission chains. Strikingly, the behavior of C suggests the important fact that the observation provided by [7] that in non-human primates it is possible to observe evolution of structure (tetris-like figures) could be an indirect effect of the increasing of connectivity.

As shown in fig. 3 (bottom), over time the participants prefer to slowly put activated cells closer to each other. More precisely, a smooth increase in D vs. time can be noticed. Two domains can be observed. First, D increases abruptly for t < 3, indicating that for this domain the participants prefer to put activated cells together. Second, a more stable phase is attained for t > 3.

Zoom In Zoom Out Reset image size

We now shift our focus from general observations about cell connectivity patterns to a general view of simplification in cognition. As shown in fig. 4, the energy-like function E over transmission chains exhibited a global tendency of smooth decreasing. As in the previous subsection, two domains can be observed for the evolution of E. First, E decreases abruptly for t < 7, which is an evidence for an initial phase of energy minimization. Second, a stable phase is attained for t > 7. The inverse relationship between the average connectivity and the energy-like function might provide some evidence for three main facts. Firstly, non-human participants might prefer connected cell patterns of lower energy; secondly, it is plausible to think that the appearance of tetris-like structures is an indirect consequence of energy minimization; and maybe more importantly, thirdly, non-human primates simplify efforts while they participate in cultural evolution experiments. Although this kind of minimization seems obvious, it is important to remark the influence of this fact on the observed structured behavior.

Zoom In Zoom Out Reset image size

What is the influence of the energy-like minimization on the grid of cells? To answer this question, we studied a simple algorithm, based on several related models of discrete dynamics (see, for example, [18–20]; for a general overview of this kind of models, see [21]). Our interest is to provide a quantitative view of the collective phenomena emerging from the interactions of individual cells as elementary units in trial matrices. The energy-like function E is minimized with the following algorithm. Consider an initial cell grid A of size $N \times N$. N cells of A are activated, while $N \times N - N$ are deactivated. At each time step, the matrix A is modified by randomly changing the state (activated or deactivated) of some pairs of cells with opposite states, and the new A matrix is accepted if the energy-like function E is lowered. With this procedure, we ensure that the number of activated cells remains constant.

We provide a simple definition of a tetris-like structure. Consider the matrix A of size $N \times N$ with N activated cells. T is a tetris-like structure if 1) T is formed by N cells; and T is connected. A group of activated cells G is connected if each neighbor's cell is activated.

If the hypothesis that the appearance of structured behavior over transmission chains is a consequence of energy minimization were valid, tetris-like structures should appear. As shown in fig. 5, the algorithm provides a simple way to minimize the energy-like function E. We need to remark an important fact: all trajectories defined by the algorithm minimized the energy-like function, so we can expect the formation of tetris-like structures. This is true indeed, as shown in fig. 6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In this paper, we proposed a bridge between cultural evolution in non-human primates and Statistical Mechanics techniques. We focused on measuring the emergence of tetris-like structures as described by the experiments of [7]. The most important measures studied were the average connectivity and the energy-like function, which pointed to a fine-grained characterization of the formation of tetris-like structures.

This study's results suggest an inverse relationship between the tendency of activated cells to become closer to each other and the minimization of (cognitive) efforts. The most important fact arising from this observation is twofold: Firstly, it is plausible to think that the appearance of tetris-like structures is an indirect consequence of energy minimization; and, secondly, we may hypothesize that non-human primates simplify their efforts while they participate in cultural evolution experiments. If the previous idea is confirmed, a couple of pending questions are: What are the cognitive pressures and constraints that lead to the appearance of behavioral structures? What is the role of reward in this process of structure formation?

To go in depth on low-energy configurations for each trial, we count the proportion of tetromino squares of activated cells over transmission chains. As shown in fig. 7, the proportion of squares varies from 0.05 to 0.2. Underlying this measure, it is plausible to suggest that the participants minimized cognitive (and maybe not motor) efforts.

Zoom In Zoom Out Reset image size

An interesting way to answer these pending questions would be to investigate how in idealized agent-based scenarios agents can negotiate the formation of behavioral structures [22–24], constrained by simple cognitive pressures. Current work is investigating an agent-based game, in which players are equipped with simple forms of working memory while developing structures in cultural evolution experiments.

Future work could involve a fascinating related question within the Iterated Learning paradigm: Is it possible to observe energy-like minimization in a population of human participants? For example, as described above, in [5] human participants must learn and reproduce a miniature language. The participants are asked to learn a language made up of written labels for visual stimuli. As usual, the output of a participant is the input language for the next participant. We may ask: Is there some kind of optimization of a quantity (related to the energy-like function studied here)? Are there conserved quantities? The answer to these questions is not obvious. At first glance, future work should propose energy-like functions for the emergence of linguistic structure in cultural evolution frameworks. A good start point to face this problem is the literature related to optimization of linguistic units or principles (see, for example, [25–29]). With this, we could explore the hypothesis that the emergence of linguistic structure is strongly related to the minimization of some form of cognitive effort.

Data availability statement: The data that support the findings of this study are openly available at the following URL/DOI: https://royalsocietypublishing.org/doi/suppl/10.1098/rspb.2014.1541.