Saúl Huitzil

A random Boolean network is a minimal model of a gene regulatory circuit. Each node is a gene that is either on or off (expressed or not), with k randomly chosen regulators and a random Boolean rule that decides its next state from theirs. Every gene updates at the same time, one step per frame.

The graph on the left is the network: bright nodes are genes that are on, faint ones are off, and the edges are regulatory interactions. The grid on the right is the state trajectory — one row per time step, one column per gene, green for on. After an irregular transient the same block of rows starts repeating: that cycle is an attractor, the set of states the network settles into.

The numbers D(s0, s1) … D(s3, s4) average, over many random starting states, the fraction of genes that flip at each step. They are a quick read on the dynamical regime: values near zero describe a frozen, ordered network, values near 0.5 mean that half the genes change at every step, as in the chaotic regime. The k slider moves you between those regimes — k = 1 tends to freeze, k = 2 is the classic critical case, and larger k becomes chaotic.

The Vicsek model simulates the collective behaviour of self-propelled particles — birds or fish in a flock, or cells in a tissue. Each particle updates its velocity to the average velocity of its neighbours within a certain distance, and then moves in the updated direction.

It matters for living systems because it shows how coordinated behaviour can emerge in a group without a leader or central control, and it helps explain how cells in a tissue coordinate to form patterns during development.

Force-directed graph drawing visualises complex network structures — social, transport or biological. Nodes are treated as particles and edges as springs, and the algorithm simulates the forces between them (Hooke's law among others) until the energy of the system is minimised, producing a layout that is easy to read.

It is useful because it uncovers hidden patterns in large networks: groups of highly connected nodes (communities or clusters), network hierarchies, and the key players that hold a structure together — often in a way non-experts can interpret at a glance.

The active elastic model is a minimal description of self-propelled agents that interact through attraction, repulsion and alignment using elastic interactions. It has a simple mechanical realisation and applies to real systems such as active cell membranes, robotic swarms or animal groups.

Agents are connected to their neighbours by linear springs placed a distance R in front of their centres of rotation. Depending on R, the elastic interactions produce mainly attraction–repulsion or mainly alignment.

A hands-on version of the model above, running directly in the page: watch how the collective state changes as the geometry of the springs between agents is varied.