Liquid Brains and Agent Soups

A unified mathematical skeleton for systems where who talks to whom is itself part of what the system computes.

The central idea is simple but underappreciated: most real computational systems — brains, ant colonies, robot swarms, multi-agent AI — don't operate on a fixed wiring diagram. The connections change as the agents move, update their internal states, or learn. This paper formalizes that observation into a single framework called an agent soup. Each agent has an internal state and a position (or other dynamical variable), emits an output signal, and receives inputs from neighbors. The key move is treating the connectivity matrix — who influences whom, and how strongly — as a full dynamical variable with its own equations of motion, not a static backdrop.

The formalism is deliberately minimal. Four numbers of equations cover it: agent state evolves driven by weighted inputs from neighbors; the weight matrix itself evolves driven by agent states; outputs are a function of agent state; inputs are a weighted sum of neighbors' outputs. That's the whole skeleton. What makes it useful is that an enormous range of systems drop out as special cases by choosing different state spaces and update rules. The paper works through four "computational classes": Boolean agents doing majority voting on a time-varying graph; oscillating agents doing Kuramoto synchronization while moving (connecting to "swarmalators"); neural-style agents with continuous vector states and Hebbian-like connectome updates; and full algorithmic agents that maintain compressed world-models, evaluate objectives, and plan actions — with social interaction implemented as agents exchanging compressed model fragments ("gossip as model exchange").

One of the more interesting conceptual moves is the distinction between structural and effective connectivity. A system can have a completely fixed anatomical wiring — neurons don't move — and still exhibit liquid-like computation if the effective communication pathways change rapidly with dynamical state. Oscillatory phase alignment, gain modulation, and envelope correlations can gate which anatomical connections are actually used at any moment. The brain is the canonical example: solid scaffold, liquid routing. The paper formalizes this as $G_e(t) = f(G_s, x(t), \theta)$, where the structural graph $G_s$ is fixed but the effective graph $G_e(t)$ fluctuates with the system's instantaneous dynamical state $x(t)$.

The paper is primarily a technical note establishing vocabulary and formalism rather than reporting new empirical or simulation results. It doesn't prove theorems about what agent soups can compute, nor does it benchmark the framework against alternatives. What it does do is provide a clean common language that lets you see Kuramoto oscillators, liquid-brain ant colonies, neural ODEs, and active-inference-style algorithmic agents as instances of the same underlying structure — which is genuinely useful for anyone trying to reason across those literatures simultaneously.