Crossing the Functional Desert: Critical Cascades and a Feasibility Transition for the Emergence of Life

TL;DR

The paper addresses how persistent, structured functional organization can emerge in exponentially branching configuration spaces where unguided exploration is memoryless. It proposes functional percolation, a substrate-agnostic mechanism based on threshold cascades on networks, to enable persistent input-output mappings. Near a critical connectivity threshold, cascades become system-spanning yet discriminative, allowing biased accumulation without long-term memory or replication. This framework provides a dynamical route to crossing combinatorial feasibility barriers and yields testable predictions for synthetic and biological networks.

Abstract

The origin of life poses a problem of combinatorial feasibility: How can persistent functional organization arise in exponentially branching assembly spaces when unguided exploration behaves as a memoryless random walk? We show that nonlinear threshold-cascade dynamics in connected interaction networks provide a minimal, substrate-agnostic mechanism that can soften this obstruction. Below a critical connectivity threshold, cascades die out locally and structured input-output response mappings remain sparse and transient-a "functional desert" in which accumulation is dynamically unsupported. Near the critical percolation threshold, system-spanning cascades emerge, enabling persistent and discriminative functional responses. We illustrate this transition using a minimal toy model and generalize the argument to arbitrary networked systems. Also near criticality, cascades introduce structural and functional persistence, directional bias, and weak dynamical path-dependence into otherwise memoryless exploration, allowing biased accumulation over short coherence timescales. This connectivity-driven transition-functional percolation-requires only generic ingredients: interacting units, nonlinear thresholds, influence transmission, and non-zero coherence times. The mechanism does not explain specific biochemical pathways, but it identifies a necessary dynamical regime in which structured functional organization can emerge and persist, providing a physical foundation for how combinatorial feasibility barriers can be crossed through network dynamics alone.

Figures (2)

• Figure 1: Even a microscopic collection of interacting units can support functional accumulation. Minimal interaction structure already supports non-trivial functional responses. Two input nodes $A$ and $B$ (e.g., coarse-grained stimuli or activation sites) influence a downstream node $C$ through directed couplings. Node $C$ is a thresholded unit with threshold parameter $\phi$: It activates if the weighted fraction of active inputs exceeds $\phi$. For $\phi \le 0.5$, activation of either input suffices and $C$ implements an OR-like response. For $\phi > 0.5$, both inputs are required and $C$ implements an AND-like response. Because the interaction links and response rule persist over the cascade timescale, structured multi-input responses can recur over short coherence intervals within the same dynamical regime, supporting a minimal form of functional accumulation.
• Figure 2: Functional percolation as a feasibility transition in function space (schematic). As effective interaction connectivity increases, threshold-cascade dynamics organize into three regimes. Below the transition (subcritical), cascades die out locally and realizable input--output response functions are sparse and nonpersistent. Near the functional percolation threshold, non-saturating system-spanning cascades emerge, sharply expanding the accessible repertoire of persistent response functions and enabling long-range directed influence. Above the transition (supercritical), increasing connectivity strengthens collective coupling, reducing functional diversity and constraining responses toward more correlated global patterns without eliminating information flow. The near-critical regime constitutes a narrow feasible window in which accumulation in an exponentially branching space becomes possible through persistence and bias, without collapse or saturation of functional distinctions.