12  From Structure to Time: FEP and Attractors

Status: Draft

v0.3

12.1 Learning Objectives

After reading this chapter, you will be able to:

  • Understand how Structural patterns (perfect forms and invariant patterns) prepare for dynamic processes
  • Introduce the Free Energy Principle (FEP) as a unifying framework
  • Recognise how FEP connects Structural (idealised/invariant attractors) to Dynamical (time-varying attractors)
  • Understand the Markov blanket as system boundary
  • See how Structural world prepares for Dynamical world
  • Prepare for the transition to Part II: Dynamical

12.2 Introduction

In the Structural world, we’ve focused on stable modelling assumptions: graph structure, invariances, and abstract relations. This chapter shows how Structural assumptions prepare for time to enter the framework. While still working at an abstract level, we introduce concepts that bridge to the Dynamical world: the Free Energy Principle (FEP) and attractors.

12.3 Structural Summary: What We’ve Learned

12.3.1 What We Learned in Structural

In Part I (Structural), we established:

  1. The Causal Hierarchy and Three Worlds (Chapter 1): Framework of three worlds and three levels
  2. The Primary Unit: The Dyad (Chapter 2): Fundamental unit of directed dependence
  3. Pure Structural (Chapter 3): Graph theory, SCMs, identification, do-calculus, counterfactuals, transportability
  4. State-Space Models: Inferring structural patterns (perfect forms, invariant mechanisms, edge structure) from observable data (Chapter 12)
  5. Observational Methods: Learning from data using G-methods, TMLE, and experimental design (Chapters 19-21)

The Structural world is characterised by:

  • Idealised structure: Causal structure expressed as assumptions (e.g., graphs/mechanisms) that are treated as time-invariant within a modelling context
  • Invariant Patterns: Stable, environment-independent abstract relations
  • Graph Structure: Directed dependencies (edges) encoding structural assumptions and invariances
  • Abstract Relations: Still no time—relations are abstract, not yet temporal

12.3.2 Idealised and Invariant Attractors in Structural

Idealised attractors (Structural):

  • Idealised states toward which a modelled system tends in principle
  • Timeless, spaceless
  • Represent structure expressed at the level of assumptions (before specifying time-varying dynamics)

Invariant Attractors (Structural):

  • Fitness-maximizing states that are stable and environment-independent
  • Still abstract, still no time
  • Represent invariant mechanisms (stable dependencies across contexts)

Both idealised and invariant attractors belong to the Structural layer: they describe stable targets implied by a model class, before time-varying evolution is made explicit.

12.3.3 Structural Dependencies (Edges)

In the Structural world, edges encode both:

  • Idealised structure: Which variables can directly affect which others (model assumptions)
  • Invariant patterns: Stable, environment-independent relations (invariances)

Edge structure remains constant across environments and contexts, but edges remain abstract and timeless—not yet temporal.

12.4 The Free Energy Principle as Bridge

12.4.1 FEP Connects Structural to Dynamical Attractors

The Free Energy Principle (FEP) provides a unifying framework that connects Structural (perfect and invariant) to Dynamical (dynamic) attractors (Friston 2010; Friston et al. 2006). FEP states that systems minimize variational free energy, which can be understood as:

  • Surprise minimization: Systems avoid surprising states
  • Prediction error minimization: Systems minimize prediction errors
  • Active inference: Systems act to maintain expected states

FEP across worlds: - Perfect Attractors (Structural): Ideal free energy minima in perfect causal space (perfect edges) - Invariant Attractors (Structural): Stable free energy minima in structural space (invariant edges) - Dynamic Attractors (Dynamical): Time-dependent free energy minima (dynamic edges)

12.4.2 FEP as Principle Explaining Edge Evolution

FEP provides the principle that explains why systems tend toward attractor states at all levels, and how edges (prehensive relations) evolve:

  • At the Structural level: Edges encode perfect forms and invariant patterns (perfect and invariant prehensive relations)
  • At the Dynamical level: Edges encode dynamic equilibria (dynamic prehensive relations)

The FEP shows how the same principle governs edge evolution from Structural to Dynamical, connecting perfect/invariant → dynamic.

12.4.3 The Markov Blanket: System Boundaries in FEP

The Markov blanket is a fundamental concept in FEP that defines the boundary between a system and its environment (Friston 2010, 2013). In FEP, the Markov blanket separates a system into three parts:

  1. Internal states (\(\mu\)): The system’s internal variables (e.g., latent states in a state-space model)
  2. External states (\(\eta\)): The environment outside the system
  3. Markov blanket: The boundary that separates internal from external states, consisting of:
    • Sensory states (\(s\)): Observations that the system receives from the environment
    • Active states (\(a\)): Actions that the system takes to influence the environment

12.4.4 Conditional Independence Structure

The Markov blanket creates a conditional independence:

\[ \mu ⫫ \eta \mid (s, a) \]

In words: internal states and external states are conditionally independent given the Markov blanket (sensory and active states).

12.4.5 FEP and System Boundaries

The Free Energy Principle states that systems minimise variational free energy by maintaining their Markov blanket structure. The blanket defines what is “inside” the system (internal states) versus what is “outside” (external states), and the system acts to maintain this boundary structure.

World context: The Markov blanket is a Structural/Dynamical concept—it describes how systems maintain their structure through time. It bridges the Structural world (invariant system boundaries) and the Dynamical world (time-dependent boundary maintenance).

12.4.6 Connection to Three Levels of Reason

The Markov blanket structure relates to all three levels of Reason:

  • Level 1 (Association): The conditional independence \(\mu ⫫ \eta \mid (s, a)\) is an associational statement about what can be learned from observations
  • Level 2 (Intervention): Active states (\(a\)) represent interventions—actions the system takes to maintain its structure
  • Level 3 (Counterfactual): Internal states (\(\mu\)) represent the system’s beliefs, which can be used for counterfactual reasoning about alternative system states

12.5 World Comparison: Structural vs Dynamical

Aspect Structural Dynamical
Attractor Type Perfect and Invariant (ideal forms, stable patterns) Dynamic (changing, environment-dependent)
Nature Abstract relations, no time Dynamic processes, time enters
FEP Application Ideal and stable free energy minima Time-dependent free energy minima
Mechanisms Perfect and invariant mechanisms Time-dependent mechanisms
Examples Graph structure, perfect forms, invariant patterns Dynamic processes, flows

12.6 Attractor Progression: Structural → Dynamical

Perfect and Invariant Attractors (Structural): - Perfect: Ideal forms toward which systems tend in principle - Invariant: Stable, environment-independent fitness-maximizing states - Both abstract, both no temporal dimension

Dynamic Attractors (Dynamical): - Time-dependent, environment-dependent - Fitness-maximizing but changing with environment - Temporal dimension enters

The progression: From perfect and invariant attractors (Structural) to dynamic attractors that change with time and environment (Heraclitean/Whiteheadian process).

12.7 Time Enters: The Dynamical Dimension

12.7.1 What Changes: Structural Prehensive Relations → Temporal Prehensive Relations

With the transition to Dynamical, time enters the framework:

  • Structural edges → Dynamic edges: Prehensive relations become time-dependent
  • Abstract prehensive relations → Temporal prehensive relations: Relations gain temporal structure
  • Perfect/invariant mechanisms → Dynamic mechanisms: Processes evolve over time
  • Perfect/invariant attractors → Time-dependent attractors: Attractors shift with environment

12.7.2 Time Enters, But Processes Remain Abstract

Critical distinction: Time enters, but processes remain abstract (not yet material/physical):

  • Temporal but abstract: Edge structure becomes time-dependent, but still abstract
  • Time-dependent abstract processes: Processes become time-dependent, but not yet actualised into material reality
  • Not yet material/physical: The Dynamical world is temporal but abstract, not yet fully embodied

Edge structure becomes temporal, but still abstract: - Edges encode time-dependent prehensive relations - Edge mechanisms evolve over time - But edges remain abstract—not yet material/physical

12.7.3 Structural Edges → Dynamic Edges

The transition in terms of edges:

  • Perfect/invariant prehensive relations → Dynamic prehensive relations
  • Perfect/invariant edge semantics (ideal forms, stable patterns) → Dynamic edge semantics (time-dependent patterns)
  • Timeless but structured → Temporal but abstract

The edges themselves don’t change—what changes is the temporal structure of the edges, from perfect/invariant to dynamic, while remaining abstract.

12.7.4 Heraclitean Change

“Everything flows” — πάντα ῥεῖ — and “you cannot step into the same river twice” (Kirk et al. 1983). The optimum is ever-changing, but this change is still abstract—not yet material/physical actualisation.

This is the key transition: from abstract relations (Structural) to temporal but abstract processes (Dynamical). Full material/physical actualisation happens at the Dynamical → Observable transition.

12.8 Setting Up Part II: Dynamical

In Part II (Dynamical), we will explore:

  1. Deterministic Dynamics (ODEs): Dynamic processes as causal mechanisms (dynamic edges)
  2. Stochastic Dynamics (SDEs): Time-dependent processes with environmental variation (stochastic edges)
  3. Regime Switching: Multiple dynamic attractors (switching edge mechanisms)
  4. Network Models: Prehensive structure in space (spatial edge structure)
  5. Dynamics on Networks: Collective dynamic attractors (network edge dynamics)
  6. Resilience and Robustness: Stability of dynamic attractors (edge stability)

The Dynamical world introduces: - Time: Temporal dimension enters (edges become time-dependent) - Dynamic Processes: Time-dependent mechanisms (edge mechanisms evolve over time) - Dynamic Attractors: Environment-dependent, ever-changing optima (dynamic edge attractors) - Temporal but Abstract: Processes are time-dependent but still abstract, not yet material/physical

12.8.1 Emphasising Edges in Dynamical World

In the Dynamical world, we will see how:

  • Edges become temporal: From perfect/invariant prehensive relations to dynamic prehensive relations
  • Edge structure becomes time-dependent: The graph structure evolves over time
  • Edge mechanisms are dynamic: The functions encoding prehensive relations change with time
  • Edge topology remains abstract: Still abstract, but now temporal

The focus remains on edges (prehensive relations) as the fundamental units, but now they are time-dependent while remaining abstract. Full material/physical actualisation happens at the Dynamical → Observable transition.

12.9 Worked Example: Sheep System Evolving Through Worlds

12.10 Worked Example: Sheep System from Structural to Dynamical

Structural View (Part I): - Perfect forms: ideal causal structure of predator-prey relations - Invariant patterns: stable, environment-independent interactions - Abstract relations: graph structure of interactions - No time: timeless, spaceless causal structure

Dynamical View (Part II): - Dynamic mechanisms: time-dependent predator-prey dynamics - Dynamic attractors: population states that change with environment - Dynamic processes: actual population trajectories over time

The same system, now with time: from abstract structural relations to dynamic processes.

12.11 Transition to Dynamical

As we move to Part II, we transition from: - Perfect/invariant mechanisms → Dynamic mechanisms - Perfect/invariant attractors → Time-dependent attractors - Abstract relations → Dynamic processes - No time → Time enters

The Dynamical world introduces the temporal dimension while maintaining causal structure, preparing us for the Observable world where processes become fully actualised. Dynamical is made from Structural—it takes the structural forms (perfect and invariant) and dynamizes them, further embodying the inner worlds.

12.12 World Context

This chapter bridges the Structural and Dynamical worlds. We’ve completed our exploration of structural patterns (perfect forms and invariant patterns) and now introduce the Free Energy Principle and dynamic processes. This sets the foundation for understanding how time enters the framework through gradual embodiment—Dynamical is made from Structural, further embodying the inner worlds. This prepares us for the Dynamical world where dynamic attractors and temporal processes appear.

12.13 Key Takeaways

  1. Structural summary: Perfect forms and invariant patterns exist within the Structural world (perfect and invariant edges)
  2. The Free Energy Principle as bridge: FEP connects Structural (perfect/invariant) → Dynamical (dynamic) attractors (perfect/invariant → dynamic edges)
  3. Time enters: Structural prehensive relations (edges) become dynamic prehensive relations (edges)
  4. Temporal but abstract: Edge structure becomes time-dependent, but processes remain abstract (not yet material/physical)
  5. Actualisation happens later: Full material/physical actualisation happens at Dynamical → Observable transition, not in Dynamical world itself
  6. This transition prepares us for Part II: Dynamical

12.14 Further Reading