# Continuous-Time Active Inference This note explores the mathematical and cognitive foundations of continuous-time active inference, bridging [[free_energy_principle|free energy principles]] with continuous-time dynamical systems. ## Core Concepts ### Mathematical Foundations - [[variational_calculus|Variational Calculus]] framework for continuous-time optimization - [[differential_geometry|Differential Geometry]] of belief manifolds - [[path_integral|Path Integral]] formulation of belief trajectories - [[generalized_coordinates|Generalized Coordinates]] for representing continuous dynamics ### Cognitive Architecture - [[predictive_coding|Predictive Coding]] in continuous time - [[hierarchical_processing|Hierarchical Processing]] of sensory streams - [[neural_computation|Neural Computation]] implementation - [[bayesian_brain|Bayesian Brain]] perspective ## Mathematical Framework ### Free Energy in Continuous Time The [[free_energy|Free Energy]] functional in continuous time is given by: ```math F[q] = ∫ dt [⟨ln q(s(t)) - ln p(o(t),s(t))⟩_q] ``` where: - q(s(t)) is the [[variational_inference|variational density]] over states - p(o(t),s(t)) is the [[generative_model|generative model]] - ⟨·⟩_q denotes expectation under q ### Belief Dynamics [[belief_updating|Belief updating]] follows gradient flows on the [[information_geometry|information geometric]] manifold: ```math ∂_t q(s) = -κ ∂_s F[q] ``` where κ is a [[temperature_parameter|temperature parameter]] controlling update speed. ### Action Selection [[action_selection|Action selection]] minimizes [[expected_free_energy|Expected Free Energy]] through: ```math a = -∂_a G ``` where G is the path integral of expected free energy: ```math G = ∫ dt [⟨ln q(s(t)) - ln p(o(t),s(t)|π)⟩_q] ``` ## Implementation Components ### Neural Architecture 1. [[generative_model|Generative Model]]: - Observation mapping p(o|s) - Continuous dynamics p(ds/dt|s,a) 2. [[variational_inference|Recognition Model]]: - Approximate posterior q(s|o) - Gradient-based belief updating 3. [[action_selection|Action Selection]]: - [[path_integral|Path integral]] optimization - [[exploration_exploitation|Exploration-exploitation]] balance ### Computational Elements - [[neural_coding|Neural Coding]] of continuous variables - [[synaptic_plasticity|Synaptic Plasticity]] for learning - [[precision_weighting|Precision Weighting]] of prediction errors - [[hierarchical_processing|Hierarchical Message Passing]] ## Applications ### Cognitive Domains - [[motor_control|Motor Control]] and planning - [[perceptual_inference|Perceptual Inference]] - [[learning_mechanisms|Learning]] in continuous time - [[attention_mechanisms|Attention]] allocation ### Complex Behaviors - [[decision_making|Decision Making]] under uncertainty - [[skill_acquisition|Skill Acquisition]] - [[cognitive_control|Cognitive Control]] - [[metacognition|Metacognition]] ## Relationship to Discrete Models The continuous-time formulation generalizes [[active_inference_pomdp|discrete POMDP]] approaches: - State transitions become differential equations - Discrete actions become continuous control signals - Belief updates become gradient flows - [[path_integral_free_energy|Path integral free energy]] replaces discrete sums ## Implementation Considerations ### Numerical Methods - [[optimization_theory|Optimization]] techniques - Discretization schemes - Stability analysis - Error bounds ### Practical Aspects - [[neural_computation|Neural Network]] architectures - Training procedures - [[uncertainty_resolution|Uncertainty handling]] - Performance metrics ## Research Directions 1. [[information_geometry|Information Geometric]] interpretations 2. [[category_theory|Category Theoretic]] foundations 3. Connections to [[optimal_control|Optimal Control]] 4. [[statistical_foundations|Statistical]] guarantees ## See Also - [[active_inference_theory|Active Inference Theory]] - [[free_energy_theory|Free Energy Theory]] - [[predictive_processing|Predictive Processing]] - [[cognitive_architecture|Cognitive Architecture]] ## References 1. Friston, K. J., et al. (2017). Active inference, curiosity and insight. 2. Buckley, C. L., et al. (2017). The free energy principle for action and perception: A mathematical review. 3. Tschantz, A., et al. (2020). Learning action-oriented models through active inference.