This paper explores predictive coding under the free-energy principle, proposing that the brain uses hierarchical dynamical models to infer the causes of sensory input. The brain is modeled as a hierarchy of dynamical systems that encode causal structures, with perception equated to the optimization of these models to explain sensory data. The paper introduces hierarchical dynamical models, which are used to simulate brain function and demonstrate how the brain can recognize and predict sensory sequences. These models are based on a free-energy bound on model evidence, enabling the brain to invert internal models and infer causes from sensory data. The paper also discusses how these models can be implemented in neural networks, with a focus on the hierarchical structure of the brain. The authors use synthetic birdsong as an example to illustrate how hierarchical models can be used to categorize and recognize sequences of sensory input. The paper concludes that hierarchical models provide a framework for understanding how the brain processes sensory information, with implications for perception, prediction, and inference. The study highlights the importance of both structural and dynamical priors in enabling accurate perception and inference, and demonstrates how these principles can be applied to real-world sensory data. The paper also discusses the implications of these findings for understanding brain function and the role of hierarchical processing in perception and cognition.This paper explores predictive coding under the free-energy principle, proposing that the brain uses hierarchical dynamical models to infer the causes of sensory input. The brain is modeled as a hierarchy of dynamical systems that encode causal structures, with perception equated to the optimization of these models to explain sensory data. The paper introduces hierarchical dynamical models, which are used to simulate brain function and demonstrate how the brain can recognize and predict sensory sequences. These models are based on a free-energy bound on model evidence, enabling the brain to invert internal models and infer causes from sensory data. The paper also discusses how these models can be implemented in neural networks, with a focus on the hierarchical structure of the brain. The authors use synthetic birdsong as an example to illustrate how hierarchical models can be used to categorize and recognize sequences of sensory input. The paper concludes that hierarchical models provide a framework for understanding how the brain processes sensory information, with implications for perception, prediction, and inference. The study highlights the importance of both structural and dynamical priors in enabling accurate perception and inference, and demonstrates how these principles can be applied to real-world sensory data. The paper also discusses the implications of these findings for understanding brain function and the role of hierarchical processing in perception and cognition.