The provided code is part of a computational model rooted in Hierarchical Gaussian Filtering (HGF), primarily used to model learning and perception in the brain. The HGF model is a powerful tool in cognitive neuroscience for modeling an individual's perception of hidden states of the world through hierarchical layers of processing. This approach is particularly well-suited to capturing the probabilistic nature of perception and decision-making as they occur in the human brain. ### Biological Basis The HGF model reflects several fundamental principles of neuroscience: 1. **Hierarchical Processing:** - The brain processes information at multiple hierarchical levels, gradually integrating sensory inputs into perceptions, decisions, and actions. In the HGF model, this is captured by multiple "levels" in the hierarchy (`l` in the code), where each level may represent a different stage of processing in the brain. 2. **Bayesian Inference:** - The brain is thought to perform a form of Bayesian inference, updating beliefs about the world based on incoming sensory information. In the code, parameters like `mu` (mean) and `sa` (variance) at each level represent the brain's belief and uncertainty, respectively. The updating of these parameters corresponds to the Bayesian update mechanism where prior beliefs are combined with new evidence. 3. **Prediction and Prediction Error:** - Neurons in the brain often rely on predicting sensory inputs and calculating prediction errors (the difference between expected and actual inputs). This concept is embedded in the HGF framework, where hierarchical levels adjust beliefs based on prediction errors. The `mu` and `sa` values change in response to these errors, capturing the dynamic adaptation of beliefs. 4. **Neurotransmitter Dynamics:** - Although not explicitly modeled in this code, HGF models can relate to neurotransmitter dynamics such as dopamine's role in prediction error signaling. The model's parameters, such as `phi`, `m`, `ka`, `om`, and `al`, can indirectly reflect these processes, influencing learning rates and updating mechanisms. 5. **Learning under Uncertainty:** - The model accounts for the inherent uncertainty in sensory inputs, which the brain continuously deals with. The standard deviation of expectations (`sa` values) captures this uncertainty. In the plot, standard deviations are visualized to denote intervals of uncertainty, mirroring how neuronal populations might encode uncertainty. --- ### Key Aspects from the Code - **Trajectory Plotting:** The code visualizes the trajectory of belief updates over trials (`ts`) for each hierarchical level. This reflects how various parts of the brain might track and visualize evolving representations of the environment over time. - **Level-Specific Prior and Posterior:** The code distinguishes between the prior expectation (`mu_0`) and the posterior ('updated') expectation (`traj.mu`), marking the biologically relevant idea of prior knowledge and its updating through experience. - **Connection to Inputs and Responses:** The code plots both the generated inputs (`u`) and responses (`y`), directly relating to how brain models use external stimuli to inform internal belief states. In summary, the provided code aligns with fundamental biological processes related to perception and learning in the brain via hierarchical, Bayesian, and probabilistic frameworks. It captures the essence of how the brain constructs and updates internal models of the world.