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Predictive Coding — Hierarchical Prediction-Error Minimization
Implements Friston's Box 3: the canonical microcircuit for hierarchical
prediction error processing:
- Superficial Pyramidal (SG): prediction errors ε (bottom-up)
- Layer 4 (L4): state estimation
- Deep Pyramidal (IG): predictions μ (top-down)
Recognition dynamics: perception as gradient descent on free energy.
μ̇ = Dμ − ∂F/∂μ (internal state update)
ε = observed − predicted (prediction error)
Reference: Friston (2009) Box 3, Figure I
Author: Algorembrant, Rembrant Oyangoren Albeos (2026)
"""
import numpy as np
from typing import Optional
class CorticalColumn:
"""
A single cortical column implementing the canonical microcircuit.
Contains:
- Superficial granular (SG): prediction error neurons
- Layer 4 (L4): state representation neurons
- Infragranular (IG): prediction neurons (deep pyramidal)
"""
def __init__(self, size: int, level: int = 0):
"""
Args:
size: Number of units in this column.
level: Hierarchical level (0 = lowest/sensory).
"""
self.size = size
self.level = level
# State variables (generalized coordinates of motion)
self.mu = np.zeros(size) # Internal state estimates (expectations)
self.mu_dot = np.zeros(size) # Velocity of expectations
self.epsilon = np.zeros(size) # Prediction errors
self.prediction = np.zeros(size) # Top-down predictions to level below
# Precision (inverse variance) — controls gain on prediction errors
self.precision = np.ones(size) # π = 1/σ² (higher = more confident)
# Connection weights
self.forward_weights = None # Bottom-up: from level below
self.backward_weights = None # Top-down: to level below
self.lateral_weights = None # Within level
def initialize_connections(self, input_size: int, output_size: Optional[int] = None):
"""Initialize synaptic weights for this column's connections."""
self.forward_weights = np.random.randn(self.size, input_size) * 0.1
if output_size is not None:
self.backward_weights = np.random.randn(output_size, self.size) * 0.1
self.lateral_weights = np.random.randn(self.size, self.size) * 0.01
np.fill_diagonal(self.lateral_weights, 0)
class PredictiveCodingHierarchy:
"""
Full hierarchical predictive coding network.
Implements the hierarchical generative model from Friston Box 3:
- Each level generates predictions for the level below
- Prediction errors propagate upward (superficial pyramidal)
- Predictions propagate downward (deep pyramidal)
- Recognition dynamics minimize free energy via gradient descent
This is NOT variational inference in the ML sense — this is
biological free-energy minimization via neural dynamics.
"""
def __init__(self, layer_sizes: list[int], learning_rate: float = 0.01,
dt: float = 0.1, n_iterations: int = 10):
"""
Args:
layer_sizes: Sizes of each hierarchical level [sensory, ..., abstract].
learning_rate: Step size for recognition dynamics.
dt: Integration time step.
n_iterations: Number of iterations per perception step.
"""
self.n_levels = len(layer_sizes)
self.learning_rate = learning_rate
self.dt = dt
self.n_iterations = n_iterations
# Build cortical columns at each level
self.columns: list[CorticalColumn] = []
for i, size in enumerate(layer_sizes):
col = CorticalColumn(size, level=i)
self.columns.append(col)
# Initialize inter-level connections
for i in range(1, self.n_levels):
self.columns[i].initialize_connections(
input_size=layer_sizes[i-1],
output_size=layer_sizes[i-1]
)
# Level 0 has lateral connections only
self.columns[0].lateral_weights = np.random.randn(
layer_sizes[0], layer_sizes[0]
) * 0.01
np.fill_diagonal(self.columns[0].lateral_weights, 0)
def _generate_prediction(self, level: int) -> np.ndarray:
"""
Generate top-down prediction from level i to level i-1.
g(μ⁽ⁱ⁾) — the generative model mapping from higher to lower.
"""
col = self.columns[level]
if col.backward_weights is not None:
# Nonlinear generative mapping (sigmoid for bounded predictions)
hidden = np.tanh(col.mu)
return np.dot(col.backward_weights, hidden)
return np.zeros(self.columns[level - 1].size if level > 0 else col.size)
def _compute_prediction_errors(self, sensory_input: np.ndarray):
"""
Compute prediction errors at each level of the hierarchy.
ε⁽ⁱ⁾ = μ⁽ⁱ⁻¹⁾ − g(μ⁽ⁱ⁾)
Prediction error = what I observe − what I predicted.
"""
# Level 0: error between sensory input and level 1's prediction
if self.n_levels > 1:
prediction_from_above = self._generate_prediction(1)
self.columns[0].epsilon = sensory_input - prediction_from_above
else:
self.columns[0].epsilon = sensory_input - self.columns[0].mu
# Higher levels: error between current state and prediction from above
for i in range(1, self.n_levels - 1):
prediction_from_above = self._generate_prediction(i + 1)
self.columns[i].epsilon = self.columns[i].mu - prediction_from_above
def _recognition_dynamics(self):
"""
Recognition dynamics — gradient descent on free energy.
μ̇⁽ⁱ⁾ = Dμ⁽ⁱ⁾ − ∂F/∂μ⁽ⁱ⁾
The internal states update to minimize prediction error,
weighted by precision. This IS perception in the Fristonian framework.
"""
for i in range(self.n_levels):
col = self.columns[i]
# Gradient of free energy w.r.t. internal states
# ∂F/∂μ = precision-weighted prediction error + prior gradient
dF_dmu = np.zeros(col.size)
# Bottom-up: precision-weighted error from level below
if i > 0:
lower_col = self.columns[i - 1]
if col.forward_weights is not None:
# Error signal from lower level, weighted by precision
weighted_error = lower_col.precision * lower_col.epsilon
dF_dmu -= np.dot(col.forward_weights, weighted_error)
# Top-down: prediction error at this level
if i < self.n_levels - 1:
dF_dmu += col.precision * col.epsilon
# Lateral dynamics (recurrent processing within level)
if col.lateral_weights is not None:
lateral = np.dot(col.lateral_weights, col.mu)
dF_dmu -= 0.1 * lateral
# Update internal states (gradient descent on F)
col.mu_dot = -self.learning_rate * dF_dmu
col.mu += col.mu_dot * self.dt
# Bounded activation
col.mu = np.clip(col.mu, -5.0, 5.0)
def process(self, sensory_input: np.ndarray) -> dict:
"""
Run perception: minimize free energy given sensory input.
This is "seeing" — the brain settling into an interpretation
of sensory data that minimizes surprise (prediction error).
Args:
sensory_input: Raw sensory data (preprocessed by retina/V1).
Returns:
Dict with internal states, prediction errors, and free energy.
"""
free_energy_history = []
for iteration in range(self.n_iterations):
# 1. Compute prediction errors at all levels
self._compute_prediction_errors(sensory_input)
# 2. Run recognition dynamics (update internal states)
self._recognition_dynamics()
# 3. Update predictions (generative model output)
for i in range(1, self.n_levels):
self.columns[i].prediction = self._generate_prediction(i)
# 4. Compute total free energy (should decrease over iterations)
F = self._compute_free_energy()
free_energy_history.append(F)
return {
'states': [col.mu.copy() for col in self.columns],
'errors': [col.epsilon.copy() for col in self.columns],
'predictions': [col.prediction.copy() for col in self.columns],
'free_energy': free_energy_history,
'final_F': free_energy_history[-1] if free_energy_history else 0.0
}
def _compute_free_energy(self) -> float:
"""
Compute variational free energy across the hierarchy.
F ≈ Σᵢ (εᵢ)ᵀ Πᵢ εᵢ (precision-weighted sum of squared errors)
This is the Laplace approximation to true free energy.
"""
F = 0.0
for col in self.columns:
# Precision-weighted prediction error (energy term)
F += 0.5 * np.sum(col.precision * col.epsilon**2)
return float(F)
def learn(self, learning_rate: float = 0.001):
"""
Update generative model parameters (synaptic plasticity).
This slowly adjusts the top-down generative model to better
predict sensory input. Corresponds to synaptic plasticity
in the brain (much slower timescale than recognition dynamics).
"""
for i in range(1, self.n_levels):
col = self.columns[i]
lower = self.columns[i - 1]
if col.backward_weights is not None:
# Gradient of F w.r.t. backward weights
# ΔW = -lr * ε * ∂g/∂W
hidden = np.tanh(col.mu)
dg_dW = np.outer(lower.epsilon, hidden)
col.backward_weights += learning_rate * dg_dW
if col.forward_weights is not None:
# Forward weights learn from prediction errors
dW = np.outer(col.mu, lower.epsilon * lower.precision)
col.forward_weights += learning_rate * dW
def get_representation(self, level: int) -> np.ndarray:
"""Get the internal state representation at a given hierarchical level."""
return self.columns[level].mu.copy()
def get_total_surprise(self) -> float:
"""Total surprise = total precision-weighted prediction error."""
return self._compute_free_energy()
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