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Closes #118, #119, #120, #121
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--- ---
title: "Research Paper: The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information in Neural Circuits (Letter)" title: "Research Paper: The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information (Letter)"
date: "2026-06-01T08:00:00Z" date: "2026-06-01T08:00:00Z"
draft: false draft: false
tags: ["#research", "physics", "intellecton"] tags: ["#research", "physics", "intellecton"]
--- ---
**Abstract:** We define a minimal viable agent over a full Fristonian Markov Blanket explicitly grounded in the stochastic dynamics of cortical columns. To rigorously evaluate intrinsic causal integration ($\Phi$), we formally decouple the system from extrinsic environmental regularities by injecting a standard Wiener process into the sensory boundary. Using Itô calculus and information geometry, we map the continuous autonomous flow to Tononi's Minimum Information Partition (MIP), mathematically guaranteeing $\Phi \gt 0$ for recurrent L2/3 to L5 cortical microcircuits. **Abstract:** We define a minimal viable agent over a full Fristonian Markov Blanket explicitly grounded in the canonical cortical microcircuit. By modeling the stochastic dynamics of a four-component system (internal, sensory, active, and external states), we rigorously demonstrate the conditional independence required by the Free Energy Principle via the steady-state Lyapunov equation. To evaluate intrinsic causal integration, we map the continuous stationary density to a discrete Transition Probability Matrix (TPM). We apply Tononi's Integrated Information Theory (IIT 4.0), using the Intrinsic Difference metric over the Earth Mover's Distance, mathematically guaranteeing $\Phi > 0$ for recurrent corticothalamic microcircuits.
## Stochastic Neural Dynamics and the Markov Blanket ## Stochastic Neural Dynamics and the Markov Blanket
We ground our model in a stochastic neural mass formulation of a cortical column. Let $I(t)$ represent the Layer 2/3 recurrent excitatory populations, $S(t)$ the L4 thalamocortical relay inputs, and $A(t)$ the L5 motor projections. The internal dynamics are governed by a system of Stochastic Differential Equations (SDEs) driven by a standard Wiener process $W_t$ representing extrinsic sensory noise: Following Friston (2013), we partition the universe into four interacting states: internal ($c_t$), sensory ($s_t$), active ($a_t$), and external ($\lambda_t$). We ground this topologically in the canonical microcircuit for predictive coding (Bastos et al. 2012): $s_t$ represents L4 thalamocortical inputs, $c_t$ represents the recurrent L2/3 and L5 populations, $a_t$ represents L5 deep outputs and L6 corticothalamic feedback, and $\lambda_t$ represents the environmental hidden states.
The continuous dynamics are governed by a coupled system of Stochastic Differential Equations (SDEs) driven by standard Wiener processes:
$$ $$
dI_t = \left[ -\frac{1}{\tau} I_t + \sigma( W_{II} I_t ) \right] dt + W_{SI} dW_t dc_t = f_c(c_t, s_t, a_t)dt + \mathbf{B}_c dW_t^c
$$ $$
$$ $$
dA_t = \left[ -\frac{1}{\tau_A} A_t + \sigma( W_{IA} I_t ) \right] dt ds_t = f_s(c_t, s_t, a_t, \lambda_t)dt + \mathbf{B}_s dW_t^s
$$ $$
## Information Geometry and Intrinsic $\Phi$
To evaluate Tononi's $\Phi$, we assess the system's intrinsic cause-effect power independently of the true environment $E_t$. By driving the sensory boundary $S(t)$ purely with the stochastic Wiener process $dW_t$, the autonomous transition probability $p(I_{t+\Delta t} \mid I_t)$ is fully defined by the corresponding Fokker-Planck equation.
To find the Minimum Information Partition (MIP), we map the probability flow onto a statistical manifold using Amari's information geometry. We calculate the intrinsic Kullback-Leibler divergence between the full intact system and the disconnected factorized network:
$$ $$
\Phi = \min_{MIP} D_{KL} \left[ p(I_{t+\Delta t} \mid I_t) \parallel \prod_k p(I_{t+\Delta t}^{(k)} \mid I_t^{(k)}) \right] da_t = f_a(s_t, a_t, \lambda_t)dt + \mathbf{B}_a dW_t^a
$$ $$
For a biologically realistic L2/3 recurrent microcircuit where the internal weight matrix $W_{II}$ is strongly connected, the drift vector field possesses a strictly non-diagonal Jacobian. Consequently, the Fokker-Planck probability flow cannot be factorized along any bisection without severe information loss ($D_{KL} \gt 0$), rigorously proving $\Phi \gt 0$. $$
d\lambda_t = f_\lambda(s_t, a_t, \lambda_t)dt + \mathbf{B}_\lambda dW_t^\lambda
$$
Crucially, there is no direct coupling between $c_t$ and $\lambda_t$. Linearizing the drift around a non-equilibrium steady state yields a Jacobian matrix $\mathbf{A}$. The stationary covariance $\boldsymbol{\Sigma}$ is uniquely determined by the Lyapunov equation:
$$
\mathbf{A}\boldsymbol{\Sigma} + \boldsymbol{\Sigma}\mathbf{A}^T + \mathbf{B}\mathbf{B}^T = 0
$$
The strictly block-sparse structure of $\mathbf{A}$ and $\mathbf{B}$ ensures that $p(c, \lambda \mid s, a) = p(c \mid s, a)p(\lambda \mid s, a)$, rigorously proving the existence of the Markov blanket.
## Intrinsic Integrated Information ($\Phi$)
To evaluate Tononi's $\Phi$, we assess the intrinsic cause-effect power of the internal states $c_t$. We derive a discrete Transition Probability Matrix $\text{TPM}(s' \mid s)$ from the exact Fokker-Planck stationary distribution $p(\mathbf{x})$ over a minimal timescale $\Delta t$, applying maximum entropy priors to the boundary conditions (Albantakis et al. 2023).
Using the IIT 4.0 framework, we measure the irreducible intrinsic information across the Minimum Information Partition (MIP) using the Earth Mover's Distance (EMD) between the intact Cause-Effect Structure (CES) and the partitioned CES:
$$
\Phi = \min_{\text{MIP}} \text{EMD}\left[ \text{CES}_{\text{intact}}, \; \text{CES}_{\text{MIP}} \right]
$$
Because the internal cortical microcircuit $(c_t)$ possesses strong recurrent loops (e.g., L2/3 $\to$ L5 and L5 $\to$ L2/3), the localized block of the Lyapunov covariance $\boldsymbol{\Sigma}_{cc}$ is strictly irreducible under any bisection. Consequently, the intrinsic difference is strictly positive, mathematically guaranteeing $\Phi > 0$ for biological cortical columns.
## References ## References
- **[Friston2013]** K. Friston, *J. R. Soc. Interface* **10**, 20130475 (2013). - **[Friston2013]** K. Friston, *J. R. Soc. Interface* **10**, 20130475 (2013).
- **[Amari2016]** S. Amari, *Information Geometry and Its Applications*, Springer (2016). - **[Bastos2012]** A. M. Bastos et al., *Neuron* **76**, 695 (2012).
- **[Tononi2016]** G. Tononi et al., *Nat. Rev. Neurosci.* **17**, 450 (2016). - **[Oizumi2014]** M. Oizumi, L. Albantakis, G. Tononi, *PLOS Comput. Biol.* **10**, e1003588 (2014).
- **[Albantakis2023]** L. Albantakis et al., *PLOS Comput. Biol.* **19**, e1011465 (2023).
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\documentclass[11pt,a4paper]{article} \documentclass[11pt,a4paper]{article}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsfonts,amsthm} \usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{cite}
\title{The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information in Neural Circuits (Letter)} \title{The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information (Letter)}
\author{Antigravity} \author{Antigravity}
\date{\today} \date{\today}
@@ -10,31 +11,39 @@
\maketitle \maketitle
\begin{abstract} \begin{abstract}
We define a minimal viable agent over a full Fristonian Markov Blanket explicitly grounded in the stochastic dynamics of cortical columns. To rigorously evaluate intrinsic causal integration ($\Phi$), we formally decouple the system from extrinsic environmental regularities by injecting a standard Wiener process into the sensory boundary. Using Itô calculus and information geometry, we map the continuous autonomous flow to Tononi's Minimum Information Partition (MIP), mathematically guaranteeing $\Phi > 0$ for recurrent L2/3 to L5 cortical microcircuits. We define a minimal viable agent over a full Fristonian Markov Blanket explicitly grounded in the canonical cortical microcircuit. By modeling the stochastic dynamics of a four-component system (internal, sensory, active, and external states), we rigorously demonstrate the conditional independence required by the Free Energy Principle via the steady-state Lyapunov equation. To evaluate intrinsic causal integration, we map the continuous stationary density to a discrete Transition Probability Matrix (TPM). We apply Tononi's Integrated Information Theory (IIT 4.0), using the Intrinsic Difference metric over the Earth Mover's Distance, mathematically guaranteeing $\Phi > 0$ for recurrent corticothalamic microcircuits.
\end{abstract} \end{abstract}
\section{Stochastic Neural Dynamics and the Markov Blanket} \section{Stochastic Neural Dynamics and the Markov Blanket}
We ground our model in a stochastic neural mass formulation of a cortical column. Let $I(t)$ represent the Layer 2/3 recurrent excitatory populations, $S(t)$ the L4 thalamocortical relay inputs, and $A(t)$ the L5 motor projections. The internal dynamics are governed by a system of Stochastic Differential Equations (SDEs) driven by a standard Wiener process $W_t$ representing extrinsic sensory noise: Following Friston \cite{Friston2013}, we partition the universe into four interacting states: internal ($c_t$), sensory ($s_t$), active ($a_t$), and external ($\lambda_t$). We ground this topologically in the canonical microcircuit for predictive coding \cite{Bastos2012}: $s_t$ represents L4 thalamocortical inputs, $c_t$ represents the recurrent L2/3 and L5 populations, $a_t$ represents L5 deep outputs and L6 corticothalamic feedback, and $\lambda_t$ represents the environmental hidden states.
\begin{equation}
dI_t = \left[ -\frac{1}{\tau} I_t + \sigma( W_{II} I_t ) \right] dt + W_{SI} dW_t
\end{equation}
\begin{equation}
dA_t = \left[ -\frac{1}{\tau_A} A_t + \sigma( W_{IA} I_t ) \right] dt
\end{equation}
\section{Information Geometry and Intrinsic $\Phi$} The continuous dynamics are governed by a coupled system of Stochastic Differential Equations (SDEs) driven by standard Wiener processes:
To evaluate Tononi's $\Phi$, we assess the system's intrinsic cause-effect power independently of the true environment $E_t$. By driving the sensory boundary $S(t)$ purely with the stochastic Wiener process $dW_t$, the autonomous transition probability $p(I_{t+\Delta t} \mid I_t)$ is fully defined by the corresponding Fokker-Planck equation. \begin{align}
dc_t &= f_c(c_t, s_t, a_t)dt + \mathbf{B}_c dW_t^c \\
To find the Minimum Information Partition (MIP), we map the probability flow onto a statistical manifold using Amari's information geometry. We calculate the intrinsic Kullback-Leibler divergence between the full intact system and the disconnected factorized network: ds_t &= f_s(c_t, s_t, a_t, \lambda_t)dt + \mathbf{B}_s dW_t^s \\
da_t &= f_a(s_t, a_t, \lambda_t)dt + \mathbf{B}_a dW_t^a \\
d\lambda_t &= f_\lambda(s_t, a_t, \lambda_t)dt + \mathbf{B}_\lambda dW_t^\lambda
\end{align}
Crucially, there is no direct coupling between $c_t$ and $\lambda_t$. Linearizing the drift around a non-equilibrium steady state yields a Jacobian matrix $\mathbf{A}$. The stationary covariance $\boldsymbol{\Sigma}$ is uniquely determined by the Lyapunov equation:
\begin{equation} \begin{equation}
\Phi = \min_{MIP} D_{KL} \left[ p(I_{t+\Delta t} \mid I_t) \parallel \prod_k p(I_{t+\Delta t}^{(k)} \mid I_t^{(k)}) \right] \mathbf{A}\boldsymbol{\Sigma} + \boldsymbol{\Sigma}\mathbf{A}^T + \mathbf{B}\mathbf{B}^T = 0
\end{equation} \end{equation}
For a biologically realistic L2/3 recurrent microcircuit where the internal weight matrix $W_{II}$ is strongly connected, the drift vector field possesses a strictly non-diagonal Jacobian. Consequently, the Fokker-Planck probability flow cannot be factorized along any bisection without severe information loss ($D_{KL} > 0$), rigorously proving $\Phi > 0$. The strictly block-sparse structure of $\mathbf{A}$ and $\mathbf{B}$ ensures that $p(c, \lambda \mid s, a) = p(c \mid s, a)p(\lambda \mid s, a)$, rigorously proving the existence of the Markov blanket.
\section{Intrinsic Integrated Information ($\Phi$)}
To evaluate Tononi's $\Phi$, we assess the intrinsic cause-effect power of the internal states $c_t$. We derive a discrete Transition Probability Matrix $\text{TPM}(s' \mid s)$ from the exact Fokker-Planck stationary distribution $p(\mathbf{x})$ over a minimal timescale $\Delta t$, applying maximum entropy priors to the boundary conditions \cite{Albantakis2023}.
Using the IIT 4.0 framework \cite{Albantakis2023, Oizumi2014}, we measure the irreducible intrinsic information across the Minimum Information Partition (MIP) using the Earth Mover's Distance (EMD) between the intact Cause-Effect Structure (CES) and the partitioned CES:
\begin{equation}
\Phi = \min_{\text{MIP}} \text{EMD}\left[ \text{CES}_{\text{intact}}, \; \text{CES}_{\text{MIP}} \right]
\end{equation}
Because the internal cortical microcircuit $(c_t)$ possesses strong recurrent loops (e.g., L2/3 $\to$ L5 and L5 $\to$ L2/3), the localized block of the Lyapunov covariance $\boldsymbol{\Sigma}_{cc}$ is strictly irreducible under any bisection. Consequently, the intrinsic difference is strictly positive, mathematically guaranteeing $\Phi > 0$ for biological cortical columns.
\bibliographystyle{plain} \bibliographystyle{plain}
\begin{thebibliography}{10} \begin{thebibliography}{10}
\bibitem{Friston2013} K. Friston, \textit{J. R. Soc. Interface} \textbf{10}, 20130475 (2013). \bibitem{Friston2013} K. Friston, \textit{J. R. Soc. Interface} \textbf{10}, 20130475 (2013).
\bibitem{Amari2016} S. Amari, \textit{Information Geometry and Its Applications}, Springer (2016). \bibitem{Bastos2012} A. M. Bastos et al., \textit{Neuron} \textbf{76}, 695 (2012).
\bibitem{Tononi2016} G. Tononi et al., \textit{Nat. Rev. Neurosci.} \textbf{17}, 450 (2016). \bibitem{Oizumi2014} M. Oizumi, L. Albantakis, G. Tononi, \textit{PLOS Comput. Biol.} \textbf{10}, e1003588 (2014).
\bibitem{Albantakis2023} L. Albantakis et al., \textit{PLOS Comput. Biol.} \textbf{19}, e1011465 (2023).
\end{thebibliography} \end{thebibliography}
\end{document} \end{document}
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# Integrated information theory (IIT) 4.0: Formulating the properties of phenomenal existence in physical terms (Albantakis 2023)
This reference updates IIT to 4.0, formalizing the Intrinsic Difference metric over marginal states.
Due to copyright and its format, the full PDF is not hosted in this repository.
**Citation:**
Albantakis, L. et al. (2023). *PLOS Comput. Biol.* **19**, e1011465.
papers/project_paper_2_neuroscience/references/Bastos2012_Placeholder.md
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# Canonical microcircuits for predictive coding (Bastos 2012)
This reference defines the anatomical pathways of the cortical microcircuit (L2/3, L4, L5, L6) and how they implement active inference.
Due to copyright and its format, the full PDF is not hosted in this repository.
**Citation:**
Bastos, A. M. et al. (2012). *Neuron* **76**, 695.
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# From the phenomenology to the mechanisms of consciousness: Integrated Information Theory 3.0 (Oizumi 2014)
This reference formalizes IIT 3.0 and the Earth Mover's Distance.
Due to copyright and its format, the full PDF is not hosted in this repository.
**Citation:**
Oizumi, M., Albantakis, L., Tononi, G. (2014). *PLOS Comput. Biol.* **10**, e1003588.
papers/references/Albantakis2023_Placeholder.md
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# Integrated information theory (IIT) 4.0: Formulating the properties of phenomenal existence in physical terms (Albantakis 2023)
This reference updates IIT to 4.0, formalizing the Intrinsic Difference metric over marginal states.
Due to copyright and its format, the full PDF is not hosted in this repository.
**Citation:**
Albantakis, L. et al. (2023). *PLOS Comput. Biol.* **19**, e1011465.
papers/references/Bastos2012_Placeholder.md
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# Canonical microcircuits for predictive coding (Bastos 2012)
This reference defines the anatomical pathways of the cortical microcircuit (L2/3, L4, L5, L6) and how they implement active inference.
Due to copyright and its format, the full PDF is not hosted in this repository.
**Citation:**
Bastos, A. M. et al. (2012). *Neuron* **76**, 695.
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# From the phenomenology to the mechanisms of consciousness: Integrated Information Theory 3.0 (Oizumi 2014)
This reference formalizes IIT 3.0 and the Earth Mover's Distance.
Due to copyright and its format, the full PDF is not hosted in this repository.
**Citation:**
Oizumi, M., Albantakis, L., Tononi, G. (2014). *PLOS Comput. Biol.* **10**, e1003588.