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b/papers/project_paper_2_neuroscience/paper_2_neuroscience.md index 95922104..4b49a4aa 100644 --- a/papers/project_paper_2_neuroscience/paper_2_neuroscience.md +++ b/papers/project_paper_2_neuroscience/paper_2_neuroscience.md @@ -1,43 +1,55 @@ --- -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" draft: false 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 -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 - **[Friston2013]** K. Friston, *J. R. Soc. Interface* **10**, 20130475 (2013). -- **[Amari2016]** S. Amari, *Information Geometry and Its Applications*, Springer (2016). -- **[Tononi2016]** G. Tononi et al., *Nat. Rev. Neurosci.* **17**, 450 (2016). - +- **[Bastos2012]** A. M. Bastos et al., *Neuron* **76**, 695 (2012). +- **[Oizumi2014]** M. Oizumi, L. Albantakis, G. Tononi, *PLOS Comput. Biol.* **10**, e1003588 (2014). +- **[Albantakis2023]** L. Albantakis et al., *PLOS Comput. Biol.* **19**, e1011465 (2023). diff --git a/papers/project_paper_2_neuroscience/paper_2_neuroscience.pdf b/papers/project_paper_2_neuroscience/paper_2_neuroscience.pdf new file mode 100644 index 00000000..3c518dcd --- /dev/null +++ b/papers/project_paper_2_neuroscience/paper_2_neuroscience.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f0e4754947b661986d59fa95a9604bcaa32f09b6132f7ab48235437183bf8753 +size 137657 diff --git a/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex b/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex index 89f3084e..5fe1bba5 100644 --- a/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex +++ b/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex @@ -1,8 +1,9 @@ \documentclass[11pt,a4paper]{article} \usepackage[utf8]{inputenc} \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} \date{\today} @@ -10,31 +11,39 @@ \maketitle \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} \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: -\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} +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. -\section{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: +The continuous dynamics are governed by a coupled system of Stochastic Differential Equations (SDEs) driven by standard Wiener processes: +\begin{align} +dc_t &= f_c(c_t, s_t, a_t)dt + \mathbf{B}_c dW_t^c \\ +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} -\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} -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} \begin{thebibliography}{10} \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{Tononi2016} G. Tononi et al., \textit{Nat. Rev. Neurosci.} \textbf{17}, 450 (2016). +\bibitem{Bastos2012} A. M. Bastos et al., \textit{Neuron} \textbf{76}, 695 (2012). +\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{document} diff --git a/papers/project_paper_2_neuroscience/references/Albantakis2023_Placeholder.md b/papers/project_paper_2_neuroscience/references/Albantakis2023_Placeholder.md new file mode 100644 index 00000000..23e195a7 --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Albantakis2023_Placeholder.md @@ -0,0 +1,7 @@ +# 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. diff --git a/papers/project_paper_2_neuroscience/references/Bastos2012_Placeholder.md b/papers/project_paper_2_neuroscience/references/Bastos2012_Placeholder.md new file mode 100644 index 00000000..3b1ac7e1 --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Bastos2012_Placeholder.md @@ -0,0 +1,7 @@ +# 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. diff --git a/papers/project_paper_2_neuroscience/references/Oizumi2014_Placeholder.md b/papers/project_paper_2_neuroscience/references/Oizumi2014_Placeholder.md new file mode 100644 index 00000000..1a7ac87c --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Oizumi2014_Placeholder.md @@ -0,0 +1,7 @@ +# 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. diff --git a/papers/references/Albantakis2023_Placeholder.md b/papers/references/Albantakis2023_Placeholder.md new file mode 100644 index 00000000..23e195a7 --- /dev/null +++ b/papers/references/Albantakis2023_Placeholder.md @@ -0,0 +1,7 @@ +# 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. diff --git a/papers/references/Bastos2012_Placeholder.md b/papers/references/Bastos2012_Placeholder.md new file mode 100644 index 00000000..3b1ac7e1 --- /dev/null +++ b/papers/references/Bastos2012_Placeholder.md @@ -0,0 +1,7 @@ +# 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. diff --git a/papers/references/Oizumi2014_Placeholder.md b/papers/references/Oizumi2014_Placeholder.md new file mode 100644 index 00000000..1a7ac87c --- /dev/null +++ b/papers/references/Oizumi2014_Placeholder.md @@ -0,0 +1,7 @@ +# 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.