codex
50 lines
4.2 KiB
TeX
50 lines
4.2 KiB
TeX
\documentclass[11pt,a4paper]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amssymb,amsfonts,amsthm}
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\usepackage{cite}
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\title{The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information (Letter)}
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\author{Antigravity}
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\date{\today}
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\begin{document}
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\maketitle
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\begin{abstract}
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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.
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\end{abstract}
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\section{Stochastic Neural Dynamics and the Markov Blanket}
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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.
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The continuous dynamics are governed by a coupled system of Stochastic Differential Equations (SDEs) driven by standard Wiener processes:
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\begin{align}
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dc_t &= f_c(c_t, s_t, a_t)dt + \mathbf{B}_c dW_t^c \\
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ds_t &= f_s(c_t, s_t, a_t, \lambda_t)dt + \mathbf{B}_s dW_t^s \\
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da_t &= f_a(s_t, a_t, \lambda_t)dt + \mathbf{B}_a dW_t^a \\
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d\lambda_t &= f_\lambda(s_t, a_t, \lambda_t)dt + \mathbf{B}_\lambda dW_t^\lambda
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\end{align}
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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:
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\begin{equation}
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\mathbf{A}\boldsymbol{\Sigma} + \boldsymbol{\Sigma}\mathbf{A}^T + \mathbf{B}\mathbf{B}^T = 0
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\end{equation}
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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.
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\section{Intrinsic Integrated Information ($\Phi$)}
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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}.
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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:
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\begin{equation}
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\Phi = \min_{\text{MIP}} \text{EMD}\left[ \text{CES}_{\text{intact}}, \; \text{CES}_{\text{MIP}} \right]
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\end{equation}
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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.
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\bibliographystyle{plain}
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\begin{thebibliography}{10}
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\bibitem{Friston2013} K. Friston, \textit{J. R. Soc. Interface} \textbf{10}, 20130475 (2013).
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\bibitem{Bastos2012} A. M. Bastos et al., \textit{Neuron} \textbf{76}, 695 (2012).
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\bibitem{Oizumi2014} M. Oizumi, L. Albantakis, G. Tononi, \textit{PLOS Comput. Biol.} \textbf{10}, e1003588 (2014).
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\bibitem{Albantakis2023} L. Albantakis et al., \textit{PLOS Comput. Biol.} \textbf{19}, e1011465 (2023).
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\end{thebibliography}
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\end{document}
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