codex
41 lines
3.1 KiB
TeX
41 lines
3.1 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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\title{The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information in Neural Circuits (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 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.
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\end{abstract}
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\section{Stochastic Neural Dynamics and the Markov Blanket}
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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:
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\begin{equation}
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dI_t = \left[ -\frac{1}{\tau} I_t + \sigma( W_{II} I_t ) \right] dt + W_{SI} dW_t
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\end{equation}
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\begin{equation}
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dA_t = \left[ -\frac{1}{\tau_A} A_t + \sigma( W_{IA} I_t ) \right] dt
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\end{equation}
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\section{Information Geometry and Intrinsic $\Phi$}
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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.
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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:
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\begin{equation}
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\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]
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\end{equation}
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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 $\Phi > 0$.
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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{Amari2016} S. Amari, \textit{Information Geometry and Its Applications}, Springer (2016).
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\bibitem{Tononi2016} G. Tononi et al., \textit{Nat. Rev. Neurosci.} \textbf{17}, 450 (2016).
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\end{thebibliography}
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\end{document}
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