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title, date, draft, tags
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| Research Paper: The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information (Letter) | 2026-06-01T08:00:00Z | false |
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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
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:
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
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).
- [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).