A Typed, Algebraic Approach to Parsing (2019)

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Implementation of reinforcement learning in chemical reaction networks: application to phototaxis as curiosity-driven exploration

Living systems navigate environments using noisy and incomplete sensory signals. In unicellular algae, phototaxis is often modeled as a mechanistic run--tumble process driven by stimulus--response rules. However, such descriptions overlook how organisms actively sample their environment to reduce sensory ambiguity. From a minimal cognition perspective, we reframe this navigation as a subjective, information-driven sensorimotor process. To this end, we propose a framework linking a Partially Observable Markov Decision Process (POMDP) with biochemical reaction dynamics. Environmental variables are hidden, while the cell updates a minimal internal state from each observation through a memoryless Bayesian step. These internal dynamics balance orienting toward light with exploratory reorientation and can be implemented through Chemical-Reaction-Network Ordinary Differential Equations (CRN--ODEs). Our model includes a biophysical observation process for photoreception and a chemically computable polynomial bound on information gain. Using Inverse Reinforcement Learning (IRL) on 30 experimentally recorded Chlamydomonas trajectories, we infer the behavioral objective consistent with observed phototactic motion and benchmark the resulting dynamics with standard Stochastic Simulation Algorithm (SSA) baselines. Our model reproduces the empirical alignment-to-light distribution, comparable to objective SSA baselines on this dataset. Within this framework, run--tumble alternation emerges as an information-acquisition strategy: tumbling reorients the cell to sample new sensory configurations and resolve sensor ambiguity, demonstrating how intracellular biochemical networks can support adaptive information-seeking behavior in cellular navigation.

Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations

The Fokker-Planck equation is fundamental to statistical mechanics, yet in settings with multiple state variables, anisotropic (cross-) diffusion, and jumps, conventional discretizations frequently produce non-physical negative probability densities. Building on the operator approach of "A. Itkin, Pricing derivatives under Levy models. Modern finite difference and pseudo-differential operators approach, Springer, 2017, ISBN 978-1-4939-6792-6", we introduce a family of "Diagonal Frog" discretizations whose spatial operators are eventually M-matrices (EM-matrices). Although these operators lack a local M-matrix structure, positivity of the directional sub-operators emerges in the spirit of Zeno's paradox: the matrix exponential, assembled as the limit of infinitely many ever-smaller substeps, is provably nonnegative after a short transient even though no single substep is. For the mixed-derivative block, whose generator is not eventually nonnegative, positivity instead rests on a factorized resolvent solver and holds conditionally, on an explicit step-size window; discrete mass is conserved exactly by the splitting for every step size. The resulting schemes are second-order accurate in time and space and require O(m 2 N + m 3) operations per time step, where m is the dimension of the Krylov subspace used to apply the exponential. As stress tests, we solve a two-dimensional anisotropic Fokker-Planck equation in the strong cross-diffusion regime against an exact Gaussian reference, a Kramers escape problem in a double-well potential, and an advection-dominated problem, and observe that the schemes remain stable, nonnegative, and mass-conservative for a wide range of Pécklet numbers (so, don't need any flux limiter). Finally, we extend the construction to multidimensional processes and to the backward Kolmogorov equation with jumps.