Draft:Triangulation sensing

• Comment: Some of the references are hallucinated. Ca ^{talk to me!} 23:42, 1 June 2025 (UTC)

• Comment: Tried to improve the article a bit, but there's simply large swathes of it that are entirely unsourced. MWFwiki (talk) 20:57, 22 May 2025 (UTC)

• Comment: Please clean up formatting and prose to be more encyclopedic. Refer to WP:MOS Bluethricecreamman (talk) 16:27, 23 December 2024 (UTC)

• Comment: Issues have not been fixed since the last decline. LR.127 (talk) 01:18, 18 September 2024 (UTC)

• Comment: Requires complete rewrite and more references to prove notability. The Herald (Benison) (talk) 07:01, 4 February 2024 (UTC)

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Triangulation sensing refers to a mathematical principle, where multiple diffusing sources are used to accurately determine the location or movement of an object in space that receives the diffusing fluxes at small receptors located on its surface. Borrowing its name from the geometric principle of triangulation—where the position of an unknown point is established by measuring angles or distances from multiple known points—this concept underpins a variety of biological processes, ranging from cellular navigation to the spatial orientation of a moving entity.

Navigation using external cues

Many living organisms employ navigational strategies that integrate diverse environmental cues: Migratory birds, for instance, combine information from the Earth's magnetic field, and even olfactory landmarks to chart long-distance routes. Similarly, marine turtles and fish use a geomagnetic “map sense” to travel long distances.

Chemical gradients

Cells such as bacteria and immune cells uses triangulation to transform chemical gradients within their environments in positional infomation. During chemotaxis, a cell detects small differences in the concentration of signaling molecules across its surface. For example, the bacterium Escherichia coli moves toward higher concentrations of attractants by comparing the binding events at receptors distributed around its cell body.

Neuronal triangulation

Triangulation sensing also plays a critical role in the nervous system. Neurons in the hippocampus and entorhinal cortex—place cells, grid cells, and head‐direction cells—work in concert to create an internal map of the external world. Moreover, during neuronal development the growth cone at the tip of an extending axon navigates by interpreting extracellular chemical gradients, transforming these molecular fluxes into directional cues essential for accurate circuit wiring.

Physical model

At the core of triangulation sensing lies a physical model describing how diffusing molecules—such as morphogens or transcription factors—interact with cellular receptors. In this model, molecules released from a source undergo Brownian motion until they encounter small, absorbing receptor windows on a cell’s surface. The rate at which these particles arrive, or molecular flux, provides quantitative information that can be used to estimate the source’s location.

Reconstruction of gradient source location

The reconstruction of a gradient source from diffusing particles involves the following steps:

1. Arrival of Diffusing Particles – Particles are released from the source and diffuse until hitting receptors.
2. Counting Particle Arrivals – Receptors count bound particles over time, estimating the flux at each site.
3. Source Position Estimation – Combining these fluxes solves the inverse problem of Laplace's equation to pinpoint the source.
4. Reduction of Fluctuations – Accuracy improves as more receptors are added, averaging out stochastic arrival noise.

Mathematical formulation

The mathematical framework builds on diffusion theory. For N narrow absorbing windows on the surface of a sphere or disk, one imposes rapid‐binding boundary conditions at each window. The steady‐state flux at each receptor then yields a system of equations whose unknowns are the source coordinates.

Numerical and computational elements

In large‐N systems (e.g. N > 10), hybrid stochastic simulations—merging deterministic PDE solvers with random‐walk particle tracking—can dramatically cut run‐times while retaining localization precision.