Draft:Triangulation sensing

Triangulation sensing is a theory describing the computational steps of a cell containing on its surface small windows, to recover the location of a source emitting random particles in a medium. This is particularly relevant for neuron navigation in the Brain.. . Reconstructing the source location allows the cell to triangulate its position. The reconstruction steps of the gradient source from the fluxes of diffusing particles arriving to small absorbing receptors are


 * 1) Arrival of the Brownian particles to the small windows
 * 2) Counting of the particles
 * 3) Inversion of the Laplace's equation to estimate the position from the fluxes
 * 4) Possible noise reduction by applying the same procedure to several triplets

The mathematical formulation consists in considering diffusing molecules that have to bind to N narrow windows located on the surface of a three dimensional shaped object, typically a ball (in dimension 3) or a disk in dimension 2. The number N can range between 10 and 50, but can be much higher for other receptor types. Individual Brownian particles are released from a source at position $$x_0$$ outside the ball. The triangulation sensing method consists in reconstructing the source position from estimated steady-state fluxes at each narrow window for fast binding (i.e. the probability density has an absorbing boundary condition at the windows).

To reconstruct the location of a source from the measured fluxes, at least three windows are needed. Reconstructing the source location $$x_0$$ requires to invert a system equation. With $N>3$ windows, numerical procedures are used to find the position of the source $$x_0$$.

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