RESUMO
Given a directed graph of nodes and edges connecting them, a common problem is to find the shortest path between any two nodes. Here we show that the shortest path distances can be found by a simple matrix inversion: If the edges are given by the adjacency matrix A i j then with a suitably small value of γ the shortest path distances are D i j = ceil ( log γ [ ( I - γ A ) - 1 ] i j ) We derive several graph-theoretic bounds on the value of γ , and explore its useful range with numerics on different graph types. Even when the distance function is not globally accurate across the entire graph, it still works locally to instruct pursuit of the shortest path. In this mode, it also extends to weighted graphs with positive edge weights. For a wide range of dense graphs this distance function is computationally faster than the best available alternative. Finally we show that this method leads naturally to a neural network solution of the all-pairs-shortest-path problem.
RESUMO
An animal entering a new environment typically faces three challenges: explore the space for resources, memorize their locations, and navigate towards those targets as needed. Here we propose a neural algorithm that can solve all these problems and operates reliably in diverse and complex environments. At its core, the mechanism makes use of a behavioral module common to all motile animals, namely the ability to follow an odor to its source. We show how the brain can learn to generate internal "virtual odors" that guide the animal to any location of interest. This endotaxis algorithm can be implemented with a simple 3-layer neural circuit using only biologically realistic structures and learning rules. Several neural components of this scheme are found in brains from insects to humans. Nature may have evolved a general mechanism for search and navigation on the ancient backbone of chemotaxis.
