Greedy versus social: resource-competing oscillator network as a model of amoeba-based neurocomputer

Abstract

A single-celled amoeboid organism, the true slime mold Physarum polycephalum, exhibits rich spatiotemporal oscillatory behavior and sophisticated computational capabilities. The authors previously created a biocomputer that incorporates the organism as a computing substrate to search for solutions to combinatorial optimization problems. With the assistance of optical feedback to implement a recurrent neural network model, the organism changes its shape by alternately growing and withdrawing its photosensitive branches so that its body area can be maximized and the risk of being illuminated can be minimized. In this way, the organism succeeded in finding the optimal solution to the four-city traveling salesman problem with a high probability. However, it remains unclear how the organism collects, stores, and compares information on light stimuli using the oscillatory dynamics. To study these points, we formulate an ordinary differential equation model of the amoeba-based neurocomputer, considering the organism as a network of oscillators that compete for a fixed amount of intracellular resource. The model, called the “Resource-Competing Oscillator Network (RCON) model,” reproduces well the organism’s experimentally observed behavior, as it generates a number of spatiotemporal oscillation modes by keeping the total sum of the resource constant. Designing the feedback rule properly, the RCON model comes to face a problem of optimizing the allocation of the resource to its nodes. In the problem-solving process, “greedy” nodes having the highest competitiveness are supposed to take more resource out of other nodes. However, the resource allocation pattern attained by the greedy nodes cannot always achieve a “socially optimal” state in terms of the public cost. We prepare four test problems including a tricky one in which the greedy pattern becomes “socially unfavorable” and investigate how the RCON model copes with these problems. Comparing problem-solving performances of the oscillation modes, we show that there exist some modes often attain socially favorable patterns without being trapped in the greedy one.

Appendices

Appendix

General form of dynamics for star network

In general for any star network consisting of a hub and n terminals, the dynamics Eqs. 2 and 6 of the hub node 0 and the terminal nodes q (≠ 0) can be written separately as follows:

$$ \begin{aligned} \dot{{x}_{0}}= & - \frac{1} {1+n \cdot \lambda} \cdot \sum_{p=1}^{n} (\lambda - \lambda \cdot {W}_{0,p} + {W}_{0,p}) \cdot (\dot{{x}_{p}^{\ast}} -\dot{{x}_{0}^{\ast}}),\\ \dot{{x}_{q}}= & \frac{1} {1+n\cdot \lambda}\cdot \left( {\lambda}^{2}\cdot (n\cdot \dot{{x}_{q}^{\ast}} - \sum_{p=1}^{n} \dot{{x}_{p}^{\ast}}) - \left( (\lambda -1)\cdot (n\cdot \lambda + 1) \cdot {W}_{0,q}- \lambda \right)\cdot (\dot{{x}_{q}^{\ast}}-\dot{{x}_{0}^{\ast}})\right. \\ &\left.+({\lambda}^{2} - \lambda)\cdot \sum_{p=1}^{n} {W}_{0,p}\cdot (\dot{{x}_{p}^{\ast}}-\dot{{x}_{0}^{\ast}}) \right), \end{aligned} $$

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$$ \begin{aligned} \dot{{y}_{0}}= & - \frac{1}{1+n \cdot \lambda}\cdot \left( \sum_{p=1}^{n} (\lambda - \lambda \cdot {W}_{0,p} + {W}_{0,p})\cdot (\dot{{y}_{p}^{\ast}}-\dot{{y}_{0}^{\ast}})\right. \\ &\left.- \frac{(n+1)\cdot \lambda}{(n+2) \cdot L} \cdot \left( \mu \cdot \sum_{p=1}^{n} ({v}_{p}\cdot {s}_{p} - {v}_{0}\cdot {s}_{0}) - \delta \cdot \sum_{p=1}^{n} {W}_{0,p}\cdot ({v}_{p}-{v}_{0}) \right) \right),\\ \dot{{y}_{q}}= & \frac{1}{1+n \cdot \lambda}\cdot \left( {\lambda}^{2} (n\cdot \dot{{y}_{q}^{\ast}} - \sum_{p=1}^{n} \dot{{y}_{p}^{\ast}}) - \left( (\lambda -1)\cdot (n \cdot \lambda + 1)\cdot {W}_{0,q} - \lambda \right)\cdot (\dot{{y}_{q}^{\ast}}-\dot{{y}_{0}^{\ast}})\right. \\ &+ ({\lambda}^{2} - \lambda)\cdot \sum_{p=1}^{n} {W}_{0,p} \cdot (\dot{{y}_{p}^{\ast}}-\dot{{y}_{0}^{\ast}}) \\ & - \frac{\lambda \cdot \mu}{(n+2)\cdot L} \cdot \left( \frac{(n+2)\cdot (n \cdot \lambda +1)}{2}\cdot {s}_{q} \cdot {v}_{q} - (n+1)\cdot {s}_{0}\cdot {v}_{0} - \frac{(n+2) \cdot \lambda -1}{2}\cdot \sum_{p=1}^{n}{s}_{p} \cdot {v}_{p} \right)\\ & + \frac{\lambda\cdot \delta}{2\cdot (n+2) \cdot L}\cdot \left( (n+2)\cdot (n\cdot \lambda+1) \cdot {W}_{0,q} \cdot ({v}_{q} - {v}_{0}) \right.\\ &\left. \left.- ((n+2)\cdot \lambda -1)\cdot \sum_{p=1}^{n} {W}_{0,p}\cdot ({v}_{p}-{v}_{0}) \right) \right). \end{aligned} $$

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