The directed Physarum dynamics is known to solve positive linear programs: minimize @math subject to @math and @math for a positive cost vector @math . The directed Physarum dynamics evolves a positive vector @math according to the dynamics @math . Here @math is the solution to @math that minimizes the "energy" @math . In this paper, we study the non-uniform directed dynamics @math , where @math is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with @math being the identity matrix), as it allows each component of @math to react with different speed to the differences between @math and @math . Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.
The paper @cite_11 proposes a mathematical model for the behavior of the slime and argues extensively that the model is adequate. Physarum is modeled as an electrical network with time varying resistors. We have a simple graph @math with two distinguished nodes modeling the food sources. Each edge @math has a positive length @math and a positive capacity @math ; @math is fixed, but @math is a function of time. The resistance @math of @math is @math . In the electrical network defined by these resistances, a current of value 1 is forced from one of the distinguished nodes to the other. For an (arbitrarily oriented) edge @math , let @math be the resulting current over @math . Then, the capacity of @math evolves according to the differential equation where @math is the derivative of @math with respect to time.
@cite_11
2003810228 1964894406 1981530182 1987048169
We describe here a mathematical model of the adaptive dynamics of a transport network of the true slime mold Physarum polycephalum, an amoeboid organism that exhibits path-finding behavior in a maze. This organism possesses a network of tubular elements, by means of which nutrients and signals circulate through the plasmodium. When the organism is put in a maze, the network changes its shape to connect two exits by the shortest path. This process of path-finding is attributed to an underlying physiological mechanism: a tube thickens as the flux through it increases. The experimental evidence for this is, however, only qualitative. We constructed a mathematical model of the general form of the tube dynamics. Our model contains a key parameter corresponding to the extent of the feedback regulation between the thickness of a tube and the flux through it. We demonstrate the dependence of the behavior of the model on this parameter. Physarum polycephalum is a unicellular, multinucleate slime mold in the Mycetozoa group that exhibits remarkable capabilities, including the ability to construct efficient networks when foraging food [1]. The Physarum cell grows, as long as nutrition is abundant. When nutrition is limited, Physarum forms a network of interconnected veins; the veins are gel-like tubes in which the cytoplasm flows. It has been experimentally observed that, with two food sources, Physarum's tubular network often retracts to the shortest path between the food sources. Tero, Kobayashi and Nakagaki [2] proposed a model for the dynamics of the Physarum, which in the computer simulations converged to the shortest path on any initial network. We analytically prove that, under this model, the mass of the mold has to eventually converge to the shortest path in the initial network between the two food sources, independently of the structure of the initial network or of the initial mass distribution. This presentation is based on the work in [3,4] and is supported by the Italian Flagship Initiative "InterOmics" (PB.P05). References [1] Toshiyuki Nakagaki, Hiroyasu Yamada, Agota Toth: Maze-solving by an amoeboid organism. Nature 407:470 (2000) [2] Atsushi Tero, Ryo Kobayashi, Toshiyuki Nakagaki: A mathematical model for adaptive transport network in path finding by true slime mold. Journal of Theoretical Biology 244: 553-564 (2007) [3] Vincenzo Bonifaci, Kurt Mehlhorn, Girish Varma: Physarum can compute shortest paths. Journal of Theoretical Biology 309: 121-133 (2012) [4] Vincenzo Bonifaci: Physarum can compute shortest paths: A short proof. Information Processing Letters 113(1-2): 4-7 (2013) This paper explores the complexity of deep feedforward networks with linear pre-synaptic couplings and rectified linear activations. This is a contribution to the growing body of work contrasting the representational power of deep and shallow network architectures. In particular, we offer a framework for comparing deep and shallow models that belong to the family of piecewise linear functions based on computational geometry. We look at a deep rectifier multi-layer perceptron (MLP) with linear outputs units and compare it with a single layer version of the model. In the asymptotic regime, when the number of inputs stays constant, if the shallow model has @math hidden units and @math inputs, then the number of linear regions is @math . For a @math layer model with @math hidden units on each layer it is @math . The number @math grows faster than @math when @math tends to infinity or when @math tends to infinity and @math . Additionally, even when @math is small, if we restrict @math to be @math , we can show that a deep model has considerably more linear regions that a shallow one. We consider this as a first step towards understanding the complexity of these models and specifically towards providing suitable mathematical tools for future analysis. We have proposed a mathematical model for the adaptive dynamics of the transport network in an amoeba-like organism, the true slime mold Physarum polycephalum. The model is based on physiological observations of this species, but can also be used for path-finding in the complicated networks of mazes and road maps. In this paper, we describe the physiological basis and the formulation of the model, as well as the results of simulations of some complicated networks. The path-finding method used by Physarum is a good example of cellular computation.