The aim of this paper is to investigate the discrete nature of chemically reacting systems. In order to achieve our purpose we propose a systematic method to compare the discrete stochastic model of chemically reacting systems with the continuous stochastic model. We adopt the chemical master equation (CME) as the discrete stochastic model and the chemical Fokker-Planck equation (CFPE) as the continuous stochastic model. By making use of the well-known idea of approximating diffusion processes by birth-death processes, we construct a family of master equations parameterized by the degree of discreteness. This family of master equations bridges CME and CFPE. With full degree of discreteness we obtain CME and as decreasing discreteness the family of master equations converges to CFPE. Our strategy is not to study CME directly but to distinguish the properties of CME by putting CME into the family of master equations bridging CME and CFPE. We examine the usefulness of our construction by two simple examples.
An amoeboid organism, the true slime mold Physarum, has been studied actively in recent years to explore its spatiotemporal oscillatory dynamics and various computational capabilities. In the authors' previous studies, the amoeba was employed as a computing substrate to construct a neurocomputer. Under optical feedback control to implement a recurrent neural network model, the amoeba grows or withdraws its photosensitive branches by exhibiting a number of spatiotemporal oscillation modes in search of a solution to some combinatorial optimization problems. In this paper, considering the amoeba as a network of oscillators that compete for constant amounts of resources, we model the amoeba-based neurocomputer. The model generates several oscillation modes and produces not only simple behavior to stabilize a single mode but also complex behavior to spontaneously switch among different modes. To explore significances of the oscillatory dynamics in producing the computational capabilities, we establish a test problem that is a kind of optimization problem of how to allocate a limited amount of resource to oscillators so that conflicts among theoscillators can be avoided. We compare the performances of the oscillation modes in solving the problem in a bottom-up manner.