To implement the dynamics of the Hamiltonian in Eq. (15) one can map it into a spin representation via Jordan-Wigner transformation, as described in Figure 2, H e ⟶ JWT ∑ i, j, k, … ∈ { x, y, z } g i j k ⋯ ( σ i 1 ⊗ σ j 2 ⊗ σ k 3 ⋯ ).
Formally, the dynamics and their roles do not depend on the particular neurophysiological mechanisms that implement the dynamics within the CPG.
In the sections that follow we first present the Boolean network model that we use to implement the dynamics of the genetic networks, and the well known mean-field results that predict the existence of a phase transition from ordered to chaotic dynamics in this model.
The full details of our numerical scheme for implementing the interface dynamics are given in Appendix 3.
The numerical method for implementing the interface dynamics can be based upon that, for an infinite domain, with a specific choice for the paths Γ defining this integral.
We computed the phase transition displayed in Fig. 2(b) by implementing the Boolean dynamics in networks with scale-free output topology and Poisson input topology.
Implementing the specific stochastic dynamics for the whole parameter space of more than 260.000 models, however, goes beyond the scope of this study and merits future investigation.
To the best of our knowledge, these have not yet been identified; however, a first step towards a continuous description of this GRN is to implement the Glass dynamics in the network [29].
Due to this lack of information, to implement the Boolean dynamics on the GRN of these three organisms we used biased random Boolean functions generated with a gene expression probability p inferred from microarray experiments.
Finally, a numerical case study is performed to assess the effects of initial CO2 droplets on the plume dynamics, implementing the observation data taken from the experiments to a two-fluid CO2 plume evolution model.
