1 3a-1

Firing Brain Model(Effects of Eroder)

  • Santana, Ramos, and Toom (2015)

  • Cirillo, Nardi, and Spitoni (2018)

  • Szaban (n.d.)

  • Gacs and Torma (2018)

  • In this part, it uses a two-dimensional CA, entailing a square grid of N × N cell with 3 possible states( ready(G), firing(F) and resting(R)). Initilaly, probability of firing status assumes to be 0.3, ready status assumes to be 0.7 while resting status assumes to be 0. The Transition Rule Function TRF depends on cell state around its neighborhood. The cell status is updating done synchronously or asynchronous mode at each next time step. For the von Neumann neighborhood, the evolution of a CA with the von Neumann neighborhood describes as

  • This model implements with a two-dimensional cellular automata model under a periodic boundary condition. Each grid cells interact over the top-and-bottom and left-and-right boundary. That is Von Neumann neighborhood problem.

This model simultation uses NxN grid structure under period boundary with 3 states(firing, read and resting ):

1.1 Grid Model Setting: code and color

  • state Ready (G): 0 ( yellow in color)
  • state Firing (F): 1 ( red in color)
  • state Resting(R): 2 ( green in color)
  • Grid Dim : \(40 \mathrm{x} 40\)
  • t: time
  • r: radius
  • sh: horizontal location
  • sv: vertical location

Thi simulation is governed by following 3 rules for 3 states at time t :

  • first rule : If its any exact two neighbors are firing, a ready neuron fires on the next time step:

  • second rule: A firing neuron foreces to go to the resting state on the next time step:

  • third rule: A resting neuron foreces to go to the ready state on the next time step:

1.2 Periodic Boundary Condition

  • States = [F, G, R]
  • States(sh,sv+N,t) = States(sh,sv,t)= (sh,sv-N,t)
  • States(sh+N,sv,t) = States(sh,sv,t)= (sh-N,sv,t)

1.3 Simulation Setting

  • grid_N = 40
  • prob_firing_state_t0 = 0.3 = \(\gamma\)
  • prob_read_state_t0 = 0
  • prob_resting_state_t0 = \(0.7 = 1- \gamma\)
  • all_other_neuron_t0 = read_state
  • \(0\leq sv, sh \leq grid\_N= 40\)

1.4 Von Neumann Neighborhood

1.5 Simulation: simultaneously update

1.5.1 Case I: prob_firing_state_t0 = 0.3

1.5.1.1 Simulation 01:

1.5.1.2 Simulation 02

1.5.1.3 Simulation 03

1.5.1.4 Simulation 04

1.5.1.5 Simulation 05

1.5.1.6 Simulation 06

1.5.1.7 Simulation 07

1.6 Results

1.6.1 Case I: prob_firing_state_t0 = 0.3, simulation: 100 times

1.6.1.1 No. of Ready Vs time_step

1.6.1.2 No. of Firing Vs time_step

1.6.1.3 No. of Resting Vs time_step

1.6.2 Case II: prob_firing_state_t0 = 0.1, simulation: 100 times

1.6.2.1 No. of Ready Vs time_step

1.6.2.2 No. of Firing Vs time_step

1.6.2.3 No. of Resting Vs time_step

1.6.3 Case III: prob_firing_state_t0 = 0.5, simulation: 100 times

1.6.3.1 No. of Ready Vs time_step

1.6.3.2 No. of Firing Vs time_step

1.6.3.3 No. of Resting Vs time_step

1.6.4 Case IV: prob_firing_state_t0 = 0.7, simulation: 100 times

1.6.4.1 No. of Ready Vs time_step

1.6.4.2 No. of Firing Vs time_step

1.6.4.3 No. of Resting Vs time_step

1.6.5 Case V: prob_firing_state_t0 = 0.9, simulation: 100 times

1.6.5.1 No. of Ready Vs time_step

1.6.5.2 No. of Firing Vs time_step

1.6.5.3 No. of Resting Vs time_step

1.7 Conclusion:

  • Three states of 3-state Von Neumann Cellular Automata in 2-d configuration of CA will generate the effect of eroder outcome provided that time step is sufficient large.
  • The simulations outcomes indicate that
    • larger the initial probability of firing, shorter the time-step to reach the total eroder effect (100% becomes ready-state)
  • Three states of 3-state Von Neumann Cellular Automata in 2-d configuration of CA is similar to spatial epidemics discrete SIR model.

2 3a-2

    1. Move forward at a rate of one cell per time step, while preservingthe same shape:
      • Block , Beehive, Loaf , Boat and Tub
    1. Move forward at a rate of one cell per time step, launching other shapes behind them:
      • Glider
    1. Move forward at a rate of less than one cell per time step, whilereturning to the same shape after some period:
      • Light-weightspaceship (LWSS), Middle-weightspaceship (LWSS)
    1. Stay stationary but oscillate periodically:
      • Blinker(Period 2), Toad(Period 2),Beacon(Period 2), Pulsar(Period 3)

3 3b

3.1 3b-1: Uniform Bernoulli measures of 1-dim of PCA configuration

  • Definition and Setting:
    • Suceptible (S) state = 0
    • Infectious (I) state = 1
    • N =100
    • 1 dimensional of PCA of the given transfer function listed above, the events occured have only two cases:
      • the probability of being not infected(0) or suceptible is \(\gamma\)
      • the probability of being infected(I) is \(1-\gamma\)
    • therefore, 1 dimensional PCA of the given transfer function above follows binormial distribution . For detail why it follows binormial distribution, please refer the paper of Fukś (2020).

3.2 3b-2 : Binomial Distribution

  • p is the probability of success or being Infected (I)
  • \((1-p) = q\) is the probability of NOT success or being NOT Infected(NI)
  • popoulation N = 100

\[ P(A) = \sum P(\{ (e_1,\dotsc,e_N) \}) = \binom{N}{k} \cdot p^kq^{N-k} \]

3.3 3b-3 : Design spatial epidemic SEIR model

  • using an \(N x N\) grid with periodic boundary conditions.

  • Each cell entails 3 different states:

    • Susceptible (or healthy; designated as 0 )
    • Exposed (designated as 1)
    • Infected (designated as -1)
    • Recovered(designated as +2)

  • cellular space using Von Neumann neighborhood

  • construct updated transfer function for SEIR Model

  • simulate the model with the \(\alpha\)-synchronous approach

  • initialize all necesary variables and parameters

  • loop through generations

  • update count of suceptible, Exposed, infected, Recovered neighbours for each cell one by one

  • update state of each cell based on number ofsuceptible, Exposed, infected, recovered neighbours

  • plot all states over time step by change all the parameters

Reference

Cirillo, Emilio N. M., Francesca R. Nardi, and Cristian Spitoni. 2018. “Basic Ideas to Approach Metastability in Probabilistic Cellular Automata.” In Probabilistic Cellular Automata: Theory, Applications and Future Perspectives, edited by Pierre-Yves Louis and Francesca R. Nardi, 37–51. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-65558-1_3.

Fukś, Henryk. 2020. “Orbits of Bernoulli Measures in Cellular Automata.” A. Meyers, Ed., Encyclopedia of Complexity and Systems Science, Pages 1–19. Springer, Berlin, Heidelberg, 2017, February. http://arxiv.org/abs/2002.09079v1.

Gacs, Peter, and Ilkka Torma. 2018. “Stable Multi-Level Monotonic Eroders,” September. http://arxiv.org/abs/1809.09503v1.

Santana, L. H. de, A. D. Ramos, and A. Toom. 2015. “Eroders on a Plane with Three States at a Point. Part I: Deterministic.” Journal of Statistical Physics 159 (5): 1175–95. https://doi.org/10.1007/s10955-015-1226-9.

Szaban, Miroslaw. n.d. “Probabilistic 2D Cellular Automata Rules for Binary Classification.” In 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), 161–64. Gdansk: IEEE. https://doi.org/10.15439/2016f409.