# Implications of Noise on Neural Correlates of Consciousness: A Computational Analysis of Stochastic Systems of Mutually Connected Processes

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- Initialize the process state vector, $\overrightarrow{P}$, and set the initial time at 0.
- Calculate the propensities, ${a}_{k}(\overrightarrow{P})$.
- Generate a uniform random number, ${r}_{1}$.
- Compute the time for the next event, $\tau =-\frac{1}{{\sum}_{k}{a}_{k}(\overrightarrow{P})}\mathrm{ln}{r}_{1}$.
- Generate a uniform random number, ${r}_{2}$.
- Find which event is next, $I=i$, if $\frac{{\sum}_{k=1}^{i-1}{a}_{k}(\overrightarrow{P})}{{\sum}_{k}{a}_{k}(\overrightarrow{P})}\le {r}_{2}<\frac{{\sum}_{k=1}^{i}{a}_{k}(\overrightarrow{P})}{{\sum}_{k}{a}_{k}(\overrightarrow{P})}$
- Update state vector, $\overrightarrow{P}\to \overrightarrow{P}+{y}_{i}$.
- Update time, $t\to t+\tau $.
- Repeat (2)–(8).

## 3. Results

_{1}and p

_{2}processes are shown in Figure 1C and the corresponding limit cycle in the p

_{1}p

_{2}-phase plane is shown in Figure 1D. Using stochastic simulation results, we computed power spectral densities (see Figure 1E) from stochastic trajectories as well as the spectral entropy using Equation (1) as described in the Materials and Methods section. For the stochastic version of a nonlinear system (2), we found that spectral entropy was ~0.5 for both the p

_{1}and p

_{2}processes.

**A**is a hollow distance matrix [25]. Matrix

**A**defines how each process ${p}_{i}$ in the system is connected to all other processes, and we considered the following relationship between processes:

## 4. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

# code A par R=1 init p1=-0.1412 p2=-0.2316 p1’=-p2+p1*(R^2-p1^2-p2^2) p2’=p1+p2*(R^2-p1^2-p2^2) @ dt=.025, total=40, xplot=t,yplot=p1 @ xmin=0,xmax=30,ymin=-1,ymax=1 done

#code B par R=100 init p1=-14.12,p2=-23.16 # compute the sum of all event propensities x1=abs(p2) x2=x1+abs((R^2-p1*p1-p2*p2)*p1) x3=x2+abs(p1) x4=x3+abs((R^2-p1*p1-p2*p2)*p2) # choose random event s2=ran(1)*x4 y1=(s2<x1) y2=(s2<x2)&(s2>=x1) y3=(s2<x3)&(s2>=x2) y4=(s2>=x3) # time for next event tr’=tr-log(ran(1))/x4 p1’=p1-sign(p2)*y1+sign((R^2-p1*p1-p2*p2)*p1)*y2 p2’=p2+sign(p1)*y3+sign((R^2-p1*p1-p2*p2)*p2)*y4 @ bound=10000000000,meth=discrete,total=1000000,njmp=1000 @ xp=tr,yp=p1 @ xlo=0,ylo=-100,xhi=30,yhi=100 done

# code C init p1=1 p2=0 z1=0 z2=0 par eps=-1.0 p1’=eps*p2-z1-p1 z1’=p1 p2’=eps*p1-z2-p2 z2’=p2 @ dt=.025, total=40, xplot=t,yplot=p1 @ xmin=0,xmax=40,ymin=-1,ymax=1 done

# code D par eps=-1 init p1=1000,p2=0, z1=0, z2=0 # compute the sum of all event propensities x1=abs(eps*p2) x2=x1+abs(eps*p1) x3=x2+abs(z1) x4=x3+abs(z2) x5=x4+abs(p1) x6=x5+abs(p2) # choose random event# s2=ran(1)*x6 y1=(s2<x1) y2=(s2<x2)&(s2>=x1) y3=(s2<x3)&(s2>=x2) y4=(s2<x4)&(s2>=x3) y5=(s2<x5)&(s2>=x4) y6=(s2>=x5) # time for the next event tr’=tr-log(ran(1))/x6 p1’=p1+sign(eps)*sign(p2)*y1-sign(z1)*y3-sign(p1)*y5 p2’=p2+sign(eps)*sign(p1)*y2-sign(z2)*y4-sign(p2)*y6 z1’=z1+sign(p1)*y5 z2’=z2+sign(p2)*y6 @ bound=100000000,meth=discrete,total=1000000,njmp=1000 @ xp=tr,yp=p1 @ xlo=0,ylo=-1000,xhi=40,yhi=1000 done

# code E par eps=-0.1 init p1=1000,p2=0, p3=0, p4=0, z1=0, z2=0, z3=0, z4=0 # compute the sum of all event propensities x11=abs(eps*p2) x12=x11+abs(4*eps*p3) x13=x12+abs(9*eps*p4) x21=x13+abs(eps*p1) x22=x21+abs(eps*p3) x23=x22+abs(4*eps*p4) x31=x23+abs(4*eps*p1) x32=x31+abs(eps*p2) x33=x32+abs(eps*p4) x41=x33+abs(9*eps*p1) x42=x41+abs(4*eps*p2) x43=x42+abs(eps*p3) x51=x43+abs(z1) x52=x51+abs(z2) x53=x52+abs(z3) x54=x53+abs(z4) x61=x54+abs(p1) x62=x61+abs(p2) x63=x62+abs(p3) x64=x63+abs(p4) # choose random event s2=ran(1)*x64 y1=(s2<x11) y2=(s2<x12)&(s2>=x11) y3=(s2<x13)&(s2>=x12) y4=(s2<x21)&(s2>=x13) y5=(s2<x22)&(s2>=x21) y6=(s2<x23)&(s2>=x22) y7=(s2<x31)&(s2>=x23) y8=(s2<x32)&(s2>=x31) y9=(s2<x33)&(s2>=x32) y10=(s2<x41)&(s2>=x33) y11=(s2<x42)&(s2>=x41) y12=(s2<x43)&(s2>=x42) y13=(s2<x51)&(s2>=x43) y14=(s2<x52)&(s2>=x51) y15=(s2<x53)&(s2>=x52) y16=(s2<x54)&(s2>=x53) y17=(s2<x61)&(s2>=x54) y18=(s2<x62)&(s2>=x61) y19=(s2<x63)&(s2>=x62) y20=(s2>=x63) # time for next event tr’=tr-log(ran(1))/x64 p1’=p1+sign(eps)*sign(p2)*y1+sign(eps)*sign(p3)*y2+sign(eps)*sign(p4)*y3-sign(z1)*y13-sign(p1)*y17 p2’=p2+sign(eps)*sign(p1)*y4+sign(eps)*sign(p3)*y5+sign(eps)*sign(p4)*y6-sign(z2)*y14-sign(p2)*y18 p3’=p3+sign(eps)*sign(p1)*y7+sign(eps)*sign(p2)*y8+sign(eps)*sign(p4)*y9-sign(z3)*y15-sign(p3)*y19 p4’=p4+sign(eps)*sign(p1)*y10+sign(eps)*sign(p2)*y11+sign(eps)*sign(p3)*y12-sign(z4)*y16-sign(p4)*y20 z1’=z1+sign(p1)*y17 z2’=z2+sign(p2)*y18 z3’=z3+sign(p3)*y19 z4’=z4+sign(p4)*y20 @ bound=100000000,meth=discrete,total=1000000,njmp=1000 @ xp=tr,yp=p1 @ xlo=0,ylo=-1000,xhi=40,yhi=1000 done

# code F par eps=-0.0119047619 init p1=1000,p2=0, p3=0, p4=0, p5=0, p6=0, p7=0, p8=0, z1=0, z2=0, z3=0, z4=0, z5=0, z6=0, z7=0, z8=0 # compute the sum of all event propensities x11=abs(eps*p2) x12=x11+abs(4*eps*p3) x13=x12+abs(9*eps*p4) x14=x13+abs(16*eps*p5) x15=x14+abs(25*eps*p6) x16=x15+abs(36*eps*p7) x17=x16+abs(49*eps*p8) x21=x17+abs(eps*p1) x22=x21+abs(eps*p3) x23=x22+abs(4*eps*p4) x24=x23+abs(9*eps*p5) x25=x24+abs(16*eps*p6) x26=x25+abs(25*eps*p7) x27=x26+abs(36*eps*p8) x31=x27+abs(4*eps*p1) x32=x31+abs(eps*p2) x33=x32+abs(eps*p4) x34=x33+abs(4*eps*p5) x35=x34+abs(9*eps*p6) x36=x35+abs(16*eps*p7) x37=x36+abs(25*eps*p8) x41=x37+abs(9*eps*p1) x42=x41+abs(4*eps*p2) x43=x42+abs(eps*p3) x44=x43+abs(eps*p5) x45=x44+abs(4*eps*p6) x46=x45+abs(9*eps*p7) x47=x46+abs(16*eps*p8) x51=x47+abs(16*eps*p1) x52=x51+abs(9*eps*p2) x53=x52+abs(4*eps*p3) x54=x53+abs(eps*p4) x55=x54+abs(eps*p6) x56=x55+abs(4*eps*p7) x57=x56+abs(9*eps*p8) x61=x57+abs(25*eps*p1) x62=x61+abs(16*eps*p2) x63=x62+abs(9*eps*p3) x64=x63+abs(4*eps*p4) x65=x64+abs(eps*p5) x66=x65+abs(eps*p7) x67=x66+abs(4*eps*p8) x71=x67+abs(36*eps*p1) x72=x71+abs(25*eps*p2) x73=x72+abs(16*eps*p3) x74=x73+abs(9*eps*p4) x75=x74+abs(4*eps*p5) x76=x75+abs(eps*p6) x77=x76+abs(eps*p8) x81=x77+abs(49*eps*p1) x82=x81+abs(36*eps*p2) x83=x82+abs(25*eps*p3) x84=x83+abs(16*eps*p4) x85=x84+abs(9*eps*p5) x86=x85+abs(4*eps*p6) x87=x86+abs(eps*p7) x91=x87+abs(z1) x92=x91+abs(z2) x93=x92+abs(z3) x94=x93+abs(z4) x95=x94+abs(z5) x96=x95+abs(z6) x97=x96+abs(z7) x98=x97+abs(z8) x101=x98+abs(p1) x102=x101+abs(p2) x103=x102+abs(p3) x104=x103+abs(p4) x105=x104+abs(p5) x106=x105+abs(p6) x107=x106+abs(p7) x108=x107+abs(p8) # choose random event# s2=ran(1)*x108 y1=(s2<x11) y2=(s2<x12)&(s2>=x11) y3=(s2<x13)&(s2>=x12) y4=(s2<x14)&(s2>=x13) y5=(s2<x15)&(s2>=x14) y6=(s2<x16)&(s2>=x15) y7=(s2<x17)&(s2>=x16) y8=(s2<x21)&(s2>=x17) y9=(s2<x22)&(s2>=x21) y10=(s2<x23)&(s2>=x22) y11=(s2<x24)&(s2>=x23) y12=(s2<x25)&(s2>=x24) y13=(s2<x26)&(s2>=x25) y14=(s2<x27)&(s2>=x26) y15=(s2<x31)&(s2>=x27) y16=(s2<x32)&(s2>=x31) y17=(s2<x33)&(s2>=x32) y18=(s2<x34)&(s2>=x33) y19=(s2<x35)&(s2>=x34) y20=(s2<x36)&(s2>=x35) y21=(s2<x37)&(s2>=x36) y22=(s2<x41)&(s2>=x37) y23=(s2<x42)&(s2>=x41) y24=(s2<x43)&(s2>=x42) y25=(s2<x44)&(s2>=x43) y26=(s2<x45)&(s2>=x44) y27=(s2<x46)&(s2>=x45) y28=(s2<x47)&(s2>=x46) y29=(s2<x51)&(s2>=x47) y30=(s2<x52)&(s2>=x51) y31=(s2<x53)&(s2>=x52) y32=(s2<x54)&(s2>=x53) y33=(s2<x55)&(s2>=x54) y34=(s2<x56)&(s2>=x55) y35=(s2<x57)&(s2>=x56) y36=(s2<x61)&(s2>=x57) y37=(s2<x62)&(s2>=x61) y38=(s2<x63)&(s2>=x62) y39=(s2<x64)&(s2>=x63) y40=(s2<x65)&(s2>=x64) y41=(s2<x66)&(s2>=x65) y42=(s2<x67)&(s2>=x66) y43=(s2<x71)&(s2>=x67) y44=(s2<x72)&(s2>=x71) y45=(s2<x73)&(s2>=x72) y46=(s2<x74)&(s2>=x73) y47=(s2<x75)&(s2>=x74) y48=(s2<x76)&(s2>=x75) y49=(s2<x77)&(s2>=x76) y50=(s2<x81)&(s2>=x77) y51=(s2<x82)&(s2>=x81) y52=(s2<x83)&(s2>=x82) y53=(s2<x84)&(s2>=x83) y54=(s2<x85)&(s2>=x84) y55=(s2<x86)&(s2>=x85) y56=(s2<x87)&(s2>=x86) y57=(s2<x91)&(s2>=x87) y58=(s2<x92)&(s2>=x91) y59=(s2<x93)&(s2>=x92) y60=(s2<x94)&(s2>=x93) y61=(s2<x95)&(s2>=x94) y62=(s2<x96)&(s2>=x95) y63=(s2<x97)&(s2>=x96) y64=(s2<x98)&(s2>=x97) y65=(s2<x101)&(s2>=x98) y66=(s2<x102)&(s2>=x101) y67=(s2<x103)&(s2>=x102) y68=(s2<x104)&(s2>=x103) y69=(s2<x105)&(s2>=x104) y70=(s2<x106)&(s2>=x105) y71=(s2<x107)&(s2>=x106) y72=(s2>=x107) # time for the next event tr’=tr-log(ran(1))/x108 p1’=p1+sign(eps)*sign(p2)*y1+sign(eps)*sign(p3)*y2+sign(eps)*sign(p4)*y3+sign(eps)*sign(p5)*y4+sign(eps)*sign(p6)*y5+sign(eps)*sign(p7)*y6+sign(eps)*sign(p8)*y7-sign(z1)*y57-sign(p1)*y65 p2’=p2+sign(eps)*sign(p1)*y8+sign(eps)*sign(p3)*y9+sign(eps)*sign(p4)*y10+sign(eps)*sign(p5)*y11+sign(eps)*sign(p6)*y12+sign(eps)*sign(p7)*y13+sign(eps)*sign(p8)*y14-sign(z2)*y58-sign(p2)*y66 p3’=p3+sign(eps)*sign(p1)*y15+sign(eps)*sign(p2)*y16+sign(eps)*sign(p4)*y17+sign(eps)*sign(p5)*y18+sign(eps)*sign(p6)*y19+sign(eps)*sign(p7)*y20+sign(eps)*sign(p8)*y21-sign(z3)*y59-sign(p3)*y67 p4’=p4+sign(eps)*sign(p1)*y22+sign(eps)*sign(p2)*y23+sign(eps)*sign(p3)*y24+sign(eps)*sign(p5)*y25+sign(eps)*sign(p6)*y26+sign(eps)*sign(p7)*y27+sign(eps)*sign(p8)*y28-sign(z4)*y60-sign(p4)*y68 p5’=p5+sign(eps)*sign(p1)*y29+sign(eps)*sign(p2)*y30+sign(eps)*sign(p3)*y31+sign(eps)*sign(p4)*y32+sign(eps)*sign(p6)*y33+sign(eps)*sign(p7)*y34+sign(eps)*sign(p8)*y35-sign(z5)*y61-sign(p5)*y69 p6’=p6+sign(eps)*sign(p1)*y36+sign(eps)*sign(p2)*y37+sign(eps)*sign(p3)*y38+sign(eps)*sign(p4)*y39+sign(eps)*sign(p5)*y40+sign(eps)*sign(p7)*y41+sign(eps)*sign(p8)*y42-sign(z6)*y62-sign(p6)*y70 p7’=p7+sign(eps)*sign(p1)*y43+sign(eps)*sign(p2)*y44+sign(eps)*sign(p3)*y45+sign(eps)*sign(p4)*y46+sign(eps)*sign(p5)*y47+sign(eps)*sign(p6)*y48+sign(eps)*sign(p8)*y49-sign(z7)*y63-sign(p7)*y71 p8’=p8+sign(eps)*sign(p1)*y50+sign(eps)*sign(p2)*y51+sign(eps)*sign(p3)*y52+sign(eps)*sign(p4)*y53+sign(eps)*sign(p5)*y54+sign(eps)*sign(p6)*y55+sign(eps)*sign(p7)*y56-sign(z8)*y64-sign(p8)*y72 z1’=z1+sign(p1)*y65 z2’=z2+sign(p2)*y66 z3’=z3+sign(p3)*y67 z4’=z4+sign(p4)*y68 z5’=z5+sign(p5)*y69 z6’=z6+sign(p6)*y70 z7’=z7+sign(p7)*y71 z8’=z8+sign(p8)*y72 @ bound=100,000,000,meth=discrete,total=1,000,000,njmp=1000 @ xp=tr,yp=p1 @ xlo=0,ylo=-1000,xhi=40,yhi=1000 done

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**Figure 1.**Dynamic behavior of a deterministic system (2) and the corresponding stochastic system. (

**A**) Trajectories for processes p

_{1}and p

_{2}described by the system of differential equations (2), (

**B**) the limit cycle in the p

_{1}p

_{2}-phase plane, (

**C**) stochastic trajectories, (

**D**) the limit cycle describing the solution of the stochastic system, (

**E**) the normalized power spectral density characterizing the noise spectra in stochastic trajectories. The parameter R = 1 for the deterministic system (2) and R = 100 for the corresponding stochastic system.

**Figure 2.**The influence diagram for processes described by the system of equations (3). Arrow-headed lines represent a positive influence and bar-headed lines represent a negative influence of one process on another or itself. The dot-headed lines represent positive or negative influence depending on the sign of the ε parameter. Different line colors are used for tracking purposes. Red lines represent interactions between ${p}_{i-1}$, ${p}_{i+1}$, and ${p}_{i}$; green lines represent interactions between ${p}_{i}$, ${p}_{i+1}$ and ${p}_{i-1}$; blue lines wire ${p}_{i-1}$, ${p}_{i}$ with ${p}_{i+1}$.

**Figure 3.**One parameter bifurcation diagram for the system of two ${p}_{i}$ and two ${z}_{i}$ processes (

**A**), and real parts (blue curves) and imaginary parts (red curves) of eigenvalues as a function of parameter ε (

**B**). Hopf bifurcation points (HB) were obtained at ε = ±1. The solid black line indicates the values of ε for which a spiral sink solution was obtained, the dashed black line indicates the values of ε for which a spiral source solution was observed, and open circles indicate periodic solutions. Further, the spiral sink solution was confirmed by the fact that real parts of all eigenvalues are negative for $-1<\epsilon <1$ as shown in (

**B**).

**Figure 4.**The implication of noise on system dynamics depends on system size (the number of processes in the system): (

**A**,

**D**,

**G**) Stochastic trajectories for all processes ${p}_{i}$, (

**B**,

**E**,

**H**) distribution histograms for process ${p}_{2},$ and (

**C**,

**F**,

**I**) normalized power spectral densities for process ${p}_{2},$ which were obtained using systems of two, four, and eight ${p}_{i}$-processes, respectively. The power spectral density for a process ${p}_{i}$ depends on the number of processes constituting the system.

**Figure 5.**Spectral entropy decreases as a function of system size (the number of processes in the system). Open circles represent the average values of spectral entropy provided in Table 1. The solid line is a linear fit with the function displayed in the chart area.

The System of Two p_{i} Processes | The System of Eight p_{i} Processes | ||

Process Name | Spectral Entropy | Process Name | Spectral Entropy |

p_{1} | 0.5735 | p_{1} | 0.483 |

p_{2} | 0.5693 | p_{2} | 0.466 |

The average entropy value = 0.5714 | p_{3} | 0.474 | |

The System of Four p_{i} Processes | p_{4} | 0.512 | |

Process Name | Spectral Entropy | p_{5} | 0.4986 |

p_{1} | 0.539 | p_{6} | 0.4688 |

p_{2} | 0.542 | p_{7} | 0.4686 |

p_{3} | 0.5375 | p_{8} | 0.4664 |

p_{4} | 0.5343 | The average entropy value = 0.48 | |

The average entropy value = 0.538 |

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**MDPI and ACS Style**

Kraikivski, P.
Implications of Noise on Neural Correlates of Consciousness: A Computational Analysis of Stochastic Systems of Mutually Connected Processes. *Entropy* **2021**, *23*, 583.
https://doi.org/10.3390/e23050583

**AMA Style**

Kraikivski P.
Implications of Noise on Neural Correlates of Consciousness: A Computational Analysis of Stochastic Systems of Mutually Connected Processes. *Entropy*. 2021; 23(5):583.
https://doi.org/10.3390/e23050583

**Chicago/Turabian Style**

Kraikivski, Pavel.
2021. "Implications of Noise on Neural Correlates of Consciousness: A Computational Analysis of Stochastic Systems of Mutually Connected Processes" *Entropy* 23, no. 5: 583.
https://doi.org/10.3390/e23050583