# Scaling in Colloidal and Biological Networks

^{*}

## Abstract

**:**

## 1. Introduction

^{3/4}. The explanation of the value of the exponent ¾ was suggested in an influential paper by West et al. [2], who used a fractal model of branching of blood vessels serving a certain volume with the conservation of the cross-sectional area of the vessels and of the volume covered at every stage of branching. Despite its shortcomings [3,4,5,6], the fractal theory by West et al. [2] remains the main explanation of the allometric scaling exponents, and it can likely be expanded on such topics as the analysis of the ergodicity in the vascular network [7].

## 2. Scaling Concepts in Networks and Clusters

#### 2.1. Scale-Free and Small-World Networks

_{ave}.

_{c}, or the fraction of nodes, which should be kept to sustain the connectivity [16]. Thus, for an infinite network with 2 < a < 3, the critical percolation is zero, p

_{c}= 0, which means that removing randomly any fraction of the network cannot destroy it. For a finite scale-free network with N nodes, ${p}_{c}~{N}^{-1/3}$; thus, a large network such as the internet with $N>{10}^{9}$ will not be destroyed until more than 99.9% of nodes are removed in a random manner (however, it can be destroyed if high-degree nodes are removed preferentially). Note that, as opposed to the scale-free network, in an ER random network, the threshold percolation is non-zero, and it is inversely proportional to the average degree ${p}_{c}~1/{k}_{ave}$. This consideration may be applicable to the spreading of infectious diseases, such as Covid-19, when emergency closure measures of a sufficient number of public institutions can prevent disease spreading.

#### 2.2. Self-Organized Criticality and Percolation

## 3. Networks in Colloidal Science: Granular Material, Colloidal Crystals, and Droplet Clusters

#### 3.1. Granular Material

#### 3.2. Droplet Clusters

_{n}is the fraction of polygons with n sides or edges [26].

#### 3.3. Colloidal Crystals

_{n}is the statistical probability of the n-th state and N is the total number of states. The Shannon entropy is used in materials science, for example, as a surface roughness parameter characterizing informational content in the surface given by its profile [15]. The Shannon-entropy-based informational approach is also used for various other aspects of surface science, such as wetting transitions [31] and stick-slip transition [32].

## 4. Artificial Neural Networks in Surface Science

## 5. Scaling of Branching Vascular Networks

_{k}and radius r

_{k}branches into n tubes with the lengths l

_{k+1}= γl

_{k}and radii r

_{k+1}= βr

_{k}(where γ and β are constants), the volume served by the next-generation tubes and their cross-section area should conserve, which leads to two separate scaling relationships for the constants $\gamma \propto {n}^{-1/3}$ and $\beta \propto {n}^{-1/2}$. These relationships are satisfied simultaneously [7]. The area is preserved due to the constant rate of the fluid flow at different hierarchical levels. The volume is preserved, assuming that the same volume in the organism is served by blood vessels of different hierarchical levels (Figure 10).

_{0}is a certain elementary volume (e.g., volume served by a capillary), and N is the total number of branch generations. Therefore, the volume scales as $V\propto {\left(\gamma {\beta}^{2}\right)}^{-N}$. From this, the scaling dependency of the total number of thinnest capillaries as a function of volume is ${n}^{N}\propto {V}^{a}\propto {\left(\gamma {\beta}^{2}\right)}^{-Na}\propto {\left({n}^{-4/3}\right)}^{-Na}\propto {n}^{4Na/3}$ yielding a = 3/4, the well-established empirical results known as the Kleiber law, which is based on the assumption of a constant flow rate.

## 6. Cortical Neural Networks

#### 6.1. Structure and Properties of Neural Networks of the Human Brain

^{10}neurons with more than 10

^{14}synapses connecting between them. While it is extremely difficult to study such a complex network, a number of important insights have been achieved, particularly, since the early 2000s. This knowledge was obtained due to novel methods of in vivo observation of neural activity, including the electroencephalography (EEG), functional magnetic resonance imaging (fMRI), diffusion tensor imaging (DTI), two-photon excitation microscopy (TPEF or 2PEF), and positron emission tomography (PET).

^{2}. The neocortex is made of six distinct layers of neurons, and it consists of 10

^{8}cortical mini-columns with the diameter of about 50–60 μm spanning through all six layers, with about 100 neurons in each mini-column (Figure 11). Although the functionality of the microcolumns (and their very existence) is being debated by some researchers, the columnar structure of the neocortex is widely accepted by most neuroscientists. The microcolumns are combined into the large hyper-columns or macro-columns, which are 300–500 μm in diameter. The hyper-columns have a roughly hexagonal shape, and each column is surrounded by six other columns. Each hyper-column, by some estimates, may include 60–80 microcolumns [36,37,38].

^{7}synapses, has been presumably obtained by 2020 [40].

^{10}neurons and 10

^{4}synapses per neuron yielding n = 5 × 10

^{13}. These numbers are consistent with the idea that at least a three-level hierarchy exists formed by nodes as neurons, hypercolumns, and modules. The reduction of 10

^{10}neurons and 10

^{14}synapses to a depth of three levels and a diameter of D = 12 is viewed as a simplification [42].

_{B}of size l

_{B}that are necessary to cover the network, ${N}_{B}~{l}_{N}^{-d}$. According to the estimates, the values are β = 0.247 and d = 3.7 ± 0.1 [43].

#### 6.2. Current Hypotheses on the Formation of the Cortical Network

^{10}neurons linked by more than 10

^{14}synaptic connections, while the number of base pairs in a human genome is only 0.3 × 10

^{10}. Therefore, it is impossible that the information about all synaptic connections is contained in the DNA. Currently, two concepts, namely, the Protomap hypothesis and the Radial Unit hypothesis, which complement each other, are employed to explain the formation of the neo-cortex. Both hypotheses were suggested by Pasko Rakic [44].

^{3}times, while its thickness did not change significantly. This is explained by the Radial Unit Hypothesis of cerebral cortex development, which was first described by Pasko Rakic [44,45,46]. According to this hypothesis, the cortical expansion is the result of the increasing number of radial columnar units. The increase occurs without a significant change in the number of neurons within each column. The cortex develops as an array of interacting cortical columns or the radial units during embryogenesis. Each unit originates from a transient stem cell layer. The regulatory genes control the timing and ratio of cell divisions. As a result, an expanded cortical plate is created with the enhanced capacity for establishing new patterns of connectivity that are validated through natural selection [36].

_{2}(N) transcription factors is sufficient to encode the connection, if transcription factors are combined with what they called the biological operators.

_{A}receptors (these are receptors of γ-Aminobutyric acid or GABA, the major neurotransmitter). The mean size of receptor networks in a synapse followed a power-law distribution as a function of receptor concentration with the exponent 1.87 representing the fractal dimension of receptor networks. The results suggested that receptor networks tend to attract more receptors to grow into larger networks in a manner typical for SOC systems that self-organize near critical states.

#### 6.3. Dynamics of Cortical Networks

- Signals from one location to another may follow any of a number of pathways in the system. This provides the redundancy and resilience.
- Actions may be initiated at various nodal loci within a distributed system rather than at one particular spot.
- Local lesions within a distributed system usually may degrade a function, but not eliminate it completely.
- The nodes are open to both externally induced and internally generated signals.

^{4}(mouse to whale), as it will be discussed in more detail in consequent sections. Random removals of nodes from scale-free networks have negligible effects; however, lesions of hubs are catastrophic. Examples in humans are coma and Parkinson’s disease from small brain stem lesions [42].

#### 6.4. Default Mode Network and Its Presumed Relations to Cognition

#### 6.5. Information and Entropic Content of the Network

_{2}(N) bits of information, with the total information in all neurons kN·log

_{2}(N). For large organisms, this would significantly exceed the information contained in DNA (Table 3). Thus, 3.1·10

^{21}bits of information would be required to characterize the human brain; for comparison, some estimates indicate that human brain memory capacity is 10

^{15}–10

^{16}bit, while the human genome has about 10

^{11}pairs of nucleotides.

## 7. Time Scale of Neuronal Activities

#### 7.1. The Critical Flicker Fusion Thresholds

^{4}times from a mouse to a whale, mammalian brains tend to operate at almost the same time scales. This can be called the law of conservation of the characteristic time scale. There are two approaches to the characterization of the time scale of brain activity of different creatures: studying brain waves (rhythms of oscillation) and investigations of the critical flicker fusion (CFF) thresholds. The CFF is defined as the frequency of an intermittent light, at which the light appears steady to a human or animal observer (similar to frames in the cinema). It has been hypothesized that the ability of an animal to change their body position or orientation (manoeuvrability) is related to the ability to resolve temporal features of the environment and eventually to the CFF [67]. Manoeuvrability usually decreases with increasing body mass.

^{n}(where f is the frequency and n is the power exponent) noise spectrum when measured at long-scale ranges. As we have discussed in the preceding sections, such a distribution spectrum is a signature of SOC. However, when specific brain activities are considered, such as concentrating on particular features, moving or orienting in space or various cognitive functions, particular oscillation frequencies become dominant, and the spectrum deviates from the 1/f or 1/f

^{n}statistics, showing peaks at some characteristic frequencies. These frequencies of various rhythm classes do not vary significantly with the size of the brain [68].

#### 7.2. Brain Rhythms and Scaling

## 8. Immune Networks and Artificial Neural Networks

^{10}–10

^{11}. The mass of immune cells in a human is on the order of 1 kg, which is somewhat comparable with the brain mass, at least, by the order of magnitude. Both the immune and nervous systems respond to external stimuli, and they both possess memory [73].

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Small-world network concept. While some nodes have connections only with their neighbors, edges connecting with remote nodes provide short geodesic paths.

**Figure 2.**The sand-pile conceptual model of self-organized criticality (SOC). The pile tends to have a slope angle defined by the friction between grains. Adding one new grain to the pile may have no effect (grain is at rest) or it may cause an avalanche. The magnitude and frequency of avalanches are inversely related (based on [15]).

**Figure 3.**Percolation. (

**a**) A 2D torus model with a two active neighbors local update rule (if there are two active neighbors, the edge becomes active), showing that the first four iteration steps at the eighth step all sites become active [18]. (

**b**) A typical dependency of the correlation length on the shear load for an avalanche. At the critical value of the load, τ

_{0}, the correlation length approaches the infinity. (

**c**) With the increasing normal load, the size of slip zone spots (black) increases. A transition to the global sliding is expected when the correlation length approaches infinity.

**Figure 5.**Apollonian packing and a corresponding graph (the concept based on [22]).

**Figure 6.**A hexagonally ordered 2D droplet cluster levitating over a water surface. The size of the frame is 0.75 mm (credit: Dr. A. Fedorets, based on [26]).

**Figure 7.**Schematic of colloidal particles forming small clusters (concept based on [28]).

**Figure 8.**Experimental probability distributions of 8-bond (

**right part**) and 7-bond (

**left part**) structures. Each point is representing each distinguished structure of a colloidal cluster (based on data from [28]).

**Figure 9.**The architecture of Artificial Neural Networks (ANNs) used for the determination of the contact angle (based on [33]).

**Figure 10.**Branching of a vascular network; three levels are shown (based on [7]).

**Figure 11.**Neurons and their parts forming a network. (

**a**) The arrangement of neurons, dendrites, and axons in vertical modules of the striate cortex of the macaque monkey. (

**b**) The arrangement of the apical dendrites of pyramidal cells in the cortex showing the six layers (I to VI). The cells in layers II to V (red), VI (green), and (IV) (blue, no dendrites) are shown. (

**c**) Columns built of dendrites and axons (based on [36]).

**Figure 12.**Hypotheses of brain network formation by SOC. (

**a**) The neocortex network evolves after the birth toward regions of criticality. Once the critical regions (black) are established, the connectivity structure remains essentially unchanged, but it can adjust close to critical regions (based on [51]). (

**b**) Schematic (log-log scale) showing distribution of pyramidal axon tree size. The power law is a typical footprint of scale-free organization [42].

**Figure 14.**The effect of (

**a**) body mass (gram) and (

**b**) temperature-corrected mass-specific resting metabolic rate (qWg) on the critical flicker fusion (CFF) shows that the CFF increases with the metabolic rate but decreases with body mass (based on [67]).

**Figure 16.**Scaling relationship between the brain diameter (cm) and the ratio of white and gray matter.

**Table 1.**Structures of eight-bond colloidal clusters and their magnitudes of probability distribution and Zipf-Law distribution (data from [28]).

A | B | C | D | E | F | G | H | |
---|---|---|---|---|---|---|---|---|

Structures | ||||||||

Probability | 0.35 ± 0.1 | 0.23 ± 0.06 | 0.11 ± 0.05 | 0.10 ± 0.02 | 0.07 ± 0.01 | 0.08 ± 0.02 | 0.07 ± 0.01 | 0.05 ± 0.0 |

Zipf-Law | 0.36 | 0.19 | 0.14 | 0.10 | 0.08 | 0.07 | 0.06 | 0.05 |

**Table 2.**Structures of seven-bond colloidal clusters and their magnitudes of probability distribution and Zipf-Law distribution (data from [28]).

A | B | C | D | E | F | G | H | |

Structure | ||||||||

Probability | 0.20 ± 0.05 | 0.17 ± 0.03 | 0.18 ± 0.03 | 0.09 ± 0.03 | 0.07 ± 0.01 | 0.05 ± 0.01 | 0.05 ± 0.02 | 0.03 ± 0.001 |

Zipf Law | 0.21 | 0.16 | 0.14 | 0.10 | 0.09 | 0.07 | 0.05 | 0.04 |

I | J | K | L | M | N | O | P | |

Structures | ||||||||

Probability | 0.04 ± 0.002 | 0.025 ± 0.002 | 0.045 ± 0.002 | 0.026 ± 0.002 | 0.015 ± 0.001 | 0.005 ± 0.0 | 0.005 ± 0.0 | 0.01 ± 0.001 |

Zipf Law | 0.03 | 0.02 | 0.02 | 0.02 | 0.01 | 0.007 | 0.007 | 0.01 |

**Table 3.**Number of neurons, synapses, and corresponding information content of some organisms (based on [49]).

Organism | Neurons, N | Synapses, k | Bits Per Neuron, k·log_{2}(N) | Bits Per Connectome, kN·log_{2}(N) | Transcription Factors, T | Transcription Factors Information, N·T |
---|---|---|---|---|---|---|

C. elegans | 302 | 6398 | 3818 | 1.3·10^{6} | 934 | 2.8·10^{5} |

Drosophila fruit fly | 10^{5} | 10^{7} | 2.3·10^{6} | 2.3·10^{11} | 627 | 6.3·10^{7} |

Mouse | 7.1·10^{6} | 1.3·10^{11} | 2.6·10^{8} | 1.9·10^{15} | 1457 | 1.0·10^{10} |

Cat | 7.6·10^{8} | 6.1·10^{12} | 3.2·10^{10} | 2.5·10^{19} | 887 | 6.7·10^{11} |

Human | 8.1·10^{9} | 1.6·10^{14} | 3.8·10^{11} | 3.1·10^{21} | 1391 | 1.1·10^{13} |

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**MDPI and ACS Style**

Nosonovsky, M.; Roy, P.
Scaling in Colloidal and Biological Networks. *Entropy* **2020**, *22*, 622.
https://doi.org/10.3390/e22060622

**AMA Style**

Nosonovsky M, Roy P.
Scaling in Colloidal and Biological Networks. *Entropy*. 2020; 22(6):622.
https://doi.org/10.3390/e22060622

**Chicago/Turabian Style**

Nosonovsky, Michael, and Prosun Roy.
2020. "Scaling in Colloidal and Biological Networks" *Entropy* 22, no. 6: 622.
https://doi.org/10.3390/e22060622