Neuronal Fractal Dynamics

Abstract

Synapse formation is a unique biological phenomenon. The molecular biological perspective of this phenomenon is different from the fractal geometrical one. However, these perspectives are not mutually exclusive and supplement each other. The cornerstone of the first one is a chain of biochemical reactions with the Markov property, that is, a deterministic, conditional, memoryless process ordered in time and in space, in which the consecutive stages are determined by the expression of some regulatory proteins. The coordination of molecular and cellular events leading to synapse formation occurs in fractal time space, that is, the space that is not only the arena of events but also actively influences those events. This time space emerges owing to coupling of time and space through nonlinear dynamics. The process of synapse formation possesses fractal dynamics with non-Gaussian distribution of probability and a reduced number of molecular Markov chains ready for transfer of biologically relevant information.

Appendix

If one assumes that the spatial variable x and the temporal variable t are coupled to each other in a linear manner into a single, complex spatiotemporal variable θ such that

$$ \theta =\mu x+t $$

(9.1)

then the Gompertz function, the function of probability distribution P(θ), the anharmonic potential U(θ), and the diffusion coefficient D are all related to each other through the one-dimensional differential operator (9.2). This operator contains the function of probability distribution [21]:

$$ -\frac{1}{D}\frac{\partial^2P\left(\theta \right)}{\partial {\theta}^2}+\frac{D}{4}P\left(\theta \right)+U\left(\theta \right)P\left(\theta \right)=0 $$

(9.2)

This linear coupling of variables can also be defined as a function with both spatial and temporal fractal dimensions. It is known from experimental data from the in vitro cellular system of P19/RAC 65 that the number of cells (or their volume) also changes in time t according to the Gompertzian function f(t) [19]. The volume of the spheroid V is given by the equation:

$$ V={V}_kF\left({t}_0\right){e}^{a\left(1-{e}^{- bt}\right)}={V}_kf\left({t}_0\right){at}^{b_t}={V}_0{at}^{b_t} $$

(9.3)

where V_k is the mean volume of a single cell, n stands for the number of cells in the spheroid, and the Gompertzian function can be fitted with the fractal function f(t) = at^b with extremely high accuracy; for a coefficient of the nonlinear regression R >> 0.95 for n ≥ 100 pairs of coordinates), a stands for the scaling coefficient, b_t is the temporal fractal dimension, and t is scalar time.

The volume V of the spheroid can also be expressed as a function of the scalar geometrical variable x (i.e., the radius of a family of the concentric spheres covering the entire spheroid) by Eq. (9.4):

$$ V={a}_1{x}^{b_s} $$

(9.4)

where a₁ stands for the scaling coefficient, b_s is the spatial fractal dimension after scalar time t₁, and x is the scalar, geometrical variable, which locates an effect in space.

Hence, we get Eq. (9.5):

$$ V={a}_1{x}^{b_{\textrm{s}}}={V}_0{at}^{b_t}={a}_0{x}^{b_{{\textrm{s}}_0}}{at}^{b_t} $$

(9.5)

where a, a₀, and a₁ stand for the scaling coefficients, b_t is the temporal fractal dimension, b_s0 and b_s are the spatial fractal dimensions after time t₀ and t, respectively, and x is the geometrical variable.

Finally, Eq. (9.6) relates space and time in which proliferation, differentiation, and synapse formation occur. This equation defines the geometrical variable x as a function of the scalar time t and both temporal and spatial fractal dimensions

$$ \ln x=\frac{1}{b_{\textrm{s}}-{b}_{{\textrm{s}}_0}}\ln \frac{a_0a}{a_1}+\frac{b_t}{b_{\textrm{s}}-{b}_{{\textrm{s}}_0}}\ln t $$

(9.6)

where t stands for scalar time, x is the geometrical variable, b_s is the spatial fractal dimension, and b_t is the temporal fractal dimension.

1.1 Entropy and Dynamics of Synapse Formation in Fractal Time Space

It is worth noticing that the assumed Markov model of molecular interactions within differentiating neurons implies at least three important consequences. First, the entropy (i.e., missing information) H_M of the Markov chain of the coupled molecular reactions is always lower than the entropy of a set of random and independent biochemical reactions H_R. Indeed, entropy is defined as the expected value of missing information H_p:

$$ {H}_{\textrm{P}}=H(X)=-\sum \limits_{j=1}^N{p}_j\log {p}_j $$

(9.7)

where p = (p₁, p₂,...p_j), jε N, is the probability density function over a generic variable X, and if p_j = 0, then H_p = 0 and log is a natural logarithm providing a unit of measure.

Hence, the conditional entropy H(X_k|Y_k-1) of the X_k reaction stands for which conditional information is determined when the state Y_k-1 = i, is given by the following equation:

$$ H\left({X}_k\left|{Y}_{k-1}\right.=i\right)=-\sum \limits_j{p}_{ij}\log {p}_{ij} $$

(9.8)

The conditional entropy of the Markov chain H_C is given by (9.9):

$$ {H}_{\textrm{C}}=H\left({X}_k\left|{X}_{k-1}\right.\right)=-\sum \limits_i{p}_i\sum \limits_j{p}_{ij}\log {p}_{ij} $$

(9.9)

Finally, we get Eq. (9.10) for the n first steps of the Markov chain X₁, X₂,..., X_n from (9.7), (9.8), and (9.9), the principle of additivity of independent random events, and from the analog principle for the conditional probabilities, we get:

$$ \begin{aligned} {H}_{\textrm{M}}&=H\left({X}_1\right)+\sum \limits_{k=2}^{k=n}H\left\langle {X}_k|{X}_{k-1}\right\rangle =-\sum {p}_j\log {p}_j+\left(n-1\right){H}_{\textrm{C}}<{nH}_{\textrm{P}}\\ &=-\sum \limits_j{p}_j\log {p}_j={H}_{\textrm{R}} \end{aligned} $$

(9.10)

Second, the Gompertzian dynamics of molecular cellular growth can be normalized, i.e., the growth dynamics of various tissue systems can be described by a single normalized Gompertz function f_N(t) (9.11). In fact, this normalized Gompertz function is both a dynamics function f_N(t) and a probability function p_N(t) (for details, see Waliszewski [19]).

$$ {f}_N(t)={e}^{-{e}^{\left(- bt\right)}}={p}_N(t) $$

(9.11)

Consider a coupling of probability function p_N(t) and anti-probability function –log p_N(t), where r = b:

$$ \frac{dp_N(t)}{dt}=-{rp}_N(t)\log {p}_N(t) $$

(9.12)

This equation defines the relationship between entropy H(t) and the normalized Gompertzian dynamics of growth p_N(t):

$$ {p}_N(t)=\int \frac{\partial {p}_N(t)}{\partial t} dt=-r\int {p}_N(t)\log {p}_N(t) dt= rH(t) $$

(9.13)

Finally, from (9.11) and (9.12), we get (9.14):

$$ H{(t)}_{\textrm{Gompertz}}=\frac{1}{b}{e}^{-{e}^{\left(- bt\right)}} $$

(9.14)

According to the Shannon theorem, of all the continuous distribution densities for which the standard deviation exists and is fixed, the Gaussian, (i.e., normal) distribution has the maximum value of entropy H:

$$ {H}_{\textrm{Gauss}}=-\int\limits_{-\infty}^{\infty}\frac{e^{-\frac{t^2}{2{\sigma}^2}}}{\sqrt{2{\pi \sigma}^2}}\log \frac{e^{-\frac{t^2}{2{\sigma}^2}}}{\sqrt{2{\pi \sigma}^2}} dt=\frac{1}{2}\log \left(2\pi e{\sigma}^2\right) $$

(9.15)

In the case of growing supramolecular cellular system such as neurons, entropy or missing information H(t) is a function of time related to the dynamic function of growth in fractal space time. For b = 1, both the normalized Gompertz function (9.11) and the entropy function (9.14) overlap each other. However, b << 1 for the majority of cellular systems. The distribution of probability is in such cases is non-Gaussian.

Third, there is a relationship between the number of elements in the Markov chain and entropy. If M_p(n) stands for a number of Markov chains of length n with the total probability p, 0 < p < 1, then there exists the same limit for each probability p that equals entropy H:

$$ \underset{n\to \infty }{\lim}\frac{\log {M}_p(n)}{n}=H $$

(9.16)

If the total number of states of the supramolecular cellular system equals 2^m, then the number of molecular reactions interconnected in the Markov chains of length n is 2^nm. It is clear from (9.16) that only 2^nH molecular Markov chains with probability 1-ε, ε > 0 will be involved in the transfer of biologically relevant information.