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, those perspectives are not mutually exclusive and supplement each other. A 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 expression of some regulatory proteins. The coordination of molecular and cellular events leading to the synapse formation occurs in fractal time-space, that is, the time-space that is not only the arena of events but also influences those events actively. That 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

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

(8.1)

then the Gompertz function, the function of probability distribution P(θ), the anharmonic potential U(θ), and the diffusion coefficient D are related each other through the one-dimensional differential operator (8.2). This operator contains the function of probability distribution [19]:

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

(8.2)

This linear coupling of variables can also be defined as a function with both spatial and temporal fractal dimension. 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 Gompertz function f(t) [17]. A volume of the spheroid V is given by equation:

$$ V={V}_kF\left({t}_0\right){e}^{a\left(1-{e}^{-bt}\right)}={V}_kf\left({t}_0\right)a{t}^{b_t}={V}_0a{t}^{b_t} $$

(8.3)

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

The volume V of the spheroid can also be expressed as a function of scalar geometrical variable x (i.e., a radius of a family of the concentric spheres covering the entire spheroid) by Eq. (8.4):

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

(8.4)

in which a ₁ stands for a scaling coefficient, b _s is a spatial fractal dimension after scalar time t ₁, and x is a scalar, geometrical variable, which locates an effect in space.

Hence, we get Eq. (8.5):

$$ V={a}_1{x}^{b_s}={V}_0a{t}^{b_t}={a}_0{x}^{b_{s_0}}a{t}^{b_t} $$

(8.5)

in which a, a ₀, and a ₁ stand for the scaling coefficients, b _t is the temporal fractal dimension, b _s0 and b _s are the spatial fractal dimensions after time t ₀ and t, respectively, and x is a geometrical variable.

Finally, Eq. (8.6) relates space and time in which proliferation, differentiation, and synapse formation occurs. This equation defines the geometrical variable x as a function of the scalar time t and both temporal and spatial fractal dimension:

$$ 1nx=\frac{1}{b_s-{b}_{s_0}} \ln \frac{a_0a}{a_1}+\frac{b_t}{b_s-{b}_{s_0}}1 nt $$

(8.6)

in which t stands for scalar time, x is 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 to notice that the assumed Markov model of molecular interactions within differentiating neurons implies at least three important consequences. First, entropy (i.e., missing information) H _M of such the Markov chain of the coupled molecular reactions is always lower than entropy of the set of random and independent biochemical reactions H _R . Indeed, entropy is defined as the expected value of missing information H _p :

$$ {H}_P=H(X)=-{\displaystyle \sum}_{j=1}^N{p}_j \log {p}_j $$

(8.7)

in which p = (p ₁,p ₂,…p _j ), jε N, is a probability density function over a generic variable X, and if p _j  = 0, then H _p  = 0, 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:

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

(8.8)

The conditional entropy of the Markov chain H _C is given by (8.9):

$$ {H}_C=H\left({X}_k\Big|{X}_{k-1}\right)=-{\displaystyle \sum}_i{p}_i{\displaystyle \sum}_j{p}_{ij} \log {p}_{ij} $$

(8.9)

Finally, we get Eq. (8.10) for the n first steps of the Markov chain X ₁, X ₂,…, X _n from (8.7), (8.8), and (8.9), the principle of additivity of independent random events, and from the analog principle for the conditional probabilities:

$$ \begin{array}{l}{H}_M=H\left({X}_1\right)+{\displaystyle \sum}_{k=2}^{k=n}H\left({X}_k\Big|{X}_{k-1}\right)=-{\displaystyle \sum }{p}_j \log {p}_j+\left(n-1\right){H}_C<n{H}_P\\ {}\kern2.5em =-{\displaystyle \sum}_j{p}_j \log {p}_j={H}_R\end{array} $$

(8.10)

Second, Gompertz dynamics of molecular cellular growth can be normalized, i.e., growth dynamics of various tissue systems can be described by a single normalized Gompertz function f _N (t) (8.11). In fact, this normalized Gompertz function is both a dynamics function f _N (t) and a probability function p _N (t) (see for details [17]):

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

(8.11)

Consider a coupling of probability function p _N (t) and antiprobability function −log p _N (t), in which r = b:

$$ \frac{d{p}_N(t)}{dt}=-r{p}_N(t) \log {p}_N(t) $$

(8.12)

This equation defines a relationship between entropy H(t) and the normalized Gompertz dynamics of growth p _N (t):

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

(8.13)

Finally, from (8.11) and (8.12), we get (8.14):

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

(8.14)

According to 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:

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

(8.15)

In the case of growing supramolecular cellular system such as neuron, entropy, or missing information, H(t) is a function of time related with dynamic function of growth in fractal space-time. For b = 1 both the normalized Gompertz function (8.11) and the entropy function (8.14) overlap each other. However, b<<1 for the majority of cellular systems. The distribution of probability is in those cases 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 the Markov chains of the length n with the total probability p, 0<p<1, there exists the same limit for each probability p that equals entropy H:

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

(8.16)

If a total number of states of the supramolecular cellular system equal 2^m, then the number of molecular reactions interconnected in the Markov chains of the length n is 2^nm. It is clear from (8.16) that only 2^nH molecular Markov chains with probability 1−ε, ε>0 will be involved in transfer of biologically relevant information.