**1. Introduction**

Cellulose is defined as a natural homopolymer which is obtained mainly from living organisms including fungi, bacteria, algae, and animals [1, 2]. The cellulose is constructed by repetition of d-glucose units in which two glucose units are linked together by β(C1 → C4)-glycosidic bond [1] as shown in **Figure 1**. Additionally, degree of polymerization of cellulose highly depends upon the raw material; for native cellulose it can be up to 15,000 [3]. Cellulose has become one of the most widely used natural polymers because of its various fascinating physical and structural properties along with its biocompatibility and wide range of availability [4]. The several magnificent properties of cellulose arise due to the presence of multiple inter- and intrahydrogen bonding interactions as illustrated in **Figure 2**. The cellulose is an unbranched homopolymer because it exists in semicrystalline form containing both crystalline and amorphous phases [5].

It is clear from **Figure 1** that cellulose consisted of two types of hydroxyl (─OH) substituents, one is primary hydroxyl substituent and the other is secondary hydroxyl substituent [5].

The discovery of cellulose was made in late 1830s by Anselme Paven and Hyatt. In the year 1870, manufacturing company took a positive outlook in cellulose

**Figure 1.** *Chemical structure of cellulose.*

**Figure 2.** *Pictorial presentation of inter- and intramolecular H bonding in cellulose.*

research and produced first cellulosic thermoplastics [4, 6, 7]. The research of cellulose modification has been developing, from one process to another, trying to find as the most eco-friendly and eco-efficient method as possible from the time when it was discovered as cellulose is the most abundant natural material on earth.


A log-normal distribution of activities in a different context, when there are inhomogeneous degree distributions or other quenched noise in the system, have been reported in different studies (Roxin et al., 2011; Mizuseki and Buzsáki, 2013; Buzsáki and Mizuseki, 2014). Outside the neuroscience literature it has been claimed that log-normality could be an emergent feature of competition among subgroups of individuals (Halloy, 1998; Halloy and Whigham, 2005). For a balanced random network with the same in-degree for all neurons, however, to the best of our knowledge, it is the first time that this statistical behavior is reported. We believe that this emergent feature is tightly related to the large-scale dynamics of the interacting excitatory and inhibitory populations. The state dependence of the transition rates, and the doubly stochastic nature of the Markov system that follows from it, might provide a formal explanation for this result.

#### **3.7. UNIVERSALITY IN BALANCED NETWORKS OF DIFFERENT SIZE**

The network under study is a strongly connected network with identical in-degrees for all neurons. This implies that each neuron receives inputs from a fixed fraction of each population in the network. To keep the mean and the variance of the activity in the network limited, when the size *N* of the network tends to infinity, we scaled the synaptic weights by √ 1 *N* . We studied the network size effect on the variance of fluctuations and the nullclines of the reduced dimension system for 3 different networks of total size 7500, 12,500, and 20,000 neurons where in all these cases 20% of the neurons were inhibitory. The corresponding EPSP amplitude was 0.13, 0.1, and 0.08, respectively. To see whether there is any universal feature in the population dynamics of such networks, we normalized the spike counts of each population by the size of the population. This helps us to study dynamics of the fraction of the active neurons irrespective of the size of the network. To check the size invariance property of the ARM model, we rescaled equation (8) by the corresponding population size for the excitatory and inhibitory population. Introducing new variables *Xe* = *Se Ne* and *Xi* = *Si Ni* , equation (8) could be rewritten in the following form

$$\begin{cases} \dot{X}\_{\varepsilon}(t) &= \exp\left(c\_{0} + \iota\_{1}^{\prime}X\_{\varepsilon}(t) + \iota\_{2}^{\prime}X\_{i}(t)\right)\left(\boldsymbol{\upchi}dt - X\_{\varepsilon}(t)\right) \\ &- (\boldsymbol{\upbeta} + \boldsymbol{\upchi})X\_{\varepsilon}(t) \\ \dot{X}\_{i}(t) &= \exp\left(c\_{0} + \iota\_{1}^{\prime}X\_{\varepsilon}(t) + \iota\_{2}^{\prime}X\_{i}(t)\right)\left(\boldsymbol{\upchi}dt - X\_{i}(t)\right) \\ &- (\boldsymbol{\upbeta} + \boldsymbol{\upchi})X\_{i}(t) \end{cases} (12)$$

In equation (12), the coefficients *c*- 1 and *c*- 2 depend on the population sizes. The variables *X* are confined between 0 and 1 and comparing the dynamics of the networks with different sizes is reduced to a comparison between transition rates α, β, and γ . In our study, we were not able to identify an exact relationship between these parameters and the single neuron and connectivity parameters. However, **Figure 10** shows that these parameters change such that the equation is invariant to network size.

As illustrated in **Figure 10**, the nullclines have the same shape irrespective of the size of the network, but the variability that is reflected in the total visited area of the state space decreases as the network size increases. The suggested active refractory model can reproduce the vector field of the spiking network simulation for different network sizes, provided that the parameters of the model are correctly estimated. Therefore, our conclusion is that the nonlinearity in the interaction between populations for stationary input and the heavy-tail (positively skewed) distribution of the activity are both universal properties of strongly connected networks of finite size.

#### **3.8. IS THE STATIC NONLINEAR TRANSFER FUNCTION RESPONSIBLE FOR THE NONLINEAR DYNAMICS?**

Some studies of network dynamics assume that the stationary input-output transfer function of a neuron could provide a good description even for the time dependent activity of the network (Ledoux and Brunel, 2011; Pernice et al., 2012; Tetzlaff et al., 2012; Ostojic, 2014). The firing rate of a neuron in these models is typically approximated by the differential equation

$$\text{tr}\dot{r}\_i(t) = -r\_i(t) + \mathbf{F}(\mathbf{r}(t))\tag{13}$$

where **F** is the static nonliner function and **r** is the vector of the firing rate of other neurons in the network. This model, however, can be a good approximation for the temporal dynamics of the network only at large time scales, when the network activity is filtered over time and fluctuations have relatively low amplitude. Under these conditions, a linear approximation of the dynamics does provide a good match with the data. In our study, however, the time scale was chosen based on the time scale of the autocorrelation function of the network activity such that a wide range of frequencies contributes to the fluctuations. For *dt* = 0.1 ms, we demonstrate that the neural static nonlinearity cannot follow the delicate nonlinearity of the isoclines (**Figure 11A**) and the log-normality of the spike counts (**Figures 11B,C**). The nullclines of this model, after transforming the rates to spike counts, are almost linear functions of the respective spike counts (**Figure 11A**). Comparing the cumulative distributions of the spike counts generated from SNN, ARM and the input-output nonlinearity in equation (13) illustrates the discrepancy between the latter and the former. In fact, assuming that the instantaneous rates are equal to the stationary rates results in an overdispersed spike count distribution that does not match with that of SNN. Therefore, this assumption is obviously not correct for small time bins. These results show that system (13) is not valid on small time scales. On the other hand, as shown in the previous sections, the ARM model reproduces statistics and dynamics similar to the SNN data.

### **4. DISCUSSION**

In this study, we highlighted some properties of the large scale dynamics of finite-size balanced random networks in the inhibition dominated regime. It was shown that a linear system with additive Gaussian white noise cannot capture the statistics and dynamics of the two neuronal populations, and a more sophisticated model that represents the nonlinearity of the interactions and the statistics of the activity in a self-consistent way was suggested. We showed that a two state Markov process that the states of which reflect the coarse-grained membrane potential of a neuron, along with appropriate state-dependent transition probabilities, can reproduce the dynamics and statistics of a finite size network in the stationary balanced state in a satisfactory way. The state-dependent transition probabilities were all inferred from the

numerical simulation of a spiking network of leaky integrate-andfire neurons with stationary input. The approach was based on a reconstruction of the mean-field dynamics flow. The deterministic dynamics was complemented by a state-dependent stochastic component, assuming independent spiking in all neurons. This essentially leads to a binomial noise model, approximating Poisson statistics in very small time bins. On the level of the meanfield, the Active-Refractory Markov (ARM) model resembled the vector field extracted from the data with high fidelity. The population correlation functions of the spiking network simulation and the stochastic implementation of the model shared the same dynamical behavior.

We do not claim that the temporal dynamics of the finite size system is Markovian but a Markov model provides a good approximation. In general, the transition probabilities of the model depend on an unbounded history of the network activity (Cessac, 2011). However, we showed that a Markov model with a carefully chosen time scale is able to reflect some major statistical and dynamical properties of a balanced random network with fixed in-degrees. In the first part of the paper we explored the Markov properties of the network dynamics, and the size of time bins was chosen based on the similarity between the power spectra of a Markov model and the spiking network simulation. Then, we specifically considered a two-state Markov model for each neuron and estimated the state dependent transition probabilities from the simulated data. A self-consistent description that also accounts for the fluctuations of the finite-size system was obtained.

As outlined in detail in the Results section, there is a distinct similarity between the mean-field equations derived from the ARM model and the well-known Wilson-Cowan model (Wilson and Cowan, 1972) for the joint dynamics of excitatory and inhibitory neuronal populations. In the Wilson-Cowan model, the population response function, which gives the expected proportion of active neurons in a population as a function of the overall excitation in the system, derives from a cumulative unimodal density function and therefore typically has a sigmoidal shape. Using the Fokker-Planck equation to obtain the stationary distribution of the membrane potential and neglecting the absolute refractory time, a unimodal distribution will emerge. As discussed in this paper, the integral of the density function gives the transition probability from the refractory to the active state, the rate α in our model, as it reflects the local probability flux between the refractory and the active state. The local behavior of this function, for a wide range of the membrane potential between the threshold and reset, is well approximated by an exponential function and this approximation is quite good in comparison with spiking network simulations. A more precise result might come out if the CDF of the solution of equation (9) with free parameters is used for α. As shown in **Figure 4**, the CDF is a sigmoid function due to its unimodality. It is important to mention that the parameters of the model do not directly come from the CDF, but they should be estimated from the simulated data. An exponential function for the monotonically increasing CDF provides a good approximation for a wide range of network parameters and external input levels, as long as the network activity is asynchronous and irregular. Another comparison between the two models could be done on the kernel that is applied on the total excitation level in the network. In Wilson and Cowan (1972) an exponential kernel was used, which is justified by the impulse response of a leaky integrate and fire neuron.

The self-consistent noise model is an important feature of the model suggested in this article. The noise distribution is specific for each state, due to a deterministic dependency of the transition rates on the most recently generated spikes. Surprisingly, the state-dependent noise is such that it creates a strongly skewed distribution of the spike counts, similar to what is observed in the numerical simulations of spiking networks. The variance of the fluctuations generated by the model is, however, slightly smaller than the variance of the spike counts in the spiking network simulations. This is probably caused by the symmetric and relatively narrow state-dependent distribution of the noise in the excitatory population (**Figure 5A**). As mentioned in the result section, this problem is due to the assumption of independent spiking of individual neurons in each pool, implying the emergence of binomial distributions to describe the size of the neuronal populations that undergo a transition. As the variance of the outcome will be proportional to the total number of available neurons in each state, the variance of the increments will be directly proportional to the population size. One way of coping with this problem would be to create a correlation between the noise distributions in the two populations. Another way would be to consider the spike count distributions of the excitatory population given the spike counts of the inhibitory one. This way the heavy tail (positively skewed) distribution will come out automatically.

The nonlinear isoclines of the vector field provide strong evidence for a nonlinear interaction between the two neuronal populations. In particular the nullcline of the excitatory population exhibits very nonlinear behavior in the regime of small spike counts. Close to the fixed point of the flow, the excitatory nullcline changes direction. This property of the network can only be explained by a nonlinear model. It was already shown by Wilson and Cowan (1972) that different shapes of the nullclines emerge due to the asymmetry between excitation and inhibition, and because of their different signs in the argument of the population response function. In our model, however, the parameters of the two differential equations are the same, but different signs of the coefficients of the excitatory and inhibitory spike counts are enough to yield different shapes of the nullclines. A similar shape of the nullclines was in fact obtained in a model of the thalamo-cortical response transformations in the Barrel cortex of rodents (Pinto et al., 2003). We also checked whether the shape of the nullclines is invariant with regard to the size of the network. To compare the nullclines of networks of different sizes, however, it was necessary to normalize the activities of both populations. After normalization, the nullclines of networks with different sizes were almost identical, and it seems justified to claim that the nonlinear interaction is a universal property of strongly connected balanced networks. Since our method is based on the reconstruction of the dynamic flow, we conclude that our model can successfully capture the dynamics due to its size-invariance property.

It has been shown in van Vreeswijk and Sompolinsky (1996, 1998) that balanced random networks are extremely fast in following the temporal dynamics of an external input. This means among other things that the power spectral density of "spontaneous" network activity has a very broad frequency range and, as a result, the autocorrelation functions of the populations decay sharply. Our suggested model was successful in reproducing this fast temporal dynamics. Experimental evidence and theoretical studies suggest that dynamic responses of neocortical neurons are much faster when multiplicative input noise is imposed, compared to the case of additive noise (Lindner and Schimansky-Geier, 2001; Silberberg et al., 2004; Boucsein et al., 2009). We hypothesize that the multiplicative nature of the self-generated noise in balanced random network could also contribute to fast neuronal responses and sharp autocorrelation functions. Furthermore, an emergent property of balanced networks in the asynchronous irregular state is the high correlation between excitatory and inhibitory population activity, which was also successfully retrieved by our model. This is due to the fact that the same mean-field provides input to both populations. In other words, log (α) is a linear function of excitatory and inhibitory activity at any given point in time, with a positive weight for excitation and a negative weight for inhibition. This type of dependency makes a major contribution to correlating excitatory and inhibitory spike counts.

Recently, some studies have featured heavy-tail distributions of various phenomena in the brains of different species, supposedly a robust and important aspect of cortical computation (for a review see Buzsáki and Mizuseki, 2014), often approximated by log-normal distributions. Neurons in the auditory cortex (Deweese and Zador, 2008) and in the hippocampus and enthorinal cortex (Mizuseki and Buzsáki, 2013) of rats exhibit a log-normal distribution of firing rates. In Mizuseki and Buzsáki (2013) it is also shown that during the bursting activity state of the network, the fraction of all recorded neurons that fire a spike, for either stimulus-evoked or spontaneous activity, display a log-normal distribution. This, in turn, might be attributed to the log-normal distribution of the synaptic weights. Similarly, high density micro-electrode recordings of the human brain during sleep, in combination with a separation of excitatory and inhibitory cellular activities based on the spike wave-forms, has shown a log-normal distribution of firing rates (Peyrache et al., 2012). A theoretical study (Roxin et al., 2011) has shown that for a randomly connected network of excitatory and inhibitory LIF neurons with a random number of external inputs for each neuron and random synaptic efficacies, using an exponential *f*-*I* curve to describe the single neuron dynamics and a Gaussian distribution of inputs to each cell, a log-normal distribution of neural firing rates arises. In our study, however, all neurons had the same statistical inputs without any quenched variability, resulting in identical firing rates for all neurons in the network. However, the distribution of the spike counts for a large simulation time with small time bins was shown to be well fitted by a log-normal distribution. The suggested model was also able to represent this emergent statistical property of the network. This might emerge due to the exponential dependency of the transition probability α with suitably chosen parameters. In the theory of complex systems, the log- normal distribution is considered to be a universal statistical property of many natural systems (Halloy, 1998; Halloy and Whigham, 2005; Kobayashi et al., 2011). In particular, Kobayashi et al. (2011) showed that if the history of each component of the system defines its present state, log-normality becomes emergent. This could be tested on a network of multiplicatively interacting point processes that mimic the behavior of LIF neurons (Cardanobile and Rotter, 2010, 2011). However, a rigorous analysis shows that the Gibbs distribution is a unique invariant probability measure of the system under stationary input (Cessac, 2011). It might be interesting to investigate the underlying reason for the emergence of the log-normal distribution analytically.

We would like to stress once more the importance of choosing the right time scale for the ARM model. Due to the Markov assumption, the time bin must be chosen such that the influence of the past activity of the network on the transition rates (its memory) is minimal. This time scale is typically small, and we observe that a wide range of the power spectrum of the spiking activity is preserved. We conclude that the time dependent dynamics of the network is quite accurately captured. It was also shown in our paper that the typical input-output static nonlinearity of a neuron is neither suited to reproduce the nonlinear activity of the network, nor the log-normal distribution of the spike counts.

Markov models have been suggested before as models of the temporal dynamics of finite size networks (Soula and Chow, 2007; El Boustani and Destexhe, 2009; Cessac, 2011; Buice and Chow, 2013a). In Soula and Chow (2007), an approach similar to ours was proposed assuming a statistically homogeneous network of excitatory neurons. The difference to our work is that an active neuron is by definition a neuron that emits a spike, therefore only one transition probability from the silent to the active state is needed. This probability is proportional to the steady-state firing rate of the neuron. Also, a full-blown theoretical framework was introduced to calculate the first and second moment of fluctuations in the network. Our model, however, is different in the sense that the interaction between the excitatory and inhibitory population is taken into account, and that a high correlation between the activities of the two populations is preserved. Moreover, a heavytail distribution of the activity emerges as a result of the dynamical interaction between excitation and inhibition. However in Soula and Chow (2007) the distribution of the single population activity is symmetric due to the lack of inhibitory population. Our model on the other hand considers a neuron in the active state if its membrane potential is above some unspecified value of the membrane potential between threshold and rest. It takes the effect of leak into account by assuming a potential transition from the active to the refractory state without emitting a spike. El Boustani and Destexhe (2009) followed the same approach in continuous time for a sparse random network of excitatory and inhibitory neurons. Assuming quasi-stationarity, they derived the first two moment equations of the activity using the static input-output transfer function of a typical neuron in the network. However, Poisson statistics for each neuron was assumed and the transition function was calculated based on the mean activity of the network; therefore, the resultant distribution of the activity was Gaussian.

The main underlying assumption of the ARM model is the two-state Markovian single-neuron dynamics. For an appropriate choice of the time step, it was shown in the present study that the statistics and dynamics of recurrent networks of leaky integrateand-fire neurons can be captured by the model. However, we neglected the role of absolute refractoriness in the dynamics of the membrane potential. Refractoriness shapes the low frequency range of the population dynamics (Mar et al., 1999; Spiridon and Gerstner, 1999). It could be modeled by introducing a chain of refractory states and thereby, increasing the dimensionality of the model (Toyoizumi et al., 2009). In the ARM model, absolute and relative refractoriness together, effectively, will have a wide distribution (for a relevant study, see Deger et al., 2010). Another aspect of our study is that synaptic transmission delays were neglected in the model as well as in our network simulations. In general, the transition probabilities depend on an unbounded past (Cessac, 2011) and delayed feedback makes the system non-Markovian (Vidybida and Kravchuk, 2013). It might be interesting to investigate whether a delayed α rate operating on the current pool of refractory neurons can represent the dynamics of the spiking network simulations. Furthermore, the important assumption of the ARM model is that all neurons statistically behave the same, because they all have the same in-degree. This allows us to reduce the dimensionality of the large scale dynamics and come up with a simple two state stochastic model of the system. If there is any inhomogeneity in the system, this model will not be a good candidate. Maybe for a network with homogeneous subpopulations, each component could be modeled by the ARM presented in this paper, with suitable parameters.

Population density methods are promising approaches for dimensionality reduction in dynamic networks. Including finitesize effects and correlations in the model, however, is a challenge. Deterministic density equations describing the temporal dynamics of finite-size networks were derived by using an eigenfunction expansion of the Fokker-Planck equation (Mattia and Del Giudice, 2002, 2004). Using a stochastic and deterministic approach, Buice and Chow recently suggested a mean-field equation and moment hierarchies of a density equation to obtain corrections arising from the finite size of the system and from correlations which are basically due to heterogeneities in the system (Buice and Chow, 2013a). For a homogeneous network, using the effective action approach of field theory (Buice and Chow, 2013c) and system size expansion around the mean-field density function (Buice and Chow, 2013b), they derived moment equations leading to the dynamics of mean and covariance in the network. Our suggestion for future exploration of the field is that a nonlinear set of Fokker-Planck equations for each population might be recast in the form of ARM model suggested in this paper. The more general model will have nonlinear and state dependent transition parameters that could be analytically derived and the stochastic behavior will emerge as a result of the finite size of the system.

Finally, to extend the model to cover the more general case of non-stationary and time dependent input, it is necessary to investigate the precise role of the external input in the ARM model. There are at least two possibilities: it could either be reflected in γ , or it could be included in the transition rate from the refractory to the active state, α. We suggest that the same data analysis method that we applied in this study might also help in determining the role of external time-dependent input in the model. Furthermore, it is possible to test whether this model can capture the dynamics of more than two interacting populations. However, this will be more challenging, as the dynamics of these type of networks are not necessarily stationary in time. Particularly, under certain conditions, switching dynamics between populations might arise (Litwin-Kumar and Doiron, 2012).

## **AUTHOR CONTRIBUTIONS**

Mathematical analysis and numerical experiments of this study were conceived and designed by Fereshteh Lagzi and Stefan Rotter. Numerical simulations and data analysis were performed by Fereshteh Lagzi, and supervised by Stefan Rotter. The paper was written by Fereshteh Lagzi and Stefan Rotter.

#### **ACKNOWLEDGMENTS**

We thank Fatihcan Atay and Björn Schelter for helpful discussions. We thank the developers of the simulation software NEST (see http://www.nest-initiative.org) and the maintainers of the BCF computing facilities for their assistance throughout this study. We would like to acknowledge the use of the computing resources provided by the Black Forest Grid Initiative. This work was supported by the German Ministry of Education and Research (BFNT Freiburg∗Tübingen, grant 01GQ0830) and the German Research Foundation (DFG, grant EXC 1086). The article processing charge was covered by the open access publication fund of the University of Freiburg.

#### **REFERENCES**

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 25 May 2014; accepted: 20 October 2014; published online: 03 December 2014.*

*Citation: Lagzi F and Rotter S (2014) A Markov model for the temporal dynamics of balanced random networks of finite size. Front. Comput. Neurosci. 8:142. doi: 10.3389/ fncom.2014.00142*

*This article was submitted to the journal Frontiers in Computational Neuroscience. Copyright © 2014 Lagzi and Rotter. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## **APPENDIX**

#### **A.1. PROPERTIES OF A LINEAR SYSTEM WITH GAUSSIAN NOISE**

In this section we highlight the main properties of a linear system and we will show that the temporal dynamics of finite size networks does not suggest a linear model. To demonstrate this, we consider a general two-dimensional linear system and will use two sources of Gaussian white noise to drive the excitatory and the inhibitory population. Then, we will discuss the features of this model due to the linear nature of the system and those which emerge due to the external drive. We assume in the following that *ye*(*t*) and *yi*(*t*) represent the instantaneous firing rates of the excitatory and inhibitory population. We consider an arbitrary communication delay in population interactions, which need not to be identical. Without loss of generality, we assume zero initial conditions for both populations. A coupled twodimensional linear system in its general form is represented in the following way

$$
\dot{\mathbf{y}}\_{\varepsilon}(t) = a\_{\varepsilon\varepsilon}\mathbf{y}\_{\varepsilon}(t-d) + a\_{\varepsilon i}\mathbf{y}\_{i}(t-d) + a\_{\varepsilon}I\_{\varepsilon}(t)
$$

$$
\dot{\mathbf{y}}\_{i}(t) = a\_{i\varepsilon}\mathbf{y}\_{\varepsilon}(t-d) + a\_{i\overline{i}}\mathbf{y}\_{i}(t-d) + a\_{i}I\_{i}(t) \tag{A1}
$$

Taking the Laplace transform of the above equation results in

$$sY\_{\varepsilon}(s) = a\_{\varepsilon\varepsilon}e^{-s d}Y\_{\varepsilon}(s) + a\_{\varepsilon i}e^{-s d}Y\_{i}(s) + a\_{\varepsilon}I\_{\varepsilon}(s)$$

$$sY\_{i}(s) \ = a\_{i\varepsilon}e^{-s d}Y\_{\varepsilon}(s) + a\_{i\varepsilon}e^{-s d}Y\_{i}(s) + a\_{i}I\_{i}(s) \tag{A2}$$

which in matrix form is

$$\begin{aligned} \begin{bmatrix} Y\_{\epsilon}(s) \\ Y\_{i}(s) \end{bmatrix} &= \begin{bmatrix} a\_{\epsilon\epsilon}e^{-sd} - s & a\_{\epsilon i}e^{-sd} \\ a\_{i\epsilon}e^{-sd} & a\_{i\bar{\imath}}e^{-sd} - s \end{bmatrix}^{-1} \begin{bmatrix} -a\_{\epsilon}I\_{\epsilon}(s) \\ -a\_{\bar{\imath}}I\_{i}(s) \end{bmatrix} \\ &= \frac{1}{G(s)} \begin{bmatrix} a\_{\bar{\imath}\epsilon}e^{-sd} - s & -a\_{\epsilon i}e^{-sd} \\ -a\_{i\epsilon}e^{-sd} & a\_{\epsilon e}e^{-sd} - s \end{bmatrix} \begin{bmatrix} -a\_{\epsilon}I\_{\epsilon}(s) \\ -a\_{\bar{\imath}}I\_{i}(s) \end{bmatrix}, \end{aligned} \tag{A3}$$

where

$$G(s) = \left(a\_{il}e^{-sd} - s\right)\left(a\_{ee}e^{-sd} - s\right) + a\_{ic}a\_{ci}e^{-2sd}$$

is called the characteristic equation of the linear system and appears in the denominator of both *Ye*(*s*) and *Yi*(*s*). The peaks in the power/amplitude spectra show the location of the zeros of the characteristic equation of the system. Therefore, the poles of the two components are identical and result in the same location of peaks in the power spectrum of *ye*(*t*) and *yi*(*t*). This statement is true in the more general case, when the inputs are different and non-Gaussian, since the two inputs play a role in shaping the power spectrum of both signals. The conclusion that we draw from this analysis is that if the power or amplitude spectra of two mutually interacting signals do not have the same peaks, then the interaction between them cannot be linear.

Equation (A3) shows that a coupled system as such behaves like a low-pass filter and for white noise input, the output would just be a filtered version of the noise. Since the distribution of the input is symmetric about its mean, the distribution of the output is also symmetric. Therefore, a non-symmetric distribution of the output must be interpreted as either a sign of nonlinearity of the system or the non-Gaussian nature of the input.

#### **A.2. NON-IDENTICAL POLES IN THE AMPLITUDE SPECTRUM OF EXCITATORY AND INHIBITORY SPIKE COUNTS**

A balanced network of excitatory and inhibitory population, with the same characteristics of the network introduced in the Methods section, but with a synaptic delay of 1.5 ms and the total simulation time of 100 s, was simulated. The amplitude spectra of the excitatory and the inhibitory population are illustrated in **Figure A1**. In the low frequency regime of the dynamics, the peaks of the excitatory and inhibitory amplitude spectra are identical. However, for the high frequency part of the spectra the locations of the poles of the two populations are slightly shifted with respect to each other. This fact shows that the dynamics of the system that describe the temporal activity of the population interactions cannot be linear.

**FIGURE A1 | Amplitude spectral density of spike counts with a bin size of 0***.***1 ms and** *d* **= 1***.***5 ms.** The entire spectrum of excitatory spike counts in bins of 0.1 ms is plotted in blue. The power spectrum of inhibitory spike counts under the same condition is plotted in red. Due to less number of neurons in the inhibitory population, its amplitude is lower than that of excitation. Dashed lines are plotted at frequencies corresponding to the peakes in the power spectra. For low frequencies, the peaks of the two spectra are the same. In the high frequency range the peaks in the amplitude spectrum of the inhibitory spike counts are slightly shifted with respect to those of excitatory ones.


**10. Concluding remarks**

Deodorizer distillate is an excellent source of valuable compounds such as phytosterols and tocopherols. Numerous procedures have been described to isolate bioactive compounds from soybean oil deodorizer distillates to improve the value and the quality of this byproduct. All these procedures can be grouped in four generic categories: classic methods such as crystallization and precipitation, chemical and enzymatic modification, molecular distillation, and supercritical fluid extraction.

Crystallization seems successful as a simple and efficient process to remove and concentrate sterols and tocopherols from SODD. However solvent based processes are expensive, unattractive and less environmentally friendly, resulting in a scarce and expensive final product.

To increase the separation efficiency of the compounds of interest from SODD, esterification and/or transesterification reactions are usually carried out prior to the purification or fractionation procedure. Hence, the utilization of enzymes, for instance, makes easier the separation of tocopherols from SODD by converting sterols to steryl esters, acylglycerols to free fatty acids and free fatty acids to fatty acid methyl or ethyl esters (FAMEs or FAEEs). Then, it is easier to separate the new product mixture by distillation or supercritical fluid extraction. The main difficulties of the enzymatic processes are the numerous parameters involved such as moisture content, enzyme concentration, time, temperature, ratio of the reactants, stability, recovery and reutilization of the enzyme preparation, among others. However, it is possible to separate the sterol esterification and ethyl esterification in time or space to optimize each of these reactions independently, thereby minimizing costs or improving the yield of the desired final reaction products.

Among the great variety of processes that have been patented for the purification of the compounds of the SODD, only the processes of esterification of fatty acids and acylglycerols with methanol or ethanol followed by high vacuum distillation, have been developed on a commercial scale for the concentration of tocopherols

Regarding supercritical fluid fractionation SODD is not adequate feed material to work with SC-CO2 for tocopherol enrichment, owing to its poor SC-CO2 solubility. So, to concentrate tocopherols from SODD, pre-treatment of the raw material is needed to obtain the primary tocopherols concentrate with improve solubility in SC-CO2.

Alternatively, deodorizer distillates have also non-food applications, such as biodiesel or can be used mixed with the fuel oil to fire the steam boilers. The use of deodorizer distillate instead of refined vegetable oils is an important alternative as a feedstock for biodiesel production.

Two main degradation products from sterols can be observed in SODD, namely dehydration and oxidation products. The degree of sterol dehydration is mainly influenced by deodorization temperature giving rise to a variable concentration of steradienes in the distillate. The content of oxidized sterols in deodorization distillate fractions from edible oil refining processes fluctuates depending on both the temperature applied during vacuum distillation and the breakdown and transformation of oxidized sterols into other unidentified degradation products. Finally, formation of oxidized sterols can be partially prevented by the high amounts of natural antioxidants in acid oil obtained from physical refining distillate. The current knowledge on the possible biological effects of oxyphytosterols is limited and further research to clarify the possible impact of oxyphytosterols on human health is needed.


*2.4. Bivariate Spatial Association*

Lee (2001) developed a bivariate spatial association based on univariate spatial correlation analysis to examine the spatial association between bivariate observations [38]. The bivariate spatial association is defined as follows:

$$I\_{kl}^{\dot{i}} = Z\_k^{\dot{i}} \sum\_{j=l}^n W\_{\hat{i}j} Z\_l^{\dot{i}}$$

where *Wij* denotes the spatial weight matrix, *Zi k* = - *xi k* − *xk* /*σk*, *Zi k* = [*xj k* − *xl*]/*σl*, *xi k* is the observation *k* at location *i*, *xj l* is the observation *l* at location *j*, and *σk* and *σl* denote the variance of *xk* and *xl*, respectively.

As with spatial correlation analysis, the bivariate spatial association visualizes the results using a Moran's *I* scatterplot and the bivariate LISA cluster map [39]. We used this method to examine the spatial association between newly increased construction land and newly increased polluting enterprises.


*2.5. Aging*

Mitochondrial dysfunction has been shown to lead to oxidative stress due to the generation of excessive reactive oxygen species, triggering senescence and age-related diseases. Since cell lines with MITOL deficiency induces cellular senescence [27], we also focused on the association between reduced MITOL function and senescence and age-related diseases in vivo. In our in vivo analysis, mice lacking MITOL in a skin-specific manner showed significant signs of aging, including gray hair and hair loss. These results and MITOL downregulation during aging suggest that MITOL may block the induction of physiological aging. We hoped that drugs that upregulate MITOL expression could inhibit or delay aging, and in a collaborative study with pharmaceutical companies, we screened drugs that upregulate MITOL mRNA from a Chinese medicine library using cultured human keratinocytes. We found that extracts of *Coptis japonica* and *Phellodendron amurense* in a Chinese medicine library upregulated MITOL mRNA approximately threefold (Patent No: P2019-52145A). When berberine, common compound of *Coptis japonica* and *Phellodendron amurense*, was mixed with drinking water, the expression of mRNA and protein of MITOL increased in various organs and tissues, including skin. To investigate the anti-aging activity of berberine, mice treated with it were irradiated with ultraviolet light to induce the formation of wrinkles in the skin, and a significant inhibitory e ffect on this was observed compared with that in control mice. In addition, with collaboration research, we found that the mice that took berberine for more than 1 year were significantly less likely to show signs of aging, such as skin hair loss and thickening of the epidermis, than control mice. Because the anti-aging e ffect of berberine on mice deficient of MITOL specifically in the skin was not confirmed, anti-aging e ffects are induced especially via the upregulation of MITOL expression levels. These results strongly indicate that MITOL has the potential to be a target for anti-aging drugs. In the future, it is anticipated that new MITOL activators other than berberine will be identified and applied to various aging-related diseases such as neurodegenerative diseases and heart failure. Interestingly, the expression of MITOL has been observed to gradually decrease with aging in many organs such as the brain and heart of mice. Moreover, Alzbase, a database for gene dysregulation in Alzheimer's disease (AD), suggested that the expression level of the mitol/march5 gene was significantly reduced in the brain of AD patients. If the worsening of AD is linked to the reduced MITOL expression, the compounds we have identified may exert a good therapeutic e ffect on AD. Therefore, we speculate that the decreased degradation of denatured proteins caused by decreased expression of MITOL may be one of the factors exacerbating the development of neurodegenerative diseases with aging (Figure 2).

**Figure 2.** Relativity between MITOL and aging. Since MITOL gradually decrease following aging, upregulation of MITOL by a Chinese herb might be able to e ffect anti-aging.

### **3. Regulation of Mitochondrial Dynamics by MITOL**


**1. Introduction**

Due to more than 200 years of industrialization and to the use of dangerous substances in many production processes, the countries across the world are facing the problem of soil, sediment, and water matrices contamination. Contaminated sites, namely environmental burdens, generally resulted in past and also arise nowadays from the manufacturing, storage, use, and disposal of hazardous chemicals and materials. It is now widely recognized that polluted sites pose threats to human health and the environment.

Polychlorinated biphenyls (PCBs) represent an environmental concern due to their hydrophobicity and toxicity. Although the production of PCBs has been banned and their use heavily restricted, they still pose an environmental problem due to their presence in old electrical transformers, capacitors, landfills, and in contaminated soil and sediments mainly in the areas around the former production facilities [1, 2]. Their physical and chemical properties such as thermal and chemical stability, resistance to degradation, and general inertness contribute to their persistence in the environment [3]. PCBs represent potential health risks for living organisms due to their lipophilic nature, bioaccumulation, and potential carcinogenic properties [4]. Hydroxylated PCBs (HPCBs), known PCB metabolites, have been detected in human serum samples and wildlife blood samples [5]. Numerous adverse health effects in human have been associated with these compounds. HPCBs are capable of mimicking a thyroid hormone, thyroxin [6], and may generate reactive oxidative species and cause DNA damage. Studies performed with the individual PCB congeners show that PCB toxicity and biodegradability are structure related as well [7].

Many conventional and sustainable remediation techniques have been invented to destroy hazardous organic pollutants [8]. The finding that both Gram-negative bacteria, such as *Achromobacter, Alcaligenes, Burkholderia, Commamonas, Pseudomonas*, and Gram-positive bacteria, such as *Bacillus, Corynebacterium*, and *Rhodococcus*, are able to degrade some PCB congeners opened the door to implement biological technologies. Bioremediation technologies using degradation capacity of microorganisms, mainly bacteria, have been seen ecological and economical alternative approach to physicochemical processes to eliminate diffusive contamination of persistent organic pollutants (POPs) in various environmental matrices, e.g., soil, sediments, and sludges. Bioremediation is an attractive, generally low-cost, innovative technology that is a sustainable approach to clean up organic compounds from contaminated areas. Bioremediation represents a perspective and prospective technique for treatment of polluted environments which involves usage of microorganisms and/or plants for pollutant biodegradation or biotransformation. The technology can be performed as natural attenuation or employed as an assisted bioremediation: biostimulation (addition of nutrients and inducers to fortify and stimulate the growth and metabolism of indigenous microorganisms), and bioaugmentation (introduction of indigenous or suitable exogenous bacteria to enhance biodegradation of relevant pollutant) [9–13]. However, successful soil bioaugmentation requires not only application of the individual bacterial strain or a bacterial consortium with the required degradation ability but also of the microorganisms able to survive in the adverse environment [14–17]. Poor survival of the inoculated microorganisms (usually bacteria) and low bioavailability of the hydrophobic carbon source are usually the main obstacles to the successful inoculum amendment. Moreover, the bottleneck for the successful catabolism of a recalcitrant hydrophobic compound is most often not the nature of the biochemical pathway for its degradation, but the overcoming of the endogenous and exogenous stress associated to the utilizing conditions. Although many bacteria have ability to metabolize, e.g., PCBs, high concentrations of these chemicals act as environmental stress factor and inhibit cell survival and then ability to metabolize these pollutants. If bacterial strains wanted to survive, they had to develop efficient adaptation mechanisms in the adverse environment [18, 19].

For the purpose to select the degradation-effective and adverse environment-resistant bacterial strain from 11 environmental isolates, obtained from the PCB-historically contaminated sediment and identified using molecular-biological methods [20], our research was focused on the study of adaptation mechanisms and responses of bioaugmented bacteria during the biological treatment of water and sediment matrices contaminated with PCBs. Since PCBs are highly hydrophobic, they may efficiently cross cell membrane through free diffusion. The effects of PCBs, chlorobenzoic acids (CBAs, PCB-biodegradation end products), biphenyl, and terpenes (the potential inducers of PCB degradation) on bacterial cytoplasmic membrane were determined [10, 11, 15, 19]. Only the resistant bacteria that possess the appropriate enzymes may play a major role in bioremediation technologies.


**2.1. Soil texture, porosity and density**

The textural class is the first parameter that defines soil properties, and is determined by the relative percentage of the three major soil compounds: sand, silt and clay, defined by the respective particle diameter size.

Clayey grounds are characterized by particles of a diameter of less than 0.002 mm, constituted by flinty minerals with different capacities to inflate in the presence of water and to contract in dry conditions, forming cracks which are typical of vertisols. Clayey soils show a low water permeability and high plasticity, which can induce stagnation phenomena and root asphyxia in wet conditions; while in the dry state it has notable tenacity and cohesion.

Silty soils have elementary particles of greater dimensions than the clayey ones and, unlike these, they have greater difficulty in reaching a glomerular structure. The lower structural stability of silty soils causes a low macroporosity and a great bulk density, that determine conditions of low aeration, low permeability and water stagnation in the profile. This type of soil is subject to loosening conditions with greater facility in comparison to clayey soils. However, they have a greater tendency to pulverization in the dry state, and the formation of mud in the wet state.

Sandy soils have particles of a diameter between 2 and 0.02 mm; they are characterized by high permeability of rainwater, and fast mineralization of organic matter.

The Olive tree responds best to soil textures with balanced proportions of sand, silt and clay. Soils that are primarily sandy do not have good nutrient or water-holding capacities, but they do provide good aeration and olives do well, especially when water is available and the crop is properly fertilized to satisfy its mineral requirements. The soil should not contain too much clay to avoid limiting air circulation and to prevent soil management problems. The soil particles should aggregate in granules or crumbs to make the soil porous; this is ensured by sufficient quantities of organic matter and rational soil management to prevent compacting and erosion. Soil is composed of solid particles (mineral and organic matter) of different sizes, usually bound together into aggregates by organic matter, mineral oxides, and charged clay particles. The number and size of pores vary considerably among soils exhibiting different organic matter content, texture and structure and cultivation techniques have a great effect on bulk density and porosity: any management practice that increases organic matter will increase the granular structure of the soil, increase the pore space, and decrease the bulk density (Gisotti, 1988; Giordano, 1999; Hao et al., 2008).


**Abstract**

This Chapter seeks to look at people's perspectives on Covid 19, Fake news and the vaccination drive in South Africa. The Chapter looks at the advent of Covid 19 and the various government initiatives to combat the spread in South Africa. The Chapter will also seek to look at fake news and its impact on the populace as citizens try to make informed choices towards vaccinating against Covid 19 infections. The chapter will thus look at the information warfare and its attempt to influence or derail public action. Comments from study participants will be looked at as conclusions and recommendations are drawn.

**Keywords:** people's perspectives, Covid 19, fake news, vaccination, social media


**Table 1.**

inadequate knowledge of the environment and over exploitation of soil resources [25]. Sustainable crop production can be undermined or constrained by poor soil management as crop performance is highly related to soil quality.


*2.4.1 Dialysis-associated hypoxemia*

During dialysis arterial PaO2 decreases by 5–20 mmHg reaching nadir at 30–60 minutes then resolves within 60–120 minutes after discontinuation of dialysis. This is usually of no significance unless patient has pre-existing cardiopulmonary disease [14].

In high-risk patients, preventive measures include intradialytic use of oxygen, conventional bicarbonate dialysate, and biocompatible membranes. Further optimizing haematocrit and performing sequential isolated UF followed by HD may reduce the likelihood of hypoxemia.