to instability in a dynamical system [2.74, 2.118, 2.163]. While local unstable behavior on a state-by-state basis may be chaotic, the macro-state instability of valve formation may be inherently random at a global scale. Alternatively, there may be intermittent chaotic or random unstable states [2.74] during the morphogenetic process. Because of this, a test for randomness [2.34] is necessary.
Thus far, our dynamical system model has been analyzed at equilibrium. However, to detect possible random instability, we need to consider the possible non-equilibrium consequences of such behavior in this system. Non-equilibrium may be a long-term phenomenon in contrast to short-term steady-states present in a dynamical system, for example, as time oscillations create Turing effects of diffusion driven instability [2.6]. Toy models have been used to link time intervals over multiple levels enabling the application of equilibrium dynamics to non-equilibrium behavior [2.96]. They have great potential in modeling complicated biological processes [2.2]. Non-equilibrium dynamics can be assessed in terms of constrained information loss or an increase in entropy [2.65].
As entropy increases over longer and longer sequences for a given dynamical system, entropy blocks or sections that define the structure of these sequences do not characterize the entire system. What remains is randomness that is not taken into consideration and given as the source entropy rate
Maximizing the rate of variation in the entropy values during valve formation involves converting Boltzmann entropies of symmetry states to Kolmogorov-Sinai (KS) entropies based on probability [2.97]. KS entropies are time-averaged Shannon entropies over joint probability space [2.147] and are used to construct a PDF as a maximum entropy distribution [2.65, 2.137]. Randomness distributed over a probability distribution involves maximum entropy with regard to a PDF.
Evaluating entropy S using the PDF of associated probabilities involves first derivatives of S as entropy rates [2.26, 2.74] corresponding to bandwidth in the histogram used to determine the PDF [2.137]. For the expected value of the probabilities associated to
The sum of the positive Lyapunov exponents is a maximization of the possible states of KS entropy, and as such, KS entropy indicates degree of randomness [2.26].
The Lyapunov exponents for KS entropy of a sequence L(t) as a random function over a probability distribution for the αth probability [2.74] are
2.3 Results
2.3.1 Symmetry Analysis
For analysis, 151 centric diatom images (Figure 2.6a) were background masked, rotated, and overlain to calculate entropy values using the equation for Boltzmann entropy. From this, 94 external valves and 19 forming valves, along with Cyclotella meneghiniana valves: 12 replicates of SClemtbx1800, 13 abnormal valves, 6 forming valves, and 2 initial valves. Entropy filtering applied to the original images enabled detection of those structures that contribute to the measurement of symmetry (Figure 2.6b). Tilt and slant were calculated for each image. Tilt ranged from 0.725 to 0.785, and slant ranged from 1.392 to 1.496° for all images. Tilt correction was applied to each image (Figures 2.6c, d), and slant was corrected via rotation. All initial entropy values were tilt-corrected via tilt entropy to produce final entropy values.
For each image, final entropy (hereafter simply called “entropy”) values were plotted against the number of rotations, resulting in R2 = 0.0063, which indicates that number of rotations was not correlated to entropy values (Figure 2.7). Each entropy value was converted to symmetry using the equation for Boltzmann entropy. Entropy vs. symmetry was plotted for each image, and the result was a negative exponential relation between the two variables. An example is plotted for image ProvBay5_12lx450, Arachnoidiscus ehrenbergii and depicted in Figure 2.8. Measurement error was calculated as 0.000153 with 0.1068% bias.
The optimum number of rotations that gives the minimum entropy value was tested for those taxa with valve face features that were not easily divisible into equally spaced partitions. Image SClemtbx1800, Cyclotella meneghiniana, was tested with 3, 4, 5, 7, 13, 14, 15, 17, 18, 24, 27, 31, and 64 rotations, where 64 represents the actual number of marginal chambered striae in the image. Entropy values were plotted against number of rotations, and from a least-squares regression, no trend was discernable between the minimum entropy value and number of rotations (Figure 2.9). That is, an increasingly larger number of rotations used did not correlate to any improvement in obtaining a minimum entropy value (R2 = 0.1097). Fourteen rotations were used for taxa lacking distinguishing valve face features.
For all image entropies, a histogram and PDF was constructed and depicted a Gaussian distribution (Figure 2.10). The distribution has a slightly long left tail and is therefore slightly positively skewed. A CDF of all entropies was also constructed (Figure 2.11).
Average symmetry was determined and plotted for each taxon (Figure 2.12). A least-squares regression was performed that resulted in a best-fit curve of wtaxon = 0.9051 S + 15.654, and the coefficient of determination was 0.9474. Average taxon symmetry was determined and plotted for all external valves (Figure 2.13). A least-squares regression was performed that resulted in a best-fit curve of wtaxon = 0.9457S + 15.473, and the coefficient of determination was 0.9451. External and forming valve average symmetries were compared for ten taxa. From the bar graph, Asteromphalus heptactis, Asteromphalus imbricatus, Asteromphalus shadboltianus, Asteromphalus vanheurckii, Asterolampra marylandica, Cyclotella meneghiniana, and Glyphodiscus stellatus had higher external valve symmetry than forming valve symmetry (Figure 2.14). Triceratium