Читать книгу Diatom Morphogenesis - Группа авторов - Страница 36

Lyapunov exponents enable the assessment of randomness with respect 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 where L is the number of sequence steps [2.26]. One way to test for randomness is to consider the non-deterministic behavior of the valve formation simulated system as a simple renewal process [2.35, 2.74, 2.115]. In a study of morphology and inheritance, Cyclotella meneghiniana offspring cell structure was analyzed for partial renewal with regard to the mother cell and complete renewal in terms of genetic or epigenetic inheritance [2.131], so that valve formation was considered to be a renewal process.

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 associated Lyapunov exponents are found via Lagrange multipliers partitioned on the function and given as a maximum entropy probability distribution as [2.65]. With constraint , maximum entropy with respect to Lyapunov exponents is [2.65, 2.67]. Over a probability distribution, , where is a positive Lyapunov exponent.

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 , where DKS entropy is the density matrix of first partial derivatives from KS entropy where the diagonal elements are probability values and tis the time step ofL. Using from Boltzmann entropy, Lyapunov exponents from KS entropy become and from Pesin’s identity [2.110], . When SK-S = 0, the rate of prior information loss is equal to the rate of new information created, and stability is evident [2.35]. As A and is no longer a constant when changes in initial conditions approach zero, and randomness occurs [2.35]. For a measure of randomness, , and .