# Entropy-Complexity Characterization of Brain Development in Chickens

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## Abstract

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**PACS classifications:**02.50.-r; 05.45. Tp;87.19.La

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Methods

_{max}= −10 wavelet resolution levels with a spline cubic mother wavelet. (2) After that, a cleaning and stationary signal is obtained by reconstruction (inverse wavelet transform) using the resolution wavelet levels corresponding to the frequency range 0.5–32.0 Hz. (3) The wavelet frequency bands at which the saccadic movement frequency appear, as well as their time localization were identified. The corresponding wavelet coefficients were reduced so that they present a contribution below the noise-signal level, and the corresponding frequency band is reconstructed. (4) Finally, the complete signal is obtained by superposition of the all wavelet reconstructed frequency bands [25].

#### 2.2. Shannon Entropy, Fisher Information Measure and MPR Statistical Complexity

_{t}; t = 1, ···, M}, a set of M measures of the observable and the associated pdf, given by P ≡{p

_{j}; j = 1, ···, N} with ${\sum}_{j=1}^{N}{p}_{j}=1$, and N the number of possible states of the system under study, the Shannon’s logarithmic information measure (Shannon entropy) [26] is defined by:

_{j}, will actually take place. Our knowledge of the underlying process, described by the probability distribution, is maximal in this instance. In contrast, this knowledge is commonly minimal for a uniform distribution P

_{e}= {p

_{j}= 1/N, ∀ j = 1, ···, N}. The Shannon entropy S is a measure of “global character” that is not too sensitive to strong changes in the pdf taking place in a small region. Such is not the case with the Fisher information measure [27,28]:

^{2}, then:

_{max}. On the other hand, when the system under study lies in a very disordered state, one gets an almost flat pdf and S ~ S

_{max}, while F ~ 0. Of course, S

_{max}and F

_{max}are, respectively, the maximum values for the Shannon entropy and Fisher information measure. One can state that the general behavior of the Fisher information measure is opposite to that of the Shannon entropy [34].

_{max}= S[P

_{e}] = lnN, (0 ≤ H

_{S}≤ 1) and the disequilibrium

_{J}defined in terms of the Jensen–Shannon divergence. That is,

_{0}, a normalization constant (0 ≤

_{J}≤ 1), are equal to the inverse of the maximum possible value of [P, P

_{e}]. This value is obtained when one of the components of the pdf, P, say p

_{m}, is equal to one and the remaining p

_{j}are equal to zero. The Jensen–Shannon divergence, which quantifies the difference between two (or more) probability distributions, is especially useful to compare the symbolic composition between different sequences [39]. Note that the above introduced statistical complexity depends on two different probability distributions, the one associated with the system under analysis, P, and the uniform distribution, P

_{e}. Furthermore, it was shown that for a given value of H

_{S}, the range of possible C

_{JS}values varies between a minimum C

_{min}and a maximum C

_{max}, restricting the possible values of the statistical complexity in a given entropy-complexity plane [40]. Thus, it is clear that important additional information related to the correlational structure between the components of the physical system is provided by evaluating the statistical complexity measure.

#### 2.3. The Bandt–Pompe Approach to the pdf Determination

_{t}< x

_{t}

_{+}

_{k}should not depend on t [1]).

_{t}; t = 1, ···, M} with embedding dimension D > 1 (D ∈ ℕ) and embedding time delay τ (τ ∈ ℕ). We are interested in “ordinal patterns” of order (length) D generated by (s) ↦ (x

_{s}

_{− (}

_{D}

_{−1)}

_{τ}, x

_{s}

_{− (}

_{D}

_{−2)}

_{τ}, ···, x

_{s}

_{−}

_{τ},x

_{s}), which assigns to each time s the D-dimensional vector of values at times s, s − τ, ···, s − (D − 1)τ. Clearly, the greater the D–value, the more information on the past is incorporated into our vectors. By “ordinal pattern” related to the time (s), we mean the permutation π = (r

_{0}, r

_{1}, ···, r

_{D}

_{−1}) of [0, 1, ···,D − 1] defined by x

_{s}

_{−}

_{rD}

_{−1}

_{τ}≤ x

_{s}

_{−}

_{rD}

_{−2}

_{τ}≤ ··· ≤ x

_{s}

_{−}

_{r}

_{1}

_{τ}≤ x

_{s}

_{−}

_{r}

_{0τ}. In order to get a unique result, we set r

_{i}< r

_{i}

_{−1}if x

_{s}

_{−}

_{ri}= x

_{s}

_{−}

_{ri}

_{−1}. This is justified if the values of x

_{t}have a continuous distribution, so that equal values are very unusual. Thus, for all the D! possible permutations π of order D, their associated relative frequencies can be naturally computed by the number of times this particular order sequence is found in the time series divided by the total number of sequences.

_{i}must be seriously taken into account in evaluating the sum in Equation (4). The pertinent numerator can be regarded as a kind of “distance” between two contiguous probabilities. Thus, a different ordering of the pertinent summands would lead to a different Fisher information value. In fact, if we have a discrete pdf given by P = {p

_{i}, i = 1, ···,N}, we will have N! possibilities for the i-ordering.

## 3. Results and Discussion

_{max}and C

_{min}, respectively, as functions of the normalized Shannon entropy [40]. A closer look reveals that our results are in the chaotic attractors zone, corresponding to the localization of a dissipative chaotic system. As can be appreciated from the contrast of our results with the one being shown in [42], in which the Bandt–Pompe pdf was evaluated considering D = 6 (pattern length), τ = 1 (time lag), our results correspond to the localization of a chaotic dissipative system: the Tinkerbell map X-component [42].

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Normalized permutation Shannon Entropy versus the maturation week (average values and standard deviation estimated considering the 24 chickens). The used pdf Bandt–Pompe parameters are D = 6 and τ = 1. (A) Left frontal electrode. (B) Right frontal electrode. (C) Left posterior electrode. (D) Right posterior electrode.

**Figure 2.**Permutation Fisher information versus the maturation week (average values and standard deviation estimated considering the 24 chickens). The used pdf Bandt–Pompe parameters are D = 6 and τ = 1. (A) Left frontal electrode. (B) Right frontal electrode. (C) Left posterior electrode. (D) Right posterior electrode.

**Figure 3.**MPRpermutation statistical complexity versus the maturation week (average values and standard deviation estimated considering the 24 chickens). The used pdf Bandt–Pompe parameters are D = 6 and τ = 1. (A) Left frontal electrode. (B) Right frontal electrode. (C) Left posterior electrode. (D) Right posterior electrode.

**Figure 4.**Causality entropy-complexity plane H × C: localization of the mean values. We used an embedding dimension of D = 6 and a time lag of τ = 1 to estimate the pdf associated with the Bandt–Pompe methodology. (A) Left frontal electrode. (B) Right frontal electrode. (C) Left posterior electrode. (D) Right posterior electrode.

**Figure 5.**Localization in the causality entropy-complexity plane of the EEGs results presented in this work. Note that our results are localized in a chaotic dissipative behavior zone [42].

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Montani, F.; Rosso, O.A.
Entropy-Complexity Characterization of Brain Development in Chickens. *Entropy* **2014**, *16*, 4677-4692.
https://doi.org/10.3390/e16084677

**AMA Style**

Montani F, Rosso OA.
Entropy-Complexity Characterization of Brain Development in Chickens. *Entropy*. 2014; 16(8):4677-4692.
https://doi.org/10.3390/e16084677

**Chicago/Turabian Style**

Montani, Fernando, and Osvaldo A Rosso.
2014. "Entropy-Complexity Characterization of Brain Development in Chickens" *Entropy* 16, no. 8: 4677-4692.
https://doi.org/10.3390/e16084677