On the validation of Newcomb-Benford law and Weibull distribution in neuromuscular transmission
OOn the validation of Newcomb-Benford law and Weibulldistribution in neuromuscular transmission
A J da Silva a , ∗ , S Floquet b , D O C Santos c and R F Lima d a Instituto de Humanidades, Artes e Ciências, Universidade Federal do Sul da Bahia, , Itabuna, Bahia. Brazil b Colegiado de Engenharia Civil, Universidade Federal do Vale do São Francisco, , Juazeiro, Bahia, Brazil c Departamento de Fisiologia e Farmacologia, Faculdade de Medicina, Universidade Federal do Ceará, , Fortaleza, Ceará, Brazil
A R T I C L E I N F O
Keywords :Neuromuscular TransmissionElectrophysiologyNewcomb-Benford LawWeibull distribution
A B S T R A C T
The neuromuscular junction represents a relevant substrate for revealing important biophysicalmechanisms of synaptic transmission. In this context, calcium ions are important in the synapsemachinery, providing the nervous impulse transmission to the muscle fiber. In this work, we care-fully investigated whether intervals of spontaneous electrical activity, recorded in seven differentcalcium concentrations, conform Newcomb-Benford law. Our analysis revealed that electricaldischarge of neuromuscular junction obeys the expected values for Newcomb-Benford law forfirst and second digits, while first-two digits do not perfectly follows the law. We next examinedprevious theoretical studies, establishing a relation between the law and lognormal and Weibulldistributions. We showed that Weibull distribution is more appropriate to fit the intervals ascompared to lognormal distribution. Altogether, the present findings strongly suggest that spon-taneous activity is a base-scale invariant phenomenon.
1. Introduction
The neuromuscular junction (NMJ) is responsible for communicating electrical impulses from the motor neuron tothe skeletal muscle, yielding muscular contraction [1]. The terminal formed by the NMJ constitute a well-studied caseof chemical synapse. Facility of tissue extraction and stereotyped electrical response of nervous activity, representsome of the advantages of using NMJ in biophysical research.Bernard Katz led most of the pioneering work on the mechanistic basis of neuromuscular transmission [2]. Makinguse of electrophysiological recordings, Katz and Fatt discovered spontaneous small subthreshold depolarization, calledminiature end-plate potential (MEPP) [3]. These peculiar signals were further interpreted as due to a single vesiclefusion with the membrane terminal, configuring the vesicular hypothesis. Most importantly, NMJ emerged as an at-tractive substrate for combining mathematical modelling with empirical protocols. This enabled to elucidate severalmechanisms involved in neurotransmitter transmission. For instance, studies revealed that MEPPs are no longer con-stant in size or temporal distribution. These investigations associated MEPPs occurrence as governed by Gaussian andPoisson statistics [4]. Curiously, Katz already had attempted to point out weaknesses in the Poissonian predictions. Infact, Poisson and Gaussian models, commonly used to access the quantal nature of neurosecretion, require uniformityand stationarity assumptions [5]. However, to take into account such noticeable deviation from the early studies, amore general mathematical structure for explaining MEPP firing dynamics was developed. This approach allowed toreveal scale invariance or fractality embedded in several NMJ preparations [6, 7, 8].In a previous work, we showed the existence of long-range correlations associated to MEPP discharge at the NMJof mouse diaphragms [9]. We demonstrated that q -Gaussian distributions are more accurate in describing MEPPamplitudes as compared to the Gaussian function. These results suggest that spontaneous secretion of neurotransmittersexhibits scale invariance and long-range correlations. Furthermore, in two independent studies, Robinson and Van derKloot used Gamma, Weibull and lognormal functions to examine MEPP amplitudes [10, 11]. In addition, two reportsanalysing chemical synapses from the brain, concluded that quantal statistics can be better understood by presumingPascal and Weibull distributions, respectively [5, 12, 13]. On the other hand, other authors also advocate that thelognormal distribution is appropriate to study interbursts harvested from brain synapses [14, 15]. However, to ourknowledge, Weibull statistics has not been applied to analyse MEPP intervals. ∗ Corresponding author [email protected],[email protected] (A.J.d. Silva)
ORCID (s):
A J da Silva et al.:
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Page 1 of 17 a r X i v : . [ q - b i o . S C ] F e b alidation of Newcomb-Benford law and Weibull distribution at NMJ In living organisms, ions are responsible for mediating several physiological functions. Among many ionic species,one can highlight potassium, which is responsible for the resting membrane potential. In addition, sodium is crucialfor triggering the action potential generation, while calcium ions ( 𝐶𝑎 ) deserve attention due to their vital function intriggering the neurotransmission, also acting as a second messenger in molecular signalling [16]. Since the discoveryof MEPPs, systematic research has revealed how extracellular calcium concentration ( [ 𝐶𝑎 ] 𝑜 ) modify their frequency.Yet, after invade the nerve terminal 𝐶𝑎 from extracellular ambient interacts with proteins within the synaptic terminalthat are responsible for exocytosis. Manipulation of [ 𝐶𝑎 ] 𝑜 also allows the modulation of the degree of neuronalplasticity as a function of vesicular dynamics. Thus, the experimental protocol assuming [ 𝐶𝑎 ] 𝑜 modulation is asuitable manner to verify if machinery behind the MEPP time series follows the first digit phenomenon, also to studyif Weibull or longonormal distributions are a more appropriate approach to adjust the MEPP intvervals.The anomalous numbers law was pioneered and documented by the polymath Simon Newcomb in 1881, but onlyafter 57 years were his observations revisited and popularised thanks to Frank Benford, who analysed different typesof data [17, 18]. This intriguing numerical phenomenon is now known as Newcomb-Benford law (NBL). This coun-terintuitive law is among the several power or scaling laws found in many physical systems. In mathematical terms,the probabilities for occurrences of first digit are inferred by: 𝑃 ( 𝐷 = 𝑑 ) = log ( 𝑑 ) , 𝑑 ∈ {1 , , ..., (1)It is important to mention that second digit analysis has also been shown to be relevant in NBL validation. Forexample, Diekmann successfully identified that articles published in the American Journal of Sociology are well de-scribed by taking the second digit into account [19]. Also, Mebane argued that this digit is relevant for detection ofelection frauds. Probabilities for the appearance of a second digit are quantified by: 𝑃 ( 𝐷 = 𝑑 ) = ∑ 𝑑 =1 log ( 𝑑 𝑑 ) , 𝑑 ∈ {0 , , ..., (2)Although not frequently used, first-two digits analysis was applied by Nigrini and Miller to investigate accountspayable data and hydrometric statistics [20, 21]. Its functional form is presented below: 𝑃 ( 𝐷 𝐷 = 𝑑 𝑑 ) = log ( 𝑑 𝑑 ) , 𝑑 𝑑 ∈ {10 , , ..., (3)Beyond the heuristic formulation, rigorous mathematical descriptions have been proposed to explain why certaindata conforms to the first digit phenomenon. For instance, Pinkham argued that if NBL obeys some universal distribu-tion, then this law should be scale invariant to the units chosen [22]. In Physiology, relationships that depend on spatialinvariance can have profound morphological implications, being documented in heart, lung and brain [23, 24, 25, 26].In these schemes, if a certain data follows NBL, it must exhibit a base invariance profile [27]. Finally, Hill contributedwith a rigorous statistical pillar, inserting the law as a branch of modern probability theory, showing that NBL is relatedto scale invariance [27]. In this sense, investigations have examined how closely a dynamical system fulfils NBL andhow this law is remarkable for distinguishing noise from chaos [28, 29].Subsequent studies also showed a relationship between NBL and lognormal and Weibull distributions. Accordingto Rodriguez, given certain conditions, datasets that conform to NBL also exhibit adjustments with a lognormal dis-tribution [30]. On the other hand, Cuff et al. pointed out that given particular conditions, NBL is close to the Weibulldistribution [31]. Motivated by these authors and by the lack of studies examining the statistics of MEPP intervals,we decide to assume both functions in our work. Among many confirmations in different fields, NBL was verifiedin physical constants, number of cells in colonies of the cyanobacterium, alpha decay half-lives, and fraud-detection[32, 33, 34, 35]. It is important mention that NBL is the only distribution that is not derived from stationary processes.This observation is particularly suitable in time series analysis from physiological data [28]. In Physiology, NBL con-firmation involves dynamical transitions in cardiac models and brain electrical activity [36, 37]. Nevertheless, eventhough this law has been attested in different biological systems, it remains to be verified at the synaptic level. Thus, the A J da Silva et al.:
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Table 1
Electrophysiological parameters obtained in the recordings for each [ 𝐶𝑎 ] 𝑜 . [ 𝐶𝑎 ] 𝑜 Number of Number Membrane Frequency(mM) neurojunctions (n) MEPPS (N) potential (mV) ( MEPP/s) . .
14 ± 763 .
88 59 .
08 ± 9 .
85 0 .
35 ± 0 . . .
82 ± 1063 .
11 −66 .
26 ± 12 .
31 0 .
44 ± 0 . . .
07 ± 810 .
16 −64 .
15 ± 6 .
34 0 .
88 ± 0 . . .
14 ± 242 .
36 −60 .
73 ± 8 .
95 0 .
89 ± 0 . . .
71 ± 356 .
64 −67 .
10 ± 7 .
26 1 .
10 ± 0 . . .
15 ± 668 .
98 −70 .
90 ± 8 .
85 1 .
32 ± 0 . . .
00 ± 1063 .
11 −71 .
44 ± 13 .
85 2 .
10 ± 1 . primary goal of this work is to investigate if MEPP time series follows the numerical predictions established by NBL.To rigorously address this issue, physiological [ 𝐶𝑎 ] 𝑜 , such as values above and below the physiological level, willbe assumed in order to modulate the MEPP frequency. In this sense, this strategy also gives rise to the opportunity: (a)assess if NBL is valid only in the physiological concentration of [ 𝐶𝑎 ] 𝑜 or if it is independent of [ 𝐶𝑎 ] 𝑜 ; (b) checkhow robust is the conformity as function of the ionic manipulation; (c) test the hypotheses of other authors, stating thatclassical NBL is more frequently realized in nature; (d) verify if MEPP time series conform to NBL, then demonstratewhich distribution, Weibull or the lognormal, is the most appropriate statistics to investigate the intervals.
2. Materials and Methods
The experimental procedures here adopted were approved by the Animal Research Committee (CEUA - UFC, pro-tocol 130/2017). Roughly speaking, adult mice were euthanized by cervical dislocation, diaphragms were extractedand inserted into a physiological artificial fluid (Ringer solution). The following successive [ 𝐶𝑎 ] 𝑜 were used (mM):0.6, 1.2, 1.8, 2.4, 4.8, 10 and 15. The external composition for [ 𝐶𝑎 ] 𝑜 = (0 . . mM contained the followingconcentrations (mM): NaCl (137), NaHCO (26), KCl (5), NaH PO (1.2), glucose (10), and MgCl (1.3). The ex-ternal solution for [ 𝐶𝑎 ] 𝑜 = (10 − 15) mM contained (mM): NaCl (137), NaHCO (12), KCl (5), NaH PO (0.3),glucose (10), and MgCl (1.3). In both solutions pH was adjusted to 7.4 after gassing with 95% O - 5% CO . Tis-sues were left bathing in the solution for 30 minutes before the electrophysiological recordings began, minimizing themechanic trauma of their extraction. Next, tissues were gently transferred to a recording chamber continuously bathedwith artificial solution at 𝑇 = 24 ± 1 ◦ 𝐶 . Standard intracellular recording technique was performed to monitor thefrequency of spontaneous MEPP by inserting a micropipette at the chosen muscle fiber. We employed borosilicateglass microelectrodes with resistances of 8 - 15 M Ω when filled with 3 M KCl in electrophysiological acquisitions.Strathclyde Electrophysiology Software (John Dempster, University of Strathclyde), R Language [38], Origin (Origin-Lab, Northampton, MA), and MATLAB (The MathWorks, Inc., Natick, MA) were employed for electrophysiologicalacquisition and data analysis.The experimental paradigm used here afforded a rigorous analysis, where 125,565 MEPP intervals were collectedfrom 78 experiments, with the interval number varying between 213-4123 events. Table 1 brings a statistical summarytaking into account each concentration. Figure 1 shows a representative segment from a typical recording in three [ 𝐶𝑎 ] 𝑜 . The statistical resume given by figure 2 attests that on average, when [ 𝐶𝑎 ] 𝑜 increases there is a rising inthe MEPP discharge rate. In summary,beyond confirming results obtained in other studies, these control experimentsguarantee a reliable analysis for all subsequent work [39]. Despite displaying asymmetric distributions, a number of phenomena, apparently do not follow NBL. Examplescorroborating observations include seismic activity, distribution of the Discrete Cosine Transform, quantized JPEGcoefficients, and cognition experiments [40, 41, 42]. To overcome this problem, among the generalizations proposed byseveral authors, one can highlight the theoretical description introduced by Pietronero et al. , mathematically described
A J da Silva et al.:
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Amplitude (mV)
T i m e ( m s )
Amplitude (mV) [ C a + 2 ] o = 1 . 8 m M [ C a + 2 ] o = 1 5 m M T i m e ( m s )
Amplitude (mV)
Figure 1:
Intervals of electrophysiological recordings showing MEPPs for three different ionic concentrations ( [ 𝐶𝑎 ] 𝑜 =0.6, 1.8, 15 mM ). as below: 𝑃 ( 𝑛 ) = ∫ 𝑛 +1 𝑛 𝑁 − 𝛼 𝑑𝑁 (4)or by the differential equation: 𝑑𝑃 ( 𝑁 ) 𝑑𝑁 = 𝑁 − 𝛼 (5) A J da Silva et al.:
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C o n c e n t r a t i o n ( m M )
Frequency (MEPP/s)
Figure 2:
Statistical resume showing how [ 𝐶𝑎 ] 𝑜 influences MEPP frequency. Table 2
Expected proportions for NBL occurrence.
Digit 0 1 2 3 4 5 6 7 8 9Position in number 1st - 0.30103 0.17609 0.12494 0.09691 0.07918 0.06695 0.05799 0.05115 0.045762nd 0.11968 0.11389 0.10882 0.10433 0.10031 0.09668 0.09337 0.09035 0.08757 0.08500
Solving eq. (5) results in an 𝛼 -logarithm: 𝑃 𝛼 ( 𝑛 ) = 11 − 𝛼 [ ( 𝑛 + 1) (1− 𝛼 ) − 𝑛 (1− 𝛼 ) ] (6) = 𝑛 (1− 𝛼 ) ln 𝛼 ( 𝑛 + 1 𝑛 ) (7)According to eq. (7), defined as generalized NBL (gNBL), more frequent first digits than expected by NBL implies 𝛼 > , while 𝛼 < means a first digit frequency below the predicted percentage. As expected, when 𝛼 = 1 NBL isfully recovered. Taking 𝑛 = 𝑑 equation (7) is rewritten as: 𝑃 𝛼 ( 𝑑 ) = 𝑑 𝛼 ln 𝛼 ( 𝑑 + 1 𝑑 ) , (8)From the approach developed by Pietronero, it is also possible to obtain expressions for the second digit: 𝑃 𝛼 ( 𝑑 ) = ∑ 𝑑 =1 ( 𝑑 𝑑 ) 𝛼 ln 𝛼 ( 𝑑 𝑑 + 1 𝑑 𝑑 ) , (9)One could consider gNBL probability for the first-two digits as follows: 𝑃 𝛼 ( 𝑑 𝑑 ) = [( 𝑑 𝑑 + 1 ) 𝛼 − ( 𝑑 𝑑 ) 𝛼 ] 𝛼 , 𝑑 𝑑 ∈ {10 , , , ..., , (10)normalized for each 𝛼 value.Table 2 shows each proportional frequency for the first and the second digits. A J da Silva et al.:
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Table 3
Ranges of conformity for first, second, and first-two digits.
First Digit Second Digit First-two DigitsRange Conclusion Range Conclusion Range ConclusionMAD . to . Close conformity . to . Close conformity . to . Close conformity . to . Acceptable conformity . to . Acceptable conformity . to . Acceptable conformity . to . Marginal conformity . to . Marginal conformity . to . Marginal conformityabove . Nonconformity above . Nonconformity above . NonconformitySSD to Close conformity to Close conformity to Close conformity to Acceptable conformity to Acceptable conformity to Acceptable conformity to Marginal conformity to Marginal conformity to Marginal conformityabove
Nonconformity above Nonconformity above Nonconformity
An important issue in NBL analysis is to select a convenient statistical method for measuring statistical significancebetween empirical data and law predicted values. For instance, many investigators traditionally assume both 𝜒 and 𝑍 tests, although both manifest the "excess of power" problem. Thus, to circumvent this effect, Nigrini and Kossovskysuggested the mean absolute deviation (MAD) and sum of squares difference (SSD), respectively [35, 43]. We de-cided to use both methods in order to enhance the robustness of our statistical analysis, avoiding misinterpretations orsuperficial conclusions. In this framework, the use of both MAD and SSD enabled Slepkov et. al to uncover NBLconformity in answers to every end-of-chapter question in introductory physics and chemistry textbooks [44]. Thus,in mathematical form, MAD is defined as: 𝑀𝐴𝐷 = 𝑛 ∑ 𝑖 =1 | 𝐴𝑃 𝑖 − 𝐸𝑃 𝑖 | 𝑛 (11)where AP is the actual proportion and EP is the expect proportion. In addition, SSD is calculated with the followingexpression: 𝑆𝑆𝐷 = 𝑛 ∑ 𝑖 =1 ( 𝐴𝑃 𝑖 − 𝐸𝑃 𝑖 ) × 10 (12)Once again, AP is the actual proportion and EP is the expected proportion. Table 3 presents the conformance rangefor MAD and SSD analysis. Briefly speaking, Weibull and lognormal distributions were fitted to histograms of MEPP intervals applying themethod of the maximum likelihood for parameter estimation. The Weibull probability distribution function (pdf) isgiven by equation (13): 𝑓 𝑤𝑏 ( 𝑥 | 𝑟, 𝑏 ) = 𝑏𝑟 ( 𝑥𝑟 ) 𝑏 −1 exp(−( 𝑥 ∕ 𝑟 ) 𝑏 ) (13)For 𝑥 > , and 𝑟, 𝑏 ≥ . And the lognormal pdf is given by the following: 𝑓 𝑙𝑛 ( 𝑥 | 𝜇, 𝜎 ) = 1 𝑥𝜎 √ 𝜋 exp ( − (log( 𝑥 ) − 𝜇 ) 𝜎 ) (14)Where 𝑥, 𝜎 > .The likelihood function is the joint probability density, associated with each distribution function, of n identicallydistributed and independent observations, 𝑥 , ..., 𝑥 𝑛 , where n is also the sample size. This function gives the probabilitythat a set of observations is best described by a parameter set, 𝜃 , ..., 𝜃 𝑠 , from a pdf: A J da Silva et al.:
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Page 6 of 17alidation of Newcomb-Benford law and Weibull distribution at NMJ ( 𝜃 , ..., 𝜃 𝑠 | 𝐱 ) = 𝑓 ( 𝑥 , ..., 𝑥 𝑛 | 𝜃 , ..., 𝜃 𝑠 ) = 𝑛 ∏ 𝑖 =1 𝑓 ( 𝑥 𝑖 | 𝜃 , ..., 𝜃 𝑠 ) (15)The maximum likelihood estimator for a parameter 𝜃 𝑖 in the parameter set is given by: ̂𝜃 𝑖 ( 𝐱 ) = argmax 𝜃 𝑖 ( 𝜃 , ..., 𝜃 𝑠 | 𝐱 ) (16)or by maximizing the logarithm of the likelihood function (or log-likelihood function)[45].We applied this method to find the parameters from Weibull and lognormal distributions with the highest proba-bility of describing the data. After that, we selected the best model describing the data using, as criteria the AkaikeInformation Criterion: 𝐴𝐼𝐶 = −2 ln ( ( ̂𝜃 , ..., ̂𝜃 𝑠 | 𝐱 ) + 2 𝑘 (17)where k is the number of parameters of the model to be selected, and the Bayesian Information Criterion [46]: 𝐵𝐼𝐶 = −2 ln ( ( ̂𝜃 , ..., ̂𝜃 𝑠 | 𝐱 ) + 𝑘 ln( 𝑛 ) (18)
3. Results
The results summarized in tables 4 and 5, suggests that the MEPP interval generally follows NBL predictions. Weadopted the mean value as a reference value to conclude the existence of conformity. Taling the average value as refer-ence, experimental data studied with MAD and SSD, considering NBL for first and second digit, provided ambiguousresults given by different conformance degrees. When assuming MAD, for first-two digits results, both NBL e gNBLpointed to non-conformities, while in analysis performed with SSD all data achieved at least a marginal conformity.Figure 3 presents a summary for statistical analysis, considering [ 𝐶𝑎 ] 𝑜 at physiological levels, showing both NBLe gNBL statistical descriptions. A visual inspection enables to observe an excellent agreement of experimental dataand predicted values. According to our calculations, taking the first digit, gNBL gives a slightly better conformity for [ 𝐶𝑎 ] 𝑜 < . mM, while above this concentration (with exception of [ 𝐶𝑎 ] 𝑜 = 15 mM), both laws seem to performequivalently. Moreover, NBL and gNBL are equivalent for second digit analysis, with exception of [ 𝐶𝑎 ] 𝑜 = 2 . mM, which SSD pointed a better gNBL corformity. In summary, these findings show that MEPP time series followthe first and second digit phenomenon. On the other hand, experimental data followed the first-two digits proportionsonly when SSD was assumed or for [ 𝐶𝑎 ] 𝑜 ≥ mM. These results suggest that MEPP time series does not perfectlyconforms NBL or gNBL for first-two digits levels.The assumption of gNBL gave 𝛼 values associated to each concentration. On average, for [ 𝐶𝑎 ] 𝑜 = 0 . mM weobtained 𝛼 < , which can be attributed to smaller time series, since during low [ 𝐶𝑎 ] 𝑜 administration, the MEPPfiring is attenuated. In contrast, [ 𝐶𝑎 ] 𝑜 = 1 . mM provided 𝛼 = 1 , showing that considering the standard deviation, atleast in physiological concentrations, NBL analysis is fully accomplished. Furthermore, data for [ 𝐶𝑎 ] 𝑜 = (2 . . mM resulted 𝛼 ≠ for the three digits analysis, showing that gNBL is more appropriate in both concentrations. Finally, [ 𝐶𝑎 ] 𝑜 = (10−15) mM gave 𝛼 ≅ 1 , again demonstrating that NBL is suitable to investigate MEPP discharge at higherfrequencies. We analysed a set of 78 electrophysiological recordings, verifying which distribution, Weibull or lognormal, bestfitted those data. Applying both Akaike and Bayesian information criterion, we find that 76 recordings were best fittedby the Weibull distribution. The only exceptions were two recordings collected at [ 𝐶𝑎 ] 𝑜 = 1 . mM and [ 𝐶𝑎 ] 𝑜 = 15 mM, respectively. A table with statistical data from fittings is presented in the supplementary material. Figure 4 showsinterval histograms fitted by Weibull and lognormal distributions for three different [ 𝐶𝑎 ] 𝑜 . A J da Silva et al.:
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F i r s t d i g i t
Relative frequency
N B L p r e d i c t i o n E x p e r i m e n t a l d a t a [ C a ] o = 1 . 8 m M Relative frequency
F i r s t d i g i t [ C a ] o = 1 . 8 m M g N B L E x p e r i m e n t a l d a t a (a) (b) N B L p r e d i c t i o n E x p e r i m e n t a l d a t a [ C a ] o = 1 . 8 m M Relative frequency
S e c o n d d i g i t
Relative frequency
S e c o n d d i g i t [ C a ] o = 1 . 8 m M g N B L E x p e r i m e n t a l d a t a (c) (d) N B L p r e d i c t i o n E x p e r i m e n t a l d a t a [ C a ] o = 1 . 8 m M Relative frequency
F i r s t - t w o d i g i t s
Relative frequency
F i r s t - t w o d i g i t s [ C a ] o = 1 . 8 m M g N B L E x p e r i m e n t a l d a t a (e) (f) Figure 3:
Statistical summary (n=14) for physiological [ 𝐶𝑎 ] 𝑜 , using NBL and gNBL.
4. Discussion
In the present work, successful application of NBL and gNBL strengthen previous evidences showing that MEPPdynamics is governed by scale invariance laws. To prudently tackle this problem, [ 𝐶𝑎 ] 𝑜 was adjusted in sevenconcentrations, where conformity with first, second, and first-two digit pattern was investigated. Our results stronglysuggest that spontaneous secretion of neurotransmitters, in the mammalian diaphragm, obeys Benford’s law predictions A J da Silva et al.:
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Table 4
Statistical resume of conformity analysis, where the conformance level is concluded taking the mean value as reference.
Concentration(mM) First DigitNBL - MAD gNBL - MAD NBL - SSD gNBL - SSD
Value Conformity Value Conformity Value Conformity Value Conformity0.6 . . Marginal . . Acceptable . . Marginal . . Acceptable1.2 . . Marginal . . Acceptable . . Acceptable . . Acceptable1.8 . . Acceptable . . Acceptable . . Acceptable . . Acceptable2.4 . . Acceptable . . Acceptable . . Acceptable . . Acceptable4.8 . . Acceptable . . Acceptable . . Acceptable . . Acceptable10 . . Acceptable . . Acceptable . . Acceptable . . Acceptable15 . . Acceptable . . Close . . Acceptable . . AcceptableConcentration(mM) Second DigitNBL - MAD gNBL - MAD NBL - SSD gNBL - SSDValue Conformance Value Conformance Value Conformance Value Conformance0.6 . . Acceptable . . Acceptable . . Marginal . . Marginal1.2 . . Close . . Close . . Marginal . . Marginal1.8 . . Close . . Close . . Acceptable . . Acceptable2.4 . . Acceptable . . Acceptable . . Marginal . . Acceptable4.8 . . Close . . Close . . Acceptable . . Acceptable10 . . Close . . Close . . Acceptable . . Acceptable15 . . Close . . Close . . Acceptable . . AcceptableConcentration(mM) First-two DigitsNBL - MAD gNBL - MAD NBL - SSD gNBL - SSDValue Conformance Value Conformance Value Conformance Value Conformance0.6 . . Nonconformity . . Nonconformity . . Marginal . . Marginal1.2 . . Nonconformity . . Nonconformity . . Marginal . . Marginal1.8 . . Nonconformity . . Nonconformity . . Acceptable . . Acceptable2.4 . . Nonconformity . . Nonconformity . . Marginal . . Marginal4.8 . . Nonconformity . . Nonconformity . . Marginal . . Acceptable10 . . Acceptable . . Acceptable . . Acceptable . . Acceptable15 . . Marginal . . Marginal . . Acceptable . . Acceptable
Table 5
Statistical analysis for 𝛼 parameter for each [ 𝐶𝑎 ] 𝑜 . [ 𝐶𝑎 ] 𝑜 (mM) First Digit Second Digit First-two Digits0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in a very satisfactory manner. It is important to highlight that a visual inspection of all experimental data alreadyshowed the typical asymmetric shape, characterized by a long tail and heavy skewness toward the smaller digits.Despite this common feature shared by all data, curiously certain recordings did not reach statistical conformity. Thisraised a question about what physiological reason could be associated with these unexpected findings. It is well knownthat membrane potential fluctuations impose changes in MEPP rate. In fact, membrane depolarizations increase, whilehyperpolarization decreases the frequency of these events. Moreover, there is a 𝐶𝑎 oscillation within the synapticterminal, which is modulated by the extracellular content of this ion [47]. Such oscillations will reflect in a periodicalfluctuation of MEPP discharge as well, which could not only favours short intervals or, in NBL framework, smaller A J da Silva et al.:
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Page 9 of 17alidation of Newcomb-Benford law and Weibull distribution at NMJ N o r m a li z ed F r equen cy -4 LognormalWeibull [Ca ] o = 0.6 mM N o r m a li z ed F r equen cy -4 LognormalWeibull [Ca ] o = 1.8 mM (a) (b) N o r m a li z ed F r equen cy -3 LognormalWeibull [Ca ] o = 15 mM (c) Figure 4:
Weibull and lognormal fits for MEPP interval distributions for three representative [ 𝐶𝑎 ] 𝑜 . The adjustedparameters from the best fitted distributions (Weibull) are: r = 3125.813 and b = 1.070 (0.6 mM), r = 1910.256 and b= 1.022 (1.8 mM), r = 506.781 and b = 0.925 (15 mM). digits. In particular, the fluctuations described above also may be manifested in 𝛼 ≠ , attested by deviations observedfrom the classical NBL prediction. Although we indicate probable biological scenarios for observing nonconformityand deviations from the classical NBL in certain data, further studies are required to investigate this interpretation.Regulatory mechanisms for neurotransmitter releasing require a complex network of molecular cascades, beingregulated by extracellular ions. Exocytosis rate may be exacerbated by increasing [ 𝐶𝑎 ] 𝑜 in the Ringer solution.According to our in vitro research, we did not observe a relation between [ 𝐶𝑎 ] 𝑜 and conformity level. From theseobservations, one can formulate: could in vitro results be extrapolated to in vivo conditions? Indeed, an importantdebate in electrophysiology concerns to question if electrical activity extracted from in vitro recordings corresponds tothe in vivo conditions. Among the disadvantages we find that in vitro produce mechanical stress in tissues, introducingelectrical artefacts and biochemical changes. On the other hand, this category also has many advantages such as theisolation of the tissue or cell, allowing easy local pharmacological manipulation, elimination of afferent contributionfrom other body areas, among others. For these reasons, isolated diaphragm tissue is also amenable to NBL verifi-cation when pH and temperature are locally modified. Many reports documented that electrical activity of synapticterminals is dependent on thermal and acidification level [48]. Both parameters also govern MEPP frequency as well.An important physiological consequence of temperature elevation is the acceleration of the vesicular fusion rate, re-flected in the MEPP frequency increment. In the future, we expect to expand this study in order to address how both A J da Silva et al.:
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Page 10 of 17alidation of Newcomb-Benford law and Weibull distribution at NMJ physiological parameters may interfere in the NBL conformance.The gradual increment in [ 𝐶𝑎 ] 𝑜 evoked the expected elevation of MEPP frequency, allowing us to verify thevalidity of NBL/gNBL for different sample size intervals or physiological conditions. This procedure uncovered caseswhere the statistical analysis gave different levels of conformity or did not even indicate conformity. Thus, our workrevealed a rich statistical scenario. We had already pointed to a physiological substrate to explain the conformanceheterogeneity. In mathematical terms, a complementary answer may be achieved reasoning around gNBL. In general,independently of the digit, using gNBL gave more frequent conformity than NBL. Due to this heterogeneous scenario,one can advocate that the best strategy for studying MEPP time series consists in adopting both NBL and gNBL inconjunction with MAD and SSD tests. We also recommend to consider performing a first, second, and first two-digitscalculations in order to achieve a more careful conclusion.A relationship is reported between NBL and long range correlation phenomenon described in terms of the nonex-tensive theory proposed by Tsallis [49]. In this framework, Shao and Ma carried out a theoretical study associatingNBL in the nonextensive context. These authors stated that NBL confirmation in different systems is theoreticallyexpected, at least, in systems obeying nonextensive statistics. Relative to NMJ, Silva et al. showed that MEPP his-tograms are better understood when adjusted with long tail functions, for instance, a q-Gaussian distribution [9]. Stillaccording to this study, application of Detrended Fluctuation Analysis (DFA) also strengthened the scenario for theexistence of long-range correlations in MEPP intervals. Since DFA allows detection of scale-invariance embeddedin time series data, the present results merge with these previous reports. Yet, the relationship between NBL is wellstudied with other distribution functions such as lognormal and Weibull distributions [50]. Also motivated by theirutility in neurophysiology, we decide to perform simulations assuming both functions. Our experimental preparationand theoretical design confirmed a theoretical study showing that Weibull distribution is the more suitable functionthan the lognormal distribution.The findings also represent an important empirical support in favour of a close rela-tion between NBL and the Weibull function. Consequently, another important result, is that the Weibull distributioncharacterized the best statistical modelling for describing MEPP time series. Li offered an intrinsic plasticity model,demonstrating that the probability distribution of the neuronal firing is better explained using the Weibull distribution[51]. Along with this results, Weibull function also appeared as very useful statistics for investigating the neuronalfiring of primary olfactory system and single locus coeruleus neurons [52, 53]. Our results reinforce the importanceof Weibull statistics in the quantitative analysis in context of communication between nerve and muscle. Thus, a nextstep concerns to employ Weibull statistics to quantify the plasticity mechanisms at the diaphragm NMJ.Finally, Bormashenko asked why NBL is frequently observed in statistical data [54]. According to his view, likeNBL, many systems entropically governed are described by a logarithm dependence. This argument may explainwhy the first digit phenomenon is so frequently observed in different systems and conditions, including the resultshere described. It is worth mentioning that previous work, carried out in amphibian NMJ, also reported the intervalsdistribution described by a logarithmic behaviour [8]. Another intriguing question was also formulated by Lemonsand Kossovsky, which asked why there are more small things in the world than large things [55, 43]. In keeping withthese authors one can paraphrase: Why does short MEPP interval, given by the abundance of first digit, prevail amongthe other ones? In the NMJ diaphragm there are thousands of crowded vesicles, with a diameter of 50 nanometers,sharing a confined space. When combined, high density of vesicles and reduced spatial dimension, certainly favours ahigher likelihood for vesicle fusion, corroborating for short MEPP intervals existence. Thus, NBL/gNBL descriptionsrepresent an elegant methodology for assessing interesting features of the probabilistic nature of neurotransmission[56]. Furthermore, shorter intervals still represent a cellular mechanism to avoid receptor recruitment from the cellularmembrane. Indeed, Saitoe et al. showed that absence of the glutamate receptor in Drosophila is directly relatedto lacking spontaneous transmitter release. Strengthening this reflection, McKinney et al. also demonstrated thatminiature synaptic events are required in order to maintain dendritic spines in hippocampal slice. Prompted by thesefindings, one can hypothesize that base or scale invariance are important requisites in preventing the synapse collapse.Finally, the present results also reinforce that MEPP activity organizes itself in time as scale-invariant phenomenon.However, how exocytotic machinery can be orchestrated into a base/scale-invariant behaviour will be the focus offurther research.
5. Concluding remarks
To the best of our knowledge, this is the first work suggesting that spontaneous synaptic transmission obeys NBL.In addition, MEPP intervals showed conformity with the NBL independtly of [ 𝐶𝑎 ] 𝑜 . In this context, NBL remained A J da Silva et al.:
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Page 11 of 17alidation of Newcomb-Benford law and Weibull distribution at NMJ valid in hypercalcemia and hypocalcemia conditions. We next showed that, when compared to lognormal distribution,Weibull statistics is more appropriate to adjust MEPP intervals from diaphragm electrophysiological recordings. Wehope to extend the present research including both excitatory and inhibitory synapses in the brain. Moreover, it couldalso be relevant to examine a possible existence of the anomalous number phenomenon from NMJ of non-mammalianspecies and pathological tissues, such as during administration of toxins and drugs.
6. Acknowledgements
We are in debt to Alex Ely Kossovsky for his discussion and valuable help during the preparation of this work. Theauthors would like to thank the Multi-User Facility of Drug Research and Development Center of Federal Universityof Ceará for technical support. We also to thank Dr. Barbara Piechocinska for reading the manuscript.
CRediT authorship contribution statement
A J da Silva:
Conceptualization of this study, Experimental measurements, Data analysis, Original draft prepa-ration, Writing.
S Floquet:
Data analysis, Original draft preparation.
D O C Santos:
Data analysis, Original draftpreparation.
R F Lima:
Experimental measurements.
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Page 13 of 17 upplementary Information
On the validation of Newcomb-Benford law and Weibull distribution inneuromuscular transmission
A. J. da SilvaInstituto de Humanidades, Artes e CiˆenciasUniversidade Federal do Sul da BahiaItabuna, [email protected],[email protected]. FloquetColegiado de Engenharia CivilUniversidade Federal do Vale do S˜ao FranciscoJuazeiro, Bahia, BrazilD. O. C. SantosInstituto de Humanidades, Artes e CiˆenciasUniversidade Federal do Sul da BahiaItabuna, Bahia. BrazilR. F. LimaDepartamento de Fisiologia e FarmacologiaFaculdade de Medicina, Universidade Federal do Cear´aFortaleza, Cear´a, Brazil
October 22, 2019
The following table gives results regarding ample sizes, model selection criteria values(AIC - Akaike Information Criteria and BIC - Bayesian Information Criteria). It gives alsoadjusted model parameter values for the best fitted distribution.
Table 1: Statistical summary of data used in the paper.Concentration(mM) SampleSize AICWeibull AIClognormal BICWeibull BIClognormal BestDistribution Best DistributionParameters0.6 463 8354.20 8428.67 8362.47 8436.95 Weibull r=3125.813 b=1.0700.6 213 3879.31 3886.06 3886.03 3892.79 Weibull r=3330.704 b=1.0310.6 252 4501.81 4551.81 4508.87 4558.87 Weibull r=2772.104 b=1.008 Validation of Newcomb-Benford law and Weibull distribution at NMJA J da Silva et al.:
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Page 14 of 17 able 1: Statistical summary of data used in the paper.Concentration(mM) SampleSize AICWeibull AIClognormal BICWeibull BIClognormal BestDistribution Best DistributionParameters0.6 289 5092.28 5145.52 5099.61 5152.85 Weibull r=2512.852 b=1.0600.6 571 10471.90 10537.06 10480.59 10545.75 Weibull r=3357.899 b=0.8890.6 1620 29526.83 29747.62 29537.61 29758.40 Weibull r=3339.153 b=1.0040.6 1494 25364.14 25534.81 25374.76 25545.42 Weibull r=1788.477 b=1.0040.6 681 12533.68 12578.14 12542.73 12587.19 Weibull r=3469.084 b=0.8920.6 1019 17824.60 17940.18 17834.45 17950.04 Weibull r=2328.031 b=1.0200.6 624 11605.64 11704.70 11614.51 11713.57 Weibull r=4060.891 b=1.0300.6 1621 28769.18 29018.18 28779.96 29028.96 Weibull r=2637.961 b=1.0120.6 2432 42351.50 42716.07 42363.09 42727.67 Weibull r=2237.991 b=1.0170.6 2471 42945.34 43223.34 42956.97 43234.96 Weibull r=2170.007 b=0.9850.6 1050 20045.18 20223.53 20055.09 20233.45 Weibull r=5129.077 b=0.9981.2 775 12933.46 13006.33 12942.76 13015.63 Weibull r=1561.943 b=1.0281.2 585 10064.30 10104.34 10073.04 10113.09 Weibull r=1956.114 b=0.9551.2 244 4553.50 4595.41 4560.50 4602.41 Weibull r=4251.110 b=1.0761.2 473 7974.68 7937.22 7983.00 7945.53 lognormal µ =6.648 σ =1.3781.2 1835 32990.05 33236.22 33001.08 33247.25 Weibull r=2959.899 b=1.0111.2 2472 42957.54 43301.31 42969.17 43312.94 Weibull r=2198.702 b=1.0181.2 3468 57915.47 58369.68 57927.77 58381.98 Weibull r=1572.010 b=1.0251.2 2632 45410.54 45778.68 45422.29 45790.43 Weibull r=2044.036 b=0.9921.2 978 18056.48 18194.19 18066.25 18203.96 Weibull r=3741.418 b=0.9951.2 2369 40983.87 41328.65 40995.41 41340.19 Weibull r=2115.485 b=1.0181.2 1195 21235.47 21437.29 21245.64 21447.47 Weibull r=2602.688 b=0.9541.8 730 11882.86 11948.62 11892.04 11957.81 Weibull r=1245.394 b=0.9791.8 869 13905.27 14020.06 13914.80 14029.60 Weibull r=1123.562 b=1.0601.8 2181 30673.86 30776.52 30685.24 30787.89 Weibull r=429.317 b=1.0701.8 928 14439.41 14501.21 14449.08 14510.88 Weibull r=869.668 b=0.9781.8 2973 41789.38 41929.36 41801.37 41941.35 Weibull r=432.438 b=1.0941.8 642 10976.21 11056.26 10985.14 11065.19 Weibull r=1910.256 b=1.0221.8 1349 23613.66 23719.39 23624.07 23729.80 Weibull r=2308.445 b=0.9851.8 1162 19750.02 19865.06 19760.14 19875.18 Weibull r=1797.833 b=0.9951.8 1685 29734.03 29953.06 29744.89 29963.92 Weibull r=2461.072 b=0.9681.8 1009 18263.11 18362.41 18272.95 18372.24 Weibull r=3160.157 b=1.0241.8 2048 35343.02 35565.73 35354.27 35576.98 Weibull r=2074.024 b=1.0221.8 3060 50346.97 50673.32 50359.02 50685.38 Weibull r=1385.304 b=1.0181.8 1510 26980.77 27147.15 26991.41 27157.79 Weibull r=2798.198 b=1.0091.8 2436 41190.27 41449.69 41201.87 41461.28 Weibull r=1732.496 b=1.0082.4 602 10709.15 10798.61 10717.95 10807.41 Weibull r=2714.169 b=1.0352.4 498 8357.89 8402.87 8366.31 8411.29 Weibull r=1624.869 b=1.0132.4 624 10019.69 10082.47 10028.57 10091.35 Weibull r=1134.855 b=1.0202.4 671 11654.46 11733.78 11663.47 11742.80 Weibull r=2209.671 b=1.0442.4 1037 15891.29 15995.06 15901.18 16004.95 Weibull r=799.759 b=1.0552.4 883 13977.95 14036.58 13987.51 14046.14 Weibull r=1020.370 b=1.034 Validation of Newcomb-Benford law and Weibull distribution at NMJA J da Silva et al.:
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Page 15 of 17 able 1: Statistical summary of data used in the paper.Concentration(mM) SampleSize AICWeibull AIClognormal BICWeibull BIClognormal BestDistribution Best DistributionParameters2.4 1138 16833.20 16903.86 16843.27 16913.93 Weibull r=615.306 b=1.0634.8 1008 16735.31 16853.06 16745.14 16862.90 Weibull r=1525.433 b=1.0714.8 910 14376.92 14471.08 14386.55 14480.70 Weibull r=1026.688 b=1.0854.8 741 11871.27 11931.32 11880.49 11940.54 Weibull r=1107.344 b=1.0054.8 1864 27586.85 27690.66 27597.91 27701.72 Weibull r=609.204 b=1.0304.8 1021 16084.53 16170.36 16094.39 16180.21 Weibull r=964.116 b=0.9914.8 1116 17199.21 17293.53 17209.25 17303.56 Weibull r=825.880 b=1.0294.8 1122 17480.60 17598.72 17490.65 17608.76 Weibull r=913.554 b=1.06710.0 2757 41243.63 41506.01 41255.47 41517.86 Weibull r=668.122 b=1.05810.0 1824 27574.87 27751.12 27585.89 27762.14 Weibull r=705.565 b=1.00310.0 2048 32842.47 33024.05 32853.72 33035.30 Weibull r=1122.269 b=1.01410.0 3724 55012.21 55228.03 55024.66 55240.47 Weibull r=603.834 b=1.04010.0 2991 44266.03 44481.18 44278.04 44493.19 Weibull r=605.018 b=1.01410.0 2809 41915.24 42163.77 41927.12 42175.65 Weibull r=653.808 b=1.05210.0 2819 42048.05 42204.54 42059.94 42216.43 Weibull r=650.601 b=1.04710.0 2178 33611.51 33759.74 33622.88 33771.11 Weibull r=839.966 b=1.04110.0 1228 20352.49 20521.83 20362.71 20532.06 Weibull r=1480.594 b=1.03510.0 1511 24425.58 24576.64 24436.22 24587.28 Weibull r=1201.893 b=1.02410.0 2342 35473.26 35676.76 35484.78 35688.28 Weibull r=733.248 b=1.05710.0 2216 34118.73 34306.74 34130.13 34318.14 Weibull r=825.766 b=1.04310.0 2703 40540.11 40687.59 40551.91 40699.40 Weibull r=673.718 b=1.03315.0 3213 42608.54 42701.69 42620.69 42713.84 Weibull r=289.022 b=1.08215.0 2242 31343.30 31463.93 31354.73 31475.36 Weibull r=414.211 b=1.08415.0 2294 33283.42 33296.53 33294.89 33308.01 Weibull r=539.385 b=1.08115.0 2328 35617.21 35820.59 35628.71 35832.09 Weibull r=780.920 b=1.02715.0 445 7299.81 7358.86 7308.01 7367.06 Weibull r=1356.476 b=1.03515.0 1925 30183.29 30332.90 30194.42 30344.02 Weibull r=942.651 b=1.02315.0 3464 41886.88 41892.97 41899.19 41905.27 Weibull r=159.048 b=1.05215.0 2277 33071.01 33135.13 33082.47 33146.59 Weibull r=506.781 b=0.92515.0 4123 52549.98 52154.28 52562.63 52166.92 lognormal µ =4.723 σ =1.20115.0 1064 17080.36 17137.50 17090.30 17147.44 Weibull r=1101.935 b=0.95615.0 1517 24585.73 24659.27 24596.38 24669.92 Weibull r=1179.773 b=0.93515.0 1880 29561.57 29784.76 29572.65 29795.84 Weibull r=970.410 b=1.039 Validation of Newcomb-Benford law and Weibull distribution at NMJA J da Silva et al.: