Brain electrical activity obeys Benford's law

Engineering nonlinear epileptic biomarkers using deep learning and Benford's law

In this study, we designed two deep neural networks to encode 16 features for early seizure detection in intracranial EEG and compared them and their frequency responses to 16 widely used engineered metrics to interpret their properties: epileptogenicity index (EI), phase locked high gamma (PLHG), time and frequency domain Cho Gaines distance (TDCG, FDCG), relative band powers, and log absolute band powers (from alpha, beta, theta, delta, low gamma, and high gamma bands). The deep learning models were pretrained for seizure identification on the time and frequency domains of 1 s, single-channel clips of 127 seizures (from 25 different subjects) using "leave-one-out" (LOO) cross validation. Each neural network extracted unique feature spaces that were interpreted using spectral power modulations before being used to train a Random Forest Classifier (RFC) for seizure identification. The Gini Importance of each feature was calculated from the pretrained RFC, enabling the most significant features (MSFs) for each task to be identified. The MSFs were extracted to train another RFC for UPenn and Mayo Clinic's Seizure Detection Kaggle Challenge. They obtained an AUC score of 0.93, demonstrating a transferable method to identify and interpret biomarkers for seizure detection.

Characterizing Human Cell Types and Tissue Origin Using the Benford Law

Processing massive transcriptomic datasets in a meaningful manner requires novel, possibly interdisciplinary, approaches. One principle that can address this challenge is the Benford law (BL), which posits that the occurrence probability of a leading digit in a large numerical dataset decreases as its value increases. Here, we analyzed large single-cell and bulk RNA-seq datasets to test whether cell types and tissue origins can be differentiated based on the adherence of specific genes to the BL. Then, we used the Benford adherence scores of these genes as inputs to machine-learning algorithms and tested their separation accuracy. We found that genes selected based on their first-digit distributions can distinguish between cell types and tissue origins. Moreover, despite the simplicity of this novel feature-selection method, its separation accuracy is higher than that of the mean-expression level approach and is similar to that of the differential expression approach. Thus, the BL can be used to obtain biological insights from massive amounts of numerical genomics data—a capability that could be utilized in various biomedical applications, e.g., to resolve samples of unknown primary origin, identify possible sample contaminations, and provide insights into the molecular basis of cancer subtypes.

Generalized Benford's Law as a Lie Detector

The first significant (leftmost nonzero) digit of seemingly random numbers often appears to conform to a logarithmic distribution, with more 1s than 2s, more 2s than 3s, and so forth, a phenomenon known as Benford’s law. When humans try to produce random numbers, they often fail to conform to this distribution. This feature grounds the so-called Benford analysis, aiming at detecting fabricated data. A generalized Benford’s law (GBL), extending the classical Benford’s law, has been defined recently. In two studies, we provide some empirical support for the generalized Benford analysis, broadening the classical Benford analysis. We also conclude that familiarity with the numerical domain involved as well as cognitive effort only have a mild effect on the method’s accuracy and can hardly explain the positive results provided here.

Elucidating tissue specific genes using the Benford distribution

BACKGROUND

The RNA-seq technique is applied for the investigation of transcriptional behaviour. The reduction in sequencing costs has led to an unprecedented trove of gene expression data from diverse biological systems. Subsequently, principles from other disciplines such as the Benford law, which can be properly judged only in data-rich systems, can now be examined on this high-throughput transcriptomic information. The Benford law, states that in many count-rich datasets the distribution of the first significant digit is not uniform but rather logarithmic.

RESULTS

All tested digital gene expression datasets showed a Benford-like distribution when observing an entire gene set. This phenomenon was conserved in development and does not demonstrate tissue specificity. However, when obedience to the Benford law is calculated for individual expressed genes across thousands of cells, genes that best and least adhere to the Benford law are enriched with tissue specific or cell maintenance descriptors, respectively. Surprisingly, a positive correlation was found between the obedience a gene exhibits to the Benford law and its expression level, despite the former being calculated solely according to first digit frequency while totally ignoring the expression value itself. Nevertheless, genes with low expression that exhibit Benford behavior demonstrate tissue specific associations. These observations were extended to predict the likelihood of tissue specificity based on Benford behaviour in a supervised learning approach.

CONCLUSIONS

These results demonstrate the applicability and potential predictability of the Benford law for gleaning biological insight from simple count data.