Multiscale entropy analysis of retinal signals reveals reduced complexity in a mouse model of Alzheimer’s disease

The data analyzed in this work were described with more details in a previous publication of our group21, and thus we provide here only a brief description of how the recordings were obtained. The entropy/complexity analyses of \(\mu\)ERG signals presented here have not been the subject of previous analyses or publications, and thus we provide a more detailed description of the entropy tools used in this work.

Animal care and use

All experimental procedures followed protocols approved by the bioethics committee of Universidad de Santiago de Chile (report #457), following the international guidelines on animal handling and manipulation and the Chilean National Agency for Research and Development (ANID) bioethics and biosecurity standards. All experiments and methods were conducted in accordance with the ARRIVE guidelines. 5xFAD mice (Jackson laboratory, Bar Harbor, Maine, USA) and WT mice (B6SJLF1/J) were used in this study21. These animals were maintained in Universidad de Valparaíso, with a time routine of 12:12 hrs light/dark cycle, with controlled temperature, and water and food ad libitum. The animals were grouped into young subjects (2–3 months-old), age at which the animals start exhibiting amyloid-\(\beta\) brain aggregation, and adult subjects (6–7 months-old), which present thorough accumulation of plaques in the brain and behavior alterations42.

\(\mu\)ERG recordings from mice retina

The protocol for \(\mu\)ERG recordings was performed using a Multi-Electrode Array (MEA) (USB256, Multichannel Systems GmbH, Reutlingen, Germany) with 252 electrode channels and a sampling rate of 20 kHz, which recorded the responses of a small piece of the retina (\(n=\)13, 12, 11 and 11 for WT-young, 5xFAD-young, WT-adult and 5xFAD-adult, respectively, taken from 9, 9, 8 and 9 animals, respectively, with no more than two pieces of retina from the same animal). In brief, all the recordings were stored in a computer for offline analysis. Before the experiments, the animals were dark-adapted for 30 min, and then profoundly anesthetized with Isofluorane (Baxter, Deerfield, Illinois, USA) and euthanized. Eyes were removed and enucleated under dim red light, and eyecups were maintained in Ames medium (Sigma-Aldrich, San Luis, Missouri, USA) at 32 C and pH 7.4 and oxygenated with a mixture of 95% O2 & 5% CO2. Small pieces of the retina were positioned on an o-ring (MWCO-25000, Spectrumlabs, Rancho Dominguez, California, USA) with polylysine treatment (Product P4707, Sigma-Aldrich, San Luis, Missouri, USA).The retina was stimulated with different visual stimuli created on MATLAB software (Natick, Massachusetts, USA), projected on it using a LED projector (PB60G-JE, LG, Soul, South Korea), mounted in an inverted microscope (Eclipse T200, Nikon, Minato, Tokyo, Japan). The average radiance of each stimulus was 70 nW/mm\(^2\) (Newport Corporation, Irvine, California, USA) with a spectral content ranging between 460 and 520 nm (Ocean Optics Inc, Dunedin, Florida, USA). All stimuli thoroughly covered the piece of the retina.

We applied two visual stimulation protocols consisting of 21 repetitions each of: 1) a chirp stimulus (CS) protocol, and 2) a natural image (NI) protocol. The CS protocol consisted of an ON-light flash (3s) and an OFF-dark period (3s) followed by a sinusoidal light stimulus of increasing frequency (1 to 10 Hz) and fixed maximum intensity, and followed by a sinusoidal light stimulus of 1 Hz with increasing intensity. Each repetition of the CS protocol lasted \(\sim\) 35 s (\(\sim\) 12 min total). The NI protocol consisted of a sequence of images recorded in natural habitat and presented at a refresh rate of 60 fps21. Each repetition of the NI protocol lasted \(\sim\) 11 s (\(\sim\) 5 min total). In the experiments, we used a fotodiode (PDA100A-EC, Thorlabs, Newton, New Jersey, United States) to control the retinal illumination stimulation (Fig. 1).

Channels were selected using the Neuroexplorer software (Plexon, Inc, Dallas, TX, USA) and according to a quality criterion (QI)21, and selected channels were averaged to obtain a single \(\mu\)ERG signal for each piece of retina stimulated with the CS and NI protocols. The median number of selected channels (± 25–75% interquartile range) for the WT-young, 5xFAD-young, WT-adult and 5xFAD-adult groups was: 197.5 (± 64), 206 (± 116), 186 (± 78) and 203.5 (± 99), respectively. The averaged \(\mu\)ERG signal was LP-filtered at 30 Hz, then subsampled at 200 Hz, and for each stimulation protocol, the recorded \(\mu\)ERG signal was averaged over 21 repetitions. Only animals for which QI \(\le\) 0.45 were selected in this study, regardless of sex (Table 3).

Spectrum and coherence

We calculated the power spectral density (PSD) of signals for frequency-domain characterization by means of Welch’s method48. This method estimates PSD as an average of the periodogram of multiple overlapping segments, thereby reducing noise in PSD estimation at the cost of frequency resolution. We used a Hann window of size 256 and an overlap of 128. In addition, for the CS protocol, we calculated the coherence, \(C_{xy}\), between the chirp stimulus and the \(\mu\)ERG response by means of the cross-spectral density, \(P_{xy}\), and the PSD of each signal:

$$\begin{aligned} C_{xy} = \frac{|P_{xy} |^2}{P_x\cdot P_y} \end{aligned}$$


Here, \(P_{xy}\) can be viewed as the PSD of the cross-correlation signal, and it is estimated using Welch’s method. For both PSD and coherence calculation we used the Scipy Python library v1.6.2.

Multiscale entropy

In this study, we used the concept of entropy as a measure of signal uncertainty or disorder. The sample entropy49, SampEn, is a regularity statistics so that higher SampEn values are assigned to less predictable, more irregular time series. Given a time series X of length N, SampEn counts, for each subset sequence i of length m, the number of sequences of the same length that are at a distance smaller than r from the sequence i. Let this number be \(U_i^m(r)\), then:

$$\begin{aligned} \begin{aligned} SampEn(N,m,r) = \ln {\frac{\sum _{i=1}^{N-m}{U_i^{m} (r)}}{\sum _{i=1}^{N-m}{U_i^{m+1} (r)}}}\\ \end{aligned} \end{aligned}$$


Here, \(U_i^{m}(r)\) excludes the comparison of sequence i with itself. SampEn overcomes some difficulties of other entropy measures, namely, the need of many samples, and sensitivity to noise. SampEn resembles the so-called approximate entropy, ApEn, but it is a less biased statistic since, differently to ApEn, it eliminates the self-matching in the counting process. However, SampEn shows high sensitivity to parameter selection since, when comparing a pair of subsets, a Heaviside function is applied as a similarity metric after taking the distance – the contribution of such a pair is 1 if the distance between the subsets is less than r, and 0 otherwise. In contrast, the fuzzy entropy (FuzzyEn)36 uses a fuzzy degree of similarity, \(D_{ij}^m\), which instead of being binary, depends on the distance between the pairs, \(d_{ij}^m\), through a continuous, concave function, typically quadratic exponential:

$$\begin{aligned} \begin{aligned} D_{ij}^m = \exp ( -(d_{ij}^m/r))^2. \end{aligned} \end{aligned}$$


The MSE method was proposed to account for the multiple time scales present in physiological processes22,37. Under this framework, a metric of entropy, e.g., the FuzzyEn, is calculated for the original time series, i.e., time scale 1, and for coarse-grained time series constructed by averaging as many samples as the current scale, in non-overlapping windows. For example, for time scale 2, the corresponding coarse-grained series is of length N/2, and each sample corresponds to the average of two consecutive samples of the original time series. This coarse-graining procedure is illustrated in Fig. 6 for two types of noise and for an electrophysiological signal. The coarse-grained time series appears less irregular for larger scales due to averaging for uniform noise. However, pink noise (a complex signal example) and the example \(\mu\)ERG signal exhibit a degree of irregularity that appears to persist across all scales.

Figure 6
figure 6

Illustration of the multiscale entropy (MSE) procedure. (a) Effect of the coarse-graining procedure on uniform noise (left), pink noise (middle) and an example \(\mu\)ERG signal (right). Original signals (scale 1, top) were low-pass filtered at 30 Hz (sampling frequency 200 Hz), and the coarse-grained signals were calculated for four different scales: 10, 20, 30 and 40. The irregularity of the uniform noise signal decays for scale \(>10\), whereas for the other two signals, the main irregular features persist across all scales, reflecting a more complex underlying structure. (b) MSE curves calculated for the signals in (a). The entropy for the scales shown in (a) are shaded. Note that for scales \(>10\), the entropy of uniform noise decays monotonically, whereas for pink noise and the \(\mu\)ERG signal, it increases.

Note that when coarse-graining a time series for a given scale \(\tau\), the resulting reduced series depends on the choice of the starting position for the initial window, so that there are \(\tau\) possible coarse-grained resulting series. In the composite MSE (CMSE), the entropies of all these coarse-grained time series corresponding to scale \(\tau\) are calculated and then averaged to obtain a single entropy value for \(\tau\). This procedure ensures more reliable results for larger scales50. Further, in the refined CMSE (RCMSE), the logarithm for entropy calculation is taken after computing the number of similar sequences, which reduces the probability of obtaining undefined entropy and increases the accuracy of entropy estimation51. In this study, we calculated the RCMSE of \(\mu\)ERG responses with the FuzzyEn using the Neurokit2 toolbox for Python52. For the NI protocol, the shortest one, each \(\mu\)ERG response comprised 2, 200 samples. Since the FuzzyEn demonstrated strong consistency in entropy estimation with a time series of 50 samples or greater36, we calculated the RCMSE up to scale \(2200 / 50 = 44\). For FuzzyEn calculation, we used \(m=2\) and \(r=0.2\).


More complex signals have consistently higher entropy values across different time scales22,37,53, and accordingly, the area under the MSE curve has been used to quantify complexity27,31,37. We calculated a complexity index, \(C_i\), as the cumulative sum of the RCMSE for scales \(20-44\), noting that (1) at the time scale of 20 and sampling frequency of 200 Hz, frequency components should concentrate to \(\le 10\) Hz, (2) \(\mu\)ERG responses strongly attenuate for sinusoidal light stimuli \(>10\) Hz11, and (3) coherence for the CS protocol decayed for frequencies of \(\sim\) 10 Hz and above.

Statistical analysis

Given the size of the datasets, no assumptions of data normality were made, and therefore we report median and interquartile range throughout, and we performed nonparametric statistical tests. For the complexity index calculated for the four groups, WT-young (\(n=13\)), WT-adult (\(n=11\)), 5xFAD-young (\(n=12\)) and 5xFAD-adult (\(n=11\)), we performed group-to-group comparisons by means of the Mann–Whitney U rank test provided with the Scipy Python library v1.6.2. Significance level was set at \(\alpha = 0.05\).

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