Ecology

Stability analysis of the deterministic modelSolving (left( x(t) times left( r_{p}x(t)^{(alpha )}left( 1-big (frac{x(t)}{K}big )^{beta }right) – nx(t)^{(delta )} right) right) =0), we obtain two stable and one unstable equilibrium points for the model. One stable equilibrium is trivial, i.e., (x(t)=0), another stable equilibrium point being the non-zero satisfying (left( r_{p}x(t)^{(alpha )}left( 1-big (frac{x(t)}{K}big )^{beta }right) – nx(t)^{(delta )} right) =0). Figure 1a shows three different equilibrium points of the model. In addition to the equilibrium, the model has two inflection points (Fig. 1a). At these inflection points the absolute growth rates are minimum and maximum. The density vs relative proliferation rate (RPR) profile of the model shows that the model can attain negative RPR for a positive cell density, suggesting that the model can portray the Allee phenomenon (Fig. 1b). Figure 1c,d portray the proliferation and decay phases, respectively through the model.Figure 1Growth dynamics of the proposed model: (a) Absolute proliferation rate (APR) profile considering (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99) and (delta =0.2); (b) RPR profiles for different n and other same constant model parameters; (c) Cell population survive for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99) and (delta =0.2) with the initial cell density 0.1; (d) The population goes to extinction for the initial cell density 0.06 with the same constant parameters.Full size imageThe solution of the deterministic model finally provides two theorems.
Theorem 1

(x^{*}approx K -Kleft( frac{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )-sqrt{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )^{2}-2 left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) nK^{delta }}}{left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) }right)) is the conditional MSSCD for the intercellular-interaction-induced proliferative cells. The conditional threshold density for cell-proliferation upon interaction is (x^{*}=K -Kleft( frac{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )+sqrt{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )^{2}-2 left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) nK^{delta }}}{left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) }right)) (proof is in the supplementary information).
Allee and cooperation models are the only extended logistic law other than our model to provide a threshold population size for growth or proliferation. Our proposed model is superior to the Allee and cooperation model as it can detect the conditional threshold cell density for proliferation and regulate the density by its different parameters. For example, One may reduce the conditional threshold density by either regulating the interaction between growth-inhibiting molecules and cells ((delta)) or reducing the inhibiting molecule concentration (n).The conditional MSSCD from Theorem 1 is lower than the carrying capacity of the conventional logistic model due to growth-inhibiting molecules; it provides the expected cell density during culture in a given environment. Theorem 1 also states the set of parameters to control the cell proliferation and get the desired density during such cell cultures. A further question arises knowing this set of parameters: which one of the parameters in the expression is crucial in terms of application purpose? Since the (r_{p}) is the constant proliferation rate for a given cell line, controlling the conditional MSSCD is not possible through (r_{p}). We simulate the distribution of conditional MSSCD for other parametric planes to answer this question. For this, we use the parameter values obtained from the data.

Theorem 2

The RPR is maximum at the cell density (x^{*}= K-Kleft( frac{r_{p}beta K^{alpha -1}+ndelta K^{delta -1}}{2r_{p}alpha beta K^{alpha -1}+r_{p}beta (beta -1)K^{alpha -1}+ndelta (delta -1)K^{delta -1}}right)) for the concave downward profile under the condition (r_{p}alpha (alpha -1){x^{*}}{}^{(alpha -2)}-frac{r_{p}}{K^{beta }}(alpha +beta )(alpha +beta -1){x^{*}}{}^{(alpha +beta -2)}-ndelta (delta -1){x^{*}}{}^{(delta -2)}n) (see the supplementary information). The cell population sustain with any positive initial cell density x(t) and try to stabilize at (x(t)= K(1-frac{n}{r_{p}})^frac{1}{beta }). Therefore, bimodality vanishes and unimodality is observed for the case (alpha =delta) (r_{p} >n). The RPR profile will be concave downward always with the maximum RPR value is at the inflection point (x(t)= K(frac{(r_{p}-n)alpha }{r_{p}(alpha +beta )})^frac{1}{beta }). The deterministic potential function in this case is (U(x)=-Big [(r_{p}-n)frac{x^{(alpha +2)}}{(alpha +2)}-frac{r_{p}}{K^{beta }}frac{x^{(alpha +beta +2)}}{(alpha +beta +2)} Big ]). The minima of this effective potential function will be at (x(t)= K(1-frac{n}{r_{p}})^frac{1}{beta }) which is the maximum stable cell density for (r_{p} >n).
Parameter estimationThe density-RPR and time-density fitting to the scratch assay datasets show a lower RSS for our model than the logistic one for each of the three seeding conditions. The estimated parameters from the RPR fitting through the grid-search are in Table 2. Although the RSS for the RPR fitting of the seeding 2 is very low, the data itself is too scattered in both the upper and lower range for the small cell density. Therefore, there is a chance that regardless of the low RSS value, the fitting for seeding 2 may not reflect the actual estimates of the parameters with the bias in the data set (Fig. 2b). Nevertheless, the density-RPR fittings to the other two seeding density datasets do not suffer from bias (Fig. 2a,c).Table 2 Estimated model parameters from density-RPR fitting of our model.Full size table
Figure 2Our proposed model best fitted the cell density-RPR datasets for all of the seeding conditions generated through the grid-search method.Full size image
Jin et al.1 suggested that their two phase logistic model may share similarities with the Allee effect. However, they did not fit the Allee model stating seeding 2 and 3 were large enough seeding densities. We calculated the conditional threshold density, conditional MSSCD, density at the minimum and maximum RPR for the model from our estimated parameters (Table 3). The conditional threshold cell density calculated from our estimated parameters confirms that the smallest initial seeding density of the dataset was greater than the conditional threshold cell density.Table 3 Calculated cell densities from estimated parameters from our model fitting.Full size tableFigure 3 compares the portrayal of the data through our model with the fitting by Jin et al.1. The blue dashed line is the time-series fitting of the proposed model, and the red-colored line is the time-series fitting of the logistic model to the scratch assay data sets in the Fig. 3. The carrying capacity values are unexpectedly very high in the logistic fit, keeping the model near the exponential phase for the entire dataset. Thus the overall and two phase logistic fits are unrealistic compared to the highest cell density observed in the assay. Also, logistic fitting of the RPR profiles to the data after 18 h does not capture the whole scenario. The green solid and the violet dashed line represent the logistic time-density fit after and before 18 h density profiles respectively. The orange-colored lines in the Fig. 3 are the expected population density as per estimated parameters from the RPR fitting after 18 h data sets. Table 4 enlists all parameters for a comparison between logistic and our model fitting.Figure 3Time series solution of the proposed model and logistic law with comparative RSS for all three seeding conditions.Full size imageTable 4 Logistic model fitting with the Jin et al.1 estimates used in Fig. 3 with the specific colors.Full size tableTrends in cell densities under deterministic set upThe (r_{p}) is fixed for a cell line among all the determining parameters of the conditional MSSCD. n and K vary together with the culture media, flask, and environmental setup. On the other hand, the (alpha), (beta), and (delta) vary together with intercellular-interactions and cellular-interaction with growth-inhibitory molecules, which depend on the medium’s initial cell density per well and fluidity. We observe that the distribution of the conditional MSSCD depends more on the K than the n. There is a chance of overproliferation in the deterministic setup under low n but high K. The cells may die under high n. The cell density at maximum RPR also depends more on K than n (Fig. 4). So the cells should be cultured in the larger flask to achieve maximum proliferativeness.Figure 4The distribution of conditional MSSCD and cell density at maximum RPR in n-K parametric plane.Full size imageThe conditional MSSCD depends more on (beta) than (alpha) (Fig. 5a). The cells may tend to overproliferate under both high (alpha) and (beta). The conditional MSSCD does not exist for a high (delta) and low (beta) depending more on (delta) than (beta). The cells may overproliferate only under a high (beta) and low (delta) (Fig. 5b). The conditional MSSCD also depends more on (delta) than (alpha) showing mostly underproliferation of cells in the (delta ~-alpha) parametric plane. Therefore, the proliferation can be controlled via regulating the interaction between the growth-inhibitory molecules and cells followed by density-regulation through contact-inhibition and cell-cell cooperation (Fig. 5c).Figure 5The distribution of the conditional MSSCD in parametric plane of regulators in the growth law: (a) dependence of the conditional MSSCD on (alpha) and (beta) parameters; (b) dependence of the conditional MSSCD on (delta) and (beta) parameters; (c) dependence of the conditional MSSCD on (alpha) and (delta) parameters.Full size imageThe new cell fitness measure, i.e. cell density at maximum RPR depends more on the (alpha) than the (beta) (Fig. 6a). The cells achieve maximum RPR at a great cell density under the high value of these two parameters. Figure 6b,c suggest that cell density depends only a little on the (delta) under high (alpha) and (beta). Under the low value of these two regulators, a high (delta) always reduces the cell density attaining the maximum RPR, resulting a poor cell-fitness.Figure 6The distribution of cell density at maximum RPR in parametric plane of regulators in the growth law: (a) dependence on (alpha) and (beta) parameters; (b) dependence on (alpha) and (delta) parameters; (c) dependence on (delta) and (beta) parameters.Full size imageStochastic model analysisOur proposed stochastic model (3) can be compared with the general stratonovich stochastic differential equation (frac{dx}{dt}=f(x)+g_{1}(x)epsilon (t)+g_{2}(x)Gamma (t)). Comparing it with our proposed stochastic model we obtain (g_{1}(x)=-x^{delta +1}) and (g_{2}(x)=1). Using the help of47, we get noise induced drift (A(x)=r_{p}x^{alpha +1}left( 1-Big (frac{x}{K}Big )^{beta } right) -nx^{(delta +1)}+D(delta +1)x^{(2delta +1)}-lambda sqrt{DQ}(delta +1)x^{delta }) and noise induced diffusion coefficient (B(x)=Dx^{(2delta +2)}-2lambda sqrt{DQ}x^{(delta +1)}+Q). The cell density at long run can be obtained from the steady state probability density function (SSPDF). The analytical expression of the SSPDF is obtained from the Fokker-Planck equation. The Fokker-Planck equation is (frac{partial P(x, t)}{partial t} =- frac{partial big [ A(x) P(x, t)big ]}{partial x}+ frac{partial ^{2} big [B(x) P(x, t)big ]}{partial x^{2}}), where P(x,t) is the probability density function of the cell population at the time point t. Solving the Fokker-Planck equation we get the SSPDF as (P_{st} (x)= frac{N^{prime }}{B(x)} exp left( int _{x} frac{A(x^{prime })}{B(x^{prime })} dx^{prime }right)) with the normalizing constant (N^{prime }). The value of (N^{prime }) can be obtained from (int _{0}^{infty } P_{st} (x)dx=1).This SSPDF (P_{st} (x)) helps to understand the validity of the proposed stochastic model. Since the number of the data points is too low to fit the stochastic model to the data directly, validation of the stochastic model is challenging in this case. The dataset we used is a time series with 15 data points with three replicates only. An experiment must have many replicates to have a sample with a large sample size so that the SSPDF of cell densities obtained from theoretical findings can be validated with the real observation of cell densities at the steady state. Such datasets with many replicates are rare.So, we generate 2000 sample paths with the help of numerical simulation based on stochastic model 3. We use the parameter values estimated from the fittings of the deterministic model to the seeding condition 1, and we consider some particular values for the two noise intensities and correlation strength ((lambda)) to get a simulated dataset. To achieve the stationary state, we consider sufficiently large time points, and the cell densities at the final time point are used as the data set for the stationary state. We compare the frequency density of cell densities at steady-state of a simulated dataset of 2000 sample paths with the SSPDF obtained from the analytical solution. This comparison shows that the cell density distribution at the steady state matches the steady state probability density function obtained analytically (Fig. 7).In addition, we illustrated the time series generated with the help of stochastic model 3 through numerical technique (Fig. 8). We have plotted the time series data thus obtained for each of the three seeding conditions and in the same figure we also plotted the observed cell densities. The red dots (o) represent the original/experimental dataset of Jin et al.1. The blue dots ((*)) represent the simulated dataset obtained from the stochastic model. This Fig. 8 clarifies our claim that the proposed stochastic model is in good agreement with the actual observation.Figure 7The histogram shows the distribution of cell densities at steady state under additive and multiplicative noises. The blue curve is the SSPDF. The function SSPDF and the distribution of cell densities matches to each other.Full size imageFigure 8The red dots (o) in each sub-figures represent the experimental data of Jin et al.1. The blue dots ((*)) are obtained from the stochastic model (3) considering: (a) The seeding 1 estimated model parameters with (D= 0.002), (Q= 0.06) and (lambda = 0.4). (b) The seeding 2 estimated model parameters with (D= 0.01), (Q= 0.15) and (lambda = 0.6). (c) The seeding 3 estimated model parameters with (D= 0.002), (Q= 0.2) and (lambda = 0.4).Full size imageFigures 7 and 8 suggest that the stochastic model is valid. So the model can be further analyzed to meet the first objective. Differentiating (P_{st} (x)), we obtain (frac{dP_{st} (x)}{dx}=frac{N^{prime }}{[B(x)]^2} exp left( int frac{A(x)}{B(x)}dx right) left( A(x)-frac{dB(x)}{dx} right)) and (frac{d^{2}P_{st} (x)}{dx^{2}}= frac{N^{prime }}{[B(x)]^{2}}exp left( int frac{A(x)}{B(x)}dx right) left( frac{dA(x)}{dx}-frac{d^{2}B(x)}{dx^{2}} right) +frac{N^{prime }}{[B(x)]^{2}} left( A(x)-frac{dB(x)}{dx} right) exp left( int frac{A(x)}{B(x)}dx right) frac{A(x)}{B(x)}-frac{2}{[B(x)]^3}N^{prime } exp left( int frac{A(x)}{B(x)}dx right) left( A(x)-frac{dB(x)}{dx} right) frac{dB(x)}{dx}). At the extrema of the SSPDF, we must have (frac{dP_{st} (x)}{dx}=0) i.e. (left( A(x)-frac{dB(x)}{dx} right) =0).

Theorem 3

(x^{*}approx K-K left( frac{nK^{delta +1}+D(delta +1) K^{2delta +1}-lambda sqrt{DQ}(delta +1)K^{delta }}{beta r K^{alpha +1}+n(delta +1) K^{(delta +1)}+D(delta +1) (2delta +1)K^{(2delta +1)}-lambda sqrt{DQ}delta (delta +1)K^{delta }} right)) is the conditional MSSCD due to the correlated additive and multiplicative noises under the condition (r_{p}(alpha +1)x^{*}{}^{alpha }-frac{r_{p}}{K^{beta }}(alpha +beta +1)x^{*}{}^{(alpha +beta )} -n(delta +1)x^{*}{}^{delta }-D(delta +1)(2delta +1)x^{*}{}^{(2delta )}+lambda sqrt{Dalpha }delta (delta +1)x^{*}{}^{(delta -1)} < 0) (proof is in the supplementary information). Figure 9 visualizes the effect of noise strength and correlation strength on the conditional MSSCD. The conditional MSSCD increases with the additive noise strength (Q) and decreases with the multiplicative noise strength (D) when the other model parameters are fixed (Fig. 9a). There is a high chance of overproliferation for a low D and a high Q (Fig. 9a). Again, there is a high chance of extinction for the low Q and high D. The conditional MSSCD depends more on D than (lambda) (Fig. 9b), and more on (lambda) than Q (Fig. 9c). The conditional MSSCD increases with (lambda) and Q; there is a high chance of overproliferation for high (lambda) and Q. The extinction risk of cells from the culture increases with low (lambda) and Q.Figure 9The change in the conditional MSSCD value for different noise strengths and correlation strength using the parameters estimated for seeding 1: (a) the conditional MSSCD values in (D-Q) noise strength plane with highest correlation ((lambda =1)); (b) the conditional MSSCD values in (D-lambda) noise plane with (Q=0.01); (c) the conditional MSSCD values in (Q-lambda) noise plane with (D=0.01).Full size imageDue to the difficulty and complicated expression of the analytical expression of the SSPDF, we use numerical simulation to study the steady-state behavior in the long run under correlated noises. We draw a histogram of the cell densities based on 500 normal sample paths at the final time points. We use seeding 1 fitting estimates as the initial parameter values for this simulation. The cell population is stable and steady at either 0 cell density or at the conditional MSSCD. The distribution is symmetric around the conditional MSSCD for (lambda =1) (Fig. 10a). There is a loss in the symmetry for the decreasing (lambda). For (lambda =0.5), there is a mode at the zero states with another mode at conditional MSSCD (Fig. 10b). The histogram shows a bi-modality for low values of (lambda). The mode at the zero state is highest for (lambda =0) (Fig. 10c). Therefore, the extinction chance increases for zero noise correlation between the additive and the multiplicative noises.Figure 10Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (D=0.01), (Q=0.01), and variable correlation between additive and multiplicative noises: (a) (lambda =1), (b) (lambda =0.5) and (c) (lambda =0).Full size imageThe sustainability of the cell population depends on the strength of the two noises, like the correlation strength between them. For the zero strength multiplicative noise, the population has the mode at around the conditional MSSCD value (Fig. 11). Therefore, the population sustains in this case and tries to stabilize at the conditional MSSCD value. For (D=0.02), there is a bimodality, where the highest mode is at the zero cell density. For (D=0.05), we observe only one mode at (x=0). Therefore, with the increasing values of the multiplicative noise strengths (D), the chance of extinction increases for (lambda =0.5), (Q=0.01), and other constant model parameters for the seeding condition 1. Similar things happen for increasing Q values considering (D=0.01), (lambda =0.5), and other constant model parameters (Fig. 12).Figure 11Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (lambda =0.5), (Q=0.01), and variable strength of multiplicative noise: (a) (D=0.05), (b) (D=0.02) and (c) (D=0).Full size imageFigure 12Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (lambda =0.5), (D=0.01), and variable correlation between multiplicative noise: (a) (Q=0.05), (b) (Q=0.02) and (c) (Q=0).Full size image Remark 5 We have previously discussed the scenario for (alpha =delta) for deterministic case in Remark 4. It is important to understand the scenario under stochastic case too. For (alpha =delta) the proposed stochastic model 3 becomes (frac{dx(t)}{dt}=r_{p}x(t)^{(alpha +1)}left( 1-big (frac{x(t)}{K}big )^{beta }right) - nx(t)^{(alpha +1)}-x(t)^{(alpha +1)} epsilon (t)+ Gamma (t)). For this stochastic model (g_{1}(x)=-x^{alpha +1}) and (g_{2}(x)=1). We get, (A(x)=r_{p}x^{alpha +1}left( 1-Big (frac{x}{K}Big )^{beta } right) -nx^{(alpha +1)}+D(alpha +1)x^{(2alpha +1)}-lambda sqrt{DQ}(alpha +1)x^{alpha }) and (B(x)=Dx^{(2alpha +2)}-2lambda sqrt{DQ}x^{(alpha +1)}+Q). The extrema of the SPDF (big (x(t)=x^{*}big )) must satisfy the growth equation (r_{p}{x^{*}}^{alpha +1}-frac{r_{p}}{K^{beta }}(x^{*})^{alpha +beta +1}-n(x^{*})^{alpha +1}-D(alpha +1)(x^{*})^{2alpha +1}+lambda sqrt{D~Q}(alpha +1)(x^{*})^{alpha }=0). Therefore, for (alpha =delta) the conditional MSSCD is (x^{*}= K-Kfrac{nK^{(alpha +1)}+D(alpha +1)K^{(2alpha +1)}-lambda sqrt{DQ}(alpha +1)K^{alpha }}{beta r_{p}K^{(alpha +1)}+nK^{(alpha +1)}(alpha +1)+D(alpha +1)(2alpha +1)K^{(2alpha +1)}-alpha lambda sqrt{DQ}(alpha +1)K^{alpha }}) under the condition ((r_{p}-n)(alpha +1)(x^{*})^{alpha }-frac{r_{p}}{K^{beta }}(alpha +beta +1)(x^{*})^{(alpha + beta )}-(alpha +1)(2alpha +1)D(x^{*})^{2alpha }+lambda sqrt{DQ}(alpha +1)alpha (x^{*})^{(alpha -1)} More

SubjectsThe study comprised 34 men (19–26 years, mean 23 years; body mass index 19–31 kg/m2, mean 24 kg/m2) with no physical disabilities or mental disorders, including color blindness and claustrophobia30,31. All subjects were informed of the aims, the study procedure, and the financial compensation for participation, and were asked to follow the rules of the study. Prior to each experiment, subjects underwent short-term starvation31,54 (18–20 h; no food except pure water after lunch (12–1 pm) or dinner (6–7 pm), no later than 1 pm or 7 pm, respectively, one the day before the test), no medical treatments, and normal sleep (at least 6 h, between 10 pm the day before the test day to 8 am on the test day)31. Prior to starting each experiment, subjects were stabilized on a chair for ~ 10 min in a room next to the testing room. Based on an assessment with a pre-experiment questionnaire and the first blood glucose level, measured before starting the experiment (see “Geomagnetic orientation assay” section below), subjects who had not followed these rules were not allowed to take the test on the day and the test was postponed. The study was approved by the Institutional Review Board of Kyungpook National University and all the procedures followed the regulations for human subject research. Informed consent was obtained from all subjects.Modulation of GMFThe ambient GMF in the testing room had a total intensity 45.0 μT, inclination 53°, and declination − 7° (Daegu city, Republic of Korea); the total intensity of 50.0 μT in our previous study31 was changed due to a reconstruction of the building; 45.0 μT was maintained throughout the period of this study. To provide the subjects with various GMF-like magnetic fields (i.e., by modulating of total intensity, inclination, or direction of magnetic north), the coil system from our previous studies6,7,31 was used. It comprised three double-wrapped, orthogonal, rectangular Helmholtz coils (1.890 × 1.890 m, 1.890 × 1.800 m, and 1.980 × 1.980 m for the north–south, east–west, and vertical axes, respectively), electrically-grounded with copper mesh shielding. The subject was seated on a rotatable plastic chair with no metal components, at the center of the three-dimensional coils with his head positioned in the middle space of the vertical axis of the coils. To modulate the geomagnetic north, each pair of coils was supplied with direct current from a power supply (MK3003P; MKPOWER, Republic of Korea). The magnetic field was measured using a 3-axis magnetometer (MGM 3AXIS; ALPHALAB, USA); the field homogeneity at the position of the subject’s head was found to be 95%. The testing room was shielded by a six-sided Faraday cage comprising 10 mm thick aluminum plates, and was grounded during the entire experiment40. Background electromagnetic noise was measured inside the coils at the start and the end of each experimental day. It was attenuated by the Faraday cage more than 200-fold over the range from 500 Hz to 100 MHz as described in detail in our previous study31. The 60 Hz power frequency magnetic field was no more than 2 nT (3D NF Analyzer NFA 1000; Gigahertz Solutions, Germany). All electronic devices were placed outside the Faraday cage during the experiments, with the exception of the switch button module for GMF modulation and the antenna for generating the oscillating magnetic fields. The temperature experienced by the subjects was maintained at 25 ± 0.5 °C (Data logger 98,581; MIC Meter Industrial, Taiwan) by air circulation through the honeycomb on the ceiling of the Faraday cage31.Geomagnetic orientation assayAdopting a two-alternative forced choice (2-AFC) paradigm33,34, a geomagnetic orientation assay was conducted similar to our previous study31. Experiments were performed at 09:30–11:30 am or 1:00–5:00 pm (local time, UTC + 09:00) (each experiment: 50 min–1 h 10 min; mean ≈ 1 h, which was shorter by approximately 30 min than that in the previous study: 1 h 20 min–1 h 40 min; mean ≈ 1 h 30 min). Depending on the experiment, starved or unstarved subjects were tested individually. Prior to each experiment, the subjects were asked to remain with their heads facing the front, with eyes closed and earmuffs on during the experiment. In particular, they were asked to concentrate on sensing, if they could, the ambient geomagnetic north during the association phase, and to use the sensed information, depending on the experiment, to orient toward one of the two modulated magnetic norths (0°/180° for magnetic north–south axis or 90°/270° for magnetic east–west axis, rotated clockwise with respect to the ambient geomagnetic north) during the test phase. Subjects were instructed to avoid distracting thoughts and to think immediately “which direction is modulated magnetic north?” whenever they were distracted during the test phase, or felt they were being biased by experiences from earlier experiments. While seated on the rotatable chair, the subject’s blood glucose level was measured shortly before the first session and immediately after each session with eyes open except in the ‘dark’ experiment (Accu-Chek Guide; Roche, Germany)31. If the determined value before the first session varied by more than 15% relative to the mean (Table S2)31, the experiment was postponed and repeated at a later date (approximately 2% of experiments). The subjects were stabilized with eyes closed for 2 min before the first trial in the absence of visual, auditory, olfactory, and haptic sensory cues. For the ‘dark’ experiment (light intensity ≈ 0 lx), subjects wore home-made ‘blind’ goggles and were stabilized with eyes closed for 5 min55,56, and then asked whether they could see any light. If they could, the goggles were adjusted to prevent leakage of light, and the subject then had another 5 min of stabilization with eyes closed before starting the experiment. The subjects were illuminated with light from a filtered/non-filtered diffused light-emitting diode, depending on the experiment (Table S1). The home-made filter goggles were worn throughout the experiment, including the association and test phase, when required. The goggles contained glass filters (Tae Young Optics, Republic of Korea) to provide the eyes with particular wavelengths of light (Spectrometer USB4000-UV-VIS, Ocean Optics, USA) (Fig. S1). Each experiment consisted of 16 sequential trials for ‘no-association’ and ‘food-association’. For the food-association, a subject facing toward the ambient geomagnetic north was gently provided with a chocolate chip31 on his right palm by an experimenter, and given 30 s to eat it, while during no-association trials, food was not provided during the association phase. After a subsequent 5 s interval, the experimenter gently touched the subject’s right thenar area using a paper rod, as the cue to start the test. One of the two modulated magnetic north directions, depending on the experiment, was randomly provided 3 s before the cue for the test. Each of the modulated magnetic north directions was provided eight times for the no-association and food-association sessions. Subjects were informed of the nearly equal probability for each of the modulated magnetic north directions before each experiment. With the touch cue, subjects were asked to rotate freely toward any direction (clockwise or counterclockwise) by themselves (1–4 cycles of two-thirds rotation) and try to sense the direction of the modulated magnetic north during a 1 min period. Rotation was allowed within the rotation angle (− 30° to 210° for the magnetic north–south axis or − 120° to 120° for the east–west axis, depending on experiments, with respect to the ambient magnetic north), which was confined by the plastic stool (Fig. 1A) touching the left or right ankle of the subjects. When subjects determined the direction of the magnetic north, they stopped rotating to face toward the direction and lifted their right hand to indicate the direction to the experimenter. The direction was measured by the experimenter at 10° intervals using the scale on the walls of the Faraday cage31. A prerequisite for correct orientation was that the subject indicated the direction within the range of 30° to the both sides with respect to the magnetic cardinal directions, which was instructed to the subjects before each experiment. When the direction the subjects indicated was out of the 30° range, the trial was not included in the data and was repeated (approximately 0.63% of trials). Before the subsequent trial, the subject was gently rotated to face toward the ambient geomagnetic north and then rested for 5 s. For the ‘dark’ experiment, subjects were asked whether they could see any leaked light immediately after the last measurement of blood glucose level at the end of experiment. If the subject could see leaked light, the experiment was nullified and repeated later on (approximately 3% of experiments; 2/68). All experiments were performed in a double-blind fashion. The experimenter who conducted the orientation assay knew whether a subject was starved or not, wearing filter goggles, and food-associated or not, but did not know the random magnetic north sequences that were controlled by the personal computer (PC) system. Another experimenter responsible for analyzing the data did not know whether the subject was starved or not, the experimental conditions, including light wavelengths, or whether an oscillating magnetic field had been provided to the subjects. Thus, none of the experimenters were aware of all the subject information and data during the experiments and data analysis. The correct orientation rate was calculated by (the number of correct orientation trials/total number of trials) (raw data, Appendix S3). All the subjects participated in all the experiments performed in random order with an interval of at least 3 days between experiments. After each experiment, the subjects were asked to answer a post-experiment questionnaire about whether they closed their eyes when required during the entire period of the experiment. In cases when a subject did not maintain closed eyes, the experiment was repeated (approximately 1% of experiments). For each subject, a preliminary experiment on the “magnetic north–south axis” was conducted twice (unstarved and starved for each) with no goggles for adaption to the experimental procedure. These data were not included in the results.Experiments with oscillating magnetic fieldsExperiments with oscillating magnetic fields were performed using the standard geomagnetic orientation assay described above. To produce the oscillating magnetic fields, oscillating currents from a function generator (AFG3021; Tektronix, USA. For each magnetic field, sweep of 500 ms; interval of 1 ms. See Fig. S6A) were amplified (ENI 2100L RF power amplifier; Bell Electronics, USA) and fed into a calibrated coil antenna (30 cm diameter, 6509 loop antenna; ETS-LINDGREN, USA) mounted on a wooden frame, comprised of a single winding of coaxial cable. The oscillating magnetic fields were measured daily, before the first and after the last experiment of the day, using a spectrum analyzer (SPA-921TG; Com-Power, USA) with a calibrated loop antenna (48 cm diameter, AL-130R; Com-Power, USA) and a calibrated magnetometer (Probe HF 3061, NBM-550; Narda, Germany). Magnetic field intensities were measured on the glabella of the subjects; variations in intensity between subjects due to different seating heights were less than 10% of the average values (Table S3). The function generator and amplifier were placed outside the Faraday cage, and switched on during the dummy load control experiments with no signal from the PC system. The band widths of the monochromatic magnetic fields, i.e., 1.260 and 1.890 MHz were 0.020 and 0.019 MHz (“average”, √3 kHz), respectively, at the bottoms of the peaks. During the test phase, the maximum values of magnetic noise on the glabella of subjects including the dummy load did not exceed the following values: (1) 5 Hz–9 kHz; 2 nT/√ 2 kHz of “average” and 8 nT/√ 9 kHz of “max-hold” (0.05 nT/√ 2 kHz of “average” and 5 nT/√ 9 kHz of “max-hold” in the dummy load) (3D NF Analyzer NFA 1000; Gigahertz Solutions, Germany); (2) 9 kHz–500 kHz; 5 nT/√ 3 kHz of “average” and 8 nT/√ 3 kHz of “max-hold” (≈ 0 nT/√ 3 kHz of “average” and ≈ 1 nT/√ 3 kHz of “max-hold” in the dummy load) (the AL-130R antenna) (Fig. S6C); and (3) 500 kHz–30 MHz; 0.006 nT of 3.780 MHz harmonic in the 1.260 MHz, 0.03 nT of 5.670 MHz harmonic in the 1.890 MHz, and ≈ 0 nT in the dummy load (/√ 10 kHz of “average”) (Fig. S6B), and 0.15 nT/√ 10 kHz of “max-hold” at the same frequencies above and ≈ 0 nT in the dummy load (the AL-130R antenna).Statistical analysisTo determine the significance of orientation data from the 2-AFC paradigm, a one-sample t-test (test mean: 0.5), paired sample t-test, or two-sample t-test was performed using Origin software 11 (Origin, USA). To verify the reasonability of the t-tests, all data sets were checked using the Anderson–Darling test if the data follow a normal distribution (Appendix S4). To evaluate the difference between the means of two data sets when at least one of them did not show a normal distribution, the percentile bootstrap method57 was used (95% confidence interval, see Fig. S2, Appendices S1 and S2 for raw data). To analyze the blood glucose data, a paired sample t-test was used. Based on the results of previous study31, to describe different response groups of magnetic orientation in the 2-AFC paradigm, a principal component analysis36,37 was conducted on correct orientation rates by starved subjects, with no association/food-association under the full wavelength or  > 400 nm light conditions using SPSS 23 (IBM, USA). Following the principal component analysis calculation, the k-means clustering algorithm—one of the unsupervised learning methods—was used to objectively classify the groups58. 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Very high-resolution (VHR) satellite imagery allows us to survey regularly remote and large areas of the ocean, difficult to access by boats or planes. The interest in using VHR satellite imagery for the study of great whales (including sperm whales and baleen whales) has grown in the past years1,2,3,4,5 since Abileah6 and Fretwell et al.7 showed its potential. This growing interest may be linked to the improvement in the spatial resolution of satellite imagery, which increased in 2014 from 46 cm to 31 cm. This upgrade enhanced the confidence in the detection of whales in satellite imagery, as more details could be seen, such as whale-defining features (e.g. flukes).Detecting whales in the imagery is either conducted manually1,4,5,7, or automatically2,3. A downside of the manual approach is that it is time-demanding, with manual counter often having to view hundred and sometimes thousands of square kilometres of open ocean. The development of automated approaches to detect whales by satellite would not only speed up this application, but also reduce the possibility of missing whales due to observer fatigue and standardize the procedure. Various automated approaches exist from pixel-based to artificial intelligence. Machine learning, an application of artificial intelligence, seems to be the most appropriate automated method to detect whales efficiently in satellite imagery2,3,8,9.In machine learning an algorithm learns how to identify features by repeatedly testing different search parameters against a training dataset10,11. Concerning whales, the algorithm needs to be trained to detect the wide variety of shapes and colour characterising whales. Shapes and colour will be influenced by the type of species, the environment (e.g. various degree of turbidity), the light conditions, and the behaviours (e.g. foraging, travelling, breaching), as different behaviours will result in different postures. The larger a training dataset is, the more accurate and transferable to other satellite images the algorithm will be. At the time of writing, such a dataset does not exist or is not publicly available.Creating a large enough dataset necessary to train algorithms to detect whales in VHR satellite imagery will require the various research groups analysing VHR satellite imagery to openly share examples of whales and non-whale objects in VHR satellite imagery, which could be facilitated by uploading such data on a central open source repository, similar to the GenBank12 for DNA code or OBIS-Seamap13 for marine wildlife observations. Ideally clipped out image chips of the whale objects would be shared as tiff files, which retains most of the characteristics of the original image. However, all VHR satellites are commercially owned, except for the Cartosat-3 owned by the government of India14, which means it is not possible to publicly share image chips as tiff file. Instead, image chips could be shared in a png or jepg format, which involve loosing some spectral information. If tiff files are required, georeferenced and labelled boxes encompassing the whale objects could also be shared, including information on the satellite imagery to allow anyone to ask the commercial providers for the exact imagery.Here we present a database of whale objects found in VHR satellite imagery. It represents four different species of whales (i.e. southern right whale, Eubalaena australis; grey whale, Eschrichtius robustus; humpback whale, Megaptera novaeangliae; fin whale, Balaenoptera physalus; Fig. 1), which were manually detected in images captured by different satellites (i.e., GeoEye-1, Quickbird-2, WorldView-2, WorldView-3). We created the database by (i) first detecting whale objects manually in satellite imagery, (ii) then we classified whale objects as either “definite”, “probable” or “possible” as in Cubaynes et al.1; and (iii) finally we created georeferenced and labelled points and boxes centered around each whale object, as well as providing image chips in a png format. With this database made publicly available, we aim to initiate the creation of a central database that can be built upon.Fig. 1Database of annotated whales detected in satellite imagery covering different species and areas. Humpback whales were detected in Maui Nui, US (a); grey whales in Laguna San Ignacio, Mexico (b); fin whales in the Pelagos Sanctuary, France, Monaco and Italy (c); southern right whales were observed in three areas, off the Peninsula Valdes, Argentina (d); off Witsand, South Africa (e); and off the Auckland Islands, New Zealand (f). The dot size represents the number of annotated whales per location. Whale silhouettes were sourced from philopic.com (the grey and humpback whales silhouettes are from Chris Luh).Full size image More

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