External low energy electromagnetic fields affect heart dynamics: surrogate for system synchronization, chaos control and cancer patient’s health

Figures

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  FIGURE 1. Schematics of the electronic analogue for the FitzHugh-Nagumo model.

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  FIGURE 2. Evolution towards the limit cycle of an unforced solution of the FHN model with parameters ϵ=0.1 , E=0.5 and R=0.5 (A). Time series V0τ representing the evolution of the action potential (B). Representation in the V0τ,W0τ phase space highlighting the refractory region of the limit cycle (C). Phase Response Curve (PRC) superimposed to the limit cycle solution of the model. Near the refractory region, short external stimuli have little to no effect on the undriven trajectory. Before the ascending and descending parts of the action potential, an external stimulus can either advance or retard the trajectory, respectively, depending on the sign of the PRC.

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  FIGURE 3. (A). Comparison between the limit cycle solution for the undriven system (unforced) and the delayed trajectory forced by a short external stimulus of the form of Equations 2, 3 (B). Short current stimulus applied during a negative portion of the PRC. The parameters for the current are A0=5 , ωmod=12.6 and ωc=88.0 (C). Representation of the trajectories in the phase space (D). Comparison between the limit cycle solution for the undriven system (unforced) and the advanced trajectory forced by a short external stimulus of the form of Equations 2, 3 (E). Short current stimulus applied during positive portion of the PRC. The parameters for the current are A0=5 , ωmod=12.6 and ωc=88.0 (F). Representation of the trajectories in the phase space.

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  FIGURE 4. Comparison between the limit cycle solution for the undriven system (unforced) and the trajectory forced by a continuous external stimulus represented by an effective current (Equation 4), with envelope evolving as Equation 3. Parameters K=0.03 and A0=1 . (A) ωmod=3.9 and starting point of the envelope were chosen to suppress the current train in most of the positive regions of the PRC, leading to a delayed action potential. (B) ωmod=3.7 and starting point of the envelope were chosen to suppress the current train in most of the negative regions of the PRC, leading to an advanced action potential. (C) ωmod=9.0 and starting point of the envelope were chosen as to leave the action potential unaffected.

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  FIGURE 5. (A). displays Difference Equation curves xt+1 = axt (1−xt) for seven cancer patients, each characterized by a specific growth rate constant a . The intersection points of these curves with the line f(x) = x identify fixed points (orthogonal projections on the X(t) axis) and their slopes indicate the stability of these points (dashed lines). Figures (B–D) illustrate strong correlation trends: a positive correlation between survival rates and the growth rate constant a and fixed points, and a negative correlation between the slope and survival rates. Figures (E–G) further highlight these trends, showing strong positive correlations between the fixed points and the constant a , and strong negative correlations between the slope, the constant a , and the fixed points. See Supplementary Table 1S and Supplementary Figure 1S for more information. Blue bars corresponds to mean survival. Green bars corresponds to mean constant a values.

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  FIGURE 6. Comparison between non-exposure vs. exposure 15-minute tachograms recorded from a cancer patient. EMF exposure caused a delay in the heart periodic oscillations. In the time-domain, EMF interference resulted in increase in RRI duration, increase in SDNNI, decrease in RMSSD. In the frequency-domain, there is a shift for high VLF power, large increment in total power and LF/HF ratio. Graphic produced by Kubios. Note: RRI: R-R intervals; RMSSD: Square root of the mean squared differences between successive RR intervals; SDNNI: Mean of the standard deviations of RR intervals in 5-min segments; VLF: very low band peak frequencies; LF: low frequency band peak frequencies; HF: high frequency band peak frequencies.

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  FIGURE 7. (A) The boxplot displays outlier heartbeats categorized into decelerated and accelerated types, compared to the neighboring R-R interval values. The neighboring beats consist of a total of 8–10 heartbeats, divided equally before and after the outlier heartbeat. The comparison of two sequential 10-second tachograms shows (B) progressive larger RRI and (C) trajectory effect in gradient phase plot vectograms, when no EMF effect is observed, or the deceleration effect is introduced into the time series of heart beats.

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  FIGURE 8. A ROC curve for 22 cancer patients is presented (left). The Kaplan-Meier curve showed significantly better overall survival for cancer patients with constant a > 1.1758.