(a) Coupled oscillator model. Each sperm has two independent oscillators: i) stimulus function and ii) [Ca2+]i, which can be coupled through a forcing term that connects them, in our case the slope of the chemoattractant concentration gradient (ξ0). (b) Maximum relative slopes (ξmax) of the chemoattractant concentration gradient experienced by S. purpuratus (Sp) spermatozoa when exposed to f1, f2, f3, f4, f5 speract gradients. The maximum relative slopes of the chemoattractant concentration gradient experienced by L. pictus spermatozoa (Lp) toward f4 experimental regime are also shown. Note that ξmax for f2, f3, and f5, are up to 2–3 times greater than in f4, regardless of the species. (c) Experimental signal-to-noise ratios (SNR) regimes experienced by spermatozoa swimming in different gradient conditions. Note that only f2, f3 and f5 have higher SNR, compared to other gradient conditions, for which stochastic fluctuations mask the signal. This SNR calculation assumes a 10% of speract uncaging. The maximum relative slopes (ξ) are shown in log scale (d) Arnold’s tongue indicating the difference in intrinsic frequency of the internal Ca2+ oscillator of S. purpuratus spermatozoa, just before and after the speract gradient exposure. (e). Phase difference between the time derivative of the stimulus function and the internal Ca2+ oscillator of S. purpuratus spermatozoa, obtained by computing the cross-correlation function between both time series (Figure 6—figure supplement 2). (f). Phase difference between the time derivative of the stimulus function and the internal Ca2+ oscillator of S. purpuratus spermatozoa expressed in temporal delays. (d-f) Gray points represent the collated data of all f1, f2, f3, f4, f5 experimental regimes. Red, black and blue points indicate chemotactic spermatozoa (CI > 0 at second three after UV flash), located in R3, and R4 regions just before the speract gradient is established under f2, f3 and f5 experimental regimes, respectively. Magenta lines represent the transition boundary (γmin = ~ 2.6×10−3 µm−1, see also Figure 1d–f) below which no synchrony is observed, obtained from the theoretical estimates (black curves, mean of Δω) of panels (e) and (f). Green dashed lines indicate confidence intervals (mean ± standard deviation).