1. Introduction In this work we analyze the role of the light-dark cycle and constructive diversity in the dynamics of a system of circadian neurons [1]. We introduce neuronal heterogeneity in the form of quenched noise, by rescaling the individual neuronal periods by a scaling factor drawn from a normal distribution. The system response to the light-dark cycle periodicity is studied as a function of the interneuronal coupling strength, external forcing amplitude and neuronal heterogeneity. We show that the right amount of diversity helps the system to respond globally in a more coherent way to the external forcing. Our proposed mechanism for neuronal synchronization under external periodic forcing is based on heterogeneity-induced oscillators death, damped oscillators being more entrainable by the external forcing than the self-oscillating neurons. Figure 6. Synchronizing role of diversity based on stability of fixed point 1. Gonze D, Bernard S, Waltermann C, Kramer A, Herzel H., “Spontaneous Synchronization of Coupled Circadian Oscillators”, Biophys. J. 2005; 89: 120-129. Model-Equations and Parameters [1] 1.Response R of the Mean Field F We quantify the resonance effect by the spectral amplification factor R for the mean field X: 2. Order parameter or Synchrony   measures the distribution of phases of the neurons. It is ranging between 0 (no synchronization) and 1 (perfect synchronization, with all neurons in phase) Synchronization parameters Conclusions Diversity (at optimal levels; not too large, not too small) is able to improve the collective response of the neuron ensemble to the 24h cycle. The mechanism is related to the oscillator death it produces: the damped oscillators follow better the external signal than the ones self-oscillating with different periods (which in the strong coupling regime leads to fast oscillations). The system response to the light-dark cycle periodicity is studied as a function of the interneuronal coupling strength K, external forcing amplitude L 0 and neuronal heterogeneity . Figure 1 shows that by increasing diversity, parameters  and R reach a maximum, which corresponds to the value of  for which the real part of linearization eigenvalues  become negative  and also to the value of  at which  mean period of oscillations of the mean field is 24 h. Figure 2 shows the stability domain for 100 neurons as a function of (L 0, K ), in terms of maximum eigenvalues obtained by linearization. It can be seen that system stability is more sensitive to L 0 than to K. Figure 3. Contour plot of R G as a function of (L 0,  ) shows that synchrony is favored by strong light intensity, low diversity, and large coupling. Figure 4. Contour plot of the mean of the individual period of oscillations, as a function of (L 0,  ). It shows that synchrony is not always at the 24h-period of the forcing: individual periods take-off. Figure 4 shows that the 24h cycle is respected for strong light and large diversity (and small internal coupling). Figure 5. Contour plot of R as a function of (L 0,  ). It shows that log(response) to 24h forcing larger for weak light and intermediate diversity. Optimal diversity to give maximum response at 24h. Figure 6. Contour plot of the maximum eigenvalue of the linearized system as a function of (L 0,  ). It shows that system heterogeneity (and light) induces oscillator death by stabilizing the singular point. Results and Discussion Figure 1. System responses R and R G as functions of (K, L 0,  ) negative eigenvalues complex/real negative singular point pos. eig. Figure 2. Stability domain of 100 neurons as a function of (L 0, K), in terms of maximum eigenvalues obtained by ES linearization. Figure 3. Synchrony order parameter R G L0L0  K=0 K=1 Figure 4. Mean of the individual periods under forcing at 24h cycle L0L0  K=0 K=1 Figure 5. Log(R) to 24h forcing larger for weak L and intermediate   L0L0 K=0 K=1  L0L0 Synchronization properties of coupled circadian oscillators Adrian C. Murza, Niko Komin, Emilio Hernández García, Raúl Toral, IFISC, (CSIC-UIB), Campus Universitat Illes Balears, E - 07122, Palma de Mallorca, Spain K=0 K=1   L0L0 