Behavioral and physiological rhythms in the living beings are regulated by the main clock, suprachiasmatic nucleus (SCN) of the brain.^[1–5] One main function of the SCN is to entrain the body rhythms to the environmental light-darkness cycle. The SCN can entrain to not only the natural 24-h light–darkness cycle but also to the artificial non 24-h light–darkness cycles. For example, a Sudanian grass rat can entrain from 22.9 h (lower limit entrainment) to 25.3 h (higher limit entrainment), a southern flying squirrel can entrain from 23.5 h to 24.9 h, and a human being can entrain from 21.5 h to 28.6 h.^[6–9] The range between the lower limit entrainment (LLE) and upper limit entrainment (ULE) is called entrainment range, which reflects the flexibility of the SCN, i.e., the larger the entrainment range is, the larger the flexibility is. The entrainment range is often represented by the LLE in the mathematical model, provided that the LLE and the ULE are symmetrical to 24 h.^[6]

The entrainment range is one kind of collective behaviors of about 2.0 × 10⁴ SCN neurons.^[5,6] The SCN neurons can be divided into two distinct groups, i.e., the ventrolateral part (VL) and the dorsomedial part (DM), based on the different functions. The VL containing around 25% SCN neurons, and the DM composed of the rest 75% SCN neurons, differ in the sensitivity to the photic information.^[9–14] Most of the VL neurons can directly receive the photic information from the retina, while most of the DM neurons can not. The DM neurons receive the information from the VL neurons. Additionally, the coupling is directed between VL and DM due to the neurotransmitters and the directed synapses. The neurotransmitters differ between the VL and the DM.^[15–17] The VL neurons produce the vasoactive intestinal polypeptide (VIP) which is absorbed by both the VL and the DM neurons, and the DM neurons are coupled through the arganine vasopressin (AVP). The VL neurons dominate the DM neurons also because the synapses mostly project from the VL. Therefore, the coupling strength from the VL to the DM is larger than the other directions.^[5]

This difference in the sensitivity to the photic information plays a key role in the function of the SCN, such as recovery from the jet-lag. In particular, in the beginning days after the jet-lag, the VL switches to the new environmental time immediately because the VL neurons are directly sensitive to the light information, while the DM remains the previous time information. Due to the directed coupling, the DM is gradually synchronized to the VL and reaches the new environmental time in the following days.^[13,18] Another experiment is that exposed to a 22-h light–dark cycle, the dissociation between the VL and the DM emerges.^[8,11] Two periodic components of behavioral activities are observed, i.e., one component of 22-h period, which is equal to the period of the external cycle, is controlled by the VL, and the other component of about 24-h period, which is close to the endogenous period of the SCN, is regulated by the DM.

Recently, by contrary, it has been found that the light-responsive neurons are not located in the specific subregion of the SCN.^[19,20] Not only the VIP neurons receive the light information from the retina but also the AVP neurons do. Therefore, both the VL and the DM neurons are sensitive to the light information, directly. Interestingly, the study implied that the VIP neurons can be activated at low light levels whereas AVP neurons require higher light intensities. Accordingly, the sensitive strength to light information differs between the VL neurons and the DM neurons, i.e., the strength is larger in the VL neurons than the DM neurons.

In all previous modeling studies as best as we know, all the SCN neurons were assumed to be sensitive to the photic information with the same strength,^[16,21–23] or only the VL neurons are light-sensitive but the DM neurons are light-insensitive.^[9,24] Thus far, it is unknown about the effects of the difference in the sensitivity to the light-information. In the present study, we will examine the effects of this difference on the entrainment range based on a Poincaré model. The rest of this article is organized as follows. In Section 2, a Poincaré model is introduced to mimic the SCN. The numerical results and the theoretical explanations are presented in Section 3. At last, we show the conclusions and discussions in Section 4.

In the present study, the Poincaré model composed of N neuronal oscillators is introduced to mimic the SCN network exposed to an artificial external light–dark cycle.^[24–32] For each neuronal oscillator, there are two variables, i.e., x and y. The oscillators are coupled through the mean field F to form an all-to-all network. The model is described as follows:

The coupling term and the light-input term are g_iGF and l_iL sin (Ω t), respectively, where G and L are coupling strength and light intensity, respectively. The key parameters g_i and l_i depend on the region of the neurons, which represents the sensitivity to the mean field and the sensitivity to the light information for oscillator i, respectively. For simplicity, the SCN neurons are divided into two groups to represent the VL and the DM. We assume that for the neurons within the same group, the values of g_i or l_i are the same. If the neuron i is located in the first group of the SCN, i.e., 0 < i ≤ N₁, we let g_i = p and l_i = q, and if the neuron j is located in the second group of the SCN, i.e., N₁ < j ≤ N, g_j = (N − N₁p)/(N − N₁), and l_j = (N − N₁q)/(N − N₁), where the parameters p and q satisfy and . Specially, when the numbers of neurons are the same (N₁ = N − N₁) for these two groups, the parameters are simplified as g_j = 2 − p and l_j = 2 − q (N₁ < j ≤ N).

The key parameter q represents the difference in the sensitivity to the photic information between the two groups. If q = 1, there is no difference. If q > 1 (or q < 1), the sensitivity of the first group is larger than the second group (or vice versa). Similarly, p represents the direction of the coupling between the two groups. If p = 1, the coupling is undirected. If p > 1 (or p < 1), the VL dominates the DM (or vice versa). In the results, we will examine whether the difference represented by q affects entrainment range (represented by the LLE). For the numerical simulation, the parameters are set as a = 1, γ = 1, and τ = 24. Without special statement, the number is N = 2, i.e., one oscillator represents the VL and the other is DM, and the deviation is β = 0.

Figure 1 shows illustrative examples for the effects of the sensitivity q on the entrainment of the SCN exposed to the light–dark cycle of 22-h period with selected values of parameters. In panel (a), when the value is q = 0.75, both the VL and the DM are not entrained to the external cycle, because the phase difference between each group and the external cycle fluctuates over time. In panel (b), when the value increases to q = 1.25, both the VL and the DM are entrained to the external cycle, because the phase difference is stable over time. In panel (c), when the value is q = 1.75, both the VL and the DM are not entrained to the external cycle, again. Therefore, the value of q influences the entrainment of the SCN to external cycle.

Fig. 1.

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	Fig. 1. Illustrative examples for the effects of q on the entrainment of the SCN exposed to the light–dark cycle of 22-h period. The value of q is q = 0.75 in panel (a), q = 1.25 in panel (b), and q = 1.75 in panel (c). The parameters are set as G = 0.1, L = 0.05, and p = 0.25. The grey region and the white region represent the night time and day time, respectively.

In this section, we will examine the relationship of LLE to the sensitivity q with given values of G and L in Figs 2–4. Because it is unknown that whether the focused neuron is influenced by the coupling term more or the light term more, we select the coupling strength G = 0.05, 0.1, and 0.15, respectively, and the light intensity L = 0.05, 0.1, and 0.15, respectively.

Fig. 2.

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Fig. 2. The effect of the sensitivity q on the lower limit entrainment (LLE) with G = 0.1. The parameter L is 0.05, 0.1, and 0.15 in panels (a)–(c), respectively. The relationship of the LLE to q is monotonous for cases of p ≠ 1 in panel (a), and is parabolic-like for each case of p in panel (b) or panel (c). In panel (d), the relationship of the trough qc to p is shown. As an example, the trough in the case of p = 0 is indicated by qc in panel (b). Note that larger LLE corresponds to smaller entrainment range. The dashed line indicates the line of qc = 1 in panel (d).

The parameters are set as G = 0.1, and L = 0.05, 0.1, and 0.15 in Fig 2. In panel (a) with L = 0.05 (L < G), the relationship is monotonously decreasing or increasing in the cases of p ≠ 1. In particular, when p = 0.0 or p = 0.5, the relationship is negative; when p = 1.5 or p = 2, the relationship is positive; and when p = 1.0, the LLE is a constant of around 21.9 h. In panel (b) or panel (c), with L = 0.1 or 0.15 (L ≥ G), the relationship is altered to be parabolic-like, in each case of p. If q is smaller than the trough q_c, the LLE decreases with the increase of q, and if q is larger than the trough q_c, the LLE increases with the increase of q for each case of p in panel (b) or panel (c). It is visible that the values of q_c depend on the value of p in each panel of Figs. 2(a)–2(c). Consequently, the quantitative relationship of q_c to p with distinct values of L is plotted in panel (d). In the case of L = 0.05, q_c is far away from 1, i.e., q_c is around 2 if p < 1, and q_c is around 0 if p > 1; in the case of L = 0.1, q_c decreases with the increase of p which is from 1.4 to 0.6; in the case of L = 0.15, q_c is around 1.25 if p < 0.8, and q_c is around 0.75 if p > 1.2. Therefore, the function of q_c on p depends on the values of L. With the larger value of L, the function is closer to the line q_c = 1.

Next, we also examine the cases of G < 0.1 and G > 0.15, respectively. The parameters are set as G = 0.05 in Fig. 3 and G = 0.15 in Fig. 4. The following results are compared with Fig. 2. With L = 0.05, the relationship of the LLE to q is similar in Fig. 4(a), whereas the parabolic relationship of LLE to q emerges for each case of p in Fig. 3(a). With L = 0.1 or L = 0.15, there are also parabolic relationships, however, the values of trough are altered. In Figs. 3(b) and 3(c), the trough q_c is closer to 1 for each case of p, and in Figs. 4(b) and 4(c), the trough q_c is further from 1 for each case of p. Figures 3(d) and 4(d) confirm the dependence of q_c on both p and L. Therefore, from Fig. 2 to Fig. 4, we observe that the parabolic relationship exists, and the value of trough q_c is closer to 1, if G is smaller than L, i.e., the influence on the SCN neurons from the other neurons is smaller than the light-information.

Fig. 3.

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Fig. 3. The effect of the sensitivity q on the lower limit entrainment (LLE) with G = 0.05. The parameter L is 0.05, 0.1, and 0.15 in panels (a)–(c), respectively. The relationship of the LLE to q is parabolic-like for each case of p in panels (a)–(c). In panel (d), the relationship of the trough qc to p is shown. Note that larger LLE corresponds to smaller entrainment range. The dashed line indicates the line of qc = 1 in panel (d).

Fig. 4.

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Fig. 4. The effect of the sensitivity q on the lower limit entrainment (LLE) with G = 0.15. The parameter L is 0.05, 0.1, and 0.15 in panels (a)–(c), respectively. The relationship of the LLE to q is shown for each case of p in panels (a)–(c). In panel (d), the relationship of the trough qc to p is shown. Note that larger LLE corresponds to smaller entrainment range. The dashed line indicates the line of qc = 1 in panel (d).

The SCN neurons are nonidentical in the intrinsic periods.^[33] We examine whether the main results are affected by this nonidentity. Figure 5 shows the relationship of the LLE to the sensitivity q when the standard deviation is β = 0.03. It is evident that the relationship is not altered qualitatively, compared with Fig. 2. Therefore, the nonidentity does not influence the relationship, qualitatively.

Fig. 5.

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Fig. 5. The effect of the sensitivity q on the lower limit entrainment (LLE) when the intrinsic periods of the neurons are nonidentical (the standard deviation is β = 0.03). The parameter L is 0.05, 0.1, and 0.15 in panels (a)–(c), respectively. The relationship of the LLE to q is monotonous for cases of p ≠ 1 in panel (a), and is parabolic-like for each case of p in panel (b) or panel (c). In panel (d), the relationship of the trough qc to p is shown. The number of neurons are N = 100 and the coupling strength is G = 0.1. Note that larger LLE corresponds to smaller entrainment range. The dashed line indicates the line of qc = 1 in panel (d).

For simplicity, we let N be 2, i.e., one oscillator represents the first group and the other represents the second group. Equation (1) is reduced to

When all the oscillators are entrained to the external cycle, we obtain , , , and . Let θ₁ = Ω t + ϕ₁ and θ₂ = Ω t + ϕ₂. Considering the averaging method developed by Krylov and Bogoliubov as used in Refs. [6,21,24,32], the alternation of ϕ has a lower time scale than Ω t. We let α = ⟨ϕ₂⟩ − ⟨ϕ₁⟩, and we obtain

Generally speaking, Ω_LLE depends on the slower oscillator with smaller Ω. Mathematically, Ω_LLE of the SCN is achieved through the interplay between the coupling terms and light terms. The light terms are positive which contributes to Ω_LLE. ϕ₁ and ϕ₁ + α reflect the “absorption efficiency” of the light information. When ϕ₁ = ϕ₁ + α = π, where α is zero, the efficiency is perfect. However, the value of α may not be zero which depends on the coupling term. The coupling term influences Ω_LLE through the phase difference α. If α is zero, both C₁ and C₂ are zero. If α is not zero, one of C₁ and C₂ is positive and the other is negative. In the following discussions, we will discuss the influence of G_1,2 and L_1,2 on Ω_LLE, where G₁ = pG, G₂ = (2 − p)G, L₁ = qL, and L₂ = (2 − q)L, under two limited conditions.

In the present study, we examined the relationship of the entrainment range (represented by the LLE) to the sensitivity (q) to the photic information, where q ≠ 1 represents there exists difference of sensitivity between the VL and the DM. We found that the relationship depends on the coupling strength. In particular, when the coupling strength (G) is much smaller than the light intensity (L), the relationship is parabolic-like, and the maximum of the entrainment range is obtained when there is no difference (q = 1). This relationship is independent of the direction of the coupling (represented by p). When the coupling strength is much larger than the light intensity, the relationship is monotonically decreasing or increasing if there is directed coupling (p ≠ 1), and the maximum of the entrainment range is obtained when the difference is the largest (q = 2 for p < 1 and q = 0 for p > 1). When the coupling strength is close to the sensitive strength, the relationship is parabolic-like if there is directed coupling (p ≠ 1), and the maximum of the entrainment range is obtained when the difference is moderate (1 < q < 2 for p < 1 and 0 < q < 1 for p > 1).

In a recent experiment, both the VL neurons and the DM neurons are sensitive to light information, and the sensitive strength to the light information for the VL neurons is larger than the DM neurons.^[19] However, the benefits of the difference in the sensitive strength are unknown so far. In the present study, we found that this difference may lead to large entrainment range which reflects the flexibility of the SCN to the external light–dark cycle. Our finding may provide an explanation for the exitance of the difference in the sensitivity to light-information, as well as shed light on how to increase the flexibility of the SCN represented by widening the entrainment range.

For simplicity, the SCN network is assumed to be an all-to-all network represented by the mean-field, and no noise or stochasticity is considered in the circadian signaling in the present study. Additionally, the Poincaré model is a genetic model which is oversimplification. In the future, it is worth rebuilding a more realistic model for the SCN, i.e., a Goodwin model,^[34] where the noise in the circadian signaling,^[35–37] the amplitude difference between the neurons,^[38] and the network structure, such as a small-world network, a scale-free network, and a two-layer network^[39–43] are considered.