NMDAR-Mediated Ca²⁺ Increase Shows Robust Information Transfer in Dendritic Spines

Abstract

A dendritic spine is a small structure on the dendrites of a neuron that processes input timing information from other neurons. Tens of thousands of spines are present on a neuron. Why are spines so small and many? We addressed this issue by using the stochastic simulation model of N-methyl-D-aspartate receptor (NMDAR)-mediated Ca²⁺ increase. NMDAR-mediated Ca²⁺ increase codes the input timing information between prespiking and postspiking. We examined how much the input timing information is encoded by Ca²⁺ increase against presynaptic fluctuation. We found that the input timing information encoded in the cell volume (10³μm³) largely decreases against the presynaptic fluctuation, whereas that in the spine volume (10⁻¹μm³) slightly decreases. Therefore, the input timing information encoded in the spine volume is more robust against presynaptic fluctuation than that in the cell volume. We further examined the mechanism of the robust information transfer in the spine volume. We demonstrated that the condition for the robustness is that the stochastic NMDAR-mediated Ca²⁺ increase (intrinsic noise) becomes much larger than the presynaptic fluctuation (extrinsic noise). When the presynaptic fluctuation is large, the condition is satisfied in the spine volume but not in the cell volume. Moreover, we compared the input timing information encoded in many small spines with that encoded in a single large spine. We found that the input timing information encoded in many small spines are larger than that in a single large spine when presynaptic fluctuation is large because of their robustness. Thus, robustness is a functional reason why dendritic spines are so small and many.

Introduction

A spine is a small structure on the dendrites of a neuron, which processes input timing information from other neurons. Dendritic spines are characterized by their small volume and large numbers. For example, the volume of a spine on a rat hippocampal CA1 pyramidal cell is ∼0.1 μm³, which is 10⁴-fold smaller than the cell body (∼1000 μm³), and 10,000 spines are present on a single cell (1, 2, 3, 4). Why are spines so small and many? In this study, we address this issue by using a stochastic simulation model of N-methyl-D-aspartate receptor (NMDAR)-mediated Ca²⁺ increase.

The most representative receptors for spike-timing detection for synaptic plasticity on the spine are NMDARs (5, 6, 7, 8, 9). Activation of NMDARs requires the conjunctive inputs of presynaptic spiking (prespiking) and postsynaptic spiking (postspiking) (Fig. 1, A and B). Prespiking represents a glutamate release from the presynaptic terminals, and postspiking is a large increase of membrane potential via backpropagating action potentials. The prespiking opens NMDARs, and the postspiking removes the receptors’ Mg²⁺ block. Coincident prespiking and postspiking enable the high Ca²⁺ influx via NMDARs (10, 11, 12) (Fig. 2, A and B). NMDAR-mediated Ca²⁺ increase consequently induces long-term potentiation, which is thought to be the molecular and cellular basis of learning and memory (13, 14).

Figure 1.

Diagrams of NMDAR-mediated Ca²⁺ increase. (A and B) Schematic diagram of NMDAR-mediated Ca²⁺ increase is shown. Prespiking opens NMDAR, and a postspiking removes the Mg²⁺ block. (C) Block diagram of NMDAR-mediated Ca²⁺ increase is shown. C₀, C₁, C₂, O, D₀, and D₁ are the states of NMDARs. C₀, C₁, and C₂ are the closed states, O is the open state, and D₀ and D₁ are the desensitization states (see Materials and Methods). To see this figure in color, go online.

Figure 2.

NMDAR-mediated Ca²⁺ increase in the stochastic model. (A and B) Shown are the experimental results of NMDAR-mediated Ca²⁺ increase by prespiking and postspiking (10). (A) Shown is the time course of Ca²⁺ increase at the pre-postspiking interval Δt = t_post − t_pre = 10 ms. (B) Shown is the peak of the Ca²⁺ increase with the pre-postspiking interval Δt. (C–H) Shown are the stochastic simulation results of NMDAR-mediated Ca²⁺ increase in the spine volume (10⁻¹μm³) (C, E, and G) and in the cell volume (10³μm³) (D, F, and H). (C and D) Shown are the time courses of the Ca²⁺ increase in the spine volume (C) and in the cell volume (D) at the pre-postspiking interval Δt = 10 ms (the number of trials is 10). (E and F) Shown are the probability density distributions of Ca_res with the pre-postspiking interval Δt, p_c(Ca_res/Δt), in the spine volume (E) and in the cell volume (F) at Δt = 10 ms (the number of trials is 2000). (G and H) Shown are the probability density distributions of Ca_res with the pre-postspiking interval Δt, p_c(Ca_res/Δt), in the spine volume (G) and in the cell volume (H) from Δt = 0 ms to Δt = 100 ms (the number of trials is 2000). To see this figure in color, go online.

The volume of a dendritic spine is so small that the number of molecules in the spine is limited to tens to hundreds. Therefore, biochemical reactions are stochastic in the spine volume (15, 16, 17, 18, 19, 20). For example, the number of NMDARs on a spine is estimated to be only ∼20 (21, 22, 23), which increases the stochasticity in biochemical reactions, and thus the number of NMDARs opening fluctuates widely, which causes a large fluctuation in the NMDAR-mediated Ca²⁺ increase (24, 25, 26, 27, 28, 29). Intuitively, such a large fluctuation caused by the small volume of the spine is disadvantageous for information transfer. However, are there no advantageous characteristics for information transfer in the small volume of the spine?

Other representative receptors for spike-timing detection for synaptic plasticity are metabotropic glutamate receptors (mGluRs) (30, 31). We previously constructed the stochastic simulation model of mGluR-mediated Ca²⁺ increase and examined how much spike-timing information encoded to mGluR-mediated Ca²⁺ increase (19, 20). We demonstrated that although the encoded information is smaller in the smaller volume because the fluctuation of Ca²⁺ increase is larger, the encoded information does not decrease against the neurotransmitter fluctuation in the small volume like the spine compared to the large volume like the cell. Therefore, the small volume enables the robust information transfer against the neurotransmitter fluctuation. However, mGluR-mediated Ca²⁺ increase involves multiple biochemical reactions, including positive feedback and delayed negative feedback, which shows threshold response (12, 32). On the other hand, NMDAR-mediated Ca²⁺ increase is a single-molecule reaction, which shows graded response. Therefore, it remains unclear whether NMDAR-mediated Ca²⁺ increase shows a robustness similar to that of mGluR-mediated Ca²⁺ increase. Moreover, the question of why spines are many has not been examined in previous studies (19, 20).

In this study, using a stochastic simulation model of NMDAR-mediated Ca²⁺ increase, we examined the input timing information encoded to NMDAR-mediated Ca²⁺ increase. We demonstrate that the encoded information largely decreases against the presynaptic fluctuation in the cell volume (10³ μm³), whereas it slightly decreases in the spine volume (10⁻¹ μm³). Therefore, NMDAR-mediated Ca²⁺ increase shows robust information transfer in the small volume like the spine but not in the large volume like the cell. We also demonstrated that the robustness occurs when the extrinsic noise caused by the presynaptic fluctuation is much smaller than the intrinsic noise caused by the fluctuation of biochemical reactions. Next, we compared the information transfer in many small spines with that in a single large spine and demonstrated that the input timing information encoded in many small spines is larger than that in a single large spine when presynaptic fluctuation is large because of their robustness. These results indicate that robustness is a functional reason why spines are so small and many.

Materials and Methods

Stochastic simulation model of NMDAR-mediated Ca²⁺ increase

We constructed the stochastic simulation model of NMDAR-mediated Ca²⁺ increase on the basis of the literature (11, 23, 33, 34, 35, 36, 37, 38) (Fig. 1 C). Initial states and reaction constants are described in Tables S1 and S2. In this model, prespiking and postspiking are the inputs, and Ca²⁺ is the output. C₀, C₁, C₂, O, D₀, and D₁ are the states of NMDARs. C₀, C₁, and C₂ are the closed states, O is the open state, and D₀ and D₁ are the desensitization states. Mg²⁺ can bind all states of NMDARs.

Prespiking represents an increase in glutamate concentration (6 μM), and the glutamate concentration exponentially decreased with the time constant 10 ms (36). Postspiking represents the increase of the membrane potential (60 mV), and the membrane potential exponentially decreased with the time constant 6 ms (37, 38). The resting membrane potential is −65 mV. Prespiking induces the transition from the closed states to the open state, and postspiking induces the dissociation of Mg²⁺ from NMDARs.

The propensity function of Ca²⁺ influx via NMDAR was described as follows:

$$\begin{array}{c}{f}_{in}=SVR\left(-\frac{P_{NMDA}{V}_m\left[O\right]}{1-\exp \left(\varphi V_m\right)}\right)\end{array},$$ (1)

where SVR is the surface-to-volume ratio, V_m is the membrane potential, [O] is the concentration of NMDAR that is open and free from Mg²⁺, P_NMDA is the permeability of the NMDAR channel, and $\varphi$ is 2F/RT, where F is Faraday’s constant, R is the gas constant, and T is the absolute temperature (11). Here, SVR = 11.926 μm⁻¹, P_NMDA = 0.05, and $\varphi =0.0756{\text{mV}}^{-1}$(11). Furthermore, the propensity function of Ca²⁺ removal by Ca²⁺ pump was described as follows:

$$\begin{array}{c}{f}_{out}=\frac{1}{{\tau }_{Ca}}\left({\left[{\text{Ca}}^{2+}\right]}_0-\left[{\text{Ca}}^{2+}\right]\right)\end{array},$$ (2)

where [Ca²⁺]₀ is the basal concentration of Ca²⁺, and τ_Ca is the time constant (11). Here, τ_Ca = 12 ms (35).

We simulated these reactions using a stochastic simulation algorithm, the τ-leap method (39). For example, the number of the Ca²⁺ in the spine, Ca²⁺ (t + τ), is described as follows:

$$\begin{array}{c}C{a}^{2+}\left(t+\tau \right)=C{a}^{2+}\left(t\right)+R_{in}-R_{out}\end{array},$$ (3)

where R_in and R_out indicate the number of reactions of inflow and outflow, which occur in the time interval between t and t + τ. R_in and R_out obey the Poisson distribution, R_in ∼Poisson(f_inτ) and R_out ∼Poisson(f_outτ), respectively. The appropriate τ is calculated in accordance with Cao et al (39).

In general, the surface area of the membrane is proportional to the order of the square of length, whereas the volume is proportional to the cube of length, which means that, as a system’s size increases, the increasing rate of the number of membrane molecules becomes smaller than that in the cytoplasm. Hence, in the system larger than the spine, the number of NMDARs is so small that the Ca²⁺ influx is smaller. To eliminate this influence, we assumed that the number of membrane molecules is proportional to the volume (i.e., the cube of length) throughout this study.

Parameter correction

Based on parameters from the literature (33, 34), NMDARs did not effectively bind glutamate in our model. This is because our model does not consider the spatial structure of the spine. In the actual spine, the reaction between NMDAR and glutamate is limited postsynaptic density (PSD). In other words, each molecule is enriched in PSD. However, our model does not consider such localization to simplify calculation, and therefore, NMDAR and glutamate are apparently distributed uniformly in the spine rather than in PSD. Hence, the reaction between NMDAR and glutamate hardly occurred. To calibrate such localization effect, we increased the reaction constant of NMDAR, k₁ ∼ k₂₄, by fivefold. However, the number of open NMDARs was still fewer than two, which was smaller than what was reported in a previous study (27). The number of open NMDARs reported in the previous study is 3 ∼ 5. To reproduce the number of open MMDARs in the previous study, we increased the open reaction constant of NMDAR, k₅, by fivefold.

Mutual information between the input timing and Ca²⁺ increase

We defined Ca_res as the area under the curve of the time course of Ca²⁺, given by the following:

$$\begin{array}{c}C{a}_{res}=\int \left\{\left[Ca\right]\left(t\right)-C{a}_{basal}\right\}dt\end{array},$$ (4)

where Ca_basal denotes the basal concentration of Ca²⁺, which is 0.050 μM.

We measured the input timing information coded by the Ca²⁺ increase using the mutual information between the Ca_res and the pre-postspiking interval, Δt = t_post − t_pre, given by the following:

$$\begin{array}{c}I\left(C{a}_{res}|\mathit{\Delta }t\right)=\underset{\mathit{\Delta }t}{\int }{p}_{in}\left(\tau \right)\left(\underset{C{a}_{res}}{\int }{p}_c\left(c|\tau \right){\log }_2\frac{p_c\left(c|\tau \right)}{{\int }_{\mathit{\Delta }t}{p}_{in}\left(\tau \right)p_c\left(c|\tau \right)d\tau }dc\right)d\tau \end{array},$$ (5)

where p_in(Δt), the probability density distribution of Δt, follows the uniform distribution, and p_c(Ca_res/Δt), the conditional probability density distribution of Ca_res given Δt, is obtained by the stochastic simulation. Here, τεΔt and cεCa_res. To remove the bias caused by the bin width of Ca_res, the mutual information was calculated by using the method introduced by Cheong et al (40).

Results

NMDAR-mediated Ca²⁺ increase in the spine and cell volumes

We examined the time course of the Ca²⁺ increase in the spine volume and cell volume using the stochastic simulation model of the NMDAR-mediated Ca²⁺ increase (Fig. 2). Here, we denoted 10⁻¹ μm³ as the spine volume and 10³ μm³ as the cell volume. When postspiking occurred at 10 ms after prespiking, a large Ca²⁺ increase was always observed in the spine volume and in the cell volume (Fig. 2, C and D), consistent with the earlier experimental result (10). However, the time course of the Ca²⁺ increase was always the same in the cell volume, whereas it differed greatly between trials in the spine volume, indicating that the Ca²⁺ increase was deterministic in the cell volume and stochastic in the spine volume.

To quantify the variability of the Ca²⁺ increase, we examined the probability density distribution of Ca_res, which was defined by the temporal integration of Ca²⁺ concentration, subtracted by the basal Ca²⁺ concentration (see Materials and Methods). The probability density distribution of Ca_res was unimodal in both the spine and cell volumes (Fig. 2, E and F). We previously reported that the probability density distribution of mGluR-mediated Ca²⁺ increase was unimodal in the cell volume, whereas it was bimodal in the spine volume (19, 20), indicating that the probability density distribution of Ca_res in the spine volume differs between NMDAR-mediated and mGluR-mediated Ca²⁺ increase.

Next, we examined the probability density distribution of Ca_res at 2 ms pre-postspiking intervals from 0 to 100 ms. The probability density distribution of Ca_res was unimodal regardless of pre-postspiking intervals in both the spine and cell volume (Fig. 2, G and H). In addition, the mean of the probability density distribution of Ca_res changed depending on the pre-postspiking interval. For example, when postspiking occurred within 30 ms after prespiking, the mean of the probability density distribution of Ca_res became large in both the spine and cell volume.

Finally, we calculated the mutual information of the probability density distribution of Ca_res against the pre-postspiking intervals Δt and I(Ca_res/Δt) as a measure of how much information of the pre-postspiking interval is transferred to the Ca²⁺ increase (Fig. 3) (see Materials and Methods). The mutual information I(Ca_res/Δt) in the spine volume is smaller than that in the cell volume. Thus, the spine volume is more disadvantageous than the cell volume in terms of the amount of the mutual information. However, are there any advantageous characteristics for information transfer in the small volume of the spine?

Figure 3.

Mutual information between the pre-postspiking interval Δt and the Ca_res.

Robustness against presynaptic fluctuation in the spine volume

It is known that the amplitude of prespiking, the released glutamate concentration, is largely different between trials (15, 17, 41). Hereafter, we call the fluctuation of the prespiking amplitude between trials “presynaptic fluctuation.” The fluctuation of Ca²⁺ increase can be influenced by the presynaptic fluctuation as well as the stochasticity in biochemical reactions (17, 26). The former can be regarded as extrinsic noise and the latter as intrinsic noise. We examined the effect of presynaptic fluctuation on the fluctuation of the Ca²⁺ increase.

Presynaptic fluctuation is mainly composed of two fluctuations. One is the fluctuation of the vesicle size (42, 43), and the other is the fluctuation of the released vesicle number (44, 45). The distribution of the vesicle size is similar with the Gaussian distribution (42, 43), and the distribution of the released vesicle number is given by the binomial distribution (44, 45). Considering that the distribution of the amplitude of prespiking is given by the convolution between the distribution of the vesicle size and that of the released vesicle number, the distribution of the amplitude of prespiking is expected to be similar with the Gaussian distribution. Therefore, we set the probability density distribution of the amplitude of prespiking p_a(Amp_pre/μ_a,σ_a) as the Gaussian distribution given by N(Amp_pre/μ_a,σ_a), where Amp_pre denotes the amplitude of prespiking, and μ_a and ${\sigma }_a^2$ denote the mean and the SD of the amplitude of prespiking in each trial, respectively.

We examined the probability density distribution of Ca_res against the coefficient of the variation of the amplitude of prespiking, CV_a. Note that CV_a is σ_a/μ_a with μ_a = 6 μM. In the spine volume, the probability density distributions of Ca_res with CV_a = 0.0 (Fig. 4 A) and CV_a = 0.5 (Fig. 4 B) were similar; however, as the volume increased, the probability density distributions of Ca_res with CV_a = 0.0 (Fig. 4, C and E) and CV_a = 0.5 (Fig. 4, D and F) changed enormously. The variability of Ca_res remained similar regardless of CV_a in the spine volume, whereas the variability of Ca_res increased as the CV_a increased in the cell volume (Fig. S2).

Figure 4.

Probability density distributions of Ca_res against the presynaptic fluctuation. (A–F) Probability density distributions of Ca_res in the spine volume (A and B), in the intermediate volume (10¹μm³) (C and D), and in the cell volume (E and F) with the coefficient of variation of prespiking CV_a. CV_a = 0 (A, C, and E), and CV_a = 0.5 (B, D, and F). (G and H) Shown is the mutual information between the pre-postspiking interval Δt and the Ca_res with the presynaptic fluctuation CV_a in the indicated volume. (H) The mutual information is normalized by the mutual information at CV_a = 0.0 for each volume. To see this figure in color, go online.

We examined the mutual information I(Ca_res/Δt) against CV_a (Fig. 4, G and H). In the cell volume, the mutual information I(Ca_res/Δt) largely decreased as CV_a increased. In the spine volume, however, the mutual information I(Ca_res/Δt) slightly decreased as much as CV_a increased. Thus, information of the pre-postspiking interval is robustly transferred to the Ca²⁺ increase against the presynaptic fluctuation in the spine volume but not in the cell volume. Furthermore, even if Δtε[0,100] is extended into Δtε[−100,100], the similar result was obtained (Fig. S3).

Unchanging probability density distributions of Ca_res against presynaptic fluctuation in the spine volume explains the robustness

The mutual information I(Ca_res/Δt) decreased greatly as CV_a increased in the cell volume, whereas I(Ca_res/Δt) did not decrease as much as CV_a increased in the spine volume (Fig. 4, G and H). Why was the mutual information in the spine volume more robust against CV_a than that in the cell volume?

The change of the probability density distribution of Ca_res against CV_a in the spine volume was smaller than that in the cell volume (Fig. 4, A–F). Generally, when an input distribution and an output distribution do not change, the mutual information between the input and output remains the same. In other words, the change of the mutual information I(Ca_res/Δt) can be caused by the change of the probability density distribution of Ca_res against CV_a.

Therefore, we examined the change of the probability density distribution of Ca_res against CV_a in more detail, fixing the pre-postspiking interval Δt. We previously set the probability density distribution of Amp_pre, p_a(Amp_pre/μ_a,σ_a), as the Gaussian distribution N(Amp_pre/μ_a,σ_a). In addition, we obtained the probability density distribution of Ca_res against Amp_pre, p_c(Ca_res/Amp_pre), from the stochastic simulation (Fig. 5, A, C, and E). Thus, the probability density distribution of Ca_res against CV_a, p_ac(Ca_res/μ_a,σ_a) is given by the following:

$$\begin{array}{c}{p}_{ac}\left(C{a}_{res}|{\mu }_a,{\sigma }_a\right)=\underset{Am{p}_{pre}}{\int }{p}_a\left(a|{\mu }_a,{\sigma }_a\right)p_c\left(C{a}_{res}|a\right)da\\ =\underset{Am{p}_{pre}}{\int }N\left(a|{\mu }_a,{\sigma }_a\right)p_c\left(C{a}_{res}|a\right)da\end{array},$$ (6)

(Fig. 5, B, D, and F). Here, aεAmp_pre. In the spine volume, the probability density distribution of Ca_res remained similar with an increase in the CV_a (Fig. 5 B). In larger volumes, including the cell volume, the probability density distribution of Ca_res changed greatly as CV_a increased (Fig. 5, D and F).

Figure 5.

Changes of probability density distributions of Ca_res with the increase in presynaptic fluctuation. (A, C, and E) Shown are the probability density distributions of Ca_res against Amp_pre in the spine volume (A), in the intermediate volume (C), and in the cell volume (E) at the pre-postspiking interval Δt = 10 ms. Amp_pre indicates the amplitude of prespiking. (B, D, and F) Shown are the probability density distributions of Ca_res against the presynaptic fluctuation CV_a in the spine volume (B), in the intermediate volume (D), and in the cell volume (F) at the pre-postspiking interval Δt = 10 ms. (G) Shown is the CV_a dependency of x² distance between the probability density distribution of Ca_res at CV_a and that of Ca_res at CV_a = 0.0 at the pre-postspiking interval Δt = 10 ms. To see this figure in color, go online.

Next, we quantified the change of the probability density distribution of Ca_res against CV_a based on the x² distance between the probability density distribution of Ca_res at CV_a = 0.0 and the probability density distribution of Ca_res with the increase in the CV_a (Fig. 5 G). The x² distance is the index of the difference between two probability density distributions, which is given by the following:

$$\begin{array}{c}\underset{X}{\int }\frac{{\left(p_0\left(x\right)-p_1\left(x\right)\right)}^2}{p_0\left(x\right)+p_1\left(x\right)}dx\end{array},$$ (7)

where p₀(x) and p₁(x) are probability density distributions. The x² distance is zero when two distributions are the same, and it is one when two distributions are completely different. In the cell volume, the x² distance increased greatly with the increase in CV_a and was 0.87 at CV_a = 0.5. In the spine volume, the x² distance remained almost the same regardless of CV_a and was 0.042 at CV_a = 0.5. Similar results were obtained with other Δt (Fig. S4). Thus, the change of the probability density distribution of Ca_res in the spine volume was much smaller than that in the cell volume with an increase in CV_a. This result indicates that the unchanging probability density distribution of Ca_res against the presynaptic fluctuation in the spine volume, but not in the larger cell volume, is the reason for the robustness.

Larger intrinsic noise than extrinsic noise is critical for the robustness

Why was the probability density distribution of Ca_res against CV_a unchanged in the spine volume but not in the larger volume? We derived the condition for robustness, following the previous study (20). As described above, the probability density distribution of Ca_res against CV_a, p_ac(Ca_res|μ_a,σ_a) is given by Eq. 6. Here, when we define x = Amp_pre − μ_a, the relative amplitude of prespiking, Eq. 6 can be changed as follows:

$$\begin{array}{c}{p}_{ac}\left(C{a}_{res}\text{|}{\mu }_a,{\sigma }_a\right)=\underset{-\infty }{\overset{\infty }{\int }}N\left(x\text{|}0,{\sigma }_a\right)p_c\left(C{a}_{res}\text{|}{\mu }_a+x\right)dx\end{array}.$$ (8)

When x is very large or very small, N(x/0,σ_a) is also small. In particular, the probability where x is included in the range −3σ_a ≤ x ≤ 3σ_a, which is given by $\underset{-3{\sigma }_a}{\overset{3{\sigma }_a}{\int }}N(x|0,{\sigma }_a)dx=0.9974$, that is, ≈ 1. Thus, the probability of x < −3σ_a or x > 3σ_a is quite small and almost negligible. Therefore, Eq. 8 can be approximated as follows:

$$\begin{array}{c}{p}_{ac}\left(C{a}_{res}|{\mu }_a,{\sigma }_a\right)\simeq \underset{-3{\sigma }_a}{\overset{3{\sigma }_a}{\int }}N\left(x\text{|}0,{\sigma }_a\right)p_c\left(C{a}_{res}\text{|}{\mu }_a+x\right)dx\end{array}.$$ (9)

Because N(x/0,σ_a) is symmetric with respect to x = 0, Eq. 9 can be changed as follows:

$$\begin{array}{c}{p}_{ac}\left(C{a}_{res}|{\mu }_a,{\sigma }_a\right)\simeq \underset{0}{\overset{3{\sigma }_a}{\int }}N\left(x|0,{\sigma }_a\right)\left[p_c\left(C{a}_{res}|{\mu }_a+x\right)+p_c\left(C{a}_{res}|{\mu }_a-x\right)\right]dx\end{array}.$$ (10)

Here, we consider the condition where the averaged probability density distribution between the probability density distribution of Ca_res with Amp_pre = μ_a + x and that with Amp_pre = μ_a − x (i.e., $\left(1/2\right)\left[p_c\left(C{a}_{res}\text{|}{\mu }_a+x\right)+p_c\left(C{a}_{res}\text{|}{\mu }_a-x\right)\right]$) is almost the same with the probability density distribution of Ca_res with Amp_pre = μ_a, p_c(Ca_res/μ_a), under xε[0,3σ_a], given by the following:

$$\begin{array}{c}\forall x\in \left[\mathrm{0,3}{\sigma }_a\right]:p_c\left(C{a}_{res}|{\mu }_a\right)\simeq \frac{1}{2}\left[p_c\left(C{a}_{res}|{\mu }_a+x\right)+p_c\left(C{a}_{res}|{\mu }_a-x\right)\right]\end{array}.$$ (11)

When Eq. 11 is satisfied, Eq. 10 can be calculated as follows:

$$\begin{array}{c}{p}_{ac}\left(C{a}_{res}|{\mu }_a,{\sigma }_a\right)\simeq p_c\left(C{a}_{res}|{\mu }_a\right)\end{array}.$$ (12)

Considering the meaning of Eq. 12, we consider Eq. 6 when CV_a = 0.0. When CV_a = 0.0, that is, σ_a = μ_a × CV_a = 0.0, N(a/μ_a,σ_a) becomes the Dirac δ function δ(a − μ_a). Therefore, p_ac(Ca_res|μ_a,σ_a) is equal to p_c(Ca_res|μ_a) at CV_a = 0.0. Hence, Eq. 12 means that the probability density distribution of Ca_res does not change against CV_a. Therefore, Eq. 11 is the sufficient condition for the robustness.

Here, we can derive the sufficient condition in which Eq. 11 is satisfied, following the previous study (20) (see Supporting Materials and Methods, Note S1), which is given by the following:

$$\begin{array}{c}\forall x\in \left[\mathrm{0,3}{\sigma }_a\right]:\mathit{\Delta }C{a}^{\ast }\left(x\right)\ll {\sigma }_c\end{array},$$ (13)

where $x$ denotes the relative amplitude of prespiking, Amp_pre − μ_a, ΔCa^∗(x) denotes the gap between the peak of p_c(Ca_res|μ_a) and the peak of p_c(Ca_res|μ_a + x), which corresponds to extrinsic noise, and σ_c denotes the SD of p_c(Ca_res|μ_a), which corresponds to intrinsic noise. When $\mathit{\Delta }C{a}^{\ast }\left(x\right)\ll {\sigma }_c$ is satisfied for larger x, the probability density distribution of Ca_res is unchanged against larger CV_a. In other words, when the range where the extrinsic noise, ΔCa^∗(x), was much smaller than the intrinsic noise, σ_c, is wider, the probability density distribution of Ca_res is more robust against CV_a.

We examined the range of presynaptic fluctuation satisfying the condition for robustness. In the spine volume, $\mathit{\Delta }C{a}^{\ast }\left(x\right)/{\sigma }_c\ll 1$ when x was small, and ΔCa^∗(x)/σ_c increased with the increase in x (Fig. 6 A). In contrast, in the cell volume, ΔCa^∗(x)/σ_c exceeds one even when x was small (Fig. 6 A). Therefore, in the spine volume, information transfer by Ca_res is robust with a larger CV_a because ΔCa^∗(x)/σ_c was smaller than one even with a large x. In contrast, in the cell volume, information transfer by Ca_res is not robust even with a small CV_a because ΔCa^∗(x)/σ_c was larger than one even with a small x.

Figure 6.

Volume dependency of the ratio between extrinsic noise and intrinsic noise. (A) Shown is the relative amplitude of prespiking, x, dependency of ΔCa^∗/σ_c = 1 with μ_a = 6.0 μM at Δt = 10 ms. The dashed line indicates ΔCa^∗/σ_c = 1. The robustness index δ_max is defined as x giving ΔCa^∗/σ_c = 1. (B) The volume dependency of δ_max is at Δt = 10 ms. (C) Shown is the relationship between ΔCa^∗/σ_c and the x² distance between the averaged probability density distribution of the probability density distributions of Ca_res with Amp_pre = μ_a ± x and that of Ca_res with Amp_pre = μ_a at Δt = 10 ms. The red diamonds, blue circles, and black dots indicate the value obtained in the spine volume, cell volume, and intermediate volumes, respectively. To see this figure in color, go online.

We defined δ_max as x that gives ΔCa^∗(x)/σ_c = 1. δ_max provides the relative upper bound of x where $\mathit{\Delta }C{a}^{\ast }/{\sigma }_c\ll 1$. Thus, we used δ_max as the index of robustness (Fig. 6 B). A larger δ_max means greater robustness. δ_max was highest at the spine volume and decreased with the increase in volume, indicating that the spine volume gives the highest robustness.

Next, we confirmed that when ΔCa^∗(x)/σ_c is smaller than one, the averaged probability density distribution between the probability density distribution of Ca_res with Amp_pre = μ_a + x and that with Amp_pre = μ_a − x (i.e., $\left(1/2\right)\left[p_c\left(C{a}_{res}|{\mu }_a+x\right)+p_c\left(C{a}_{res}|{\mu }_a-x\right)\right]$) is almost the same with the probability density distribution of Ca_res with Amp_pre = μ_a, p_c(Ca_res|μ_a). In other words, we confirmed that Eq. 13 can be the sufficient condition for Eq. 11. We quantified the similarities between the two probability density distributions of Ca_res by the x² distance. In the spine volume (Fig. 6 C, red), most of the ΔCa^∗(x)/σ_c values were smaller than one, and the x² distance was also small, indicating that ΔCa^∗(x)/σ_c is smaller than one, and the two probability density distributions of Ca_res are quite similar in the spine volume. In contrast, in the cell volume (Fig. 6 C, blue), most ΔCa^∗(x)/σ_c values were larger than one, and the x² distances were almost one, indicating that ΔCa^∗(x)/σ_c is larger than one, and the two distributions of Ca_res are quite different. Similar results were obtained with other Δt (Fig. S6). Therefore, when ΔCa^∗(x), which corresponds with the extrinsic noise, is smaller than σ_c, which corresponds with the intrinsic noise, the probability density distribution of Ca_res does not change and becomes robust against presynaptic fluctuation.

The reason why the probability density distribution of Ca_res against CV_a is more robust in the smaller volume can be also intuitively explained as follows. The probability density distribution of Ca_res against CV_a, p_ac(Ca_res|μ_a,σ_a), is calculated by the convolution integral between the probability density distribution stemming from the extrinsic noise, p_a(Amp_pre|μ_a,σ_a), and the probability density distribution stemming from the intrinsic noise, p_c(Ca_res|Amp_pre), as described by Eq. 6. When the SD of the extrinsic noise is smaller than that of the intrinsic noise, the probability density distribution of Ca_res against CV_a does not change so much. However, if the SD of the intrinsic noise is small and is similar to Dirac δ function, the extrinsic noise has a noticeable impact to the probability density distribution of Ca_res against CV_a.

Many small spines versus a single large spine

Why are spines so many? We address this issue with regard to NMDAR-mediated Ca²⁺ increase. We showed that the mutual information between the pre-postspiking interval and the Ca²⁺ increase is more robust in the smaller volume. However, the mutual information between the pre-postspiking interval and the Ca²⁺ increase decreased as the volume decreased (Fig. 7 A). The mutual information was only 0.45 bit in the spine volume, indicating that the information transfer in the spine is insufficient to reliably determine even a binary decision such as whether prespiking and postspiking are coincident or not. However, spines are many. Around 10,000 spines have been estimated on a single cell (1, 2, 3, 4), and several spines are thought to receive the same input, raising the possibility that the input timing information is coded by the integrated Ca²⁺ increase over spines.

Figure 7.

Mutual information in many small spines versus a single large spine. (A) The volume dependency of the mutual information between the pre-postspiking interval Δt and the Ca_res. (B) The number dependency of the mutual information is shown. Their volumes are the spine volume (10⁻¹μm³). (C) Shown is the CV_a dependency of the mutual information in a single large “cell-volume” spine (10³μm³) (red) and in many small “spine-volume” spines (blue). (D) Shown is the CV_a dependency of the mutual information in spines with varying volume and number while keeping the total volume (volume × number) constant (10³ μm³). To see this figure in color, go online.

Therefore, we set the mean of the Ca²⁺ increase over multiple trials as the integrated Ca²⁺ increase in many spines and examined the mutual information between the pre-postspiking interval and the Ca²⁺ increase in many spines (Fig. 7 B). The mutual information increased with the increase in number at CV_a = 0.0. The mutual information was around 1 bit with 3–10 spines at CV_a = 0.0. This result indicates that tens of spines can cooperatively code sufficient input timing information.

We further examined the mutual information of many small spines as CV_a increased. Although the mutual information of a single large spine decreased greatly with the increase in CV_a (Fig. 7 A), the mutual information of many small spines remained the same regardless of CV_a (Fig. 7 B). This result indicates that the increase in volume increases the mutual information but loses the robustness, whereas the increase in number increases the mutual information without losing the robustness.

We compared the mutual information against the presynaptic fluctuation in many small “spine-volume” spines with that in a single large “cell-volume” spine. In the many small “spine-volume” spines, the mutual information remained the same regardless of CV_a (Fig. 7 C, blue), whereas in the single large “cell-volume” spine, the mutual information abruptly decreased with the increase in CV_a (Fig. 7 C, red). The mutual information in a single large “cell-volume” spine was equal to that in 10,000 small “spine volume” spines at CV_a = 0.0, whereas the mutual information in a single large “cell-volume” spine was smaller than that in only 10 small “spine-volume” spines at CV_a = 0.5. This result indicates that many small spines are much more efficient for information transfer than a single large spine when presynaptic fluctuation is large.

Finally, we compared the mutual information in spines with a varying volume and number while keeping the total volume (volume × number) constant (10³ μm³) (Fig. 7 D). When CV_a = 0.0, every case has almost the same mutual information. However, when CV_a is large, the mutual information in many smaller spines is larger than that in few larger spines. This result indicates that many smaller spines are more advantageous for information transfer than few larger spines when presynaptic fluctuation is large.

Discussion

In this study, we addressed the issue of why dendritic spines are small and many by using a stochastic simulation model of NMDAR-mediated Ca²⁺ increase. We found that the input timing information encoded to NMDAR-mediated Ca²⁺ increase largely decreases against the presynaptic fluctuation in the cell volume, whereas it slightly decreases in the spine volume. Therefore, the smallness of a spine enables robust information transfer of input timing information against the presynaptic fluctuation. The robustness appears when the intrinsic noise (i.e., the SD of the Ca²⁺ increase) is larger than the extrinsic noise (the change of the peak of the Ca²⁺ increase caused by the presynaptic fluctuation). The intrinsic noise in the spine volume was much larger than that in the cell volume, and thus the information transfer in the spine volume is more robust than that in the cell volume.

We also found that the manyness of spines increases the input timing information. The input timing information a single small spine can code is much less than 1 bit, but many small spines can cooperatively code sufficient input timing information. In addition, we also demonstrated that the increase in volume increases the input timing information but loses the robustness, whereas the increase in number increases the input timing information without robustness.

Finally, we compared the input timing information in spines with varying volume and number while keeping their total volume (volume × number) constant. We demonstrated that the input timing information many small spines can code are larger than that few larger spines can code when presynaptic fluctuation is large because of their robustness. Therefore, “small and many” is an advantageous strategy for information transfer in noisy environments.

We previously demonstrated that mGluR-mediated Ca²⁺ increase in the spine volume also realizes the robust information transfer (Fig. S8) (19, 20). Thus, the robust information transfer in the smaller volume is the conserved characteristic for NMDAR-mediated and mGluR-mediated Ca²⁺ increase. Importantly, the mechanism of NMDAR-mediated Ca²⁺ increase is totally different from that of mGluR-mediated Ca²⁺ increase; mGluR-mediated Ca²⁺ increase is a complex reaction of an excitable system with a positive feedback loop and delayed negative feedback, which shows threshold response, whereas NMDAR-mediated Ca²⁺ increase is a single reaction, which shows graded response. For these different mechanisms, the distributions of the Ca²⁺ increase show the different qualitative characteristics; the distribution of mGluR-mediated Ca²⁺ increase in the smaller volume becomes bimodal (Fig. S8), whereas the distribution of NMDAR-mediated Ca²⁺ increase in the smaller volume becomes unimodal (Fig. 2 G). In other words, mGluR-mediated Ca²⁺ increase codes the input timing information to the probability of the Ca²⁺ increase, whereas NMDAR-mediated Ca²⁺ increase codes the input timing information to the amplitude of Ca²⁺ increase. Despite these different coding styles, both NMDAR-mediated and mGluR-mediated Ca²⁺ increase shows the robustness in a small volume. This result indicates that the robustness depends not on structures of biochemical reactions but only small volumes. Therefore, the robust information transfer may also be realized in other biochemical reactions in spines or in small intracellular organelles, such as intracellular vesicles and mitochondria.

In addition to the robustness, mGluR-mediated Ca²⁺ increase shows efficiency in information transfer and sensitivity to lower amplitude of input (Fig. S8) (19, 20). We also examined whether NMDAR-mediated Ca²⁺ increase shows these properties (see Supporting Materials and Methods, Note S2). Although NMDAR-mediated Ca²⁺ increase shows efficiency, it does not show sensitivity (Fig. S8). This result indicates that robustness and efficiency in the previous study does not depend on structures of biochemical reactions, whereas sensitivity is specific to excitable systems.

In this study, we assumed that the input timing information is coded by the integrated Ca²⁺ increase over many spines. We can also assume that the input timing information is coded by the integrated Ca²⁺ increase in a single spine over time. Indeed, the induction of long-term potentiation and long-term depression mediated by NMDAR at excitatory synapses requires a repetitive stimulation of pre- and postspiking stimulation (5, 6, 7). Repetitive stimulation of large NMDAR-mediated Ca²⁺ increase activates calcium/calmodulin-dependent protein kinase II (CaMKII), which is thought to play an important role in learning and memory (46). Once CaMKII is activated, CaMKII can remain active several minutes after Ca²⁺ returns to basal level because of their autophosphorylation. Therefore, CaMKII is thought to integrate multiple Ca²⁺ increases over time (12). The robust and sufficient information transfer can also be realized by the integration of the Ca²⁺ increase over time, like that over many spines.

Author Contributions

T.T., M.F., and S.K. designed research. T.T. performed research. T.T., M.F., and S.K. analyzed data. T.T. and S.K. wrote the article.

Editor: Arthur Sherman.