Asymptotic Analysis of the Narrow Escape Problem in Dendritic Spine Shaped Domain: Three Dimension

# Asymptotic Analysis of the Narrow Escape Problem in Dendritic Spine Shaped Domain: Three Dimension

Hyundae Lee Department of Mathematics, Inha University, Incheon 402-751, Korea (hdlee@inha.ac.kr).    Xiaofei Li Department of Mathematics, South University of Science and Technology of China, Shenzhen, China (xiaofeilee@hotmail.com). Corresponding author.    Yuliang Wang Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR (jadelightking@qq.com).
###### Abstract

This paper deals with the three-dimensional narrow escape problem in dendritic spine shaped domain, which is composed of a relatively big head and a thin neck. The narrow escape problem is to compute the mean first passage time of Brownian particles traveling from inside the head to the end of the neck. The original model is to solve a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper we seek to transfer the original problem to a mixed Robin-Neumann boundary value problem by dropping the thin neck part, and rigorously derive the asymptotic expansion of the mean first passage time with high order terms. This study is a nontrivial generalization of the work in [14], where a two-dimensional analogue domain is considered.

## 1 Introduction

The narrow escape problem (NEP) in diffusion theory, which goes back to Lord Rayleigh [18], is to calculate the mean first passage time (MFPT) of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. NEP has recently attracted significant attention from the point of view of mathematical and numerical modeling due to its relevance in molecular biology and biophysics. The small absorbing window often represents a small target on a cellular membrane, such as a protein channel, which is a target for ions [8], a receptor for neurotransmitter molecules in a neuronal synapse [5], a narrow neck in the neuronal spine, which is a target for calcium ions [13], and so on. A main concern for NEP is to derive an asymptotic expansion of the MFPT when the size of the small absorbing window tends to zero. There have been several significant works deriving the leading-order and higher-order terms of the asymptotic expansions of MFPT for regular and singular domains in two and three dimensions [1, 2, 3, 6, 9, 10, 11, 19, 20].

In this paper we focus on the MFPT of calcium ion in the three-dimensional dendritic spine shape domain. For many neurons in the mammalian brain, the postsynaptic terminal of an excitatory synapse is found in a specialized structure protruding from the dendritic draft, known as a dendritic spine (Figure 1(a)). Dendritic spines function as biochemical compartments that regulate the duration and spread of postsynaptic calcium fluxes produced by glutamatergic neurotransmission at the synapses [16]. Calcium ions are ubiquitous signaling molecules that accumulate in the cytoplasm in response to diverse classes of stimuli and, in turn, regulate many aspects of cell function [7]. The rate of diffusional escape from spines through their narrow neck, which we call MFPT, is one factor that regulates the retention time of calcium ions in dendritic spines.

Most spines have a bulbous head, and a thin neck that connects the head of the spine to the shaft of the dendrite (Figure 1). Spines are separated from their parent dendrites by this thin neck and compartmentalize calcium during synaptic stimulation. It has been given that many organelles inside the spine head do not affect the nature of the random motion of ions, mainly due to their large size relative to that of ions [12]. So we assume the environment inside the spine is homogenous, which means the movement of the calcium ions is pure diffusion.

The NEP can be mathematically formulated in the following way. Let be a bounded simply connected domain in . Let be a cylinder with length and radius which is much smaller than the length. Connecting this cylinder with , we have the geometry for the spine (Figure 1(b)). The connection part between and is a small interface which is denoted by . Let denote the domain for the whole spine. Suppose that the boundary is decomposed into the reflecting part and the absorbing part , where is the end of the thin cylinder neck. We assume that the area of , is much smaller than the area of the whole boundary. The NEP is to calculate the MFPT which is the unique solution to the following boundary value problem,

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩Δu=−1,in Ω,∂u∂ν=0,on ∂Ωr,u=0,on ∂Ωa, (1)

where is the outer unit normal to . The asymptotic analysis for NEP arises in deriving the asymptotic expansion of as , from which one can estimate the escape time of the calcium ions. In [15, 4], the first order of the asymptotic expansion has been obtained numerically as

 u≈|Ωh|Lπε2, (2)

where denotes the volume of the spine head. In this study, we derive higher order asymptotic solution to (1) by means of the Neumann-Robin model which is proposed in [14] to deal with the narrow escape time in a two-dimensional analogue domain.

In the Robin-Neumann model the solution to the original boundary value problem (1) in the singular domain is approximated by the solution to the following boundary value problem in the smooth domain :

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩Δuε=−1,in Ωh,∂uε∂ν=0,on ∂Ωr,∂uε∂ν+αuε=β,on Γε:=∂Ωh∖¯¯¯¯¯¯¯¯¯∂Ωr, (3)

where is the spine head of mentioned in Figure 1, and is the connection part between the spine head and the spine neck. Here and are constants to be determined. We assume for some a priori constant and is sufficiently small so that . We shall apply the layer potential technique to derive the asymptotic solution to (3) as

 uε(x)≈|Ωh|παε2+|Ωh|Mπ2ε+βα−|Ωh|2π|x−x∗|

for and away from , where is a computable constant, is a fixed point in .

This study is organized as follows. In section 2, we review the Neumann function for the Laplacian in , which is a major tool for our study. In section 3, we derive the asymptotic solution for Robin-Neumann model. In section 4, we apply the Robin-Neumann boundary model to approximate the MFPT of calcium ion in dendritic spine. Numerical experiments are also given in this section to confirm the theoretical results. This study ends with a short conclusion in section 5.

## 2 Neumann function in R3

Let be a bounded domain in with smooth boundary , and let be the Neumann function for in with a given . That is, is the solution to the boundary value problem

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ΔxN(x,z)=−δz,x∈Ω,∂N∂νx=−1|∂Ω|,x∈∂Ω,∫∂ΩN(x,z)dσ(x)=0,

where is the outer unit normal to the boundary .

If , then can be written in the form

 N(x,z)=14π|x−z|+RΩ(x,z),x∈Ω,

where has weaker singularity than and solves the boundary value problem

 ⎧⎪⎨⎪⎩−ΔxR∂Ω(x,z)=0,x∈Ω,∂RΩ∂νx∣∣x∈∂Ω=−1|∂Ω|+14π⟨x−z,νx⟩|x−z|3,x∈∂Ω.

where denotes the inner product in .

If , then Neumann function on the boundary is denoted by and can be written as

 N∂Ω(x,z)=12π|x−z|+R∂Ω(x,z),x∈Ω,z∈∂Ω, (4)

where has weaker singularity than and solves the boundary value problem

 ⎧⎪⎨⎪⎩ΔxR∂Ω(x,z)=0,x∈Ω,∂R∂Ω∂νx∣∣x∈∂Ω=−1|∂Ω|+12π⟨x−z,νx⟩|x−z|3,x∈∂Ω, z∈∂Ω.

The structure of is given in [17] as

 R∂Ω(x,z)=−14πH(z)ln|x−z|+v∂Ω(x,z), (5)

where , , where denotes the mean curvature of at , and is a bounded function.

## 3 Derivation of the asymptotic expansion

The goal in this section is to derive the asymptotic expansion of to (3) as . For simplicity we assume the connection part lies in a plane. The general case where is curved can be handled with minor modifications.

###### Theorem 3.1.

The solution to the boundary value problem (3) has the following asymptotic expansion,

 uε(x)=|Ωh|παε2+|Ωh|Mπ2ε+βα−|Ωh|2π|x−x∗|+Φ(x)+O(α),

where is a constant given by

 M=∫Γ1∫Γ112π|x−z|dxdz, (6)

is a fixed point in , and is a bounded function depending only on . The remainder is uniform in satisfying dist for some constant .

Proof. By integrating the first equation in (3) over using the divergence theorem we get the compatibility condition

 ∫Γε∂uε∂νdσ=−|Ωh|. (7)

Let us define by

 g(x)=∫ΩhN(x,z)dz,x∈Ωh,

which is seen to solve the boundary value problem

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Δg=−1,in Ωh,∂g∂ν=−|Ωh||∂Ωh|,on ∂Ωh,∫∂Ωhgdσ=0. (8)

Applying the Green’s formula and using (3) and (8), we obtain

 uε(x)=g(x)+∫ΓεN∂Ωh(x,z)∂uε(z)∂νzdσ(z)+Cε, (9)

where

 Cε=1|∂Ωh|∫∂Ωhuε(z)dσ(z).

Let . Substitute the Robin boundary condition and the structure of Neumann function (5) into (9), we obtain

 βα−1αϕε(x)=g(x)+12π∫Γε1|x−z|ϕε(z)dσ(z)+∫Γεv∂Ωh(x,z)ϕε(z)dσ(z)+Cε,x∈Γε,

where . Note that the mean curvature for since is assumed to be flat.

By a simple change of variables, the above equation can be written as

 βα−1αε~ϕε(x)=g(εx)+12π∫Γ11|x−z|~ϕε(z)dσ(z)+ε∫Γ1v∂Ωh(εx,εz)~ϕε(z)dσ(z)+Cε, (10)

where , and , .

Define two integral operators by

 L[φ](x) =12π∫Γ11|x−z|φ(z)dz, L1[φ](x) =∫Γ1v∂Ωh(εx,εz)φ(z)dz.

Since is bounded, one can easily see that is bounded independently of . The integral operator is a also bounded (see the proof in Appendix A).

So we can write (10) as

 βα−1αε~ϕε(x)=g(εx)+(L+εL1)~ϕε(x)+Cε.

Collecting terms, we have

 [I+αε(L+εL1)]~ϕε(x)=εα(βα−g(εx)−Cε). (11)

Assume here and . It is easy to see that

 (I+αε(L+εL1))−1=I−αε(L+εL1)+O(α2ε2).

Noting that is in , and on , we have from (11) that

 ~ϕε(x)=εα[I−αε(L+εL1)+O(α2ε2)](~Cε+O(ε)).

where

 ~Cε:=βα−g(x∗)−Cε. (12)

By the compatibility condition (7), we can see that . Then collecting terms we have

 ~ϕε(x)=εα~Cε−(εα)2~Cε(L[1]+εL1[1])+O(αε2). (13)

Plug (13) into the compatibility condition (7), we obtain

 παε2~Cε−α2ε3~Cε∫Γ1(L[1]+εL1[1])(x)=−|Ωh|+O(αε3),

which implies

 ~Cε=(I+αεπ∫Γ1(L[1]+εL1[1])(x)+O(α2ε2))(−|Ωh|παε2+O(ε))=−|Ωh|παε2−|Ωh|π2εM−|Ωh|v∂Ω(x∗,x∗)+O(α),

where , and . Hence from (12), we have

 Cε=|Ωh|παε2+|Ωh|π2εM+|Ωh|v∂Ω(x∗,x∗)+βα−g(x∗)+O(α). (14)

Substitute into (13), we have

 ~ϕε(x)=−|Ωh|πε−α|Ωh|π2(M−πL[1])+O(α2ε),

and hence

 ∂uε(x)∂νx=ϕε(x)=1ε~ϕε(xε)=−|Ωh|πε2−α|Ωh|π2ε[M−πL[1](xε)]+O(α2). (15)

In order to obtain the solution to (3), it remains to calculate the second term in (9). Combining (4), (5) and (15) yields

 ∫ΓεN∂Ωh(x,z)∂uε(z)∂νzdσ(z) =∫ΓεN∂Ωh(x,z)[−|Ωh|πε2−α|Ωh|π2ε(M−πL[1](xε)+O(α2))]dσ(z) =−|Ωh|N∂Ωh(x,x∗)+O(αε) =−|Ωh|2π|x−x∗|+|Ωh|v∂Ωh(x,x∗). (16)

provided that for some constant .

Finally, combining(9), (14) and (16), we obtain

 uε(x)=|Ωh|παε2+|Ωh|Mπ2ε+βα−|Ωh|2π|x−x∗|+Φ(x)+O(α),

where

 Φ(x)=g(x)−g(x∗)+|Ωh|[v∂Ωh(x∗,x∗)−v∂Ωh(x,x∗)]

is a bounded function depending only on .

## 4 Application to the narrow escape problem in dendritic spine

In this section we use the asymptotic solution to the Robin-Neumann model to approximate the calcium ion diffusion time in a dendritic spine domain. Without loss of generality, let be placed along the -axis with at and at . Since , we assume is constant in each cross section of and thus is a function of only. The three-dimensional problem (1) restricted in is then approximated by the one-dimensional problem

 ⎧⎪⎨⎪⎩d2udx21=−1,0

By direct calculation we obtain the solution to the above problem as

 u(x1)=−12x21+Cx1+12L2−CL,0

where is a constant. Solution satisfies the Robin boundary condition at . Evaluating and at yields the Robin condition

 dudx1(0)+1Lu(0)=L2.

By the continuity of and on , we obtain the Robin-Neumann boundary value problem in :

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩Δuε=−1,in Ωh,∂uε∂ν=0,on ∂Ωr,∂uε∂ν+αuε=β,on Γε.

where and . Applying Theorem 3.1 we obtain the asymptotic expansion of as

 uε(x)≈|Ωh|Lπε2+|Ωh|Mπ2ε+L22−|Ωh|2π|x−x∗|, (17)

Note that the leading term coincides with (2), which is obtained in [15, 4] using numerical simulation.

### 4.1 Numerical experiments

In the rest of this section we shall conduct numerical experiments to verify the asymptotic expansion (17). We shall compare the asymptotic solution (17) with the solution to the original problem (1) obtained numerically with the finite element method. We shall also confirm the coefficients in the first two terms in (17).

For simplicity we confine ourself to the case when the connection part is a disk of radius . In this case the constant in has an explicit and elegant value (see Appendix B). The spine neck is chosen to be a cylinder with radius and length , and whose axis is parallel to the normal of . To nondimensionalize our problem, the numerical results are regarded as using consistent units throughout this section.

For the first experiment, we select the spine head as an unit ball and set . In Figure 2(a) we plot the numerical solution of on , i.e. the whole surface of the spine. Observe that the MFPT is relatively large in the spine head and decreases monotonically to zero towards the end of the spine neck. This is consistent with our intuition about the underlying physical process. In Figure 2(b) we plot the asymptotic solution according to (17) on , i.e. the surface of the spine head. Note that is relatively constant but is smaller for points closer to the connection part, which is also consistent with the physical intuition. In Figure 2(c) we plot the relative error between and , computed as , on the surface of the spine head. The maximal relative error is seen to be about .

For the next experiment, we fix the neck length and let the neck radius decreases from to in a step size of . For each value of , we compare the value of the numerical solution to the original problem (1), the numerical solution of the Robin-Neumann model (3), which is denoted by , and the asymptotic solution given in (17). In Table 1 we list the value of , as well as the relative error , at the center of the spine head. Clearly we have a good match of the solutions and small relative error for all values of .

The asymptotic solution in (17) at a fixed can be considered as a quadratic polynomial of with leading coefficients and . We now confirm these coefficients by fitting the value of by a quadratic polynomial of in Table 1. The result is plotted in Figure 3. Clearly the coefficients of the fitting polynomial matches well with those of the asymptotic solution.

Another important parameter in the asymptotic solution (17) is , the length of the spine neck. For the next experiment, we fix the neck radius and let the neck radius increases from to in a step size of . In Table 2 we list the value of , as well as the relative error , at the center of the spine head. Clearly we have a good match of the solutions and small relative error for each value of .

Finally we conduct numerical experiments on three different shapes of the spine head, which may correspond to different types of spine. The numerical solution , the asymptotic solution and the relative error between them are shown in Figure 4. The first row shows the results when the spine head is a relatively flat three dimensional domain. The second one is a relatively thin domain and the third one is a non-convex domain. From the relative error , which is described in the third column, we can see that the relative error is small enough to show that asymptotic formula is a good approximation to the MFPT.

## 5 Conclusion

In this study, we used the Robin-Neumann model to compute the NEP in a three-dimensional domain with a long neck, which is often referred as a dendritic spine domain. This is a following up paper of [14], where the Robin-Neumann model is presented to solve the NEP in a two-dimensional analogue of a dendritic spine domain. In this paper, we derived asymptotic expansion formula for three-dimensional Robin-Neumann model. Our results demonstrate that this asymptotic expansion formula could approximate the MFPT up to at least second leading order using this model, which has not been reported previously.

The work of Hyundae Lee was supported by the National Research Foundation of Korea grant (NRF-2015R1D1A1A01059357). The work of Yuliang Wang was supported by the Hong Kong RGC grant (No. 12328516) and the NSF of China (No. 11601459).

## 6 Appendix

### 6.1 Appendix A.

###### Lemma 6.1.

(Single layer potential on an surface) Let be a bounded domain with . The integral operator : defined by

 L[ϕ](x)=∫Γ11|x−z|ϕ(z)dz

is bounded, i.e.

 ∥S[ϕ]∥L∞(Γ1)≤C∥ϕ∥L∞(Γ1),

where is a constant independent of .

###### Proof.

Let denote the disk centered at with radius . For given we have

 ∫Γ11|x−z|dz≤∫Bx,21|x−z|dz=∫2π0∫201ρρdρdθ=4π.

Hence

 ∣∣∣∫Γ11|x−z|ϕ(z)dz∣∣∣≤∥ϕ∥L∞(Γ1)∫Γ11|x−z|dz≤4π∥ϕ∥L∞(Γ1),x∈Γ1.

Therefore is bounded.

### 6.2 Appendix B.

The value of

 ∫Γ1∫Γ11|x−y|dxdy

is , where is unit disk.

###### Proof.

First, fix point , draw a circle centered at with radius , where the distance between and is . Let be the distance between and .

Then we have

 ∫Γ1∫Γ11|x−y|dxdy=∫102πs∫1−s02πdrds+∫102πs∫1+s1−ssarccoss2+r2−12srdrds. (18)

When , by simple calculation, we can calculate the first term of (18) as

 ∫102πs∫1−s02πdrds=23π2.

Changing the order of and , the second term of (18) becomes

 4π∫10∫11−rsarccoss2+r2−12srdsdr+4π∫21∫1r−1sarccoss2+r2−12srdsdr. (19)

By calculations, we get the value of the first term of (19) as , and second term .

Finally, we get

 ∫Γ1∫Γ11|x−y|dxdy=163π.

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