• 沒有找到結果。

Figure 4.8: Average all the recipient unit activity changes in the ACC, InC and MI when the rat was opening the gate for a toy rat or nothing inside the box. Values in the Y-axis are normalized Z-score calculated from ACC neurons which received projection from InC (A, n = 8), MI (B, n = 6), and ACC (C, n = 8) units; InC neurons which received projection from ACC (D, n = 7), MI (E, n = 2), and InC (F, n = 14) units; MI neurons which received projection from InC (G, n = 4), ACC (H, n = 9), and MI (I, n = 12) units. The red line represents the 99 % confidence interval. Bin size = 100 ms.

Evidence from human studies implicates that the ACC and InC are involved in em-pathy [23, 36, 46, 64, 70]. According to our previous findings, neurons in the ACC and InC showed activity specifically increased prior to rescuing acts and relate to empathic driven prosocial behaviors (Wu et al., 2016). However, the neuronal flow direction and causal information remain unclear. Therefore, we utilize the proposed method to assess this issue. By means of the method, we discover causal relationship among ACC, InC and MI under the empathy-like behavior task. The results show that the activity of the project-ing neurons in InC connected with MI was increased significantly and specifically prior to the execution of prosocial-rescuing behaviors. These findings provide an important evidence to support that the empathy-related neurons in InC would convey information to MI to trigger the prosocial behaviors.

To realize the dynamics of neural circuitry under prosocial acts is crucial for elucidat-ing underlyelucidat-ing empathic mechanisms. Our data demonstrated that the means by which observing another in trap engages empathically motivated helping behaviors may relate to the increased activity of the neurons in the InC which projected to the MI. In view of the idea that the InC integrates the endogenous and exogenous information of self [17], our results would imply a greater role of the InC in perception of emotions of the others without confusion between self and others, which is an important characteristic of the empathy. In light of our findings, the InC may serve the affective-motivational compo-nent, i.e., the perspective and evaluation of subjective discomfort and response to trigger the specific prosocial acts through the neurons in the MI.

Now, back to the mathematical part of the proposed method that should be noticed.

For greedy-like algorithms, the most crucial component lies in when should we stop the iterative procedure. As Table 4.3 to Table 4.6 in Section 4.3.1 show, HDHQ performs quite satisfactorily even when the sample size is small and only includes 1 − 2 irrelevant neurons when the behavior of target neurons can not be fully explained due to the noise.

On the other hand, the HDBIC performs well only when the sample size is large enough and is too conservative when the sample size is samll. For practical use, we suggest that HDHQ can be the first stopping criterion to be used. One can also resort to the HDBIC if the information in hand is quite rich (e.g., the sample size n > 200). However, the HDHQ can play a more important role than the HDBIC since the sample size is often limited in reality.

Appendix A

Derivation of the Explicit Formula

We first denote xk = x(n − k) and yk = y(n − k) for convenience. Then for the model in (2.1), we compute matrices A and Σ by the method of Yule-Walker [54]. Since x and y are stationary, multiply (2.1) from the right by the vector x(n − 1) y(n − 1) and then take the expectation E, we have R(-1)=AR(0), where

R(0) =

 E x21

E x1y1 E x1y1

E y12

and R(−1) = E x1x2

E x2y1 E x1y2

E y1y2


Thus, we get A = R(−1)R−1(0). Alternatively, Σ can be obtained by Σ = R(0) − AR>(−1) [68]. Substituting A into Σ gives

Σ = R(0) − R(−1)R−1(0)R>(−1). (A.1) Using the same computation, we have ˜A = ˜R(−1) ˜R−1(0) and ˜Σ = ˜R(0) − ˜A ˜R>(−1) for the perturbed model in (2.4), where

R(0) =˜  E x21+ δx21+ 2x1δx1

E x1y1+ y1δx1 E x1y1 + y1δx1

E y21


R(−1) =˜  E (x1+ δx1)(x2 + δx2)

E x2y1+ y1δx2 E x1y2+ y2δx1

 E y1y2

 . Substituting ˜A into ˜Σ also gives

Σ = ˜˜ R(0) − ˜R(−1) ˜R−1(0) ˜R>(−1). (A.2) Using (A.1) and (A.2), and denoting δR(0) := ˜R(0) − R(0) and δR(−1) := ˜R(−1) − R(−1), it follows that

∆ := Σ − Σ˜

= δR(0) − δR(−1)R−1(0)R(1)

− ˜R(−1)R−1(0)δR(1)

+ ˜R(−1)R−1(0)δR(0) ˜R−1(0) ˜R(1).


By the definition of S and ˜S defined in (2.5) we know that ˜S −S = ∆2,2, the (2,2)-element of matrix ∆. Hence, we can decompose ˜S into S + ∆2,2. Annoying algebraic computation from (A.3) gives ∆2,2 = (Sy − S)I, where I is defined in (2.7), and the formula in (2.6) is obtained by denoting Θ = (Sy − S)I.

Appendix B

Derivation of the NSI using Simple Network

Here, we re-formulate the NSI using the simple network (Figure 3.1). Let u = αx+βy+γz form the BLP of w, then there exist p, {fr, r = 1, 2, . . . , p}, and {dr, r = 1, 2, . . . , p} such that wt=Pp

r=1[frut−r+ drwt−r] + t, where  is a stationary white noise possessing the smallest variance among G = span({x, y, z, v1, v2, v3}). Replacing u with the weighted trajectory, we obtain

wt =




[frut−r+ drwt−r] + t





[fr(αxt−r + βyt−r+ γzt−r) + drwt−r] + t





[αfrxt−r + βfryt−r + γfrzt−r+ drwt−r] + t.


On the other hand, fitting to data the following empirical regression wt=




[arxt−r + bryt−r + crzt−r+ grvt−r + drwt−r] + ˜t, (B.2) where grvt−r :=P3

k=1gk,rvk,t−r for convenience.

• If v is stochastically independent of x, y, z, w, then we have gr≡ 0. Since {ar}, {br}, {cr} can be obtained through Least-Squares method, comparing (B.2) with (B.1), we have




ar = α








br = β








cr = γ




fr, (B.3)

and get

α : β : γ =




ar :




br :




cr, provided




fr > 0, (B.4)

where sgn(α) = sgn(




ar), sgn(β) = sgn(




br), and sgn(γ) = sgn(





• If v is linear dependent of x, y, z, w, then gr  0 and {ar}, {br}, {cr} will be af-fected. However, since  in (B.1) possessing the smallest variance among G, taking out v does not increase the variance of ˜, therefore we still can correct the model coefficients by ruling out the useless information v.

Finally, the neuron synaptic index from x, y, z to w are defined respectively as Nx→w := |α|+|β|+|γ|α Fu→w,

Ny→w := |α|+|β|+|γ|β Fu→w, Nz→w := |α|+|β|+|γ|γ Fu→w,


where |Nx→w| + |Ny→w| + |Nz→w| = Fu→w is the GC index from the weighted trajectory u = αx + βy + γz to the target trajectory w.


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