A More Complicated Model of Motor Neural Circuit of C. elegans

Transcription

1 A More Complicated Model of Motor Neural Circuit of C. elegans Overview Locomotion of C. elegans follows a nonlinear dynamics proposed in Fang-Yen et al The dynamic equation is written as y C N t In this equation, denotes the position of the worm body along the centerline. comes from the external force transverse to the worm s body during lateral movement, by ssss is the force due to internal elasticity of the worm, where b is the modulus of the worm body. y C N t + by y ssss + b ssss v + Mss a = 0. (1) t s y b ssss v t is the load of internal viscosity, and the last term, Mss a is the torque provided by body wall muscles of the worm. According to Fang-Yen et al. 2010, internal viscosity is negligible compared to the internal elasticity, so the equation evolves into y C N t + by ssss + Mss a = 0. (2) The second derivative of Eq. 2 leads to y C ss N + b 6 y t s + 4 M a = 0. (3) 6 s 4 Assuming the curvature κ of the worm body is proportional to y ss, we will derive Eq. 4, the dynamics of the shape (curvature) of the worm body, which has been published in Wen et al κ C N t + b 4 κ s = 4 M 4 s. (4) 4 M = M a, Note that this will not lead any difference to the dynamics because it only changes the positive direction of torque that we define. In both Fang-Yen et al and Wen et al. 2012, further simplification was made by assuming the curvature and the torque are sinusoidal functions of the position and the time. Thus the dynamics will reduce to a linear equation λ κ 2π C N + b( t λ )4 κ = ( 2π λ )4 M, (5) where is the wavelength of the sinusoidal functions of curvature and torque. In Xu et al. 2017, Eq. 5 was used to predict the effect of gap junctions on the propagation of bending wave along the worm body. The conclusion was that gap junctions between B neurons and AVB neuron are very critical to equalization of the amplitude of the bending wave. And the mechanism is that the AVB/B gap junction will shift the dynamics of B neurons to a high-frequency oscillation phase, due to nonlinear properties of B neurons. However, the model is a little debatable because of Eq. 5. A straightforward question is that, what if we do not reduce Eq. 4 to a linear function? Or in other words, is the assumption, that the curvature and the torque follows a sinusoidal relationship with position, realistic or questionable? 1

2 In order to verify the conclusions in Xu et al. 2017, we developed a more complicated model using Eq. 4, and also based on previous model in Xu et al We have found that the shape of the worm, and the bending wave, are not sinusoidal. So the reduction to Eq. 5 may be not very appropriate. However, as for the effect of gap junction between AVB and B neurons, we get the same conclusion as Xu et al Methods 1. Boundary Conditions Our dynamic equation for the worm (Eq. 4) is very similar to Euler-Bernoulli beam theory. Based on Euler-Bernoulli beam theory, our work is subject to free end boundary conditions (Eq. 6). L is the length of the worm body. Since κ = conditions can be rewritten with respect to curvature (Eq. 7) into N these boundary In the discrete form, which is used for numerical simulation, we discrete the worm segments, so the boundary conditions are expressed as where the subscript denotes segments from head to tail. 2. Fourth-order Difference Quotient We used two methods to calculate the 4th-order difference quotient, which appears in Eq. 4. One is simple but coarse and used in the simple linear model (will be mentioned below). The other is complicated but with higher order of accuracy, which is used in our complex nonlinear model. In the simpler method, the 4th-order difference quotient of curvature, computed as h 2 y s 2 s=0 = 0 ; 3 y s 3 s=0 = 0 ; 2 y s 2 s=l = 0. (6) 3 y s 3 s=l = 0. y (1 + y 2 ), 3/2 κ s=0 = 0 ; κ s=l = 0. κ s s=0 = 0 ; κ s s=l = 0. (7) κ 1 = κ 2 = κ N 1 = κ N = 0, (8) 1 N 4 κ i s = 1 4 h (κ 4 i 2 4κ i 1 + 6κ i 4κ i+1 + κ i+2 ), (9) where is the step length, i.e. the length of every discrete segment. is calculated in the similar way. This is a first-order method, which means the error of the calculation is o(h). Eq. 9 can not be applied at the two ends of the worm body, but because of the existence of the boundary conditions (Eq. 8), we actually do not need Eq. 9 to be defined at the ends. The nonlinear model is much more complex than the linear model, with nonlinear dynamics of the membrane potential of neurons and interactions among neurons and muscle cells. In order to ensure the accuracy of computation, we used a 5th-order to 2 4 κ s 4 4 M s 4, is

3 calculate. For every segment, we use the Taylor expansion of 8 surrounding segments, with the origin at segment, to form a set of equations (composed of 8 equations). And then we solve example, for And the result is 4 κ i(i > 2 and i < N 1) s 4 4 κ i s 4 i > 4 and i < N 3, the equations are κ i+n κ i = 8 1 k=1 k! (nh)k, n = 4, 3, 2, 1,1,2,3,4. from the set of equation. For For other segment (several segments on two ends), the terms and coefficients are different. Using this method, the error is reduced to o(h 5 ), which increases the robustness of our numerical simulation. We test the accuracy of Eq. 10 using the very simple function,, And the error is shown in Figure 1 below. i (10) 4 κ i = 1 (7κ s 4 240h 4 i 4 96κ i κ i κ i κ i 1952κ i κ i+2 96κ i+3 + 7κ i 4 ). y = sin(x) Figure 1 (11) Error of calculating the 4th-order difference quotient with Eq. 10. The number shown in the figure is the euro divided by h 4, so we are sure the error is at most o(h 4 ). Linear Model Our linear model is basically the same as the one in Xu et al. 2017, with a parameter representing the membrane potential, which decays linearly and has an input of proprioception signal. What we change here is the form of proprioception signal, and the dynamics of the curvature. We use the parameter v to evaluate the membrane voltage, and we also assume the torque M, is proportional to v, i.e. M = M 0 v (Wen et al. 2012). So the model mainly consists of two dynamic equations: 3

4 F[k(s); s] is a functional of curvature and position. It is a normalized integral of proprioception signal (Eq. 13). s = exp( x /l)d x 0 Here l is defined as the characteristic length of proprioception. g in Eq. 12 denotes the conductance of AVB-B gap junctions (Xu et al. 2017). The mechanism of oscillation at the worm head is still unknown, so we impose a sinusoidal oscillation at the head, namely where determines the amplitude of the oscillation, and are wavelength and angular frequency of the bending wave respectively. This two parameters, depend on the viscosity η of the media. And the time constant C N in Eq. 12, is also proportional to η. Values of these parameters will be determined based on some experimental results (Fang-Yen et al. 2010, Xu et al. 2017). The initial condition of this model is is randomly chosen in the interval. The value of will not affect the simulative result. We use the simpler method to calculate the 4th-order difference quotient (mentioned above in Methods ). Then we find that the whole body will oscillate approximately sinusoidally, with the same frequency ω, but phase shift among different positions (Figure 2). We calculate the amplitude of oscillation at every segment that we divide, and compare the cases with and without AVB-B gap junctions. The result is shown in Figure 3 below. We can see that in the presence of AVB-B gap junctions, the amplitude decays much more obviously from head to tail. This is consistent with the linear model in Xu et al. 2017, but is opposite to the reality. This indicates the linear model is not correct, the dynamics of the membrane voltage is not that simple. A more complicated model is necessary for understanding the propagation of the bending wave, and the effect of the AVB-B gap junctions. Parameters in the linear model is listed below: Other parameters related to computation: Time length of simulation: T = 10s; Time step of simulation: Number of segments: τ m dv C N κ t F[k(s, t); s] = s κ(s x, t)exp( x /l)d x 0 = (1 + g)v + cf[k(s, t); s]; (12) + b 4 κ = M 4 v s 4 0. s 4 s κ(s x, t)exp( x /l)d x 0 l(1 exp( s/l)) k(l, t) = k 0 sin(2πl /λ ωt), (14) k 0 λ ω k(s,0) = k 0 sin(2πs/λ); { v(s,0) = sin(2πs/λ + ϕ v ). ϕ v [0, 2π] ϕ v (15) L = 1mm, λ = 1mm, l = 200um; g = 0 or 1; τ m = 15ms, b = 9.5 * N * m 2, η = 1Pa * s, C N = 480 * η = 480Pa * s; ω = 1.4πs 1, c = 6 * 10 4 m, M 0 = 5.5 * N * m, k 0 = 5000m 1. = 5μs; N = 50;. (13) (16) 4

5 Figure 2 Curvature change of 5 different positions from head to tail. Figure 3 Simulative result of the linear model. Amplitude of the worm body in the presence ( g = 1, shown as red curve) and in the absence ( g = 0, shown as black curve) of the AVB-B gap junctions. The left purple line indicates the position of the imposed oscillation (or namely the head). The right blue line indicates the position of 0.9 (normalized with head=0 and tail=1). In the experiments, the portion between 0.9 and 1 is not analyzed because the boundary effect is too dominant (Xu et al. 2017). Nonlinear Model Similar to the compartment in Xu et al. 2017, the nonlinear model here also gives a detailed description of dynamic properties of B motor neurons using K-Ca Hopfield- Huxley model. We used the identical nullicline (Figure 4) of neural dynamics in Xu et al But since we need to incorporate the 4th-order difference quotient into the model, we also need to divide the body wall muscle into an appropriate number of segments. Thus we need a realistic way to model the interaction between neurons and muscle cells. Here we refer to Haspel et al. 2011, which proposed a model of connectome and repetitive units of neural system and muscles of C. elegans. We make some modifications based on Haspel s model, mainly on the amount and position of neurons and muscles. We divide the whole worm body to N = 30 segments. This number is large enough to ensure the continuity of the membrane potential and the curvature. And it is also a reasonable estimation of the number of body wall muscles. There are 71 neurons connected to B neurons, so on each side, the average number is about 35. In Haspel et al. 2011, the number of muscles on either dorsal side or the ventral side is N = 30

6 lies between this two estimation, which is a very realistic assumption. And we assume that the number of muscle cells on both sides are equal, which is consistent with Haspel et al There are some overlap of neuron-muscle gap junctions, i.e. one muscle cell is governed by two adjacent neurons. This kind of overlap is not broad. Based on Haspel et al. 2011, there is at most one muscle cell having the overlap phenomenon in every repetitive unit. The number of dorsal B neurons (DB) is 7, and there are 11 ventral dorsal neurons (VB). Based on above information, we simplify the connectome of neurons and muscles as the table below. For simplicity, we do not distinguish MDR from MDL, and MVR from MVL, because they all contribute to the torque. And the weight (or namely conductance) of the neuron-muscle gap junctions, are considered to be uniform. Muscle cells Motoneuron Muscle cells Motoneuron MD01 DB01 MV01 VB01 MD02 DB01 MV02 VB01 MD03 DB01 MV03 VB01 MD04 DB01 MV04 VB01 MD05 DB01 MV05 VB01 MD06 DB01 MV06 VB01 MD07 DB01 MV06 VB02 MD07 DB02 MV07 VB02 MD08 DB02 MV08 VB02 MD09 DB02 MV08 VB03 MD10 DB02 MV09 VB03 MD11 DB03 MV10 VB03 MD11 DB03 MV10 VB04 MD12 DB03 MV11 VB04 MD13 DB03 MV12 VB04 MD14 DB03 MV12 VB05 MD15 DB03 MV13 VB05 MD15 DB04 MV14 VB05 MD16 DB04 MV14 VB06 MD17 DB04 MV15 VB06 MD18 DB04 MV16 VB06 MD19 DB04 MV16 VB07 MD19 DB05 MV17 VB07 MD20 DB05 MV18 VB07 6

7 Muscle cells Motoneuron Muscle cells Motoneuron MD21 DB05 MV19 VB07 MD22 DB05 MV19 VB08 MD23 DB05 MV20 VB08 MD23 DB06 MV21 VB08 MD24 DB06 MV21 VB09 MD25 DB06 MV22 VB09 MD26 DB06 MV23 VB09 MD27 DB06 MV24 VB09 MD27 DB07 MV24 VB10 MD28 DB07 MV25 VB10 MD29 DB07 MV26 VB10 MD30 DB07 MV27 VB10 MV27 MV28 MV29 MV30 VB11 VB11 VB11 VB11 C dv di C dv vi The dynamic equation for the membrane potential is = g L (V di E L ) g Ca m (V di )(V di E Ca ) g K n di (V di E K ) + c * K[k(x, t); i] + g(v 0 V di ); = g L (V vi E L ) g Ca m (V vi )(V vi E Ca ) g K n vi (V vi E K ) c * K[k(x, t); i] + g(v 0 V vi ); (17) Here V denotes the membrane potential, m and n are voltage dependent Ca 2+ and K + activation variables respectively. The subscript d means dorsal and v means ventral side of the worm. K[k(x, t); i] is a functional calculated for every neuron. It is the average of curvature of several muscle cells connected to the anterior neuron on the same side. An exception is that when i = 1, K[k(x, t); i] is the average of curvature of muscle cells connected to the first neuron on the same side (i.e. just the neuron itself). V 0 is the membrane potential of the AVB neuron, which is almost a constant. g is the conductance of AVB-B gap junctions. n di and n vi are both voltage dependent, following the dynamic equations below (Eq. 18). M τ c dn di τ c dn vi = n di + n (V di ); = n vi + n (V vi ). The torque is directly determined by the membrane potential of the neurons (Eq. 19). Torque is respectively calculated on the dorsal and the ventral side, then subtract 7 (18)

8 Figure 4 Nullicline analysis of motor neuron dynamics. The n- and V-nulliclines were defined as respectively. In the absence of proprioceptive coupling and AVB-B gap junctions ( g = 0), the fixed point of the membrane potential (intersect of black curve and blue curve) corresponds to a steady state; in the presence of AVB-B gap junctions, that is, g = 100pS, the system undergoes bifurcation, and the new fixed point becomes unstable (intersect of red curve and blue curve), leading to periodic oscillation. dn = 0, dv = 0, Figure 5 Oscillation of curvature at several different positions of the worm. The AVB-B gap junctions are present ( ). (Left) Original oscillation patterns (Right) Oscillation after sine fitting. g = 100pS each other, rescaled by a constant coefficient τ u dm d τ u dm v M 0. = M d + M (V d ) M (V v ); = M v + M (V v ) M (V d ); M = M 0 (M d M v ). Similar to linear model, we also impose a sinusoidal (19) 8

9 Combined with Eq. 4, and the more complex method of calculating the 4th-order difference quotient mentioned above, we can simulate the locomotion of the worm. The change of curvature, in the presence of AVB-B gap junctions ( g = 100pS), is shown in Figure 5 (Left panel). Obviously, it has more than one components of oscillation, with different frequencies. This phenomenon will be explained below. And we fit the oscillation with a sine function, which is shown in Figure 5 (Right panel). The amplitude of curvature, with and without AVB-B gap junctions, is shown in Figure 6 below. We can find that the bending wave is obviously more equalized with AVB-B gap junctions. So based on Xu et al. 2017, we can know that this nonlinear model gives the correct prediction on the effect of AVB-B gap junctions. Figure 6 Simulative result of the nonlinear model. Amplitude of the worm body in the presence ( g = 1, shown as red curve) and in the absence ( g = 0, shown as black curve) of the AVB-B gap junctions. The left blue line indicates the position of the imposed oscillation (or namely the head). The right orange line indicates the position of 0.85 (normalized with head=0 and tail=1). In the experiments, the portion between 0.9 and 1 is not analyzed because the boundary effect is too dominant (Xu et al. 2017). Here we find the boundary effect is still obvious at 0.9, so we do the cut-off at 0.85 in this model. As for the multiple oscillation components in Figure 5. We think it is the combination of proprioceptive coupling, background mechanic coupling (due to the elasticity of the worm body, the 4th-order difference quotient itself is a kind of coupling) and the high frequency self-oscillation mentioned in Xu et al The background mechanic coupling and the high frequency oscillation, are shown in Figure 7 below. Since our nullicline of neural dynamics is identical with Xu et al. 2017, the high frequency oscillation does not exist in the absence of AVB-B gap junctions. Parameters in the nonlinear model: L = 1mm, b = 9.5 * N * m 2, λ = 1mm, η = 1Pa * s, C N = 480η = 480Pa * s; τ u = 85ms, τ c = 30ms, C = 3pF, V 0 = 20mV, E Ca = 60mV, E K = 70mV, E L = 60mV; 1 m (V ) =, n 1 + exp( V V m (V ) =, M κm ) 1 + exp( V V n (V ) = ; ) 1 + exp( V V mus ) κn κmus g L = 100pS, g C a = 400pS, g K = 500pS, κ n = 20mV, V n = 55mV, κ m = 10.25mV, V m = 29mV; g = 0pS or 100pS, κ mus = 10mV, V mus = 45mV, k 0 = 4000m 1 ; ω = 1.4πs 1, c = 1.5 * m * A, M0 = 3 * 10 9 N * m. Other parameters related to computation: 9 1 1

10 Time length of simulation: Time step of simulation: T = 10s; = 0.1ms; Figure 7 Amplitude results from background mechanic coupling (Left, with impose at the head, but without proprioception and without torque) and the high frequency oscillation (Right, without impose and without proprioception but there are torques). Conclusion Our conclusion is basically the same as the conclusion in Xu et al The AVB- B gap junctions has a critical effect on equalizing the bending wave and facilitating the propagation from the wave from head to tail. Thus the tail of the worm will be fragile. This effect is due to the bifurcation of neural dynamics that the AVB-B gap junctions introduce. A self-oscillation will come out at the presence of the AVB-B gap junctions. When the head is paralyzed, we can observe the high frequency self-oscillation in the middle of the worm body, both theoretically (Figure 7) and experimentally (Xu et al. 2017). Compared to Xu et al., we describe the dynamics of muscles, and the interaction between neurons and muscles, in a more realistic way. With the Euler-Bernoulli equation, we find that the bending wave is not sinusoidal. This may be a flaw of previous models. The connectome between neurons and muscles, does not present much new information here. But it makes our model more reliable, thus providing a stronger support to the conclusion of Xu et al