Lecture Notes in Computer Science
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- Apump V (mV)
- Apump (pmol/s/cm 2 )
[Ca 2+ ] (mM) t (ms) [B]
Fig. 1. Waveforms of [A] the membrane potential and [B] [Ca 2+ ] in the case of a 1ms narrow pulse injection t (ms) [A]
-80 -60
-40 -20
0 20
V (mV)
4000 5000 6000
7000 [B]
Fig. 2. [A] A waveform of the membrane potential under a long pulse injection. [B] A typical waveform of the membrane potential of pyramidal cell in physiological experi- ments [12]. density 40μA/cm 2 ) is injected at t = 4000ms. The waveforms of the membrane potential and [Ca 2+ ] are not so different from the physiological experimental data qualitatively [1]. In contrast, as shown in Fig. 2A, in the case that a long pulse (length 1000ms, density 40μA/cm 2 ) is injected, the membrane potential keeps a resting state after one action potential is generated. Though the membrane potential is spiking continuously in the physiological experiment (Fig. 2B), the membrane potential of the model does not show such a behavior. In general, it is well known that the membrane potential of pyramidal cell is resting in the case of no external stimulus, and spiking or bursting when the external stimulus is added. The aim of this paper is a reconstruction or a parameter tuning of the model (1) which can reproduce such a behavior of the membrane potential. 3 Bifurcation Analysis The characteristics of the membrane potential vary with the change of the value of some parameters, therefore we investigate the bifurcation structure of the model (1) to estimate the values of the parameters. For the bifurcation analysis in this paper, we used the bifurcation analysis software AUTO [13]. 10 T. Ishiki et al. -70 -65
-60 -55
-50 -45
-40 -35
0 20
40 60
80 100
V (mV) I ext (µA/cm 2 ) Fig. 3. One-parameter bifurcation diagram on the parameter I ext . The solid curve denotes stable equilibria of eq. (1). 3.1
The External Stimulation Current I ext We analyze the bifurcation structure of the model to understand why the con- tinuous spiking of the membrane potential is not generated when a long pulse is injected. In order to investigate whether the spiking is generated or not when I ext
is increased, we vary the external stimulation current I ext
as a bifurcation pa- rameter. We show the one-parameter bifurcation diagram on the parameter I ext (Fig. 3) in which the solid curve denotes the membrane potential at stable equi- libria of eq. (1). The one-parameter bifurcation diagram shows the dependence of the membrane potential on the parameter I ext . There is no bifurcation point in Fig. 3. Therefore the stability of the equilibrium point does not change, and thus the membrane potential of the model keeps resting even if I ext is increased. This result means that we cannot reproduce the physiological experimental re- sult in Fig. 2B by varying I ext , thus we have to reconsider the parameter values of the model which cannot be determined by physiological experiments only. 3.2
The Maximum Conductance of Ca 2+ -Dependent Potassium Channel G K Ca The current through the Ca 2+ -dependent potassium channel is involved in the generation of spiking or bursting of the membrane potential. Therefore, we select the maximum conductance of Ca 2+ -dependent potassium channel G K Ca as a bifurcation parameter, and show the one-parameter bifurcation diagram when I ext = 10 (Fig. 4A). There are two saddle-node bifurcation points (SN1, SN2), three Hopf bifurcation points (HB1-HB3) and two torus bifurcation points (TR1, TR2). An unstable periodic solution which is bifurcated from HB1 changes its stability at the two torus bifurcation points and merges into the equilibrium point at HB3. Only in the range between HB1 and HB3, the membrane potential can oscillate. In order to investigate the dependence of the oscillatory range on the I ext
value, we show the two-parameter bifurcation diagram (Fig. 4B) in which the horizontal and vertical axes denote I ext and G
K Ca , respectively. The two- parameter bifurcation diagram shows the loci where a specific bifurcation occurs. Global Bifurcation Analysis of a Pyramidal Cell Model 11 - 60 - 50 - 40 - 30 - 20 - 10 0 HB1 HB2
SN1 SN2
HB3 TR1
TR2 2.50
2.75 3.00 3.25 3.50 3.75 4.00 V (mV) G K Ca (mS/cm 2 ) [A]
Iext =10 0 2 4 6 8 10 12
14 -15 -10 -5 0 5 10 15 20 25 30 HB [B] G K Ca (mS/cm 2 ) SN
ext (µA/cm 2 ) Fig. 4. [A] One-parameter bifurcation diagram on the parameter G K Ca
broken curves show stable and unstable equilibria, respectively. The symbols • and ◦
denote the maximum value of V of the stable and unstable periodic solutions, respec- tively. [B] Two-parameter bifurcation diagram in the (I ext , G
K Ca )-plane. In the diagram, the gray colored area separated by the HB and SN bifurcation curves corresponds to the range between HB1 and HB3 in Fig. 4A where the periodic solutions appear. Increasing I ext
, the gray colored area shrinks gradually and disappears near I ext = 25. This result means that the membrane potential of the model cannot present any oscillations (spontaneous spiking) for large values of I
ext even if we changed the value of the parameter G K Ca
3.3 The Maximum Pumping Rate of the Ca 2+ Pump
A pump
The Ca 2+ pump plays an important role in the regulation of the intracellular Ca 2+ . We also investigate the effect of varying the A pump value, which is the max- imum pumping rate of the Ca 2+ pump, on the membrane potential. Figure 5A is the one-parameter bifurcation diagram when I ext
= 10. There are two Hopf bifurcation points (HB1, HB2) and four double-cycle bifurcation points (DC1- DC4). A stable periodic solution generated at HB1 changes its stability at the four double-cycle bifurcation points and merges into the equilibrium point at HB2. Similarly to the case of G K Ca , we show the two-parameter bifurcation diagram in the plane of two parameters I ext and A
pump (Fig. 5B) in order to ex- amine the dependence of the oscillatory range between HB1 and HB2 on I ext
. In Fig. 5B, the gray colored area, where the membrane potential oscillates, shrinks and disappears as I ext
increases. The result shows that the membrane potential of the model cannot present any oscillations (spontaneous spiking) for large val- ues of I ext
even if we changed the value of A pump
similarly to the case of G K Ca . 3.4
Slow/Fast Decomposition Analysis In this section, in order to investigate the dynamics of our pyramidal cell model in more detail, we use the slow/fast decomposition analysis [14].
12 T. Ishiki et al. 0 300
500 800
1000 1300
- 60 - 50 - 40 - 30 - 20 - 10 0 10 20 30 HB1
HB2 Apump___V___(mV)'>Apump V (mV) DC1
DC2 DC3
DC4 [A]
(pmol/s/cm 2 ) Iext =10 0 200 400 600
800 1000
0 20 40 60 80 100 120 140 HB
[B]
I ext (µA/cm 2 ) Fig. 5. [A] One-parameter bifurcation diagram on the parameter A pump . [B] Two- parameter bifurcation diagram in the (I ext
, A pump
)-plane. A system with multiple time scales can be denoted generally as follows: dx dt
x ∈ R
n , y
∈ R m , (2a) dy dt = g(x, y), 1. (2b) Equation (2b) is called a slow subsystem since the value of y changes slowly while equation (2a) a fast subsystem. The whole eq. (2) is called a full system. So-called slow/fast analysis divides the full system into the slow and fast subsystems. In the fast subsystem (2a), the slow variable y is considered as a constant or a parameter. The variable x changes more quickly than y and thus x is considered to stay close to the attractor (stable equilibrium points, limit cycle, etc.) of the fast subsystem for a fixed value of y. The variable y changes slowly with a velocity g(x, y) in which x is considered to be in the neighborhood of the attractor. The attractor of the fast subsystem may change if y is varied. The problem of analysis of the dependence of attractor on the parameter y is a bifurcation problem. Thus the slow/fast analysis reduces the analysis of full system to the bifurcation problem of the fast subsystem with a slowly-varying bifurcation parameter. In the case of the pyramidal cell model (1), under the assumption that the change of the intracellular Ca 2+ concentration [Ca 2+ ] is slower than the other variables, the slow/fast analysis can be made. Thus, we consider [Ca 2+ ] as a bi- furcation parameter and eq. (1c) as a slow subsystem, and all other equations of eqs. (1a,b,d,e) are considered as a fast subsystem. We show the bifurcation dia- gram of the fast subsystem by varying the value of [Ca 2+ ] as a parameter (Fig. 6). The figure shows the stable and unstable equilibria of the fast subsystem with I ext = 0 (thick solid and broken curves, resp.), and the slow subsystem (thin curve). The point at the intersection of the equilibrium curve of the fast subsys- tem with the nullcline of the slow subsystem is the equilibrium point of the f ull system. The stability of the full system is determined whether the intersection point is on the stable or unstable branch of the equilibrium curve of the f ast subsystem. Therefore, the stability of the full system is stable in the case of Global Bifurcation Analysis of a Pyramidal Cell Model 13 -80 -70 -60
-50 -40
-30 -20
-10
0 10 0 0.0001 0.0002 0.0003 0.0004 0.0005 V (mV) [Ca 2+ ] (mM) Fig. 6. Bifurcation diagram of the fast subsystem (I ext = 0) with Ca 2+ as a bifurcation parameter and the slow-nullcline of the slow subsystem Fig. 6. In addition, when I ext is increased, the bifurcation diagram (equilibrium curve) of the fast subsystem shifts upward and the stability of the full system keeps stable and no oscillation appear, as will be shown in Fig. 7. By changing the parameter of the slow subsystem, the shape of the slow- nullcline changes and the intersection point is also shifted. First, we select some parameters of the slow subsystem. Because the Ca 2+ pump is included only in the slow subsystem, we select the A pump
and the dissociation constant K pump
which are both parameters contained in Ca 2+ pump as the parameters of the slow subsystem. Second, we change the values of A pump
and K pump
in order to change the shape of the nullcline. Figure 7A shows the slow-nullclines (thin solid or broken curves) with varying A pump and also the equilibria of the fast subsystem (thick solid and broken curves) with I ext = 0, 20 and 40. By the increase of A pump , the nullcline of the slow subsystem shifts upward, and the intersection point of the equilibrium curve of the fast subsystem (I ext = 0) with the slow-nullcline is then located at an unstable equilibrium. Therefore, the membrane potential of the full system is spiking when I ext = 0. Figure 7B is the similar diagram to Fig. 7A, where the value of K pump
is varied (A pump
is varied in Fig. 7A). By the change of K pump value, the shape of the slow-nullcline is not changed much, therefore the intersection point of the equilibrium curve of the fast subsystem with the nullcline keeps staying at stable equilibria and the full system remains stable at a resting state. Next, in Fig. 8, we show an example of spontaneous spiking induced by an in- crease of A pump (A
= 20). The gray colored orbit in Fig. 8A is the projected trajectory of the oscillatory membrane potential and the waveform is shown in Fig. 8B. Because the equilibrium curve of the fast subsystem intersects with the slow-nullcline at the unstable equilibrium, the membrane potential oscillates even though I ext
= 0. The projected trajectory of the full system follows the stable equilibrium of the fast subsystem (the lower branch of thick curve) for a long time, and this prolongs the inter-spike interval. After the trajectory passes through the intersection point, the membrane potential makes a spike. When the trajectory passes through the intersection point, the trajectory winds around the intersection point. This winding is possibly caused by a complicated nonlinear 14 T. Ishiki et al.
Iext=0
Iext=20 Iext=40
Apump=5(default) Apump=50
Apump=100 Apump=150 or or
[A] -80
-70 -60
-50 -40
-30 -20
-10 0 10 0 0.0002 0.0004 0.0006 0.0008 0.001 V (mV) [Ca 2+ ] (mM) -80
-70 -60
-50 -40
-30 -20
-10
0 10 0 0.0002 0.0004 0.0006 0.0008 0.001 Kpump=0.32 Kpump=1.2 Kpump=2.0 Kpump=3.2 Iext=0 Iext=20 Iext=40 Kpump=0.4(default) or or
V (mV) Kpump=0.2 [B]
Fig. 7. Variation of the equilibria of the fast subsystem and the nullcline of the slow subsystem, by the change of the parameters of slow subsystem: [A] A pump
, [B] K pump
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-40 -20
0 20
0 0.0005
0.001 0.0015
V (mV) [Ca 2+ ] (mM) [A]
-80 -60
-40 -20
0 20 20000 20400 20800 21200 21600 22000 V (mV) t (ms) [B]
Fig. 8. [A] An oscillatory trajectory of the full system (gray curve) with bifurcation diagram of the fast subsystem (I ext = 0, thick solid and broken curve) and the nullcline of the slow subsystem (A pump
= 20, thin curve), [B] The oscillatory waveform of the membrane potential dynamics [14] and makes the subthreshold oscillation of the membrane potential just before the spike in Fig. 8B. However, this subthreshold oscillation cannot be observed in physiological experiments of pyramidal cell (Fig. 2B). 4 Conclusion In this research, we tried to construct a model of pyramidal cells in the visual cortex focusing on Ca 2+ regulation mechanisms, and analyzed the global bifur- cation structure of the model in order to seek the physiologically plausible values of its parameter. We analyzed the global bifurcation structure of the model using the maximum conductance of Ca 2+ -dependent potassium channel (G K Ca ) and the maximum pumping rate of the Ca 2+ pump (A pump ) as bifurcation parameters. According to the two-parameter bifurcation diagrams we showed that the range where the spontaneous spiking occurs shrinks as the external stimulation current I ext
Global Bifurcation Analysis of a Pyramidal Cell Model 15 increases. Therefore, the membrane potential of the model cannot oscillate for large values of I ext
even if both values of G K Ca and A pump
were changed. We also investigated the effect of A pump and the dissociation constant K pump on the nullcline of slow subsystem based on the slow/fast decomposition analysis. If A pump
is increased, the membrane potential is spiking when I ext
= 0 because the nullcline shifts upward and the stability of the full system becomes unstable. When K pump
is varied, the membrane potential keeps a resting state because the full system remains stable. Unfortunately, no expected behavior was obtained by the change of values of the parameters considered in this paper. We have, however, demonstrated the usefulness of such nonlinear analyses as the bifurcation and slow/fast anal- yses to examine parameter values and construct a physiological model. More detailed study using the other parameters is necessary for the construction of the appropriate model as a future subject. References 1. Osanai, M., Takeno, Y., Hasui, R., Yagi, T.: Electrophysiological and optical studies on the signal propagation in visual cortex slices. In: Proc. of 2005 Annu. Conf. of Jpn. Neural Network Soc., pp. 89–90 (2005) 2. Herz, A.V.M., Gollisch, T., Machens, C.K., Jaeger, D.: Modeling single-neuron dynamics and computations: a balance of detail and abstraction. Science 314, 80– 85 (2006) 3. Hodgkin, A.L., CHuxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol(Lond) 177, 500–544 (1952) 4. Brown, A.M., Schwindt, P.C., Crill, W.E.: Voltage dependence and activation ki- netics of pharmacologically defend components of the high-threshold calcium cur- rent in rat neocortical neurons. J. Neurophysiol. 70, 1516–1529 (1993) 5. Peterson, B.Z., Demaria, C.D., Yue, D.T.: Calmodulin is the Ca 2+ sensor for Ca 2+ dependent inactivation of L-type calcium channels. Neuron. 22, 549–558 (1999) 6. Cummins, T.R., Xia, Y., Haddad, G.G.: Functional properties of rat and hu- man neocortical voltage-sensitive sodium currents. J. Neurophysiol. 71, 1052–1064 (1994) 7. Korngreen, A., Sakmann, B.: Voltage-gated K + channels in layer 5 neocortical pyramidal neurons from young rats: subtypes and gradients. J. Neurophy. 525, 621–639 (2000) 8. Kang, J., Huguenard, J.R., Prince, D.A.: Development of BK channels in neocor- tical pyramidal neurons. J. Neurophy. 76, 188–198 (1996) 9. Hayashida, Y., Yagi, T.: On the interaction between voltage-gated conductances and Ca
+ regulation mechanisms in retinal horizontal cells. J. Neurophysiol. 87, 172–182 (2002) 10. Naraghi, M., Neher, E.: Linearized buffered Ca + diffusion in microdomains and its implications for calculation of [Ca + ] at the mouth of a calcium channel. J. Neurosci. 17, 6961–6973 (1997) 11. Noble, D.: Influence of Na/Ca exchanger stoichiometry on model cardiac action potentials. Ann. N.Y, Acad. Sci. 976, 133–136 (2002)
16 T. Ishiki et al. 12. Yuan, W., Burkhalter, A., Nerbonne, J.M.: Functional role of the fast transient outward K + current I A in pyramidal neurons in (rat) primary visual cortex. J. Neurosci. 25, 9185–9194 (2005) 13. Doedel, E.J., Champeny, A.R., Fairgrieve, T.F., Kunznetsov, Y.A., Sandstede, B., Xang, X.: Continuation and bifurcation software for ordinary differential equations (with HomCont). Technical Report, Concordia University (1997) 14. Doi, S., Kumagai, S.: Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models. J. Comp. Neurosci. 19, 325–356 (2005) Appendix I total = I Na + I Ks + I
Kf + I
K −Ca
+ I CaL
+ I Ca + I leak + I
ex I Catotal = I CaL
+ I Ca − 2I ex + I
pump I Na = G Na · M1 · H1 · (V − E Na ), G Na = 13.0(mS/cm 2 ), E
Na = 35.0(mV) τ M1
0.182 V +29.5
1 −exp
[ − V +29.5 6.7 ] −
0.124 V +29.5
1 −exp
[ V +29.5
6.7 ] τ H1 (V ) = 0.5 + 1/ exp − V +124.955511 19.76147 + exp
− V +10.07413 20.03406 I Ks = G Ks · M2 · H2 · (V − E K ), G Ks = 0.66(mS/cm 2 ), E
Ks = −103.0(mV) τ M2 (V ) = 1.25 + 115.0 exp[0.026V ], (V < −50mV)
1.25 + 13.0 exp[ −0.026V ], (V ≥ −50mV) τ H2
−(V + 75.0)/48.0) 2 I Kf = G
Kf · M3 · H3 · (V − E Kf ),
Kf = 0.27(mS/cm 2 ), E
Kf = E
Ks τ M3 (V ) = 0.34 + 0.92 exp ( −(V + 71.0)/59.0) 2 τ
(V ) = 8.0 + 49.0 exp ( −(V + 37.0)/23.0) 2 I
Ca = G
K Ca · M4 · (V − E K Ca ), G K Ca = 12.5(mS/cm 2 ), E K Ca = E Ks M 4
∞ (V, [Ca
2+ ]) = ([Ca 2+ ]/([Ca
2+ ] + K
h )) · (1/(1 + exp[−(V + 12.7)/26.2])) K h = 0.15(μM) τ M4 (V ) = 1.25 + 1.12 exp[(V + 92.0)/41.9] (V < 40mV) 27.0 (V
I Ca L = P CaL
· M5 · H5 · (2F )
2 RT · V · [Ca 2+ ] exp[2V F /RT ] −[Ca 2+ ] o exp[2V F /RT ] −1 P
= 0.225(cm/ms), H5 ∞ = K Ca 4 /(K Ca 4 + [Ca 2+ ] 4 ), K Ca = 4.0(μm) τ M5 (V ) = 2.5/(exp[ −0.031(V + 37.1)] + exp[0.031(V + 37.1)]) τ H5 = 2000.0 I Ca = P Ca · M6 · H6 · (2F ) 2 R ·T · V ·
[Ca 2+ ] exp[2V F /RT ] −[Ca 2+ ] o exp[2V F /RT ] −1 P
= 0.155(cm/ms), τ H6 = 2000.0 τ M6 (V ) = 2.5/(exp[ −0.031(V + 37.1)] + exp[0.031(V + 37.1)]) I leak = V /30.0 I ex = k [Na + ] 3 i · [Ca 2+ ] o · exp s · V F
RT − [Na
+ ] 3 o · [Ca
2+ ] · exp −(1 − s) · V F RT k = 6.0 × 10 −5 (μA/cm 2 /mM
4 ), s = 0.5 I pump
= (2 · F · A
pump · [Ca
2+ ])/([Ca
2+ ] + K
pump ) A pump = 5.0(pmol/s/cm 2 ), K
pump = 0.4(μM) M i ∞
−(V − α M i
)/β M i
), i = 1, 2, 3, 5, 6 Hj ∞
− α Hj )/β Hj ), j = 1, 2, 3, 6 i 1 2 3 5 6 α M i
(mV ) −29.5 −3.0 −3.0 −18.75 18.75 β M i
6.7 10.0 10.0 7.0 7.0
j 1 2 3 6 α Hj (mV )
−65.8 −51.0 −66.0 −12.6 β Hi 7.1 12.0
10.0 18.9
Global Bifurcation Analysis of a Pyramidal Cell Model 17 C = 1.0(μF/cm 2 ), S = 3.75(/cm), k − = 5.0(/ms), k + = 500.0(/mM · ms) [Ca
2+ ] o = 2.5(mM), [Na + ] i = 7.0(mM), [Na + ]
= 150.0(mM) T and R denote the absolute temperature and gas constant, respectively. In all ionic currents of the model, the powers for gating variables (M 1, · · · ,
M 6, H1, · · · , H6) are approximately set at one in order to simplify the equations. Leak current is assumed to have no ion selectivity and follow Ohm’s law. Thus, the reversal potential of leak current is set at 0 (mV), though it might be unusual.
M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 18–26, 2008. © Springer-Verlag Berlin Heidelberg 2008 Download 12.42 Mb. Do'stlaringiz bilan baham: |
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