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

] in the case of a 1ms

narrow pulse injection

t (ms)

[A]

-80

-60

-40

-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

) is injected at t = 4000ms. The waveforms of the membrane

potential and [Ca

] 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

) 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.

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].

T. Ishiki et al.

-70

-65

-60

-55

-50

-45

-40

-35

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

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

-Dependent Potassium

Channel G

The current through the Ca

-dependent potassium channel is involved in the

generation of spiking or bursting of the membrane potential. Therefore, we select

the maximum conductance of Ca

-dependent potassium channel G

as a

bifurcation parameter, and show the one-parameter bifurcation diagram when

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

ext

value, we show the two-parameter bifurcation diagram (Fig. 4B) in which

the horizontal and vertical axes denote I

ext

and G

, respectively. The two-

parameter bifurcation diagram shows the loci where a speciﬁc bifurcation occurs.

Global Bifurcation Analysis of a Pyramidal Cell Model

HB1

HB2

SN1

SN2

HB3

TR1

TR2

2.50

2.75

3.00 3.25

3.50

3.75 4.00

(mV)

G

K

Ca

(mS/cm

2

)

[A]

Iext =10

-15 -10 -5 0

5 10 15 20 25 30

[B]

G

K

Ca

(mS/cm

2

)

SN

I

ext

(µA/cm

2

)

Fig. 4. [A] One-parameter bifurcation diagram on the parameter G

Ca

. The solid and

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

)-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

Ca

.

3.3

The Maximum Pumping Rate of the Ca

Pump

pump

The Ca

pump plays an important role in the regulation of the intracellular

. We also investigate the eﬀect of varying the A

pump

value, which is the max-

imum pumping rate of the Ca

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

, 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

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].

T. Ishiki et al.

300

500

800

1000

1300

HB1

HB2

Apump___V___(mV)'>Apump

V

(mV)

DC1

DC2

DC3

DC4

[A]

(pmol/s/cm

2

)

Iext =10

200

400

600

800

1000

20 40 60 80 100 120 140

HB

Apump

(pmol/s/cm

2

)

[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:

dt

= f (x, y),

∈ R

, y

∈ R

(2a)

= g(x, y),

(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

concentration [Ca

] is slower than the other

variables, the slow/fast analysis can be made. Thus, we consider [Ca

] 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

] as a parameter (Fig. 6).

The ﬁgure shows the stable and unstable equilibria of the fast subsystem with

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

-80

-70

-60

-50

-40

-30

-20

-10

0 0.0001 0.0002 0.0003 0.0004 0.0005

(mV)

[Ca

2+

] (mM)

Fig. 6. Bifurcation diagram of the fast subsystem (I

ext

= 0) with Ca

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

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

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

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

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

pump

= 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

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

0.0002 0.0004 0.0006 0.0008 0.001

V

(mV)

[Ca

2+

] (mM)

-80

-70

-60

-50

-40

-30

-20

-10

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]

[Ca

2+

] (mM)

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

-80

-60

-40

-20

0.0005

0.001

0.0015

(mV)

[Ca

2+

] (mM)

[A]

-80

-60

-40

-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).

Conclusion

In this research, we tried to construct a model of pyramidal cells in the visual

cortex focusing on Ca

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

-dependent potassium channel (G

) and the maximum

pumping rate of the Ca

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

increases. Therefore, the membrane potential of the model cannot oscillate for

large values of I

ext

even if both values of G

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

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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)

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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

sensor for Ca

dependent inactivation of L-type calcium channels. Neuron. 22, 549–558 (1999)

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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)

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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)

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potentials. Ann. N.Y, Acad. Sci. 976, 133–136 (2002)

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12. Yuan, W., Burkhalter, A., Nerbonne, J.M.: Functional role of the fast transient

outward K

current I

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)

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Hopf bifurcation in single neuron models. J. Comp. Neurosci. 19, 325–356 (2005)

Appendix

total

= I

+ I

−Ca

+ I

CaL

+ I

leak

+ I

Catotal

= I

CaL

+ I

− 2I

+ I

pump

= G

· M1 · H1 · (V − E

= 13.0(mS/cm

), E

= 35.0(mV)

M1

(V ) = 1/

0.182

V +29.5

−exp

[

−

V +29.5

6.7

] −

0.124

V +29.5

−exp

[

V +29.5

6.7

]

(V ) = 0.5 + 1/ exp

−

V +124.955511

19.76147

+ exp

−

V +10.07413

20.03406

= G

· M2 · H2 · (V − E

= 0.66(mS/cm

), E

−103.0(mV)

(V ) =

1.25 + 115.0 exp[0.026V ], (V <

−50mV)

1.25 + 13.0 exp[

−0.026V ], (V ≥ −50mV)

H2

(V ) = 360.0 + (1010.0 + 24.0(V + 55.0)) exp (

−(V + 75.0)/48.0)

= G

· M3 · H3 · (V − E

),

G

= 0.27(mS/cm

), E

= E

(V ) = 0.34 + 0.92 exp (

−(V + 71.0)/59.0)

τ

H3

(V ) = 8.0 + 49.0 exp (

−(V + 37.0)/23.0)

I

K

= G

· M4 · (V − E

= 12.5(mS/cm

), E

= E

M 4

∞

(V, [Ca

]) = ([Ca

]/([Ca

] + K

))

· (1/(1 + exp[−(V + 12.7)/26.2]))

= 0.15(μM)

(V ) =

1.25 + 1.12 exp[(V + 92.0)/41.9]

(V < 40mV)

27.0

(V

≥ 40mV)

= P

CaL

· M5 · H5 ·

(2F )

· V ·

[Ca

] exp[2V F /RT ]

−[Ca

]

exp[2V F /RT ]

−1

P

CaL

= 0.225(cm/ms), H5

∞

= K

/(K

+ [Ca

]

), K

= 4.0(μm)

(V ) = 2.5/(exp[

−0.031(V + 37.1)] + exp[0.031(V + 37.1)])

= 2000.0

= P

· M6 · H6 ·

(2F )

·T

· V ·

[Ca

] exp[2V F /RT ]

−[Ca

]

exp[2V F /RT ]

−1

P

Ca

= 0.155(cm/ms), τ

= 2000.0

(V ) = 2.5/(exp[

−0.031(V + 37.1)] + exp[0.031(V + 37.1)])

leak

= V /30.0

= k [Na

]

· [Ca

]

· exp s ·

V F

− [Na

]

· [Ca

]

· exp −(1 − s) ·

V F

k = 6.0

× 10

−5

(μA/cm

/mM

), s = 0.5

pump

= (2

· F · A

pump

· [Ca

])/([Ca

] + K

pump

)

pump

= 5.0(pmol/s/cm

), K

pump

= 0.4(μM)

M i

∞

= 1/(1 + exp (

−(V − α

M i

)/β

M i

i = 1, 2, 3, 5, 6

∞

= 1/(1 + exp ((V

− α

)/β

j = 1, 2, 3, 6

M i

(mV )

−29.5 −3.0 −3.0 −18.75 18.75

M i

6.7

10.0 10.0

7.0

(mV )

−65.8 −51.0 −66.0 −12.6

7.1

12.0

10.0

18.9

Global Bifurcation Analysis of a Pyramidal Cell Model

C = 1.0(μF/cm

), S = 3.75(/cm), k

−

= 5.0(/ms), k

= 500.0(/mM

· ms)

[Ca

]

= 2.5(mM), [Na

]

= 7.0(mM), [Na

]

o

= 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.

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