简化分析：SRM神经元模型——从离子通道到阈值动态

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1. The Spike Response Model 7

a) b)

0.0 20.0 40.0 60.0 80.0 100.0

t [ms]

-50.0

0.0

50.0

100.0

150.0

u [mV]

0.0 5.0 10.0 15.0 20.0

0.0

50.0

100.0

ν [Hz]

FIGURE 4. (a)

Spike train

of the Ho dgkin-Huxley mo del for constant input

current

. The mean ring rate as a function of

is shown in (b).

2.1.3 Example: Step current input

In the previous example we have seen that a constant input current

> I



generates regular ring. In this paragraph we generalize this approach and

study the resp onse of the Ho dgkin-Huxley model to a step current of the

form

(

) =

+ 

I H

(

)

(1.6)

Here

(

) denotes the Heaviside step function. At

= 0 the input changes

from a constant value

to a new constant value

+ 

; see Fig. 5

We may now ask whether spiking for

t >

0 dep ends only on the nal value

or also on the step size 

The answer to this question is given by Fig. 5

. A large step 

facilitates

the spike initiation. Even for

= 0 a spike is possible, provided that the

step size is large enough. This is an example of inhibitory reb ound: A single

spike is red, if an inhibitory current

0 is released. The letter

Fig. 5

denotes the regime where only a single spike is initiated. Rep etitive

ring (regime

) is possible for



A/cm

, but must b e triggered by

suciently large current steps.

We may conclude from Fig. 5

that there is no unique current threshold

for spike initiation: The trigger mechanism for action potentials dep ends

not only on

but also on the size of the current step 

. More generally,

it can be shown that the concept of a threshold itself is questionable from

a mathematical point of view [RE89, KpBD95]. In a mathematical sense,

the transition in Fig. 3

, that `lo oks' like a threshold is, in fact, smooth.

At a higher resolution of the input current in the regime between 6.9 and

7.0



A/cm

, we would nd a family of resp onse curves in b etween the

curves shown in Fig. 3

. For practical purp oses, however, the transition

can be treated as a threshold eect as we will see b elow.

2.1.4 Example: Stimulation by time-dep endent input

As a nal example, we stimulate the Ho dgkin -Hux ley mo del by a time-

dependent input current

(

). In the numerical implementation, the current

is generated by the following procedure. Every 2 ms, a random numb er is

drawn from a Gaussian distribution with zero mean and standard deviation

1. The Spike Response Model 9

voltage / mV

0 200 400 600 800 1000

100

time / ms

FIGURE 6. Spike train of the Hodgkin-Huxley model driven by a time depen-

dent input current. The action potentials o curr irregularly. The gure shows the

voltage

as a function of time. Taken from [Kistler et al., 1997].

2.1.5 Extensions

Using the above equations and an appropriate set of parameters, Ho dgkin

and Huxley were able to describ e an enormous amount of data from ex-

periments on the giant axon of the squid. Due to its success in this sp ecial

system, there have subsequently been several attempts to generalize the

model in order to describ e other experimental situations as well (for a re-

view see, e.g., [JNT75, BB95]).

Whereas the model had originally been designed to describe the form

and temp oral change of an action p otential during

axonal

transmission, a

set of equations completely analogous to Eqs. (1.2) to (1.4) has b een used

to describ e spike generation at the

soma

of the neuron [BDMK91, BD91,

EWL

91, RYS92, TWMM91, WBUB89, YKA89]. The main dierence is

that additional ion channels have to be included, in particular those that

account for Ca

and the slow components of the p otassium current. For

each typ e of ion channel

, a current

(

) is added. Here

is yet another variable with dynamics (1.5). The conductance parameters

, the exp onents

, the reversal potential

, as well as the functions

(

) and



(

) are adjusted to t exp erimental data. Nonlinear eects on

dendrites are describ ed analogously.

2.2 Simplication of the model: Spike Response Method (1)

The system of equations prop osed by Ho dgkin and Huxley is rather com-

plicated. It consists of four coupled nonlinear dierential equations and as

such is dicult to analyze mathematically. For this reason, several simpli-

cations of the Hodgkin-Huxley equations have b een proposed. The most

common approach reduces the set of four dierential equations to a sys-

tem of two equations [Fit61, NAY62, Rin85, RE89, AK90]. Two important

approximations are made. First, the

dynamics which has a faster time

course than the other variables (see the plot for



in Fig. 2

) is considered

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