LEDHHUNSCfl

20.0 8.00 T Figure 4.4 12.0 4.00 10.00

Graph of leakage current against time during a simulated train of impulses. to CM .OO 10.0 20.0 Figure 4.5 30.0 40.0

It was then reasoned that if the input stimulus voltage could be raised above the threshold for firing and then returned to resting potential then this might simulate a burst of action potentials. It was therefore decided to stimulate the H-H model equations with an input forcing function designed to simulate a change in calcium concentration. It is known that treatment of the squid axon with low calcium concentration bathing solutions increases the excitability of the membrane and can lead to oscillations (Noble 1966; Guttman & Barnhill 1970). The effect of the reduction in calcium concentration is an apparent shift in membrane potential (Huxley 1959). The form of the forcing function was chosen purely on the grounds of its shape, the function used having a fairly steep rise to a peak followed by a long exponential tail (see Fig. 4.6). This change in calcium concentration was translated into a change in the membrane potential according to an equation derived by Huxley

(Huxley 1959).

VSHIFT = 9.3 ln([Ca]/44)... (4.1) Where [Ca] is in mM

The resulting burst of action potentials is shown in Figure 4.8. There are several features to note. One is the acceleration of firing rate at the beginning of the burst. This is observed in recordings from bursting neurones. The variation in spike amplitude is also noted in recordings from real neurones (see Fig. 1.3, also Leng 1988). The first spike

of the burst has a lower amplitude than the succeeding spike, a feature that is also often present in experimental recordings. What is not apparent in the simulated trace is the existence of a plateau potential. R15 neurones show a pronounced depolarizing plateau on which the action potential spikes are superimposed. Recordings from supraoptic neurones show a slight dip in the baseline voltage during the period of maximum firing frequency (see Fig. 1.3). From the above experiment it appears possible to model bursting behaviour by driving the H-H model equations by a slow oscillation in calcium concentration. However such a model would be incomplete for the reasons stated above and a more sophisticated model should be sought. Such a simple model also fails to take account of various ionic currents known to be present in oxytocin-secreting cells. Connor & Stevens used a modified H-H model to simulate bursting in R15 neurones (Connor & Stevens 1971) by incorporating a transient potassium current that they had previously identified and modelled. A transient potassium current has also been recorded in the oxytocin-secreting neurone (Cobbett, Legendre & Mason 1989). This current is also incorporated into a model of bursting in R15 neurones by Smith (Smith 1978). Calcium currents have also been recorded in oxytocin-secreting cells and such cells produce reduced amplitude action potentials even in the absence of sodium currents. Calcium-dependent potassium currents have also been recorded (but not fully characterised) in supraoptic neurones (Cobbett, Legendre & Mason 1989). Smith's model (Smith 1978) incoporates such a current. Any complete model of the electrical behaviour of

the oxytocin-secreting neurone should incorporate mathematical descriptions of all the currents that have been observed. If such a model fails to demonstrate bursting behaviour and the model is known to be accurate then it may be assumed that there is another mechanism present responsible for driving the bursting activity.

Figure 4.6 shows the assumed change in calcium concentration. Figure 4.7 shows how this appears as the apparent shift in voltage. Figure 4.8 shows the resulting train of action potentials. Figure 4.9 shows the combination of membrane voltage and the apparent membrane voltage shift. Figures 4.10, 4.11 and 4.12 show the variation in the sodium activation, inactivation and potassium activation variables respectively during a the burst of action potentials. Unless stated otherwise membrane voltages are expressed in millivolts and time in milliseconds.

C a l c i u m c o n c e n t r a t i o n v o r i o t i o n d u r i n g b u r st ___________________________________________________ <c 200 80 T Figure 4.6 160 40

Graph of the postulated variation in calcium concentration (mM). «-> ftpparent m e m b r a n e v o l t q q e shi/ft. 04' C O CO. 200 120 180 40

Graph showing the variation in calcium concentration translated into an equivalent shift in membrane voltage.

b u r s t ing in m o d i f i e d H H m o d e l

Mwamww1Aw

1/vi ^

Burst of action potentials resulting from the postulated change in apparent membrane potential.

B M e m b r a n e p o t e n t i a l p l u s s h i f t v o i t a q g

> o 40 eo 120 160 200

T

Figure 4.9

This Figure shows the result of adding the apparent shift in membrane potential to the calculated membrane potential.

8 A C T I V A T I O N VARIA B L E D U R I N G BURST

/ U -

1-20 1.50

'O'. OO O'- 30 O'- 60 0-90

T «io2 Figure 4.10

Graph showing the variation in sodium activation variable during the burst.

8 I N A C T I V A T I O N D U R I N G B U R S T

0.00 0.30 O.60 0-90

T •JO2 1.20 1-50

Figure 4.11

Graph showing the variation in the sodium inactivation variable during the burst.

S P O T A S S I U M A C T I V A T I O N D U R I N G B U R S T