Figure 1.
Experimental measurement of EPSP, IPSP and SSP.
The time when EPSP reaches its peak value is denoted by .
and
represent the amplitude of EPSP and IPSP at time
, respectively.
represents the amplitude of SSP at time
. SC is the difference between the SSP and the linear sum of the individual EPSP and IPSP measured under separate excitatory and inhibitory inputs. When the excitatory input and the inhibitory input are elicited simultaneously, the response amplitude measured at the soma is found to be smaller than that of the linear sum. (modified from Ref. [20]).
Figure 2.
The spatial dependence of obtained from Eq. (9).
Given three fixed inhibitory input locations, we can parameterize in Eq. (9) by identifying
with the shunting coefficient measured in the experiment [20] to obtain the spatial dependence of
. Inset: the experimental measurements of the shunting coefficient [20] as a function of the distance between the excitatory input site and the soma for three fixed inhibitory input sites on the dendrite (marked by dashed lines).
Figure 3.
Reproduced profiles of EPSP and IPSP by the I&F model.
The thick dark (blue online) lines are produced from the I&F model, the light gray lines represent EPSP and IPSP measured in the experiment for different trials, and the thin dark (red online) lines represent the trial-averaged responses in the experiment. Parameters in the I&F model are chosen as follows, ,
ms,
ms,
,
ms, and
ms. (See Methods for details).
Figure 4.
Dendritic integration rule obtained from the I&F model (A–C) and the modified I&F model (D–F).
We choose the values of and
so that the value of
computed from Eq. (8) matches the value measured in the experiment [20]. Lines in figures indicate linear fits with slope
. (A) Ratio between SC and EPSP (SC/EPSP) plotted against IPSP (red square online) and SC/IPSP plotted against EPSP (blue circle online) at time
for the I&F model (Red online:
= 0.070; blue online:
). Inset: experimental measurement (Red online:
= 0.142; blue online:
). (B) Ratio between the mean SC and the mean EPSP (SC/EPSP) plotted against the mean IPSP (red square online), the mean SC and the mean IPSP (SC/IPSP) plotted against the mean EPSP (blue circle online) for the I&F model. Inset: experimental measurement. (C) the same as (A) but at time (1)
= 20 ms, (2)
= 40 ms, (3)
= 60 ms and (4)
= 80 ms, respectively, as marked on the figures. (D) the same as (A) but for the modified I&F model (Red online:
= 0.147; blue online:
). Inset: the same as the inset of (A). (E) the same as (C) but for the modified I&F model. (F) the same as (B) but for the modified I&F model. Inset: the same as the inset of (B).
Figure 5.
The shunting coefficient as a function of time.
With three different values of :
(thick dark line: red online),
(thick gray line: blue online),
(light gray line: green online). Here
is fixed as
. It can be seen that
is nearly independent of the parameter
.
Figure 6.
Comparison of DIF model with experiments (A–C), multiple inputs (D), Predictions (E), spatial dependence of in the I&F model (F).
(A) SSP (measured sum) vs. the linear sum between EPSP and direct somatic hyperpolarization. Here, SC is not observed. The inhibition is caused by direct injection of an inhibitory current with amplitude of −2.6 mV (red circle online) and of −1.3 mV (red square online). Inset: experimental measurement (modified from Ref. [20]). (B) SC vs. the relative time delay between IPSP and EPSP. For fixed inhibitory input site, we choose two different input sites for excitation: one corresponds to the distal dendrite (red circle online) and the other the proximal dendrite (blue square online). Inset: experimental measurement (modified from Ref. [20]). (C) SC is not affected by changing the driving force (DF) of IPSP from 0 to −10 mV. Inset: experimental measurement (modified from Ref. [20]). (D) Dendritic integration rule for two excitatory and two inhibitory inputs. The simulated E–I sum represents the SSP obtained from Eq. (12) with a coincident activation of all excitatory and inhibitory inputs, whereas the predicted E–I sum represents the somatic membrane potential obtained from Eq. (11). (E) Comparison between and
as a function of time. The remainder term is defined as
(thick gray, blue online). Inset: The asymptotic behavior
(denoted by “asymptotic”, thick dark dash line, red online) of the remainder term (denoted by “exact”, thick gray, blue online) obtained from Eq. (9). (F) spatial dependence of
in the I&F model when the inhibitory input site is fixed. Using the spatial dependence of the conductance time constants, we can obtain the result that
decreases as the distance increases between the excitatory input site and the soma. This is not consistent with the experimental observation (Inset: the same as the inset of Fig. 2).