Figure 1.
Asymptotic solutions of various orders for the two-compartment passive cable model.
Asymptotic solutions for (A) EPSP, (B) IPSP, and (C) SSP in comparison with numerical simulations of Equation (3). The blue dashed line is the first order approximation. The red circle is the second order approximation. The black solid line is the numerical solution of the full Equation (3). The stimuli are given at the location and
. Physiological parameters in the simulation can be found in the section of Materials and Methods.
Figure 2.
Description of as a function of time
and stimulus arrival difference
for a fixed pair of excitatory and inhibitory input locations.
Left, a morphological plot of the realistic neuron model. The excitatory and inhibitory input locations are indicated by arrows. Right, (lower) an IPSP arrives at the soma earlier than an EPSP. The arrival times are indicated by vertical dashed lines. (upper) The shunting coefficient remains at zero until the EPSP arrives at
.
Figure 3.
An example of EPSP, IPSP, SSP, SC, and the corresponding linear sum.
(A) The EPSP and the IPSP are elicited concurrently. Here denotes the time when EPSP reaches its peak value. (B) the IPSP is elicited
before the EPSP. The results are obtained in the realistic pyramidal neuron model simulation which is described in detail in the section of Materials and Methods. The excitatory input is given at the location
and the inhibitory input is given at the location
.
Figure 4.
Dendritic integration of a pair of excitatory and inhibitory inputs.
(A–D) Simulation results with the excitatory input given at the location and the inhibitory input given at the location
. (A) The SC amplitude is plotted against the product of EPSP amplitude and IPSP amplitude, at the time when EPSP reaches its peak, i.e.,
(Note that
and
are plotted). Varying
less than
and varying
less than
, it can be seen that
increases linearly with
. (B) Dendritic integration in the time interval
, where
. (upper)
for the goodness of the linear fitting of
vs.
at different times. (lower) The shunting coefficient
(in the unit of
) as the slope of the linear fitting is plotted at different times. The error bar indicates
confidence interval (The error bars are relatively small and are within the circles). The circle marked by red indicates the case in (A). (C–D) The same as (A–B) except that the IPSP is elicited
before the EPSP. (E–H) Experimental results with the excitatory input given at the location
and the inhibitory input given at the location
. (E–F) for concurrent inputs and (G–H) for nonconcurrent inputs that the IPSP is elicited
earlier than the EPSP.
Figure 5.
Shunting coefficient in branched dendrites measured in experiments.
The data marked by grey squares were collected from 7 neurons in our experiments, and lines connect data from the same neuron. The data marked by black squares are the average of the data marked by grey squares. The error bar indicates one standard deviation. In all figure panels, the locations of the inhibitory input (I) and excitatory inputs (E1 and E2) are marked by a blue dot and red dots, respectively. The I path is marked by green. (A) The inhibitory input I at an oblique branch: is nearly constant for two distal E1 and E2 on the same branch. (B) As in (A) except that E1 and E2 are more proximal than I.
is significantly different at E1 and E2 sites. (C) The inhibitory input I at the trunk:
is nearly constant between E1 at the trunk and E2 at the oblique branch. (D) The inhibitory input I at an oblique branch:
is significantly different between E1 and E2, where E1 is on the same branch as I and E2 is on a different branch.
Figure 6.
Dendritic integration of a pair of excitatory inputs.
(A–D) Simulation results with two excitatory inputs given at the location and
. (A) The SC amplitude is plotted against the product of the two EPSP amplitudes, at the time
when one of the EPSPs reaches its peak (Note that
is plotted). Varying
and
less than
, it can be seen that
increases linearly with
. (B) Dendritic integration in the time interval
, where
. (upper)
for the goodness of the linear fitting of
vs.
at different times. (lower) The shunting coefficient
(in the unit of
) as the slope of the linear fitting is plotted at different times. The error bar indicates
confidence interval (The error bars are relatively small and are within the circles). The circle marked by red indicates the case in (A). (C–D) The same as (A–B) except that one of the EPSPs is elicited
earlier than the other. (E–F) Our experimental result shows the nearly linear summation for (E) a pair of concurrent excitatory inputs and (F) nonconcurrent excitatory inputs with arrival time difference
, when two excitatory inputs are given at the location
and at
.
Figure 7.
Integration of a pair of excitatory inputs on passive dendrites.
Simulation results with two excitatory inputs given at the location and
. (A) The SC amplitude is plotted against the product of the two EPSP amplitudes, at the time
when one of the EPSPs reaches its peak (Note that
is plotted). (B) Dendritic integration in the time interval
, where
. (upper)
for the goodness of the linear fitting of
vs.
at different times. (lower) The shunting coefficient
(in the unit of
) as the slope of the linear fitting is plotted at different times. The error bar indicates
confidence interval (The error bars are relatively small and are within the circles). The circle marked by red indicates the case in (A). (C–D) The same as (A–B) except that one of the EPSPs is elicited
earlier than the other.
Figure 8.
Dendritic integration of a pair of inhibitory inputs.
(A–D) Simulation results. Two inhibitory inputs are given at the location and at
. (A) The SC amplitude is plotted against the product of the two IPSP amplitudes, at the time
when one of the IPSPs reaches its peak. Varying
and
less than
, it can be seen that
increases linearly with
. (B) Dendritic integration in the time interval
. (upper)
for the goodness of the linear fitting of
vs.
at different times. (lower) The shunting coefficient
(in the unit of
) as the slope of the linear fitting is plotted at different times. The error bar indicates
confidence interval (The error bars are relatively small). The circle marked by red indicates the case in (A). (C–D) The same as (A–B) except that one of the IPSPs is elicited
earlier than the other. (E–H) Experimental results with two inhibitory inputs given at the location
and at
. (E–F) for concurrent inhibitory inputs and (G–H) for nonconcurrent inhibitory inputs with arrival time difference
.
Figure 9.
Dendritic integration of multiple synaptic inputs.
(A) Distribution of 15 excitatory inputs (red dots) and 5 inhibitory inputs (blue dots) at the dendritic arbor of the realistic pyramidal neuron model. (B) One trial of membrane potential obtained by setting the arrival time of each stimulus randomly distributed from to
. The SSP (black dots) from the simulation of the realistic neuron model nearly overlaps with the SSP (red) predicted by the bilinear integration rule (25) while deviating from the trace of the direct linear summation of all postsynaptic potentials elicited separately (blue). (C) The direct linear sum (blue) and the SSP (red) predicted by rule (25) are plotted against the SSP from the simulation of the realistic neuron model. Here, the data are points on the corresponding curves from ten trials sampled uniformly from
to
. For comparison, the slope of the grey line is unity. It can be observed that the red dots fall on the grey line. This indicates that the predicted SSP is equal to the simulated SSP at any time.
Figure 10.
Graph representation of dendritic integration.
(A) A complete dendritic graph with 15 excitatory inputs (red) and 5 inhibitory inputs (blue). (B–D) Activated dendritic graph at time ,
and
, respectively. The color of an edge in (B–D) denotes the normalized SC value. Data are collected from simulations in Fig. 9.