Bilinear Rule for E-I Integration

We begin to study the spatiotemporal dendritic integration of a pair of excitatory and inhibitory inputs. An analytical derivation of the bilinear integration rule is described in the section of Derivation of the Rule. The details of the cable model used in the derivation can be found in the section of Materials and Methods. The validation of the bilinear integration rule using the realistic neuron modeling and electrophysiological experiments is described in the section of Validation of the Rule. The spatial dependence of the coefficient in the rule is described in the section of Spatial Dependence of .

Derivation of the rule.

Most neurons possess complicated dendritic morphology, however, for simplicity, we start to investigate the spatiotemporal dendritic integration rule with an idealized two-compartment passive cable model, in which a spherical soma is connected to an unbranched cylindrical dendrite with finite length and diameter . The distance between a dendritic location and the soma is denoted by . Given an excitatory input at location and at time , and an inhibitory input at location and at time , the membrane potential dynamics is governed by(3)

where is the membrane potential with respect to the resting potential on the dendrite, is the membrane capacitance per unit area, is the axial resistivity, and is the leak conductance per unit area. The excitatory and inhibitory input strengths and control the amplitude of EPSP and IPSP, respectively. The variables and denote the unitary excitatory and inhibitory conductances per unit area with their peak value normalized to unity [described by Equation (29)]. and are the corresponding reversal potentials, respectively. The values of the parameters in the model can be found in the section of Materials and Methods.

By assuming that one dendritic end is sealed and the other dendritic end connects to the spherical soma, the boundary conditions are given by(4)(5)where is the surface area of the soma. The initial condition is simply set as(6)for a neuron at its resting state.

For the physiological regime, the corresponding synaptic input strengths, and , are relatively small in the model [Equations (3)–(6)]. To be specific, for the amplitude of an EPSP less than and the amplitude of an IPSP less than , the corresponding strengths and are considered to be small. Therefore, we can represent the membrane potential as an asymptotic series in powers of and as follows:(7)

Substituting Equation (7) into Equations (3)–(6), order by order, we can obtain its asymptotic solutions . The solutions to the second order () are described below (see the section of Materials and Methods for a detailed calculation). For the zeroth order, we have(8)

The solution corresponds to the fact that the membrane potential response remains at its resting state when there is no stimuli presented (, ). The first order excitation is(9)where ‘’ denotes convolution in time, and is the Green's function of the cable equation given a impulse input (the analytical expression is presented in the section of Materials and Methods). Note that the input at can be viewed as the synaptic current when the local membrane potential is maintained at the resting state. The second order of excitation is(10)where the expression of is given in Equation (9). Note that the input at can be viewed as the synaptic current, which is the product of the local conductance with the first order local membrane potential . Similarly, we have the first and second order inhibition and :(11)(12)

For the order of , we have(13)where and are given by Equations (9) and (11), respectively. On account of the fact that (relative to the resting potential) is nearly an order of magnitude larger than , in Equation (13) can be further simplified as(14)which indicates that the nonlinear integration effect mainly originates from the outward synaptic current, i.e., , induced by the first order EPSP measured at the inhibitory input site , i.e., .

Numerical simulation of the cable model indicates that the second order asymptotic approximation is sufficiently accurate in capturing the model's solution of physiologically realistic membrane potentials, as demonstrated in Fig. 1. Therefore, if only an individual excitatory input is given [ in Equation (3)], the corresponding EPSP measured at the soma, denoted by , can be approximated by(15)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1004014.g001

Similarly, if only an individual inhibitory input is given [ in Equation (3)], the corresponding IPSP measured at the soma, denoted by , can be approximated by(16)

If both the excitatory and inhibitory inputs are given at and , the corresponding SSP measured at the soma, denoted by , can be approximated by(17)

We define the difference between the SSP, and the linear summation of EPSP and IPSP as(18)

From Equations (15), (16) and (17), we have(19)

If is set to be at the resting potential, we can show that the value of indicating hyperpolarization vanishes, while the value of stays nearly the same. Therefore, is mainly caused by the shunting effect and thus is referred to as the shunting component (SC). In our analysis, the SC is the leading order of the nonlinear integration between excitation and inhibition. If we further define the shunting coefficient as(20a)(20b)

then is nearly independent of the amplitude of EPSP and IPSP, because the input strengths and , which determine the amplitudes of EPSP and IPSP, cancel each other in both denominator and numerator in Equation (20b).

From Equation (20a), we have the following spatiotemporal dendritic integration rule(21)

The shunting coefficient is nearly independent of the amplitude of EPSP and IPSP. In addition, depends on the location of excitatory and inhibitory inputs and . For a fixed pair of excitatory and inhibitory input locations, is a function of both time and the arrival time difference between the excitatory and inhibitory input , as illustrated in Fig. 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 .

https://doi.org/10.1371/journal.pcbi.1004014.g002

Validation of the Rule.

As the bilinear integration rule (21) is derived from the idealized passive neuron model, we need to investigate its validity for a realistic neuron, which has active ion channels embedded in its tree-like dendrites.

We first perform the simulation of a biologically realistic pyramidal neuron with active channels (morphology shown in Fig. 2). The details of the model and the related computational method can be found in the section of Materials and Methods. The simulation results are summarized below.

For the case of concurrent inputs (Here, we use “concurrent” to denote the case when ), as shown in Fig. 3A, when the excitatory and inhibitory inputs are elicited concurrently at different locations on the dendritic trunk, the SSP is found to be always smaller than the linear sum of the EPSP and the IPSP when elicited separately. In this case, the bilinear integration rule (21) holds at the time when the EPSP reaches its peak value. We can vary to control the amplitude of EPSP (less than ) and vary to control the amplitude of IPSP (less than ). For fixed input strengths and , we obtain the set of time courses of the EPSP, the IPSP, and the corresponding SSP. Using 9 sets of such data with different input strengths, we find that the SC amplitude depends linearly on the product of the EPSP and IPSP amplitudes, i.e., . The excellent linear fitting in Fig. 4A shows that the slope is independent of the amplitude of EPSP and IPSP. This result is consistent with the experimental observation [3].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 .

https://doi.org/10.1371/journal.pcbi.1004014.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1004014.g004

For the same case of concurrent inputs, we then calculate at any time , instead of the peak time . We compute within the interval , where . We choose this interval because the amplitude of the EPSP is relatively large within that interval. As a consequence, we can avoid a small denominator in calculating and improve the numerical accuracy. At each fixed time , is estimated by linear regression using the same 9 sets of data used to validate the bilinear integration rule at . As demonstrated in Fig. 4B, at each time , there is a small error bar for the slope estimation and the value is very close to 1. Both facts indicate an excellent linear fitting of vs. . Therefore, the shunting coefficient is nearly independent of the amplitude of EPSP () and IPSP () at any time. As expected, the error bar for the slope estimation increases dramatically far away from the peak time due to the fact that the numerical accuracy is low when EPSP and IPSP are small, in particular, when they approach zero. However, the SC amplitude in this case is sufficiently small, and can thus be neglected. Therefore, the bilinear integration rule can naturally be considered valid with .

For the case of nonconcurrent inputs, when the onset of the inhibitory input is earlier than that of the excitatory input (Fig. 3B), our numerical results show that the bilinear integration rule (21) still holds, as shown in Fig. 4C–D. The rule is also confirmed for any excitatory and inhibitory input locations arbitrarily distributed on the dendritic trees.

We next perform electrophysiological experiments to validate our bilinear integration rule (21). The details of the experimental procedure can be found in the section of Materials and Methods. In experiments, the excitatory input is given at with the EPSP amplitude less than and the inhibitory input is given at with the IPSP amplitude less than . For the case of concurrent inputs, we found that depends linearly on at the time when EPSP reaches its peak, as shown in Fig. 4E, and at a non-peak time, as shown in Fig. 4F. Therefore, the slope is nearly independent of EPSP and IPSP amplitudes. For the case of nonconcurrent inputs, when the IPSP is elicited earlier than the EPSP, the linear relationship between and still holds, as shown in Fig. 4G–H, except that the value of the regression is smaller (0.77 to 0.81) than those in the concurrent case (0.90 to 0.99). Therefore, it can be seen from the above that, the bilinear integration rule (21) is confirmed in rat hippocampal CA1 pyramidal neurons.

Note that the bilinear integration rule is derived from an idealized passive neuron case. Interestingly, our results from the simulation and experiments demonstrate that the structure of the rule is preserved in the presence of both active channels and dendritic branches.

Spatial dependence of .

Although we have obtained the form of the bilinear integration rule (21), how the value of the shunting coefficient depends on the input location is difficult to analyze directly from its explicit form [Equation (20b)]. Here we investigate the spatial dependence of input location for , which may partially reveal the way in which encodes spatial integration information.

In our previous study [23], a spatial rule for as a function of input locations has been proposed based on a theoretical analysis of a multi-compartment cable model. Under the situation when the excitatory and inhibitory inputs are given concurrently, the spatial profile of at the time when EPSP reaches its peak is characterized by the following spatial rule: For a fixed inhibitory input location, the I path (marked by green on the dendritic trees in Fig. 5) is defined as the path between the soma and the inhibitory input; Along the I path, is predicted to increase as the distance between excitatory input location and the soma increases; for any branch (including the trunk) connecting to the I path, is predicted to be constant for all excitatory input sites on the branch.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1004014.g005

The prediction of this spatial rule is consistent with our electrophysiological experimental results as shown in Fig. 5 (also see Ref. [3]). We next use the spatial rule to explain these experimental results.

In Fig. 5A and 5B, an inhibitory input is given on an oblique branch and two excitatory inputs are given at two locations on the same oblique branch. The corresponding shunting coefficients were estimated based on Equation (20a). In our experiment, no significant difference was found between the values of when the two excitatory locations were both distal, as shown in Fig. 5A. This experimental observation can be easily understood by our spatial rule for because the two excitatory inputs are on the same branch connected to the I path. By our rule above, the shunting coefficient should be the same on such a branch.

In contrast, as shown in Fig. 5B, for two excitatory inputs at proximal locations, was experimentally found significantly smaller for the excitatory input closer to the soma. This is consistent with our rule because is predicted to be an increasing function of the distance between the excitatory input location and the soma for this case.

In Fig. 5C, an inhibitory input is given on the apical trunk and two distal excitatory inputs are given at either the trunk or a branch. For this case, the values were found to be nearly constant in our experiment. This is the case in which the two branches where the two excitatory inputs are located connect to the I path with the same branching point. Therefore, this experimental observation can be understood through our rule that all on the two branches are the same as the one at the branching point.

In Fig. 5D, an inhibitory input is given at an oblique branch and two distal excitatory inputs are given at different branches. For this case, the value of for inhibitory and excitatory inputs located at the same branch was found in our experiment to be significantly larger than the case in which inhibitory and excitatory inputs are located at different branches. For the case when inhibitory and excitatory inputs located at different branches, by our rule, equals to the value at the branching point on the I path. Because this branching point is closer to the soma than the other excitatory input location on the I path, the increase of along the I path predicted by our rule explains the experimental observation.

Our theory [23] further predicts that, in principle, it is possible for an inhibitory input located at a branch to shunt the excitatory input on other branches with a large value of . This generalizes the conclusion in Ref. [3], in which the shunting inhibition is mainly confined within the branch where the inhibitory input is located. In Ref. [3], the simulation investigation is focused on the case when inhibitory input is elicited on a branch close to soma. In this case, for excitatory inputs on different branches is found to be small in simulation. This observation is consistent with our rule. According to our rule, the value of for an excitatory input on the branch is equal to that for excitatory input on the branching point connected to the I path. Due to the branching point being very close to the soma in their simulation, can be small. In general, when the inhibitory input is elicited at a location far from the soma, for the excitatory input on some other distal dendritic branch, the distance between the branching point and soma can be large, which leads to a large value of .

Bilinear Rules for E-E & I-I Integration

So far we have addressed the dendritic integration for a pair of excitatory and inhibitory inputs. A natural question arises: how does a neuron integrate a pair of time-dependent synaptic conductance inputs with identical type?

The dendritic integration of excitatory inputs has been extensively investigated in experiments (reviewed in Ref. [1]), yet a precise quantitative characterization is still lacking. According to our idealized cable model, given a pair of excitatory inputs with input strengths and at locations and and at times and , the dynamics of the membrane potential on the dendrite is governed by the following equation:(22)

with the initial and boundary conditions the same as given in Equations (4)–(6). Similarly, we can represent its solution as an asymptotic series and solve it order by order to obtain the following bilinear integration rule:(23)

where and are EPSPs induced by two individual excitatory inputs, and is the SSP when the two excitatory inputs are present. Similar to the case of a pair of excitatory and inhibitory inputs, the shunting coefficient only depends on the excitatory input locations and the input time difference. It does not depend on the EPSPs' amplitudes. Here will still be referred to as a shunting coefficient because the origin of the nonlinear integration for the paired excitatory inputs is exactly the same as that for the paired excitatory and inhibitory inputs from the passive cable model.

The bilinear integration rule (23) is found to be consistent with the numerical results obtained using the same realistic pyramidal neuron model as the one used in the section of Bilinear Rule for E–I Integration. For a pair of excitatory inputs with their locations fixed on the dendritic trunk, the rule holds when the amplitude of each EPSP is less than . For the case of concurrent inputs, at the time when one of the EPSPs reaches its peak valueis found to be linearly dependent of , as shown in Fig. 6A. This linear relationship indicates is independent of the amplitudes of the two EPSPs. In addition, as shown in Fig. 6B, the bilinear integration rule is numerically verified in the time interval , for , within which the amplitude of EPSPs are relatively large. For the case of nonconcurrent inputs, the bilinear integration rule is also numerically verified in the same way, as shown in Fig. 6C–D.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 .

https://doi.org/10.1371/journal.pcbi.1004014.g006

In addition, we find that when the input strengths become sufficiently strong so as to make the depolarized membrane potential too large, i.e. , there is a deviation from the bilinear integration rule (23). This deviation can be ascribed to the voltage-gated ionic channel activities in our realistic pyramidal neuron model. After blocking the active channels, the rule becomes valid with a different value of for large EPSPs amplitudes, as shown in Fig. 7. However, we note that, regardless of input strengths, the amplitude of SC is always two orders of magnitude smaller than the amplitude of SSP. Therefore, the integration of two excitatory inputs can be naturally approximated by the linear summation of two individual EPSPs, i.e. . We then perform electrophysiological experiments with a pair of excitatory synaptic inputs to confirm the linear summation. As expected, this linear summation is also observed in our experiments for both concurrent and nonconcurrent input cases, as shown in Fig. 6E and 6F, respectively. Note that, the linear summation is also consistent with experimental observations as reported in Ref. [24].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1004014.g007

Similarly, for a pair of inhibitory inputs, we can arrive at the following bilinear integration rule from the cable model:(24)where and are IPSPs induced by two individual inhibitory inputs, and is the SSP when the two inhibitory inputs are present. Here, is the shunting coefficient that is independent of the IPSPs amplitudes but is dependent on the input time difference and input locations. The above bilinear integration rule (24) is consistent with our numerical results using the realistic pyramidal neuron model, as shown in Fig. 8A–D. Our electrophysiological experimental observations further confirm this rule, as shown in Fig. 8E–H.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 .

https://doi.org/10.1371/journal.pcbi.1004014.g008

Bilinear Rule for Multi-input Integration

In the previous sections, we have discussed the integration of a pair of synaptic inputs. In vivo, a neuron receives thousands of excitatory and inhibitory inputs from dendrites [2]. Therefore, we now address the question of whether the integration rule derived for a pair of synaptic inputs can be generalized to the case of multiple inputs.

Our theoretical analysis shows that, for multiple inputs, the SSP can be approximated by the linear sum of all individual EPSPs and IPSPs, plus the bilinear interactions between all the paired inputs with shunting coefficients , , and respectively (the superscript labels the synaptic inputs), i.e.,(25)

We next validate the rule (25) using the realistic pyramidal neuron model. It has been reported that, for a CA1 neuron, inhibitory inputs are locally concentrated on the proximal dendrites while excitatory inputs are broadly distributed on the entire dendrites [25]. Based on this observation, we randomly choose 15 excitatory input locations and 5 inhibitory input locations on the model neuron's dendrites (Fig. 9A). In the simulation, all inputs are elicited starting randomly from to . In order to compare Equation (25) with the SSP simulated in the realistic neuron model, we first measure , , and pair by pair for all possible pairs. We then record all membrane potential traces and induced by the corresponding individual synaptic inputs. Our results show that the SSP measured from our simulation is indeed given by the bilinear integration rule (25), as shown in Fig. 9B and 9C. In contrast, the SSP in our numerical simulation deviates significantly from the linear summation of all individual EPSPs and IPSPs.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1004014.g009

Graph Representation of Dendritic Integration

According to our bilinear integration rule (25), the dendritic integration of multiple synaptic inputs can be decomposed into the summation of all possible pairwise dendritic integration. Therefore, we can map dendritic computation in a dendritic tree onto a graph. Each dendritic site corresponds to a node in the graph and the corresponding shunting component is mapped to the weight of the edge connecting the two nodes. We refer to such a graph as a dendritic graph. The dendritic graph is an all-to-all connected graph if all stimuli are given concurrently (Fig. 10A). However, the dendritic integration for all possible pairs of synaptic inputs is usually not activated concurrently in realistic situations. For instance, if the arrival time difference between two inputs is sufficiently large, there is no interaction between them. The activated level of the nonlinear dendritic integration for a pair of synaptic inputs can be quantified by the SC amplitude—the weight of the edge in the graph. The simulation result shows that the number of activated edges at any time is relatively small on the dendritic graph (Fig. 10B–D), compared with the total number of edges on the all-to-all connected graph (Fig. 10A). Therefore, for the case of a hippocampal pyramidal neuron, the dendritic graph could be functionally sparse in time. The functional sparsity of a dendritic graph may also exist in neocortical pyramidal neurons. In vivo, a cortical pyramidal neuron receives about synaptic inputs [26]. Most of them are from other cortical neurons [27], [28], which typically fire about 10 spikes per second in awake animals [29], [30]. Thus, the neuron can be expected to receive synaptic inputs per second. The average number of synaptic inputs within (membrane potential time constants in vivo) is . The number of activated dendritic integration pairs within the interval is , which is relatively small compared with the total possible synaptic integration pairs . Therefore, the activated integrations or edges in the dendritic graph within a short time window can be indeed functionally sparse (). In general, the neuronal firing rates vary across different cell types, cortical regions, brain states and so on. Therefore, based on the above estimate, in an average sense, the graph of dendritic integration is functionally sparse.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1004014.g010

Ethics Statement

The animal-use protocol was approved by the Animal Management Committee of the State Key Laboratory of Cognitive Neuroscience & Learning, Beijing Normal University (Reference NO.: IACUC-NKLCNL2013-10).

The Cable Model

We consider an idealized passive neuron whose isotropic spherical soma is attached to an unbranched cylindric dendrite with finite length and diameter . Each small segment in the neuron can be viewed as an RC circuit with a constant capacitance and leak conductance density [11], [38]. The current conservation within a segment on the dendrite leads to(26)where is the membrane potential with respect to the resting potential on the dendrite, is the membrane capacitance per unit area, and is the leak conductance per unit area. Here, is the synaptic current given by:(27)where and are excitatory and inhibitory synaptic conductance per unit area and and are their reversal potentials, respectively. When excitatory inputs are elicited at dendritic sites and inhibitory inputs are elicited at dendritic sites, we have(28)where . For a synaptic input of type , is the input strength of the input at the location, is the arrival time of the input at the location, is the input location. The unitary conductance is often modeled as(29)with the peak value normalized to unity by the normalization factor , and with and as rise and decay time constants, respectively [38]. Here is a Heaviside function. The axial current can be derived based on the Ohm's law,(30)where is the axial resistivity. Taking the limit , Equation (26) becomes our unbranched dendritic cable model,(31)

In particular, for a pair of excitatory and inhibitory inputs with strength and received at and , and at time and , respectively, we have(32)

Similarly, for a pair of excitatory or inhibitory inputs with strengths and received at and , and at time and (), respectively, we have(33)

For the boundary condition of the cable model [Equation (31)], we assume one end of the dendrite is sealed:(34)

For the other end connecting to the soma, which can also be modeled as an RC circuit, by the law of current conservation, we have(35)where is the somatic membrane area, and is the somatic membrane potential. The dendritic current flowing to the soma, , takes the form of Equation (30) at . Because the membrane potential is continuous at the connection point(36)we arrive at the other boundary condition at :(37)

For a resting neuron, the initial condition is simply set as(38)

Green's Function

In the absence of synaptic inputs, Equation (31) is a linear system. Using a impulse input, its Green's function can be obtained from(39)with the following boundary conditions and initial condition,

For simplicity, letting , , , the solution of Equation (39) can be obtained from the following system,(40)

with rescaled boundary and initial conditions,

where . Taking the Laplace transform of Equation (40), we obtain(41)

Combining the two boundary conditions ( is thus eliminated), we have(42)where(43)whose denominator is denoted as for later discussions. For the inverse Laplace transform, we need to deal with singular points that are given by the roots of . It can be easily verified that these singularities are simple poles and is analytic at infinity. Then can be written as(44)where is a constant coefficient in the complex domain, and are the singular points. Then taking the inverse Laplace transform of Equation (44), we obtain(45)

Now we only need to solve and in Equation (45) to obtain the Green's function of Equation (40). We solve the singular points first. Defining yields(46)whose roots can be determined numerically. There are solutions for with for and Next, to determine the factors we use the residue theorem for integrals. For a contour that winds in the counter-clockwise direction around the pole and that does not include any other singular points, the integral of on this contour is given by(47)

Using Equations (42–44) and (47), we obtain(48)where(49)for . The solution of the original Green's function for Equation (39) can now be expressed as(50)

Asymptotic Analysis

We first consider the case when a pair of excitatory and inhibitory inputs are received by a neuron. Similar results can be obtained for a pair of excitatory inputs and a pair of inhibitory inputs. For the physiological regime (the amplitude of an EPSP being less than and the amplitude of an IPSP being less than ), the corresponding required input strengths and are relatively small. Therefore, given an excitatory input at location and time , and an inhibitory input at location and time , we represent as an asymptotic series in the powers of and ,(51)

Substituting Equation (51) into the cable equation (31), order by order, we obtain a set of differential equations. For the zeroth-order, we have(52)

Using the boundary and initial conditions [Equations (34), (37), and (38)], the solution is simply

(53)

For the first order of excitation , we have(54)

With the help of Green's function, the solution can be expressed as

(55)

here ‘’ denotes convolution in time. For the second order of excitation , we have(56)

Because is given by Equation (55), the solution of Equation (56) is(57)

Similarly, we can have the first and second order inhibitory solutions,(58)(59)

For the order of , we have(60)whose solution is obtained as follows,(61)

Numerical Simulation

For the numerical simulation of the two-compartment passive cable model [Equation (3)], the Crank-Nicolson method [39] was used with time step and space step . Parameters in our simulation are within the physiological regime [3], [12] with , , , , , , , . , , , and . The time constants here were chosen to be consistent with the conductance inputs in the experiment [3].

The realistic pyramidal model is the same as that in Ref. [3]. The morphology of the reconstructed pyramidal neuron includes 200 compartments and was obtained from the Duke-Southampton Archive of neuronal morphology [40]. The passive cable properties and the density and distribution of active conductances in the model neruon were based on published experimental data obtained from hippocampal and cortical pyramidal neurons [18], [19], [34], [41]–[50]. We used the NEURON software Version 7.3 [51] to simulate the model with time step .

Hippocampal Slice Preparation and Electrophysiology

The experimental measurements of summation of EPSPs or IPSPs in single hippocampal CA1 pyramidal cells in the acute brain slice followed a method described in Ref. [3], with some modifications. A brief description of modified experimental procedure is as follows. Acute hippocampal slices ( thick) were prepared from Sprague Dawley rats (postnatal day 14–16), using a vibratome (VT1200, Leica). The slices were incubated at 34°C for 30 min before transferring to a recording chamber perfused with the aCSF solution (2ml/min; 30–32°C). The aCSF contained (in mM) 125 NaCl, 3 KCl, 2 CaCl₂, 2 MgSO₄, 1.25 NaH₂PO₄, 1.3 sodium ascorbate, 0.6 sodium pyruvate, 26 NaHCO₃, and 11 D-glucose, and was saturated with gas containing 95% O₂ and 5% CO₂ (pH 7.4). Whole-cell recording was made from the soma of CA1 pyramidal cells using glass micropipettes under an upright microscope (BX51WI, Olympus) equipped with the DIC optics and an infrared camera (IR-1000E, DAGE-MTI). The intra-micropipette solution contained (in mM) 145 K-gluconate, 5 KCl, 10 HEPES, 10 disodium phosphocreatine, 4 Mg₂ATP, 0.3 Na₂GTP, and 0.2 EGTA (pH 7.3), together with fluorescent dye Alexa Fluor 488 (, Invitrogen) to visualize the dendritic trees. Pipette resistance was about 3–4 MΩ, and the access resistance during the whole-cell recording was normally less than 20 MΩ. The same method for micro-iontophoretic application of extracelluar glutamate or GABA at the apical dendrite of CA1 pyramidal cells was used to elicit rapid membrane depolarizations (EPSPs) and hyperpolarizations (IPSPs). For all three experimental configurations (EPSP-IPSP, EPSP-EPSP and IPSP-IPSP summation), two micro-iontophoretic pipettes were placed at dendritic locations and from the soma, respectively, in particular for the EPSP-IPSP summation GABA iontophoretic pipette was always placed at the more proximal location than glutamate iontophoretic pipette was placed. For each recorded cell, an electrode was placed at the soma to set the resting membrane potential to about in order to obtain a driving force of for inhibitory GABA inputs. Electrical signals of individual and summed iontophoretic responses were amplified and filtered at 3 kHz (low pass) by a patch clamp amplifier (MultiClamp 700B, Molecular Devices), digitalized (100 kHz) by an AD-DA converter (Digidata 1440A, Molecular Devices), and acquired by a pClamp 10.3 (Molecular Devices) into a computer for further analysis.

Data Processing

In order to study the dendritic integration of a pair of excitatory and inhibitory inputs, for fixed input locations and strengths, a moving average technique with time lag was first applied to smooth each individual trace of the EPSP, IPSP, and SSP recorded in our experiments. After smoothing, we measured the amplitudes of EPSP, IPSP, and SSP at different times, including those when EPSP reached its peak value, and denoted them by , , and , respectively. By varying the excitatory and inhibitory input strengths, we measured values of , and . We then constructed a scatter plot of vs. at different times. We divided the range of into approximately 10 bins and averaged all the data points within each bin. The number of bins was chosen to ensure at least 8 data points were used for averaging. However, the qualitative results were not sensitive to the number of bins (e.g, from 6 bins to 16 bins). Using the Curve Fitting Toolbox in Matlab Version 7.14, we finally fitted the averaged data points by a linear function , from which the slope was estimated together with its confidence interval. For the dendritic integration of a pair of identical type, the same data processing procedure was followed.