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Division of Neuroscience, Baylor College of Medicine, Houston, Texas 77030
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ABSTRACT |
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 Abstract
 Introduction
Methods
Results
Discussion
References
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Cook, Erik P. and Daniel Johnston. Active dendrites reduce location-dependent variability of synaptic input trains. J. Neurophysiol. 78: 2116-2128, 1997. We examined the hypothesis that dendritic voltage-gated channels can reduce the effect synaptic location has on somatic depolarization in response to patterns of short synaptic trains (referred to as location-dependent variability). Three computer models of a reconstructed hippocampal CA1 cell, each of increasing realism and complexity, were used. For each model, the goal was to identify the dendritic composition that best reduced the location-dependent variability. The first model was linear and a single parameter, dendritic membrane conductance (GDm, where Rm = 1/GDm), was varied. Surprisingly, a negative GDm minimized the location-dependent variability. Superposition of the synaptic inputs showed that, compared with passive dendrites, active dendrites increase the mean of the individual responses while decreasing the variance between synapses at different locations. Active dendrites compensate the three components of passive cable signal interference that increase with distance from the soma: the accumulation of charge on dendritic membrane capacitance, the escape of charge across synaptic and nonsynaptic dendritic membrane conductances, and the reduction in synaptic charge entry due to increased depolarization of dendrites located farther from the soma. We also found that the entire active dendritic tree contributes charge to any one active synapse. The second model contained an artificial voltage-dependent current (Iboost) added to passive apical dendrites. The optimal amount of Iboost that minimized location-dependent variability was found to be independent of the strength of individual synaptic inputs but inversely related to the synaptic duration. In the third model, realistic T-type Ca2+ and persistent Na+ channel models were added to passive dendrites and numerically fit to reproduce the effects of Iboost. Both realistic currents minimized synaptic variability. The densities for the realistic dendritic currents were not uniform but showed subtle variations and a slight reduction with distance from the soma. A heteroassociative memory network also was modeled to demonstrate the important relationship between location-dependent variability and memory recall performance. Compared with passive dendrites, active dendrites increased memory storage by reducing recall errors. These simulations demonstrate that active dendrites can minimize the cable properties of passive dendrites and enhance the soma's ability to determine the strength of the synaptic input. These models predict dendrites that minimize location-dependent variability will have an overall negative slope conductance I-V relationship that is tuned precisely.
Wilfrid Rall's (1959) passive theory of dendrites predicts that a synapse's effectiveness at the soma is dependent on its location (referred to as location-dependent variability). It, however, has long been hypothesized that dendritic voltage-gated channels may counteract this property and make all synapses electrotonically equidistant from the soma (Andersen et al. 1980
The reconstructed CA1 pyramidal cell used was provided by D. Turner and is shown in compartmentalized form in Fig. 1A. The neurophysiological modeling program NEURON was used for all simulations (Hines 1993
Dendritic fidelity in the linear model (model 1)
In the first model, the dendritic Rm was allowed to assume negative values. We therefore decided to call this the linear model. This allows a distinction between a positive dendritic Rm that leaks current and is passive and a negative Rm that injects current and is active. The goal was to determine if active dendrites produced by making Rm negative could more accurately transmit the strength of the synaptic input to the soma. Each individual synaptic input (gs) was modeled as a constant conductance of 27.2 pS,1 which was chosen to keep all simulations consistent.
Contribution of individual synapses to somatic depolarization
Our first model was linear because it contained no voltage-dependent channels and the synapses were modeled using fixed value conductances. This allowed us to use superposition to assess directly the effect a negative GDm has on individual synaptic contributions to somatic depolarization. We predicted that a model that minimized
Active dendrites have increased voltage attenuation
Figure 3 shows the peak dendritic membrane potential for each apical compartment in response to the same input pattern used in Fig. 1B. The active dendrites (GDm =
Active dendrites compensate for synaptic charge loss and accumulation
How does a negative dendritic membrane conductance minimize the effect synaptic location has on somatic response? This question is answered best by examining how charge is transferred from the synapses to the soma. Charge transfer has been recognized as an important measure of synaptic strength (Jack et al. 1975
Dendritic fidelity in the boosting model (model 2)
The second model was made more realistic than the previous model in three ways (see Table 2). First, the dendrites had a passive membrane (GDm = 0.04) on which a voltage-dependent "artificial" current was added. In reference to the "boosting hypothesis" of active dendrites, this current was named Iboost, and its I-V relationship is shown in the inset of Fig. 5A. On activation at Effects of individual synaptic strength and integration time on the amount of boosting
We wanted to understand how the strength of individual synapses and duration of the synaptic input would affect the optimal value for Gboost. The strength of individual synapses was increased (by increasing synaptic frequency to 75 Hz) or decreased (30 Hz) while the duration was kept at 100 ms, and two new sets of Pf-curves were computed. In Fig. 6A,
Using realistic voltage-gated channels to improve dendritic fidelity (model 3)
A major limitation of the simulations described above is that the artificial current Iboost was modeled using a simplified negative conductance Gboost. In the third model, T-type Ca2+ (ICaT) and persistent Na+ (INaP) channels were used to test the hypothesis that voltage-gated inward currents could serve the role of the negative conductance channel Iboost.
Heteroassociative memory network
A heteroassociative memory network was used to test the prediction that active dendrites could improve recall performance. The heteroassociative network (also known as the matrix memory network) has the synaptic connectivity shown in Fig. 8 and has been used as a model for the CA1 region of the hippocampus (McNaughton 1989
Active dendrites improve heteroassociative memory recall
Heteroassociative networks were constructed with the boosting model (model 2) and the realistic model (model 3) to examine how minimizing location-dependent variability affects recall performance. For comparison, a model containing a spiking soma and passive dendrites (with various dendritic Rm) also was used. The prediction was that networks constructed with models containing active dendrites would outperform the passive dendrite models.
The hypothesis that dendritic voltage-gated channels counteract cable properties and minimize the effect synaptic location has on somatic depolarization was addressed. Correctly setting a single conductance (either GDm or Gboost) to the appropriate value greatly reduced location-dependent variability. Dendritic T-type Ca2+ and persistent Na+ currents functioned as well as Iboost but required nonuniform channel densities. Recent experimental evidence suggests both dendritic T-type Ca2+ and persistent Na+ are capable of boosting EPSPs in hippocampal neurons (Gillessen and Alzheimer 1997 Assumptions of the model
In establishing that active dendrites could improve the fidelity of transmission of synaptic input to the soma, several important assumptions were made: 1) hippocampal pyramidal neurons are temporal integrators. The most fundamental assumption underlying this study is that synaptic input is integrated in the soma where the decision to fire an action potential is made. These simulations assumed that neurons integrate the synaptic input on the order of tens of milliseconds. Therefore, the effect of active dendrites on single EPSPs was not considered. 2) All synapses are the same. This is an oversimplification, as there are cases of specialized synaptic inputs to central neurons (e.g., the mossy fiber input to hippocampal CA3 cells). 3) Patterns of activity were selected randomly. At present it is entirely unknown how patterns of activity impinge on hippocampal neurons. 4) Only AMPA-like synapses were used. N-methyl-D-aspartate (NMDA) synaptic inputs provide a negative slope conductance and theoretically could reduce location-dependent variability. 5) Only threshold properties were explored in the models. The continuous firing properties of the models were not addressed. In vivo single-unit recordings have shown that hippocampal pyramidal neurons have low firing rates and a tendency to fire in bursts (Ranck and Feder 1973 Implications of negative dendritic slope conductances
Depending on the sign and slope of the dendritic I-V relationship, three basic "dendritic modes of operation" can be described. 1) A positive slope I-V relationship. This is the passive case in which current escapes across the membrane conductance. This mode has high location-dependent variability. 2) A negative slope I-V relationship that is stable. Here, active dendrites inject current but do not regeneratively depolarize. In this mode, location-dependent variability is reduced. 3) A negative slope I-V relationship of sufficient magnitude to produce regenerative depolarizations with location-dependent variability greatly increased. The transition point at which an active dendrite becomes regenerative depends on the relative conductances of the entire neuron (Jack et al. 1975 Other models of active dendrites
It has been suggested that voltage-dependent currents could alter the integrative properties of dendrites (Jack et al. 1975
INTRODUCTION
Abstract
Introduction
Methods
Results
Discussion
References
; Jack et al. 1975; Johnston et al. 1996; Shepherd et al. 1985). This is known as the "boosting hypothesis" of active dendrites because it implies distal inputs are amplified selectively (Crill 1996; Lorente de Nó and Condouris 1959; Rall 1970; Schwindt and Crill 1995). If we assume that the boosting hypothesis is correct, then what predictions can be made regarding the properties of dendrites that would eliminate the effects of synaptic location on somatic response? What would the dendritic I-V relationship be? And how might dendritic voltage-gated channels produce this effect?
; Magee and Johnston 1995), dendritic function has become the focus of neuronal modeling with many proposing elaborate dendritic computations (Koch et al. 1983; Mel 1993; Segev 1992; Shepherd and Brayton 1987; Woolf et al. 1991). This study departs from these models by assuming that dendrites function as high-fidelity transducers of synaptic input that accurately convert synaptic conductance into somatic depolarization independent of synaptic location. The importance of accurately transferring synaptic input to the soma becomes evident when one considers that most theories of learning and memory place the site of information storage at the synapse.
). Location-dependent variability, resulting from dendritic cable properties, reduces the amount of synaptic charge reaching the soma. First, the amount of synaptic charge injected into the dendrites decreases with distance from the soma. Next, synaptic charge traveling to the soma is reduced as charge escapes across dendritic membrane conductances and accumulates on the dendritic capacitance. The amount of location-dependent variability is confounded further by the synaptic input itself, which can alter drastically the membrane conductance of the dendritic tree (Bernander et al. 1991).
have demonstrated that active dendrites in a purkinje cell model can reduce the variability of single excitatory postsynaptic potentials (EPSPs). In this paper, the inputs to our models are short trains of synaptic input, and rather than analyzing a previously designed model, we attempt to find the optimal dendritic configuration that minimizes location-dependent variability.
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FIG. 1.
Minimizing location-dependent variability in a nonfiring CA1 pyramidal model. A: compartmentalized version of the CA1 neuron used in the simulations and the region of synaptic input. B: an example of the somatic depolarization to a 100-ms synaptic input (heavy bar) for GDm = 0.04 and 
0.026 mS/cm2. Because the soma was passive, an artificial threshold (Vth) was arbitrarily defined as 45 mV. If somatic potential reached Vth, the model was said to have fired an action potential. Number of active synapses was adjusted in each case to produce the same amount of somatic depolarization for each model. C: examples of 3 Pf curves for GDm = 0.04, 0, and 0.026 mS/cm2 (which corresponds to dendritic Rm of 25, 1, and 38.5 k
cm2, respectively). Each curve represents the probability (Pf) that the model reaches Vth vs. the total synaptic strength (Gsyn) in response to a random synaptic input pattern. One hundred different input patterns were used to generate each curve. Negative values of GDm increase the model's excitability thereby requiring a smaller Gsyn to reach Vth. 
is the range of Gsyn over which Pf increases from 5 to 95%. This is a measure of location-dependent variability. D: vs. GDm. Notice that the Pf curve corresponding to aGDm of 0.026 mS/cm2 has the smallest .
). Active dendrites that reduce the effect of location on synaptic input increased memory recall performance of the network.
METHODS
Abstract
Introduction
Methods
Results
Discussion
References
). Table 1 lists the general parameters of the model. Unless otherwise indicated, these parameters were used in all simulations. The passive parameters were chosen to provide a membrane time constant (
m) of 50 ms. To account for the lack of explicitly modeled spines in the dendrites, Rm was halved and Cm doubled in the apical and basal dendrites (Spruston et al. 1993). The basal dendrites were always passive, whereas the properties of the apical dendrites varied between models.
View this table:
TABLE 1.
General model parameters
Na = 0.05 S/cm2) and delayed-rectifier K+ (KDR = 0.012 S/cm2) conductances were assigned to the soma compartments to always ensure an action potential for the different dendritic conditions. The apical dendrites of model 2 contained the artificial voltage-dependent current, Iboost, which is illustrated in the inset of Fig. 5A. Model 3 had either T-type Ca2+ (ICaT), persistent Na+ (INaP), or both currents in its apical dendrites. The channel densities for the apical dendrites were the parameters of interest and were varied in this study. Channel descriptions are reported in the APPENDIX except for the T-type Ca2+ channel, which was previously described in Jaffe et al. (1994).
View this table:
TABLE 2.
Model specific parameters
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FIG. 5.
Voltage-dependent current (Iboost) was used to minimize location-dependent variability. A: membrane potential (Vm) at the soma in response to a 50-Hz randomly selected synaptic input pattern for passive and boosting dendrites. For the boosting dendrites, Gboost = 0.12 mS/cm2. Heavy bar indicates the time of synaptic input (100 ms). Number of active synapses per pattern was set to produce responses just above threshold. Inset: I-V relationship of Iboost, which activates at 50 mV. B: is plotted against Gboost. A Gboost of 0.12 mS/cm2 corresponds to the minimum .
functions with a time to peak of 2 ms and a peak conductance of 100 pS (see Table 2). The synaptic reversal potential was 65 mV above rest. All synaptic input was assumed to be -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA)-like. Patterns of synaptic input were produced by randomly selecting synapses to be activated. Spines were not included in the model because preliminary simulations suggested they had little effect on the electrical properties of the model.
RESULTS
Abstract
Introduction
Methods
Results
Discussion
References
0.026 mS/cm2 for the active dendrites. Notice that GDm = 0.026 produces a slow response because the dendritic membrane time constant is very large. The number of active synapses was adjusted for each value of GDm to produce 20 mV of somatic depolarization at the end of the synaptic input (100 ms). Because the linear model did not produce spikes, an artificial threshold (Vth) was assumed to be 20 mV above rest. If the somatic membrane potential reached Vth, the model was said to have fired. Whether or not a model fires an action potential in response to synaptic input was the criterion used to evaluate the location-dependent variability.
, which corresponds to the range of Gsyn that causes Pf to go from 0.05 to 0.95 (Fig. 1C). Active dendrites with a GDm of 0.026 mS/cm2 produce a Pf curve with a small compared with the model whose dendrites were passive.
is plotted in Fig. 1D. AsGDm is increased in the negative direction, the dendrites become better tuned in their ability to faithfully transmit synaptic input to the soma. The highest fidelity dendrite corresponds to the minimum at GDm = 0.026. Notice that just eliminating the leak conductance alone (by settingGDm = 0) did not minimize . When GDm assumed a value more negative than 0.027, the dendrites became unstable, resulting in regenerative dendritic depolarization and a total loss of dendritic fidelity.
also would minimize the variance between individual synaptic responses, regardless of dendritic location.
). Therefore, to see the effect of a single synapse, the reversal potential for all other synapses was set to the resting membrane potential. This procedure is not, however, equivalent to applying a single synapse to the dendrites. It is important to emphasize that all synaptic conductances of a particular input pattern remained in the dendrites, but only one had a reversal potential of 0 mV.
0.026) dendrites, Fig. 2A shows the peak somatic response of 1,000 individual synapses as a function of anatomic distance from the soma. The points in Fig. 2A were generated by applying random patterns of synaptic input and computing the individual contribution of each synapse to the soma using superposition. Each input pattern used was strong enough to cause the soma to reach Vth at the end of the 100-ms synaptic duration (as in Fig. 1B). In the passive dendrite model, the influence on the soma from synapses farther away diminished steadily. In the active dendrite model, the decrement was less. Figure 2B accentuates the differences between the two models in a histogram of the voltage responses in Fig. 2A. Compared with the passive model, the active dendrites clearly produced a narrower distribution of somatic responses. For the active dendrites, the mean response was increased whereas the variance decreased. The coefficient of variation (CV) of individual synaptic responses was computed as a function of GDm (Fig. 2C).2 Optimal active dendrites (GDm = 0.026) produced a greater than fivefoldreduction in CV when compared with passive dendrites(GDm = 0.04). This result verifies that the dendritic composition that minimized also minimized the location-dependent variability of individual synapses.
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FIG. 2.
Effect of active dendrites on individual synapses. A: superposition was used to compute the peak somatic response from 1,000 individual synapses. Peak somatic potential is plotted as a function of synaptic distance from the soma. Two models used were a passive model (+) with GDm = 0.04 and the optimal active model ( 
) with GDm = 0.026. Each data point corresponds to a single synapse, but as there were 1,000 possible synaptic locations in the dendrites, some points may overlap. B: histograms of the responses in A (binwidth = 0.005 mV). Mean response was 0.11 ± 0.026 (SD) mV for the passive dendrites and 0.29 ± 0. 012 mV for the active dendrites. C: coefficient of variation (CV) for the synaptic responses as a function of GDm. CV of the passive dendrites is >5 times higher than for the optimal active. At values of GDm < 0.027, the dendrites produced regenerative depolarizations, and thus CV was not computed.
0.026) depolarized more than the passive dendrites (GDm = 0.04). Notice that the potential was the same at the soma for both models. The increased depolarization of the active dendrites resulted from additional inward current entering the distal dendrites. From Ohm's Law, this increased current flowing toward the soma causes an increased voltage drop across Ri of the distal dendrites. A negative GDm, therefore, boosts the dendritic potential and thus increases the amount of voltage attenuation that occurs from the dendrites to the soma.
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FIG. 3.
Synaptic input depolarizes the active dendrite more than the passive. Peak dendritic depolarization is shown in response to a 100-ms synaptic input for the passive (GDm = 0.04) and active (GDm = 0.026) dendrites. Dendritic potential is illustrated as a function of distance from the soma for the passive (+) and active () models. Time at which the peak depolarization occurred was slightly different for all dendritic compartments and usually occurred within 1 ms after the synaptic input. Number of active synapses per pattern was adjusted so that the somatic potential in each model reached Vth (see Fig. 1B).
; Redman 1976). Examining charge movement provides a clearer picture of the dendrite's effect on the synaptic input than looking at either current or voltage.
Ig, Qc = Ic, and Qs = Is. This principle is illustrated for a simple three-compartment dendrite in Fig. 4A. Conservation of charge requires that the net charge transferred to the soma (Qsoma) from the apical dendrites is
where n is the number of dendritic compartments. Equation 1 can be expressed as the sum of each component of charge transfer or
(1)
where Q*g, Q*c, and Q*s represent the net charge transferred by the membrane conductance, capacitance, and synaptic conductance, respectively, for the entire dendritic tree.
(2)
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FIG. 4.
Active dendrites have uniform charge transfer. Superposition was used to determine the amount of charge movement across the dendritic membrane in response to each individual synapse of an input pattern. Charge escaping across the membrane conductance (Q*g), synaptic conductance(Q*s), and accumulating on the capacitance (Q*c), was computed by integrating the currents associated with each component for the entire dendrite during the 100-ms synaptic input. Net charge transferred to the soma (Q*soma) is equal to the sum of all charge that passes through ionic conductances less the amount that accumulates on the membrane capacitance. A: simple illustration of charge movement when activating a single synapse on compartment 1 in a passive dendrite. Except for the 1 active synapse, the direction of charge movement across the dendritic membrane is outward. B-E: charge contribution of the entire dendritic tree vs. the location of each active synapse on passive (GDm = 0.04) and active (GDm = 0.026) dendrites. B: total synaptic charge injected into the dendrites decreases as a function of synaptic location. Charge transfer for the passive (+) and active () dendrites is shown for the same 1,000 synapses used in Fig. 2. C: capacitive charge accumulation increases with synaptic location for both models. Active dendrites have a larger overall accumulation of charge on the membrane capacitance. D: passive dendrites have outward charge movement across the membrane conductance, whereas active dendrites inject charge. In both cases, the amount of charge moving across the dendritic conductance increases with the synapses distance from the soma. E: dendritic contribution (Q*dend) is the sum of charge movement and accumulation for the dendritic conductance and capacitance. Active dendrites produce a more uniform charge movement as a function synaptic location. F: entire active dendrites contribute to each synapse. For 3 activated synapses (indicated by 
), the percent charge injected (Qg) by different regions of the active dendrites is shown. Dendritic regions where the synapses are located contribute only ~50% of the total charge. Regions were determined by dividing dendritic path length from soma into 3 equal sections (proximal, middle, and distal).
0.026) dendrites of the linear model. Superposition was used with the same input patterns and 1,000 synapses as in Fig. 2. In Fig. 4B, each point represents the total synaptic charge injected (Q*s) as a function of the location of the single activated synapse. (Inward charge, as with inward current, is plotted in the negative direction.) The amount of charge each synapse injected is slightly greater in the active model than in the passive model. With both models, the amount of charge injected decreased for synapses farther from the soma. This was due to the decrease in driving force resulting from increased depolarization of the distal dendrites (as shown in Fig. 3).
50 mV, Iboost was linear with the slope of the I-V relationship equal to the conductance, Gboost. Iboost was an instantaneous function of voltage with no time dependence. The difference between adding Iboost and varying GDm as in the previous simulations, was that Iboost more closely mimics the nonlinear activation of realistic voltage-gated inward currents. This was important for developing the final and most realistic model.
). For a range of Gboost values, is plotted in Fig. 5B. As Gboost was made more negative, decreased until a minimum was reached at 0.12 mS/cm2. Gboost values more negative than 0.12 mS/cm2 caused the dendrites to become excessively excitable and increased. These results are very similar to that of the linear model. The nonlinearities of Iboost, however, prevented us from using superposition to compute the CV of individual synaptic inputs.
is compared with the original set of curves for 50 Hz (replotted from Fig. 5). It was striking to see that the optimal value of Gboost did not change for the three different input conditions. The minimum for each input condition corresponds to the same value of Gboost. Next, the effect of increasing or decreasing duration of the synaptic input was investigated. The strength of each synaptic input was maintained at 50 Hz with the duration set to 60 or 140 ms. Figure 6B illustrates that unlike synaptic strength, integration time affects the optimal value of Gboost. For a 60-ms duration, the optimal value of Gboost more negative than for the 100-ms integration time. The opposite effect was observed for a synaptic duration of 140 ms. These results are consistent with the idea that active dendrites provide charge compensation as shown in Fig. 3. The shorter the synaptic duration, the less time the dendrites have to inject the same amount of charge, thereby requiring a larger Iboost.
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FIG. 6.
Effects of individual synaptic strength and duration on the optimal value of Gboost. A: effect of varying synaptic strength on . Synaptic strength was varied by increasing or decreasing the frequency of synaptic input. Optimal value of Gboost (that produced the minimum ) did not change with synaptic strength. Number of active synapses per pattern corresponding to the minimum were 204, 125, and 82 for 30, 50, and 75 Hz, respectively. B: effect of the duration of the synaptic input on . Input duration and the optimal value of Gboost appear to be inversely related. Number of active synapses per pattern corresponding to the minimum were 162, 125, and 114 for 60, 100, and 140 ms, respectively.
for all values of Gboost (especially apparent for passive dendrites with a Gboost = 0). This was due to the number of active synapses per pattern. Increasing the number of active synapses per pattern would result in smaller variability between patterns, regardless of the dendritic composition. Figure 6B presents a more complex situation in that two parameters are varied
the number of synapses and the synaptic duration. These two parameters have opposing effects on . Because our measure of is based on Gsyn, it does not account for the change in synaptic effectiveness that accompanies a change in synaptic duration. For example, in Fig. 6B, increasing duration to 140 ms also increased the amount of charge injected for each synapse even though the individual synaptic conductances had not changed. This resulted in a decrease in that counteracted the effect of decreasing the number of active synapses (which alone would increase ).
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FIG. 7.
ICaT and INaP decrease location-dependent variability. A: density of realistic conductances as a function of distance from the soma. Densities reported correspond to the minimum . Inset: somatic voltage response to synaptic input (heavy bar) adjusted for each model to be just above threshold. Scale bar is 20 mV and 50 ms. B: plotted against the scaled channel densities of the realistic models. A density scale factor of 1 corresponds to the densities shown in A.
, so did the realistic conductances. The density in each dendritic compartment was scaled up or down with a constant multiplier, which we called a "density scale factor." is plotted in Fig. 7B for both increases and decreases in the channel densities. A density scale factor of 1 corresponds to the densities in Fig. 7A that minimized . A density scale factor of 0 corresponds to passive dendrites. Scaling the fitted densities up or down had the same effect on as varying Gboost.
; Rolls 1989). The network associates a specific pattern of input activity with a specific output pattern. Many associations can be stored simultaneously, giving the network the property of distributed memory. Potentiation of synaptic connections was the mechanism for storing associations. A simplifying assumption in our network simulations was that the input and output patterns were binary. Cells were considered either to be firing or silent. This imposed binary coding scheme allowed the performance of the network to be quantified easily.
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FIG. 8.
Demonstrating the heteroassociative memory network. A simple network consisting of 5 cells with 5 synaptic inputs per cell. Unpotentiated synapses are represented as 
and potentiated synapses as . Activity is represented by one or more action potentials (| | |). Sequence of storing and recalling 3 associative patterns of activity is illustrated. A, 1-3: storage of 3 associations. A1: first association is stored by potentiating the 4 synapses that have a coincident pre- and postsynaptic activity. A2: storage of the second association. Notice that with the storage of each additional association, more synapses are potentiated. A3: storage of the third association. B, 1-3: after the associations have been stored, network recall is tested. B1: presenting the first input pattern correctly recalls the first pattern stored. B2: applying the second input results in correct recall of the second pattern stored. B3: an error, in the form of an extra cell firing, occurs when the third input pattern is applied.
).
= 0 and therefore provide the basis with which to evaluate the realistic network's performance. Associations between the input and output patterns were stored by potentiating the appropriate synapses. The network then was tested by replaying the complete set of input patterns and observing the output to calculate PS.
cm2 to infinity. Compared with the 95% performance level of the perfect threshold model, no passive model performed well. These simulations indicate that simply eliminating the leak conductance (i.e., increasing Rm) results in only marginal improvement in performance.
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FIG. 9.
Heteroassociative network performance with passive (A) and active dendrites (B and C). Each bar represents the proportion of cells that fired correctly (PS) during recall of a 10-cell associative matrix memory network (average of 3 network simulations). Number of patterns stored for each network was set to produce PS = 0.95 in an ideal threshold model, which had no location-dependent variability (- - -). A: networks were simulated with passive dendrites with different dendritic Rm. Increasing Rm only marginally enhances performance with passive dendrites. Dendritic Rm was varied from 10 k cm2 to 
. B: networks simulated with the boosting model (model 2). Performance as a function of Gboost. With Gboost = 0.12 mS/cm2, the model's performance is near maximal. Gboost was scaled from 0.19 to 0.04 mS/cm2. Dendritic Rm for the boosting model was 25k cm2. C: network performance for the 3 realistic models (model 3). Densities used are those shown in Fig. 7A. Moving left to right, the number of active synapses per pattern and the number of patterns stored (in parenthesis), for each bar is: A, 266 (27), 161 (43), 128 (53), 112 (58), 100 (65), and 97 (67); B, 110 (59), 116 (58), 120 (56), 125 (53), 130 (52), 137 (50), and 149 (46); C, 117 (57), 124 (55), and 119 (56). Error bars are SE.
0.12 mS/cm2 gave the best network performance. (Gboost = 0.12 also minimized in Fig. 5.) Notice in Fig. 9B, that when Gboost was made less negative, the dendrites became more passive, and network performance degraded. As Gboost was made more negative than 0.12 mS/cm2, the dendrites became hyperexcitable, which also impaired network performance.
DISCUSSION
Abstract
Introduction
Methods
Results
Discussion
References
; Lipowsky et al. 1996).
). These results, however, were due to channel densities that did not provide the optimal I-V relationship in the dendrites. As demonstrated here, dendrites that are too active can amplify the effect of location on synaptic input.
in dendritic patches. For example, the authors reported that low-voltage-activated Ca2+ channels have a dendritic density of ~1 mS/cm2. The model containing only T-type Ca2+ channels had an average density of ~0.75 mS/cm2. The model that also included the persistent Na+ channels required a slightly lower average T-type Ca2+ density (~0.5 mS/cm2). The discrepancy between the model and experimental data may be attributed to the lack of depolarization-activated dendritic K+ channels (Hoffman et al. 1997). The only positive slope conductance in the dendrites of our models was the leak conductance. Adding such K+ channels to the dendrites would increase the required densities of the fitted inward currents.
, we improved on the passive dendrite model's ability to store any general set of patterns. It is interesting to imagine a different scenario where information is not only stored in the synaptic strength but also in the composition of the active dendrites. In this way, the dendrites might be tuned to further enhance the storage of information beyond that demonstrated in our network simulations.
). 6) No synaptic background noise was included; this may affect active dendrites more than passive (De Schutter 1995). And, 7) minimizing location-dependent variability of the synaptic input was the only dendritic function considered. Accurately transmitting synaptic input to the soma is one of several important functions dendrites perform. Dendrites also regulate plasticity through Ca2+ entry (Bliss and Collingridge 1993), the transmission of backpropagating action potentials (Magee and Johnston 1997; Spruston et al. 1995), and burst firing (Wong and Prince 1978). It is interesting to note, however, the model that maximized fidelity with dendritic T-type Ca2+ channels also produced burst firing.
).
). The idea that active dendrites boost weaker distal inputs has been speculated, especially as a role for active spines (Miller et al. 1985; Perkel and Perkel 1985; Shepherd et al. 1985). In general, any voltage-gated current producing a negative slope conductance (either depolarization-activated inward or depolarization-inactivated outward currents), in dendrites or spines, will amplify the synaptic input. If dendrites are to act as high-fidelity transmitters of synaptic input, the amount of amplification is critical. This study attempted to regulate carefully the amount of dendritic amplification to allow all synapses to be electrically equidistant from the soma.
; Woolf et al. 1991).
has proposed that active dendrites detect spatial clustering of synaptic activity. Although not tested here, it is likely that in addition to maximizing dendritic fidelity, the active dendrites constructed from realistic currents also would have some cluster sensitivity. The nonlinear I-V relationships of the Ca2+ and Na+ channels would allow spatially clustered inputs to inject more charge than spatially diffuse inputs.
demonstrated that dendritic K+ channels could linearize the effects of driving force. Their study differed significantly from ours in that they addressed the problem in terms of driving force versus synaptic strength. It would be impossible for any such dendritic voltage-gated K+ channels to alone reduce location-dependent variability. The authors did mention, however, that the functional properties of their K+ channel model might be replicated using a voltage-gated inward current.
). It also has been proposed that central neurons may act as coincidence detectors that integrate the synaptic input over extremely short durations of a few milliseconds (Softky and Koch 1993), though this is somewhat controversial (Shadlen and Newsome 1994). To be a coincidence detector, dendrites would be required to either have a relatively fast time constant (i.e., be very passive) or be extremely nonlinear (i.e., produce regenerative depolarizations). Both would allow dendrites to be sensitive to simultaneously arriving EPSPs. Such mechanisms are on the extreme ends (either highly passive or highly active) of the three possible modes for dendritic operation. The dendritic mechanisms proposed here to maximize dendritic fidelity do not shorten the membrane time constant (in fact the time constant is increased substantially) or initiate regenerative dendritic spikes. It is unlikely that our model of active dendrites (a model that reduces location-dependent variability) would act as a coincidence detector using a small window of integration as demonstrated in other models (Jaslove 1992; Softky 1994).
; Tsai et al. 1994; Zador et al. 1995). The methods developed to study passive dendrites, however, do not directly extend to nonlinear dendrites that contain voltage-dependent currents. The modification of existing methods or the development of new methods to explore nonlinear electrotonic structure will greatly aid in the understanding of how active dendrites influence synaptic integration.
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ACKNOWLEDGEMENTS |
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We thank M. Vollrath and N. Poolos for helpful comments on this manuscript. We also are grateful to M. Haque and R. Gray for computer and network support.
This work was supported by The Keck Center for Computational Biology and National Institutes of Health Grants MH-10475 to E. P. Cook and NS-11535, MH-44754, and MH-48432 to D. Johnston.
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APPENDIX |
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Channel kinetics
The INa model was based on voltage-clamp measurements by C. M. Colbert (unpublished observations). The IKDR model was based on similar models (Migliore et al. 1995) to produce good spike repolarization when used with the INa model. These models where produced by M. Migliore and follow the general Hodgkin-Huxley formalism. For a state variable x that represents a gating particle
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= 1/(1 + x) is the steady state value; x0 is the initial value; and x is the time constant where
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24)/10. For all simulations, Temp = 30°C, ENa = 55 mV, and EK = 91 mV.
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0n = 2; 
0n = 2; 
n = 0.04; zn = 5; V1/2n = 35; 0l = 0.08; l = 1; zl = 4; V1/2l = 45; V is membrane voltage (millivolts); F is the Faraday's constant; R is the gas constant; and T is the absolute temperature (Temp + 273.16).
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INaP model
This model has the same n as INa except V1/2n = 48.
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0m = 0.03; 0m = 0.03; m = 0.2; zm = 5; V1/2m = 32; 0h = 0.001; 0h = 0.001; h = 0; zh = 2; V1/2h = 61.
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Numerically fitting realistic currents to Iboost
A least-squares minimization algorithm called PRAXIS (Brent 1972) included with the NEURON simulation package was used to fit T-type Ca2+ and persistent Na+ currents to the artificial negative slope conductance current Iboost. The dendrites contained a passive leak (Rm = 25 kcm2) and a Gboost of 0.072 mS/cm2. Gboost was slightly less than the optimal value previously used because a uniform constant conductance synaptic input was used to provide a smooth depolarizing effect on the model. Using a synaptic train as input would have made the numerical fitting difficult because membrane potentials fluctuated with the constant change in synaptic conductance. After determining the channel densities, however, the realistic current models were tested using the standard 50-Hz synaptic trains. Replacing the artificial Iboost with a more realistic model proceeded one compartment at a time. Starting at the most distal dendritic compartment, Iboost was replaced with the type of channel (or channels) being fit. PRAXIS then was used to minimize the least-squares difference between the realistic current's I-V trajectory and that of Iboost during the last 10 ms of the synaptic input. An example of a fit using both ICaT and INaP is illustrated in Fig. A1 for a distal dendritic compartment located ~500 µm from the soma. Once a best fit had been found by PRAXIS, the conductance of the realistic current (or currents) was used as the initial condition to fit the next dendritic compartment. Only the dendritic compartment being fit contained the realistic channel models, all others contained Iboost. Goodness of fit was monitored graphically and computed as the sum of the least-squares difference of the fit for all the compartments, with each compartment being weighted as its percentage of the apical dendrites surface area. Although this method of fitting current versus voltage produced realistic active dendrites that minimized the location-dependent variability as well as Iboost, we speculate that other methods of fitting (e.g., current vs. time or even total injected charge vs. time) would have produced similar results.
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Binary heteroassociative network To allow comparison between the biophysical models, we had to compensate for the effect of different dendritic excitability. Depending on the dendritic composition, some models required a smaller Gsyn to produce an action potential. Therefore the number of patterns stored was a function of the number of active input lines per pattern. This was accomplished with the aid of a heteroassociative network constructed of simple binary threshold neurons. This network was used to determine how many patterns should be stored for a given number of active inputs. For each realistic network, the number of patterns stored was always set to produce a PS = 0.95 in the corresponding network constructed from the ideal binary units.
where T is the threshold of the unit and is equal to the number of active inputs per pattern. The ith input is represented by xi, which was equal to 0 for no activity or 1 for activity. The synaptic weight from the ith input to unit j is wij. Synaptic weights were either 0.5 for a nonpotentiated synapse or 1 for a potentiated synapse. Potentiation used a simple Hebbian rule and is illustrated inFig. 8.
(3)
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FOOTNOTES |
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1
This value is equivalent to a time-averaged conductance of a synapse modeled with a 2-ms alpha function with a 100 pS peak activated at 50 Hz (Bernander et al. 1991).
2
Because of dendritic instability, CV was not computed for GDm < 0.027.
Address for reprint requests: E. P. Cook, Division of Neuroscience, Baylor College of Medicine, 1 Baylor Plaza, Houston, TX 77030.
Received 24 February 1997; accepted in final form 26 June 1997.
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REFERENCES |
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Abstract
Introduction
Methods
Results
Discussion
References
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