Partially neglected for a long time, dendrites have been recently shown to treat synaptic input in a surprising variety of modes (Stuart et al., 2007). Indeed, experiments have revealed that dendrites are excitable and that they can generate either sodium (Golding and Spruston, 1988), NMDA (Schiller et al., 2000) or calcium (Llinas and Sugimori, 1980) spikes. One particularly striking example is found in pyramidal cells of deep cortical layers. In these cells, a coincidence between a back-propagating action potential and dendritic input can trigger voltage-sensitive ion channels situated on the apical dendrite more than 300 μm from the soma (Larkum et al., 1999, 2001). The somatic membrane potential increases only after the activation of dendritic ion channels. This often resulting in a burst of action potentials. Bursts in these cells can therefore signal a coincidence of input from the soma (down) with inputs in the apical dendrites (top). Such top-down coincidence detection is one computation that is attributed to dendritic processes. Other allegedly dendritic computations include subtraction (Gabbiani et al., 2002), direction selectivity (Taylor et al., 2000), temporal sequence discrimination (Branco et al., 2010), binocular disparity (Archie and Mel, 2000), gain modulation (Larkum et al., 2004) and self-organization of neuron networks (Legenstein and Maass, 2011). These computations rely on the dendrite acting as an excitable subunit (Polsky et al., 2004; Stuart et al., 2007).

Models of large pyramidal neurons with active apical dendrites were first described by Traub et al. (1991) for the hippocampus. This model of the large CA3 pyramidal neurons included voltage-dependent conductances on the dendrites. It is a model based on the Hodgkin-Huxley description of ion channels. Cable properties of dendrites are taken into account by segmenting the dendrite into smaller compartments. The resulting set of equations is solved numerically. A simplified version of this model was advanced by Pinsky and Rinzel (1994). They have reduced the model to a dendritic compartment and a somatic compartment connected by an effective conductance. The model has a restricted set of five ion channels and accounts for bursting of CA3 pyramidal cells.

Models specific to deep cortical cells have been described by extending the approach of Traub et al. (1991); Schaefer et al. (2003) used morphological reconstruction to define compartments. This model could reproduce the top-down coincidence detection.

Using a simplified approach similar to Pinsky and Rinzel (1994), Larkum et al. (2004) have modeled dendrite-based gain modulation. The parameters in the model could be tuned to quantitatively reproduce the firing rate response of layer 5 pyramidal cells stimulated at the soma and the dendrites simultaneously. Larkum et al. (2004) concluded that a two-compartment model was sufficient to explain the time-averaged firing rate.

A more stringent requirement for neuron model validation, however, is to predict spike times (Keat et al., 2001; Pillow et al., 2005; Jolivet et al., 2006, 2008a,b; Gerstner and Naud, 2009). Given the low spike-time reliability of pyramidal neurons, spike time prediction is compared to the intrinsic reliability (Jolivet et al., 2006). This approach can be seen as predicting the instantaneous firing rate (Naud et al., 2011). Generalized integrate-and-fire models can predict instantaneous firing rate of layer 5 pyramidal neurons with substantial precision (Jolivet et al., 2008a; Gerstner and Naud, 2009; Naud et al., 2009) in the absence of dendritic stimulation. The question remains whether a neuron model can predict the spike times of layer 5 pyramidal neurons when both the dendrites and the soma are stimulated simultaneously.

We present a study of a simplified model of layer 5 pyramidal cells of the cortex with dendrites excitable with calcium spikes (Larkum et al., 2004, 2009). Following Larkum et al. (2004), we hypothesized that we can model the soma and its apical dendrite with two compartments, without significant loss in the accuracy of spike-timing predictions. We introduce experimentally measurable impulse-response functions (Segev et al., 1995), which transfer the effect of current injected in one compartment to current reaching the other. The impulse-response functions replace the instantaneous connection used in previous two-compartment models (Pinsky and Rinzel, 1994; Larkum et al., 2004) and acts as a third, passive, compartment. Each compartment was modeled with a pair of non-linear differential equations with a small number of parameters that approximate the Hodgkin-and-Huxley equations. The predictive power of this model was tested on electrophysiological experiments where noisy current was injected in both the soma and the apical dendrite simultaneously (Larkum et al., 2004).

Methods are separated in four parts. First we present the model, second the experimental protocol, then fitting methods and finally the analysis methods.

Dual patch-clamp recordings were performed in L5 Pyramidal cells of Wistar rats (see Experimental Methods). A simplified two-compartment model (see Model Description) was fitted on the first 36 s of stimulation for all repetitions. The rest of the data (36 s) was reserved to evaluate the model's predictive power. The predictive power of the two-compartment model with active dendrites was then compared to a model without activity in the dendrites (see Section 2.1), the generalized linear passive model.

Figure 2 summarizes the predictive power of the two-compartment model. The somatic and dendritic voltage traces are well captured (Figures 2A–D). The main cause for erroneous prediction of the somatic voltage trace is extra or missed spikes (Figures 2A,B lower panels). The dendritic voltage trace of the model follows the recorded trace both in a low dendritic-input regime (Figure 2C) and in a high dendritic-input regime with dendritic “spikes” (Figure 2D). The greater spread of voltage-prediction-error (Figure 2) is mainly explained by the larger range of voltages in the dendrites (somatic voltage prediction is strictly subthreshold whereas dendritic voltage prediction ranges from −70 to + 40 mV). The interspike interval distribution is well predicted by the model (Figure 2G).

The generalized passive model does not predict as many spike times (Figure 2H). The intrinsic variability in the test set was 68% and the two-compartment model predicted 50%. The prediction falls to 36% in the absence of a dendritic non-linearity (Figure 2H).

The fitted kernels show that spike triggered adaptation is a monotonically decaying current that starts very strongly and decays slowly for at least 500 ms (Figure 3A). The back-propagating action potential is mediated by a strong pulse of current lasting 2–3 ms (Figure 3B). The coupling ϵ_ds from dendrite to soma has a maximal response after 2–3 ms and then decays so as to be slightly negative after 35 ms (Figure 3C). The coupling ϵ_sd from soma to dendrite follows qualitatively ϵ_ds with smaller amplitudes and slightly larger delays for the maximum and minimum peaks (Figure 3D), consistent with the larger membrane time-constant in the soma than in the dendrites.

The two-compartment model can reproduce qualitative features associated with the dendritic non-linearity in the apical dendrite of L5 pyramidal neurons. We study two of these features: the critical frequency (Larkum et al., 1999) and the gain modulation (Larkum et al., 2004). The first relates to the critical somatic firing frequency above which a non-linear response is seen in the soma, reflecting calcium channel activation in the dendrites. To simulate the original experiment, we force 5 spikes in the soma at different frequencies and plot the integral of the dendritic voltage. The critical frequency for initiating a non-linear increase in summed dendritic voltage is 138 Hz (Figure 4A). Pérez-Garci et al. (2006) reported a critical frequency of 105 Hz while (Larkum et al., 1999) reported 85 Hz. This appears to vary across different cells and pharmacological conditions.

The model also appears to perform gain modulation as in Larkum et al. (2004) (Figure 4B). The relation between somatic firing rate and mean somatic current depends on the dendritic excitability. The firing threshold but also the gain (or slope) of the somatic frequency vs. somatic current curve depend on the mean dendritic current. The gain modulation is attributed to a greater presence of bursts (Figure 4B) caused by dendritic calcium-current activation at higher dendritic input. The link between burst and dendritic activity is reflected in the burst- and spike-triggered average injected current (Figures 4C,D) similar to Larkum et al. (2004). The burst-triggered current is greater for the dendritic injection, whereas the spike-triggered current is larger for somatic injection. Therefore bursts signal a higher dendritic current that was concomittant with an increased somatic current. This can be interpreted as a top-down coincidence detection.

A dendrite is said active when it sustains either sodium, calcium or NMDA spikes. Our model reflects calcium spikes in the dendritic compartment, but not dendritic sodium spikes or NMDA spikes. The parameters fitted (Table 1) are in agreement with voltage-activated calcium channels. An activation sensitivity D_m of 5 mV is typical of many ion channels, and the time constant τ_x of about 50 ms is slower than the high-voltage activated calcium channel which has a time constant of about 10 ms (Gerstner et al., 2014). The current injection in the apical dendrite presumably does not solicit NMDA spikes known to occur in the apical tuft (Larkum et al., 2009).

Even if the spike-time prediction is high, the fitted parameters may differ from the real biophyical parameters for various reasons. First, the fitting method we used avoids local minima combining convex fitting procedures with exhaustive search. Even if the steps in the procedure are convex, the sequence of such steps may not be convex. Therefore the fitted parameters may reflect a local minimum. Also, the drop in coincidence rate between the training set and the test set indicate that overfitting is present. This could be avoided by using a smaller number of non-linear bases for the filters, or the use of raised cosine functions instead of the rectangular ones. Lastly, the filters and reduced model parameters may lump together different biophysical processes. The particular shape of the filter will depend on the average membrane potential and the average firing rate. This is one reason why we did not estimate the filters empirically with a separate set of experiments, but instead we fitted the model on current injection designed to imitate the natural condition.

Using a two-compartment model interconnected with temporal filters, we were able to predict a substantial fraction of spike times. The predicted spike trains achieved an averaged coincidence rate of 50%. The scaled coincidence rate obtained by dividing by the intrinsic reliability (Jolivet et al., 2008a; Naud and Gerstner, 2012) was 72%, which is comparable to the state-of-the performance for purely somatic current injection which reaches up to 76% (Naud et al., 2009). Comparing with a passive model for dendritic current integration, we found that the predictive power decreased to a scaled coincidence rate of 53%. Therefore we conclude that regenerating activity in the apical dendrite is required to properly account for the dynamics of layer 5 pyramidal cells under in-vivo-like conditions.