Abstract
This study highlights a new and powerful direct impact of the dendritic tree (the input region of neurons) on the encoding capability of the axon (the output region). We show that the size of the dendritic arbors (its impedance load) strongly modulates the shape of the action potential (AP) onset at the axon initial segment; it is accelerated in neurons with larger dendritic surface area. AP onset rapidness is key in determining the capability of the axonal spikes to encode (phase lock to) rapid changes in synaptic inputs. Hence, our findings imply that neurons with larger dendritic arbors have improved encoding capabilities. This “dendritic size effect” was explored both analytically as well as numerically, in simplified and detailed models of 3D reconstructed layer 2/3 cortical pyramidal cells of rats and humans. The cutoff frequency of spikes phase locking to modulated inputs increased from 100 to 200 Hz in pyramidal cells of young rats to 400–600 Hz in human cells. In the latter case, phase locking reached close to 1 KHz in in vivo-like conditions. This work highlights new and functionally profound cross talk between the dendritic tree and the axon initial segment, providing new understanding of neurons as sophisticated nonlinear input/output devices.
- cable theory
- dendritic modeling
- linking dendrites to axon
- spike sharpness
- spike tracking of input modulation
Introduction
Encoding rapid changes in synaptic inputs via axonal spikes is a key function of neurons and of the networks they form (Ilin et al., 2013). Pyramidal cells in rodent neocortex and hippocampus can encode input modulations of 200–400 Hz through phase locking of their output spikes to the input (Köndgen et al., 2008; Boucsein et al., 2009; Higgs and Spain, 2009; Tchumatchenko et al., 2011; Broicher et al., 2012; Ilin et al., 2013). Interestingly, recent analytical and experimental studies showed that this encoding capability depends critically on the onset rapidness of the action potential (AP) in its axonal initiation site; the faster the AP onset the better the encoding (Brunel et al., 2001; Fourcaud-Trocmé et al., 2003; Naundorf et al., 2005; Ilin et al., 2013). But what determines the AP onset rapidness at its initiation site?
Clearly, the faster the kinetics of voltage-gated Na+ ion channels in the axon the more rapid the AP onset is in the spike initiation site (SIP), located in the axon initial segment ∼40–50 μm away from the soma (Stuart and Sakmann, 1994; Palmer and Stuart, 2006; Kole and Stuart, 2012; Baranauskas et al., 2013). Here, however, we show that another surprising and powerful factor affects AP onset rapidness in the SIP and, thus, the axonal encoding. Our analytical and numerical simulations study demonstrates that increasing the dendritic membrane surface area (the dendritic impedance load) both enhances the AP onset in the axon and also shifts the cutoff frequency of the modulated membrane potential to higher frequencies. This combined “dendritic size effect” is the consequence of the decrease in the effective time constants of the neuron with increasing the dendritic impedance load.
The dendritic size effect was systematically explored in simple neuron models and in models of 3D reconstructed layer 2/3 (L2/3) pyramidal neurons from humans and rats. The impact of dendritic load on the axon was assessed using the measure ρaxon, the axon-to-(soma + dendrites) conductance ratio. ρaxon was estimated to range from 150 to 200 in L2/3 pyramids of young rats to 400–500 in mature L2/3 cells of humans. For these ρaxon values, the AP onset rapidness increased by ∼30%, from humans compared with rats, assuming fixed axonal morphology and default AP excitability (Mainen et al., 1995). The increase in both AP onset rapidness and in the cutoff frequency with increasing ρaxon, was associated with a 3.5-fold improvement in the tracking capabilities of input modulations (from 100 to 200 Hz in rats to 400– 600 Hz in humans). The dendritic size effect was further accentuated in in vivo-like (“high conductance state”) conditions.
It is typically assumed that the dendritic tree (the input region) and the axon (the output region) are separate electrical compartments, interacting with each other primarily via a two-way current flow. However, this study highlights a new, highly nonlinear, impact of the dendritic tree on the shape of the AP onset in the axon and, consequently, on the encoding capabilities of the neuron.
Materials and Methods
We have used both simplified neuron models and detailed models of 3D reconstructed layer 2/3 pyramidal neurons from rats and humans to examine the effect of the dendritic load on the spike onset rapidness in the axon initial segment (AIS) and, consequently, the ability of the neuron to track high-frequency input modulations. The simplified reference model used in Figures 1⇓–3 consists of an isopotential soma, cylindrical axon with fixed diameter (see below), and a cylindrical dendrite of variable size. In Figure 5 we examined the impact of the dendritic load on the AP generation in the AIS in 3D reconstructed L2/3 neurons from humans and rats.
Simplified neuron models.
The reference model included an isopotential soma coupled to a cylindrical axon (1 μm in diameter) and a cylindrical dendrite with variable diameter and length (see below). The first 50 μm of the axon represents the axon initial segment, AIS; it is endowed with “hot” excitable channels (see below). The next 1 mm long cylinder represents the myelinated part of the axon. The soma was modeled by a 30 × 20 μm compartment (Fig. 1, schematics at top).
The AIS is divided into 25 compartments, the soma by either 1 or 6 compartments, and the myelin and dendritic cable by 21 and 60 compartments, respectively. Simulations were performed using NEURON 7.2 (Carnevale and Hines, 2006) running on a grid of 60 Sun ×4100 AMD 64-bit Opteron dual core (240 cores in total), running Linux 2.6. We used 100 kHz sampling rate for the characterization of spike-shape parameters and 25 kHz for the tracking of high input frequency modulations (see below).
Passive properties.
The passive properties of the axon, soma, and dendrite were as follows: specific membrane capacitance Cm = 0.75 μF/cm2, axial resistance − Ra = 100 Ωcm, and membrane resistance Rm = 30,000 Ωcm2. The reversal potential for the leak current was set to −70 mV. The passive time constant of the model was 22.5 ms. The myelin-specific membrane capacitance was 0.02 μF/cm2 and the membrane resistivity was 1,125,000 Ωcm2 such that the passive membrane time constant was uniform over the whole modeled structure (22.5 ms). For the reference simplified model (see below) these parameters yielded a somatic input resistance of 101 MΩ.
Active properties.
Active conductances were modeled using Hodgkin–Huxley formalism at 37°C: INa = ḡNam3h(V − ENa), Ik = ḡkn(V − Ek) with ENa = 50 mV and Ek = −85 mV. Unless otherwise stated, the kinetics for the sodium and potassium currents were taken from Mainen et al. (1995) and downloaded from the SenseLab ModelDB database (Hines et al., 2004; http://senselab.med.yale.edu/modeldb). Maximal conductances were as in (Yu et al., 2008) ḡNa = 8000 pS/μm2 in the AIS and 800 pS/μm2 in the soma with passive dendrites. The corresponding ḡk values were as follows: 1500 pS/μm2 and 320 pS/μm2, respectively. The ratio of 1:10 soma–axon sodium channel density is an average between lower estimates (Fleidervish et al., 2010) and higher estimates (Kole et al., 2008; Kole and Stuart, 2012). With these densities, the AP was initiated in the AIS, 47 μm from the soma in all models studied (Fig. 1c). To separate the effects of spike slope rapidness and the filtering of high frequencies due to cable properties of the neuron, we used in Figure 3, e and f, different kinetics for the AP so that the onset rapidness will be kept at a fixed phase slope of 3.5/ms albeit the changes in dendritic size. To achieve this, the sodium activation time constant was adjusted in the Mainen et al. (1995) model. To account for the ultrafast kinetics (Fig. 3e, purple), we used model kinetics as described previously (Hay et al., 2013). To speed up these kinetics, the slope of the activation curve (m∞) was made steeper by shifting the slope factor from 6 to 1 mV (Brette, 2013). We also shifted the temperature-sensitive parameter, Tadj (only for the m parameter) from 10°C to 1°C. Using these kinetics, the AP was generated using ḡNa = 8000 pS/μm2 and 3600 pS/μm2, and ḡk. = 3000 pS/μm2 and 1600 pS/μm2 in the AIS and soma, respectively.
Dendritic load over the axon initial segment.
To examine the impact of dendritic load on spike onset rapidness and on the neuron model capability to track high-frequency input modulations, we fixed the excitability parameters in the modeled neuron and changed the diameter of the dendritic cylinder while adjusting its physical length, l, so that L = l/λ remains constant, λ =
Detailed compartment models of rat and human L2/3 pyramidal neurons.
Fully 3D reconstructed dendritic morphologies of rat somatosensory L2/3 were taken from Sarid et al. (2007) and of human temporal and frontal lobe were reconstructed in the labs of Mansvelder and de Kock in Amsterdam. Dendritic spines were added to the modeled dendrites as described previously (Sarid et al., 2007); spines size and density in humans and rats cells were taken from Benavides-Piccione et al. (2002). The same model axon used for the simplified structure was also used here. Importantly, although the dendritic tree of human L2/3 cells is much larger than that of the rat, the diameter of the axon seems to be rather similar (J. DeFelipe, personal communication). Dendritic sections for both human and rat morphologies were subdivided into compartments; each dendritic compartment was shorter than 20 μm.
For the three reconstructed L2/3 pyramidal cells of the rat, the total dendritic surface area (including dendritic spines) was for rat 11,058, 13,008, and 15,401 μm2 and 49,190, 62,418, and 77,478 μm2 for human (Fig. 5a). The corresponding ρaxon values were 161, 146, and 183 for rat and 293, 450, and 489 for humans.
Simulation of in vivo-like conditions.
To reproduce in vivo conditions, we decreased Rm of the dendrites, mimicking extended shunt by input synapses (Bernander et al., 1991; Rapp et al., 1992; Destexhe et al., 2003). Reducing Rm by a factor of 5 in the simplified model in Figure 3f (orange curve) increased ρaxon from 370 to 890. In Figure 5c, decreasing Rm by the same factor as for the detailed models of the three L2/3 pyramidal cells from humans increased ρaxon from 293, 450, and 489 to 783, 1340, and 1417, respectively.
Tracking of high input frequency modulations.
To measure the capability of modeled neurons to track high-frequency modulations, noisy current I(t) was injected into the models' soma. This current was composed of three components (Fourcaud-Trocmé et al., 2003; Köndgen et al., 2008; Tchumatchenko et al., 2011; Ilin et al., 2013):
I0 is the steady (DC) component, I1 is the modulated input, and Inoise is the noise component, and was generated using realization of an Ornstein—Uhlenbeck process with zero-mean, variance s2, and time correlation τnoise = 5 ms (Köndgen et al., 2008). Unless otherwise mentioned, the ratio between the amplitude of the sinusoidal component, I1, and s, the square root of the variance of Inoise was kept constant–1:1 for the different cases modeled whereas the ratio of I0 to I1 was 6 (Figs. 2⇓⇓–5).
Fourcaud-Trocmé et al. (2003) showed that the ability of theoretical models to phase lock their spikes' timing to high-frequency input modulations depends strongly on the AP onset rapidness and on the firing rate. Since various dendritic load models were used in this study, I(t) was normalized to yield 10 Hz firing rate in all models studied; namely, all the current components in Equation 1 were scaled by a similar factor to ensure the firing of 10 Hz in all models. We also examined different ratios for the different components of I(t), and also different firing rates. We also normalized the input current in different ways for the various models, including normalizing the noisy component to yield similar voltage fluctuations (SD ranging from 2 to 4 mV) in all models and scaling the amplitude of the sine input current by the input resistance ratios of the different models, etc. (data not shown). These manipulations did change the overall tracking capability of the modeled cells (Fourcaud-Trocmé et al., 2003), but they did not change the basic phenomenon (the dendritic size effect), namely the improved tracking capabilities of neurons with larger ρaxon values.
In each model, 240 current inputs of different frequencies, ranging between 0.5 and 2000 Hz, were used. Each input lasted for 30 s with sampling rate of 0.025 ms between successive points. The average simulation time was ∼10 min for the simplified models and ∼30 min for the 3D reconstructed L2/3 pyramidal cells models.
Encoding of high input frequencies by axonal spikes.
For quantifying the ability of the models to phase lock to the fluctuating input, we used a method based on Fourier transform (Tchumatchenko et al., 2011; Ilin et al., 2013). For each frequency, the strength of a vector R was calculated (Figs. 2⇓⇓–5) as explained below. The spike timing was measured when the AP at the soma crossed a predefined value (0 mV). For each input frequency we computed a vector r of spikes timing tj; r is calculated as follows:
Each tj is defined as vector of unit length with a phase shift defined as the spike time modulo the stimulus period. We note that if the values of tj are all the same (equal phase shift), then r = 1 implying a perfect locking of the spike train to the modulated input. If there is no locking at all so that the timing of the spikes is random, then r = 0.
All simulations lasted 30 s, resulting in N = ∼300 spikes per run. To compare between different models and inputs, the strength R (Figs. 2, 3) was normalized to the reference value of r at 3 Hz: R(f) = r(f)/r(3 Hz). Because R is affected by the input frequency, f, which might slightly shift among different models and cases, the results for each input were smoothed with a sliding window of size 9 (the four nearest frequencies above and below a particular frequency, f). The SD for each such a window is presented in Figures 2⇓⇓–5 as a colored cloud around the respective curve.
To evaluate R versus f for different AP phase slopes (Fig. 2d), the cutoff frequency (CF), defined previously (Fourcaud-Trocmé et al., 2003), was used. The CF is the frequency in which R decreased below 1/
For the detailed example shown in Figure 5b regarding the capability of human versus rat L2/3 neurons to track an exemplar input frequency of 500 Hz, we superimposed the corresponding raster plots (for 30,000 cycles) on one cycle of the modulated sine wave (top black trace). Based on these raster plots, the peristimulus time histograms (PSTHs) were then calculated (bottom). A sine wave of 500 Hz was fitted to the PSTH, using Levenberg–Marquardt algorithm (Press et al., 1992). The modulation depth is defined as the difference between the peak of the fitted curve and the mean firing rate (Köndgen et al., 2008); the larger this depth, the better the tracking of the modulated input by the axonal spikes.
Rapidness of AP onset.
To quantify how rapid the AP onset is, we used the phase slope measurement (Naundorf et al., 2006; McCormick et al., 2007). APs depicted by their phase plot (V vs dV/dt), and the slope of this graph at a point where the phase plot crossed a certain value (10 mV/ms), were measured in units of 1/ms. This slope was defined as the phase slope in this work.
Conductance load on the axon–ρaxon.
The main objective of this work was to study the relationship between the dendritic load, the spike shape, and the ability to track high input frequency modulations. To characterize the conductance load imposed by the dendrites on the soma, Rall (1969, 1977) defined ρ = Gdend/Gsoma, the dendrite-to-soma conductance ratio. Here we extended this definition to characterize the conductance load that the dendritic tree plus soma impose on the axon:
Gsoma, Gdend, and Gaxon are, respectively, the input conductance of the soma (when isolated), the input conductance of the dendrite (at its soma origin, when isolated), and the input conductance of the AIS, 47 μm from the soma, when the axon is isolated (Hay et al., 2013).
Results
Dendrites affect the AP onset rapidness in the axon
Two basic points are highlighted in Figure 1, using a simplified neuron model consisting of an isopotential soma coupled to two cylindrical cables, representing the axon and the dendrites (Fig. 1, schematics at top). First, albeit the uniform excitability of the AIS and the fact that the depolarizing current was injected to the model soma, the AP is not initiated at the soma but at a point 47 μm away from it, the SIP (Fig. 1a, red spike; red point in the schematics). As predicted (Coombs et al., 1957; Segev and London, 2007; Baranauskas et al., 2013), the favorable condition for AP initiation at the SIP rather than near the soma results from the partial electrical decoupling of the SIP from the large impedance load (“current loss”) imposed by the soma and dendrites on the AIS (Rall, 1967). This was recently validated by direct experimental measurements (Bekkers and Haüsser, 2007; Baranauskas et al., 2013). Second, that at the SIP, the onset of the AP is the slowest (Fig. 1a,b, red lines and in insets; c, arrow; see also McCormick et al., 2007; Yu et al., 2008). Indeed, the AP phase slope (Fig. 1c) becomes increasingly larger (the AP onset becomes faster) when the AP backpropagates from the SIP to the soma; this backpropagation makes the AP onset particularly steep (“kinki”) at the soma (Fig. 1c, blue dot).
AP onset rapidness in the axon initial segment. Schematic at top depicts the model used, consisting of a spherical soma (blue) coupled to a passive cylindrical dendrite and a cylindrical axon. The latter starts with of 50 μm long, uniformly excitable AIS, followed by a myelinated cylinder (see Materials and Methods). a, Brief, 5 ms, depolarizing current pulse was injected to the soma; the resultant AP is recorded in three locations (color coded as in the schematics). The AP peaked first at a distal point, 47 μm from the soma; this is the SIP (red). Inset, Zoom in at the AP onset, which is less rapid in the SIP (red). b, AP phase plot at the corresponding locations coded by colors in a. The initial part of the phase plot is enlarged in the inset. The AP onset rapidness (the phase slope) was defined as the slope at the point where the phase slope was 10 mV/ms (horizontal dotted line in the inset). The phase slope is minimal at the SIP (red line in inset) and becomes very steep at the soma (blue line). c, AP phase slope in the AIS as a function of distance from the soma.
In Figure 2, the AP onset rapidness at the axon was measured following a depolarizing somatic current. This current was composed of three components: DC, a modulated sine wave, and a noisy current (Fig. 2a, near electrode; b, bottom; see Materials and Methods). The injected current gave rise to a firing rate of ∼10 Hz (Fig. 2b, top trace), with SD of membrane potential fluctuations ranging between 2 and 4 mV (Fig. 2b, top). We focused on how the impedance load that the dendritic cable imposed on the axon affects the AP onset rapidness; this load is characterized by the parameter ρaxon (see definition in Materials and Methods) and is related to the size/surface area of the dendritic tree. Figure 2c shows that the rapidness of the AP onset (the AP phase slope; see Materials ad Methods) at the axon, and in particular at the SIP (dashed square and inset), depends on ρaxon; the larger the dendritic load the faster the AP onset. For the default excitable kinetics used hereby (Mainen et al., 1995), the AP phase slope at the SIP increased from 3.5 to 4.5/ms (∼30% increase) for ρaxon ranging from 20 to 400, respectively (Fig. 2e, black curve, left y-axis). Note that the SIP remains at the same location independently of ρaxon (Fig. 2c, arrows). Note also that the relationship between ρaxon and AP onset rapidness is steeper when faster Na+ channels kinetics were used (data not shown).
Dendritic load, AP onset rapidness, and the tracking of high-frequency modulation by axonal spikes. a, Schematics showing four neuron models with increasing dendritic load (increasing ρaxon; see Materials Methods). The model with gray dendrite (ρaxon = 190) was used in Figure 1. b, Top, Voltage response of the model with ρaxon = 190 to a fluctuating current injected in the soma (schematic electrode in a). The corresponding current input was composed of a DC current (lower traces, green line of 0.2 nA), a modulated sine wave, and a noisy current input, as shown in the lower traces (see Materials and Methods). c, AP phase slope as a function of distance from the soma for the four neuron models. Input current was normalized to yield similar firing rate of 10 Hz in all four models (see Materials and Methods). Colors are as the dendrites in the respective schematics. AP was initiated at the same location (the SIP) in all four models (red points). Inset, zoom in at the SIP; AP onset becomes more rapid with increased dendritic load. d, Models' capability to track high-frequency modulations via axonal spikes, as assessed using the vector strength R. Colors are as in a. Cutoff frequency is shifted to the right with increasing the dendritic load. e, Correlation between ρaxon and AP onset rapidness (black curve, left x-axis) and the cutoff frequency (green curve, right y-axis). Note the 30% increase in AP inset rapidness and the >3-fold improvement of encoding input fluctuations via axonal spikes with increasing ρaxon.
How does this change in AP onset rapidness with ρaxon affect the encoding capability of the axon (Brunel et al., 2001; Fourcaud-Trocmé et al., 2003; Naundorf et al., 2005; Ilin et al., 2013)? Figure 2d depicts the vector strength, R, as a function of the frequency of input modulations, for the four ρaxon values as in Figure 2a. R is a measure for the quality of encoding of the input modulations by the axonal spikes (Fourcaud-Trocmé et al., 2003; Köndgen et al., 2008; Tchumatchenko et al., 2011; Ilin et al., 2013; see Materials and Methods). In the present study it was normalized so that R = 1 denotes the tracking performance at input modulation of 3 Hz (the reference frequency). The quality of encoding deteriorates as the R value decreases toward zero. Indeed, by increasing the frequency of the modulated input, the quality of the encoding is reduced in all cases. However, increasing ρaxon shifts the CF to the right–improving the encoding capability of input modulations by the axonal spikes (Fig. 2d, color dots on respective curves). Increasing ρaxon from 12 to 370 is accompanied with more than threefold increase in the CF, from 100 to 320 Hz, respectively (Fig. 2e, green curve and right axis). This implies that the tracking capability of input modulations is dramatically improved with increased ρaxon.
The theoretical basis for the impact of dendritic load on axonal encoding
What is the theoretical basis for the increase in spike onset rapidness with increasing the dendritic load? Also, are there additional factors affecting the dependence of the encoding on ρaxon? Figures 3 and 4 tackle these questions via the analysis of the voltage transients in the passive “ball and two sticks” neuron model. Figure 3a depicts the voltage response in the axon (47 μm from the soma, corresponding to the SIP in the active case) to a step current injected at that point. The larger ρaxon, the faster the voltage rise time. For ρaxon = 12 (blue line) the voltage transient reaches 64% of its maximum at t = τm (the membrane time constant) whereas for ρaxon = 370 (magenta) it reaches 90% of its maximum at t = τm. This decrease in the effective system time constant with increasing ρaxon is expected from Rall's passive cable theory (Rall, 1969, 1977; Jack et al., 1975, and see below).
The theoretical basis for the dendritic size effect. a, Voltage response in the passive model depicted in Figure 2 schematics to current step injected at the SIP. As ρaxon increases, the transient voltage response becomes faster. Vertical dashed line depicts one membrane time constant. Color codes as in Figure 2a. b, Frequency response for the corresponding passive cases shown in a. Note the significant right shift in the cutoff frequency with increasing ρaxon. Lower dashed line is for soma alone, whereas the upper dashed curve line is for an infinite cylinder. Note the difference in scales of x- and y-axes as compared with Figure 2. c, d, As in a and b, but the current was injected at the soma and voltage response was measured at the AIS. e, Vector strength for the active model with a fixed ρaxon = 370, but with increasing the activation rate of Na+ conductance in the AIS to a required target phase slope as denoted. f, As in e, but with a fixed phase slope of 3.5/ms and with varying ρaxon values.
The relationship between axial current loss (dendritic load) and tracking of input modulation. Top, Schematics of modeled neurons. a, An RC circuit (red) representing the membrane compartment at the SIP, with a membrane capacitor (cSIP) and a resistor (gSIP). The membrane time constant, τm = cSIP/gSIP. The axial resistor (ga) represents the path by which axial current is lost from the SIP toward the soma/dendritic region (blue). b, Equivalent circuit to that drawn in a, now with the two resistors, gSIP and ga drawn in parallel. This makes it clear why the effective time constant of this circuit, τeff = cSIP/(gSIP + ga), is shorter than τm. c, Voltage response to current step onset and offset in the isolated SIP (red) and in the model shown in b (which includes axial resistance). Voltage buildup during current injection and (its mirror image) voltage decay following the end of current injection. Both the buildup and the decay are faster for the case with the shorter effective time constant (purple). The fast voltage buildup (and decay) is associated with the current loss from the SIP. d, Normalized vector strength (phase locking as function of the frequency of input modulations) for the two corresponding isopotential models with active currents as in Figure 2b. Shorter membrane time constant leads to improvement in tracking of high-frequency modulations.
Figure 3b shows the effect of increasing ρaxon in the frequency domain. As ρaxon increases the cutoff frequency of the modulated input shifts to larger frequencies. For small ρaxon values (blue line) the neuron behaves as low-pass filter, comparable (at medium range of input frequencies) to an isopotential RC neuron (Fig. 3b, lower dashed line); it is strongly attenuated for input frequencies already around 100 Hz (CF, horizontal dashed line). In contrast, for large ρaxon values (magenta), the impedance retains high values even for input frequencies above 1 KHz, better than in the case of an infinitely long cylindrical cable (depicted by the upper dashed line).
Similar results, but less marked, are depicted in Figure 3c and d; now the current is injected in the model soma and the voltage response is measured in the SIP. Note that, compared with the SIP, the soma “suffers” less current sink due to the dendritic load. Therefore the effect of ρaxon in both time and frequency domains is less marked at the soma compared with the case depicted in Figure 3a and b. Still, also at the soma the filtering of the modulated voltage is less marked with increasing ρaxon. Note that, for a given passive system, the time constants that govern voltage transients are similar in all locations, but their relative weights change (the coefficients for the respective “equalizing time constants” are location dependent; Rall, 1969, 1977; Eqs. 4–9, and see below).
The theoretical explanation for the dendritic size effect shown in Figure 3, a–d, is provided in Figure 4 and in Equations 4–9. The red circle at the top of Figure 4 represents the SIP; its membrane is modeled as an RC circuit, consisting of a capacitor, cSIP, and a resistor, gSIP (Fig. 4a) with corresponding time constant, τm = cSIP/gSIP. The axial resistor (ga) represents the resistance along the path through which longitudinal current flows (is lost) from the SIP toward the soma/dendritic region (blue). The equivalent circuit for this case consists the two resistors, gSIP and ga, drawn in parallel (Fig. 4b). This explains why the effective time constant of the combined (RC + axial) circuit, τeff = cSIP/(gSIP + ga), is smaller than τm. This is indeed the case when comparing the red to the purple traces in Figure 4c depicting the voltage response to current step for an isolated SIP (red) in the model shown in Figure 4b. Voltage buildup during current injection and (its mirror image) voltage decay following the end of current injection. Both the buildup and the decay are faster for the case with the shorter effective time constant (purple). The fast voltage buildup (and decay) is associated with the current loss from the SIP. Figure 4d shows the normalized vector strength (phase locking as function of the frequency of input modulations) for the two corresponding isopotential models with active currents as in the soma of the models used in Figure 2b. As expected, shorter membrane time constant leads to improvement in tracking of high-frequency modulations.
The above intuitive explanation is based on analytical grounds following Rall's cable theory.
The 1D passive cable equation can be expressed as follows:
where λ is the space constant, λ =
Rall (1969, 1977) showed that the general solution for Equation 4 could be written as an infinite sum of decaying exponentials:
The coefficients, Ci, depend only on location x and on the initial/boundary conditions and not on time. τ0 = τm is the membrane time constant and τ1, τ2, … are location independent for a given system and are termed “equalizing” time constants (Rall, 1969). These time constants govern the rate of buildup and decay of the voltage following perturbation (e.g., current injection) at any given location in a given cable (see below).
For the case of a cylinder with both ends sealed, the equalizing time constants in Equation 5 are
where L is the electrotonic length (l/λ) of the cylindrical cable; l is its physical length of the cylinder.
To explain what happens to the equalizing time constants (namely, to the transient voltage response) when a “conductance load” (soma + dendrites) is imposed at the somatic end of the cylindrical axon, we considered the simpler case whereby the axon is loaded only by a cylindrical cable at its “somatic” end. This cylindrical extension represents the load through which axial current is lost from the SIP toward that extended cable.
From Equation 6 it is easy to see that increasing L by a factor of 2 adds an additional equalizing time constant between each pair of the original time constants corresponding to the shorter cylinder (so that now the time constants are more closely spaced). For example, with the membrane parameters we have used above we get for L = 1, τ0 = 25 ms, τ1 = 2.07 ms, τ2 = 0.56 ms, τ3 = 0.25 ms, τ4 = 0.14 ms and for L = 2, τ0 = 25 ms, τ1 = 6.49 ms, τ2 = 2.07 ms, τ3 = 0.97 ms, τ4 = 0.56 ms. It can be seen that for L = 2 we get two new equalizing time constants compared with the corresponding time constants for L = 1. Note that these additional τi in the longer cable come with their corresponding coefficients Ci in Equation 5. These additional time constants and their corresponding coefficients are responsible for the faster buildup (and decay) attenuation of the (normalized) voltage for the longer cable.
For a case of a cylinder with both ends sealed, the voltage response to a step current at one end (X = 0) of the cylinder is (Rall, 1977)
Where I is the steady current input and R∞ is the input resistance for similar cylinder but of an infinite length. In cable of length L with both ends sealed, the input resistance is RN = R∞ coth L. Namely, V(0, ∞) = IRN = IR∞ coth L. Normalizing Equation 7 with respect to the input resistance we get the following:
This is a monotonous function approaching 1 for t → ∞.
Replacing τn in Equation 8 from Equation 6 one gets the following:
It can be shown by plotting the left side of Equation 9 as a function of L that, indeed, as L increases the rate by which the voltage grows to its normalized value increases (data not shown; Jack et al., 1975, their pp 168). Thus, with increasing the load on the somatic side of the axon, the voltage buildup at the SIP in the axon becomes faster.
Finally, we note that the passive case analyzed above is highly relevant for understanding the impact of the dendritic load on AP onset rapidness at the SIP. What ignites the AP onset at the SIP is the generation of local active inward Na+ current. The faster the effective passive system time constant, the faster the charging of the local membrane capacitance by the “injected” inward Na+ current and, thus, the faster the buildup of the spike onset at the SIP (Fig. 4; see Discussion). The other effect of reducing the effective system time constant is that the filtering (dampening) of voltage modulation with frequency is less marked, enabling the modulated voltage response at the SIP to retain high values also for high input frequencies. Together, these two effects (the increase in AP onset rapidness and the right shift of the modulated input itself with input frequency) explain why larger ρaxon gives rise to improved capability to track input modulations via the axonal spikes.
The two mechanisms involved in the dendritic size effect
Figure 3, e and f, is aimed to discern the impact of the above two mechanisms: the change in AP onset rapidness and the change in low-pass filtering properties with ρaxon. In Figure 3e, the dendritic load was kept constant at ρaxon = 370 and the AP onset rapidness was manipulated by changing the kinetics of the Na+ conductances at the AIS (see Materials and Methods). Increasing the AP phase slope from 3.5 to 4.5/s, the CF increased from 220 to 320 Hz, respectively (Fig. 3e, green vs magenta curves). Changing the Na+ kinetics to be very rapid while keeping ρaxon = 370, the axonal spikes could track successfully very fast (1 KHz) input modulation (Fig. 3e, purple line; Ilin et al., 2013). The impact on the complementary mechanism governing the tracking of fluctuated input is assessed in Figure 3f, whereby the AP phase slope is kept constant at 3.5/ms when changing ρaxon (by manipulating Na+ kinetics for each case; see Materials and Methods). Increasing ρaxon from 12 (blue curve) to 370 (green curve) shifted the CF from 100 to 220 Hz (compare with the corresponding cases depicted by the two extreme points in Fig. 2e, green line; there the CF shifted from 100 to 320 Hz). Figure 3, e and f, thus demonstrates that the reduction in the effective system time constant with increasing in ρaxon has indeed two discernible effects: (1) increasing the AP onset rapidness and (2) reducing in the attenuation of the modulated voltage. Together, these two effects profoundly improve the encoding of the modulated input by the axonal spikes when the dendritic impedance load is increased.
Encoding of fast input modulations in rodents versus human pyramidal cells
Based on the above results we predicted that neurons with large dendritic surface area (larger ρaxon) are superior at tracking fast input modulations as compared with neurons with small dendritic membrane areas, assuming identical membrane properties for the dendritic tree and identical axonal properties (both morphological and excitable). This prediction was examined in Figure 5, in which 3D reconstructed layer 2/3 pyramidal cells from both humans and rats were modeled (see Materials and Methods). The morphology of these modeled cells is depicted in Figure 5a and their passive parameters and morphological features are summarized in the Materials and Methods. Figure 5b depicts the tracking capability of human versus rat L2/3 neurons for an exemplar input frequency of 500 Hz. One cycle of the modulated sine wave is shown on the top (black trace) and the corresponding raster plot (for 30,000 cycles) for the blue (rat) and red (human) depicts the spikes' firing times for the modeled neurons. The corresponding PSTHs are shown in the lower frames. It is clear that the human L2/3 neurons are superior in tracking the 500 Hz modulated input (compare peak-to-peak modulation depth, R, of red (5.7 Hz) vs blue (3.5 Hz) curves in the PSTH).
Encoding of input modulations via axonal spikes in human versus rat L2/3 pyramidal cells. a, The modeled human (red) and rat (blue) L2/3 3D reconstructed pyramidal cells. Scale bar, 200 μm; see details in Materials and Methods. b, Raster plot in human (red, left) and rat cells (blue, right) for 500 Hz input modulation (60 s input duration or 30,000 sine-wave cycles as shown on top). APs were emitted in only a few such input cycles. The improved tracking (time locking) capability of human cells can be clearly seen in the corresponding PSTHs on the lower frames. c, Vector strength in human (red) and rats (blue) as a function of the frequency of input modulations. The case of in vivo condition for the human cells is also depicted; in this case the leak conductance of the dendrites was increased to take into account the shunting effect of excitatory and inhibitory synapses bombarding the dendritic tree.
The tracking of input modulation in rats versus humans L2/3 pyramidal cells, as a function of input frequency, is depicted in Figure 5c. The CF is shifted to the right for human cells, from 195 ± 7.5 Hz in rats (blue) to 349 ± 8.7 Hz in humans (red; p value < 0.0001). The latter tracking capability is improved further in the in vivo-like conditions up to 515 ± 14.7 (p < 0.0001). In this case, the specific membrane resistivity, Rm, of the modeled L2/3 human cell's dendritic trees was reduced by a factor of 5, mimicking the effect of synaptic bombardment (shunt) over the dendritic surface (Bernander et al., 1991; Rapp et al., 1992; Destexhe et al., 2003; Broicher et al., 2012). This effectively increased ρaxon (to 800–1400 for the three modeled human cells; see Materials and Methods) and has shifted the cutoff frequency further to the right (Fig. 5c, rightmost orange curve). As a whole, Figure 5 demonstrates the superiority of L2/3 human neurons or, in general, of neurons with large dendritic load in tracking high-frequency modulation by the axonal spikes.
We conclude that, surprisingly, the dendritic tree strongly affects the onset rapidness of the AP at its axonal initiation point. For fixed excitable parameters in the axon, the larger the dendritic load, the faster the AP onset at the axon initial segment. Increasing the dendritic load also shifts the cutoff frequency (the passive filtering of the modulated input) to higher frequencies. Together, these two mechanisms give rise to the dendritic size effect, whereby the tracking capability of high-frequency input modulation by axonal spikes is significantly improved in neurons with larger dendritic surface area.
Discussion
This work highlights new and functionally profound cross talk between the dendritic tree and the axon initial segment. It is typically assumed that the two-way interaction between the dendritic and axonal compartments is via current flow/voltage spread between them. In this view, the synaptic and the active dendritic charge flows into the axon where it generates (or not) a train of output APs. In turn, in some neuron types, the axonal AP backpropagates to the dendrites (e.g., the BAC phenomenon; Larkum et al., 1999). However, here we show that the cable properties of the dendritic tree directly affect the shape of AP onset at the SIP, ∼40–50 μm from the soma (Brette, 2013). Because AP onset rapidness at the SIP critically determines the encoding capability of neurons (Fourcaud-Trocmé et al., 2003; Ilin et al., 2013), our study shows that dendrites directly affect, in a highly nonlinear way, the encoding capability of the axon. This conclusion is robust under a wide range of input noise levels, of spike firing rates, and of axonal excitability (data not shown). This is also valid for a variety of normalization of the input current that we have used to compensate for models of different ρaxon/input resistance values (see Materials and Methods).
Dendrites shaping AP onset in the axon
The shape of the AP, in particular its onset kinetics, is classically thought to be dependent solely on the properties of the voltage-gated ion channels involved in AP generation mechanisms. In contrast, here we show that morphological factors/dendritic cable properties are crucial in shaping the AP onset in the axon. This results from the large impedance load that the soma + dendrites impose on the thin AIS (Rall, 1969). This large impedance load (captured in the present study by the parameter ρaxon; Hay et al., 2013) implies that a significant fraction of the current generated by the active ion channels in the axon is lost axially to the soma and dendrites. One outcome of this large current sink is that the AP does not start near the soma, but rather at the SIP away from the soma, where it is partially electrically decoupled from the soma/dendritic current sink (Segev and London, 2007; Baranauskas et al., 2013; Ma and Huguenard, 2013). The other outcome of the huge dendritic load imposed on the axon is that the AP onset at the AIS becomes faster with increasing ρaxon.
The latter effect could be understood by solving the cable equation for the corresponding simpler case, in which on the end of an axon an equivalent cylinder was added, representing the soma and the dendrite load (Eqs. 4–9). The longer that extended cylinder is, the larger conductance load imposed on the axon (ρaxon increases). As L of the extended cylindrical cable increases the equalizing time constants are more closely spaced, and the effective time constant is shorter, giving rise to a faster voltage buildup (following current step) and voltage decay (at the end of current injection; Jack et al., 1975). This effect can be easily understood using the simplified case depicted in Figure 4b, demonstrating that the larger ga (the larger the axial current loss from the axon to the soma), the faster the effective system time constant is.
During AP onset (AP “foot”), the voltage-gated Na+ conductance is only very partially activated and, consequently, the passive cable properties of the neuron dominants the time course of the AP foot. In other words, the rise time of the AP onset is susceptible to the effective passive system time constant, namely, to the dendritic impedance load. This explains why changes in the dendritic surface area (in ρaxon) powerfully mold the AP onset rapidness at the AIS (Fig. 2). For the AP parameters used in Figure 2 (Mainen et al., 1995), the AP phase slope shifted from 3.5 to 4.5/ms (30% increase) when ρaxon increased from 12 to 370, respectively. We found that, with ultrafast Na+ kinetics (steep activation curve and shorter activation time constant), the AP slope shifted from 2.9 to 13.3/s (450% change), respectively (data not shown). Importantly, recent experimental work has directly demonstrated the dependence of the shape of the AP onset on the dendritic load (Bekkers and Haüsser, 2007).
Dendritic size effect and the encoding of fast input modulations by axonal spikes
In an impressive theoretical study, Fourcaud-Trocmé et al. (2003) showed that the onset dynamics of the AP critically determines the capability of the neuron to track fast input modulation. Intuitively, sharp AP enables the axon to react fast enough (be time locked) to fast modulation in the (noisy) membrane voltage, whereas a “sluggish” AP will not react fast enough and, in the extreme case, it will “ignore” the underlying voltage modulations and fire at arbitrary time with respect to the input modulations. This effect is clearly seen in the work of Mainen and Sejnowski (1995), in which the first spike in the output spike train following an abrupt (steep) step depolarizing input, is tightly time locked to the current onset.
Our results imply that neurons with large and/or shunted dendrites (large ρaxon), have improved capability for encoding of (time locking to) fast varying inputs by axonal spikes. This is because the decrease in the effective time constant with large ρaxon gives rise both to rapid AP onset and to an increase in the (passive) cutoff frequency of the modulated input itself (Fig. 3b). Indeed, our work predicts that in the in vivo condition (“high conductance state” conditions; Bernander et al., 1991; Rapp et al., 1992; Destexhe et al., 2003), the tracking capabilities of fast modulation in the input for human L2/3 cells, as assessed by the cutoff frequency, should attain values close to 1 KHz (Fig. 5c, in vivo case). The prediction that neurons with “shunted” dendrites can better track fast input modulations was recently demonstrated experimentally (Broicher et al., 2012).
Dendritic size effect - functional implications and experimental predictions
For concluding remarks, we first note that several previous studies hinted at the impact of dendritic size on the characteristics of axonal spiking. Most notably is the work of Mainen and Sejnowski (1996), demonstrating that the firing pattern at the axon is strongly shaped by the size of the dendritic tree. However, in this case the dendrites were excitable and increasing dendritic size also affected the overall excitability of the modeled neuron. Another recent study (Hay et al., 2013) showed that, to retain the firing characteristics of a given electrical class of neurons (e.g., L5 thick, tufted pyramidal cells), axonal excitability must be coregulated with the impedance load imposed by the dendrites. The larger the dendrites, the denser should be the excitable channels in the AIS. In this context it is important to question whether the axonal diameter should be considered as fixed, as we have assumed in the present work. If the axon diameter is proportionally larger with larger dendrites, then ρaxon might remain constant albeit changes in dendritic size. However, to the best of our knowledge, the axonal diameter of cortical pyramidal neurons with large versus small dendrites remains rather constant at ∼1 μm (J. DeFelipe, personal communication).
This study provides several experimental predictions. First, that AP onset in the axon is more rapid in neurons with larger dendritic surface areas. Measuring accurately the AP onset in the axon is hard and rare (Kole et al., 2007, 2008; Baranauskas et al., 2013), but recent optical techniques may overcome these difficulties. A simple experiment would be to shunt the soma [using dynamically clamp; (Broicher et al., 2012) or via uncaging GABA on the dendritic tree] and to record AP onset with high sampling rate in the axon. Another prediction is that a network of neurons with large dendrites (e.g., a network of thick tufted L5 cortical pyramidal cells) would, collectively, track input modulations better than networks of similar size consisting of neurons with smaller dendrites (e.g., of L4 spiny stellate cells in the neocortex). A discussion of the correspondence between single neuron's tracking accuracy and that of the network could be found previously (Fourcaud-Trocmé et al., 2003; Tchumatchenko et al., 2011). Clearly, phase synchrony of the network has far reaching functional consequences such as facilitating binding and segmentation of natural images and for plasticity-based temporal mechanisms, such as spike timing-dependent plasticity.
Why should neurons with larger dendrites be better in encoding fast input modulations? Neurons with a large dendritic tree typically receive larger numbers of synapses; e.g., L2/3 pyramidal cells of rats receive ∼15,000 synapses compared with 30,000 in L2/3 of humans (Defelipe, 2011). Broadly speaking, it seems reasonable to assume that neurons with larger numbers of input synapses are computationally more important. If this was the case, then one would want to enhance the encoding capability of neurons receiving large numbers of synapses, namely of neurons with large dendritic trees.
Footnotes
This work was supported by the Gatsby Charitable Foundation, the German Israeli Foundation, the HUNA Foundation, the Blue Brain Project, the HU-Max Planck Center, and by the Dutch Brain Foundation (Hersenstichting Nederland HSN2010(1)-09) to C.P.J.d.K. We thank Mike Gutnick, Ilya Fleidervish, Mickey London, and Albert Gidon for their most constructive comments on this work. We thank Guilherme Testa-Silva for biocytin loading of the human neurons.
The authors declare no competing financial interests.
- Correspondence should be addressed to Idan Segev, The Hebrew University of Jerusalem, Jerusalem, 91904 Israel. idan{at}lobster.ls.huji.ac.il
