^{*}

Edited by: Yuguo Yu, Fudan University, China

Reviewed by: Lianchun Yu, Lanzhou University, China; Thomas Launey, RIKEN Brain Science Institute (BSI), Japan

*Correspondence: Xile Wei

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Neural computation is performed by transforming input signals into sequences of action potentials (APs), which is metabolically expensive and limited by the energy available to the brain. The metabolic efficiency of single AP has important consequences for the computational power of the cell, which is determined by its biophysical properties and morphologies. Here we adopt biophysically-based two-compartment models to investigate how dendrites affect energy efficiency of APs in cortical pyramidal neurons. We measure the Na^{+} entry during the spike and examine how it is efficiently used for generating AP depolarization. We show that increasing the proportion of dendritic area or coupling conductance between two chambers decreases Na^{+} entry efficiency of somatic AP. Activating inward Ca^{2+} current in dendrites results in dendritic spike, which increases AP efficiency. Activating Ca^{2+}-activated outward K^{+} current in dendrites, however, decreases Na^{+} entry efficiency. We demonstrate that the active and passive dendrites take effects by altering the overlap between Na^{+} influx and internal current flowing from soma to dendrite. We explain a fundamental link between dendritic properties and AP efficiency, which is essential to interpret how neural computation consumes metabolic energy and how biophysics and morphologies contribute to such consumption.

^{+}entry

Cortical pyramidal cells have powerful abilities to process incoming signals, which are metabolically expensive. They adopt diverse patterns of APs to encode information and perform computation. This is a primary process that consumes energy within a pyramidal cell (Sengupta et al.,

In cortical pyramidal cells, the AP is initiated in the axon initial segment (AIS; Stuart et al., ^{+} and K^{+}. According to their concentration gradients, Na^{+} flows into the cell and K^{+} out of the cell. When membrane depolarization reaches a threshold level (i.e., AP threshold), inward Na^{+} current becomes self-sustaining and vast number of Na^{+} ions flow into the cell. It effectively depolarizes membrane potential and results in the fast upstroke of AP. Strong depolarization activates K^{+} current and lets K^{+} ions exit the cell. The efflux of K^{+} hyperpolarizes membrane potential, which generates the falling phase of AP. To re-establish ion gradients and maintain signaling, the Na^{+}/K^{+} pump extrudes Na^{+} ions and imports K^{+} ions during each AP (Kandel et al., ^{+}/K^{+} pump hydrolyses one ATP when it imports two K^{+} ions to the cell and extrudes three Na^{+} ions out of the cell.

The metabolic energy consumed by an AP is tightly related to the entry of Na^{+} ions into the cell (Carter and Bean, ^{+} influx is confined to the rising phase of the AP and K^{+} efflux to its falling phase, there would be perfect energy efficiency. However, the kinetics of voltage-gated channels causes the overlap between Na^{+} and K^{+} currents (Crotty et al., ^{+} influx less efficient in generating membrane depolarization, thus inflating energy cost. The complete separation of opposite currents decreases energy expenditure to close to the minimum possible, which increases Na^{+} entry efficiency. The extent of the overlap between opposite currents determines the efficiency of both Na^{+} entry and metabolic energy, which is highly variable among neurons. It is shown that channel types, densities and kinetics (Crotty et al., ^{+} entry efficiency through altering the overlap of opposite currents. To interpret the energy cost in pyramidal cells, it is essential to understand the efficiency of Na^{+} entry during an AP and its relationship with biophysical properties and morphology of the cell.

Dendrites are the primary sites for receiving signals in cortical pyramidal cells (Spruston, ^{+} current produces sublinear integration (Hu et al., ^{+} entry? Do they affect AP efficiency also through altering the overlap of Na^{+} and K^{+} currents in the soma/axon? What properties of the dendrites are contributory factors for producing an energy efficient AP?

Here we attempt to answer these questions by numerical simulations of biophysically-based models. We develop three two-compartment models to describe passive and active dendrites. The excess Na^{+} entry ratio (Carter and Bean, ^{+} influx is used for AP depolarization. By relating dendritic properties to the internal current flowing from the soma to the dendrite, and by identifying how such current overlaps with Na^{+} influx in the soma, we explain how passive and active dendrites participate in the energy efficiency of APs in cortical pyramidal cells.

The biophysically-based two-compartment models are used in our simulations. Such type of model is the minimal structure to capture the interactions between dendrites and soma/axon in cortical pyramidal cells. One chamber represents apical dendrites, and the other one describes the soma plus the AIS. APs are initiated and recorded in latter chamber. We develop three models to quantify how the passive and active properties of the dendrites affect the energy efficiency of APs.

Our starting model is derived from the Pinsky-Rinzel (PR) model (Pinsky and Rinzel, ^{+} and outward K^{+} currents in somatic chamber. The current-balance equations for model I are described by

where _{S} and _{D} are the transmembrane potentials of soma and dendrite. _{D} (in μA/cm^{2}) is the input current applied to activate neurons. _{c} (in mS/cm^{2}). _{SD} = _{c}(_{S} − _{D})/

The gating variables, including activation and inactivation variables for inward _{Na} (i.e., _{K} (i.e.,

where _{x} and β_{x} are

In model I, the activation variable _{Na} is replaced by its steady state _{∞} = α_{m}/(α_{m} + β_{m}) (Wang, _{Na} = 55mV, E_{K} = −80mV, E_{SL} = −65mV, and E_{DL} = −65mV are the reversal potentials for relevant channels. This model is used to simulate the effects of varying morphological parameter _{c} on the energy efficiency of APs, i.e., the passive properties of dendritic chamber.

To determine how active dendrites affect AP efficiency, we introduce an inward Ca^{2+} current _{Ca} into the dendritic chamber of model I, and develop another two-compartment model, i.e., model II. The Ca^{2+} current is given by (Mainen and Sejnowski,

Here _{Ca} = 140mV, which are modified from the PR like models (Pinsky and Rinzel,

The transition rate for gating variable

The kinetics of gating variable ^{2+} current in dendritic chamber affects AP efficiency.

By introducing an outward current _{KAHP} into the active dendrite of model II, we derive model III. _{KAHP} is a voltage-independent, Ca^{2+}-activated K^{+} current, and activating it causes spike-frequency adaptation (SFA) on slow timescales. This inhibitory current is described by (Pinsky and Rinzel,

Here the maximal conductance is

where time constant is τ_{q} = 800ms, and steady-state function is _{∞} = α_{q}/(α_{q} + β_{q}). The transition rates for _{KAHP} are α_{q} = min(0.00002[Ca], 0.01) and β_{q} = 0.001. [Ca] is the intracellular Ca^{2+} concentration, and its kinetics follows

This model is used to simulate the effects of activating hyperpolarizing current in dendritic chamber on AP efficiency.

We apply excess Na^{+} entry ratio to quantify the efficiency of Na^{+} entry during an AP. Following Carter and Bean (^{+} entry Q_{total} during the AP to the minimal Na^{+} load Q_{min} necessary for producing the voltage change of the AP. For a spike train recorded in our simulations, an AP is required to begin and end below or closest to the resting potential and cross at least 0 mV at maximum (Hasenstaub et al., ^{+} load Q_{total} per spike is calculated by integrating the Na^{+} current curve over the duration of the AP (Carter and Bean, _{total} = ∫ _{Na}(_{min} necessary to produce the depolarization of the AP is calculated as Q_{min} = C_{m}Δ_{S}, where C_{m} is the membrane capacitance. As mentioned in Introduction, an AP is initiated when enough membrane depolarization accumulates to bring _{S} to reach spike threshold. After that, inward Na^{+} current becomes self-sustaining to result in a positive feedback loop and generate the rising phase of the AP. To calculate the minimum charge Q_{min}, we measure Δ_{S} as the change of somatic voltage from spike threshold (d_{S}/d_{S}/d^{+} entry ratio calculated in this way has been widely applied to describe the metabolic efficiency of the APs in different kinds of cells (Attwell and Laughlin, ^{+} entry is confined to the depolarizing phase of the spike, and there are less overlaps between inward and outward currents, corresponding to a relatively efficient AP. On the contrary, a higher value of Na^{+} entry ratio indicates that more of metabolic energy is devoted to the reversal of ion exchanges, which corresponds to an inefficient AP. The temporal overlap of inward _{Na} and outward _{K} (i.e., Q_{overlap}) is measured as the difference between the total Na^{+} load during an AP and the associated depolarizing component of the Na^{+} load (Crotty et al.,

Earlier studies (Crotty et al., ^{+}/K^{+}-ATPase hydrolyses one ATP per three Na^{+} extruded and two K^{+} imported. Based on this fact, they measure the amount of Na^{+} (or K^{+}) ions consumed in the AP. The total Na^{+} (or K^{+}) load is then converted to the number of ATP molecules by using the 3:1 (or 2:1) stoichiometry of the Na^{+}/K^{+}-ATPase. Thus, there is a direct relationship between total Na^{+} load Q_{total} during an AP and its energy cost. Following Sengupta et al. (_{total} to define the energy consumption of an AP in our simulations. Note that such definition of AP cost is not accurate, but it does not alter our predictions about how dendrites affect the metabolic efficiency of somatic APs.

The shape of the simulated APs is characterized by their height and half-width. The height is determined by measuring the difference in somatic voltage _{S} from the peak to the most negative voltage reached after the AP (Carter and Bean,

All simulations of the two-compartment models are performed in MATLAB environment. The aforementioned dynamical equations are integrated numerically by using ode23 solver, with a time resolution of 0.001 ms. The computer code for model simulations in present study will be available for public download under the ModelDB section of the Senselab database (

Our first step is to examine the effects of passive properties of the dendrites. A simple two-compartment model is adopted to simulate APs generated in the soma of cortical pyramidal cells, i.e., model I (Figure _{D} to activate model I and simultaneously record APs generated in somatic chamber. As parameter

AP shape and metabolic efficiency vary with dendrite area in model I. _{S} recorded from the soma with different values of ^{+} load Q_{total} during an AP is plotted as a function of _{total} reaches a maximum with moderate value of ^{+} load Q_{min} increases as a function of ^{+} entry ratio decreases as a function of ^{+} load Q_{overlap} during an AP is plotted as a function of _{total}, Q_{overlap} also reaches a maximum with moderate value of

We adopt excess Na^{+} entry ratio to quantify the metabolic efficiency of the recorded APs, and examine how dendrite area affects this quantity. It is shown that the total Na^{+} load Q_{total} during an AP increases at first and then decreases with parameter _{total}, the minimal Na^{+} load Q_{min} needed to generate the upstroke of the AP increases monotonically in the observed range of _{total}/Q_{min}, we find that the excess Na^{+} entry ratio decreases with parameter ^{+} entry ratio and facilitates to reduce AP efficiency. With large dendrite area (i.e., small ^{+} entry ratio shows that Na^{+} influx is inefficiently used for the depolarization of relevant AP. Here more of the Na^{+} influx during an AP is devoted to the reversal of ion exchanges. These simulations indicate that a small Na^{+} entry ratio does not correspond to low energy cost for individual spikes. The metabolically efficient APs with large dendritic chamber may arise from other factors. We also calculate the temporal overlap Q_{overlap} between Na^{+} and K^{+} currents during the repolarizing component of the AP, which has been shown to be a determinant of metabolic efficiency. Unfortunately, we fail to identify a relationship between overlap load and Na^{+} entry ratio. We find that the overlap of inward Na^{+} and outward K^{+} currents during an AP reaches a maximum with moderate values of ^{+} influx during the APs to result in inefficient use of Na^{+} entry.

To determine how dendrite area affects AP efficiency, we examine the ionic currents underlying the recorded APs in model I neuron. With low morphological parameter _{Na} and _{K} are both relatively weak during an individual spike (Figure _{Na}. This effectively increases AP size and relevant total Na^{+} load, thus increasing its metabolic cost. Once _{Na} or _{K}. Unlike two active channels, the internal current _{SD} flowing out of the soma shows a marked decrease within the whole range of _{SD} results in an overlap between with Na^{+} influx during the depolarizing phase of the AP. Such temporal overlap is in effect similar to the overlap between _{Na} and _{K} during the falling phase, which increases total Na^{+} charge required for membrane depolarization. Decreasing the intensity of such inhibitory current effectively reduces Na^{+} load. That is why total Na^{+} load during an AP shows slight decrease once ^{+} faces less competition as it depolarizes membrane potential _{S}. Then, _{Na} is able to become self-sustaining at a more hyperpolarized voltage (Figure ^{+} load needed to produce the voltage change of the spike increases with parameter ^{+} influx and outward _{SD} during the depolarizing phase of an AP decreases as ^{+} influx to achieve its depolarization, which results in a lower excess Na^{+} entry ratio. These simulations indicate that increasing dendrite area increases the outward level of internal current _{SD}, which results in significant overlap between with Na^{+} influx and then decreases the efficiency of Na^{+} entry during an AP.

Inefficient Na^{+} entry arises from increased overlap between _{SD} and _{Na} as dendrite area increases. _{Na} (red), _{K} (blue), and _{SD} (green) underlying the APs with different values of parameter _{SD} and somatic voltage _{S} (i.e., _{SD}-_{S} curve) for each AP shown in _{Na}-_{S} curves. _{S}/d_{S} for the APs. Blue dots in _{S} at which d_{S}/d

Coupling conductance _{c} between two compartments is another passive property that controls internal current _{SD} in two-compartment model. In this section, we adopt model I to simulate the effects of varying _{c} on the metabolic efficiency of APs (Figure _{c}, model I neuron generates periodic spike trains to constant input _{D} (Figure _{c} for different values of _{c} with small dendrite area, such as _{c} connecting two chambers has little effects on AP shape.

AP shape varies with coupling conductance between chambers in model I. _{c} in model I with passive dendrite. _{c}, which are indicated on the top of each panel. Data are shown for _{c} for different values of _{c} for each value of _{c} in Figure _{D} used in the case of varying dendrite area is 3 μA/cm^{2}. Such higher current injection directly controls the excitability of two-compartment models and alters the shape of output APs.

Figure ^{+} load, the minimal Na^{+} load, the overlap load between Na^{+} and K^{+} currents, and the Na^{+} entry ratio during an AP as _{c} varies. With different values of _{c}. As coupling conductance increases, the total Na^{+} load during an AP is increased (Figure ^{+} load is reduced (Figure ^{+} entry ratio, is monotonically increased with _{c} (Figure ^{+} influx. This indicates that the generation of individual spike requires more energy and becomes metabolically inefficient with high coupling conductance. Once _{c} exceeds 2 mS/cm^{2}, total Na^{+} load, minimal Na^{+} load and Na^{+} entry ratio during an AP all show slight changes as _{c} increases. Similar to parameter _{overlap} and Na^{+} entry ratio (Figures _{c} is not through altering the overlap load between Na^{+} and K^{+} currents during the repolarizing phase of APs. It is worth noting that the passive dendrite with large values of _{c} has little effects on somatic APs and their underlying currents. Then, four items for relevant APs show little changes with coupling conductance (Figure

Increasing coupling conductance between chambers reduces AP efficiency in model I. ^{+} load Q_{total}, ^{+} load Q_{min}, ^{+} load Q_{overlap}, and ^{+} entry ratio are respectively plotted as a function of _{c} in model I. The value of ^{2}. Since applied _{D} alters the final output of model I neuron, the items for an AP examined here are different from those with same _{c} in Figures

The simulations of varying dendrite area may lead one to hypothesize that it is the increase of outward current _{SD} that is the primary effect in the increase of Na^{+} entry ratio with each AP at higher coupling conductance. To test this hypothesis, we depict _{Na}, _{K}, and _{SD} associated with the APs. It is shown that _{Na} and _{K} both change slightly as coupling conductance _{c} varies (Figure _{SD} becomes progressively more prominent with _{c} (Figure ^{+} current and makes it become self-sustaining at a more depolarized voltage (Figure ^{+} load needed to produce the upstroke of AP decreases with _{c}. Further, the presence of outward _{SD} during the depolarizing phase of AP leads to the overlap between with Na^{+} influx. Under this condition, model I neuron has to import more Na^{+} ions to compete with _{SD} and generate the fast upstroke of APs. Then, the total Na^{+} load during an AP is increased and corresponding Na^{+} entry ratio gets larger. Therefore, the increase in excess Na^{+} entry induced by increasing coupling conductance is largely owing to the increased overlap between outward _{SD} and Na^{+} influx during the depolarizing phase of the AP.

Inefficient APs arise from increased overlap between _{SD} and _{Na} as coupling conductance increases. _{Na} (red), _{K} (blue), and _{SD} (green) underlying the APs with different values of coupling conductance, which have been indicated on the top of the panels. Morphological parameter is _{SD}-_{S} curve for each AP shown in _{S}/d_{S} for corresponding AP. Blue dots indicate where APs are initiated. _{c} with different values of

With a simple two-compartment model, we have simulated how the passive properties of the dendrite modulate the energy efficiency of APs. Our next step is to examine the effects of active currents in dendrites. To achieve this goal, we introduce a voltage-dependent Ca^{2+} current _{Ca} to the passive dendrite of model I and create model II (Figure _{D} is applied to activate slow _{Ca} and trigger APs. Activating active Ca^{2+} channel results in a regenerative response in dendritic chamber (Figure ^{2+} spike. Such all-or-none event in dendritic chamber leads the neuron to generate a burst of high-frequency APs at the onset of input _{D}. In this case, the spike train recorded in somatic chamber is no longer periodic (Figure _{Ca} is activated. In particular, the height and half-width of the AP both decrease at first and then increase during the course of dendritic spike (Figures ^{2+} spike and increases the intensity of internal current _{SD}, thus effectively enhancing the modulations of somatic APs.

Activating Ca^{2+} current in dendritic chamber results in dendritic spike in model II. ^{2+} current _{Ca} is introduced to the dendritic chamber of model I. _{S}, dendritic voltage _{D}, Ca^{2+} current _{Ca} and internal current _{SD} are plotted against time. With ^{2+} spike.

We use Na^{+} entry ratio to quantify the metabolic efficiency of the APs associated with dendritic Ca^{2+} spike. It is found that the total Na^{+} load (Figure ^{+} load (Figure _{Ca} is activated. Two items both decay down before _{Ca} reaches its peak value and then increase in the second phase of dendritic spike. That is, the metabolic cost per spike is significantly reduced by the activation of inward _{Ca}. Interestingly, Na^{+} entry ratio also first quickly decreases to a minimal value and then slowly rises to a lower plateau level (Figure ^{2+} spike makes APs become more efficient to use Na^{+} influx to generate their depolarization, thus increasing their metabolic efficiency.

Dendritic Ca^{2+} spike increases metabolic efficiency of APs. ^{+} load Q_{total}, ^{+} load Q_{min}, and ^{+} entry ratio are respectively computed for each AP during the activation of _{Ca} with _{S} (gray) and _{Ca} (pink) recorded in the model II with each value of

Plots of _{Na}, _{K}, _{SD}, and _{Ca} underlying individual spikes reveal that the efficient APs triggered by dendritic Ca^{2+} spike are also owing to the modulation of internal current _{SD} (Figure _{D} is forced to reach a threshold voltage, slow inward _{Ca} is activated and then a broader Ca^{2+} spike is initiated in dendritic chamber. Such regenerative event at the slow timescale results in a prolonged local depolarization of _{D}, which effectively decreases the outward level of _{SD} and even switches its direction from outward to inward (Figures _{SD} still flows out of the soma, such as in the last phase of dendritic spike, decreasing its intensity reduces its overlap with Na^{+} influx, thus increasing AP efficiency. When _{SD} flows into the soma, the overlap between it and Na^{+} influx during the upstroke of the AP disappears. Instead, the inward _{SD} cooperates with Na^{+} influx to contribute to the depolarization of somatic membrane, thus effectively increasing the efficiency of Na^{+} entry. Meanwhile, the depolarizing _{SD} induced by the activation of _{Ca} also significantly reduces the intensity of _{Na} and _{K}, especially the former (Figure ^{+} load per spike is significantly reduced during dendritic Ca^{2+} spike. As a result, activating _{Ca} in the dendrite of model II neuron reduces the energy cost of somatic APs and makes them metabolically efficient.

Activating _{Ca} makes the overlap between _{SD} and _{Na} disappear and increases AP efficiency. _{Na} (red), _{K} (blue), _{Ca} (pink) and _{SD} (green) underlying the 2nd, 8th, 40th, and 170th APs.

Apart from inward active currents, there are also active currents flowing out of the dendrites, which mainly hyperpolarize dendritic membrane voltage. To determine how these inhibitory currents affect AP efficiency, we introduce a Ca^{2+}-activated K^{+} current _{KAHP} into the dendritic chamber of model II and create model III (Figure _{KAHP} occurs at a slower timescale than the fast dynamics of APs, which includes a form of negative feedback to cell excitability. Here, we examine the Na^{+} entry efficiency of the simulated APs as _{KAHP} is activated.

Activating _{KAHP} in dendritic chamber results in SFA in model III. ^{2+} current _{Ca} and a Ca^{2+}-activated outward K^{+} current _{KAHP} are introduced to the dendritic chamber of model I. _{S}, _{KAHP}, _{Ca}, and _{SD} are plotted against time. With _{KAHP} is activated, the firing rate decays down to a lower steady-state level, and model III neuron generates SFA.

Similar to above simulations, a constant input _{D} is applied to activate _{KAHP} and evoke APs. We find that the activation of _{KAHP} in dendritic chamber reduces the firing rate to a lower steady-state level (Figure _{KAHP} increases with co-occurring APs (Figure _{KAHP}, whereas it increases the intensity of this current at the subthreshold voltages. Such manipulation also increases the amplitude of internal current _{SD}. In this case, activating _{KAHP} with small dendritic chamber has less effect on the shape of somatic APs (Figures _{SD}, increasing dendrite area reduces the intensity of inward _{Ca}, which competes with outward _{KAHP} in dendritic chamber to result in distinct modulations of AP shape. In particular, the AP half-width shows opposite evolutions with the activation of _{KAHP} at different values of _{KAHP} is close to 0 μA/cm^{2} during the first AP, which begins to take effects in the second AP. In contrast, inward _{Ca} is relatively strong in the first AP, and gradually decays as _{KAHP} activates. The non-linear competition between _{KAHP} and _{Ca} leads to the marked decrease in AP height and half-width.

We measure the Na^{+} entry ratio per AP during the time course of SFA. The results show that activating _{KAHP} in dendritic chamber increases the total Na^{+} load per spike (Figure ^{+} load (Figure _{Ca} is activated. This arises from the interactions of _{KAHP} with different intensities of _{Ca} induced by changing dendrite area. By calculating Q_{total}/Q_{min} for each AP, we show that the Na^{+} entry ratio increases as firing rate is reduced (Figure _{KAHP} in the dendrite makes APs become inefficient to use Na^{+} entry to generate their depolarization, thus reducing their metabolic efficiency.

Metabolic efficiency of the AP is reduced as _{KAHP} is activated. ^{+} load Q_{total}, ^{+} load Q_{min}, and ^{+} entry ratio are respectively calculated for each AP during the activation of _{KAHP} with _{S} (gray) and _{KAHP} (pink) recorded in the model III with each value of _{KAHP} is totally unactivated during the 1st AP.

We examine the ionic currents underlying the recorded spike trains with _{KAHP} during each AP is much lower compared to _{Na}, _{K}, or _{SD}, even when it is sufficiently activated (Figure _{SD} (Figures _{SD} and Na^{+} influx during an AP becomes progressively more prominent as _{KAHP} is activated. The augment in their temporal overlap leads corresponding AP to import more Na^{+} ions for achieving somatic depolarization (Figure ^{+} influx is employed to compete with outward _{SD}, which effectively increases the excess Na^{+} entry ratio (Figure _{KAHP} in the dendrites reduces the efficient use of Na^{+} entry and makes APs metabolically inefficient.

Activating _{KAHP} increases the overlap between _{SD} and _{Na} and reduces AP efficiency. _{Na} (red), _{K} (blue), _{KAHP} (pink) and _{SD} (green) underlying the 2nd, 4th, 6th, and 26th APs. _{SD}-_{S} curves for four APs. Activating _{KAHP} in dendritic chamber increases the outward level of _{SD} during the rising phase of the APs (see inset), which overlaps with _{Na} and reduces efficiency of Na^{+} entry.

Moreover, the effects of adaptation currents on APs have been shown to be dependent on current stimulus (Prescott et al., _{KAHP} as dendritic input is varied. We measure height and half-width, total and minimal Na^{+} load, and Na^{+} entry ratio for simulated APs. It is found that increasing dendritic input makes _{KAHP} stronger and extends its activation procedure (Figure _{KAHP} with strong _{D} produces larger effects on AP height and half-width (Figure ^{+} load and minimal Na^{+} load per spike (Figure ^{+} entry ratio (Figure _{KAHP}. In particular, our predictions are reproducible in the cases of different dendritic inputs. That is, activating inhibitory _{KAHP} in dendritic chamber increases both energy cost (Figure ^{+} entry ratio (Figure _{D} to dendritic chamber simultaneously increases the intensity of inward _{Ca}, especially during the initial APs after the onset of injection. Since outward _{KAHP} is relatively weak in this phase, _{Ca} dominates the outcome of their competition, which results in the marked decrease in AP shape, total Na^{+} load, minimal Na^{+} load and Na^{+} entry ratio. After that, _{Ca} decays and then the activated _{KAHP} dominates the outcome of their competition, which results in SFA. Thus, we neglect the decrease in each item when we examine the effects of activating _{KAHP} on AP efficiency with different stimulus.

Activation of _{KAHP} by strong dendritic input results in inefficient APs. ^{2}, and 3.5 μA/cm^{2}. _{S}, _{KAHP}, and _{Ca} are plotted against time. During the interval of 1000 ms, model III neuron respectively generates 17, 39, and 57 APs. _{D}. ^{+} load Q_{total} (top) and minimal Na^{+} load Q_{min} (bottom) of each AP. ^{+} entry ratio of each AP. Note that three intensities of _{D} are all below the threshold for activating _{Ca} to trigger dendritic Ca^{2+} spike. Once _{D} reaches that threshold, the activated _{Ca} dominates the outputs of model III neuron, and the SFA will disappear.

Our simulations develop three two-compartment biophysical models to describe the intrinsic properties of the dendrites and reproduce the APs initiated in cortical pyramidal cells. The relationships between the dendritic properties, the energy efficiency of APs, and the currents underlying relevant AP are determined. The excess Na^{+} entry ratio is applied to quantify the efficiency with which Na^{+} influx is used for AP formation. These calculations allow us to identify how passive and active properties of the dendrites modulate AP efficiency in each model.

Our approach is to forward engineer simple point-neuron models to better understand how the passive and active dendrites affect the energy efficiency of somatic/axonal APs. We create the models only as complicated as required to reproduce the phenomena of interest. Note that the morphology and biophysics of real dendrites is extremely complicated and particular to specific cells. Creating a high-dimensional biophysical model to capture these non-linearities is reasonably straightforward. However, it may fail to provide a deeper insight than the physiological experiments upon which is based. In our forward engineered models, we exclude the extraneous details of pyramidal cells and reduce the number of core parameters to workable proportions. Such reduced point-neuron models allow us to simulate the core process of interest and gain a greater understanding of fundamental principles than biophysically realistic, extensive models.

We first examine the effects of two passive properties with our biophysical models. One relevant parameter is the ratio of dendrite area to total membrane area, and the other one is the internal coupling conductance connecting chambers. By systematically varying them within the model, we find that increasing dendrite area or coupling conductance both result in a marked increase in the internal current flowing between two chambers. This is an outward current flowing out of the soma, which overlaps with Na^{+} influx during the upstroke of the AP. Increasing the intensity of such inhibitory current makes the overlap more significant. Then, the Na^{+} entry efficiency of relevant AP is reduced. It has been shown that the voltage-gated channels or pumps must fit into a limited membrane area (Faisal et al.,

We also simulate the metabolic efficiency of APs associated with active dendrites. Two types of dendritic channels are examined. One is inward Ca^{2+} current _{Ca}, and the other one is outward Ca^{2+}-dependent K^{+} current _{KAHP}. Their activations both occur at slower timescales than the fast dynamics of spike initiation, which allows us to observe how they modulate AP efficiency in a recorded spike train. Our simulations show that activating active current in dendrites can either enhance or reduce the excess Na^{+} entry ratio of somatic AP, depending on whether it is depolarizing (inward) or hyperpolarizing (outward). The activation of inward _{Ca} results in a local depolarization in dendritic chamber and evokes dendritic spike. Such event effectively decreases the outward level of internal current _{SD} and controls it to flow into the soma. In this case, internal current _{SD} is not to overlap and compete with Na^{+} influx but to cooperate with it to depolarize somatic membrane, thus increasing AP efficiency. On the contrary, activating _{KAHP} hyperpolarizes dendritic membrane and increases the outward level of _{SD}. Such event results in more overlap of inhibitory _{SD} and Na^{+} influx, thus decreasing AP efficiency.

The existence of inward active currents (including Na^{+}, NMDA and Ca^{2+}) in dendrites endows them with powerful ability of synaptic integration (Spruston, ^{2+} spike triggers a burst of APs in the soma/axon and switches the firing mode of the cell to bursting (Williams and Stuart, ^{2+} current in apical dendrites makes Na^{+} entry become efficiently used by APs for their depolarization, thus facilitating the effective utilization of metabolic energy. Our simulations suggest that the supralinear integration operated by dendritic Ca^{2+} spike is a contributory factor for the metabolically efficient coding by cortical pyramidal cells. Note that we only consider slow Ca^{2+} current in our simulations. The effects of other active channels, such as NMDA or Na^{+}, need to be examined in future work.

Outward _{KAHP} in dendritic sites is an ionic mechanism for causing SFA. In fact, the _{KAHP} in soma/axon can also lead the cell to adapt its spike frequency (Benda et al., ^{+} current _{M} (Brown and Adams, ^{+}-activated K^{+} current _{KNa} (Wang et al., _{M} or _{KAHP} in soma/axon directly leads to the overlap between with Na^{+} influx during the depolarizing phase of AP, effectively increasing the Na^{+} load to achieve depolarization, and thus resulting in an inefficient AP with higher energy consumption. Our present study finds that the activation of _{KAHP} in dendritic chamber takes effects in a different way. It directly increases the outward level of internal current _{SD}, which overlaps with Na^{+} influx in the soma and reduces AP efficiency. Even so, the similar outcomes of their activation indicate that the presence of slow inhibitory currents in dendrites, soma or axon makes an AP less efficiently use Na^{+} entry for its depolarization. Note that SFA is a common strategy used by neurons to encode signals, which is ubiquitous in the central nervous system (Sharpee et al.,

Two-compartment model is the minimal neuronal unit for capturing the interaction between dendrites and soma/axon, which has been widely used to describe the input-output transfer of single pyramidal cell. In particular, our earlier studies (Yi et al.,

Experimental and computational approaches have been used to determine the energy efficiency of APs. A major determinant of AP efficiency is the overlap of inward and outward currents (Sengupta et al., ^{+} or K^{+} (Hasenstaub et al., ^{+} and outward K^{+} currents to determine AP efficiency. For a specific cell, AP efficiency also varies across different parts (Alle et al., ^{+} and K^{+} currents than soma, resulting in higher efficiency. Such difference is largely owing to the prevalent Kv1.1/1.2 channels (Kole et al., ^{+} kinetics (Schmidt-Hieber and Bischofberger, ^{+} and outward internal current. Thus, the dendrites not only have marked and strong impacts on the final output of neuronal computation, which also affect the energy consumption and efficiency of somatic/axonal APs.

In our simulations, there is significant overlap between inward Na^{+} and outward K^{+} currents during the APs. It arises from the simultaneous activation of two active currents. Under this condition, Na^{+} enters the cell at the same time that K^{+} exits the cell. These fluxes mostly cancel each other, which increase total Na^{+} entry needed to formulate the APs. Such temporal overlap occurs during the repolarization of the AP. Except for delayed rectifier K^{+}, _{SD} is another current flowing out of the soma in our two-compartment models. This internal current mainly appears during the upstroke of APs, which effectively leads to unnecessary Na^{+} influx. The passive and active dendrites participate in somatic APs through controlling the _{SD}. Calculating the overlap Na^{+} load during the repolarization is unable to measure how outward _{SD} overlaps with Na^{+} influx. Instead, we quantify total Na^{+} entry during the AP relative to the minimal charge necessary for its depolarization. The Na^{+} entry ratio calculated in this way effectively measures how efficiently an AP uses Na^{+} influx to produce its depolarization, which takes the temporal overlap of _{SD} and _{Na} into consideration. In fact, both overlap Na^{+} load and Na^{+} entry ratio are effective measures for determining the potential biophysical causes for the variability in metabolic efficiency of APs among neurons. Based on our simulations, we suggest using Na^{+} entry ratio to measure the AP efficiency when there are outward currents in its depolarizing phase.

Experiments have recorded a myriad of APs with a wide variety of shapes (height and width). An earlier study by Carter and Bean (^{+} entry efficiency among neurons primarily arises from different AP shapes rather than Na^{+} channel kinetics. But Sengupta et al. (^{+} entry ratio or AP shape alone is unable to explain the effects of dendritic properties on AP efficiency. This is comparable to the prediction of Sengupta et al. (

There are some limitations in our model and technical considerations. First, our simulations only examine the effects of varying one core parameter. Future work should focus on how their possible combinations affect and maximize AP efficiency in pyramidal cells. Second, the morphology and active channels in the dendrites are very complicated for a real cell. Including them in the biophysically-based models to describe their relationships with AP efficiency will surely facilitate our interpretation of the energy expenditure of neuronal computation. Finally, the present study only focuses on the energy efficiency of single AP and not formally simulates the information coding by relevant model neurons. How passive and active dendrites participate in information transmission and then affect its energy efficiency should be examined in following works.

With biophysical models, we have obtained basic principles about how passive and active dendrites affect the Na^{+} entry efficiency of somatic/axonal APs. Our results emphasize that they are all potential factors for the variability in AP efficiency between pyramidal neurons. By relating dendritic properties to the overlap between Na^{+} influx and internal current, we provide an interpretable insight into their effects. Determining their contributions to the AP efficiency is a first but necessary step toward a mechanistic understanding of how single cell consumes metabolic energy to perform computation. Our models and predictions can be used to examine how other biophysics and morphologies of the dendrites affect spike efficiency. Such examinations are essential for deeply interpreting how these subcellular processes participate in the information processing of neurons and neural circuits.

Conceived and designed the work: GY, JW, XW, BD. Performed the simulations: GY. Analyzed and interpreted the data: GY, JW. Wrote the paper: GY, JW, XW.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by grants from the National Natural Science Foundation of China (Nos. 61372010, 61471265, and 61601320), and the China Postdoctoral Science Foundation (No. 2017T100158).

^{+}current in a vertebrate neurone

^{+}channel inactivation kinetics on metabolic energy costs of action potentials

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^{+}action potentials in distal and terminal dendrites of rat neocortical pyramidal neurons

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