Our starting model is derived from the Pinsky-Rinzel (PR) model (Pinsky and Rinzel, 1994). We block active currents in its dendritic chamber, and create a simple model with passive dendrite, i.e., model I. To generate APs, we include inward Na⁺ and outward K⁺ currents in somatic chamber. The current-balance equations for model I are described by

where V_S and V_D are the transmembrane potentials of soma and dendrite. I_D (in μA/cm²) is the input current applied to activate neurons. Cm=1 μF/cm2 is the membrane capacitance. p and 1 − p are two morphological parameters, which respectively describe the proportion of cell area taken up by the soma and the dendrite. Two compartments are separated by an internal coupling conductance g_c (in mS/cm²). I_SD = g_c(V_S − V_D)/p is the internal current flowing from the soma to the dendrite along internal conductance. Other ionic currents included in model I are

The gating variables, including activation and inactivation variables for inward I_Na (i.e., m and h) and activation variable for outward I_K (i.e., n), satisfy following first-order kinetics

where x ∈ {m, h, n}. Here the details of each current follow the descriptions by Wang (1998). For each gating variable, the transition rate α_x and β_x are

In model I, the activation variable m of I_Na is replaced by its steady state m_∞ = α_m/(α_m + β_m) (Wang, 1998). g¯Na=45mS/cm2, g¯K=18 mS/cm2, gSL=0.1 mS/cm2, and gDL=0.1mS/cm2 are the maximum conductances associated with the currents. E_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 p and coupling conductance g_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 Ca²⁺ current I_Ca into the dendritic chamber of model I, and develop another two-compartment model, i.e., model II. The Ca²⁺ current is given by (Mainen and Sejnowski, 1996)

Here g¯Ca=0.8mS/cm2 and E_Ca = 140mV, which are modified from the PR like models (Pinsky and Rinzel, 1994; Wang, 1998; Park et al., 2005). Kinetics of activation variable s and inactivation variable c obeys

The transition rate for gating variable s and c are

The kinetics of gating variable s and c is the same as that described by Mainen and Sejnowski (1996). This model is used to examine how activating Ca²⁺ current in dendritic chamber affects AP efficiency.

By introducing an outward current I_KAHP into the active dendrite of model II, we derive model III. I_KAHP is a voltage-independent, Ca²⁺-activated K⁺ current, and activating it causes spike-frequency adaptation (SFA) on slow timescales. This inhibitory current is described by (Pinsky and Rinzel, 1994; Park et al., 2005)

Here the maximal conductance is g¯KAHP=5mS/cm2. Kinetics of activation variable q is given by

where time constant is τ_q = 800ms, and steady-state function is q_∞ = α_q/(α_q + β_q). The transition rates for I_KAHP are α_q = min(0.00002[Ca], 0.01) and β_q = 0.001. [Ca] is the intracellular Ca²⁺ 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 (2009), it is defined as the ratio of total actual Na⁺ 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., 2010). In this case, all of the active ionic currents take place during the defined interval of the AP (Crotty et al., 2006). Total Na⁺ load Q_total per spike is calculated by integrating the Na⁺ current curve over the duration of the AP (Carter and Bean, 2009; Hasenstaub et al., 2010; Sengupta et al., 2010, 2013; Yu et al., 2012), i.e., Q_total = ∫ I_Na(t)dt. Minimum charge Q_min necessary to produce the depolarization of the AP is calculated as Q_min = C_mΔV_S, where C_m is the membrane capacitance. As mentioned in Introduction, an AP is initiated when enough membrane depolarization accumulates to bring V_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 ΔV_S as the change of somatic voltage from spike threshold (dV_S/dt = 20mV/ms) to the peak of the AP (dV_S/dt = 0mV/ms) (Carter and Bean, 2009; Yu et al., 2012; Ju et al., 2016). The excess Na⁺ 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, 2001; Alle et al., 2009; Carter and Bean, 2009; Sengupta et al., 2010; Moujahid and d'Anjou, 2012; Yu et al., 2012; Niven, 2016; Yi et al., 2016a). It is shown that a lower value means most of Na⁺ 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 I_Na and outward I_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., 2006; Sengupta et al., 2010; Moujahid and d'Anjou, 2012).

Earlier studies (Crotty et al., 2006; Hasenstaub et al., 2010; Howarth et al., 2012; Sengupta et al., 2013) determine the energy cost of an AP by the amount of ATP molecules expended in the spike duration. It is known that the Na⁺/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. (2010), we use Q_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 V_S from the peak to the most negative voltage reached after the AP (Carter and Bean, 2009; Sengupta et al., 2010; Yu et al., 2012). The half-width is measured as the spike width at half the AP height.

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 (https://senselab.med.yale.edu/modeldb/enterCode.cshtml?model=230329).

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 1A). There are no active currents in its dendritic chamber. This model includes a morphological parameter p that describes the proportion of soma area, and we use it to characterize dendritic geometry. In our two-compartment models, increasing p means there is a decrease in dendrite area. We apply constant input I_D to activate model I and simultaneously record APs generated in somatic chamber. As parameter p varies, the spike trains are always periodic (Figure 1B), but the AP shape alters. Specifically, the height and half-width of the AP both increase with parameter p (Figure 1C), i.e., decreases with dendrite area.

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 p (Figure 1D, top). This means that the metabolic cost per AP is low with large dendrite area, i.e., energy efficient. The APs with moderate size of dendritic chamber are metabolically inefficient, since they have high energy cost. Unlike Q_total, the minimal Na⁺ load Q_min needed to generate the upstroke of the AP increases monotonically in the observed range of p (Figure 1D, bottom). By calculating Q_total/Q_min, we find that the excess Na⁺ entry ratio decreases with parameter p (Figure 1E). That is, increasing dendrite area in model I neuron increases its Na⁺ entry ratio and facilitates to reduce AP efficiency. With large dendrite area (i.e., small p), the high Na⁺ 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 p (Figure 1F). Therefore, we predict that increasing dendrite area may alter other outward currents to overlap with Na⁺ 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 p, the soma is much smaller than the dendrite. Accordingly, the intensities of I_Na and I_K are both relatively weak during an individual spike (Figure 2A, p = 0.1). This results in the APs with small height and short half-width (Figures 1B,C), which effectively decreases the metabolic cost of relevant AP. In this case, the small shape of the AP is the primary factor for its high energy efficiency. As parameter p gets larger, soma area is increased, which reduces the constraint on ionic channels used for signaling. Then, the activation level of active currents gets higher, especially for inward I_Na. This effectively increases AP size and relevant total Na⁺ load, thus increasing its metabolic cost. Once p exceeds 0.5, the soma dominates cell excitability, and there are little changes in either I_Na or I_K. Unlike two active channels, the internal current I_SD flowing out of the soma shows a marked decrease within the whole range of p (Figure 2A, bottom). The presence of outward I_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 I_Na and I_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 p exceeds 0.5. With high values of p, there is less outward current prior to AP initiation (Figure 2B, top), and inward Na⁺ faces less competition as it depolarizes membrane potential V_S. Then, I_Na is able to become self-sustaining at a more hyperpolarized voltage (Figure 2B, bottom), and results in a lower AP threshold (Figures 2C,D). Combined with increased AP peak, the minimal Na⁺ load needed to produce the voltage change of the spike increases with parameter p. Further, the overlap between Na⁺ influx and outward I_SD during the depolarizing phase of an AP decreases as p is increased. This makes APs become more effective to use Na⁺ 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 I_SD, which results in significant overlap between with Na⁺ influx and then decreases the efficiency of Na⁺ entry during an AP.

Coupling conductance g_c between two compartments is another passive property that controls internal current I_SD in two-compartment model. In this section, we adopt model I to simulate the effects of varying g_c on the metabolic efficiency of APs (Figure 3A). In the observed range of g_c, model I neuron generates periodic spike trains to constant input I_D (Figure 3B). The AP shape shows similar evolutions with g_c for different values of p (Figures 3C,D). Note that the height and half-width of the AP is almost independent of the coupling conductance g_c with small dendrite area, such as p = 0.8. In this case, the soma is much larger than the passive dendrite and dominates the output APs of model I. Thus, increasing g_c connecting two chambers has little effects on AP shape.

Figure 4 shows the total Na⁺ load, the minimal Na⁺ load, the overlap load between Na⁺ and K⁺ currents, and the Na⁺ entry ratio during an AP as g_c varies. With different values of p, these items show similar trends in the observed range of g_c. As coupling conductance increases, the total Na⁺ load during an AP is increased (Figure 4A) while the minimal Na⁺ load is reduced (Figure 4B). Their ratio, i.e., excess Na⁺ entry ratio, is monotonically increased with g_c (Figure 4D), and relevant AP becomes less efficient to use Na⁺ influx. This indicates that the generation of individual spike requires more energy and becomes metabolically inefficient with high coupling conductance. Once g_c exceeds 2 mS/cm², total Na⁺ load, minimal Na⁺ load and Na⁺ entry ratio during an AP all show slight changes as g_c increases. Similar to parameter p, there is no obvious relationship between Q_overlap and Na⁺ entry ratio (Figures 4C,D). Thus, the modulations of AP efficiency with coupling conductance g_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 p is so small that increasing g_c has little effects on somatic APs and their underlying currents. Then, four items for relevant APs show little changes with coupling conductance (Figure 4, p = 0.8).

The simulations of varying dendrite area may lead one to hypothesize that it is the increase of outward current I_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 I_Na, I_K, and I_SD associated with the APs. It is shown that I_Na and I_K both change slightly as coupling conductance g_c varies (Figure 5A, center). However, internal current I_SD becomes progressively more prominent with g_c (Figure 5A, bottom), especially during the rising phase of the spike. The presence of such inhibitory current at the subthreshold potentials antagonizes inward Na⁺ current and makes it become self-sustaining at a more depolarized voltage (Figure 5B), which results in a higher AP threshold (Figure 5C). Combined with falling AP peak, the minimal Na⁺ load needed to produce the upstroke of AP decreases with g_c. Further, the presence of outward I_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 I_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 I_SD and Na⁺ influx during the depolarizing phase of the AP.

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 Ca²⁺ current I_Ca to the passive dendrite of model I and create model II (Figure 6A). Constant input I_D is applied to activate slow I_Ca and trigger APs. Activating active Ca²⁺ channel results in a regenerative response in dendritic chamber (Figure 6B), which is referred to as dendritic Ca²⁺ 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 I_D. In this case, the spike train recorded in somatic chamber is no longer periodic (Figure 6B, top). Meanwhile the AP shape also varies as I_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 6C,D). Increasing dendrite area extends Ca²⁺ spike and increases the intensity of internal current I_SD, thus effectively enhancing the modulations of somatic APs.

We use Na⁺ entry ratio to quantify the metabolic efficiency of the APs associated with dendritic Ca²⁺ spike. It is found that the total Na⁺ load (Figure 7A) and the minimal Na⁺ load (Figure 7B) during each AP show similar trends as I_Ca is activated. Two items both decay down before I_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 I_Ca. Interestingly, Na⁺ entry ratio also first quickly decreases to a minimal value and then slowly rises to a lower plateau level (Figure 7C). It is indicated that dendritic Ca²⁺ spike makes APs become more efficient to use Na⁺ influx to generate their depolarization, thus increasing their metabolic efficiency.

Plots of I_Na, I_K, I_SD, and I_Ca underlying individual spikes reveal that the efficient APs triggered by dendritic Ca²⁺ spike are also owing to the modulation of internal current I_SD (Figure 8). Specifically, once V_D is forced to reach a threshold voltage, slow inward I_Ca is activated and then a broader Ca²⁺ spike is initiated in dendritic chamber. Such regenerative event at the slow timescale results in a prolonged local depolarization of V_D, which effectively decreases the outward level of I_SD and even switches its direction from outward to inward (Figures 6B, 8B). When I_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 I_SD flows into the soma, the overlap between it and Na⁺ influx during the upstroke of the AP disappears. Instead, the inward I_SD cooperates with Na⁺ influx to contribute to the depolarization of somatic membrane, thus effectively increasing the efficiency of Na⁺ entry. Meanwhile, the depolarizing I_SD induced by the activation of I_Ca also significantly reduces the intensity of I_Na and I_K, especially the former (Figure 8B, center). Under this condition, the height of relevant AP is reduced (Figure 6C) and its half-width gets narrow (Figure 6D). Then, the total Na⁺ load per spike is significantly reduced during dendritic Ca²⁺ spike. As a result, activating I_Ca in the dendrite of model II neuron reduces the energy cost of somatic APs and makes them metabolically efficient.

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 Ca²⁺-activated K⁺ current I_KAHP into the dendritic chamber of model II and create model III (Figure 9A). The activation of I_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 I_KAHP is activated.

Similar to above simulations, a constant input I_D is applied to activate I_KAHP and evoke APs. We find that the activation of I_KAHP in dendritic chamber reduces the firing rate to a lower steady-state level (Figure 9B, top), i.e., SFA occurs. During the course of SFA, I_KAHP increases with co-occurring APs (Figure 9B, center). Once it is sufficiently activated, the firing rate no longer decreases and reaches a steady state. Increasing dendrite area produces little effects on the peak of I_KAHP, whereas it increases the intensity of this current at the subthreshold voltages. Such manipulation also increases the amplitude of internal current I_SD. In this case, activating I_KAHP with small dendritic chamber has less effect on the shape of somatic APs (Figures 9C,D). Unlike I_SD, increasing dendrite area reduces the intensity of inward I_Ca, which competes with outward I_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 I_KAHP at different values of p (Figure 9D). Further, there is a marked decrease in AP shape between the first two APs with p = 0.4. This is because outward I_KAHP is close to 0 μA/cm² during the first AP, which begins to take effects in the second AP. In contrast, inward I_Ca is relatively strong in the first AP, and gradually decays as I_KAHP activates. The non-linear competition between I_KAHP and I_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 I_KAHP in dendritic chamber increases the total Na⁺ load per spike (Figure 10A), thus increasing the metabolic cost of relevant AP. The minimal Na⁺ load (Figure 10B) with different values of p show opposite trends as I_Ca is activated. This arises from the interactions of I_KAHP with different intensities of I_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 10C). This indicates that activating outward I_KAHP in the dendrite makes APs become inefficient to use Na⁺ entry to generate their depolarization, thus reducing their metabolic efficiency.

We examine the ionic currents underlying the recorded spike trains with p = 0.4 (Figure 11A). We find that the intensity of outward I_KAHP during each AP is much lower compared to I_Na, I_K, or I_SD, even when it is sufficiently activated (Figure 11B). But activating such inhibitory current in dendritic chamber facilitates the hyperpolarization of its membrane voltage and results in slight increase in the outward level of internal current I_SD (Figures 11B,C). Then, the overlap between outward I_SD and Na⁺ influx during an AP becomes progressively more prominent as I_KAHP is activated. The augment in their temporal overlap leads corresponding AP to import more Na⁺ ions for achieving somatic depolarization (Figure 10A), which adds an additional metabolic cost. In this case, more of Na⁺ influx is employed to compete with outward I_SD, which effectively increases the excess Na⁺ entry ratio (Figure 10C). Thus, activating outward I_KAHP in the dendrites reduces the efficient use of Na⁺ entry and makes APs metabolically inefficient.

Moreover, the effects of adaptation currents on APs have been shown to be dependent on current stimulus (Prescott et al., 2006; Prescott and Sejnowski, 2008; Benda et al., 2010; Yi et al., 2015b). Here we use model III neuron to simulate how the metabolic efficiency of APs depends on the activation of I_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 I_KAHP stronger and extends its activation procedure (Figure 12A, center). Then, activating outward I_KAHP with strong I_D produces larger effects on AP height and half-width (Figure 12B), total Na⁺ load and minimal Na⁺ load per spike (Figure 12C), as well as excess Na⁺ entry ratio (Figure 12D). Even so, these items show similar trends with the activation of I_KAHP. In particular, our predictions are reproducible in the cases of different dendritic inputs. That is, activating inhibitory I_KAHP in dendritic chamber increases both energy cost (Figure 12C, top) and Na⁺ entry ratio (Figure 12D) per spike, which makes relevant AP metabolically inefficient. Note that applying strong I_D to dendritic chamber simultaneously increases the intensity of inward I_Ca, especially during the initial APs after the onset of injection. Since outward I_KAHP is relatively weak in this phase, I_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, I_Ca decays and then the activated I_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 I_KAHP on AP efficiency with different stimulus.

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.