Abstract
Dendritic Ca^{2+} spike endows cortical pyramidal cell with powerful ability of synaptic integration, which is critical for neuronal computation. Here we propose a twocompartment conductancebased model to investigate how the Ca^{2+} activity of apical dendrite participates in the action potential (AP) initiation to affect the firing properties of pyramidal neurons. We have shown that the apical input with sufficient intensity triggers a dendritic Ca^{2+} spike, which significantly boosts dendritic inputs as it propagates to soma. Such event instantaneously shifts the limit cycle attractor of the neuron and results in a burst of APs, which makes its firing rate reach a plateau steadystate level. Delivering current to two chambers simultaneously increases the level of neuronal excitability and decreases the threshold of inputoutput relation. Here the backpropagating APs facilitate the initiation of dendritic Ca^{2+} spike and evoke BAC firing. These findings indicate that the proposed model is capable of reproducing in vitro experimental observations. By determining spike initiating dynamics, we have provided a fundamental link between dendritic Ca^{2+} spike and output APs, which could contribute to mechanically interpreting how dendritic Ca^{2+} activity participates in the simple computations of pyramidal neuron.
Introduction
Pyramidal neurons are common cell types found in the cerebral cortex and hippocampus of mammalian brain^{1,2,3}. Their structures are characterized by a pyramidal shaped soma and extended apical and basal dendritic trees. This kind of nerve cells have powerful capability of processing information, which could effectively and precisely transform incoming signals into specific patterns of action potential (AP) output. During this procedure, their dendrites play a particularly vital role, since they are the predominant receiving sites for synaptic signals^{1,4,5,6,7,8}. The vast branches of dendritic tree endow a pyramidal cell with distinctive morphological feature, which disperse the primary input locations. It is known that APs usually occur in the initial segment of the axon. Due to such spatial arrangement, the apical dendrites have to deliver input signals to the site of AP initiation. Their function is not solely to receive information from connected input cells and transmit it to the axon. Each dendritic branch is also a basic signalling unit for integrating synaptic inputs^{4,6,7,8,9,10,11}, which determines how the receiving signals propagate to the axon. Such nonlinear integration operated by dendrites has a profound influence on neuronal and cortical computation^{1,2,4,5,6,7,8,9,10}.
The dendrites of pyramidal cells rely on their intrinsic nonlinearities, including voltagegated channels and complex morphology, to integrate synaptic signals^{4,5,6,7,8,9,10,11}. The active ionic channels in their apical dendrites are particularly important in synaptic integration. A common channel is the voltagedependent Ca^{2+} current that flows into the cell^{1,4,5,6,7,12,13,14}. The activation of its conductance could cause a thresholddependent, allornone regenerative response in dendrites, which is often referred to as dendritic Ca^{2+} spike^{4,7,11,14,15,16,17}. The existence of active Ca^{2+} channel in apical dendrites make pyramidal neurons operate in either global or twostage integration mode^{18,19}. For simple global integration mode, input signals directly contribute to AP output by triggering excitatory postsynaptic potentials (EPSPs) that spread to the AP initiation zone. In latter integration mode, the synaptic input directly activates the Ca^{2+} channel in dendrites and triggers dendritic spikes^{7,14,15,16,17,20}, which propagates forward to the axon where the global integration occurs^{18,19}. Such integration lies at the heart of neural computation, which is tightly related to coincidence detection^{1,16,21,22}, orientation tuning^{22}, binding of synaptic signals from brain areas^{23}, and enhancing stimulus selectivity^{24}. Understanding how it participates in AP output is therefore fundamental to understanding how relevant circuits function in cortical computation of mammalian brain.
Earlier studies have extensively explored the dendritic Ca^{2+} activities and their effects on neuronal firing behaviors with in vitro approaches. It is found that the synaptic inputs at different sites of dendrite^{16,17,25}, the backpropagating APs^{17,26,27}, and the local NMDA spikes^{7,28,29} are all important determinants for activating Ca^{2+} conductance and triggering dendritic Ca^{2+} spike. This regenerative event at apical dendrites can boost distal synaptic inputs and enhance synaptic efficacy, which is hypothesized as the main biological mechanism for propagating synaptic inputs at the distal tuft to the soma of layer 5 pyramidal neurons^{15,16,17,28}. It is usually characterized by a steep change followed by a plateau in the subthreshold inputoutput transformations conferred by dendrites. Further, the additional inward current associated with Ca^{2+} spike provides a strong local depolarization in dendritic membrane, which can enhance the somatic/axonal AP outputs. In particular, it can significantly increase the gain of inputoutput relation of pyramidal neuron, which triggers a burst of APs in the soma/axon and switches the firing mode of the cell to bursting^{23,26,30,31,32}. However, it is still not well understood how dendritic Ca^{2+} spike participates in AP initiation to influence the somatic/axonal output.
In addition to above in vitro observations, there are also modeling studies that focus on the dendritic Ca^{2+} activity^{21,33,34,35,36,37}. Most of them use biophysically realistic neurons that are modeled in NEURON or GENESIS to understand the mechanism underlying the generation and propagation of dendritic Ca^{2+} spike. Such complex multicompartment models are not sufficiently simple to allow one to uncover the dynamical or biophysical basis for AP initiation related to dendritic Ca^{2+} activity. There are also studies that use two compartments to model pyramidal neurons with dendritic Ca^{2+} channel. Such simple pointneuron models have been adopted to study their firing patterns^{38,39,40}, spiketiming predictions^{41}, spike timingdependent plasticity^{42}, and spikefrequency adaptation^{43}. However, it has attracted little attention about the somatic/axonal AP initiation associated with dendritic Ca^{2+} spike. Further, Larkum et al.^{26} have used a twocompartment integrateandfire (IF) model to reproduce the gain modulation of pyramidal cell induced by topdown dendritic Ca^{2+} spike. However, in IF model an AP is explicitly generated when its membrane voltage reaches a predefined threshold^{44}. That is, the IF model is unable to reproduce how inward and outward ionic currents interact at the subthreshold potentials to initiate AP. Therefore, it is still largely unknown that how dendritic Ca^{2+} spike affects the AP initiation of individual pyramidal cells.
Here we develop a fivedimensional (5D) twocompartment model (as shown in Fig. 1a) by introducing Ca^{2+} current into the passive dendrite of a reduced PinskyRinzel (PR) model^{45,46}. Contrary to IF model, the generation of APs in our model implicitly results from the dynamical interactions of inward Na^{+} and outward K^{+} currents at the subthreshold voltages^{45,46,47}. With this model, we have systematically investigated the firing behaviors of the pyramidal neuron with passive and active dendrite to the input current injected at different sites of the neuron. The dynamical basis for relevant AP initiation is determined with phase plane and bifurcation analysis. Our simulations indicate that the proposed model here is able to reproduce a variety of in vitro experimental observations of pyramidal neurons, which is also amenable to both dynamical analysis and efficient simulation.
Results
Somatic input is unable to trigger dendritic Ca^{2+} spike in the absence of dendritic input
We first investigate the spiking properties of the 5D model neuron to somatic input I_{S}. The current I_{D} injected at dendritic chamber is absent, i.e., . Figure 1b–e give the inputoutput relation and corresponding dynamical basis of AP initiation in the case of . We find that the neuron is unable to generate APs when (Fig. 1b and d). At these small values of I_{S}, the neuron exist in quiescent state and its somatic membrane potential V_{S} is eventually stabilized at a subthreshold voltage. With , the neuron generates repetitive APs. In this case, the average firing rate f_{S} increases with input I_{S} from 0 Hz. The relation between f_{S} and I_{S} (i.e., f_{S} − I_{S} curve) is continuous (Fig. 1d).
Figure 1c illustrates the dynamical basis of AP initiation associated with the observed behaviors shown in Fig. 1b and d. With , the V_{S} and wnullclines intersect at three points in phase plane (left panel, Fig. 1c). Since the leftmost intersection is a stable node, all of V_{S} trajectories converge to this equilibrium and the neuron does not fire APs. Increasing I_{S} shifts V_{S}nullcline upwards, while does not alter the position of wnullcline. In this case, the distance between stable node and unstable saddle decreases. With , these two equilibriums coalesce and annihilate each other. At the same time, a stable limit cycle is generated in phase plane (Fig. 1c). Since the stable node corresponding to resting state no longer exists, V_{S} trajectory jumps to the limit cycle attractor and the neuron starts to fire tonic spikes. The transition from resting to repetitive spiking occurs through a saddlenode on invariant circle (SNIC) bifurcation of equilibrium (Fig. 1e), which corresponds to the continuous curve.
In the absence of I_{D}, the firing rate f_{S} of the model evoked by I_{S} does not change with the maximum conductance of dendritic Ca^{2+} current (Fig. 1f). With , there is no external driver activating Ca^{2+} conductance and thus dendritic Ca^{2+} spike is missing. Here I_{S} only activates the Na^{+} channel in soma. Although the Na^{+}APs can be backpropagated to dendrite, such bottomup input is unable to drive V_{D} to reach the threshold voltage for activating Ca^{2+} current (Fig. 1g). Under these conditions, increasing produces no effects on the spiking behaviors of the neuron to somatic input.
Apical input activates dendritic Ca^{2+} current and results in burst of somatic APs
In this section, we investigate how 5D model neuron responds to apical input injected at dendrite. Here somatic input I_{S} is absent, i.e., . The firing behaviors of the model is determined with different values of , which are summarized in Fig. 2.
With , the dendritic Ca^{2+} current is blocked and the dendrite becomes passive. In this case, the neuron exists in quiescent state with and there are still no APs generated. Once I_{D} exceeds , repetitive APs are initiated in somatic chamber (left panel, Fig. 2a). The threshold of I_{D} here is much larger than that of I_{S} directly injected to soma. This is because the dendritic compartment in the model serves as a current sink allowing only part of the input current to invade the soma. From Fig. 2c, one can find that the f_{S} − I_{D} curve is continuous when blocking dendritic Ca^{2+} current, which indicates that the neuron is able to fire lowfrequency APs to injected current I_{D}. As I_{D} is increased, the slope of f_{S} − I_{D} curve (i.e., inputoutput gain) is obviously reduced. Further, the evoked spike trains are always repetitive and the instantaneous firing rate f_{inst} remains constant with time (top panel, Fig. 2b). The f_{inst} in our study is calculated based on the reciprocal of relevant ISI in each spike train.
With or , there is active Ca^{2+} current in dendritic chamber. We find that the threshold value of I_{D} for triggering somatic APs does not change with , which is still . However, the time course of the spike train is no longer repetitive (center and right panels, Fig. 2a). At the onset of I_{D}, the model generates a burst of highfrequency spikes, which then slowly transits to repetitive spiking with low frequency. From Fig. 2b, one can observe that the instantaneous firing rate f_{inst} first quickly increases to a peak value and then slowly decays to a lower plateau level. Under these conditions, the f_{S} − I_{D} curve with is no longer continuous, and the neuron becomes unable to maintain lowfrequency spiking. As shown in Fig. 2c, the average firing rate f_{S} immediately jumps to a high value once I_{D} reaches the threshold for triggering APs. After that, the slope of curve (i.e., inputoutput gain) with or both changes little as I_{D} is increased. These simulations indicate that the activation of dendritic Ca^{2+} current boosts excitatory input I_{D} and facilitates the generation of somatic APs, i.e., an active dendritic integration occurs.
Dynamical basis for the burst of APs associated with dendritic Ca^{2+} spike
In previous section, we have determined the spiking properties of 5D model neuron stimulated by apical input I_{D} alone. Our next step is to uncover the AP initiating dynamics associated with these behaviors.
When blocking dendritic Ca^{2+} current (i.e., ), the dendrite becomes passive. Its membrane voltage V_{D} only oscillates repetitively along with V_{S} and there are no dendritic spikes evoked (left panels, Fig. 3a). Here the internal current I_{DS} between two chambers is also repetitive, which transmits depolarizing input I_{D} from dendrite to soma. Under these conditions, the intersection between V_{S} and wnullclines and corresponding limit cycle attractor both remain unchanged from one AP to the next (Fig. 3b). As a result, the neuron generates repetitive spike trains.
With , the onset of apical input I_{D} depolarizes membrane voltage V_{D} and V_{S}. When V_{D} exceeds a threshold value, the Ca^{2+} conductance is activated and then a broader Ca^{2+} spike is initiated in the dendrite (right panels, Fig. 3a). When this event occurs, I_{Ca} first rapidly falls to a minimum value and then slowly rises to a steadystate plateau level. Note that the negative sign of I_{Ca} means this current is inward. Since Ca^{2+} flows into the dendritic cell, such Ca^{2+} spike evokes a prolonged obvious depolarization of dendritic membrane voltage. Under these conditions, there is an obviously depolarizing sink in internal current I_{DS}, which coincides with dendritic Ca^{2+} spike. That is, the presence of dendritic Ca^{2+} spike boosts apical input as it spreads to soma. Due to such active integration, a constant input of is amplified to a nonperiodic current I_{DS} with a maximum intensity around to invade the soma. Such depolarizing internal current shifts V_{S}nullcline upwards instantaneously in the first rapid phase of Ca^{2+} spike, and forces two nullclines to interact at an unstable fixed point (Fig. 3c). Then all of the V_{S} trajectories converge to limit cycle attractor, and the neuron generates initial APs. Before I_{Ca} reaches peak value, the amplitude of depolarizing I_{DS} continues to increase. It drives unstable fixed point upwards and the limit cycle attractor moves upwards accordingly, which makes firing rate f_{inst} further increase. In the second phase of dendritic Ca^{2+} spike, I_{Ca} slowly gets weak and the amplitude of depolarizing I_{DS} becomes to decrease. In this case, the unstable fixed point and relevant limit cycle attractor both moves downwards (Fig. 3c), which leads to a decay of firing rate f_{inst} to its steadystate plateau level. Therefore, the model generates a burst of somatic APs when dendritic Ca^{2+} spike is initiated. Once such local spike is evoked, the amplitude of dendritic voltage V_{D} and internal current I_{DS} with a specific value of both varies little with apical input I_{D}. Then, the inputoutput gain with or changes little in the observed range of I_{D}.
Figure 4 shows the AP initiating dynamics associated with the firing behaviors of the neuron as Ca^{2+} conductance is increased from to . Increasing results in more somatic APs during the course of dendritic Ca^{2+} spike (Fig. 4a and b). It is known that the intensity of I_{Ca} is proportional to , and increasing its conductance extends the duration of dendritic Ca^{2+} spike (Fig. 4c). Then, the depolarizing current I_{DS} induced by dendritic spike becomes progressively more prominent with . Such stronger depolarizing current accelerates the spike initiation in somatic chamber, and drives neuron to fire more APs at a given value of I_{D}. Then, the average firing rate f_{S} increases as Ca^{2+} conductance is increased (Fig. 4d). However, varying does not alter the kinetics or voltagedependency of Ca^{2+} current. Then, the threshold of apical input I_{D} for activating dendritic Ca^{2+} channel or evoking somatic APs remains unchanged as is increased (Fig. 4d). From Fig. 4e, one can find that blocking dendritic Ca^{2+} current results in a SNIC bifurcation of the equilibrium, which endows the neuron with a continuous curve (Fig. 2c). Introducing dendritic Ca^{2+} channel extends the stable limit cycle to the value of I_{D} below the bifurcation point of equilibrium, followed by unstable limit cycle. In this case, the equilibrium loses its stability via a saddlenode (SN) bifurcation, which corresponds to the transitions of resting to tonic spiking. When such kind of bifurcation occurs, the neuron fails to fire lowfrequency APs^{48}, and its curve becomes discontinuous (Fig. 2c). Unlike equilibrium, the limit cycle transits from unstable to stable via saddle homoclinic orbit (SHO) bifurcation (Fig. 4e). As I_{D} is increased, the unstable limit cycle disappears via a subcritical SHO bifurcation, and the stable limit cycle appears via a supercritical SHO bifurcation. The presence of Ca^{2+} current in dendrite increases the dimension of the system, and makes the SHO bifurcations of limit cycle attractor occur at the value of I_{D} below SN bifurcation. What’s more, the SHO bifurcation also occurs at different values of apical input as is increased (Fig. 4e). All these modulations of firing behavior and corresponding spike initiating dynamics with are due to dendritic Ca^{2+} activity, which endows dendrites with the powerful ability to actively integrate excitatory inputs.
Coincident somatic and dendritic inputs facilitate dendritic Ca^{2+} spike
Here we determine the firing properties of the 5D model neuron evoked by the stimulation of coincident I_{S} and I_{D}, which are summarized in (I_{S}, I_{D}) parameter space for , and . One can find that the threshold value of input I_{S} for triggering APs in somatic chamber decreases linearly with dendritic input I_{D} (Fig. 5a–c). This is because that introducing positive I_{D} results in a depolarizing current transmitted to somatic chamber. Such current increases the level of neuronal excitability and makes it more prone to generate APs to somatic stimulus. Then, the rheobase of I_{S} decreases with dendritic input I_{D}.
Varying Ca^{2+} conductance produces no effects on the rheobase of I_{S} in the observed range of I_{D}, which only determines whether there is dendritic Ca^{2+} spike. When blocking Ca^{2+} current (i.e., ), the dendrite is passive and there is no dendritic Ca^{2+} spike in (I_{S}, I_{D}) parameter space. Here the f_{S}−I_{S} curve is always continuous in the range of (Fig. 5a), which is generated by a SNIC bifurcation of equilibrium (Fig. 5d). With or , there is dendritic Ca^{2+} spike elicited in (I_{S}, I_{D}) parameter space once I_{D} reaches a threshold value. Such event causes firing rate f_{S} quickly to converge to its plateau level at the onset of I_{S} (Fig. 5b and c). Then, the f_{S} − I_{S} curve becomes discontinuous, which is generated via a SN bifurcation of equilibrium (Fig. 5e). Further, introducing dendritic Ca^{2+} current changes the bifurcation of limit cycle from SNIC to SHO, and makes it occur at another value of I_{S} below the bifurcation of equilibrium. Meanwhile, increasing is also able to alter the value of I_{S} for causing the SHO bifurcation of limit cycle (Fig. 5e).
We also find that the threshold values of apical input I_{D} for evoking dendritic Ca^{2+} spike in (I_{S}, I_{D}) parameter space are significantly lower than those in the absent of I_{S}. This indicates that the bottomup input is conducive to the initiation of dendritic Ca^{2+} spike. Particularly, with some moderate values of I_{D}, the Ca^{2+} current is unable to be activated by I_{S} above and close to the bifurcation point of equilibrium (Fig. 6a). Here only somatic input with sufficient intensity can force V_{D} to reach the threshold for initiating dendritic Ca^{2+} spike (Fig. 6b). It is known that the firing frequency f_{S} is an increasing function of I_{S}. Thus, the backpropagation of spiking behavior with high frequency contributes to activating Ca^{2+} spike, which is also referred to as BAC firing^{17,23,26}. These results demonstrate that the dendritic Ca^{2+} spike is the outcome of the interaction between backpropagated APs and excitatory synaptic inputs (Fig. 6c). Such prolonged regenerative spikes elicited in apical dendrites can be forward propagated to the AP initiation zone to modulate the final output of the neuron.
Discussion
Our simulations have shown that injecting current to soma alone makes the 5D model neuron generate continuous inputoutput relation through a SNIC bifurcation of equilibrium. Here the bottomup input from soma to dendrite is unable to activate dendritic Ca^{2+} conductance. Thus, varying Ca^{2+} conductance produces no effects on the output APs. Injecting current to apical dendrite alone results in a distinct inputoutput relation. When blocking Ca^{2+} channel, the inputoutput relation is still continuous, and the bifurcation structures of equilibrium and limit cycle both remain the same. When there is active Ca^{2+} channel in dendrite, the apical input with sufficient intensity is able to activate Ca^{2+} conductance and trigger a prolonged Ca^{2+} spike. This event boosts depolarized input as it spreads to soma, and facilitates the initiation of somatic APs. Under this condition, the neuron generates a burst of highfrequency APs during the course of dendritic Ca^{2+} spike. Then, the curve becomes discontinuous, and the firing rate quickly reaches a plateau level. These simulations demonstrate that the topdown information received by passive or active dendrite modulates the output APs in a distinct way, which depends critically on the site of synaptic inputs.
The firing rate of the 5D model neuron evoked by conjunct inputs to dendrite and soma is summarized in (I_{S}, I_{D}) parameter space. It is shown that simultaneously injecting constant current to two chambers shifts the threshold of curve to a lower value. When blocking dendritic Ca^{2+} current, the topdown input arriving at apical dendrite only increases the excitability of the neuron and reduces the rheobase of somatic input, which does not alter the shape of inputoutput relation. For the dendrite with Ca^{2+} channel, the topdown input with sufficient intensity triggers Ca^{2+} spike. This event results in a burst of APs in soma and significantly increases firing frequency, which leads to a discontinuous inputoutput relation. Here the timing of burst is able to detect whether there are coincident somatic and dendritic inputs. In fact, such burst pattern associated with dendritic Ca^{2+} spike has been observed in previous experimental^{26,30,31,32} and modeling^{35,37,39,40} reports. We have not only reproduced it with a simplified conductancebased model, but also determined the dynamical basis of relevant spike initiation with phase plane analysis. These investigations could contribute to uncovering how the Ca^{2+} activity in apical dendrites participates in neuronal computation.
It is shown that somatic input decreases the threshold value of apical input for triggering dendritic spike. This indicates that the backpropagating APs are conducive to the initiation of dendritic Ca^{2+} spike, i.e., BAC firing occurs. The simulations with our simple 5D model suggest that the generation of BAC firing arises from the interactions between backpropagating spikes and dendritic excitatory input, which is in accordance with previous predictions^{17,23,26,27}. During this procedure, the backpropagating APs play a crucial role in connecting two zones of spike initiation, which enables the integration of synaptic inputs to be disturbed in space and time. Meanwhile, BAC firing is a common mechanism for pyramidal neuron to associate conjunct somatic and dendritic inputs^{25}. Without BAC firing, the dendritic input has much less effects on the firing behavior than somatic input. Once it occurs, the firing rate and spike timing is dominated by the input received by apical dendrites. Thus, this event completely alters the relative importance of synaptic inputs to the cell. Twocompartment model is the minimum individual unit of the neuron to capture such complex phenomenon. Each chamber has its own mechanism for spike initiation, which enables the neuron to integrate synaptic inputs simultaneously in two separated regions. Thus, it can effectively reproduce the BAC firing. Further, earlier in vitro experiment^{26} has predicted that the distal dendritic inputs lead to a higher gain and higher variability of the spike train than somatic input. This phenomenon is missing in our simulations, which is because that the injected currents here are all deterministic and do not include noise. Larkum et al.^{26} have shown that the noisy components in current injection are the dominant factor for relevant gain modulation, since they significantly alter the initial slope of inputoutput relation.
The dendritic Ca^{2+} spikes in our simulations are triggered by current steps injected to the dendrite. Its roles are discussed in augmenting the influence of dendritic current flowing from the dendrite to the soma over the Na^{+}APs. That is, the influences of relevant Ca^{2+} spike are studied in the presence of background activity. This makes the argument that Ca^{2+} spike augments dendritic current problematic. To determine how dendritic Ca^{2+} spike influences somatic APs by itself, we use apical input to generate a brief pulse to trigger a single dendritic Ca^{2+} spike and repeat above simulations. As expected, the inward current associated with dendritic Ca^{2+} spike provides a strong local depolarization that boosts apical input. The resulting sustained depolarization spreads to the soma and causes a burst of highfrequency APs (Fig. 7). These modulatory effects are similar to that evoked by current steps. It indicates that a Ca^{2+} spike in apical dendrites without additional dendritic input influences the initiation of somatic APs just the same as those with dendritic inputs.
Many Ca^{2+}models include the dynamics of intracellular Ca^{2+} concentration ^{37,38,39,43,49,50,51,52,53}, which is tightly related to the Ca^{2+} influx through voltagegated channels. It has been used in modeling^{37,39,40} and experimental^{15,16} studies to characterize dendritic Ca^{2+} spike. Here we introduce the dynamics of and two types of Ca^{2+}activated K^{+} current to the dendrite of our 5D model (see Methods for model specification). By injecting current step to apical dendrite, we repeat the simulations to test whether dendritic Ca^{2+} spike has effects comparable with those described above. It is shown that the neuron generates periodic bursting behavior to constant apical input after introducing Ca^{2+} concentration and K^{+} currents (Fig. 8a), which makes the difference between burst and tonic spiking more distinguishable. As expected, each dendritic Ca^{2+} spike has similar effects on the initiation of somatic APs in more biophysically realistic model (Fig. 8) and in 5D model (Figs 2 and 7). This suggests that the simplifications inherent in our 5D twocompartment model do not compromise the applicability of our findings to biophysically realistic conditions.
As a common cell type in mammalian brain, pyramidal neurons have been studied with theoretical approaches that incorporate dendritic Ca^{2+} channel in multicompartmental models^{21,33,34,35,36,37}. These complex models may express more than 10 voltagegated channels, which are nonhomegenously distributed along the somatodendritic axis. Using biophysically realistic, highdimensional neuron models is reasonably straightforward. But they may fail to provide greater insights into the mechanism underlying AP initiation than the experiments upon which they are based, since they include so many extraneous details. There are also theoretical studies using simple twocompartment models to describe the Ca^{2+} activity of dendrites for pyramidal cell^{38,39,40,41,42,43}. However, none of them has provided a satisfied interpretation of how dendritic Ca^{2+} spike participates in the initiating dynamics of somatic/axonal APs. Unlike earlier models, our model starts simple and excludes extraneous details, which is made only as complex as required to reproduce the phenomena of interest. It enables one to perform phase plane and bifurcation analysis on how Ca^{2+} spikes initiated in apical dendrites affect the global integration of the neuron. With our simple model, one can visualize and interpret how twostage integration mode occurs in pyramidal cells. Even so, our predictions and corresponding interpretations require validation with complex models and experiments.
In summary, the current study addresses the importance of Ca^{2+} spike of apical dendrites in affecting the firing behaviors of twocompartment neurons during different sites of current injection. Our simulations provide a deep and interpretable insight into the connection between dendritic Ca^{2+} spike and firing pattern by relating them to somatic AP initiation. Determining how dendritic Ca^{2+} activities and input locations modulate the cellular responses is a pivotal first step toward uncovering how the Ca^{2+} activity of active dendrite participates in neuronal computation. The simplified twocompartment model proposed here is able to capture the complex phenomena of pyramidal neurons in experiments, which could be used to obtain a mechanistic understanding about how relevant circuits participate in cortical computation.
Methods
Simulations are based on the twocompartmental models of cortical pyramidal neuron, which are the reduced version of PR model. One compartment represents the apical dendritic zone, and the other one represents the basal integration zone around the soma plus axonal initial segment. From apical integration zone, the input signals transmitted to soma can be processed via dendritic Ca^{2+} spike. Twocompartment neuron is the simplest structure to capture such spatial inputs.
Our starting model is derived by introducing a voltagedependent Ca^{2+} current to the passive dendrite of a simple twocompartment model proposed in our earlier studies^{45,46}. The right panel of Fig. 1a shows the schematic representation of the twocompartment neuron, which is a 5D model. There are three ionic currents in somatic compartment, which are inward Na^{+} current (I_{Na}), outward K^{+} current (I_{k}), and passive leak current (I_{SL}). Here two active currents, i.e., I_{Na} and I_{k}, are responsible for generating APs. For dendritic chamber, there are two ionic currents, which are inward Ca^{2+} current (I_{Ca}) and passive leak current (I_{DL}). The somatic and dendritic chambers are connected by an internal coupling conductance . The dynamics of their membrane potential V_{S} and V_{D} are governed by the following currentbalance equations
where is the membrane capacitance, is a morphological parameter. is the internal current that flows through conductance g_{c} and connects two chambers. I_{S} and I_{D} are two input currents respectively injected at soma and dendrite, which are used to stimulate the neuron.
Three voltagedependent currents included in somatic chamber are
where is the steadystate activation function for inward Na^{+} channel. w is the activation variable for slow K^{+} current, which is governed by the following differential equation
Here and are respectively the steadystate activation function and time constant of this slow current. , , are the maximum conductances associated with the currents, and , , are their relevant reversal potentials. Unless otherwise stated, , , , , and .
Two voltagedependent currents used in dendrite are
where , , , and Ca^{2+} maximum conductance is varied as explained in Results. The dynamics of Ca^{2+} current is governed by an activation variable n and an inactivation variable h, which are characterized by the following equations
and are the time constant of the firstorder kinetics for variable n and h. Their steadystate functions are
The kinetics of dendritic Ca^{2+} channel here is the same as that described by Larkum et al.^{26}. Unless otherwise stated, our stimulations are all based on this 5D model.
To make above 5D twocompartment model more biophysically realistic, we introduce the dynamics of intracellular Ca^{2+} concentration and two types of Ca^{2+}activated K^{+} current to its dendrite (see Fig. 8). The K^{+} channels include shortduration Ca^{2+}dependent K^{+} current (I_{KC}) and longduration Ca^{2+}dependent K^{+} current (I_{KAHP}), which are commonly distributed in the dendrites of pyramidal cells^{37,39,40}. Their activations are both related to the dynamics of . After introducing I_{KC} and I_{KAHP}, the membrane equation for dendritic voltage V_{D} becomes
The details of I_{KC} and I_{KAHP} follow the descriptions by Pinsky and Rinzel^{39}, which are , and . Values of the maximum conductance are and . The kinetics of their activation variable c and q obeys
where
The kinetics for intracellular Ca^{2+} concentration [Ca] follows
That is, the [Ca] is increased proportionally to Ca^{2+} influx I_{Ca}^{37,39}. All other parameters in this more biophysically realistic model are the same as in the 5D model.
Note that the dynamics of [Ca] is not considered in the simple 5D model. As mentioned in Introduction, dendritic Ca^{2+} spike is the voltage transient caused by the activation of dendritic Ca^{2+} conductance^{5,6,7,11,23}. It is a local regenerative response involving the positive feedback loop between dendritic voltage and Ca^{2+} influx^{6,7}. Using a voltagedependent current I_{Ca} is sufficient to reproduce such regenerative dendritic response and its dependence on Ca^{2+} dynamics^{26}. By excluding the extraneous details, we propose a biophysical model complex enough to reproduce dendritic Ca^{2+} spike yet simple enough for characterizing its effects on the initiating dynamics of somatic APs. Further, the results in Fig. 8 show that ignoring the dynamics of Ca^{2+} concentration [Ca] does not alter our predictions about how dendritic spike affects the initiating dynamics of somatic APs.
Finally, the twocompartment models are integrated in MATLAB using numerical integrator ode23, with a time resolution of 0.01 ms. The phase plane and bifurcation analyses are performed with the publicly available software package XPPAUT^{54}.
Additional Information
How to cite this article: Yi, G. et al. Action potential initiation in a twocompartment model of pyramidal neuron mediated by dendritic Ca^{2+} spike. Sci. Rep. 7, 45684; doi: 10.1038/srep45684 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
This work was supported by grants from the National Natural Science Foundation of China (Nos. 61372010, 61471265 and 61601320), the China Postdoctoral Science Foundation (No. 2015M580202), and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130032110065).
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School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
 Guosheng Yi
 , Jiang Wang
 , Xile Wei
 & Bin Deng
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Contributions
G.S.Y., J.W. and X.L.W. conceived and designed the research, G.S.Y. and J.W. performed the simulations, G.S.Y., J.W. and X.L.W. wrote the paper. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Xile Wei.
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