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Edited by: Florentin Wörgötter, University Goettingen, Germany

Reviewed by: Hermann Cuntz, Ernst Strüngmann Institute in Cooperation with Max Planck Society and Frankfurt Institute for Advanced Studies, Germany; Walter Senn, University of Bern, Switzerland

*Correspondence: Yansong Chua, Institute for Advanced Simulation (IAS-6) and Institute of Neuroscience and Medicine (INM-6) and JARA BRAIN Institute I, Forschungszentrum Jülich GmbH, Leo-Brandt-Straße, 52428 Jülich, Germany

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.

Modeling the layer 5 pyramidal neuron as a system of three connected isopotential compartments, the soma, proximal, and distal compartment, with calcium spike dynamics in the distal compartment following first order kinetics, we are able to reproduce

Recent studies have shown that dendritic spikes play a significant role in the functions of neurons

Building on the above

It is notable that these hypotheses on the functional role of calcium spikes are based predominantly on observations from

The layer 5 pyramidal neuron has been extensively modeled in NEURON and latest studies try to fit neuron parameters to emulate experimental findings (Hay et al.,

Although these complex neuron models are useful for understanding the biophysical mechanisms of the generation and propagation of calcium spikes, they are less helpful for investigating their functional role, as they are neither sufficiently tractable to allow theoretical analysis nor sufficiently simple to permit efficient simulation in large networks. In this article, we address this problem by developing a neuron model that is amenable to both analysis and efficient simulation whilst still reproducing key

As an initial step, we introduce a three compartment neuron model. The three compartments represent the soma [with basal dendrites], proximal [apical dendrite with oblique proximal dendrites], and distal [distal bifurcation point and tuft dendrites]. The calcium spike is modeled using first order kinetics in the distal compartment (see 2.1). We show that this neuron model is able to reproduce a variety of

Although this initial model already represents a considerable simplification over complex biophysical models, it still contains a large number of dynamic variables and is not analytically tractable. We therefore examine the behavior of this parameterized model under different

As the reduced model is tractable, we are able to analytically obtain the mean contribution of the calcium spike to the somatic membrane potential, while accounting for the background fluctuation (3.4). This analytical form is robust to different levels of fluctuations, whereas first order approximation using linear response theory is not. Beyond the analysis of the contributions of calcium spikes to a neuron's firing activity, this study paves the way to combined theoretical and large-scale numerical investigations of the functional role of calcium spikes and the relationship of correlated synaptic activity to neuronal firing patterns in cortical networks.

The layer 5 pyramidal neuron has previously been represented as a two-compartment point neuron model, in which the calcium spike is modeled using first order kinetics in the distal compartment (Larkum et al., _{l}_{e} = _{i} = 0. The synaptic conductances are modeled as alpha functions, i.e., the time course of the conductance evoked by an incoming spike at _{s} is the synaptic time constant controlling the rise time of the alpha function. The constants _{pd} and _{sp} are the conductances across the distal-proximal and soma-proximal compartments, and _{ca} using first order kinetics
_{ca} the calcium conductance, and _{slope} > 0 and _{slope} < 0. _{∞} and _{∞} are the respective functions depending on the distal voltage ^{d} in a sigmoidal shape, determining the asymptotic values toward which

The neuron spikes when the somatic membrane potential crosses the adaptive threshold Θ_{ad}. At this point the threshold is increased by Θ_{+} from which it relaxes exponentially to Θ_{base} with a time constant of τ_{th}. Additionally, a refractory period _{ref} = 2ms is applied to the somatic compartment, during which the somatic leak term ^{s}_{l} is set to 150nS, as compared to a leak value of 10nS otherwise. This is to emulate the membrane potential coming down to a resting potential value of approximately −60mV from a peak value _{peak} = 30mV to which it jumps upon threshold crossing.

To model the effects of a back-propagating action potential in a three-compartment neuron model, we initialize alpha-shaped currents, with dynamics analogous to the synaptic conductances described above (2), first in the proximal compartment and then in the distal compartment. Specifically, an alpha current ^{p}_{AP} with maximum amplitude ^{p}_{AP} and rise time τ^{p}_{AP} is initialized in the proximal compartment 1ms after the spike, and a current ^{d}_{AP} with maximum amplitude ^{d}_{AP} and rise time τ^{d}_{AP} is initialized in the distal compartment 2ms after the spike. The complete parameters of the model are given in Table

The neuron model is fitted so as to reproduce experimental results illustrated in Figures 1C–E of Larkum et al. (_{base}, the back-propagating currents ^{p}_{AP} and ^{d}_{AP} to 0 and disable the calcium current _{ca}. We then fit the leak conductances and capacitance parameters of the neuron model, with search space for capacitance from 50 to 250 pF. Each of the parameters ^{d}, ^{p}, and ^{s} is being varied independently, as are the leak conductances ^{d}_{l}, ^{p}_{l}, and ^{s}_{l} from the range 10–50 nS. The fitting criteria are defined by Figure 1C of Larkum et al. (_{ca} turned off, a hyperpolarizing step current of −0.2nA lasting 50ms at the proximal compartment followed 30ms later by a beta current with time constants 5 and 1 ms of amplitude 2.2nA at the distal compartment does not initiate any action potentials. Only neuron parameter sets (leak conductances and capacitances) that fulfill the above requirements are considered for the next step of fitting.

In step two, _{ca} is enabled and we fit the parameters controlling the calcium dynamics (τ_{m}, τ_{h}, _{half}, _{half}, _{ca}, and _{ca}, with a search space for these parameters of ±20% around those values used in Larkum et al., _{ad} (both the jump amplitude Θ_{+} and time constant τ_{th}) and _{AP} amplitudes (τ_{AP} are set to default values of 1 ms) so as to produce the same number of action potentials (three and two, respectively) as in Larkum et al. (_{ad} and _{AP} as parameters for the neuron model. This parameter set can be found in Table

The numerical simulations are performed with the NEST simulator (Gewaltig and Diesmann,

In some experiments, coincident inputs are applied to a fraction of the excitatory synapses in the distal compartment according to a multiple interaction process as defined in Kuhn et al. (

In this section, we first show that a three-compartment model using first order kinetics to obtain the calcium current can reproduce key experimental results and so is an appropriate choice of reference model to evaluate the reduced model developed in this study (Section 3.1). In Section 3.2 we then investigate the response of the model to precise and imprecise synchrony impinging on the distal compartment whilst the neuron receives stochastic input, in order to identify in which regime it is a reasonable approximation to replace the first order kinetics with a fixed waveform triggered by a voltage threshold. We develop this reduced model in Section 3.3 and analyze it in Section 3.4, showing that this simplified calcium dynamics allows us to obtain the voltage excursion at the soma due to a calcium spike analytically.

We first investigate whether the three-compartment model with calcium currents modeled using first order kinetics, as described in Section 2.1, is capable of reproducing key experimental phenomena. Using the three step procedure detailed in Section 2.2, we identify a set of parameters for which the neuron model is able to reproduce qualitatively the experimental results presented in Figures 1C–E of Larkum et al. (

The simulation results for these parameters are shown in Figure

These results demonstrate that the three-compartment neuron model with calcium currents modeled by first order kinetics is able to reproduce key experimental results capturing the interaction of calcium spikes with action potentials. We hence conclude that the model is an appropriate choice of reference model against which the reduced model can be evaluated.

While first order kinetics is able to account for a variety of experimental findings, as shown in the previous section, unfortunately it is not analytically tractable. Consequently, we can only determine the contribution of the calcium current to the somatic membrane potential, and thus the firing behavior of the neuron, by numerical simulation. Ideally, we would like to replace the first order kinetics with something more amenable to further analysis, such as a threshold triggered fixed waveform. We therefore investigate the response of the first order kinetics model to fluctuating input, to determine under what conditions such a simplification would be an appropriate abstraction. In the following we refer to a transient calcium current as a calcium spike, and its contribution to the somatic membrane potential as a calcium somatic potential.

We first take a closer look at the dynamics of the three-compartment neuron with first order kinetics receiving noisy input with and without synchronous inputs, as shown in Figure

% of distal excitatory synapses receiving correlated inputs | 10.0 | 20.0 | 0.0 | 0.0 |

Copy probability (pair-wise correlation) | 0.3 | 0.5 | – | – |

Excitatory synaptic weight | 0.6nS | 0.6nS | 5.0nS | 12.0nS |

Inhibitory synaptic weight | 1.0nS | 1.0nS | 21.2nS | 54.4nS |

As the calcium spike has been hypothesized as the mechanism by which the layer 5 pyramidal neuron detects coincident inputs (Larkum et al.,

We next stimulate the neuron model with different magnitudes of synchronous input, without the background fluctuations. If the synaptic current has a short time constant compared to the membrane voltage, (Section 2.1) suggests that the amplitude of the membrane voltage deflection mainly depends on the temporal integral of the synaptic conductance. We hence use this parameter to measure the strength of a synaptic input. For small integral conductances Figure

Due to the relation of the peak membrane potential to the integral conductance, this threshold-like behavior can alternatively be observed with respect to this input parameter, as shown in Figure _{∞}(^{d}(_{∞}(^{d}(^{d}(

To investigate if the reduction to a threshold-triggered waveform holds true even for full compartmental neuron models such as those in Hay et al. (

In the previous section we investigated the response of the neuron to precisely synchronous inputs. In particular, we are interested in synchronous inputs with low background fluctuation as in Figure

The histogram of maximum amplitudes of calcium distal potential is shown for σ = 1 ms in Figure

Averaging over all calcium events, Figure

The reduction of the mean calcium potential shown in Figure

In the previous sections, we have demonstrated that the calcium spike modeled using first order kinetics can be approximated with a threshold-triggered fixed waveform in the regime of weak fluctuating input with occasional large synchronous events. The waveform is obtained empirically from the mean calcium spike of the neuron model with first order kinetics, which depends on τ_{e} and the background fluctuations, as illustrated in Figure _{Ca} is then obtained from the mean of calcium currents triggered by the large synchronous inputs. The waveform needs to be obtained empirically as it is very sensitive to different τ_{e}, as shown in Figure

To determine the calcium threshold, we systematically vary the time constant of the post-synaptic response (we denote by 0 ms the δ impulse, and by 0.2–5.0 ms alpha conductances with the respective time constant) and determine the minimum conductance required to trigger a calcium spike in the first order kinetics model. As the maximum amplitude of a full calcium spike is much higher than that of sub-threshold calcium currents and is effectively independent of input amplitudes and time constants (see Figure _{ca} exceeds 1100pA. Having established the minimum conductance required to trigger a calcium spike for each synaptic time constant, we then determine the corresponding membrane potential threshold to be that at which

_{e} = 5 ms. _{0} at which the calcium potential slope _{e}. ^{d} (red) against _{∞}(_{d}) ∈ [0, 1] (blue) against _{∞}(_{d}), derived from Equation 3) denotes the theoretical threshold for calcium spikes triggered by δ pulse synaptic inputs.

We next investigate analytically the emergence of the threshold-like behavior to understand its origin and to find the parameters that determine the threshold voltage. A threshold-like behavior becomes apparent from Figures

Solving for the steady-state voltage, we get
_{m} ≪ τ_{h}, and τ_{m} is close to 0. Consequently, just after the input current we can take an adiabatic approach for _{∞} and

We use this steady state fixed point to approximate the calcium spike threshold at the distal compartment. We determine the threshold numerically by plotting ^{d} against _{∞} against

An alternative definition of the voltage threshold is the maximum membrane potential reached by the EPSP for the smallest synaptic input that triggers a full calcium spike (see Figure _{e} fall within a relatively small range of −21.8 to −22.5 mV (see Figure _{half} = −21 mV and _{half} = −24mV, values that are close to the thresholds obtained above.

_{e}. _{e} = 0.4ms.

The threshold obtained for the second definition ranges from 8 to −26 mV, with the biggest discrepancy between the two definitions for τ_{e} ≤ 0.5 ms. This can be understood by considering that for small τ_{e}, synaptic conductances have to be large so as to allow the first order kinetics of

As it is not _{e} = 0.4ms is shown in Figure

# excitatory synapses per compartment | 2000 |

# inhibitory synapses per compartment | 500 |

Rate of Poisson input per synapse (spikes/s) | 1.0 |

Rate of mother process (spikes/s) | 1.0 |

% of distal excitatory synapses receiving inputs from mother process | 30.0 |

Figure _{e}, but for larger values the two approximations give very similar results. This can be understood from the δ input analysis. Firstly, the dynamics of _{m}]. This effect is best observed in the case of small τ_{e}.

Time constant of excitatory synapses τ_{e} (ms) |
0.2 | 0.4 | 1.0 | 2.0 |

Excitatory synaptic weight (nS) | 2.9 | 1.5 | 0.6 | 0.4 |

Inhibitory synaptic weight (nS) | 1.0 | 1.0 | 1.0 | 2.2 |

A similar method of comparison using the number of calcium spikes triggered instead of the firing rate yielded the same results (data not shown). This can also be intuitively understood by considering that in the low fluctuating regime with large synchronous inputs, calcium spikes are only triggered by sufficiently large synchronous events. This means that synchronous inputs must result in an EPSP of a certain minimum amplitude for a calcium spike to be triggered. In a low fluctuating regime, this minimum amplitude is close to the EPSP amplitude of the minimal calcium spike triggering conductance, i.e., the second definition of the threshold. We therefore conclude that the definition based on the maximum EPSP amplitude is more appropriate for our activity regime of interest.

Next, we obtain a calcium spike threshold of −25mV (as defined by maximum EPSP amplitude following a beta-shaped current stimulus to the distal compartment with time constants of 5.0 and 1.0ms) and the corresponding waveform of the calcium current. We use the threshold and the waveform to set up the simplified calcium dynamics, thus effectively removing the eight dynamic variables required for the first order kinetics to obtain a reduced three-compartment neuron model. All other neuron parameters remain unchanged. We then assess the ability of the reduced model to reproduce key experimental findings, using the same stimuli and protocols described in 3.1 and Figure

In the previous sections, we showed how to arrive at the threshold and waveform in the simplified model. In this section we first use the simplified model to derive the analytical form of the somatic membrane potential excursion due to a calcium spike treating the calcium current as a small perturbation to linear order. As it turns out that this result only agrees well with simulation when background fluctuations are small, in Section 3.4.2 and Section 3.4.3 we derive two alternative forms which also agree well in the case of large background fluctuations.

Using the empirical values of the mean calcium spike obtained earlier from Figure ^{p}_{AP} = ^{d}_{AP} = 0 and without the contribution due to the calcium spike _{ca} = 0 we get with the definition of the 3 by 3 matrix
^{x}〉. If the system is perturbed with a calcium spike at the distal compartment, we can write the perturbed solution as a small deflection ϵ from the stationary state 〈_{ca} refers to the calcium fixed waveform and the functions ϕ_{1, …, 3} can in principle be obtained as the matrix exponential in Equation (8).

However, this first order approximation works well for small fluctuating inputs and underestimates the calcium potential under large fluctuating inputs, as shown in Figure

^{d} and excitatory conductances ^{d}_{e} as a function of excitatory synaptic weights. ^{d} and excitatory conductances ^{d}_{e} as a function of excitatory synaptic weights.

The deviation of the theoretical approximation from simulated results are hardly affected by the amplitude of the calcium spike (see Figure

From (9), we observe that synaptic conductances act as multiplicative noise on the voltage contribution of a calcium spike. The covariance between membrane potential and excitatory conductance increases with synaptic weight, as shown in Figure _{e}(_{e}(_{e}, _{e}(_{e}, _{e}, _{e})σ(_{e}, effectively reducing the driving force and therefore the contribution of excitatory conductances to the membrane potential. In the following, we derive alternative forms for the excursion of the somatic membrane potential following a calcium spike to improve the fit to the simulated results for large fluctuating inputs.

Accounting for the correlations between membrane potential and fluctuating synaptic inputs, we arrive at a better theoretical approximation. To illustrate the main effect, it is sufficient to consider a single compartment and a δ-shaped calcium current _{ca} = _{ei}(_{e}(_{i}(_{eff} is a constant to be determined. Hence, integrating the latter equation as above, we get

By assuming that the driving force can be expressed by the effective conductance

The calculation above shows that the effective leak of the neuron model in the fluctuation regime (where conductances and membrane potential are correlated) can be corrected by considering the integral of the calcium potential. This correction is semi-analytic as we need to determine the calcium potential through simulation to obtain a better theoretical approximation. For our three compartment model we obtain from Equation (1) the coupled set of first order differential equations

Integrating both sides of Equation (12) from

Replacing the left hand side by an effective term, _{eff} obtained above can then be used in place of the respective 〈_{e}〉 + 〈_{i}〉 in (8) to obtain the corrected calcium potential approximation. If we assume that the conductances in Equation (10) can be sufficiently well-approximated by their mean, then the above consideration yields

The semi-analytical approach requires the integral of the calcium potential for approximating the potential waveform, which must be obtained empirically. We now develop a purely analytical approach capable of approximating the waveform of the calcium potential. From Richardson and Gerstner (_{0} = _{l} + 〈_{e}〉 + 〈_{i}〉, _{e} and τ_{i} are the excitatory and inhibitory synaptic time constants. ^{2}_{e} and σ^{2}_{i} are the variances of the excitatory and inhibitory conductances. Rearranging, we obtain

As we are interested in a small deviation of the membrane potential from the value _{0} resulting from static conductances, 〈_{0} + δ_{0}≃ 〈V〉. Substituting and rearranging (13), we get

Hence the effective conductance follows as the factors multiplying the mean membrane potential

The analytically obtained _{eff} for a single compartment can be applied analogously to each of the three compartments in our neuron model so as to obtain

The semi-analytical and analytical results are shown in Figure

The deviation of the theoretical predictions from the simulation results with threshold-triggered calcium spikes can be explained by the fact that when a calcium spike is threshold-triggered, the mean excitatory and inhibitory conductances deviate from their stationary values as shown in Figure _{e}(_{i}(

The semi-analytical result makes use of the integral of the calcium potential to obtain an effective leak conductance _{eff}, which is a time average considering the whole time course of the calcium potential. Hence, the simulated and semi-analytically obtained calcium potential have the same integral, by construction. Consequently, the jump at the onset of the calcium potential is averaged over the entire time course of the calcium potential, resulting in an overall slight over-estimation of the simulation results as shown in (Figure

In this study we have shown that a three compartment neuron model with calcium dynamics modeled using first order kinetics is able to reproduce a variety of experimental findings gathered using _{e}. Using analysis and simulation, we extracted the parameters of the threshold and waveform from the first order kinetics model. The threshold depends on the time constant τ_{m} of the calcium activating function as well as the membrane time constant. It turns out that the threshold that agrees best with simulation is empirically obtained from the maximum EPSP amplitude (without contribution from calcium spikes) evoked by the minimal conductance required to trigger a full calcium spike (see Figure

The reduction of the first order kinetics to a threshold-triggered additive current is of a similar computational complexity as a common integrate-and-fire type model neuron. A difference is, though, that the model needs to store the empirically obtained waveform of the calcium current generated by the first-order kinetics. In terms of performance, this reduction is a considerable gain, which is evident by comparing the run times for simulating the three different variants of the model whilst applying a step current of 100pA to the soma for 100 s: (i) The model by Hay et al. (

In the three compartment neuron model, the calcium spike is modeled at the distal compartment. As we are taking a reductionist approach in modeling the interacting dynamics of the calcium spike and the action potential generation with the aim to reproduce experimental results of action potentials and calcium spikes, we do not concern ourselves with microscopic properties of layer 5 pyramidal neurons, such as the distribution of calcium ion channels. These important characterizations remain open for future research.

The experimental results that we aim to reproduce in our model can be summarized as follows:

A large synaptic current at the distal compartment triggers a calcium spike that in turn triggers a burst of action potentials.

The distal current required for eliciting a calcium spike is considerably reduced if an action potential co-occurs.

A calcium spike does not result in a burst of action potentials if a hyper-polarizing current arrives at the proximal compartment.

While these by no means capture all experimental results of the calcium spike in the layer 5 pyramidal neuron [for instance, one very important result is that of critical frequency (Larkum et al.,

In our analytical treatment to obtain the mean calcium somatic potential, a calcium threshold that deviates from the mean distal membrane potential results in an initial jump in the calcium somatic potential (see Figure

Equipped with the presented reduced neuron model, in future work we may proceed to address earlier experimental findings in which a calcium spike generates burst of action potentials when triggered by coincident somatic and distal inputs (Larkum et al.,

Further work involving a network of these neurons can then be pursued to better understand its emergent properties. For example, different neurons are known to target different parts of the layer 5 pyramidal neuron. From Figure

In Larkum et al. (

Our approach of introducing dendritic spikes in iso-potential compartment neuron models uses a relatively small number of variables (and thus can be simulated with moderate effort) and is not limited to single-cell studies but can be scaled up to understand the behavior of large-scale network models of neurons with dendritic spikes. This could lead to new insights, as typically studies of the interaction of structure and activity in large-scale neuronal networks have been limited to point neuron models (e.g., Kunkel et al.,

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.

Partly supported by the Helmholtz Association through the Helmholtz Alliance on Systems Biology, the Initiative and Networking Fund of the Helmholtz Association and the Helmholtz young investigator group VH-NG-1028, also by the Next-Generation Supercomputer Project of MEXT, EU Grant 269921 (BrainScaleS) and EU Grant 604102 (Human Brain Project, HBP). All network simulations carried out with NEST (

The Supplementary Material for this article can be found online at: