PLoS Computational Biology

Public Library of Science

Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses

Volume:
16,
Issue: 4

DOI 10.1371/journal.pcbi.1007175

Abstract

Neurons are extended cells with multiple branching dendrites, a cell body and an axon. In an active neuronal network, neurons receive vast numbers of incoming synaptic pulses throughout their dendrites and cell body that each exhibit significant variability in amplitude and arrival time. The resulting synaptic input causes voltage fluctuations throughout their structure that evolve in space and time. The dynamics of how these signals are integrated and how they ultimately trigger outgoing spikes have been modelled extensively since the late 1960s. However, until relatively recently the majority of the mathematical formulae describing how fluctuating synaptic drive triggers action potentials have been applicable only for small neurons with the dendritic and axonal structure ignored. This has been largely due to the mathematical complexity of including the effects of spatially distributed synaptic input. Here we show that in a physiologically relevant, low-firing-rate regime, an approximate level-crossing approach can be used to provide an estimate for the neuronal firing rate even when the dendrites and axons are included. We illustrate this approach using basic neuronal morphologies that capture the fundamentals of neuronal structure. Though the models are simple, these preliminary results show that it is possible to obtain useful formulae that capture the effects of spatially distributed synaptic drive. The generality of these results suggests they will provide a mathematical framework for future studies that might require the structure of neurons to be taken into account, such as the effect of electrical fields or multiple synaptic input streams that target distinct spatial domains of cortical pyramidal cells.

Due to their extended branching in both dendritic and axonal fields many classes of neurons are not electrically compact objects, in that the membrane voltage varies significantly throughout their spatial structure. A case in point are the principal, pyramidal cells of the cortex that feature a long apical dendritic trunk, oblique dendrites, apical tuft dendrites and a multitude of basal dendrites. Excitatory synapses are typically located throughout the dendritic arbour [1], while inhibitory synapses are clustered at specific regions depending on the presynaptic cell type [2]. These cells also differ morphologically not only between different layers, but also between cells in the same layer and class [3, 4]. Many cortical cells *in vivo* fire rarely and irregularly due to the stochastic and balanced nature of the synaptic drive [5, 6]. Despite the apparent irregular firing of single neurons, computational processes are understood to be distributed across the population [7, 8] with the advantage that encoding information at a low firing rate can be energy efficient [9].

The arrival of excitatory and inhibitory synaptic pulses increases or decreases the postsynaptic voltage as well as increasing the conductance locally for a short time. Together with the spatio-temporal voltage fluctuations caused by the distributed synaptic bombardment typical of *in vivo* conditions, the increase in membrane conductance affects the integrative properties of the neuron, with reductions of the effective membrane time constant, electrotonic length constant and overall input resistance of neuronal substructures [10–12].

How different classes of neurons integrate stochastic synaptic input has been a subject of intense experimental [7, 13, 14] and theoretical [15–19] focus over the last 50 years. The majority of analytical approaches have approximated the cell as electrotonically compact and focussed on the combined effects of stochastic synaptic drive and intrinsic ion currents on the patterning of the outgoing spike train. Such models usually utilize an integrate-and-fire (IF) mechanism with some variations, and have been analysed using a Fokker-Planck approach [20–23] in the limit of fast synapses. However, this approach becomes unwieldy when synaptic filtering is included (though see [24–26]). One approximate analytical methodology, applicable to the low-firing-rate limit driven by filtered synapses is the level-crossing method of Rice [27]. In this approach, which has already been applied to compact neurons [28, 29], a system without post-spike reset is considered with the rate that the threshold is crossed from below treated as a proxy for the firing rate. The upcrossing rate and firing rate for a model with an integrate-and-fire mechanism will be similar when the rate is low, such that the effect of the previous reset has faded into insignificance by the time of the next spike.

Due partly to the sparsity of the experimental data required for model constraint, but also because of the mathematical complexity involved, few analytical results mapping from distributed stochastic synaptic input to the output firing rate for neurons with dendritic structure have followed the early work of Tuckwell [30–32]. Nevertheless there is increasing interest in the integrative and firing response of spatial neuron models [33, 34], neurons subject to and generating electric fields [35–37], and the effect of axonal load and position of the action-potential initiation region [34, 38–42]. As well the simulation-based approach using multi-compartmental reconstructions, a number of recent studies have combined analytical simplifications or reductions of cables coupled to non-linear or spike generating models. These include Morris-Lecar model neurons coupled by a quasi-active dendrite for studying the synchrony of dendritic oscillators [43], and the reduction of ball-and-stick neuron models with exponential integrate-and-fire mechanisms to investigate the effect of electric fields in the presence of synaptic drive applied at a point on the dendrite [37]. Reductions of complex dendritic arbours to a few compartments have also provided valuable insight to the role of structure in filtering high-frequency input. Using constraints to intracellular recordings, an analytical treatment of a two-compartment Purkinje neuron model has provided a mechanistic explanation for the 200 Hz resonance in the firing rate response [44]. The effects of dendritic filtering have also been included in network models studying synchronization of coupled neurons [45], and the firing rates of excitatory neuronal populations in the mean-field limit [46].

At the same time, recent advances in optogenetics and multiple, parallel intracellular recordings have made experimental measurement and stimulation of *in vivo* -like input at arbitrary dendritic locations feasible [47–51]. This potential for model constraint suggests it is timely for a concerted effort to extend the analytical framework developed for compact models driven by stochastic synapses to neurons with dendrites, soma and axon in which the voltage fluctuates in both space and time.

Here we present an analytical framework for approximating the firing rate of neurons with a spatially extended structure in a physiologically relevant low-rate regime [52–54]. To illustrate the approach we applied it to simple but exemplary neuronal geometries with increasing structural features—multiple dendrites, soma and axon—and investigated how various morphological parameters including the electrotonic length, axonal radius, number of dendrites and soma size affect the firing properties.

The cable equation for the voltage *V*(*x*, *t*) in a dendrite of constant radius *a* and axial resistivity *r*_{a} with leak and synaptic currents has the form [55, 56]

${c}_{\text{m}}\frac{\partial V}{\partial t}={g}_{L}({E}_{L}-V)+{g}_{s}(x,t)({E}_{s}-V)+\frac{a}{2{r}_{\text{a}}}\frac{{\partial}^{2}V}{\partial {x}^{2}}$

where $2\pi a{\Delta}_{x}{\tau}_{s}\frac{\partial {g}_{s}}{\partial t}=-2\pi a{\Delta}_{x}{g}_{s}(x,t)+{\gamma}_{s}{\tau}_{s}\sum _{\left\{{t}_{sk}\right\}}\delta (t-{t}_{sk}).$

Here {${N}_{s}=2\pi a{\Delta}_{x}{\varrho}_{s}{r}_{s}{\Delta}_{t}.$

Note that for a Poisson process the variance will also be For a high synaptic-arrival rate we can approximate the Poissonian impulse train by a Gaussian random number with mean *N*_{s}/Δ_{t} and standard deviation $\sqrt{{N}_{s}}/{\Delta}_{t}$ (this is an extension to spatio-temporal noise of the approach taken in [20]). We introduce ${\psi}_{k}^{i}$ as a zero-mean, unit-variance Gaussian random number that is drawn independently for each time step *i*Δ_{t} and each spatial position *k*Δ_{x} . Dividing Eq (2) by the unit of membrane area allows us to write

${\tau}_{s}\frac{\partial {g}_{s}}{\partial t}\approx -{g}_{s}+{\tau}_{s}{\gamma}_{s}{r}_{s}{\varrho}_{s}+{\tau}_{s}{\gamma}_{s}\sqrt{\frac{{\varrho}_{s}{r}_{s}}{2\pi a{\Delta}_{x}{\Delta}_{t}}}{\psi}_{k}^{i},$

where the right-hand side should be interpreted as having been discretized over time, with a time step Δ$\begin{array}{ccc}\hfill \u27e8\xi (x,t)\u27e9=0& \text{and}& \u27e8\xi (x,t)\xi ({x}^{\prime},{t}^{\prime})\u27e9=\delta (x-{x}^{\prime})\delta (t-{t}^{\prime})\hfill \end{array}$

and also note that in the steady state 〈${c}_{\text{m}}\frac{\partial \u27e8V\u27e9}{\partial t}=0=g(E-\u27e8V\u27e9)+\frac{a}{2{r}_{\text{a}}}\frac{{\partial}^{2}\u27e8V\u27e9}{\partial {x}^{2}}$

with $\begin{array}{ccc}\hfill {\tau}_{v}=\frac{{c}_{\text{m}}}{g}& \phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}& \lambda =\sqrt{\frac{a}{2g{r}_{\text{a}}}}.\hfill \end{array}$

For the fluctuating component we assume that the product ${c}_{\text{m}}\frac{\partial {v}_{F}}{\partial t}\approx -g{v}_{F}+{g}_{sF}({E}_{s}-\u27e8V\u27e9)+\frac{a}{2{r}_{\text{a}}}\frac{{\partial}^{2}{v}_{F}}{\partial {x}^{2}}.$

Rescaling synaptic variables
$s=\frac{{g}_{sF}}{g}({E}_{s}-\u27e8V\u27e9),\phantom{\rule{1.em}{0ex}}{\sigma}_{s}=\frac{{\gamma}_{s}}{2g}({E}_{s}-\u27e8V\u27e9)\sqrt{\frac{{\varrho}_{s}{r}_{s}{\tau}_{s}}{2\pi a\lambda}}\phantom{\rule{1.em}{0ex}}$

results in the following form for the synaptic equation
${\tau}_{s}\frac{\partial s}{\partial t}=-s+2{\sigma}_{s}\sqrt{\lambda {\tau}_{s}}\phantom{\rule{0.166667em}{0ex}}\xi (x,t).$

where the steady-state condition d〈${\tau}_{v}\frac{\partial v}{\partial t}=\mu -v+{\lambda}^{2}\frac{{\partial}^{2}v}{\partial {x}^{2}}+s.$

Here The morphologies explored in this paper are shown in Fig 1a and feature boundary conditions in which multiple dendrites and an axon meet at a soma. To account for these conditions we first define the axial current *I*_{a} in a cable, writing it in terms of the input conductance of an infinite cable *G*_{λ} = 2*πa*λ*g*,

${I}_{a}(x,t)=-\lambda {G}_{\lambda}\frac{\partial v}{\partial x}.$

For a sealed end at ${\frac{\partial v}{\partial x}|}_{x=0}=0.$

When the cable is unbounded and semi-infinite in extent, as shown by two small parallel lines in Fig 1a, we apply the condition that the potential must be finite at all positions,
$\left|v\right(x,t\left)\right|<\infty ,\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}x,t.$

For other cases, multiple ($\sum _{k=1}^{n}{\lambda}_{k}{G}_{{\lambda}_{k}}{\frac{\partial {v}_{k}}{\partial {x}_{k}}|}_{{x}_{k}=0}=0,$

where ${\tau}_{0}\frac{\text{d}{v}_{0}}{\text{d}t}=-{v}_{0}+\sum _{k=\alpha ,1}^{n}{\rho}_{k}{\lambda}_{k}{\frac{\partial {v}_{k}}{\partial {x}_{k}}|}_{{x}_{k}=0},$

where the subscript 0 denotes somatic quantities and the neurite dominance factor The cable equations for each neurite with a threshold-reset mechanism were numerically simulated by implementing the Euler-Maruyama method by custom-written code in the Julia language [64]. We discretized space and time into steps Δ_{x} and Δ_{t}, with *v* and *s* measured at half-integer spatial steps and the derivative ∂*v*/∂*x* at integer spatial steps. Hence, denoting *k* as the spatial index and *i* as the temporal index such that (*x*, *t*) = (*k*Δ_{x}, *i*Δ_{t}), $v((k+{\textstyle \frac{1}{2}}){\Delta}_{x},i{\Delta}_{t})={v}_{k+1/2}^{i}$ and $\partial v/\partial v(k{\Delta}_{x},i{\Delta}_{t})={\partial}_{x}{v}_{k}^{i}$. The numerical algorithm used to generate *v* and *s* was therefore as follows

$\begin{array}{ccc}\hfill {v}_{k+1/2}^{i+1}& =& {v}_{k+1/2}^{i}+\frac{{\Delta}_{t}}{{\tau}_{v}}[\mu -{v}_{k+1/2}^{i}+\frac{{\lambda}^{2}}{{\Delta}_{x}}({\partial}_{x}{v}_{k+1}^{i}-{\partial}_{x}{v}_{k}^{i})+{s}_{k+1/2}^{i}],\hfill \\ \hfill {\partial}_{x}{v}_{k}^{i+1}& =& \frac{{v}_{k+1/2}^{i+1}-{v}_{k-1/2}^{i+1}}{{\Delta}_{x}},\hfill \\ \hfill {s}_{k+1/2}^{i+1}& =& {s}_{k+1/2}^{i}+\frac{{\Delta}_{t}}{{\tau}_{s}}(-{s}_{k+1/2}^{i}+2{\sigma}_{s}\sqrt{\frac{\lambda {\tau}_{s}}{{\Delta}_{x}{\Delta}_{t}}}{\psi}_{k}^{i}),\hfill \end{array}$

where ${\psi}_{k}^{i}$ denotes a zero-mean, unit-variance Gaussian random number. The code used to generate the figures is provided in the supporting information. When the approximation of an infinite or semi-infinite neurite was required, the length Before examining more complex spatial models with multiple dendrites, soma and axon, we first review the subthreshold properties of a single closed dendrite driven by fluctuating, filtered synaptic drive. We then illustrate how the upcrossing method can be applied to spatial models by interpreting the results for the closed dendrite as either a long dendrite with a nominal soma at one end or as two long dendrites meeting at a nominal soma. More complex neuronal geometries are then considered including those with multiple dendrites, axon and an electrically significant soma. The parameter ranges used are given in Table 1.

The dendrites considered here are driven by distributed, filtered synaptic drive. For reasons of analytical transparency, excitatory and inhibitory fluctuations are lumped into a single drive term *s*(*x*, *t*), though it is straightforward to generalize the synaptic fluctuations to two distinct processes. The fluctuating component of the synaptic drive obeys the following equation

${\tau}_{s}\frac{\partial s}{\partial t}=-s+2{\sigma}_{s}\sqrt{\lambda {\tau}_{s}}\phantom{\rule{0.166667em}{0ex}}\xi (x,t)$

parametrized by a filter time constant ${\tau}_{v}\frac{\partial v}{\partial t}=\mu -v+{\lambda}^{2}\frac{{\partial}^{2}v}{\partial {x}^{2}}+s,$

The time constant $C(x,\eta )=\frac{\text{cosh}((L-x)\sqrt{\eta}/\lambda )\text{cosh}(x\sqrt{\eta}/\lambda )}{\sqrt{\eta}\text{sinh}(L\sqrt{\eta}/\lambda )}.$

Hence in terms of this function ${\sigma}_{v}^{2}\left(x\right)=\frac{2{\sigma}_{s}^{2}{\tau}_{s}}{{\tau}_{v}}\left\{C\right(x,1)-C(x,\kappa \left)\right\},$

where ${\sigma}_{\dot{v}}^{2}\left(x\right)=\frac{2{\sigma}_{s}^{2}}{{\tau}_{v}{\tau}_{s}}C(x,\kappa ).$

Note that the second term in the voltage-variance equation, Eq (21), and the variance of the voltage rate-of-change feature a second, shorter length constant $\lambda /\sqrt{\kappa}$ that is a function of the ratio of voltage to synaptic time constants. As expected, Fig 2a shows decreasing λ leading to a lower overall variance as well as a faster decay to the bulk properties from the boundaries. We also see from Note that for the cases where λ/*L* ≪ 1, which is physiologically relevant for the high-conductance state, the influence of the boundary at *L* is negligible at *x* = 0 and at the midpoint there is little influence from either boundary. With this in mind, the morphologies treated in this paper comprise neurites that are treated as semi-infinite in length.

Full analytical solution of the partial differential Eqs (18) and (19) when coupled to the integrate-and-fire mechanism does not appear straightforward, even for the simple closed-dendrite model. However, a level-crossing approach developed by Rice [27] and exploited in many other areas of physics and engineering, such as wireless communication channels [65], sea waves [66], superfluids [67] and grown-surface roughness [68] has previously been applied successfully to compact neuron models [28, 29] and can be extended to spatial models. The method provides an approximation for the mean first-passage time for any Gaussian process in which the mean 〈*v*〉, standard deviation *σ*_{v}, and rate-of-change standard deviation ${\sigma}_{\dot{v}}$ are calculable. The upcrossing rate is the frequency at which the trajectory of *v* without a threshold-reset mechanism crosses *v*_{th} from below (i.e. with $\dot{v}\phantom{\rule{-0.166667em}{0ex}}>\phantom{\rule{-0.166667em}{0ex}}0$). Example voltage-time traces for the model with and without threshold are compared in Fig 1b. This approach provides a good approximation to the rate with reset when the firing events are rare and fluctuation driven, making it applicable to the physiological low-rate firing regime. The upcrossing rate can be derived by considering the rate at which the voltage *v*(*x*, *t*) crosses threshold *v* with a positive “velocity” $\dot{v}$ therefore

${r}_{\text{uc}}={\int}_{0}^{\infty}\text{d}\dot{v}p(\dot{v},v)\dot{v}=p\left(v\right){\int}_{0}^{\infty}\text{d}\dot{v}p\left(\dot{v}\right|v)\dot{v}.$

Then using the fact that in the steady state $\dot{v}$ and ${r}_{\text{uc}}=\frac{1}{2\pi}\frac{{\sigma}_{\dot{v}}}{{\sigma}_{v}}\text{exp}(-\frac{{({v}_{\text{th}}-\u27e8v\u27e9)}^{2}}{2{\sigma}_{v}^{2}})$

where the statistical measures of the voltage are those at the trigger point The method is now applied to a neuron with a single long dendrite and nominal soma (the trigger point *x* = 0 = *x*_{th}) of negligible conductance so that the end can be considered sealed. This corresponds to a section extending from a sealed end of the closed-dendrite model considered above, in the limit that *L* /λ → ∞ (Fig 1a(ii)). The variances have already been calculated for the general case (Eqs (21) and (22)) so for *x*_{th} = 0 we have

$\begin{array}{ccc}\hfill \phantom{\rule{-5.69054pt}{0ex}}{\sigma}_{v}^{2}=\frac{2{\sigma}_{s}^{2}{\tau}_{s}}{{\tau}_{v}}\phantom{\rule{-0.166667em}{0ex}}(1-\sqrt{\frac{{\tau}_{s}}{{\tau}_{s}+{\tau}_{v}}})& \text{and}& {\sigma}_{\dot{v}}^{2}=\frac{2{\sigma}_{s}^{2}}{{\tau}_{s}{\tau}_{v}}\phantom{\rule{-0.166667em}{0ex}}\sqrt{\frac{{\tau}_{s}}{{\tau}_{s}+{\tau}_{v}}}\phantom{\rule{-0.166667em}{0ex}}.\hfill \end{array}$

Substitution of these variances into Eq (24) yields the upcrossing approximation to the firing rate for this geometry.A second interpretation of the closed dendrite model is to place the trigger position in the middle *x*_{th} = *L*/2 and then, in the limit *L* /λ → ∞ consider the halves as two dendrites with statistically identical properties radiating from a nominal soma (Fig 1a(iii)), again with negligible conductance. Taking these limits of the closed dendrite Eqs (21) and (22) for this case generates variances that happen to be exactly half that of the one-dendrite case

$\begin{array}{ccc}\hfill {\sigma}_{v}^{2}=\frac{{\sigma}_{s}^{2}{\tau}_{s}}{{\tau}_{v}}(1-\sqrt{\frac{{\tau}_{s}}{{\tau}_{s}+{\tau}_{v}}}\phantom{\rule{4pt}{0ex}})& \text{and}& {\sigma}_{\dot{v}}^{2}=\frac{{\sigma}_{s}^{2}}{{\tau}_{s}{\tau}_{v}}\sqrt{\frac{{\tau}_{s}}{{\tau}_{s}+{\tau}_{v}}},\hfill \end{array}$

where here we have written the functional dependence of The upcrossing and firing rates as a function of *μ* for the two models are compared in Fig 3, with the deterministic firing rate also shown (this is equivalent to the deterministic rate of the leaky integrate-and-fire model). Note that we keep *τ*_{v} and λ constant across the range of *μ* since these parameters would change little across the range of mean synaptic drive we investigate and it allows us to isolate the dependence of the firing rate on just one parameter. The upcrossing rate provides a good approximation to the full firing rate at low rates in the < 5Hz range (see Fig A for in-depth analysis in terms of dimensionless parameters). In this way the upcrossing rate for spatio-temporal models provides a similar approximation to the firing rate as the Arrhenius form derived by Brunel and Hakim [20] for the white-noise driven point-like leaky integrate-and-fire model.

Compared with the one-dendrite model, we see from Fig 3b and 3d that the firing rate for the two-dendrite model is significantly lower in the subthreshold regime but converges to the same value when *μ* goes above threshold. This illustrates that even simple differences in morphology affect stochastic and deterministic firing very differently. In addition Fig 4a shows that the firing rate is unaffected by the value of λ chosen, confirming by simulation the λ-independence of the firing rate. Furthermore when we choose the same value of *σ*_{v} for the one and two-dendrite models, then both the upcrossing rate and the simulated firing rates are the same, as seen in Fig 4b.

However, despite the independence of λ, the firing-rate profile for this toy model is distinct to that for the point-like leaky integrate-and-fire model, for which the variances are ${\sigma}_{v}^{2}\phantom{\rule{-0.166667em}{0ex}}\propto \phantom{\rule{-0.166667em}{0ex}}{\tau}_{s}/({\tau}_{s}+{\tau}_{v})$ and ${\sigma}_{\dot{v}}^{2}\phantom{\rule{-0.166667em}{0ex}}\propto \phantom{\rule{-0.166667em}{0ex}}1/\left[{\tau}_{s}({\tau}_{s}+{\tau}_{v})\right]$ [29]. This indicates that spatial structure by itself decreases the variance while increases derivative variance by a factor $\sqrt{1+{\tau}_{v}/{\tau}_{s}}$. The variances also differ in their dependence on *τ*_{v} and *τ*_{s} from two-compartmental models [69].

Next, we consider a dendrite connected to an axon at *x*_{1} = 0 = *x*_{α} , as shown in Fig 1a(iv), where dendritic and axonal quantities are denoted by subscripts 1 and *α*, respectively. This differs from the previous two-dendrite model as the axon receives no synaptic drive, so *μ*_{α} = 0 and *s*_{α}(*x*_{α}, *t*) = 0. Furthermore, intrinsic membrane properties of the axon (*τ*_{α}, λ_{α} ) differ from the dendrite due to the smaller axonal radius and lack of synapse-induced increased membrane conductance [11, 12]. Since *μ*_{α} = 0 we omit the subscript on the mean dendritic drive, *μ*_{1} = *μ*. Taking the reasonable assumptions that the per-area capacitance and leak conductance are the same in the axon as the soma, we can calculate *τ*_{α} in terms of *τ*_{1} given the mean level of synaptic drive (see Eqs S39, S41). Unlike the previous models, the mean is no longer homogeneous in space due to the lack of synaptic drive in the axon. Defining ${\tilde{f}}_{1}\left(\omega \right)$ as the input admittance of the dendrite relative to the whole neuron

${\tilde{f}}_{1}\left(\omega \right)=\frac{{G}_{{\lambda}_{1}}{\gamma}_{1}}{{G}_{{\lambda}_{1}}{\gamma}_{1}+{G}_{{\lambda}_{\alpha}}{\gamma}_{\alpha}}=\frac{{g}_{1}^{2}{\lambda}_{1}^{3}{\gamma}_{1}}{{g}_{1}^{2}{\lambda}_{1}^{3}{\gamma}_{1}+{g}_{\alpha}^{2}{\lambda}_{\alpha}^{3}{\gamma}_{\alpha}},$

where ${\gamma}_{j}=\sqrt{1+i\omega {\tau}_{j}}$, we can show that the mean in the axon is given by (see Eqs S13 and S19)
$\u27e8v\left({x}_{\alpha}\right)\u27e9=\mu {e}^{-{x}_{\alpha}/{\lambda}_{\alpha}}{\tilde{f}}_{1}\left(0\right).$

It is important to note that, unlike in the one and two-dendrite models, Eq (28) implies that it is now possible for the neuron to still be in the subthreshold firing regime when First we set the action-potential trigger position at *x*_{th} = 0 and evaluated the effect of the axon by comparing the firing rate for the model with an axon, *r*_{axon}, to the firing rate of the one-dendrite model with a sealed end (∂*v*/∂*x* = 0) at *x* = 0, *r*_{sealed} (effectively an axon with zero conductance load). We also kept the noise amplitude *σ*_{s} rather than the voltage standard deviation *σ*_{v} fixed as we wished to see how the axon changes the variance of fluctuations at the trigger position. The relative firing rate was defined as *r*_{axon}/*r*_{sealed}. The ratio of the axonal to dendritic radius *a*_{α}/*a*_{1} was varied and the relative firing rate calculated, with *a*_{α}/*a*_{1} = 0 being equivalent to no axon present. As illustrated in Fig 5a, the addition of a very low conductance or relatively thin axon significantly reduces the firing rate. This effect arises because the magnitude of ${\tilde{f}}_{1}\left(\omega \right)$ decreases at all frequencies for a larger radius ratio, which can be understood by recalling that ${\lambda}_{j}\propto \sqrt{{a}_{j}}$, Eq (7).

For cortical pyramidal cells, action potentials are typically triggered around *x*_{th} = 30*μ* m down the axon in the axon initial segment [70–72]. It is straightforward to investigate the effect of moving the trigger position down the axon using the upcrossing approach. Interestingly, when *x*_{th} > 0, a non-monotonic relationship between the firing rate and radius ratio *a*_{α}/*a*_{1} became apparent (see Fig 5b), with the peak ratio of ∼0.25 being similar to that between the axonal initial segment and apical dendrite diameter in pyramidal cells [41, 73]. This is caused by a non-monotonic dependence of both 〈*v*〉 and ${\sigma}_{v}^{2}$ on *a*_{α}/*a*_{1} for *x*_{th} > 0 with each peaking at intermediate values. Intuitively, this can be understood from the definition of λ_{α}, which increases as $\sqrt{{a}_{\alpha}}$. Thus the decay length of voltage fluctuations that enter the axon from the dendrite increases, increasing both 〈*v*〉 and ${\sigma}_{v}^{2}$ at *x*_{th}. On the other hand, a larger λ_{α} increases the input conductance of the neuron, which, conversely, decreases 〈*v*〉 and ${\sigma}_{v}^{2}$. For smaller λ_{α} the decay length effect is more significant, whereas for larger λ_{α} the increase in input conductance plays a larger role.

We now consider a case with multiple dendrites and an axon radiating from a nominal soma (Fig 1a(v)). The dendrites are labelled 1, 2, …, *n* with the axon labelled *α* as before. The dendrites have identical properties with independent and equally distributed synaptic drive. As in the previous case with the dendrite and axon, we kept the synaptic strength *σ*_{s} fixed as we changed the number of dendrites. An immediate consequence of multiple dendrites is that, since *μ* > 0 the mean voltage in the axon increases as more dendrites are added, with each contribution summing linearly,

$\u27e8{v}_{\alpha}\left({x}_{\alpha}\right)\u27e9=\sum _{k=1}^{n}\u27e8{v}_{\alpha k}\left({x}_{\alpha}\right)\u27e9,$

where 〈${\tilde{f}}_{n}\left(\omega \right)=\frac{{G}_{{\lambda}_{1}}{\gamma}_{1}}{n{G}_{{\lambda}_{1}}{\gamma}_{1}+{G}_{{\lambda}_{\alpha}}{\gamma}_{\alpha}}=\frac{{g}_{1}^{2}{\lambda}_{1}^{3}{\gamma}_{1}}{n{g}_{1}^{2}{\lambda}_{1}^{3}{\gamma}_{1}+{g}_{\alpha}^{2}{\lambda}_{\alpha}^{3}{\gamma}_{\alpha}},$

it can be shown that when all dendrites have identical mean input drive $\u27e8v\left({x}_{\alpha}\right)\u27e9=n\mu {e}^{-{x}_{\alpha}/{\lambda}_{\alpha}}{\tilde{f}}_{n}\left(0\right).$

Thus we can see that as When it is the fluctuations that contribute significantly for firing (i.e. smaller *μ* or λ_{α}) then a reduction in variance from adding more dendrites will decrease the firing rate; however, when the mean is more significant (larger *μ* or λ_{α} ) then the firing rate will increase as the number of dendrites increases. An example of the former case is shown in Fig 6a for λ_{α} = 100*μ* m, while an example of the latter is seen in Fig 6b for λ_{α} = 150*μ* m. The transition between these regimes can be seen in Fig 6c, which shows how the value of *n* that maximizes the firing rate, *n*_{max}, increases with *μ* and *a*_{α}/*a*_{1}. Physiologically, the reduction in variance is not simply the fact that adding dendrites increases cell size and thus input conductance, but that the relative conductance of each input dendrite to the total conductance decreases. Given that the total input conductance for *n* dendrites and an axon is

${G}_{\text{in}}\left(n\right)=n\left(2\pi {a}_{1}{\lambda}_{1}{g}_{1}\right)+2\pi {a}_{\alpha}{\lambda}_{\alpha}{g}_{\alpha},$

we can test this idea by scaling λ${\tilde{f}}_{n}\left(\omega \right)=\frac{{\textstyle \frac{1}{n}}{g}_{1}^{2}{\lambda}_{1}^{3}(n=1){\gamma}_{1}}{{g}_{1}^{2}{\lambda}_{1}^{3}(n=1){\gamma}_{1}+{g}_{\alpha}^{2}{\lambda}_{\alpha}^{3}{\gamma}_{\alpha}}.$

Since the integrands for the variances are proportional to $|{\tilde{f}}_{n}{\left(\omega \right)|}^{2}$, Eq (S38), this shows that ${\sigma}_{v}^{2}$ and hence the firing rate for fixed λFinally, we also notice better agreement between the upcrossing rate and the simulated firing rate than the infinite dendrite case for the same output firing rates. Intuitively, this is due to the additional filtering from the spatial distance between the dendrite and trigger position along the axon.

We now consider the case illustrated in Fig 1a(vii), where the electrical properties of the soma are non-negligible with its lumped capacitance and conductance providing an additional complex impedance at the point where the axon and dendrites meet. This has the somatic boundary condition we gave earlier in Eq (16) and we recall that the subscript 0 denotes somatic quantities. For simplicity, and as neither section receives synaptic drive in our model, we will let the somatic time constant be the same as the axonal time constant, so *τ*_{0} = *τ*_{α}. Note that somatic drive can be straightforwardly added in this framework, as the variance contribution from the resultant fluctuations would add linearly. This would not qualitatively change the nature of the results we present here that focus on the effects of somatic filtering on transfer of dendritic stimulation to the trigger point in the axon. As the ratio of dendritic to somatic input conductance (*ρ*_{1} , see Materials and methods) tends to infinity, the model without somatic drive converges to the dendrite and axon model with a nominal soma, allowing a clearer comparison between the two models.

For an electrically significant soma the integrand for the variance has the same form as before, Eq (S38), but $\tilde{f}$ now depends on the neurite dominance factor *ρ*,

$\phantom{\rule{1.em}{0ex}}{\tilde{f}}_{n0}\left(\omega \right)=\frac{{G}_{{\lambda}_{1}}{\gamma}_{1}}{{G}_{0}{\gamma}_{0}^{2}+n{G}_{{\lambda}_{1}}{\gamma}_{1}+{G}_{{\lambda}_{\alpha}}{\gamma}_{\alpha}}=\frac{{\rho}_{1}{\gamma}_{1}}{{\gamma}_{0}^{2}+n{\rho}_{1}{\gamma}_{1}+{\rho}_{\alpha}{\gamma}_{\alpha}},\phantom{\rule{1.em}{0ex}}{\gamma}_{0}^{2}=1+i\omega {\tau}_{0}.$

Thus for large Next, we calculated the effect of axonal load on the firing rate when we have an electrically significant soma, extending the results for the case of a nominal soma (Fig 5a). As with the nominal soma case before, we calculated the firing rate at *x*_{th} = 0 with an axon and electrically significant soma, *r*_{axon}, and the firing rate of a dendrite with the same size soma without an axon, *r*_{no axon} (Fig 1a(vi)). For each somatic size, we adjusted *σ*_{s} so that the firing rate for a negligible axon, *a*_{α}/*a*_{1} = 0, was fixed at 1Hz. This was done to account for the soma’s effect on the firing rate we observed earlier and we are thus solely focusing on the effect of the axonal admittance load. As we increase *a*_{α}/*a*_{1} from zero, Fig 7b shows that *r*_{axon}/*r*_{no axon} decreases more rapidly with increasing *a*_{α}/*a*_{1} for larger *ρ*_{1} (smaller soma). This means that, in comparison to Fig 5a, the axonal load had a lower relative effect on the firing rate in the presence of a soma. This is in line with what we should expect by looking at ${\tilde{f}}_{n0}$; lower *ρ*_{1} increases the relative magnitude of ${G}_{0}{\gamma}_{0}^{2}$ in the denominator of ${\tilde{f}}_{n0}$ as compared with the axonal admittance term of ${G}_{{\lambda}_{\alpha}}{\gamma}_{\alpha}$.

Finally, we looked at how an electrically significant soma affects the dependence of the firing rate on the number of dendrites. By varying *ρ*_{1} and the number of dendrites *n* , Fig 7c shows that the non-monotonic dependence of the firing rate on dendritic number *n* is robust in the presence of a soma. Fig 7d illustrates that the number of dendrites that maximizes the firing rate is greater for lower *ρ*_{1} and higher *μ*. We have discussed previously why the value of *n* that maximizes firing increases with *μ* as the increase in mean from additional dendrites becomes more significant for the firing rate. Decreasing *ρ*_{1} increases the value of *n* that maximizes firing because the relative increase in conductance by adding another dendrite is smaller when the fixed somatic conductance is larger.

This study demonstrated how the spatio-temporal fluctuation-driven firing of neurons with dendrites, soma and axon can be approximated using the upcrossing method of Rice [27]. Despite being reduced models of neuronal structures, they provide an analytical description of a rich range of behaviours. For the one and two-dendrite models, the firing rate was shown to be independent of the electrotonic length constant; given that the length constant sets the range over which synaptic drive contributes to voltage fluctuations, this result is surprising. However, a dimensional argument extends this independence to any model in which semi-infinite neurites are joined at a point and share the same λ (any other properties without dimensions of length can be different in each neurite). The level-crossing approach provided a good approximation for the firing rate for these simple dendritic neuron models in the low-rate limit. Beyond this limit, simulations suggest that there is a universal functional form for the firing rate when parametrized by *σ*_{v} that is independent of both λ and the number of dendrites radiating from the nominal soma. This functional form, for coloured noise and in the white-noise limit, merits further mathematical analysis as it is distinct to that of the point-like integrate-and-fire model.

Extending the study to multiple dendrites, we showed that the firing rate depends non-monotonically on their number: adding more dendrites driven by fluctuating synaptic drive can, for a broad parameter range, decrease the fluctuation-driven firing rate. Dendritic structure has been previously shown to influence the firing rate for deterministic input [74, 75]. However, apart from the work of Tuckwell [30–32], analytical studies of stochastic drive in extended neuron models have largely focussed on a single dendrite with drive typically applied at a single point [36, 39] rather than distributed over the dendrite, or as a two-compartmental model [44]. This study demonstrates that in the low-rate regime, the upcrossing approximation allows for the analytical study of spatial models that need not be limited to a single dendrite nor with stochastic synaptic drive confined to a single point, but distributed as is the case *in vivo*.

Including axonal and somatic conductance loads demonstrated their significant effect on the firing rate—even relatively small axonal loads caused a marked reduction. Furthermore, the non-monotonic dependency of the firing rate on dendrite number was also shown to be affected by axonal radius and somatic size, demonstrating that the upcrossing method can be used to examine how structural differences in properties affect the firing rate of complex, composite, spatial neuron models.

An advantage of the level-crossing approach is it can be straightforwardly extended to include a variety of additional biophysical properties affecting neuronal integration of spatio-temporal synaptic drive. An example of this would be the inclusion of non-passive effects arising from voltage-gated currents such as *I*_{h} [76]. For many scenarios, particularly in the high-conductance state [77], the spatio-temporal response can be approximated as quasi-linear, allowing the voltage mean and variances to be calculated via Green’s functions using existing theoretical machinery, such as sum-over-trips on neurons [78–80]. The approach can also be extended to examine the dynamic firing-rate response to weakly modulated drive. This has already been done for point-neurons using the upcrossing method [29, 81, 82] and would only necessitate calculating the linear-response of voltage means and variances in the non-threshold case. However, for significant membrane non-linearities [83, 84] that are not sufficiently mitigated by the high-conductance state [77], the upcrossing framework developed here, predicated on Gaussian voltage fluctuations and linearity, will be inadequate. Non-linear dendritic properties—such as back-propagating action potentials or dendritic sodium spikes—support a broad variety of additional computational functions that cannot be captured by passive or quasi-active models (see [85] for a case in point). Development of a quantitative framework that includes these non-linear properties will be challenging; however, it is hoped that the linear regime considered here will provide a foundation for further work towards that end.

It can be noted that the mathematical constructions used here for the inclusion of space within neuronal structure share similarities to the framework developed for the stochastic neural field [86, 87] that models the spread of activity at the tissue scale. In the context of the neocortex, the spatial voltage variability along the principal apical dendrites of pyramidal cells would be normal to the activity spreading throughout the transverse cortical sheet. A hybrid theory might be considered which includes both these spatial mechanisms and would be an interesting topic for further study.

In summary, the extension of the upcrossing approach to spatially structured neuron models provides an analytical in-road for future studies of the firing properties of extended neuron models driven by spatio-temporal stochastic synaptic drive.

AU Larkman. . Dendritic Morphology of Pyramidal Neurones of the Visual Cortex of the Rat: 111. Spine Distributions.

H Markram, M Toledo-Rodriguez, Y Wang, A Gupta, G Silberberg, C Wu. . Interneurons of the neocortical inhibitory system.

A Larkman, A Mason. . Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. I. Establishment of cell classes.

GR Holt, WR Softky, C Koch, RJ Douglas. . Comparison of discharge variability in vitro and in vivo in cat visual cortex neurons.

G Silberberg, M Bethge, H Markram, K Pawelzik, M Tsodyks. . Dynamics of Population Rate Codes in Ensembles of Neocortical Neurons.

M London, A Roth, L Beeren, M Häusser, PE Latham. . Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex.

B Sengupta, AA Faisal, SB Laughlin, JE Niven. . The Effect of Cell Size and Channel Density on Neuronal Information Encoding and Energy Efficiency.

LJ Bindman, T Meyer, CA Prince. . Comparison of the electrical properties of neocortical neurones in slices in vitro and in the anaesthetized rat.

WR Holmes, CD Woody. . Effects of uniform and non-uniform synaptic ‘activation-distributions’ on the cable properties of modeled cortical pyramidal neurons.

D Paré, E Shink, H Gaudreau, A Destexhe, EJ Lang. . Impact of Spontaneous Synaptic Activity on the Resting Properties of Cat Neocortical Pyramidal Neurons In Vivo.

H Köndgen, C Geisler, S Fusi, XJ Wang, HR Lüscher, M Giugliano. . The dynamical response properties of neocortical neurons to temporally modulated noisy inputs in vitro.

J Doose, G Doron, M Brecht, B Lindner. . Noisy Juxtacellular Stimulation In Vivo Leads to Reliable Spiking and Reveals High-Frequency Coding in Single Neurons.

AN Burkitt. . A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties.

N Brunel, V Hakim. . Fast global oscillations in networks of integrate-and-fire neurons with low firing rates.

B Lindner, L Schimansky-Geier. . Transmission of Noise Coded versus Additive Signals through a Neuronal Ensemble.

N Fourcaud-Trocmé, D Hansel, C Van Vreeswijk, N Brunel. . How spike generation mechanisms determine the neuronal response to fluctuating inputs.

MJE Richardson. . Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive.

N Brunel, S Sergi. . Firing frequency of leaky integrate-and-fire neurons with synaptic current dynamics.

N Brunel, FS Chance, N Fourcaud, LF Abbott. . Effects of synaptic noise and filtering on the frequency response of spiking neurons.

AK Alijani, MJE Richardson. . Rate response of neurons subject to fast or frozen noise: From stochastic and homogeneous to deterministic and heterogeneous populations.

T Tchumatchenko, A Malyshev, T Geisel, M Volgushev, F Wolf. . Correlations and synchrony in threshold neuron models.

HC Tuckwell, JB Walsh. . Random currents through nerve membranes—I. Uniform poisson or white noise current in one-dimensional cables.

HC Tuckwell. . Analytical and simulation results for the stochastic spatial FitzHugh-Nagumo model neuron.

M Rudolph, A Destexhe. . A Fast-Conducting, Stochastic Integrative Mode for Neocortical Neurons *In**Vivo*.

KH Pettersen, H Lindén, T Tetzlaff, GT Einevoll. . Power Laws from Linear Neuronal Cable Theory: Power Spectral Densities of the Soma Potential, Soma Membrane Current and Single-Neuron Contribution to the EEG.

F Aspart, J Ladenbauer, K Obermayer. . Extending Integrate-and-Fire Model Neurons to Account for the Effects of Weak Electric Fields and Input Filtering Mediated by the Dendrite.

J Ladenbauer, K Obermayer. . Weak electric fields promote resonance in neuronal spiking activity: Analytical results from two-compartment cell and network models.

G Eyal, HD Mansvelder, CP de Kock, I Segev. . Dendrites impact the encoding capabilities of the axon.

C O’Donnell, MCW van Rossum. . Spontaneous Action Potentials and Neural Coding in Unmyelinated Axons.

MS Hamada, S Goethals, SID, VriesR Brette, MHP Kole. . Covariation of axon initial segment location and dendritic tree normalizes the somatic action potential.

MWH Remme, M Lengyel, BS Gutkin. . The role of ongoing dendritic oscillations in single-neuron dynamics,

PC Bressloff, S Coombes. . Synchrony in an array of integrate-and-fire neurons with dendritic structure.

J Inglis, D Talay. . Mean-field limit of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component.

A Malyshev, R Goz, JJ LoTurco, M Volgushev. . Advantages and limitations of the use of optogenetic approach in studying fast-scale spike encoding.

L Ferrarese, et al . Dendrite-Specific Amplification of Weak Synaptic Input during Network Activity In Vivo.

JA Tiroshi L and Goldberg. . Population dynamics and entrainment of basal ganglia pacemakers are shaped by their dendritic arbors.

K Koch, J Fuster. . Unit activity in monkey parietal cortex related to haptic perception and temporary memory.

T Hromádka, MR DeWeese, AM Zador. . Sparse Representation of Sounds in the Unanesthetized Auditory Cortex.

G Buzsáki, K Mizuseki. . The log-dynamic brain: how skewed distributions affect network operations.

WR Softky and C Koch. The Highly Irregular Firing of Cortical Cells Is Inconsistent with Temporal Integration of Random EPSPs

MN Shadlen and WT Newsome. The variable discharge of cortical neurons: implications for connectivity, computation, and information coding.

A Compteet al . Temporally irregular mnemonic persistent activity in prefrontal neurons of monkeys during a delayed response task.

A Manwani, C Koch. . Detecting and Estimating Signals in Noisy Cable Structures, I: Neuronal Noise Sources.

G Buzsáki, A Kandel. . Somadendritic Backpropagation of Action Potentials in Cortical Pyramidal Cells of the Awake Rat.

DN Hill, Z Varga, H Jia, B Sakmann, A Konnerth. . Multibranch activity in basal and tuft dendrites during firing of layer 5 cortical neurons in vivo.

L Yang, MS Alouini. . Level crossing rate over multiple independent random processes: An extension of the applicability of the Rice formula.

F Shahbazi, S Sobhanian, MRR Tabar, S Khorram, G Frootan, H Zahed. . Level crossing analysis of growing surfaces.

Y Shu, A Duque, Y Yu, B Haider, DA McCormick. . Properties of action-potential initiation in neocortical pyramidal cells: evidence from whole cell axon recordings.

MHP Kole, et al . Action potential generation requires a high sodium channel density in the axon initial segment.

F Höfflin, et al . Heterogeneity of the Axon Initial Segment in Interneurons and Pyramidal Cells of Rodent Visual Cortex.

ZF Mainen, TJ Sejnowski. . Influence of dendritic structure on firing pattern in model neocortical neurons.

A van Ooyen, J Duijnhouwer, MWH Remme, J van Pelt. . The effect of dendritic topology ion firing patterns in model neurons.

M Puelma Touzel, F Wolf. . Complete Firing-Rate Response of Neurons with Complex Intrinsic Dynamics.

JM Fellous, M Rudolph, A. Destexhe, TJ Sejnowski. Synaptic background noise controls the input/output characteristics of single cells in an in vitro model of in vivo activity.

S Coombes, Y Timofeeva, CM Svensson, GJ Lord, K Josić, SJ Cox, et al . Branching dendrites with resonant membrane: a “sum-over-trips” approach.

L Yihe, Y Timofeeva. . Response functions for electrically coupled neuronal network: a method of local point matching and its applications.

T Tchumatchenko, F Wolf. . Representation of Dynamical Stimuli in Populations of Threshold Neurons.

T Tchumatchenko, C Clopath. . Oscillations emerging from noise-driven steady state in networks with electrical synapses and subthreshold resonance.

G Stuart, N Spruston, B Sakmann, M Hausser. Action potential initiation and backpropagation in neurons of the mammalian CNS.

Górskiet al . Dendritic sodium spikes endow neurons with inverse firing rate response to correlated synaptic activity

Citing articles via

Tweets

https://www.researchpad.co/tools/openurl?pubtype=article&doi=10.1371/journal.pcbi.1007175&title=Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses&author=Robert P. Gowers,Yulia Timofeeva,Magnus J. E. Richardson,Hugues Berry,&keyword=&subject=Research Article,Biology and Life Sciences,Cell Biology,Cellular Types,Animal Cells,Neurons,Neuronal Dendrites,Biology and Life Sciences,Neuroscience,Cellular Neuroscience,Neurons,Neuronal Dendrites,Biology and Life Sciences,Cell Biology,Cellular Types,Animal Cells,Neurons,Nerve Fibers,Axons,Biology and Life Sciences,Neuroscience,Cellular Neuroscience,Neurons,Nerve Fibers,Axons,Biology and Life Sciences,Cell Biology,Cellular Types,Animal Cells,Neurons,Biology and Life Sciences,Neuroscience,Cellular Neuroscience,Neurons,Biology and Life Sciences,Cell Biology,Cellular Types,Animal Cells,Neurons,Neuronal Dendrites,Dendritic Structure,Biology and Life Sciences,Neuroscience,Cellular Neuroscience,Neurons,Neuronal Dendrites,Dendritic Structure,Biology and Life Sciences,Cell Biology,Cellular Types,Animal Cells,Neurons,Neuronal Dendrites,Neurites,Biology and Life Sciences,Neuroscience,Cellular Neuroscience,Neurons,Neuronal Dendrites,Neurites,Biology and Life Sciences,Physiology,Electrophysiology,Membrane Potential,Action Potentials,Medicine and Health Sciences,Physiology,Electrophysiology,Membrane Potential,Action Potentials,Biology and Life Sciences,Physiology,Electrophysiology,Neurophysiology,Action Potentials,Medicine and Health Sciences,Physiology,Electrophysiology,Neurophysiology,Action Potentials,Biology and Life Sciences,Neuroscience,Neurophysiology,Action Potentials,Biology and Life Sciences,Anatomy,Nervous System,Synapses,Medicine and Health Sciences,Anatomy,Nervous System,Synapses,Biology and Life Sciences,Physiology,Electrophysiology,Neurophysiology,Synapses,Medicine and Health Sciences,Physiology,Electrophysiology,Neurophysiology,Synapses,Biology and Life Sciences,Neuroscience,Neurophysiology,Synapses,Biology and Life Sciences,Cell Biology,Cellular Types,Animal Cells,Neurons,Ganglion Cells,Pyramidal Cells,Biology and Life Sciences,Neuroscience,Cellular Neuroscience,Neurons,Ganglion Cells,Pyramidal Cells,

© 2020 Newgen KnowledgeWorks | Privacy & Cookie Policy | Powered by: Nova