Linearization of excitatory synaptic integration at no extra cost

Abstract

In many theories of neural computation, linearly summed synaptic activation is a pervasive assumption for the computations performed by individual neurons. Indeed, for certain nominally optimal models, linear summation is required. However, the biophysical mechanisms needed to produce linear summation may add to the energy-cost of neural processing. Thus, the benefits provided by linear summation may be outweighed by the energy-costs. Using voltage-gated conductances in a relatively simple neuron model, this paper quantifies the cost of linearizing dendritically localized synaptic activation. Different combinations of voltage-gated conductances were examined, and many are found to produce linearization; here, four of these models are presented. Comparing the energy-costs to a purely passive model, reveals minimal or even no additional costs in some cases.

Appendices

Appendix: A

The passive model is supplemented by VGCs. Our interest is in the synaptic integration process so we focus our attention on the physiological subthreshold membrane potential range that lies between resting potential and a nominal, action potential initiation threshold. In the case of pyramidal neurons and for this voltage-range, the A-type potassium (g _{K A} ), the persistent sodium (g _NaP ), and the hyperpolarization-activated mixed cation (g _h ) conductances are the most relevant. Previous work demonstrated that, in a steady-state analysis, individual active conductances can adjust to linearity the amplitude of individual post-synaptic potentials over limited voltage-ranges (Morel and Levy 2007, 2009).

A-type potassium conductance (g _{K A} )

Two sets of activation and inactivation curves were considered. One parameterization is referred to as K_A(H) and was obtained from Hoffman et al. (1997) data from CA1 dendrites less than 100 μm from soma. The distal parameters from the same authors did not produce significantly different results and so are not reported here. The other parameterization is referred to as K_A(B) and was obtained from Bekkers (2000) data from neo-cortex layer V cells. These experimentally obtained activation and inactivation curves are fitted with single Boltzmann functions in order to obtain the voltage-dependence of each potassium conductance g_KA(B) and g_KA(H). The reversal potential for all g _{K A} conductances is the potassium Nernst potential, E _K , assumed to be -95 mV. The equations governing this conductance are,

$$ I_{KAx}=g_{KAx}(V_{m}) \cdot (V_{m}-E_{K}) $$

(4)

$$ g_{KAx}=\bar{g}_{KAx} \cdot n \cdot l $$

(5)

$$ \frac{dn}{dt} = \frac{n_{inf}-n}{\tau_{n}} $$

(6)

$$ \frac{dl}{dt}=\frac{l_{inf}-l}{\tau_{l}} $$

(7)

$$ n_{inf}=\frac{1}{1+exp[(V^{n}_{1/2}-V_{m})/k_{n}]} $$

(8)

$$ l_{inf}=\frac{1}{1+exp[(V^{l}_{1/2}-V_{m})/k_{l}]} $$

(9)

Values for the Bekkers parametarization K_A(B) are as follows: half-maximum activation voltage $V_{1/2}^{n}= -24.5$ mV; half-maximum inactivation voltage $V_{1/2}^{l}= -72.3$ mV; activation slope factor k _n = + 16.9 mV; and inactivation slope factor k _l = − 5.9 mV.

Values for the Hoffman parametarization K_A(H) are as follows: half-maximum activation voltage $V_{1/2}^{n}= + 11$ mV; half-maximum inactivation voltage V1/2l = − 56 mV; activation slope factor k _n = + 18 mV; and inactivation slope factor k _l = − 8 mV.

The activation and inactivation time constants for both parameterizations are τ _n = 1 ms and τ _l = 5 ms respectively and are from Hoffman et al. (1997).

In the above, $\bar {g}_{KAx}$ is given in terms of the ion channel density (S/cm²), while parameters given in the Results section are the corresponding maximal available conductance (nS).

Persistent sodium conductance (g _NaP )

A set of activation and inactivation curves, and associated time constants τ _m and τ _q , were obtained from Magistretti and Alonso (1999) and Agrawal et al. (2001). This set is referred to as the NaP parameterization and is used to model this conductance. The experimentally obtained activation curve, fitted with a single Boltzmann function (V1/2m = − 37.6 mV and slope factor k _m = − 7.4 mV), was extracted from Agrawal et al. (2001) Fig. 4. The experimental inactivation curve, also fitted with a single Boltzmann function (V1/2q = − 48.8 mV and slope factor k _q = 10 mV), is from Magistretti and Alonso (1999) Fig. 4. The inactivation time constant, τ _q = 6000 ms is taken to be voltage-independent and is from the − 55 mV value of Magistretti and Alonso (1999) Fig. 8a. According to the same authors, the activation is almost instantaneous so τ _m is set equal to NEURON’s integration time step dt = 0.025 msec. These curves are then combined with maximal conductance $\bar {g}_{\textit {NaP}}$ to obtain voltage-dependent conductance g _NaP . The reversal potential for this conductance is the sodium Nernst potential, E _{N a} , assumed to be + 55 mV. The equations governing this conductance are,

$$ I_{NaP}=g_{NaP}(V_{m})(V_{m}-E_{Na}) $$

(10)

$$ g_{NaP}=\bar{g}_{NaP} \cdot m \cdot q $$

(11)

$$ \frac{dm}{dt}=\frac{m_{inf}-m}{\tau_{m}} $$

(12)

$$ \frac{dq}{dt}=\frac{q_{inf}-q}{\tau_{q}} $$

(13)

$$ m_{inf}=\frac{1}{1+exp[(V_{1/2}^{m}-V_{m})/k_{m}]} $$

(14)

$$ q_{inf}=\frac{1}{1+exp[(V_{1/2}^{q}-V_{m})/k_{q}]} $$

(15)

Hyperpolarization-activated mixed-cation conductance (g _h )

Parameters for this non-inactivating, slowly deactivating, mixed-cation inward current (I _h ) were obtained from the work of Magee (1998) on dendrites of CA1 pyramidal cells. The conductance g _h is the product of the steady-state activation curve (Boltzmann fit to experimental data with V_1/2 = − 82 mV and k = 8.8 mV) with the maximum available conductance $\bar {g}_h$. The reversal potential for this channel, E _h , is + 1 mV as per (Magee 1998) Table 1, and the activation time constant, τ _h , is reported as 17 ms. The equations governing this conductance are,

$$ I_{h}=g_{h}(V_{m})(V_{m}-E_{h}) $$

(16)

$$ g_{h}=\bar{g}_{h} \cdot p $$

(17)

$$ \frac{dp}{dt}=\frac{p_{inf}-p}{\tau_{h}} $$

(18)

$$ p_{inf}=\frac{1}{1+exp[(V_{1/2}-V_{m})/k]} $$

(19)

Appendix: B

Table 4 shows the maximum available conductance $\bar {g}$ for the VGCs. Passive conductance g _{l e a k} = 1.03 nS is common to all models. After analyzing a range of excitatory synaptic conductances (g _{s y n} range), synaptic conductances required for each model to reach − 50 mV from rest within the different time intervals are given (g _{s y n} , g _{i n h s y n} ). The 80 msec models correspond to those in Table 3. The depolarization (ΔV _{s o m a} ) per 100 pS in the range of constant slope is reported. The coefficient of determination R² for all the models is greater than 0.98. See Methods for explanations of synaptic activity and additional conductance to maintain rest. For the passive model, the g _{s y n} (nS), g _{i n h s y n} (nS) are 1.00 and 0.69 at 80 msec, 1.60, 0.95 at 40 msec, 2.96, 1.33 at 20 msec, and 6.27, 1.76 at 10 msec.

Note that for four of these models, the total ratio of Na/K currents are very close to the 3:2 Na-K ATPase pump ratio (e.g. 3:2 is equivalent to 1.5:1, which is very close to the ratio of 1.4 obtained by three of the models). Refer to Table 6.

Table 6 Ratios of sodium to potassium ions for selected models

Appendix: C

Figure 6 shows an example of the time evolution of the somatic membrane potential for one of the NaP-enhanced models presented above. This NaP model has levels of dendritic conductances such that threshold is reached in 40 ms when g _{s y n} = 1.26 nS and g _{i n h s y n} = 0.82 nS. See Table 4 for additional details. Not all models depolarize in the same fashion.

Fig. 6

Somatic membrane potential as a function of time for one of the NaP models used in this work. The dendritic conductances are such that the model depolarizes from rest to threshold in 40 ms when g _{s y n} = 1.26 nS and g _{i n h s y n} = 0.82 nS. See Table 4 for details

Appendix:: D

Equation (1) shows a non-linear relationship between inhibitory and excitatory conductances. However, once the changes in driving force with membrane potential are considered, the inhibitory and excitatory currents can be seen to increase proportionally over the voltage range relevant to this study. Figure 7 shows an example of time evolution of the synaptic currents over a typical time interval. During that same time interval, the membrane potential changes from rest to threshold and thus the inhibitory driving force is increasing while the excitatory driving force is decreasing. Equation (1) insures that the inhibitory current does not overwhelm excitation over the voltage range under consideration.

Fig. 7

Time evolution of synaptic currents as the membrane potential depolarizes from rest to a nominal threshold in the 40-ms NaP model. Inhibitory synaptic current is the black line and the excitatory current is the gray line. Currents are seen to increase proportionally with each other. Here g _{s y n} = 1.27 nS and g _{i n h s y n} = 0.82 nS

Excitatory and inhibitory synapses turn on at the same time and their respective effects are combined. As a consequence, increasing the amount of inhibition increases the required amount of active excitatory synaptic conductance. The models exhibit a certain level of robustness across different levels of inhibition in that linearization can still be observed. Figure 8 demonstrates this characteristic by showing that changes in the amount of inhibition have only a small impact on the linear range or the depolarization per synapse. Note that the vertical axis has been truncated to accentuate the distinction between lines.

Fig. 8

The models exhibit a certain level of robustness across different levels of inhibition. Shown here is the impact of changing the amount of inhibition on linearization in the 80-ms NaP-alone model. The solid thin black line represents the lowest amount of inhibition (scaling constant of 2 in Eq. (1)) while the bottom heavy solid black line represents the level of inhibition used in this work. VGC levels have been adjusted to maintain the same quality of linear range. Note that the vertical scale has been truncated to accentuate the distinction between lines

Appendix: E

As stated in the Discussion, there are several ways to produce linearized synaptic integration using VGCs. Evolution through natural selection is expected to have some criteria for selecting one combination of VGCs over another. This work focuses on one particularly important criterion: the metabolic cost. However, the presence of VGCs must also impact the time constant of the neuron. This is explored in Table 7, which shows the time constant and metabolic cost for each of the models presented in this study.

The modeling shows that there is no simple relationship between time constant and cost. There is a general tendency for faster neurons to cost less, but this certainly has exceptions. Table 7 allows comparison between models but does not give clear indications that one model would be preferential when time constants are considered. All costs shown are total metabolic costs (also shown in Table 5). All values of g _{s y n} are as shown in Table 4.

Table 7 Metabolic cost and time constant for all models

Although not the focus of the present work, some of the models presented here display another type of linearity. Our interest in linearity is motivated by interpulse interval coding at the spike generator, which here is equivalent to the soma. Applying the analysis of Singh and Levy (2017), which is based on a more complex model neuron, Fig. 9 shows the inverse of the time interval needed for the somatic potential to go from rest to threshold against the level of excitatory synaptic activity. Two models are shown, the passive model (square markers) and the NaP-alone model that reaches threshold in 40 ms when g _{s y n} = 1.26 nS (round markers). See Table 4 for details of the NaP model.

Figure 9 shows, over durations ranging from 18 ms to 346 ms, one kind of temporal linearity. Specifically, there is a linearity when inverse time is plotted as an function of increasing synaptic excitation (Singh and Levy 2017), a desirable property of interpulse interval codes (Levy et al. 2016). The addition of VGCs enhances the range of excitation which approximates this inverse linear effect. Linear fits to the models show a higher value of R² for the NaP model data (0.9990 for NaP vs. 0.9918 for passive).

Fig. 9

VGCs extend the linear relationship between inverse time to reach nominal threshold as a function of intensity of synaptic activation. The models shown are the same NaP-alone model (round markers) and passive model (square markers) used in this study. Linear fits to the data and associated values of R² are also shown. The NaP model has levels of dendritic conductances such that it will reach threshold from rest in 40 ms when g _{s y n} = 1.27 nS and g _{i n h s y n} = 0.82 nS