Dissecting estimation of conductances in subthreshold regimes

Abstract

We study the influence of subthreshold activity in the estimation of synaptic conductances. It is known that differences between actual conductances and the estimated ones using linear regression methods can be huge in spiking regimes, so caution has been taken to remove spiking activity from experimental data before proceeding to linear estimation. However, not much attention has been paid to the influence of ionic currents active in the non-spiking regime where such linear methods are still profusely used. In this paper, we use conductance-based models to test this influence using several representative mechanisms to induce ionic subthreshold activity. In all the cases, we show that the currents activated during subthreshold activity can lead to significant errors when estimating synaptic conductance linearly. Thus, our results add a new warning message when extracting conductance traces from intracellular recordings and the conclusions concerning neuronal activity that can be drawn from them. Additionally, we present, as a proof of concept, an alternative method that takes into account the main nonlinear effects of specific ionic subthreshold currents. This method, based on the quadratization of the subthreshold dynamics, allows us to reduce the relative errors of the estimated conductances by more than one order of magnitude. In experimental conditions, under appropriate fitting to canonical models, it could be useful to obtain better estimations as well even under the presence of noise.

Appendices

Appendix A: Mathematical model of the piramidal cell

The first model we use is the one given in Wang (1998), which describes the behaviour of a neuron that has two compartments, the dendrite and the soma plus the axonal initial segment. In our study we only consider the somatic compartment:

$$C_{m} \frac{d V}{d t} = -I_{L} -I_{Na}-I_{K} - I_{Ca} - I_{AHP} + I_{app} - I_{syn}, $$

where C _m is the capacitance, I _{s y n} the synaptic current, I _{a p p} the applied current and I _L , I _{i o n} the leak and the respective ion currents which are described by equations

$$\begin{array}{@{}rcl@{}} &&I_{L} = g_{L} (v-V_{L}), \\ &&I_{Na} = g_{Na} m_{\infty}^{3} (v) h (v-V_{Na}), \\ &&I_{K} = g_{K} n^{4}(v-V_{K}), \\ &&I_{Ca} = g_{Ca}ml_{\infty}(v-V_{Ca}), \\ &&I_{AHP} = g_{AHP}\frac{c}{c + K_{D}}(v-V_{K}), \\ &&I_{LTS}=g_{LTS} m_{LTS,\infty}^{3} h_{LTS} (v-V_{Ca}), \end{array} $$

where V _{i o n} and g _{i o n} represent the specific ion reversal potentials and maximal conductances, respectively, c is the intracellular calcium concentration [C a ²⁺] and K _D represents a growth factor of the I _{A H P} current. The variables h and n are gating variables governed by first-order kinetics of type

$$ \dot{w}= \frac{dw}{dt}= \phi [\alpha_{w}(v)(1-w)-\beta_{w} (v) w] = \phi \frac{w_{\infty} (v) - w}{\tau_{w}(v)} $$

(12)

The m-type variables are considered to be at the steady-state m l = m l _∞ (v), m = m _∞ (v). More precisely, the functions describing the gating dynamics are given by:

$$\begin{array}{@{}rcl@{}} &&w_{\infty} (v) = \alpha_{w} (v)/(\alpha_{w} (v) + \beta_{w} (v)), \\ &&\tau_{w} (v) = 1/(\alpha_{w} (v) + \beta_{w} (v)), \\ &&\alpha_{h} (v) = 0.07\, \exp(-(v+50)/10),\\ &&\beta_{h} (v) = 1/(1+ \exp(-0.1(v+20))), \\ &&\alpha_{n} (v) = -0.01\, (v+34)/(\exp(-0.1(v+34))-1) ,\\ &&\beta_{n} (v) = 0.125\, \exp(-(v+44)/25), \\ &&\alpha_{m} (v) = -0.1\, (v+33)/(\exp(-0.1(v+33))-1),\\ &&\beta_{m} (v) = 4 \, \exp(-(v+58)/12),\\ &&ml_{\infty} (v) = 1/(1+\exp(-(v+20)/5)) \end{array} $$

and, for the LTS current,

$$\begin{array}{l} m_{LTS, \infty}(v) = 1/(1+\exp(-(v+65)/7.8)),\\ dh_{LTS}/dt = (\phi_{LTS} (h_{LTS, \infty} (v) - h_{LTS} (v)))/\tau_{LTS} (v), \end{array} $$

where

$$\begin{array}{l} h_{LTS, \infty}(v) = 1/(1+\exp((v+81)/11)), \\ \tau_{LTS}(v) = h_{LTS, \infty}(v) \exp((v+162.3)/17.8). \end{array} $$

The intracellular calcium concentration c = [C a ²⁺] is assumed to be governed by a leaky-integrator

$$ dc/dt = - \alpha I_{Ca} - c/\tau_{Ca}, $$

(13)

where τ _{C a} is the time constant and α is proportional to the membrane area divided by the volume below the membrane.

The biophysical parameters are:

$$\begin{array}{l} \mathrm{Conductances\ } (\text{mS}/\text{cm}^{2}): g_{L} = 0.1,\ g_{Na}=45,\\ \qquad g_{K} = 18,\ g_{Ca} = 1.0,\ g_{AHP} = 5.0,\ g_{LTS} = 0.5; \\ \mathrm{Reversal\ potentials\ } (\text{mV}): V_{L} = -65,\ V_{Na}=55,\\ \qquad V_{K} = -80,\ V_{Ca} = 120.0,\ V_{E} = 0,\ V_{I} = -80;\\ \mathrm{Capacitance\ } (\mu\text{F}/\text{cm}^{2}): C_{m} = 1; \\ \text{Non-dimensional\ constants\ }: \phi=4,\ \phi_{LTS} = 2; \\ \mathrm{Other\ constants\ }: \alpha = 0.002 \,\mu\text{M}(\text{ms}\, \mu \text{A})^{-1} \text{cm}^{2}, \\ \qquad \tau_{Ca} = 80~\text{ms},\ K_{D} = 30.0~\mu\text{M}. \end{array} $$

Finally, we take I _{a p p} ∈ [−1, 1] μA/cm² (see Appendix B in Guillamon et al. 2006 for a justification of this choice).

Appendix B: Mathematical model of the stellate cell

The second model we use is the one given in Rotstein et al. (2006) by considering only the persistent sodium current (I _{N a P} ) and a fast-component h-current (I _h ), so the spiking currents are supposed inactivated. Except for those ionic currents, the remainder parameters of the model follow the same equations given in Appendix A. Then, the ionic currents are described as

$$\begin{array}{@{}rcl@{}} &&I_{NaP}=g_{p} p_{\infty}(v) (v-V_{Na})\\ &&I_{h}=g_{h} r_{f}(v) (v-V_{h}) \end{array} $$

where g _p and g _h are the maximal conductances, V _{N a} and V _h the reversal potentials, and p _∞ and r _f are the gating variables, all of them for the persistent sodium current and the h-current, respectively.

Note that the gating variable of I _{N a P} has been approximated, since it is evolving a fast time scale, by the adiabatic approximation p(v) = p _∞ (v). On the other hand, the gating variable r _f is supposed to be governed, as in Appendix A, by first-order kinetics of type (12) where ϕ = 1. The functions defining p _∞ , r _{f, ∞} and $\tau _{r_{f}}$ are, respectively,

$$\begin{array}{@{}rcl@{}} &&p_{\infty} (v) = 1/(1 + \exp(-(v+38)/6.5)), \\ &&r_{f, \infty} (v) =1/(1 + \exp((v+79.2)/9.78)), \\ &&\tau_{r_{f}} (v) = 0.51/(\exp((v-1.7)/10) + \exp(-(v+340)/52))) + 1, \end{array} $$

The biophysical parameters through this model are:

$$\begin{array}{l} \mathrm{Conductances\ } (\text{mS}/\text{cm}^{2}): g_{L} = 0.5,\ g_{p}=0.5,\ g_{h}=1.5; \\ \mathrm{Reversal\ potentials\ } (\text{mV}): V_{L} = -65,\ V_{Na}=55, \\ \qquad V_{h} =-20,\ V_{E} = 0,\ V_{I} = -80;\\ \mathrm{Capacitance\ } (\mu \text{F}/\text{cm}^{2}): C_{m} = 1; \\ \text{Non-dimensional\ constants\ }: \phi=1. \end{array} $$

Finally, we take I _{a p p} ∈ [−4, −3] μA/cm².