The robustness of phase-locking in neurons with dendro-dendritic electrical coupling

Abstract

We examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance ($g_c$) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of $g_c$ are calculated from the experimentally measured coupling coefficient ($CC$) based on a single-compartment description of a neuron, and therefore may be severe underestimates of $g_c$. With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of $CC$, which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.

Appendices

Appendix A: Somatic model and system parameters

We used a model of a cortical inhibitory fast spiking interneuron due to Erisir et al. (1999) to describe our somatic dynamics. The model equations and parameters are

$$\begin{aligned} C_m \frac{d v}{d t}&= - g_{Na}m^3 h (v(t)-E_{Na}) - g_K n^2(v(t)-E_K)-g_{K_s} n_s^4(v(t)-E_K)\\&\quad -g_L(v(t)-E_L) +I \\ \frac{dm}{dt}&= \alpha _m(v)(1-m)-\beta _m(v)m\\ \frac{dh}{dt}&= \alpha _h(v)(1-h)-\beta _h(v)h\\ \frac{dn}{dt}&= \alpha _n(v)(1-n)-\beta _n(v)n\\ \frac{dn_s}{dt}&= \alpha _{n_s}(v)(1-n_s)-\beta _{n_s}(v)n_s \end{aligned}$$

where

$$\begin{aligned} \alpha _m(v)&= \frac{40.0 (75.5-v)}{\exp ((75.0-v)/(13.5))-1.0} \qquad \beta _m(v)=\frac{1.2262}{\exp (v/42.248)}\\ \alpha _h(v)&= \frac{0.0035}{\exp (v/24.186)} \qquad \qquad \qquad \beta _h(v)=\frac{-0.017 (51.25+v)}{\exp (-(52.25+v)/5.2)-1.0}\\ \alpha _n(v)&= \frac{(95.0-v)}{\exp ((95.0-v)/11.8)-1.0} \qquad \beta _n(v)=\frac{0.025}{\exp (v/22.22)}\\ \alpha _{n_s}(v)&= \frac{-0.014 (44.0+v)}{\exp (-(44.0+v)/2.3)-1.0} \qquad \beta _{n_s}(v)=\frac{0.0043}{\exp ((44.0+v)/34.0)} \end{aligned}$$

and

$$\begin{aligned} C_m&= 1.0 \upmu \mathrm{F/cm}^2 \quad g_{Na}=112.5~\mathrm{mS/cm}^2 \quad g_K = 225 mS/cm^2\\ g_{K_s}&= 0.225~\mathrm{mS/cm}^2 \quad g_L= 0.25~\mathrm{mS/cm}^2 \quad E_{Na}= 74.0~\mathrm{mV}\\ E_{K}&= -90.0~\mathrm{mV} \quad E_{L}= -70.0 \mathrm{mV} \end{aligned}$$

The cable parameters we hold constant are

$$\begin{aligned} g_{LD} = 0.2~~\mathrm{mS/cm}^2 \quad d = 0.002~\mathrm{cm} \quad R_i = 0.1~\mathrm{k}\Omega \,\mathrm{cm}\\ \end{aligned}$$

Appendix B: Coupling coefficient

Here we derive the directional coupling coefficient for two electrically coupled ball-and-stick neurons with heterogeneity in dendritic length. The equations are given by

$$\begin{aligned}&C_m\frac{\partial \bar{v}_j}{\partial t} = \frac{a}{2R_i} \frac{\partial ^2 \bar{v}_j}{\partial x^2} -g_{LD}(\bar{v}_j-E_{LD})\\&C_m \frac{\partial \bar{v}_j}{\partial t}(0,t) = - g_L(\bar{v}_j(0,t)-E_L)+ I +\frac{ a^2}{d^2 R_i} \frac{\partial \bar{v}_j}{\partial x} (0,t)\\&\frac{\pi a^2}{R_i} \frac{\partial \bar{v}_j}{\partial x} (L_j,t) = g_c (\bar{v}_k(L_k,t) - \bar{v}_j(L_j,t)). \end{aligned}$$

If a current $I$ is applied to the soma of cell $j$, the steady-state equations are then

$$\begin{aligned}&\frac{d^2 v_j}{d x^2} = \frac{2R_ig_{LD}}{a} (v_j-E_{LD})=\frac{1}{\lambda ^2} (v_j-E_{LD})\\&\frac{ a^2}{d^2 R_i} \frac{d v_j}{d x} (0)= g_L(v_j(0)-E_L)- I \\&\frac{d v_j}{d x} (L_j) = \frac{g_c R_i}{\pi a^2} (v_k(L_k) - v_j(L_j)), \end{aligned}$$

where $v_j(x)=\bar{v}_j(x,\infty ), \ j,k =1,2; j \ne k$. The solutions of the above system are given by

$$\begin{aligned}&v_j(x)=A_j \sinh \left(\frac{x}{\lambda }\right)+B_j \cosh \left(\frac{x}{\lambda }\right)+E_{LD}\\&v_k(x)=A_k \sinh \left(\frac{x}{\lambda }\right)+B_k \cosh \left(\frac{x}{\lambda }\right)+E_{LD}, \end{aligned}$$

where

$$\begin{aligned}&B_j=\frac{g_{LD}}{g_L}\varepsilon A_j + F +\frac{I}{g_L} \\&B_k=\frac{g_{LD}}{g_L}\varepsilon A_k + F\\&A_j = \frac{g}{\lambda } \frac{\gamma _k}{\alpha _j \alpha _k-\left(\frac{g}{\lambda }\right)^2 \gamma _k \gamma _j}\Bigg (\frac{g}{\lambda } \bigg [F+\frac{I}{g_L}\bigg ]\cosh \left(\frac{L_j}{\lambda }\right) - F\beta _k \Bigg ) \nonumber \\&\qquad +\frac{\alpha _k}{\alpha _j \alpha _k-\left(\frac{g}{\lambda }\right)^2 \gamma _k \gamma _j}\Bigg (\frac{g}{\lambda }F \cosh \left(\frac{L_k}{\lambda }\right) - \bigg [F+\frac{I}{g_L}\bigg ]\beta _j \Bigg ) \\&A_k = \frac{g}{\lambda } \frac{\gamma _j}{\alpha _j \alpha _k-\left(\frac{g}{\lambda }\right)^2 \gamma _k \gamma _j}\Bigg (\frac{g}{\lambda }F \cosh \left(\frac{L_k}{\lambda }\right) - \bigg [F+\frac{I}{g_L}\bigg ]\beta _j \Bigg )\nonumber \\&\qquad +\frac{\alpha _j}{\alpha _j \alpha _k-\left(\frac{g}{\lambda }\right)^2 \gamma _k \gamma _j}\Bigg (\frac{g}{\lambda } \bigg [F+\frac{I}{g_L}\bigg ]\cosh \left(\frac{L_j}{\lambda }\right) - F\beta _k\Bigg ), \end{aligned}$$

and

$$\begin{aligned}&\alpha _j=\bigg [\frac{1}{\lambda } + \frac{g_{LD}}{g_L} \frac{\varepsilon g}{\lambda } \bigg ]\cosh \left(\frac{L_j}{\lambda }\right)+\bigg [\frac{g}{\lambda } + \frac{g_{LD}}{g_L} \frac{\varepsilon }{\lambda } \bigg ]\sinh \left(\frac{L_j}{\lambda }\right)\\&\beta _j=\frac{1}{\lambda }\sinh \left(\frac{L_j}{\lambda }\right)+\frac{g}{\lambda }\cosh \left(\frac{L_j}{\lambda }\right)\\&\gamma _j=\sinh \left(\frac{L_j}{\lambda }\right)+\frac{g_{LD}}{g_L}\varepsilon \cosh \left(\frac{L_j}{\lambda }\right)\\&F= E_L-E_{LD}\\&g = \frac{g_c R_i \lambda }{\pi a^2}\\&\varepsilon =\frac{a^2}{d^2 R_i g_{LD} \lambda }, \end{aligned}$$

and $\alpha _k$, $\beta _k$, and $\gamma _k$ are defined similarly. The coupling coefficient is then $\frac{\Delta v_k(0)}{\Delta v_j(0)}$, i.e., the difference in the steady state potentials for the two somata when current $I$ is injected into the soma of cell $j$ and no current is injected into either cell. Let the coefficients $\tilde{A}_{j}, \tilde{A}_{k}, \tilde{B}_{j}$ and $\tilde{B}_{k}$ correspond to the steady state potentials when no current is injected into either cell (i.e., set $I=0$ in the equations for ${A}_{j}, {A}_{k}, {B}_{j}$ and ${B}_{k}$). Then the directional coupling coefficient that measures the DC attenuation from cell $j$ to cell $k$ is

$$\begin{aligned} CC_k&= \frac{\Delta v_k(0)}{\Delta v_j(0)} = \frac{B_k-\tilde{B}_k}{B_j-\tilde{B}_j}\nonumber \\&= \frac{\frac{g_{LD}}{g_L}\varepsilon \frac{g}{\lambda } \rho \left(-\gamma _j \beta _j+\alpha _j\cosh \left(\frac{L_j}{\lambda }\right)\right)}{\frac{g_{LD}}{g_L}\varepsilon \rho \left(\left(\frac{g}{\lambda }\right)^2\gamma _k \cosh \left(\frac{L_j}{\lambda }\right)-\alpha _k\beta _j\right) +1 }, \end{aligned}$$

where

$$\begin{aligned} \rho =\frac{1}{\alpha _k \alpha _j -\left(\frac{g}{\lambda }\right)^2\gamma _k \gamma _j}. \end{aligned}$$

If $L_j=L_k=L$, then we arrive at the coupling coefficent for two identical ball-and-stick neurons

$$\begin{aligned} CC=\frac{2\varepsilon g_{LD} g}{g_L+\sinh \left(2\frac{L}{\lambda }\right)[\varepsilon g_{LD}+2g g_L] + \cosh \left(2\frac{L}{\lambda }\right)[2\varepsilon g g_{LD}+ g_L]}. \end{aligned}$$

Furthermore, if $\frac{L}{\lambda }=0$, we arrive at

$$\begin{aligned} CC&= \frac{\varepsilon g_{LD} g }{g_L+\varepsilon g_{LD} g}\\&= \frac{g_c}{\pi d^2 g_L +g_c}, \end{aligned}$$

which is the coupling coefficient assuming two coupled single-compartment neurons Bennett (1977).