A Trade-Off Between Dendritic Democracy and Independence in Neurons with Intrinsic Subthreshold Membrane Potential Oscillations

Abstract

Membrane potential oscillations are ubiquitous in neurons and have been proposed to underly important neuronal computations. As a paradigmatic example, the periodic spatial tuning of stellate cells from medial entorhinal cortex neurons is thought to be generated by the interference patterns arising from multiple, independent dendritic oscillators, each controlled by direction-selective input. We analyzed how multiple dendritic oscillators embedded in the same neuron integrate inputs separately and determine somatic membrane voltage jointly. We found that the interaction of dendritic oscillations leads to phase locking, which sets an upper limit on the time scale for independent input integration. Factors that increase this time scale also decrease the influence that the oscillations exert on somatic voltage. In stellate cells, inter-dendritic coupling dominates and causes these cells to act as single oscillators. Our results suggest a fundamental trade-off between local and global processing in dendritic trees integrating ongoing signals.

Appendix

1.1 A.1 Derivation of Phase Locking Time Constant and Stable Phase-Locked Solution

We performed a mathematical analysis to determine the dynamics of two subthreshold dendritic oscillators. We determined how fast phase locking of dendritic oscillators occurs as a function of the oscillator properties and the properties of the membrane segment connecting the oscillators, and what is the phase difference ϕ between the oscillators in the stable phase-locked solution. Consider a system of two identical oscillators with natural frequency f (in Hz) that are coupled via a cable of length l (in cm), with oscillator i = 1,2 located at x = 0 and x = l, respectively (see Figs. 21.2a and 21.4a). The membrane potential V _i (t) (in millivolts) of each dendritic oscillator is described by a sinusoidal function:

$$ {V}_{i}(t)={\overline{V}}_{\text{dend}}\mathrm{cos}(\omega(t+{\theta}_{i}))+{V}_{\text{R}},$$

(21.1)

with dendritic oscillator amplitude ${\overline{V}}_{\text{dend}}$ (in millivolts), angular frequency ω = 2π f, phase shift θ _i , and resting membrane potential V _R. We consider the oscillators are weakly coupled (i.e., the interactions only affect the oscillators’ phases). We can then write the changes in the phase shifts of the oscillators as

$$ {\dot{\theta}}_{i}={\varepsilon}Z(t){p}_{i}(t),$$

(21.2)

where the positive parameter ε ≪ 1, and Z(t) is the infinitesimal PRC which is assumed to be identical for both oscillators. It describes the change of the oscillator’s phase shift in response to an infinitesimally small and short perturbation at a particular phase (Izhikevich, 2007). Here we consider $Z(t)=-Q\mathrm{sin}(\omega t)$, where Q is the amplitude of the PRC (in seconds per millivolt). Note that when $Q=1/\omega{\overline{V}}_{\text{dend}}$ we obtain the PRC of Andronov–Hopf oscillators, the minimal dynamical system to produce the sinusoidal limit cycle oscillations in (21.1). The perturbations p _i (t) result from the axial currents that flow between the cable and oscillator i. The passive properties of the cable are determined by membrane time constant τ (in seconds), leak reversal potential E _L (in millivolts), and length constant λ (in centimeters), giving the cable an electrotonic length $L=l/\lambda$. The cable also expresses a voltage-dependent conductance with reversal potential E _m. The dynamics of this conductance are determined by a single gating variable m(x,t) with activation function m _∞ (V ) and time constant τ _m(V ). The equations governing the membrane potential V (x,t) and gating variable m(x,t) along the cable are

$$ \begin{array}{ll}{\tau}\frac{\partial }{\partial{t}}V(x,t)={\lambda}^{2}\frac{{\partial }^{2}}{\partial {x}^{2}}V(x,t)-(V(x,t)-{E}_{\text{L}})-{\gamma}_{\text{m}}m(x,t)(V(x,t)-{E}_{\text{m}}),\\ {\tau}_{\text{m}}(V(x,t))\frac{\partial }{\partial t}m(x,t)={m}_{\infty }(V(x,t))-m(x,t),\end{array}$$

(21.3)

where γ _m is the ratio of the maximal conductance of the active current to the passive membrane conductance. In order to determine the perturbations to the oscillators we need to solve (21.3) with the oscillators at the ends of the cable giving the periodically forced end conditions of the cable. For this, we first linearize (21.3) about the membrane voltage V _R around which the membrane potential oscillates, leading to the quasi-active approximation for the cable (Sabah and Leibovic, 1969; Koch, 1984; Remme and Rinzel, 2011). We define U(x,t) as the difference between the oscillating solution and the resting membrane potential V _R, i.e., U(x,t) ≡ V (x,t) − V _R and we define w(x,t) analogously as w(x,t) ≡ m(x,t) − m _∞ (V _R). The equations describing the quasi-active cable read

$$ \begin{array}{l}{\tau}\frac{\partial }{\partial t}U(x,t)={\lambda}^{2}\frac{{\partial }^{2}}{\partial{x}^{2}}U(x,t)-{\gamma}_{\text{R}}U(x,t)-{\gamma}_{\text{m}}({V}_{\text{R}}-{E}_{\text{m}})w(x,t),\\ {\tau}_{\text{m}}\frac{\partial}{\partial t}w(x,t)= \frac{\partial }{\partial V}{m}_{\infty }({V}_{\text{R}})U(x,t)-w(x,t),\end{array}$$

(21.4)

where ${\gamma}_{\text{R}}=1+{\gamma}_{\text{m}}{m}_{\infty }({V}_{\text{R}})$ and τ _m =τ _m(V _R). Using the oscillators from (21.1) as the periodically forced end conditions for (21.4) we can write the solution as

$$ U(x,t)={\overline{V}}_{\text{dend}}.\text{Re}\left[\frac{\mathrm{sinh}\left(b\left(L-x/\lambda\right)\right)}{\mathrm{sinh}(bL)}{e}^{i\omega(t+{\theta}_{1})}+\frac{\mathrm{sinh}\left(bx/\lambda\right)}{\mathrm{sinh}(bL)}{e}^{i\omega(t+{\theta}_{2})}\right],$$

(21.5)

where Re[z] is the real part of the complex number z and where

$$b=\sqrt{{\gamma}_{\text{R}}+\frac{\mu}{1+{(\omega{\tau}_{\text{m}})}^{2}}+i\omega\left(\tau-\frac{\mu{\tau}_{\text{m}}} {1+{(\omega{\tau}_{\text{m}})}^{2}}\right)},$$

(21.6)

with

$$ \mu={\gamma}_{\text{m}}({V}_{\text{R}}-{E}_{\text{m}})\frac{\partial}{\partial V}{m}_{\infty }({V}_{\text{R}}).$$

(21.7)

Note that the sign of μ determines whether the active current is restorative (μ > 0) or regenerative (μ < 0; the current is passive when μ = 0). We can now show that the perturbation to oscillator i = 1 reads

$$ {p}_{1}(t)=\frac{\partial }{\partial x}U(0,t)=\frac{{\overline{V}}_{\text{dend}}}{\lambda}\times \text{Re}\left[\frac{b}{\mathrm{sinh}(bL)}{e}^{i\omega(t+{\theta}_{2})}-\mathrm{coth}(bL){e}^{i\omega(t+{\theta}_{1})}\right].$$

(21.8)

We want to describe the evolution of the phase difference $\phi(t)={\theta}_{2}(t)-{\theta}_{1}(t)$. For this we first need to determine the phase interaction function H _i (ϕ) that describes the average effect of perturbation p _i (t) on the phase of oscillator i over a cycle of period $T=2\pi/\omega$. For oscillator i = 1 this interaction function reads

$$ {H}_{1}(\phi)=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}Z}(t){p}_{1}(t+\phi)dt=\frac{Q{\overline{V}}_{\text{dend}}}{2\lambda}\rho\mathrm{sin}(\omega \phi+\xi)+\vartheta,$$

(21.9)

where

$$\rho=\left|\frac{b}{\mathrm{sin h}(bL)}\right|,$$

(21.10)

$$ \xi=\mathrm{arg}\left[\frac{b}{\mathrm{sinh}(bL)}\right],$$

(21.11)

and ν is a constant, and where |z| and arg[z] are the absolute value and the angle of the complex number z, respectively. Since we consider two identical oscillators the interaction function ${H}_{2}(\phi)={H}_{1}(-\phi)$. We now obtain an equation describing the evolution of the phase difference between the two oscillators:

$$\dot{\phi}=\dot{\theta}_{2}-\dot{\theta}_{1} =\varepsilon({H}_{1}(-\phi)-{H}_{1}(\phi))=-\varepsilon\frac{Q{\overline{V}}_{\text{dend}}} {\lambda}\rho\mathrm{cos}(\xi)\mathrm{sin}(\omega \phi)$$

(21.12)

The fixed points of this differential equation (i.e., the points where $\dot{\phi}=0$) are $\phi=k \cdot T/2$, where k is an integer. The stable fixed points are those points where $d\dot{\phi}/d\phi<0$. The synchronous solution ϕ = 0 is thus stable when cos(ξ) > 0. When this solution is stable the anti-phase solution $\phi=T/2$ is unstable and vice versa.

Close to the stable fixed point of (21.12), the term sin(ω ϕ) is approximately linear and the phase locking time constant can be written as

$${\tau}_{\text{lock}}=\frac{\lambda}{\varepsilon\omega Q{\overline{V}}_{\text{dend}}}\frac{1}{\rho\left|\mathrm{cos}\xi\right|}.$$

(21.13)

The parameter ε can be described in terms of cable and oscillator parameters:

$$ \varepsilon=\frac{\pi{d}^{2}}{4{R}_{\text{i}}{C}_{\text{m}}a}=\frac{\pi d{\lambda}^{2}}{a\tau}$$

(21.14)

where R _i is the intracellular resistivity (in Ω cm), a is the surface area of one oscillator (in cm²), and C _m is the specific membrane capacitance (in μF/cm²). The second equality uses $\lambda=\sqrt{d/4{R}_{\text{i}}{g}_{\text{L}}}$ and $\tau={C}_{\text{m}}/{g}_{\text{L}}$, where g _L is the specific membrane conductance (in S/cm²). Hence we can write the phase locking time constant as

$$ {\tau}_{\text{lock}}=\frac{a\tau}{\pi d\lambda \omega Q{\overline{V}}_{\text{dend}}}\frac{1}{\rho\left|\mathrm{cos}\xi\right|}.$$

(21.15)

1.2 A.2 Amplitude of the Voltage Oscillations Along the Cable

Using (21.5) we can easily obtain explicit expressions for the amplitude of the voltage oscillations along the cable as a function of the cable parameters. For synchronized dendritic oscillators (i.e., ϕ = 0) the oscillation amplitude $\overline{V}(x)$ at any point x along the cable of length l is

$$\overline{V}(x)={\overline{V}}_{\text{dend}}\left|\frac{\mathrm{sinh}(b(L-x/\lambda))+\mathrm{sinh}(bx/\lambda)}{\mathrm{sinh}(bL)}\right|.$$

(21.16)

Considering the voltage at the middle of the cable as the “somatic” voltage ${V}_{\text{soma}}(t)=U(l/2,t)$, we find the somatic oscillation amplitudes for synchronized oscillators:

$$ {\overline{V}}_{\text{soma}}=\overline{V}\left(\frac{l}{2}\right)=\frac{{\overline{V}}_{\text{dend}}}{\left|\mathrm{cosh}(bL/2)\right|}.$$

(21.17)