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Subthreshold Resonance and Membrane Potential Oscillations in a Neuron with Nonuniform Active Dendritic Properties

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The Computing Dendrite

Part of the book series: Springer Series in Computational Neuroscience ((NEUROSCI,volume 11))

Abstract

Synaptic input to neurons is subject to cell-intrinsic filtering. In the subthreshold membrane potential range this filtering can have either low-pass or resonant characteristics and thereby have a key role in the frequency-dependent information flow in neuronal networks. Experimental classification of neurons as resonant versus nonresonant is usually based on somatic measurements, which, as we demonstrate here, may not accurately reflect neuronal filter properties because of nonuniform distributions of active membrane processes. Using cable theory, we identify conditions under which dendritic currents, in particular I h, can generate somatic resonances. We find that even a strong dendritic resonance may not be detectable somatically in pyramidal cells with a high density of HCN channels in the distal parts of the dendrites. In addition, we show that noise-driven membrane potential oscillations caused by dendritic resonance can propagate to the soma where they can be recorded in the absence of somatic resonance.

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Acknowledgements

 This work was funded by the German Federal Ministry of Education and Research (Grants No. 01GQ0901, No. 01GQ1001A, and No. 01GQ0972), DFG (Grants No. SFB 618 and No. GRK1589), and the Einstein Foundation Berlin.

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Correspondence to Susanne Schreiber .

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Appendix

Appendix

1.1 A.1 Model Parameters

Leak-conductance g L = 0.09 mS/cm2, capacitance C = 1 μF/cm2, passive membrane time constant \(\tau _{\mathrm{m}} = C/g_{\mathrm{L}} = 11\) ms. Length of passive dendritic cable l = 900 μm, length of the active distal end l d = 100 μm, dendritic diameter d = 2 μm, surface area of the distal end \(S_{\mathrm{d}} =\pi dl_{\mathrm{d}} = 628\,\mu\)m2, surface area of the soma S s = 1,257 μm2. Axial resistivity R a = 200 Ω cm, space constant \(\lambda = \sqrt{d/(4R_{\mathrm{a} } g_{\mathrm{L} } )} = 527\,\mu\)m.

1.2 A.2 h-Channel Kinetics

The equations describing the h-current are based on Spain et al. (1987), see also Bernander et al. (1994) and Hutcheon et al. (1996b):

$$\begin{array}{rl} {I}_{\text{h}} & ={\overline{g}}_{\text{h}}(0.8{h}_{\text{f}}+0.2{h}_{\text{s}})(V-{E}_{\text{h}})\\ {\tau}_{\text{f}}\frac{d{h}_{\text{f}}}{dt} & = {h}_{\infty }(V)-{h}_{\text{f}}\\ {\tau}_{\text{s}}\frac{d{h}_{\text{s}}}{dt} & = {h}_{\infty }(V)-{h}_{\text{s}}\\ {h}_{\infty }(V) & = \frac{1}{1+\mathrm{exp}((V+82)/7)}.\\ \end{array}$$

Here, peak conductance of the active distal dendritic segment \( {\overline{g}}_{\text{h}}=3.8\) mS/cm2, reversal potential \( {E}_{\text{h}}=-43\) mV, fast time constant τ f = 40 ms, and slow time constant τ s = 300 ms. The steady-state activation curve h (V ) is shown in Fig. 20.1b. Note that in the case of an active soma \( {\overline{g}}_{\text{h}}=1.9\) mS/cm2 to ensure the same number of h-channels as in the distal dendritic end.

1.3 A.3 Equations Describing the LRC Circuit

When an isopotential compartment containing a leak current and the h-current described above is linearized around a holding voltage V R, we obtain the equations describing an LRC electric circuit (Fig. 20.1c):

$$ \begin{array}{l}C\frac{dV}{dt}=-\frac{V}{R}-{I}_{\text{f}}-{I}_{\text{s}}+{I}_{\text{inj}}\\ {L}_{\text{f}}\frac{d{I}_{\text{f}}}{dt}=-{r}_{\text{f}}{I}_{\text{f}}+V\\ {L}_{\text{s}}\frac{d{I}_{\text{s}}}{dt}=-{r}_{\text{s}}{I}_{\text{s}}+V.\end{array}$$

The resistances and inductances are computed as

$$ \begin{array}{l}R =\frac{1}{{g}_{\text{L}}+{\overline{g}}_{\text{h}}{h}_{\infty }({V}_{\text{R}})}\\ {r}_{\text{f}} =\frac{1}{0.8{\overline{g}}_{\text{h}}({V}_{\text{R}}-{E}_{\text{h}})\frac{\partial }{\partial V}{h}_{\infty }({V}_{\text{R}})}\\ {r}_{\text{s}}=\frac{1}{0.2{\overline{g}}_{\text{h}}({V}_{\text{R}}-{E}_{\text{h}})\frac{\partial }{\partial V}{h}_{\infty }({V}_{\text{R}})}\\ {L}_{\text{f}}={r}_{\text{f}}{\tau}_{\text{f}}\\ {L}_{\text{s}}={r}_{\text{s}}{\tau}_{\text{s}}.\end{array}$$

For our default parameters we have R = 4,019 Ω cm2, r f = 3,381 Ω cm2, r s = 13,522 Ω cm2, L f = 135 H cm2, and L s = 4,057 H cm2.

1.4 A.4 Calculation of the Local and Transfer Impedances

The impedance amplitude profile of the spatially extended neuron equals the absolute value of the (complex-valued) transfer function (i.e., the impulse response function in the frequency domain). Here, we provide the transfer functions for the passive cable with a passive soma attached at x = 0 and the active distal dendritic segment at x = l. The current is injected at the distal dendritic end. The passive cable equation (20.1) can be expressed in the frequency domain as

$$ \frac{{d}^{2}\tilde{V}(x,w)}{d{x}^{2}}-{g}^{2}(w)\tilde{V}(x,w)=0$$
(20.4)

with ω = 2π f (where frequency f is in Hz) and with propagation constant

$$\displaystyle{{ \gamma }^{2}(\omega ) = \frac{1 + i\omega \tau _{\mathrm{m}}} {{\lambda }^{2}}. }$$

The boundary conditions defined by (20.2) and (20.3) with an impulse current I inj = δ(t) in the distal dendritic segment can be written in the frequency domain as

$$\begin{array}{rlrlrl} \frac{1} {r_{\mathrm{a}}} \frac{\partial \tilde{V }(0,\omega )} {\partial x} & =\gamma _{\mathrm{s}}(\omega )\tilde{V }(0,\omega ) &\end{array}$$
(20.5)
$$\begin{array}{rlrlrl} \frac{1} {r_{\mathrm{a}}} \frac{\partial \tilde{V }(l,\omega )} {\partial x} & = 1 -\gamma _{\mathrm{d}}(\omega )\tilde{V }(l,\omega ), &\end{array}$$
(20.6)

where for the passive soma \(\gamma _{\mathrm{s}}(\omega ) = S_{\mathrm{s}}\left (i\omega C + g_{\mathrm{L}}\right )\) and for the active dendritic segment \(\gamma _{\mathrm{d}}(\omega ) = S_{\mathrm{d}}\left (i\omega C + g_{\mathrm{L}} +\bar{ g}_{\mathrm{h}}h_{\infty }(V _{\mathrm{R}}) + (1/(r_{\mathrm{f}} + i\omega L_{\mathrm{f}})) + (1/(r_{\mathrm{s}} + i\omega L_{\mathrm{s}}))\right )\). By solving (20.4) with boundary conditions given by (20.5) and (20.6) we obtain the transfer function of the neuron model

$$\displaystyle{ \tilde{G}(x,\omega ) =\tilde{ V }(x,\omega ) = A\cosh (\gamma \,x) + B\sinh (\gamma \,x), }$$
(20.7)

where coefficient \(A - A(\omega) = \gamma \, K and \,B \,= \,B\,(\omega) = r_{a}\gamma \, K ,\), with

$$K = K (\omega) = \frac{r_{a}}{r_{a}\gamma(\gamma_{s} + \gamma_{d}) cosh(\gamma \,\ell) + (\gamma_{s}\,\gamma_{d}\,\gamma_{a}^{2} + \gamma^{2}) sinh(\gamma \ell)}$$

and γ =γ(ω), γ s = γ s(ω), γ d = γ d(ω). To compute the dendritic input impedance or the transfer impedance between the distal dendritic end and the soma one must let x = l or x = 0 in (20.7), respectively. When the current input is injected somatically we have \(A = (\gamma \, cosh,(\gamma \ell) + r_{a}\gamma_{d}\, sinh(\gamma \ell)) K and B = - (r_{a}\gamma_{d} cosh(\gamma \ell) + \gamma \, sinh(\gamma \,\ell))K\) and one can compute the somatic input impedance by letting x = 0. Input and transfer impedances for the neuron with active soma and/or passive distal dendritic segment can be obtained by setting γ s and γ d appropriately.

1.5 A.5 Computation of the Impedances from the Simulations

To compute somatic and dendritic input impedances, we inject a so-called ZAP current I ZAP(t) = I 0sin(2π f(t)t), with frequency \( f(t)={f}_{\text{m}}t/2T\), input amplitude I 0 = 0.01 nA, maximum frequency f m = 100 Hz, and stimulus length T = 150 s. At the same location we measure the membrane potential V (t) and compute the impedance as \( \tilde{G}(f)=\text{FFT}(V(t))/\text{FFT}({I}_{\text{ZAP}}(t))\), where FFT is the fast Fourier transform. To compute transfer impedances and power spectra, we inject a white noise current (with a duration of 100 s and standard deviation of 0.1 nA) at the distal dendritic end and measure the somatic voltage V (t). Impedance amplitude profile is determined as \( |\tilde{G}(f)|=\text{FFT}(V(t))/\text{FFT}({I}_{\text{noise}}(t))\).

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Zhuchkova, E., Remme, M.W.H., Schreiber, S. (2014). Subthreshold Resonance and Membrane Potential Oscillations in a Neuron with Nonuniform Active Dendritic Properties. In: Cuntz, H., Remme, M., Torben-Nielsen, B. (eds) The Computing Dendrite. Springer Series in Computational Neuroscience, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8094-5_20

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