Predictions of the Contribution of HCN Half-Maximal Activation Potential Heterogeneity to Variability in Intrinsic Adaptation of Spiral Ganglion Neurons

Abstract

Spiral ganglion neurons (SGNs) exhibit a wide range in their strength of intrinsic adaptation on a timescale of 10s to 100s of milliseconds in response to electrical stimulation from a cochlear implant (CI). The purpose of this study was to determine how much of that variability could be caused by the heterogeneity in half-maximal activation potentials of hyperpolarization-activated cyclic nucleotide-gated cation (HCN) channels, which are known to produce intrinsic adaptation. In this study, a computational membrane model of cat type I SGN was developed based on the Hodgkin-Huxley model plus HCN and low-threshold potassium (KLT) conductances in which the half-maximal activation potential of the HCN channel was varied and the response of the SGN to pulse train and paired-pulse stimulation was simulated. Physiologically plausible variation of HCN half-maximal activation potentials could indeed determine the range of adaptation on the timescale of 10s to 100s of milliseconds and recovery from adaptation seen in the physiological data while maintaining refractoriness within physiological bounds. This computational model demonstrates that HCN channels may play an important role in regulating the degree of adaptation in response to pulse train stimulation and therefore contribute to variable constraints on acoustic information coding by CIs. This finding has broad implications for CI stimulation paradigms in that cell-to-cell variation of HCN channel properties are likely to significantly alter SGN excitability and therefore auditory perception.

Appendix

For the equations that model the current and channel kinetics of the Na_v, K_v, and KLT channels, please refer to the Appendix in Negm and Bruce (2014). Here are the equations describing the voltage-gated activity of the HCN(r) and HCN(q,s) channel models are supplied, shifted by cV _1/2 standard deviations as functions of the relative membrane potential (σ _x , where x is the channel particle; refer to Table 1 for individual values).

HCN(r) channel model:

The ionic current follows

$$ {I}_{\mathrm{h},r}(t)={\gamma}_{\mathrm{h}}{N}_{r1}(t)\left[{V}_{\mathrm{m}}(t)-{E}_{\mathrm{h},r}\right] $$

(6)

where γ _h is the single-channel conductance, E _h,r is the reversal potential, V _m(t) is the membrane potential at time t, and ${N}_{r_1}$(t) is the number of channels in the fully open, conducting state governed by the kinetic Markov chain state transition diagram

$$ {r}_0\underset{\beta_r}{\overset{\alpha_r}{\rightleftharpoons }}{r}_1 $$

(7)

where transition rates α _r and β _r , calculated by (19) and (20), are dependent on the relative membrane potential (V) and are functions of the activation function (r _∞) and time constant (τ _r ) below

$$ {r}_{\infty }(V)=\frac{1}{1+5.879 \exp \left[\left(V-c{\sigma}_r\right)/7\right]} $$

(8)

$$ {\tau}_r(V)=4.17+\frac{758.8 \exp \left[\left(V-c{\sigma}_r\right)/14\right]}{1+9.199 \exp \left[13\left(V-c{\sigma}_r\right)/84\right]} $$

(9)

where c extends from −4 to 4.

HCN(q,s) channel model:

The ionic current follows

$$ {I}_{\mathrm{h},\left(q,s\right)}(t)={\upgamma}_{\mathrm{h}}\left[{N}_{q_2}(t)+{N}_{s_1}(t)\right]\left[{V}_{\mathrm{m}}(t)-{E}_{\mathrm{h},\left(q,s\right)}\right] $$

(10)

where ${N}_{q_2}$(t) and ${N}_{s_1}$(t) are the number of channels in the fully open, conducting states governed by the parallel kinetic Markov chain state transition diagram

$$ \begin{array}{l}{q}_0\underset{\beta_q}{\overset{2{\alpha}_q}{\rightleftharpoons }}{q}_1\underset{2{\beta}_q}{\overset{\alpha_q}{\rightleftharpoons }}{q}_2\hfill \\ {}{s}_0\underset{\beta_s}{\overset{\alpha_s}{\rightleftharpoons }}{s}_1\hfill \end{array} $$

(11)

where transition rates α _q , β _q , α _s , and β _s, calculated by (19) and (20), are dependent on the relative membrane potential and are functions of the activation functions (q _∞, s _∞) and time constants (τ _q , τ _s ) below

$$ {q}_{\infty }(V)=\frac{1}{{\left(1+9.104 \exp \left[\left(V-c{\sigma}_q\right)/12.36\right]\right)}^{1/2}} $$

(12)

$$ {s}_{A,\infty }(V)=\frac{0.6628}{1+17.09 \exp \left[\left(V-c{\sigma}_s\right)/4.883\right]} $$

(13)

$$ {s}_{B,\infty }(V)=\frac{1-0.6628}{1+3648 \exp \left[\left(V-c{\sigma}_s\right)/3.927\right]} $$

(14)

$$ {s}_{\infty }(V)=\frac{s_{A,\infty }(V)-{s}_{B,\infty }(V)}{0.5551729} $$

(15)

$$ {\tau}_q(V)=\frac{60.98 \exp \left[\left(V-c{\sigma}_q\right)/21.48\right]}{1+2.107 \exp \left[\left(V-c{\sigma}_q\right)/12.19\right]} $$

(16)

$$ {\tau}_s(V)=\frac{632.3 \exp \left[\left(V-c{\sigma}_s\right)/20.23\right]}{1+7.925 \exp \left[\left(V-c{\sigma}_s\right)/13.44\right]}. $$

(17)

Neuron-specific channel modifications

The original channel time constants for KLT: τ _w and τ _z (Rothman and Manis 2003a) and HCN: τ _r (Rothman and Manis 2003b); τ _q and τ _s (Liu et al. 2014b) were divided by their respective thermal scaling coefficients k _w , k _z , k _r , k _q , and k _s to adjust the temperature to 37 °C where

$$ {k}_x={Q}_{10,x}^{\left(T-{T}_0\right)/10} $$

(18)

and x is the channel particle and Q _10,x (see Table 1 for channel-specific values) represents the rate gain for two temperature-dependent biological processes separated by 10 °C (Cartee 2000). T ₀ represents the original temperature whereas T is the current temperature. The transition rates were then computed as

$$ {\alpha}_x(V)={x}_{\infty }(V)/{\tau}_x(V) $$

(19)

$$ {\beta}_x(V)=\left[1-{x}_{\infty }(V)\right]/{\tau}_x(V) $$

(20)

with the steady-state activation functions (x _∞) and time constants (τ _x ).