## 1. Introduction

## 2. Methods

^{2}[28]. The modified Hodgkin–Huxley model consisted of four types of ion channels of sodium (Na) and high- and low-threshold potassium (KH, KL) and cation channels (h) with stochastic ion channel models. KH, KL and h channels play an important role in helping stabilize rhythmic responses to sinusoidal functions; the KH channel accelerates a repolarization phase; the KL channel shortens the membrane time constant; and the h channel stabilizes the resting membrane potential by lowering the membrane resistance [28,29]. The electrical equivalent circuit of the neuron model is shown in Figure 1. The transmembrane potentials, V

_{m}(t), the difference between extracellular and intracellular potentials, can be expressed by Kirchhoff’s current law as a function of time:

_{rest}, was set at −65 mV, and where:

_{x}(t) denote the number of open channels of each type (x = Na, KL, KH, n, p, h). Each ion channel follows the discrete-state Markov processes with eight states for sodium channels, ten states for low-threshold potassium channels, six (3 × 2) states for high-threshold potassium channels and two states for cation channels. These channels were implemented by the channel number tracking algorithm [23]. Equation (1) was solved numerically in terms of V

_{m}(t) with the Runge–Kutta method [30].

_{Na}= 55 mV, E

_{K}= −70 mV, E

_{h}= −43 mV, and E

_{L}= −65 mV, according to the literature [5]. The single conductance was expressed as follows:

_{soma}= 4 π(d/2)

^{2}, ${\overline{g}}_{Na}=3.03\times 1000\phantom{\rule{0.2em}{0ex}}\text{nS}$, ${\overline{g}}_{KL}=3.03\times 200\phantom{\rule{0.2em}{0ex}}\text{nS}$, ${\overline{g}}_{KH}=3.03\times 150\phantom{\rule{0.2em}{0ex}}\text{nS}$, ${\overline{g}}_{h}=3.03\times 20\phantom{\rule{0.2em}{0ex}}\text{nS}$, ${\overline{g}}_{L}=3.03\times 2\phantom{\rule{0.2em}{0ex}}\text{nS}$, ρ

_{Na}= 6 channels/μ m

^{2}, ρ

_{K}= 2 channels/μ m

^{2}, i.e., ${N}_{Na}^{\mathit{max}}=8312$, ${N}_{KL}^{\mathit{max}}=2771$, ${N}_{KH}^{\mathit{max}}=2771$, ${N}_{h}^{\mathit{max}}=2771.$ The number of channels was varied to investigate how the randomness of ion channels (intrinsic fluctuations) could affect information transmission, being set at ${N}_{Na}^{\mathit{max}}=8312({\rho}_{Na}=6)$, 27,708 (ρ

_{Na}= 20), 83,126 (ρ

_{Na}= 60) and 166,253 (ρ

_{Na}= 120), given that the neuron model was asymptotically close to being deterministic as the number of channels increased.

_{ANF}(t) ascending from the primary auditory nerve fiber was modeled as a filtered inhomogeneous Poisson process, assuming that refractory period was small enough, as:

_{e}= 0 mV, and the synapse conductance waveform was the alpha function expressed as:

_{e}= 0.1 ms [28] and the excitatory synaptic equilibrium potential was set at E

_{e}= 0 mV. The excitatory synaptic conductance, ${\overline{g}}_{e}$, was set such that the excitatory synaptic current could become supra-threshold stimuli, i.e., the firing efficiency (FE), defined by the number of spike firings divided by the number of stimulus presentations, could take a value of 1.0, accordingly ${\overline{g}}_{e}=36\phantom{\rule{0.2em}{0ex}}\mathrm{nS}$ [28]. This is because the objective of the present study is to investigate how spontaneous random activity can better enhance information transmission when the excitatory synaptic current stimulus is supra-threshold. We note that inhibitory synapses are not incorporated in this study, because pre-synaptic vesicle secretion is assumed to elicit post-synaptic spikes, although GABAergic and glycinergic synapses have been found in the descending pathway from sub-nuclei of the SOC to the cochlear nucleus [31].

_{ANF}(t) in Equation (4) denotes the counting process of an inhomogeneous Poisson process [32], described by the following intensity function:

_{spon}was varied as 5, 10, 25, 50, 75, 100 and 125 s

^{−1}, λ

_{c}= 100, 200, 400 or 800 s

^{−1}and f = 220, 880, 1760 or 3520 Hz, so as to investigate the dependency of sound intensity (i.e., λ

_{c}), as well as frequency characteristics within the human speech spectrum. In representing units, we note that the intensity or rate of temporal events randomly occurred according to an inhomogeneous Poisson process has units of s

^{−1}, i.e., counts per unit time, while the frequency of periodic sinusoidal function has units of Hz, i.e., cycle per second.

_{ANF}(t), sampled from 30 realizations with the different sinusoidal phases generated by pseudo-random numbers between 0 and 2π. p(T

_{i}) and p(T

_{i}|I

_{ANF}(t)) as a function of discrete intervals (bin), T

_{i}, denote the probability of the ISI and conditional probability of the ISI on the I

_{ANF}(t), whereas in practical situations, the probability as a function of T

_{i}was estimated from the ISI histogram with a bin width of 0.5 ms. E[ ] denotes the expectation operation, but in practical situations, 30 samples were averaged to calculate the noise entropy given the 30 sample realizations of I

_{ANF}(t).

## 3. Results

_{c}= 200 s

^{−1}and f = 220 Hz.

_{spon}= 5 (left), 25 (middle) and 100 (right) s

^{−1}, λ

_{c}= 200 s

^{−1}and f = 220 Hz.

_{spon}, were varied to 5, 10, 25, 50, 75, 100 and 125 s

^{−1}.

_{spon}= 5 (left), 25 (middle) and 100 (right) s

^{−1}at λ

_{c}= 200 s

^{−1}, f = 220 Hz. The ISI histogram in the middle trace of Figure 4 is similar to that observed in the cat primary auditory nerve [37] and may comprise both random and periodic ISIs, since the intensity function in Equations (6) and (7) is composed of not only sinusoidal functions, ${\tilde{\lambda}}_{sinusoid(t)}$, but a constant value, λ

_{spon}, that generates the spontaneous random activity being modeled as a homogeneous Poisson process. However, the ISI histogram in the left of Figure 4 is only composed of periodic ISIs.

_{spon}in which the intensity of sinusoidal functions, λ

_{c}, is varied to 100, 200, 400 and 800 s

^{−1}at f = 220 Hz.

_{c}= 100, 200 and 400 s

^{−1}, while the sharpness of convex curves tended to decrease as λ

_{c}increased. This implies a resonance phenomenon that depends on the spontaneous rates, like those shown in [26]. The maximum value of the information rates around λ

_{spon}= 25–50 s

^{−1}decreased as λ

_{c}increased. Eventually, the resonance curve disappeared in which λ

_{c}= 800, suggesting that spontaneous random activity did not work at all. The results show that random spontaneous activity can better help enhance information rates at a smaller intensity of sinusoidal functions, λ

_{c}, i.e., a weaker sound signal.

_{c}= 200 s

^{−1}, showing that the sharpness of convex curves tends to decrease as the frequency of sinusoidal functions increases. Eventually, the information rates decrease monotonically when f = 3520 Hz, due possibly to the phase-locked responses disappearing.

## 4. Discussion

_{spon}observed in the present study seems like a supra-threshold stochastic resonance (SSR) [38–41] as a function of the intensity of a Gaussian white noise, because the current applied is a supra-threshold stimulus. The synaptic current could be approximated by the Gaussian white noise; if the synaptic current having a smaller value of the time constant were small, the frequency of synaptic vesicle secretion would be large, such that the voltage fluctuation could not be so large (it is diffusion approximation: [42–46]). However, the structure or configuration of the neuron model in this sort of resonance phenomenon is quite different from that of the regular SSR, since the spherical bushy neuron model used is constructed by a single neuron, unlike a population or an array of neurons in the regular SSR. If the spontaneous random activity of the supra-threshold stimulus can be regarded as an extrinsic fluctuation, this sort of resonance phenomenon that depends the spontaneous rate in a single neuron structure can be classified into the SSR phenomena. Furthermore, this sort of resonance phenomenon with the supra-threshold stimulus seems to be categorized into the regular SR phenomenon, which helps enhance a weak sub-threshold stimulus, because this phenomenon can significantly enhance information transmission when the intensity of sinusoidal functions is set at a smaller value, i.e., a weaker sound level, as is shown in Figure 5.

## 5. Conclusions

## Conflicts of Interest

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**Figure 1.**Electrical equivalent circuit of a spherical bushy neuron model with four kinds of stochastic ion channels.

**Figure 2.**An illustrative example of a membrane potential (

**top**), an excitatory synaptic conductance with an alpha function (

**middle**) and an intensity function of the inhomogeneous Poisson process modulated by a sinusoidal function (

**bottom**) as a function of time.

**Figure 3.**An intensity function (

**top**), raster plots (

**middle**) and the estimated spike firing rates (bottom) as a function of time at λ

_{spon}= 5 (

**left**), 25 (

**middle**) and 100 (

**right**) s

^{−1}at λ

_{c}= 200 s

^{−1}, f = 220 Hz. The post-stimulus time histograms (PSTHs) were generated with a bin width of 0.1 ms in the bottom traces.

**Figure 4.**Inter-spike interval histogram (ISIH) at λ

_{spon}= 5 (

**left**), 25 (

**middle**) and 100 (

**right**) s

^{−1}at λ

_{c}= 200 s

^{−1}, f = 220 Hz. The ISIHs were generated with a bin width of 0.5 ms.

**Figure 5.**Information rate as a function of spontaneous spike rate in which λ

_{c}is set to 100 s

^{−1}(blue), 200 s

^{−1}(cyan), 400 s

^{−1}(magenta) and 800 s

^{−1}(red) at f = 220 Hz.

**Figure 6.**Information rate as a function of spontaneous spike rate in which f is set to 220 Hz (blue), 880 Hz (cyan), 1760 Hz (magenta) and 3520 Hz (red) at λ

_{c}= 200 s

^{−1}.

**Figure 7.**Information rate as a function of spontaneous spike rate in which the number of sodium channels is set to 8312 (blue), 27,708 (cyan), 83,126 (magenta) and 166,253 (red) at f = 220 Hz and λ

_{c}= 200 s

^{−1}.

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