# Revealing Spectrum Features of Stochastic Neuron Spike Trains

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

#### 2.1. Spike Train PSD Model

#### 2.2. Test Data

^{2}. All these values are considered supra-threshold, i.e., they induce a sustained firing mode even in the absence of internal noise [42]. The higher ${I}_{0}$ the more frequent the induced firing activity will be. Three values for the membrane patch size were considered: (11, 100, 300) μm

^{2}. Assuming constant spatial densities for the sodium (60 N

_{Na}/μm

^{2}) and potassium (18 N

_{k}/μm

^{2}) channels, patch area determines the channel noise level, which is inversely proportional to its square root [43,44]. The model was run in the C ++ environment using the explicit Euler integration method with time step ${T}_{s}$ equal to 10 μs. In our simulation conditions this numerical method is sufficiently accurate, although higher order methods may be necessary when considering time dependent ${I}_{0}$ [45]. The observation interval $N{T}_{s}$ is equal to 2 s, resulting in a spectrum frequency resolution of 0.5 Hz.

## 3. Results

#### 3.1. Spectral Features of Stochastic Neuron Spike Trains

_{0}= 20 μA/cm

^{2}(panel a), I

_{0}= 10 μA/cm

^{2}(panel b), I

_{0}= 7 μA/cm

^{2}(panel c). In each panel, spectra obtained for the three patch areas are compared with the deterministic behavior (purple line).

^{2}.

^{2}and 300 μm

^{2}results for the first five peaks are reported in Figure 2, whereas, in the case of 11 μm

^{2}, only three peaks were faithfully detectable.

#### 3.2. Estimation of Neuronal Spectral Features

^{2}(panel a), 10 μA/cm

^{2}(panel b), and 7 μA/cm

^{2}(panel c), and three patch areas: 11 μm

^{2}(blue lines), 100 μm

^{2}(red lines), and 300 μm

^{2}(yellow lines). We can notice that for higher ${I}_{0}$ values the main peaks were narrower and taller and centered at lower ISI values. Even the patch area affected the ISI distribution: the smaller the area the wider and smaller the main peak, centered at lower ISIs. For ${I}_{0}$ = 7 μA/cm

^{2}and patch area 300 μm

^{2}(yellow line in Figure 3c), the current density was inside the bifurcation range, (6.3–9.8) μA/cm

^{2}, and a reduced noise level determined the typical multimodal distribution, in line with results presented in [46].

_{0}= 20 μA/cm

^{2}) and the highest patch area (300) μm

^{2}, i.e., when the deterministic behavior dominated. In this case, the frequency of the first peak of ${\mathsf{\mu}}_{1}$ ${f}_{1\phantom{\rule{0.166667em}{0ex}}{\mathsf{\mu}}_{1}}$, the mean firing rate $\lambda $, and $1/\mathrm{mode}\{\mathsf{\Theta}\}$ could all be used as good estimates of the position of the main peak in the spike train spectrum. In the other cases, when the stochastic behavior became more relevant, $\lambda $ underestimated the frequency of the main peak and $1/\mathrm{mode}\{\mathsf{\Theta}\}$ overestimated it.

_{0}= 20 μA/cm

^{2}and patch area = 100 μm

^{2}. Panels on the right side of Figure 4 show the expected values of such random variables versus frequency, for the same values of p. As evidenced by looking at panels b, d, and f, ${\mathsf{\mu}}_{p}({\omega}_{k})$ presents cosinusoidal terms which oscillated and decreased faster and faster with frequency as p index increased. As an example, for $p=3$, ${\mathsf{\mu}}_{5}$ was almost completely damped for f > 300 Hz. Therefore, for high p values, the contribution to the summation of (12) was significant only in the low frequency range. Furthermore the sum of these ${\mathsf{\mu}}_{p}$, which oscillated faster at low frequencies for high p values, contributed mainly to the almost constant plateau typical of the spike train spectra at lower frequencies than the main peak, whereas at highest frequency tended to zero.

^{2}and I

_{0}= 20 μA/cm

^{2}, 10 μA/cm

^{2}, and 7 μA/cm

^{2}, significant differences were evident only in the low frequency range. Indeed, as shown in the Figure 4, in the low frequency range the ${\mathsf{\mu}}_{p}({\omega}_{k})$ with high p were dominant. However, the peak positions almost coincided, as quantitatively shown in Table 2 for all simulated conditions and the first 5 peaks. For the smallest patch area (11 μm

^{2}), only the first three peaks were reported as already mentioned in the comment to Figure 2.

_{0}= 10 μA/cm

^{2}and patch = 100 μm

^{2}. When summation was limited to the first 5 terms ($P=5$), except for fast oscillations below 60 Hz, the PSD model ${S}_{u}^{P}[k]$ was a good approximation of the FFT spectrum; when all terms of summation were considered ($P=[\overline{M}]-1$), the FFT spectrum was very accurately captured in the whole frequency range.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Difference between spectrum peaks of deterministic and stochastic neuron for all currents and patches.

**Figure 3.**Probability densities ${f}_{\mathsf{\Theta}}(TH)$ for I

_{0}= (20, 10, 7) μA/cm

^{2}; and patch = (11, 100, 300) μm

^{2}.

**Figure 4.**Probability density of ${\chi}_{p}({\omega}_{k})$ are shown in panels (

**a**,

**c**,

**e**) for $p=1$, 2, and 3, respectively. ${\mathsf{\mu}}_{p}$, expected value of ${\chi}_{p}({\omega}_{k})$, are shown in panels (

**b**,

**d**,

**f**) for $p=1$, 2, and 3, respectively. In all panels I

_{0}= 20 μAcm

^{2}, and patch = 100 μm

^{2}.

**Figure 5.**Prediction of frequencies of power spectra peaks with ${\mathsf{\mu}}_{1}$ (red solid line). Panels (

**a**–

**c**) report, respectively, power spectra for I

_{0}= (20, 10, 7) μA/cm

^{2}, patch = 100 μm

^{2}.

**Figure 6.**Comparison of power spectra, for I

_{0}= 10 μA/cm

^{2}, patch = 100 μm

^{2}, obtained by means of FFT (blue solid line) and (12) with (

**a**) $P=5$ (green dashed line) and (

**b**) $P=[\overline{M}]-1=138$ (cyan dashed line).

**Table 1.**Frequency of the first peak of Fast Fourier Transform (FFT) (${f}_{1\phantom{\rule{0.166667em}{0ex}}\mathrm{PSD}}$), frequency of the first peak of ${\mathsf{\mu}}_{1}$, mean firing rate, and inverse of interspike interval (ISI) mode.

Current Density | Patch | ${\mathit{f}}_{\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{PSD}}$ | ${\mathit{f}}_{\mathbf{1}\phantom{\rule{0.166667em}{0ex}}{\mathsf{\mu}}_{\mathbf{1}}}$ | $\mathit{\lambda}$ | $\mathbf{1}/\mathbf{mode}\{\mathsf{\Theta}\}$ |
---|---|---|---|---|---|

[$\mathsf{\mu}$A/cm${}^{\mathbf{2}}$] | [$\mathsf{\mu}$m${}^{\mathbf{2}}$] | [Hz] | [Hz] | [Hz] | [Hz] |

11 | 92.5 | 90.5 | 87.9 | 96.2 | |

20 | 100 | 89.5 | 88.5 | 88.6 | 91.0 |

300 | 89.5 | 89 | 89.3 | 91.0 | |

11 | 78 | 76.5 | 72.1 | 82.5 | |

10 | 100 | 73 | 72 | 69.6 | 76.9 |

300 | 72.5 | 72 | 70.7 | 75.2 | |

11 | 73 | 71 | 65.4 | 75.4 | |

7 | 100 | 66.5 | 65.5 | 59.8 | 69.2 |

300 | 65 | 64.5 | 57.9 | 66.4 |

**Table 2.**Error and mean absolute error between peak frequencies of FFT based PSD and ${\mathsf{\mu}}_{1}$.

Peaks | 1 | 2 | 3 | 4 | 5 | ||
---|---|---|---|---|---|---|---|

Current Density | Patch | Error | Mean Abs | ||||

[$\mathsf{\mu}$A/cm${}^{\mathbf{2}}$] | [$\mathsf{\mu}$m${}^{\mathbf{2}}$] | [Hz] | Error [Hz] | ||||

11 | 2 | 0 | 2.5 | 1.5 | |||

20 | 100 | 1 | 1 | 0.5 | 1 | 1 | 0.9 |

300 | 0.5 | 0.5 | 0.5 | 1 | 0.5 | 0.6 | |

11 | 1.5 | 0.5 | −1.5 | 1.17 | |||

10 | 100 | 1 | 1.5 | 1 | 0 | 1.5 | 1.0 |

300 | 0.5 | 1.5 | 1.0 | 0.5 | 0.5 | 0.8 | |

11 | 2 | 1.0 | 1.5 | 1.5 | |||

7 | 100 | 1.0 | 1.0 | 0.5 | 0 | 1.5 | 0.8 |

300 | 0.5 | 1.0 | 1.0 | 0.5 | 0.5 | 0.7 |

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**MDPI and ACS Style**

Orcioni, S.; Paffi, A.; Apollonio, F.; Liberti, M. Revealing Spectrum Features of Stochastic Neuron Spike Trains. *Mathematics* **2020**, *8*, 1011.
https://doi.org/10.3390/math8061011

**AMA Style**

Orcioni S, Paffi A, Apollonio F, Liberti M. Revealing Spectrum Features of Stochastic Neuron Spike Trains. *Mathematics*. 2020; 8(6):1011.
https://doi.org/10.3390/math8061011

**Chicago/Turabian Style**

Orcioni, Simone, Alessandra Paffi, Francesca Apollonio, and Micaela Liberti. 2020. "Revealing Spectrum Features of Stochastic Neuron Spike Trains" *Mathematics* 8, no. 6: 1011.
https://doi.org/10.3390/math8061011