# 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

- Johnson, D.H. Point process models of single-neuron discharges. J. Comput. Neurosci.
**1996**, 3, 275–299. [Google Scholar] [CrossRef] - Mino, H. The Effects of Spontaneous Random Activity on Information Transmission in an Auditory Brain Stem Neuron Model. Entropy
**2014**, 16, 6654–6666. [Google Scholar] [CrossRef] [Green Version] - Rinzel, J.; Ermentrout, B. Analysis of neural excitability and oscillations. In Methods in Neuronal Modeling: From Ions to Networks, 2nd ed.; Koch, C., Segev, I., Eds.; MIT Press: Cambridge, MA, USA, 1998; Chapter 7. [Google Scholar]
- Gammaitoni, L.; Hänggi, P.; Jung, P.; Marchesoni, F. Stochastic resonance. Rev. Mod. Phys.
**1998**, 70, 223–287. [Google Scholar] [CrossRef] - Moss, F. Stochastic resonance and sensory information processing: A tutorial and review of application. Clin. Neurophysiol.
**2004**, 115, 267–281. [Google Scholar] [CrossRef] - Mino, H.; Durand, D.M. Enhancement of information transmission of sub-threshold signals applied to distal positions of dendritic trees in hippocampal CA1 neuron models with stochastic resonance. Biol. Cybern.
**2010**, 103, 227–236. [Google Scholar] [CrossRef] [PubMed] - Bensaid, S.; Modolo, J.; Merlet, I.; Wendling, F.; Benquet, P. COALIA: A Computational Model of Human EEG for Consciousness Research. Front. Syst. Neurosci.
**2019**, 13. [Google Scholar] [CrossRef] [PubMed] - Modolo, J.; Legros, A.; Beuter, A. The next move in neuromodulation therapy: A question of timing. Front. Comput. Neurosci.
**2015**, 8. [Google Scholar] [CrossRef] [Green Version] - Modolo, J.; Legros, A.; Thomas, A.W.; Beuter, A. Model-driven therapeutic treatment of neurological disorders: Reshaping brain rhythms with neuromodulation. Interface Focus
**2011**, 1, 61–74. [Google Scholar] [CrossRef] - Engel, A.K.; Fries, P.; Singer, W. Dynamic predictions: Oscillations and synchrony in top-down processing. Nat. Rev. Neurosci.
**2001**, 2, 704–716. [Google Scholar] [CrossRef] - Hutchison, W.D.; Dostrovsky, J.O.; Walters, J.R.; Courtemanche, R.; Boraud, T.; Goldberg, J.; Brown, P. Neuronal oscillations in the basal ganglia and movement disorders: Evidence from whole animal and human recordings. J. Neurosci. Off. J. Soc. Neurosci.
**2004**, 24, 9240–9243. [Google Scholar] [CrossRef] - Buzsáki, G.; Draguhn, A. Neuronal oscillations in cortical networks. Science
**2004**, 304, 1926–1929. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Czanner, G.; Sarma, S.V.; Ba, D.; Eden, U.T.; Wu, W.; Eskandar, E.; Lim, H.H.; Temereanca, S.; Suzuki, W.A.; Brown, E.N. Measuring the signal-to-noise ratio of a neuron. Proc. Natl. Acad. Sci. USA
**2015**, 112, 7141–7146. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gray, C.M.; Singer, W. Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Natl. Acad. Sci. USA
**1989**, 86, 1698–1702. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Buzsáki, G. Rhythms of the Brain; Oxford University Press: Oxford, UK, 2006. [Google Scholar] [CrossRef] [Green Version]
- Jarvis, M.R.; Mitra, P.P. Sampling properties of the spectrum and coherency of sequences of action potentials. Neural Comput.
**2001**, 13, 717–749. [Google Scholar] [CrossRef] [Green Version] - Moran, A.; Bar-Gad, I. Revealing neuronal functional organization through the relation between multi-scale oscillatory extracellular signals. J. Neurosci. Methods
**2010**. [Google Scholar] [CrossRef] [PubMed] - Andres, D.S.; Cerquetti, D.F.; Merello, M. Turbulence in Globus pallidum neurons in patients with Parkinson’s disease: Exponential decay of the power spectrum. J. Neurosci. Methods
**2011**, 197, 14–20. [Google Scholar] [CrossRef] [PubMed] - Mazzoni, A.; Broccard, F.D.; Garcia-Perez, E.; Bonifazi, P.; Ruaro, M.E.; Torre, V. On the Dynamics of the Spontaneous Activity in Neuronal Networks. PLoS ONE
**2007**, 2, e439. [Google Scholar] [CrossRef] [Green Version] - Bédard, C.; Kröger, H.; Destexhe, A. Does the 1/f Frequency Scaling of Brain Signals Reflect Self-Organized Critical States? Phys. Rev. Lett.
**2006**, 97, 118102. [Google Scholar] [CrossRef] [Green Version] - Mitaim, S.; Kosko, B. Adaptive stochastic resonance. Proc. IEEE
**1998**. [Google Scholar] [CrossRef] - Paffi, A.; Camera, F.; Apollonio, F.; D’Inzeo, G.; Liberti, M. Restoring the encoding properties of a stochastic neuron model by an exogenous noise. Front. Comput. Neurosci.
**2015**, 9, 1–11. [Google Scholar] [CrossRef] [Green Version] - Orcioni, S.; Paffi, A.; Camera, F.; Apollonio, F.; Liberti, M. Automatic decoding of input sinusoidal signal in a neuron model: Improved SNR spectrum by low-pass homomorphic filtering. Neurocomputing
**2017**, 267, 605–614. [Google Scholar] [CrossRef] - Lindner, B.; Schimansky-Geier, L.; Longtin, A. Maximizing spike train coherence or incoherence in the leaky integrate-and-fire model. Phys. Rev. E
**2002**, 66, 031916. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Droste, F.; Lindner, B. Exact results for power spectrum and susceptibility of a leaky integrate-and-fire neuron with two-state noise. Phys. Rev. E
**2017**, 95, 012411. [Google Scholar] [CrossRef] [Green Version] - Ozer, M.; Perc, M.; Uzuntarla, M.; Koklukaya, E. Weak signal propagation through noisy feedforward neuronal networks. NeuroReport
**2010**, 21, 338–343. [Google Scholar] [CrossRef] [PubMed] - Paffi, A.; Apollonio, F.; D’Inzeo, G.; Liberti, M. Stochastic resonance induced by exogenous noise in a model of a neuronal network. Netw. Comput. Neural Syst.
**2013**, 24, 99–113. [Google Scholar] [CrossRef] [PubMed] - Goychuk, I.; Hänggi, P. Nonstationary stochastic resonance viewed through the lens of information theory. Eur. Phys. J. B
**2009**, 69, 29–35. [Google Scholar] [CrossRef] [Green Version] - Wiesenfeld, K.; Jaramillo, F. Minireview of stochastic resonance. Chaos Interdiscip. J. Nonlinear Sci.
**1998**, 8, 539–548. [Google Scholar] [CrossRef] - Voronenko, S.O.; Stannat, W.; Lindner, B. Shifting Spike Times or Adding and Deleting Spikes—How Different Types of Noise Shape Signal Transmission in Neural Populations. J. Math. Neurosci.
**2015**, 5, 1. [Google Scholar] [CrossRef] [Green Version] - Orcioni, S.; Paffi, A.; Camera, F.; Apollonio, F.; Liberti, M. Automatic decoding of input sinusoidal signal in a neuron model: High pass homomorphic filtering. Neurocomputing
**2018**, 292, 165–173. [Google Scholar] [CrossRef] - Biagetti, G.; Crippa, P.; Orcioni, S.; Turchetti, C. Homomorphic Deconvolution for MUAP Estimation From Surface EMG Signals. IEEE J. Biomed. Health Informatics
**2017**, 21, 328–338. [Google Scholar] [CrossRef] - Abbott, L. Lapicque’s introduction of the integrate-and-fire model neuron (1907). Brain Res. Bull.
**1999**, 50, 303–304. [Google Scholar] [CrossRef] - Izhikevich, E.M. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting; MIT Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Mainen, Z.; Sejnowski, T. Reliability of spike timing in neocortical neurons. Science
**1995**, 268, 1503–1506. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Halliday, D.M.; Rosenberg, J.R. Time and Frequency Domain Analysis of Spike Train and Time Series Data. In Modern Techniques in Neuroscience Research; Windhorst, U., Johansson, H., Eds.; Springer: Berlin/Heidelberg, Germany, 1999; pp. 503–543. [Google Scholar]
- Brillinger, D.R. Comparative aspects of the study of ordinary time series and of point processes. In Developments in Statistics; Krishnaiah, P.R., Ed.; Academic Press: Cambridge, MA, USA, 1978; pp. 33–133. [Google Scholar]
- Wald, A. Some Generalizations of the Theory of Cumulative Sums of Random Variables. Ann. Math. Stat.
**1945**, 16, 287–293. [Google Scholar] [CrossRef] - Rubinstein, J. Threshold fluctuations in an N sodium channel model of the node of Ranvier. Biophys. J.
**1995**, 68, 779–785. [Google Scholar] [CrossRef] [Green Version] - Mino, H.; Rubinstein, J.T.; White, J.A. Comparison of Algorithms for the Simulation of Action Potentials with Stochastic Sodium Channels. Ann. Biomed. Eng.
**2002**, 30, 578–587. [Google Scholar] [CrossRef] [PubMed] - James, F. Monte Carlo theory and practice. Rep. Prog. Phys.
**1980**, 43, 1145–1189. [Google Scholar] [CrossRef] - Izhikevich, E.M. Dynamical Systems in Neuroscience; The MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Schneidman, E.; Freedman, B.; Segev, I. Ion Channel Stochasticity May Be Critical in Determining the Reliability and Precision of Spike Timing. Neural Comput.
**1998**, 10, 1679–1703. [Google Scholar] [CrossRef] [Green Version] - White, J.A.; Rubinstein, J.T.; Kay, A.R. Channel noise in neurons. Trends Neurosci.
**2000**, 23, 131–137. [Google Scholar] [CrossRef] - Ostrovskii, V.Y.; Karimov, T.I.; Solomevich, E.P.; Kolev, G.Y.; Butusov, D.N. Numerical Effects in Computer Simulation of Simplified Hodgkin-huxley Model. In Proceedings of the 24th International Conference on Oral and Maxillofacial Surgery, ICoMS’19, Rio de Janeiro, Brazil, 21–24 May 2019; ACM Press: New York, NY, USA, 2019; pp. 92–95. [Google Scholar] [CrossRef]
- Rowat, P.F.; Greenwood, P.E. The ISI distribution of the stochastic Hodgkin-Huxley neuron. Front. Comput. Neurosci.
**2014**, 8, 111. [Google Scholar] [CrossRef] [Green Version] - Kostal, L.; Lansky, P.; Stiber, M. Statistics of inverse interspike intervals: The instantaneous firing rate revisited. Chaos Interdiscip. J. Nonlinear Sci.
**2018**, 28, 106305. [Google Scholar] [CrossRef] [Green Version] - Dummer, B.; Wieland, S.; Lindner, B. Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity. Front. Comput. Neurosci.
**2014**, 8. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Matzner, A.; Bar-Gad, I. Quantifying Spike Train Oscillations: Biases, Distortions and Solutions. PLoS Comput. Biol.
**2015**, 11, e1004252. [Google Scholar] [CrossRef] [PubMed]

**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 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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