### Abstract

The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs. 1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed. 2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation. 3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

Original language | English |
---|---|

Pages (from-to) | 127-145 |

Number of pages | 19 |

Journal | Kybernetik |

Volume | 15 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1974 Sep |

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### ASJC Scopus subject areas

- Biophysics
- Medicine(all)

### Cite this

*Kybernetik*,

*15*(3), 127-145. https://doi.org/10.1007/BF00274585

**On some properties of stochastic information processes in neurons and neuron populations - Mathematical model approach.** / Matsuyama, Yasuo; Shirai, Katsuhiko; Akizuki, Kageo.

Research output: Contribution to journal › Article

*Kybernetik*, vol. 15, no. 3, pp. 127-145. https://doi.org/10.1007/BF00274585

}

TY - JOUR

T1 - On some properties of stochastic information processes in neurons and neuron populations - Mathematical model approach

AU - Matsuyama, Yasuo

AU - Shirai, Katsuhiko

AU - Akizuki, Kageo

PY - 1974/9

Y1 - 1974/9

N2 - The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs. 1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed. 2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation. 3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

AB - The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs. 1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed. 2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation. 3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

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U2 - 10.1007/BF00274585

DO - 10.1007/BF00274585

M3 - Article

C2 - 4853437

AN - SCOPUS:0016163838

VL - 15

SP - 127

EP - 145

JO - Biological Cybernetics

JF - Biological Cybernetics

SN - 0340-1200

IS - 3

ER -