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**Bifurcation analysis of a neural network model.**
*(English)*
Zbl 0737.92001

Summary: This paper describes the analysis of the neural network model by H. R. Wilson and J. D. Cowan [Biophys. J. 12, 2-24 (1972)]. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inbibitory populations of neurons. We analyze the dependence of the model’s behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied.

In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.

In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.

### MSC:

92B20 | Neural networks for/in biological studies, artificial life and related topics |

34C23 | Bifurcation theory for ordinary differential equations |

### Keywords:

neural network model; evolution of average activities; excitatory; inhibitory; phase diagram; long-period oscillations; stationary state; limit cycle### Software:

LINLBF
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\textit{R. M. Borisyuk} and \textit{A. B. Kirillov}, Biol. Cybern. 66, No. 4, 319--325 (1992; Zbl 0737.92001)

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### References:

[1] | Arnold VI (1978) Supplementary chapters of the theory of ordinary differential equations. Nauka, Moscow (in Russian) |

[2] | Baird B (1989) A bifurcation theory approach to the programming of periodic attractors in networks models of olfactory cortex. In: Touretzky S (eds). Advs Neural Inf Process Syst 1:459–467 |

[3] | Balabaev NK, Lunevskaya LV (1978) Motion along a curve in an n-dimensional space. FORTRAN algorithms and programs. Issue 1. Pushchino, Acad Sci USSR (in Russian) |

[4] | Bautin NN, Leontovitch EA (1976) Methods and techniques of the qualitative analysis of dynamical systems on the plane. Nauka, Moscow (in Russian) |

[5] | Bogdanov RI (1976) Versal deformation of a singular point of the vector field on the plane for a nonzero eigenvalue. In: Works of IG Petrovsky Seminar, Issue 2. Moscow State University Press, Moscow:37–65 (in Russian) |

[6] | Borisyuk RM (1981) Stationary solutions of a system of differential equations depending on parameters. FORTRAN algorithms and programs. Issue 6. Acad Sci USSR, Pushchino (in Russian) |

[7] | Borisyuk RM (1990) Interacting neural oscillators can imitate the selective attention. In: Holden AV, Kryukov VI (eds) Neurocomputers and attention. Manchester University Press, Manchester (in press) |

[8] | Borisyuk RM, Kirillov AB (1982) A qualitative study of a model of neural network consisting of two homogeneous populations. Preprint. Acad Sci USSR, Pushchino (in Russian) |

[9] | Borisyuk RM, Urzhumtseva LM (1990) Dynamical regimes in a system of interacting neural oscillators. In: Holden AV, Kryukov VI (eds) Neural Networks: theory and architecture. Manchester University Press, Manchester (in press) |

[10] | Bragin AG, Sbitnev VI (1980) The septal input to the CA3 hippocampal field. 1. Identifying the states. Biofizika 25:190 (in Russian) |

[11] | Buhmann J, von der Malsburg C (1991) Sensory segmentation by neural oscillators. In: International Joint Conference on Neural Networks – 1991, 2:603–608 · Zbl 0746.92003 |

[12] | Carpenter GA, Grossberg S (1987) A massively parallel architecture for a self-organizing neural pattern recognition machine. Cornput Vision Graph Image Process 37:54–115 · Zbl 0634.68089 |

[13] | Freeman WJ, Scarda CA (1985) Spatial EEG patterns, nonlinear dynamics and perception: the Neo-Sherringtonian view. Brain Res Rev 10:147–175 |

[14] | Gantmakher PhR (1967) The theory of matrices. Nauka, Moscow (in Russian) · Zbl 0925.60011 |

[15] | Gray CM, Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc Natl Acad Sci USA 86:1698–1702 |

[16] | Hopfield J (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA, 81:3088–3092 · Zbl 1371.92015 |

[17] | Khibnik AI (1979) Periodic solutions of a system of differential equations. FORTRAN algorithms and programs. Issue 5. Acad Sci USSR, Pushchino (in Russian) |

[18] | Khibnik AI (1990a) Using TRAX: A tutorial to Accompany TRAX, a program for simulation and analysis of dynamical systems. Exeter Software, Setauket, NY |

[19] | Khibnik AI (1990b) LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three. In: Roose D (eds) Continuation and bifurcations: numerical techniques and applications. Kluwer, London, New York · Zbl 0705.34001 |

[20] | Khibnik AI and Shnol EE (1982) Software for qualitative analysis of differential equations. Acad Sci USSR, Pushchino (in Russian) |

[21] | Kuznetsov UA (1983) One-dimensional separatrix of the system of ordinary differential equations with parameters. FORTRAN algorithms and programs. Issue 8. Acad Sci USSR, Pushchino (in Russian) |

[22] | Levitin W (1989) TRAX: Simulation and analysis of dynamical systems. Version 1. 1. Exeter Software, Setauket, NY |

[23] | Nicolis JS (1985) Chaotic dynamics of information processing with relevance to cognitive brain functions. Kybernetes 14:167–172 |

[24] | Sbitnev VI, Gulayev AA and Bragin AG (1982) Dynamical models of the functional organization of hippocampus. In: Memory and learing mechanisms. Nauka, Moscow (in Russian) |

[25] | Schuster HG and Wagner P (1990) A model for neuronal oscillations in the visual cortex. Biol Cybern 64:77–85 · Zbl 0709.92012 |

[26] | Shimizu H, Yamaguchi Y and Satoh K (1988) Holovision: a semantic information processor for visual perception. In: Kelso JAS, Mandel AJ, Shlesinger MF (eds) Dynamic patterns in complex systems. World Scientific, Signapore |

[27] | Wilson HR and Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:2–24 |

[28] | Zarhin YuG and Kovalenko VN (1978) Finding solutions of a system of two algebraic equations. FORTRAN algorithms and programs. Issue 2. Acad Sci USSR, Pushchino (in Russian) |

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