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Multiple cusp bifurcations.

Eugene M. Izhikevich¹

¹Center for Systems Science and Engineering, Arizona State University, Tempe, USA

Neural Networks : the Official Journal of the International Neural Network Society

|March 29, 2003

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Networks of neurons near cusp bifurcations exhibit complex dynamics. These networks can function as multiple attractor or globally stable systems based on parameter choices, revealing novel neural network behaviors.

Area of Science:

Computational Neuroscience
Dynamical Systems Theory

Background:

Bistability and hysteresis are fundamental properties in neural dynamics.
Cusp bifurcations offer a simplified model for studying these phenomena in neural networks.

Purpose of the Study:

To investigate the complex dynamics of weakly connected neuron networks near cusp bifurcations.
To demonstrate the transformation of these networks into a canonical model.
To explore the potential operational modes of the canonical model.

Main Methods:

Mathematical analysis of neural network dynamics near cusp bifurcations.
Application of continuous change of variables to derive a canonical model.
Parameter analysis to determine network operational states.

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Main Results:

Weakly connected neuron networks near cusp bifurcations with adaptation exhibit complex dynamics.
A canonical model was derived through variable transformation.
The canonical model demonstrated the capacity to function as both a multiple attractor and a globally asymptotically stable neural network.

Conclusions:

The study simplifies the analysis of complex neural network dynamics.
The canonical model provides a versatile framework for understanding neural computation.
Parameter selection critically determines the network's dynamic behavior and function.