Theoretical Study of Oscillator Neurons in Recurrent Neural Networks

Neurons in a network can be both active or inactive. Given a subset of neurons in a network, is it possible for the subset of neurons to evolve to form an active oscillator by applying some external periodic stimulus? Furthermore, can these oscillator neurons be observable, that is, is it a stable oscillator? This paper explores such possibility, finding that an important property: any subset of neurons can be intermittently co-activated to form a stable oscillator by applying some external periodic input without any condition. Thus, the existing of intermittently active oscillator neurons is an essential property possessed by the networks. Moreover, this paper shows that, under some conditions, a subset of neurons can be fully co-activated to form a stable oscillator. Such neurons are called selectable oscillator neurons. Necessary and sufficient conditions are established for a subset of neurons to be selectable oscillator neurons in linear threshold recurrent neuron networks. It is proved that a subset of neurons forms selectable oscillator neurons if and only if the real part of each eigenvalue of the associated synaptic connection weight submatrix of the network is not larger than one. This simple condition makes the concept of selectable oscillator neurons tractable. The selectable oscillator neurons can be regarded as memories stored in the synaptic connections of networks, which enables to find a new perspective of memories in neural networks, different from the equilibrium-type attractors.