On the piecewise analysis of networks of linear threshold neurons

The computational abilities of recurrent networks of neurons with a linear activation function above threshold are analyzed. These networks selectively realise a linear mapping of their input. Using this property, the dynamics as well as the number and the stability of stationary states can be investigated. The important property of the boundedness of neural activities can be guaranteed by global inhibition. If used together with self-excitation, the global inhibition gives rise to a multi stable winner-take-all (WTA) mechanism. A condition for a neuron to be a potential winner of the competing dynamics is derived. The network becomes a largest input selector when the self-excitation is marginal.Slowing down the global inhibition produces oscillations. The study of oscillations of random networks suggests that all cyclic trajectories of linear threshold networks are a result of the existence of partitions with undamped linear oscillations. Chaotic dynamics were never encountered in computer simulations and perhaps do not exist at all in small networks.