Thesis - Department of Electronic & Computer Engineering

Although there is no saturation in the activation function,f , for the WTA networkwith self-excitatory connections, the important property of the boundedness of neuralactivities can be guaranteed by global inhibition.• Order preserving;No matter what the initial conditions of the neural activities are, at steady states,the state potentials of the neurons are ordered in the same order as the external inputstrengths. Consider any two neurons in the network, if their initial state potentials arein the same order as their input strengths, this order is then invariant during the wholedynamic evolution (see Lemma 2 in the appendix); If their initial state potentials are inthe reverse order of their input strengths, their state potentials will eventually becomein the same order as their external input strengths under the condition that the relativedifference of the two inputs is greater than the resolution of the networkδ(SeeLemma 3 in the appendix). If the same condition holds, independent of the initial conditions,at steady states, the neuron receiving the larger external input remains active.The state potential of neuron receiving the smaller external input is below zero, correspondingto no activity (See Lemma 4 in appendix). This implies the initial-conditionindependentproperty.Theorem 1: If I π1 > I π2 > … > I πNand v π1 ( 0) ≥ v π2 ( 0) ≥ … ≥v πN ( 0), thenv π1() t > v π2() t > … > v πN() t , for all t > 0 .Proof: The proof is directly implied from Lemma 2 in the appendix. If the initialstate potentials are in the same order as the input strengths, this order is then invariantduring the whole dynamic evolution.Theorem 2: If I π1 > I π2 > … > I πN, and δ 1≥ δ , then there exists T < ∞ , such that∀t> T, v π1() t > v π2() t > … > v πN() t .11