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Two "correlation games" for a nonlinear network with Hebbian excitatory neurons and anti-Hebbian inhibitory neurons
A companion paper introduces a nonlinear network with Hebbian excitatory (E)
neurons that are reciprocally coupled with anti-Hebbian inhibitory (I) neurons
and also receive Hebbian feedforward excitation from sensory (S) afferents. The
present paper derives the network from two normative principles that are
mathematically equivalent but conceptually different. The first principle
formulates unsupervised learning as a constrained optimization problem:
maximization of S-E correlations subject to a copositivity constraint on E-E
correlations. A combination of Legendre and Lagrangian duality yields a
zero-sum continuous game between excitatory and inhibitory connections that is
solved by the neural network. The second principle defines a zero-sum game
between E and I cells. E cells want to maximize S-E correlations and minimize
E-I correlations, while I cells want to maximize I-E correlations and minimize
power. The conflict between I and E objectives effectively forces the E cells
to decorrelate from each other, although only incompletely. Legendre duality
yields the neural network