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Forrest, E. J. L. van Hemmen Sonderforschungsbereich 123, Universitat Heidelberg, D-6900 Heidelberg, Fed. Rep. of Germany Abstract : In this paper some new techniques to analyze nonlinear neural networks are reviewed. A neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through a non-linear operation. The original Hopfield model is linear whereas, for instance, clipped synapses constitute a nonlinear model. We examine the statistical mechanics of a nonlinear neural network with finitely many patterns and arbitrary synaptic kernel, study the information retrieval, and show how the abundantly present spurious states which are a consequence of the nonlinearity can be eliminated.

The system does not always recall a memory correctly. Surprisingly, after one MeSS it looks as if the dynamics drives the system in the right direction but after 12 more steps (the missing pictures have been indicated by three dots) we land in a wrong valley (HOI =112). The frequency of such an event is mther low, though. 35 For clipped synapses, the retrieval states bifurcate fIrst. The maximal eigenvalue A. 12) according to whether q is odd or even. 80. 14) Directly below Tc the retrieval states are the only ones which are stable [10].

This is precisely the condition under which the considerations of Sec. 2 hold; cf. Ref. 14. We can easily recover the above results from Secs. 2 - 4. 9) one finds that all nonzero eigenvalues are identical: ')... p = 2 for all p with 1p 1 odd. So ')... states bifurcate from zero at Tc = 2-q x 2 J 1 = 2, the retrieval = 2 l-q J, and they are stable down to T = O. However, all the other pure states belonging to ')... pwith 1 p 1 odd also branch off at Tc and they also are stable =O. There are 2 q-l of these pure states and for each of them there is a component with opposite sign of u p and, hence, of the order parameter m (x) = up v p (x); cf.