Dynamical equivalence in a class of nonlinear neural network models

The dynamical behavior of nonlinear, dilute scalar product models is investigated using generating functional techniques. In the first part of the paper (Sects. 2–4), bond averages are performed by a cumulant expansion of the interaction term. The coefficients of this expansion are calculated by diagrammatic means. Thereby special attention is paid to the role of correlations between bonds and dynamical initial conditions. The resulting averaged problem only depends on a finite number of real parameters; they thus label equivalence classes of models which behave identically at all times (including transients) on the dynamical average. In particular, the previous suggestion of Sompolinsky that nonlinearity and dilution of the bonds act just like additional stochastic bond contributions if the stored patterns are ±1-variables is confirmed for the exact dynamics at all times. However, if the ±-symmetry of the pattern distribution is removed, additional memory effects with respect to the system activity are found. In the second part (Sects. 5–6), the dynamical mean field theory of the averaged problem is derived as usual by functional saddle point integration. The resulting self-consistent singlenode problem is discussed qualitatively; as a new feature, it contains quenched disorder, introduced by the abovementioned memory effects. Finally, some remarks on the solution of particular cases are made.