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Exactly solvable neural network models with asymmetric weights are rare, andexact solutions are available only in some mean-field approaches. In thisarticle we find exact analytical solutions of an asymmetric spin-glass-likemodel of arbitrary size and we perform a complete study of its dynamical andstatistical properties. The network has discrete-time evolution equations,binary firing rates and can be driven by noise with any distribution. We findanalytical expressions of the conditional and stationary joint probabilitydistributions of the membrane potentials and the firing rates. The conditionalprobability distribution of the firing rates allows us to introduce a newlearning rule to store safely, under the presence of noise, point and cyclicattractors, with important applications in the field of content-addressablememories. Furthermore, we study the neuronal dynamics in terms of thebifurcation structure of the network. We derive analytically examples of thecodimension one and codimension two bifurcation diagrams of the network, whichdescribe how the neuronal dynamics changes with the external stimuli. Inparticular, we find that the network may undergo transitions among multistableregimes, oscillatory behavior elicited by asymmetric synaptic connections, andvarious forms of spontaneous symmetry-breaking. On the other hand, the jointprobability distributions allow us to calculate analytically the higher-ordercorrelation structure of the network, which reveals neuronal regimes where,statistically, the membrane potentials and the firing rates are eithersynchronous or asynchronous. Our results are valid for networks composed of anarbitrary number of neurons, but for completeness we also derive the networkequations in the mean-field limit and we study analytically their localbifurcations. All the analytical results are extensively validated by numericalsimulations.

Pattern Storage, Bifurcations and Higher-Order Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model