@article{
    SiOr03a,
    author = "{\v{S}}{\'{\i}}ma, Ji{\v{r}}{\'{\i}} and Orponen, Pekka",
    note = "",
    title = "Continuous-Time Symmetric {H}opfield Nets are Computationally Universal",
    url = "http://neco.mitpress.org/cgi/content/abstract/15/3/693",
    journal = "Neural Computation",
    number = "3",
    abstract = "We establish a fundamental result in the theory of computation by continuous-time dynamical systems, by showing that systems corresponding to so called continuous-time symmetric Hopfield nets are capable of general computation. As is well known, such networks have very constrained, Liapunov-function controlled dynamics. Nevertheless, we show that they are universal and efficient computational devices, in the sense that any convergent synchronous fully parallel computation by a recurrent network of $n$ discrete-time binary neurons, with in general asymmetric coupling weights, can be simulated by a symmetric continuous-time Hopfield net containing only $18n+7$ units employing the saturated-linear activation function. Moreover, if the asymmetric network has maximum integer weight size $w_{\max}$ and converges in discrete time $t^*$, then the corresponding Hopfield net can be designed to operate in continuous time $\Theta(t^*/\varepsilon)$, for any $\varepsilon>0$ such that $w_{\max}2^{12n}\leq\varepsilon2^{1/\varepsilon}$. In terms of standard discrete computation models, our result implies that any polynomially space-bounded Turing machine can be simulated by a family of polynomial-size continuous-time symmetric Hopfield nets.",
    month = "March",
    volume = "15",
    supersedes = "SiOr01a",
    flags = "public",
    year = "2003",
    keywords = "dynamical systems, continuous time, neural networks, Hopfield model, computational power",
    pages = "693--733"
}