Global dynamics in Hopfield’s models


By its potential applicability in real situations, in the last decades, there has been a growing interest of studying mathematical models of neuroscience. One of the most important models in this eld is due to Hop eld and has the expression

(1)   \begin{equation*} u_i'(t)=-u_i(t)+\sum_{j=1}^sw_{ij}f(u_j(t-\tau_{ij})),\quad 1\leq i \leq s. \end{equation*}

With this system we model a network of s neurons where u_i(t) represents the voltage on the input of neuron i at time t, \tau_{ij} denote the synaptic delays and f\colon\mathbb R\to\mathbb R is the neuron activation function. The matrix W = (w_{ij}) measures the connection strengths between the neurons.
The main goal in our talks is to understand the dynamical behaviours of all trajectories in (1), speci cally, we provide an analytic study of global stability, multistability, and bifurcation. Our strategy is to link some dynamical behaviors of (1) with a discrete system in nite dimension. As we will see, our criteria of multistability and global stability are optimal. On the other hand, with our approach we derive criteria of bifurcation without using hard computations of characteristic values or center manifolds. It is remarkable that we can consider systems with non monotone activation functions. Comparing with the little progress of the global dynamics in this setting, remarkable developments have been done in the case of monotone networks where generic convergence is guaranteed by the classical theory of monotone systems.
This is a joint work with E. Liz.


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