We consider a Hopfield network designed to model sequential activation of memory items. In this model stable equilibrium states correspond to learned patterns, representing concepts stored in the memory. Passage through a sequence of concepts from one to the next has been called latching dynamics . It has been conjectured that synaptic depression, that is weakening of synaptic connections due to the depletion of neuro-transmitter, is the biological mechanism of the transitions between the concepts. In our recent work we show that in the Hopfield network, extended to include a model of synaptic depression, latching dynamics can be approximated by heteroclinic chains . In this talk we discuss the conditions for the existence of heteroclinic chains using singular perturbation theory (dynamic bifurcation): we define a singular limit and determine necessary and sufficient conditions for the existence of heteroclinic chains.
Sequences given by heteroclinic chains are unidirectional and require an increase in the synaptic weights consistent with the order of the items in the chain. We also discuss a generalization to excitable chains, characterized by a transition involving a saddle-sink pair of equilibria, with the noise enabling the trajectories to exit the basin of attraction of the sink and follow an unstable separatrix of the saddle. Such chains are the subject of our recent work . Interestingly, excitable chains can be bidirectional and do not require an increase in the synaptic weights.
This is joint work with Carlos Aguilar and Frédéric Lavigne from Bases, Corpus, Langage, Université Nice Sophia Antipolis (UNS), Nice, France, and Pascal Chossat and Elif Köksal Ersöz from Project Team MathNeuro, INRIA-CNRS-UNS, 2004 route des Lucioles, 06902 Valbonne, Nice, France.
 Aguilar C, Chossat P, Krupa M, Lavigne F (2017) Latching dynamics in neural networks with synaptic depression. PLoS ONE 12 (8): e0183710.
 Aguilar C, Chossat, P, Köksal Ersöz, E., Krupa M, Lavigne F (2019) Neuronal mechanisms for sequential activation of memory items: dynamics and reliability. Preprint.
 Lerner I. and Shriki O. (2014). Internally and externally driven network transitions as a basis for automatic and strategic processes in semantic priming: theory and experimental validation.
Front.Psychol. 5:314. doi:10.3389/fpsyg.2014. 00314.