Purkinje local circuits with delays: mathematical conditions of stability for learning and retrieval

Mathematical conditions of stability of learning and retrieval by a Purkinje local circuit, the Purkinje unit, are investigated in the case of delay between any two neurons. The model used takes into account anatomical and physiological constraints: (i) real connectivity; (ii) specific activatory and inhibitory synaptic properties; and (iii) anatomical hierarchical structure. The Purkinje unit is defined in terms of the topology and the geometry of the network, i.e. the anatomical connectivity and the distance between given granule cells and a Purkinje cell. The neurons are assumed to be linear. The network of Purkinje units is general with regard to the number of elements, cells and synapses. For this linear model of a Purkinje unit with delay, we have obtained the conditions of stability for learning and retrieval in two cases: (i) for a single Purkinje unit, the condition is an inequality between the synaptic efficacies which occur in the granule cells-Golgi cell loop; and (ii) for a two-unit system, i.e. two coupled Purkinje units, a strong condition independent of the delay is obtained. This condition includes the granule cell-Golgi cell loop of the two units. We show that the condition of stability for the two-unit system implies the stability of each of the units. This work allows us to define the Purkinje unit in terms of the stability of its function: the physiological process of learning and retrieving must be stable within the anatomical structure that supports this physiological process. It is thus shown that a parameter of a biological nature, the ratio r between the delays of propagation from one unit to another and inside the granule cell-Golgi cell loop, governs the behaviour of the two-unit system. Another result, obtained in the framework of our theory of the functional organization [Chauvet, G. A. (1993). Phil. Trans. R. Soc. Lond. B, 339, 425-444], shows that the association of two unstable units provides a stable unit for certain values of the delay. These results constitute a basis for an eventual interpretation of the coordination of movement by means of a network of non-linear Purkinje units.