The geometry of neuronal recruitment

We address the question of whether or not a periodic train of excitatory synaptic inputs recruits an excitable cell, such that it fires repeatedly, or does not recruit a cell, such that it fails to fire, possibly after some transient. In particular, we study the scenarios of one or two inputs per period; in the latter case, the degree of synchrony of the inputs is a crucial factor in recruitment. We establish rigorous geometric conditions that pinpoint the transition between recruitment and non-recruitment as the degree of synchrony between input pairs, or other input parameters, is varied. These conditions can be used to determine whether a particular temporal relation between periodic input pairs leads to recruitment or not and to prove, in certain parameter regimes, that recruitment can only occur when the inputs are sufficiently closely synchronized. The concepts in this paper are derived for both the integrate-and-fire neuron and the theta neuron models. In the former, the location in phase space of the unique fixed point of a relevant two-dimensional map determines firing, while in the latter, it is the existence or lack of existence of a fixed point of the map which does so. These results are discussed in the context of recruitment of cells into localized activity patterns.