Communicated by Carl van Vreeswijk What Matters in Neuronal Locking? zyxw Wulfram Gerstner zyxwvu Pkysik-Department der TU Miincken, 0-85747 Garcking bei Miincken, Germany J. Leo van Hemmen Physik-Department der TU Miinchen, 0-85747 Garching bei Miinchen, Germany* Department of Mathematics, University of Chicago, Chicago, IL 60637 USA Jack D. Cowan Department of Mathematics, University of Chicago, Chicago, IL 60637 USA Exploiting local stability, we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under stan- dard conditions, a necessary and, in the limit of a large number of interacting neighbors, also sufficient condition is that the postsynaptic potential is increasing in time zyxw as the neurons fire. If the postsynaptic potential is decreasing, oscillations are bound to be unstable. This is a kind of locking theorem and boils down to a subtle interplay of axonal delays, postsynaptic potentials, and refractory behavior. The theorem also allows for mixtures of excitatory and inhibitory interactions. On the basis of the locking theorem, we present a simple geometric method to verify the existence and local stability of a coherent oscillation. zy 1 introduction Coherence may be defined as being “united in relationship” for most vertebrate neurons, meaning a temporal relationship in that they fire in unison. As such, it is another way of saying that neurons get locked. Once the proposal appeared that coherent oscillations may exist in bio- logical neural systems (Eckhorn et al. 1988; Gray and Singer 1989; Gray et al. 1989; Engel et al. 1991a, 1991b; Eckhorn et al. 1993; Gray 1994), locking phenomena attracted a considerable amount of interest and spurred quite a few people to explain or disprove the very existence of coherent oscil- latory activity. Different authors have used differing models, which vary in several aspects, as do the assumptions and the results. Some models show perfect locking, others partial locking or no locking at all. Some use excitatory interactions, some exploit inhibitory ones, and others use a mixture. In this paper, we present a unifying framework that allows one to derive exact conditions for the existence and stability of coherent *permanent address Nvuval Coniputatiorr zyxwvuts 8, 1653-1676 (1996) zyxwv @ 1996 Massachusetts Institute of Technology