Instantaneous Non-Linear Processing by Pulse-Coupled Threshold Units

Contemporary theory of spiking neuronal networks is based on the linear response of the integrate-and-fire neuron model derived in the diffusion limit. We find that for non-zero synaptic weights, the response to transient inputs differs qualitatively from this approximation. The response is instantaneous rather than exhibiting low-pass characteristics, non-linearly dependent on the input amplitude, asymmetric for excitation and inhibition, and is promoted by a characteristic level of synaptic background noise. We show that at threshold the probability density of the potential drops to zero within the range of one synaptic weight and explain how this shapes the response. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled networks of threshold units.Author Summary: Our work demonstrates a fast-firing response of nerve cells that remained unconsidered in network analysis, because it is inaccessible by the otherwise successful linear response theory. For the sake of analytic tractability, this theory assumes infinitesimally weak synaptic coupling. However, realistic synaptic impulses cause a measurable deflection of the membrane potential. Here we quantify the effect of this pulse-coupling on the firing rate and the membrane-potential distribution. We demonstrate how the postsynaptic potentials give rise to a fast, non-linear rate transient present for excitatory, but not for inhibitory, inputs. It is particularly pronounced in the presence of a characteristic level of synaptic background noise. We show that feed-forward inhibition enhances the fast response on the network level. This enables a mode of information processing based on short-lived activity transients. Moreover, the non-linear neural response appears on a time scale that critically interacts with spike-timing dependent synaptic plasticity rules. Our results are derived for biologically realistic synaptic amplitudes, but also extend earlier work based on Gaussian white noise. The novel theoretical framework is generically applicable to any threshold unit governed by a stochastic differential equation driven by finite jumps. Therefore, our results are relevant for a wide range of biological, physical, and technical systems.