Response of the instantaneous firing rate to high frequency inputs

Nicolas Fourcaud, David Hansel, Carl van Vreeswijk, and Nicolas Brunel

Neurophysique et Physiologie du Systeme Moteur - CNRS - Universite Paris 5 - France

In sensory systems, neuronal populations have to track rapidly fluctuating inputs. We explain here how ionic current dynamics leading to spike emission is a limiting factor of this ability. Previous analytical studies have investigated the response of the instantaneous firing rate of the leaky integrate-and-fire (LIF) model to noisy oscillatory inputs at high frequency f. The firing rate temporal modulation A(f) decreases as 1/sqrt(f) with a phase lag L(f) of 45 degrees in presence of white noise while it stays finite with no phase lag with temporally correlated noise. In contrast, numerical simulations of several conductance-based models reveal a different behavior: A(f) decays as 1/f and L(f) tends to 90 degrees. To explain this qualitatively different behavior, we introduced and analytically investigated a family of 1-variable models which incorporate active properties. The 'quadratic' integrate-and-fire neuron, which describes the subthreshold dynamics of a large class of neurons near the firing onset, is a particular model in this family. However, it cannot account for the 1/f behavior: A(f) decays as 1/f2 and L(f) tends to 180 degrees. Another model in the family is the 'exponential' integrate-and-fire (EIF) neuron, in which a simplified sodium current with an instantaneous exponential voltage-dependent activation is responsible for spike generation. For white as well as for temporally correlated noise, we show analytically that, in the EIF model, A(f) decreases in 1/f and L(f) tends to 90 degrees. The stationary and dynamical properties of the EIF model matches well the properties of the simulated conductance-based models for input frequencies up to about 1000 Hz. At large noise, the firing rate response is a low pass filter, with a cutoff frequency that can be determined analytically. The cutoff frequency is given by the largest of two quantities: (i) the inverse of the membrane time constant (ii) a cutoff frequency proportional to the background firing rate and to the sharpness of the activation curve of the 'active current', and inversely proportional to the slope of the f-I curve.