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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.