Abstract
The firing activity of intracellularly stimulated neurons in cortical slices has
been demonstrated to be profoundly affected by the temporal structure of the
injected current (Mainen & Sejnowski, 1995). This suggests that the timing features of the neural response
may be controlled as much by its own biophysical characteristics as by how a
neuron is wired within a circuit. Modeling studies have shown that the interplay
between internal noise and the fluctuations of the driving input controls the
reliability and the precision of neuronal spiking (Cecchi et al., 2000; Tiesinga, 2002; Fellous, Rudolph, Destexhe, & Sejnowski, 2003). In order to investigate this
interplay, we focus on the stochastic leaky integrate-and-fire neuron and
identify the Hölder exponent H of the integrated input as the key mathematical property dictating the
regime of firing of a single-unit neuron. We have recently provided numerical
evidence (Taillefumier & Magnasco, 2013) for the existence of a phase transition when 
becomes less than the statistical Hölder exponent
associated with internal gaussian white noise (H=1/2). Here we describe the theoretical and numerical framework devised for
the study of a neuron that is periodically driven by frozen noisy inputs with
exponent H>0. In doing so, we account for the existence of a transition between two
regimes of firing when H=1/2, and we show that spiking times have a continuous density when the
Hölder exponent satisfies H>1/2. The transition at H=1/2 formally separates rate codes, for which the neural firing probability
varies smoothly, from temporal codes, for which the neuron fires at sharply
defined times regardless of the intensity of internal noise.