A Transition to Sharp Timing in Stochastic Leaky Integrate-and-Fire Neurons Driven by Frozen Noisy Input

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 [Formula: see text] 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.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

“…Noise is well known to play a constructive role in the neural encoding of natural stimuli, leading to increased reliability or regularity of neuronal firing in single neurons [ 265 , 266 ] and across populations [ 267 ]. From a mathematical perspective it is natural to consider how noise may affect the reduction to a phase oscillator description.…”

First Passage Time Densities through Hölder curves

“…The analysis in Taillefumier and Magnasco (2014) as well as the proof of Theorem 1.1 uses extensively the work by Cannon Cannon (1984) on the heat equation with the moving boundary.…”

“…which can be regarded as an integral equation for F X r (ds). In [7], the equation (1.8) is studied when X is Hölder continuous with exponent greater than 1/2. It is proved that there exists a unique continuous function q such that…”

“…To conclude that F X r (ds) = q(s)ds, one still needs that F X r (ds) is absolutely continuous with respect to Lebesgue measure. The analysis in [7] as well as the proof of Theorem 1 uses extensively the work by Cannon [2] on the heat equation with the moving boundary.…”

First passage leapovers of Lévy flights and the proper formulation of absorbing boundary conditions

“…Figure2. The MFPT(47) of free Lévy flight (blue line) on the interval [0, 2] for (α, β, γ, D) = (3/2, β, 0, 1) starting at x 0 = L = 1, as a function of the skew parameter β. Numerical simulations (blue squares) using the Euler-Maruyama method[15,9] with dt = 10 −3 averaged over 10 4 repetitions agree with taking the value k = 0 in the expression (43) instead of the average of the left and right limits (yellow line).…”