Faithful Representation of Stimuli with a Population of Integrate-and-Fire Neurons

Abstract

We consider a formal model of stimulus encoding with a circuit consisting of a bank of filters and an ensemble of integrate-and-fire neurons. Such models arise in olfactory systems, vision, and hearing. We demonstrate that bandlimited stimuli can be faithfully represented with spike trains generated by the ensemble of neurons. We provide a stimulus reconstruction scheme based on the spike times of the ensemble of neurons and derive conditions for perfect recovery. The key result calls for the spike density of the neural population to be above the Nyquist rate. We also show that recovery is perfect if the number of neurons in the population is larger than a threshold value. Increasing the number of neurons to achieve a faithful representation of the sensory world is consistent with basic neurobiological thought. Finally we demonstrate that in general, the problem of faithful recovery of stimuli from the spike train of single neurons is ill posed. The stimulus can be recovered, however, from the information contained in the spike train of a population of neurons.

Reconstruction, Identification and Implementation Methods for Spiking Neural Circuits

“…generated by an IF neuron (Lazar, 2004, Lazar andPnevmatikakis, 2008a). These results have been extended for inputs belonging to the more general shift-invariant spaces (SIS) (Gontier and Vetterli, 2014).…”

“…One example is the integrate-and-fire (IF) neuron, which represents one of the most common models of the spiking neuron (Lapicque, 1907, Tuckwell, 1988. Lazar and Pnevmatikakis (2008a) have developed conditions under which a bandlimited input of the IF neuron can be reconstructed perfectly. The result has been extended to other TEMs, such as populations of IF neurons Pnevmatikakis, 2008a, Lazar andSlutskiy, 2014c), IF neurons with refractory period (Lazar, 2004), leaky IF (LIF) neurons (Lazar, 2005), Hodgkin-Huxley neurons (Lazar, 2010) and asynchronous sigma-delta modulators (ASDMs) (Lazar and Tóth, 2004b).…”

“…Reconstruction algorithms have been designed for the bandlimited inputs of linear (Lazar and Pnevmatikakis, 2008a) as well as nonlinear (Lazar and Slutskiy, 2014c) filter banks in cascade with ensembles of ideal IF neurons, ideal IF neurons with refractory period (Lazar, 2004), leaky IF (LIF) neurons (Lazar, 2005), populations of LIF neurons with random thresholds , as well as populations of Hodgkin-Huxley neurons (Lazar, 2007(Lazar, , 2010. Florescu and Coca (2015) have introduced a new reconstruction algorithm for ideal IF neurons that redefines IF time encoding as a uniform sampling problem.…”

“…Equation (2.3) is known as the t-transform of the ideal IF neuron. Lazar and Pnevmatikakis (2008a) have proven that for any input u ∈ PW Ω the t-transform can be expressed as…”

“…As in nonuniform sampling theory, where the samples at points {x k } k∈Z represent the projection of a continuous function on {g Ω (· − x k )} k∈Z , an important problem in encoding with ideal IF neurons is establishing conditions for which {φ T u k } k∈Z forms a frame on PW Ω . The following lemma addresses this issue (Lazar and Pnevmatikakis, 2008a).…”