We consider the problem of reconstructing finite energy stimuli encoded with a population of spiking leaky integrate-and-fire neurons. The reconstructed signal satisfies a consistency condition: when passed through the same neuron, it triggers the same spike train as the original stimulus. The recovered stimulus has to also minimize a quadratic smoothness optimality criterion. We formulate the reconstruction as a spline interpolation problem for scalar as well as vector valued stimuli and show that the recovery has a unique solution. We provide explicit reconstruction algorithms for stimuli encoded with single as well as a population of integrate-and-fire neurons. We demonstrate how our reconstruction algorithms can be applied to stimuli encoded with ON-OFF neural circuits with feedback. Finally, we extend the formalism to multi-input multi-output neural circuits and demonstrate that vector-valued finite energy signals can be efficiently encoded by a neural population provided that its size is beyond a threshold value. Examples are given that demonstrate the potential applications of our methodology to systems neuroscience and neuromorphic engineering. 1. Introduction Formal spiking neuron models, such as integrate-and-fire (IAF) neurons, encode information in the time domain [1]. Assuming that the input is bandlimited with a known bandwidth, a perfect recovery of the stimulus from the train of spikes is possible provided that the spike density is above the Nyquist rate [2]. Using results from frame theory [3] and statistics [4], these findings were extended to (i) bandlimited stimuli encoded with a population of IAF neurons with receptive fields modeled as linear filterbanks [5], (ii) multivariate (e.g., space-time) bandlimited stimuli encoded with a population of IAF neurons with Gabor spatiotemporal receptive fields [6], and (iii) sensory stimuli encoded with a population of leaky integrate-and-fire (LIF) neurons with random thresholds [7]. These results are based on the key insight that neural encoding of a stimulus with a population of LIF neurons is akin to taking a set of measurements on the stimulus. These measurements or encodings can be represented as projections (inner products) of the stimulus on a set of sampling functions. Stimulus recovery therefore calls for the reconstruction of the encoded stimuli from these inner products. These findings have shown that sensory information can be faithfully encoded into the spike trains of a neural ensemble and can serve as a theoretical basis for modeling of sensory systems (e.g., auditory, vision)
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