Sparse spike coding in an asynchronous feed-forward multi-layer neural network using matching pursuit

In order to account for the rapidity of visual processing, we explore visual coding strategies using a one-pass feed-forward spiking neural network. Following the work of Van Rullen and Thorpe [9], which constructs a spike code for progressive retinal transmission using a wavelet-like transform and rank order coding, we extend this model to arbitrary linear generative models by constructing lateral interactions. This method uses a Matching Pursuit scheme —recursively detecting in the image the best matches to the elements of a dictionary and then subtracting them— and which may similarly define a visual spike code. In fact, in the case of the retina and of the primary visual area (V1), the absolute value of the coefficients obeys to regularities across natural images, inducing a strategy to decode spikes’ information using a rank-order scheme. Moreover, this transform could be used with large and arbitrary dictionaries, so that we may define an overcomplete representation which may define an efficient sparse spike coding scheme in arbitrary multi-layered architectures. Extensions of this method of computation by events lead to the emergence of V1-like receptive fields by using a simple learning scheme but also to bottom-up attentive mechanisms. 1 Toward an efficient dynamical representation 1.1 How to break the code of Vision? Between neuroscience and neuromorphic engineering, our goal is to understand possible spike coding strategies. In the particular case of Vision, faced Preprint submitted to Elsevier Preprint Dec. 11, 2002 with the light influx from the physical world, what are the strategies in a perceptual system to extract the relevant features necessary to a given goal? The physiology of the neurons, the architecture of the visual system and the statistics of the light inputs are as many constraints on the visual system, and a key challenge is to “break” the code of vision. The particular efficiency of a strategy, e.g. for a animal to categorize preys and predators as quickly as possible, is a main constraint from Evolution on the visual system, and particularly, experiences of ultra-rapid categorization [7] in humans and monkeys showed that the visual system could distinguish high-level categories in as short as 150ms, urging us to move from the analogy of the primary visual system with classical image processing strategies to a dynamical neural network model. In order to understand biological Vision but also to implement neuromorphic systems, we’ll therefore study the flow of information in a feed-forward neural network model of primary visual processing and at first, to gain advantage over the speed of retinal processing, the code should convert the analog intensities into an asynchronous ’wave front’ of spikes in less than 20ms, the most ’important’ spikes being fired first. 1.2 Analog to spike coding in the Retina As described in Van Rullen and Thorpe [9], let us first define our model retina with a set of neurons, the ganglion cells (GCs), sensitive at different spatial scales to the local contrast of the image intensity detected at the photoreceptors. First, the dendrite of a neuron i may be characterized by its weight vector φi over its receptive field usually defined by its position ~r and scale σ: i = {~r, σ} as a dilated, translated and sampled Mexican Hat (or DOG) filter (see [3, pp. 77]) placed uniformly over the dyadic scales. This architecture therefore forms a dyadic wavelet-like transform [3] of the image. In fact, we generally write as in [1] the activity at the soma of the neuron as the usual dot product in the Hilbert space H. Ci :=< I, φi >= ∑ ~l∈Ri I(~l).φi(~l) where I(~l) is the luminosity at pixel ~l and Ri is here the receptive field of the neuron i. Also, instead of differentiating ON or OFF cells, we’ll consider for simplicity that each neuron i is assigned a polarity pi which is either +1 or −1, so that the coefficients are rectified (i.e. |Ci| = pi.Ci).