A Dual Algorithm for Olfactory Computation in the Locust Brain
We study the early locust olfactory system in an attempt to explain its wellcharacterized structure and dynamics. We first propose its computational function as recovery of high-dimensional sparse olfactory signals from a small number of measurements. Detailed experimental knowledge about this system rules out standard algorithmic solutions to this problem. Instead, we show that solving a dual formulation of the corresponding optimisation problem yields structure and dynamics in good agreement with biological data. Further biological constraints lead us to a reduced form of this dual formulation in which the system uses independent component analysis to continuously adapt to its olfactory environment to allow accurate sparse recovery. Our work demonstrates the challenges and rewards of attempting detailed understanding of experimentally well-characterized systems.
NOTE: Due to the range in reviews, we were asked to discuss this paper. As a result of that discussion, we'd like the authors to address the three following points:1) How are the firing rates (or some other physiological measure of activity) mapped to the variables y and in particular lambda. Lambda - in the current formulation - has no bounds. How does this affect the interpretation. 2) There is a need for "negative reciprocal" connectivity between the PNs and LNs (this comes out of the use of B and B_transpose) in Equation 12. Similarly there is a need for the same symmetry between the measurements of the sparse vector x and the projections up to the KCs. Is there evidence of this symmetry in the locus anatomy or physiology? Can the model work without it? 3) So far the main evidence that this model describes the olfactory circuit in locust is that the odor input is sparse in terms of molecules per input, somewhere along the way the sparse signal is measured by the A matrix such that it is projected into a lower-dimensional, denser representation. (It should be clarified where this happens... we believe it happens on the level of the ORs such that each OR responds to several molecules and each molecule binds with several ORs). Then there is some processing in this lower-dimension via the B matrix (PNs and LNs) which is then projected back up into a higher-dimension through A to the KCs. We feel that while there is some structure of the biological circuit in the model, this paper needs to provide more evidence. Is there evidence of gradient descent dynamics? Is there evidence in the plasticity/processing between LNs and PNs that can be mapped to the form of ICA presented here? THE TEXT BELOW IS MY ORIGINAL REVIEW:Firstly, I love seeing a theoretical paper which attempts to model a very particular circuit. Typically, we see attempts to create general neural networks or to formally characterize systems which are not well characterized from a biological/experimental point of view. Here we have the opposite, which seems like a more fruitful approach, that is if one's goal is to understand nervous systems.In this paper, the authors derive a mathematical model of processing in the locust olfactory system. The input is a bit vector representing an odor where each entry encodes the presence or absence of particular molecules. Because many ORNs each with the same OR converge on a single glomeruli, the input can be taken as noise-free. In the low-dimensional glomerular layer, the mixed signals from the ORNs interact with each other via local interneurons and PNs. Finally, the PNs then project up into the higher-dimensional space of KCs where each KC codes for a particular molecule in the input.The authors first describe the problem of recovering the sparse, high-dimensional input via the KCs as a MAP optimization, but then reformulate the problem in the compressed sensing context. Next, the dual of the compressed sensing problem is presented as a model since it can more easily be interpreted in terms of the locust nervous system. This model is then changed again to further take knowledge about the biological circuit into account by ensuring that any dynamics happen on the M-dimensional signal. This requires the circuit to learn an M-dimensional subspace of the full N-dimensional space to operate in which is implemented using ICA.This paper is highly original in the way it brings together the compressed sensing problem, ICA, and a particular neural circuit. Also, the presentation of the model is clear and easy to follow. However, I would like a clearer explanation of the biological circuit and how it maps back to the model. Is it that each OR can bind with several molecules such that the glomeruli each represent a mixture of molecules? Then the PNs readout the glomeruli and also take part in the dynamics which ultimately allows the readout by the KCs which sample a mixture of PNs? The basic question is where exactly does the mixing happen. After reading this paper and reviewing references, I'm still not completely certain. I think where I get lost is whether each OR binds with several or only one molecule. It would be helpful to really nail down how the model and biology map to each other since this paper's strongest aspect is that it models a particular circuit. Maybe to make room you can condense the sections where you present models that are ultimately not used.As it stands, the current model is very interesting and I believe that with some clarifiations it is motivated by the biology. However, the real key to making this a significant paper would be to show that the dynamics of the model (either the gradient descent dynamics or something about the ICA dynamics) match the dynamics of the true circuit. So far you've shown that this model can perform sparse regression/compressed sensing and that it has some structural similarities to the biological circuit. I'd like to see a figure that compares a simulation of this model with the true circuit even if it's only that individual neuron's responses match physiological data. Or maybe there is evidence of an ICA implementation in the locust olfactory system? Do they dynamics of ICA seem comparable to any behavioral measurements from locust? Or maybe the ability to discriminate odors wouldl show that performance was more like the reduced dual than the full dual circuit? I would love to see something like this addressed in a revised version of this paper or in a future paper.