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5 odorsSampling\fFigure 4: Average time course of log(p(s)) (left) and p(s) (right, same as in Fig. 3). For the varia-\ntional algorithm, the activity of the neurons codes for log probability (relative to some background\nto keep \ufb01ring rates non-negative). For this algorithm, the drop in probability of the non-presented\nodors from about e\u22125 to e\u221212 corresponds to a large drop in \ufb01ring rate. For the sampling based\nalgorithm, on the other hand, activity codes for probability, and there is almost no drop in activity.\n\nThere are two ways to determine which. One is to note that for the variational algorithm there is\na large drop in the average activity of the neurons coding for the non-present odors (Fig. 4 and\nSupplementary Figure 2). This drop could be detected with electrophysiology. The other focuses on\nthe present odors, and requires a comparison between the posterior probability inferred by an animal\nand neural activity. The inferred probability can be measured by so-called \u201copt-out\u201d experiments\n[11]; the latter by sticking an electrode into an animal\u2019s head, which is by now standard.\nThe two algorithms also make different predictions about the activity coding for concentration. For\nthe variational approach, activity, \u03b10j, codes for the parameters of a probability distribution. Im-\nportantly, in the variational scheme the mean and variance of the distribution are tied \u2013 both are\nproportional to activity. Sampling, on the other hand, can represent arbitrary concentration distri-\nbutions. These two schemes could, therefore, be distinguished by separately manipulating average\nconcentration and uncertainty \u2013 by, for example, showing either very similar or very different odors.\nUnfortunately, it is not clear where exactly one needs to stick the electrode to record the trace of the\nolfactory inference. A good place to start would be the olfactory bulb, where odor representations\nhave been studied extensively [12, 13, 14]. For example, the dendro-dendritic connections observed\nin this structure [4] are particularly well suited to meet the symmetry requirements on wij. We note\nin passing that these connections have been the subject of many theoretical studies. Most, however,\nconsidered single odors [15, 6, 16], for which one does not need a complicated inference process\nAn early notable exception to the two-odor standard was Zhaoping [17], who proposed a model\nfor serial analysis of complex mixtures, whereby higher cortical structures would actively adapt the\nalready recognized components and send a feedback signal to the lower structures. Exactly how her\nnetwork relates to our inference algorithms remains unclear. We should also point out that although\nthe olfactory bulb is a likely location for at least part of our two inference algorithms, both are\nsuf\ufb01ciently complicated that they may need to be performed by higher cortical structures, such as\nthe anterior piriform cortex, [18, 19].\n\nFuture directions. We have made several unrealistic assumptions in this analysis. For instance,\nthe generative model was very simple: we assumed that concentrations added linearly, that weights\nwere binary (so that each odor activated a subset of the olfactory receptor neurons at a \ufb01nite value,\nand did not activate the rest at all), and that noise was Poisson. None of these are likely to be exactly\ntrue. And we considered priors such that all odors were independent. This too is unlikely to be true \u2013\nfor instance, the set of odors one expects in a restaurant are very different than the ones one expects\nin a toxic waste dump, consistent with the fact that responses in the olfactory bulb are modulated\nby task-relevant behavior [20]. Taking these effects into account will require a more complicated,\nalmost certainly hierarchical, model. We have also focused solely on inference: we assumed that\nthe network knew perfectly both the mapping from odors to odorant receptor neurons and the priors.\nIn fact, both have to be learned. Finally, the neurons in our network had to implement relatively\ncomplicated nonlinearities: logs, exponents, and digamma and quadratic functions, and neurons had\nto be reciprocally connected. Building a network that can both exhibit the proper nonlinearities\n(at least approximately) and learn the reciprocal weights is challenging. While these issues are\nnontrivial, they do not appear to be insurmountable. We expect, therefore, that a more realistic\nmodel will retain many of the features of the simple model we presented here.\n\n8\n\n020406080100\u221210\u221250Time [ms]L3 odorsVariational02040608010000.51Time [ms]\u03bb3 odorsSampling\fReferences\n[1] J. Fiser, P. Berkes, G. Orban, and M. Lengyel. Statistically optimal perception and learning:\nfrom behavior to neural representations. Trends Cogn. Sci. (Regul. Ed.), 14(3):119\u2013130, Mar\n2010.\n\n[2] R. Vincis, O. Gschwend, K. Bhaukaurally, J. Beroud, and A. Carleton. Dense representation\n\nof natural odorants in the mouse olfactory bulb. Nat. Neurosci., 15(4):537\u2013539, Apr 2012.\n\n[3] Jeff Beck, Katherine Heller, and Alexandre Pouget. Complex inference in neural circuits with\n\nprobabilistic population codes and topic models. In NIPS, 2012.\n\n[4] W. Rall and G. M. Shepherd. Theoretical reconstruction of \ufb01eld potentials and dendrodendritic\n\nsynaptic interactions in olfactory bulb. J. 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Kiani and M. N. Shadlen. Representation of con\ufb01dence associated with a decision by\n\nneurons in the parietal cortex. Science, 324(5928):759\u2013764, May 2009.\n\n[12] G. Laurent, M. Stopfer, R. W. Friedrich, M. I. Rabinovich, A. Volkovskii, and H. D. Abarbanel.\nOdor encoding as an active, dynamical process: experiments, computation, and theory. Annu.\nRev. Neurosci., 24:263\u2013297, 2001.\n\n[13] H. Spors and A. Grinvald. Spatio-temporal dynamics of odor representations in the mammalian\n\nolfactory bulb. Neuron, 34(2):301\u2013315, Apr 2002.\n\n[14] Kevin Cury and Naoshige Uchida. Robust odor coding via inhalation-coupled transient activity\n\nin the mammalian olfactory bulb. Neuron, 68(3):570\u2013585, 2010.\n\n[15] Z. Li and J. J. Hop\ufb01eld. Modeling the olfactory bulb and its neural oscillatory processings.\n\nBiol Cybern, 61(5):379\u2013392, 1989.\n\n[16] Y. Yu, T. S. McTavish, M. L. Hines, G. M. Shepherd, C. Valenti, and M. Migliore. Sparse\ndistributed representation of odors in a large-scale olfactory bulb circuit. PLoS Comput. Biol.,\n9(3):e1003014, 2013.\n\n[17] Z. Li. A model of olfactory adaptation and sensitivity enhancement in the olfactory bulb. Biol\n\nCybern, 62(4):349\u2013361, 1990.\n\n[18] Julie Chapuis and Donald Wilson. Bidirectional plasticity of cortical pattern recognition and\n\nbehavioral sensory acuity. Nature neuroscience, 15(1):155\u2013161, 2012.\n\n[19] Keiji Miura, Zachary Mainen, and Naoshige Uchida. Odor representations in olfactory cortex:\ndistributed rate coding and decorrelated population activity. Neuron, 74(6):1087\u20131098, 2012.\n[20] R. A. Fuentes, M. I. Aguilar, M. L. Aylwin, and P. E. Maldonado. Neuronal activity of mitral-\ntufted cells in awake rats during passive and active odorant stimulation. J. Neurophysiol.,\n100(1):422\u2013430, Jul 2008.\n\n9\n\n\f", "award": [], "sourceid": 995, "authors": [{"given_name": "Agnieszka", "family_name": "Grabska-Barwinska", "institution": "Gatsby Unit, UCL"}, {"given_name": "Jeff", "family_name": "Beck", "institution": "Gatsby Unit, UCL"}, {"given_name": "Alexandre", "family_name": "Pouget", "institution": "University of Geneva"}, {"given_name": "Peter", "family_name": "Latham", "institution": "Gatsby Unit, UCL"}]}