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The brain efficiently performs nonlinear computations through its intricate networks of spiking neurons, but how this is done remains elusive. While nonlinear computations can be implemented successfully in spiking neural networks, this requires supervised training and the resulting connectivity can be hard to interpret. In contrast, the required connectivity for any computation in the form of a linear dynamical system can be directly derived and understood with the spike coding network (SCN) framework. These networks also have biologically realistic activity patterns and are highly robust to cell death. Here we extend the SCN framework to directly implement any polynomial dynamical system, without the need for training. This results in networks requiring a mix of synapse types (fast, slow, and multiplicative), which we term multiplicative spike coding networks (mSCNs). Using mSCNs, we demonstrate how to directly derive the required connectivity for several nonlinear dynamical systems. We also show how to carry out higher-order polynomials with coupled networks that use only pair-wise multiplicative synapses, and provide expected numbers of connections for each synapse type. Overall, our work demonstrates a novel method for implementing nonlinear computations in spiking neural networks, while keeping the attractive features of standard SCNs (robustness, realistic activity patterns, and interpretable connectivity). Finally, we discuss the biological plausibility of our approach, and how the high accuracy and robustness of the approach may be of interest for neuromorphic computing.

Deep feedforward and recurrent neural networks have become successful functional models of the brain, but they neglect obvious biological details such as spikes and Dale's law. Here we argue that these details are crucial in order to understand how real neural circuits operate. Towards this aim, we put forth a new framework for spike-based computation in low-rank excitatory-inhibitory spiking networks. By considering populations with rank-1 connectivity, we cast each neuron's spiking threshold as a boundary in a low-dimensional input-output space. We then show how the combined thresholds of a population of inhibitory neurons form a stable boundary in this space, and those of a population of excitatory neurons form an unstable boundary. Combining the two boundaries results in a rank-2 excitatory-inhibitory (EI) network with inhibition-stabilized dynamics at the intersection of the two boundaries. The computation of the resulting networks can be understood as the difference of two convex functions and is thereby capable of approximating arbitrary non-linear input-output mappings. We demonstrate several properties of these networks, including noise suppression and amplification, irregular activity and synaptic balance, as well as how they relate to rate network dynamics in the limit that the boundary becomes soft. Finally, while our work focuses on small networks (5-50 neurons), we discuss potential avenues for scaling up to much larger networks. Overall, our work proposes a new perspective on spiking networks that may serve as a starting point for a mechanistic understanding of biological spike-based computation. [ABSTRACT FROM AUTHOR]

Table of contents A1 Functional advantages of cell-type heterogeneity in neural circuits Tatyana O. Sharpee A2 Mesoscopic modeling of propagating waves in visual cortex Alain Destexhe A3 Dynamics and biomarkers of mental disorders Mitsuo Kawato F1 Precise recruitment of spiking output at theta frequencies requires dendritic h-channels in multi-compartment models of oriens-lacunosum/moleculare hippocampal interneurons Vladislav Sekulić, Frances K. Skinner F2 Kernel methods in reconstruction of current sources from extracellular potentials for single cells and the whole brains Daniel K. Wójcik, Chaitanya Chintaluri, Dorottya Cserpán, Zoltán Somogyvári F3 The synchronized periods depend on intracellular transcriptional repression mechanisms in circadian clocks. Jae Kyoung Kim, Zachary P. Kilpatrick, Matthew R. Bennett, Kresimir Josić O1 Assessing irregularity and coordination of spiking-bursting rhythms in central pattern generators Irene Elices, David Arroyo, Rafael Levi, Francisco B. Rodriguez, Pablo Varona O2 Regulation of top-down processing by cortically-projecting parvalbumin positive neurons in basal forebrain Eunjin Hwang, Bowon Kim, Hio-Been Han, Tae Kim, James T. McKenna, Ritchie E. Brown, Robert W. McCarley, Jee Hyun Choi O3 Modeling auditory stream segregation, build-up and bistability James Rankin, Pamela Osborn Popp, John Rinzel O4 Strong competition between tonotopic neural ensembles explains pitch-related dynamics of auditory cortex evoked fields Alejandro Tabas, André Rupp, Emili Balaguer-Ballester O5 A simple model of retinal response to multi-electrode stimulation Matias I. Maturana, David B. Grayden, Shaun L. Cloherty, Tatiana Kameneva, Michael R. Ibbotson, Hamish Meffin O6 Noise correlations in V4 area correlate with behavioral performance in visual discrimination task Veronika Koren, Timm Lochmann, Valentin Dragoi, Klaus Obermayer O7 Input-location dependent gain modulation in cerebellar nucleus neurons Maria Psarrou, Maria Schilstra, Neil Davey, Benjamin Torben-Nielsen, Volker Steuber O8 Analytic solution of cable energy function for cortical axons and dendrites Huiwen Ju, Jiao Yu, Michael L. Hines, Liang Chen, Yuguo Yu O9 C. elegans interactome: interactive visualization of Caenorhabditis elegans worm neuronal network Jimin Kim, Will Leahy, Eli Shlizerman O10 Is the model any good? Objective criteria for computational neuroscience model selection Justas Birgiolas, Richard C. Gerkin, Sharon M. Crook O11 Cooperation and competition of gamma oscillation mechanisms Atthaphon Viriyopase, Raoul-Martin Memmesheimer, Stan Gielen O12 A discrete structure of the brain waves Yuri Dabaghian, Justin DeVito, Luca Perotti O13 Direction-specific silencing of the Drosophila gaze stabilization system Anmo J. Kim, Lisa M. Fenk, Cheng Lyu, Gaby Maimon O14 What does the fruit fly think about values? A model of olfactory associative learning Chang Zhao, Yves Widmer, Simon Sprecher,Walter Senn O15 Effects of ionic diffusion on power spectra of local field potentials (LFP) Geir Halnes, Tuomo Mäki-Marttunen, Daniel Keller, Klas H. Pettersen,Ole A. Andreassen, Gaute T. Einevoll O16 Large-scale cortical models towards understanding relationship between brain structure abnormalities and cognitive deficits Yasunori Yamada O17 Spatial coarse-graining the brain: origin of minicolumns Moira L. Steyn-Ross, D. Alistair Steyn-Ross O18 Modeling large-scale cortical networks with laminar structure Jorge F. 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