Arithmetic Computing via Rate Coding in Neural Circuits with Spike-triggered Adaptive Synapses Sushrut Thorat Department of Physics Indian Institute of Technology, Bombay Maharashtra, India 400076 Email: sushrutthorat94@gmail.com Absract-We present spiking neural circuits with spike-time dependent adaptive synapses capable of performing a variety of basic mathematical computations. These circuits encode and process information in the spike rates that lie between 40- 140 Hz. The synapses in our circuit obey simple, local and spike-time dependent adaptation rules. We demonstrate that our circuits can perform the fundamental operations - addition, subtraction, multiplication and division, as well as other non-linear transfor mations such as exponentiation and logarithm for time dependent signals in real-time. We show that our spiking neural circuits are tolerant to a high degree of noise in the input variables, and illustrate its computational capability in an exemplary signal estimation problem. Our circuits can thus be used in a wide variety of hardware and software implementations for navigation, control and computation. I. INTRODUCTION One of the fundamental questions of neuroscience is how basic computational operations could be performed by neural circuits that encode information in the temporal domain. Math ematical operations such as addition and multiplication are central to signal processing, and hence to neural computations. Addition and subtraction are essential for pattern recognition. These operations can alter the number of neurons required to reach threshold and inluence the ability to distinguish diferent patterns of activation [1]. Multiplicative operations occur in a wide range of sensory systems. Computations such as looming stimulus detection in the locust visual system [2], and auditory processing in the barn-owl nervous system [3], rely on multiplication operations. In this paper, we model these operations using Spiking Neural Networks (SNNs) [4], to unravel potential mechanisms underlying these computations in biological systems. The goal of our study is to mathemati cally describe the connections in a system, and build SNNs to perform the required mathematical operations. These circuits can then be used to build complex networks to model higher order neural functions, and intricate control systems. There have been various attempts at building SNNs to perform basic mathematical operations. Koch and Seveg de tailed the role of single neurons in information processing [8]. Srinivasan and Bernard proposed a mechanism for multiplica tion which involved detecting coincident arrival of spikes [5], but coincidence detection cannot be used to perform division. Tal and Schwartz used a Log transfer function, generated by creating a refractory period in the Leaky-Integrate-and-Fire neurons, and correlated multiplication with the addition of the 978-1-4799-1959-8/15/$31.00 @2015 IEEE Bipin Rajendran Department of Electrical Engineering Indian Institute of Technology, Bombay Maharashtra, India 400076 Email: bipin@ee.iitb.ac.in logarithms, but they did not calculate the exact multiplication [6]. Inspired by the barn-owl auditory processing capabilities, Nezis & van Rossum built a SNN for multiplication by approximating multiplication by the minimum function, and using a power-law transfer function [7]. All the approaches used voltage to spike-rate (pre-synaptic) non-linear transfer functions to achieve multiplication. In this paper, we adopt a novel strategy to perform math ematical operations using SNNs. We develop small SNNs with linear transfer functions, and spike-time dependent plastic synapses using simple weight adaptation rules to perform these operations. We introduce a new token of information which we call Spike Info, s, to be used to encode information in place of Spike Rate, as required towards the linearisation of spike response to stimuli. We will use the linearized neuron model to develop circuits to perform linear operations such as addition and subtraction. For multiplication and division operations, we use exp(1og 81 ±log 82 ) , where 81 and 82 are the input spike in fos. A similar computational scheme exists in the locust visual system [2], although it uses pre-synaptic non-linear transfer functions for the logarithm and exponentiation operations. We then detail SNNs for performing transformations such as logarithm, exponentiation, multiplication, and division. Note that all these operations are in the spike rate domain, and they work in real-time for biologically plausible signals. We then assess the performance of the developed SNNs, especially its noise resilience, and illustrate its applicability in an exemplary signal estimation problem related to echolocation in real-time. The formalism and the circuits we have developed here can be used to compose large spiking neural circuits for software as well as power eicient hardware implementations. II. SPIKING NEURAL NETWORK IMPLEMENTATION We use the Adaptive Exponential Integrate and Fire (AEIF) neuron model [9] [14] for modeling the dynamics of the neurons, with the parameters chosen to mimic regular spiking ( RS) neurons. The dynamics of the membrane potential V(t) , in response to synaptic current Is y n and applied current Iext is described by the equations, dV ( V(t) - V ) et = -gL(V(t) - EL) + gL�T exp �T T - U(t) + Iext(t) + Is y n(t), (la) dU TWt = a(V(t) - EL) - U(t), (lb)