1. Modeling The quadratic integrate and fire follows from a reduced form (1) where F(V) is a voltage dependant function which aims to capture the voltage dependant current flow in active membranes. I is the applied current and dV/dt is the time varying membrane voltage. Substituting V 2 for F(V) leads to the one dimensional system (2) which is the topological normal form for saddle node bifurcation. A reset condition is required or else the solution would increase to infinity, thus (3) If V  V Peak, then V => V Reset Introduction Implementation of neuronal networks in silico can facilitate understanding of biological systems and result in hardware systems with similar performance to biological systems [1]. Hardware in silico systems; can perform specific operations faster than general purpose hardware, are more easily interfaced to biological systems, enable robotic systems. Robotic systems can complement simulation, i.e. complex physical laws need not be simulated and modeling may be verified through observing the operation of the robot. Three General Types of Neural Models: Conductance models based upon biophysical properties of the membrane. Integrate and Fire (IF) and Leaky Integrate and Fire (LIF) models. Mathematical models derived based upon bifurcation analysis. Of these, the quadratic integrate and fire model (QIF) is the simplest model capable of producing true spiking behavior [2]. The objective of this work was to design a circuit from discrete components that implemented a quadratic integrate and fire model of a spiking neuron. Design Approach Key Points True spiking behavior Mathematically tractable Biologically relevant testing and extraction System Diagram for a Reset Controller The system diagram for a reset control system [3]. Reset (R) occurs when the output reaches a set threshold. MATLAB Discrete Time Simulation MATLAB code employed a Euler approximation method to solve (2). MATLAB simulations were used to validate the model and to investigate the effect of the iteration time step. Circuit Implementation Circuit implemented from system diagram. Integrator adapted from [4]. Hysteretic Reset Controller The hysteresis in the reset allows the internal integrator node to fully discharge. Discharge rate, and thus falling edge waveform shape are set by the integrator capacitor and Rds(on) and the drain resistance of the switch. Acknowledgments We thank David Tauck for providing invaluable expertise, laboratory space and equipment. Conclusions We have presented a mathematically based hardware neural implementation that displays behavior consistent with excitable membranes. A discrete circuit implementation makes this available to experimentalist in robotics, neuroscience and control systems research. The circuit was designed using single rail capable components and circuit topologies to allow single supply operation. Single supply operation is easily obtained by tying all Vss terminals to ground. Further improvements are possible by using the outlined approach to reduce component count and improve circuit performance. An analog circuit implementation of a quadratic integrate and fire neuron Eric Basham, David W. Parent Department of Electrical Engineering, San Jose State University, San Jose, California 95192 For further information Please contact basham.eric@gmail.com. MATLAB code, simulation files, circuit board files and BOM are available upon request. Key References [1]L. Smith, "Implementing Neural Models in Silicon," in Handbook of Nature-Inspired and Innovative Computing, 2006, pp. 433-475. [2]E. M. Izhikevich, Dynamical systems in neuroscience : the geometry of excitability and bursting. Cambridge, Mass.: MIT Press, 2007. [3]O. Beker, C. Hollot, Y. Chait, and H. Han, "Fundamental properties of reset control systems," Automatica, vol. 40, pp. 905-915, 2004. [4]G. Brisebois, "Instrumentation amp makes noninverting integrator," EDN, p. 2, sept. 19, 2002. Results Biological Relevance Bursting response recorded intracellularly from a current clamped neuron. This technique can be used to extract circuit parameters. SPICE Simulation of Nonlinear Circuit Response to spiking and ramp inputs shows the circuit operates as an integrator with a single stable equilibrium Response to pulsed input demonstrating system bistability. Bistability occurs when the reset is above the second stable equilibrium. Circuit Measurement Recorded circuit spiking Spiking rate evaluation of circuit response. A nearly linear relationship between frequency and injected current demonstrates the circuit is operating as a class I excitable system Σ ∫ R b * y(t) x(t) { { { multiplier integrator reset