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
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;
Design Approach
Results
Key Points
Biological Relevance
• True spiking behavior
• Mathematically tractable
• Biologically relevant testing and extraction
System Diagram for a Reset Controller
• 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.
b
Σ
x(t)
R
∫
y(t)
Bursting response recorded intracellularly from a
current clamped neuron. This technique can be used
to extract circuit parameters.
SPICE Simulation of Nonlinear Circuit
*
The system diagram for a reset control system [3].
Reset (R) occurs when the output reaches a set
threshold.
MATLAB Discrete Time Simulation
Response to spiking and ramp inputs shows the
circuit operates as an integrator with a single stable
equilibrium
• 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.
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.
Response to pulsed input demonstrating system
bistability. Bistability occurs when the reset is
above the second stable equilibrium.
Circuit Measurement
Circuit Implementation
multiplier
Modeling
The quadratic integrate and fire follows from a
reduced form
(1)
dV
 F (V )  I
dt
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 V2 for F(V) leads to the one
dimensional system
(2)
dV
2
V  I
dt
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
VPeak, then V => VReset
integrator
reset
{
{
{
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.
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
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.
Key References
Acknowledgments
[1] L. Smith, "Implementing Neural Models in Silicon," in Handbook of Nature-Inspired and Innovative
Computing, 2006, pp. 433-475.
We thank David Tauck for providing invaluable expertise, laboratory
space and equipment.
[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.
For further information
Please contact basham.eric@gmail.com. MATLAB code, simulation
files, circuit board files and BOM are available upon request.