Scaling Issues for VLSI Implementations of Biologically
Accurate Neurons and Central Pattern Generators
Daniel DeBolt
Yong-Bin Kim
Joseph Ayers
Electrical and Computer Engineering
Northeastern University
Boston, Massachusetts
ddebolt@ece.neu.edu
Electrical and Computer Engineering
Northeastern University
Boston, Massachusetts
ybk@ece.neu.edu
Marine Science Center
Northeastern University
Nahant, Massachusetts
lobster@neu.edu
Abstract— Research on the biomimetic control of robots has
been recently progressing. This approach allows lifelike and
robust movement by mimicking the control mechanisms of
rhythmic motions seen in behaving animals. Adapting these
control models to micro-robots presents challenges. Operating
nano-scale CMOS circuits at biologically appropriate
frequencies is challenging in analog designs due primarily to
capacitor sizing limitations. An analysis of the limitations of lowpower neural models is given and future alternative solutions to
this problem are suggested for the creation of CPGs in low
power CMOS circuits for micro-robotic control.
I.
muscles that have channelrhodopsin genes inserted to render
them sensitive to blue light, grown on an organic LED to
mediate excitation/contraction coupling. Engineered sensor
cells responsive to rampamiacin or light will guide locomotion
direction and behavior. An analog CPG network will act as the
central nervous system while a chemical battery powers the
entire robot.
INTRODUCTION
The rhythmic behavior of animals is mediated by central
pattern generator (CPG) circuits in the central nervous system
[1]. A CPG is a network of neurons and synapses that controls
the underlying patterns of walking, swimming, breathing and
other motor functions [2]. These patterns can be generated in
the absence of sensory feedback and patterned neuronal input
from the brain [3]. The biological mechanisms are limited to
endogenous pacemaker, half center, recurrent cyclic inhibition
and bursting eletrotonic syncitial models [4]. Invertebrate
CPG networks are relatively simple and technically accessible
in vitro and thus subject to detailed cellular analysis [5].
A. Central Pattern Generators in Robotics
Utilizing central pattern generators for control of robotic
locomotion has been an active research topic [6]. Mimicking
the locomotion of animal models allows a robot to robustly
deal with obstacles and collisions in a rapidly changing
environment [7]. Researchers have created several iterations
of CPG based Lobster and Lamprey robots for aquatic mine
and object detection [8][9]. Neuromorphic sensors were used
as inputs and nitinol wires were used as artificial muscles
which could be controlled similarly to the way live muscles
are in natural organisms [10]. The above mentioned examples
share a similar control method: a high level finite state
machine sending commands to various CPGs for locomotion.
B. The Cyberplasm Research
This paper investigates the scaling issues about
biologically accurate neurons and central pattern generation
for the Cyberplasm program that is an effort to apply
principles of synthetic biology to the development of
biomimetic micro-robots through a collaboration of specialists
at several institutions [11]. The robot will use synthetic
This project was supported by the US National Science Foundation under
grant CBET-0943345
Figure 1. Figure 1. Diagram of the Cyberplasm robot. A:Synthic muscles,
B:Electronic Nervous System CPG, C:Environment sensors, D:Battery, E:
Kapton chassis.
II.
NEURAL MODELS
A. Overview and Lower Order Models
The goal of the present work is to form a micro central
pattern generator for swimming [9]. Our alternatives are to
form a CPG of computed neurons, or to utilize a CPG based
on aVLSI analog computer based on nonlinear dynamical
models of bursting neurons [12]. Izhikevich outlines in [13]
the differences between various neural models and the 20
identified firing outputs of each model. The simplest model is
the one dimensional “Integrate and Fire” model. A number of
currents charge and discharge a capacitor to obtain the
integrated output voltage. The firing and resetting of the
neuron occurs when a certain threshold voltage, Vth is reached.
The current voltage equation of a capacitor is shown in (1)
while a leaky integrate and fire models equation is shown in
(2). Parameter u is the leakage coefficient.
𝑖(𝑡) = 𝐶

𝑑𝑥
𝑑𝑡
=𝐼−𝑢∙𝑥
𝑑𝑉
𝑑𝑡

𝑥 ← 𝑉𝐿 𝑤ℎ𝑒𝑛 𝑥 = 𝑉𝑡ℎ 


B. The Hindmarsh-Rose Neural Model
The “Hindmarsh-Rose” neural model describes the action
potential of a biological realistic neuron. This model consists
of three coupled differential equations, one of which models
the membrane potential (x) with the other two considered as
the spiking (recovery current, y) and bursting (adaption
current, z) variables [14]. The addition of a fourth equation
models even slower calcium exchange dynamics which give
rise to increased regions of chaotic operation and is covered in
[15]. Figure 2 is an example of output of the HR neural
equations when I=3.024 given in (3-5).
𝑑𝑥

𝑑𝑡
= 𝑎𝑦 + 𝑏𝑥 2 − 𝑐𝑥 3 − 𝑑𝑧 + 𝐼
𝑑𝑦

= 𝑒 − 𝑓𝑥 2 − 𝑦

= 𝜇(𝑆(𝑥 + ℎ) − 𝑧)

𝑑𝑡
𝑑𝑧

𝑑𝑡
TABLE I.
a
1
f
5.0128
b
3
µ
0.0021

HINDMARSH ROSE COEFFICIENTS
c
1
S
3.966
d
0.99
h
1.605
e
1.01
I
0-3.024
This model has been constructed utilizing discrete
components in [15] and proposed in CMOS [16]. An
alternative computed method using a discrete time twodimensional map obtains similar dynamics to the HindmarshRose model and is the basis of our macro swimming robot [9].
The model in its simplest form uses only 3 multiplies and 5
additions per update which is computationally simpler than
numerical integration of the differential equations listed
previously [17]. This allows map based neurons to be
massively scaled on simple hardware. Another advantage is
that highly complex maps require annealing of the synaptic
weights which can be done easily in software [12].
on the order of 0.3–10.0 seconds are common while fast
spiking neurons are on the order of 1–10 milliseconds [18]. A
capacitor is used to store the state variable in analog designs.
The speed at which this capacitor charges and discharges sets
the neurons overall output time constant. A 1 pF capacitor
being charged by a 10 nA current charges to 1 V in 0.0001
seconds, much faster than the desired spike frequency.
Reference [19] creates a spiking neuron in CMOS at a realistic
output frequency of 33 Hz. This was accomplished using
subthreshold operation and by locating the state capacitor offchip. This allows a larger capacitor to be used than what is
available (and costly in terms of area) in standard CMOS
process.
Many proposed CPG models simply operate too fast to
control an electromechanical system such as a micro-robot.
For example, a CPG circuit for rhythmic chewing is proposed
in [20]. The time between chewing is shown to be 0.2 µS; this
is orders of magnitude faster than its biological equivalent due
in part to the sizing of the 60 pF capacitor. Reference [21]
proposes a neuron based on the Izhikevich model. The burst
length reported is on the order of a few microseconds, which
is much faster than the hundreds of milliseconds of its
biological counterpart. This increase in speed is suitable for
computational neural networks such as machine vision or
learning but operates too quickly for operation of a physical
electromechanical device. It is evident that capacitor sizing is
the main limiting factor for biologically accurate analog
CMOS implementations.
B. Analog Computer Scaling
The Hindmarsh-Rose implementations in [15] and [16] are
both based on the principles of analog computers and solve the
coupled differential equations using integration. The basic
building blocks of analog computers are summing junctions
and integrators constructed with operational amplifiers [22].
Figure 3 and equation (6) show an integrator op-amp
configuration and governing equation; Vo is the initial state
voltage of the system.
C
R
Vin
Vout
+
Figure 3. A standard operational amplifier based integrator.

Figure 2. Matlab simulation of the Hindmarsh Rose neuron with I=3.024
III.
SCALLING CHALLENGES
A. Capacitor Sizing and Time Scales
Time scales seen in neurobiology are much slower than
those seen in most CMOS applications. Burst length times are
𝑉out = −
1 𝑡
∫ 𝑉
𝑅𝐶 0 in
𝑑𝑡 + 𝑉o 

A particular challenge of designing analog computers was
insuring the solution did not saturate any op-amps to the
supply rails and a solution was found in a timely manner. For
example, solving problems that use large values or long time
periods such as predator-prey models require scaling of the
equations to be solved within the limits of an analog
computer. There are two types of scaling; one is output
magnitude scaling and the other is time scaling. Scaling is
accomplished by replacing a variable with a coefficient and
new representation variable [22]. Plugging in t=T/Ts and
x=X·xm (similarly for variables y and z) into (3-5) yields the
magnitude and time scaled HR equations (7-9). Values less
than one reduce the magnitudes or time scale, the opposite for
values greater than one.

𝑑𝑋
𝑑𝑇
=
1
𝑇𝑠
(𝑎
𝑦𝑚
𝑥𝑚
𝑌 + 𝑏𝑥𝑚 𝑋 2 − 𝑐𝑥𝑚 2 𝑋 3 − 𝑑

𝑍+

𝑑𝑌


𝑧𝑚
𝑥𝑚
=
𝑑𝑇
𝑑𝑍
𝑑𝑇
=
1
𝑇𝑠
1
𝑇𝑠
(𝑒
1
𝑦𝑚
(𝜇 (𝑆 (
−𝑓
𝑥𝑚
𝑧𝑚
1
𝑥𝑚
𝐼)

𝑥𝑚 2
𝑦𝑚
𝑋+ℎ
2
𝑋 − 𝑌)
1
𝑧𝑚
) − 𝑍))


antagonistic oscillation while adjacent segments on the same
side exhibit a phase delay that varies with period [9]. This
simplifies the creation of complex patterns by loosely
following the pacemaker model using simple voltage
controlled oscillators and phase delay oscillators. Integrate and
fire neurons will be inhibited by these oscillators and sensors
acts as the excitatory inputs. The DC average is taken as the
output [23]. Figure 5 shows the possible layout for a simple
swimming pattern generator. Neurons A-D are inhibited
(shown as a bubble at the inputs) by the oscillator. The phase
delay propagates the motion down the spine of the robot. Not
shown are the environmental inputs from sensors to speed up /
slow down neuron spiking thus altering the DC output.
Looking at equations (8) the only coefficient in front of the
Y variable is 1/Ts, this allows a simple explanation of the RC
ratio issue. Implementing this single term using the op-amp
integrator, shown in figure 3, yields the ratio shown in (10). If
C=1pF and Ts=2.2E-3, R must be 2.2 GΩ but only 5 MΩ if
Ts=5E-6. The heavier the time scaling, the more reasonable
RC ratios become but at the expense of not running at
biological speeds.
1

𝑇𝑠
=
1
𝑅𝐶
𝑇
→ 𝑅 = 𝑠
𝐶

Figure 5. An oscillator example for a simple swimming pattern generator.
Figure 4. Comparasion of time scaling factors, the first with no scaling,
second set to the scaling used in [15], the third set to that used in [16].
Implementation [16] is operating roughly 400 times faster
than [15] due completely to the time scaling involved. This is
another limitation of capacitors sizing in CMOS. Because the
capacitor was fixed to 3 pF, the time had to be scaled heavily
to make the resistor inputs to the integrator reasonable to
implement (in the range of a few mega-ohms). This limitation
is not seen in the discrete component version as leaded
ceramic capacitors come in a large range of values. Figure 4
outlines the difference between the given implementations.
IV.
FUTURE RESEARCH DIRECTIONS
A. Ulitizing Lower Order Models and Oscillators
Actual swimming systems involve multiple CPGs in
several body segments. The CPGs on the two sides of the
midline are typically coordinated among themselves by
reciprocal inhibitory connections to generate a rigid
B. Model Simplification
A novel approach to implementation of the Hindmarsh
Rose neuron equations in CMOS was proposed by [24]. The
group performed an in-depth analysis of 2D and 3D neuron
bursting structure and proposed a method to simplify the
implementation of the HR neuron. Modeling of a 2D neuron
in CMOS was accomplished with two types of voltage to
current loads. One was a current output Schmitt trigger and
the other was a quadratic load, both of which charge a
capacitor. This structure represents the x and y variables in the
original equations and only requires a single capacitor. The z
or adaption term was implemented with an integrator circuit.
Another advantage is that the capacitors all have a common
connection for easier off-chip placement. Also because there is
more control over the currents involved, deep subthreshold
design can reduce the size of capacitances needed by
operating in the low nano-amp regime. Figure 6 shows the
Simulink model while figure 7 compares the model to solved
HR equations at differing inputs. The updating and
implementing in low voltage CMOS is currently being
investigated.
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Figure 6. Simulink simulation of the updated model proposed in [24].
[13]
[14]
[15]
[16]
[17]
[18]
Figure 7. Comparasion of numericaly solved HR equations to the
simulation model shown in figure 6 for severial inputs.
V.
CONCLUSION
A general introduction to neural models and CPG control
being used in robotics is presented. Behavioral time constants
can be quite long and therefore require the use of large
capacitors. This temporal scaling aspect is often overlooked in
many papers in literature but is important for accurate
locomotion control. Utilizing analog computers to solve
differential equations also requires scaling in magnitude and
time. The small capacitances allowed in CMOS limit the
application to fast running neurons unless they are placed off
chip. Reduced order oscillators and models are the future
direction as subthreshold design can be employed to possibly
reduce the capacitance sizes needed
REFERENCES
[1]
[2]
[19]
Pearson, K. G.;, "Common principles of motor control in vertebrates
and invertebrates," Annu Rev Neurosci, vol. 16, pp. 265-97, 1993.
Selverston, A. I.;, "A Neural Infrastructure for Rhythmic Motor
Patterns," Cellular and Molecular Neurobiology, Vol 25, No 2, 2005.
[20]
[21]
[22]
[23]
[24]
Delcomyn, F.;, "Neural basis of rhythmic behavior in animals,"
Science, vol. 210, pp. 492-8, 1980.
Selverston, A. I.;, "Are central pattern generators understandable?," The
Behavioral and Brain Sciences, vol. 3, pp. 535-571, 1980.
Selverston, A. I.; Moulins, M.;, The Crustacean Stomatogastric System.
Berlin and Heidelberg: Springer-Verlag, 1987.
Ayers, J.; Davis, J.; Rudolph, A.;, Neurotechnology for Biomimetic
Robots. Cambridge, MA: MIT Press, 2002.
Blustein, D.; Ayers, J.;, "A conserved network for control of arthropod
exteroceptive optical flow reflexes during locomotion," Lecture Notes
in Artificial Intelligence, vol. 6226, pp. 72-81, 2010.
Ayers, J.; Witting, J.;, "Biomimetic Approaches to the Control of
Underwater Walking Machines," Phil. Trans. R. Soc. Lond. A, vol.
365, pp. 273-295, 2007.
Westphal, A.; Rulkov, N.; Ayers, J.; Brady, D.; Hunt, M.;, "Controlling
a Lamprey-Based Robot with and Electronic Nervous System," Smart
Structures and Materials, vol. in press., 2010.
Witting, J.; Ayers, J.; Safak, K.;, "Development of a biomimetic
underwater ambulatory robot: advantages of matching biomimetic
control architecture with biomimetic actuators," Proc. Spie, vol. 4196,
pp. 54-61, 2003.
http://www.cyberplasm.net
Ayers, J.; Rulkov, N.; Knudsen, D.; Kim, Y.B.; Volkovskii, A.;
Selverston, A;, ``Controlling Underwater Robots with Electronic
Nervous Systems," Special Issue on Biologically Inspired Robots,
Applied Bionics and Biomechanics, Vol. 7, No. 1, March 2010, pp. 5767.
Izhikevich, E.M.; , "Which model to use for cortical spiking neurons?,"
Neural Networks, IEEE Transactions on , vol.15, no.5, pp.1063-1070,
Sept. 2004
Hindmarsh, J.L.; Rose, R.M.; , ”A Model of Neuronal Bursting using
Three Coupled First Order Differential Equations”, Proceedings of the
Royal Society of London, pp.87-102, 1984
Pinto, R. D.; Varona, P.; Volkovskii, A. R.; Szucs, A.; Abarbanel, H.
D.; Rabinovich,M. I.;, "Synchronous behavior of two coupled
electronic neurons," Phys Rev E., vol. 62, pp. 2644-56, Aug 2000
Lee, Y.J.; Lee, J.; Kim, K.K.; Kim, Y.B.;, "A Low Power CMOS
Electronic Central Pattern Generator Design for Biomimetic
Underwater Robot," Elsevier Neurocomputing Journal, Vol 71, Issue 13, December 2007, pp. 284-296.
Rulkov, N.F; , "Modeling of spiking-bursting neural behavior using
twodimensional map.," Phys Rev E, vol. 65, 041922, 2002.
Shepherd, G.M.;, Neurobiology: Third Edition. New York, NY: Oxford
University Press, 1994.
Alvado, L.; Tomas, J.; Renaud-Le Masson, S.; Douence, V.; , "Design
of an analogue ASIC using subthreshold CMOS transistors to model
biological neurons," Custom Integrated Circuits, 2001, IEEE
Conference on. , vol., no., pp.97-100, 2001.
Hasan, S.M.R.; Xu, W.L.;, "Low-voltage analog current-mode Central
Pattern Generator circuit for robotic chewing locomotion using
130nanometer CMOS technology," Microelectronics, 2007. ICM 2007.
Internatonal Conference on , vol., no., pp.155-160, 29-31 Dec. 2007.
Wijekoon, J.H.B.; Dudek, P.; , "Spiking and Bursting Firing Patterns of
a Compact VLSI Cortical Neuron Circuit," Neural Networks, 2007.
IJCNN 2007. International Joint Conference on , vol., no., pp.13321337, 12-17 Aug. 2007.
Blum, J.J.;, Introduction to Analog Computation. Harcourt Brace
Jovanovich, Inc., 1969.
Tenore, F.; Etienne-Cummings, R.; Lewis, M.A.; , "A programmable
array of silicon neurons for the control of legged locomotion," Circuits
and Systems, 2004. ISCAS '04. Proceedings of the 2004 International
Symposium on , vol.5, no., pp. V-349- V-352 Vol.5, 23-26 May 2004.
Merlat, L.; Silvestre, N.; Merckle, J.; , "A Hindmarsh and Rose-based
electronic burster," Microelectronics for Neural Networks, 1996.,
Proceedings of Fifth International Conference on , vol., no., pp.39-44,
12-14 Feb 1996.