Analog VLSI Model of Interseginental 
Coordination With Nearest-Neighbor Coupling 
Girish N. Patel 
girish@ ece.gatech.edu 
Jeremy H. Ho!leman 
jeremy @ece.gatech.edu 
Stephen P. DeWeerth 
steved @ ece.gatech.edu 
School of Electrical and Computer Engineering 
Georgia Institute of Technology 
Atlanta, Ga. 30332-0250 
Abstract 
We have a developed an analog VLSI system that models the coordina- 
tion of neurobiological segmental oscillators. We have implemented and 
tested a system that consists of a chain of eleven pattern generating cir- 
cuits that are synaptically coupled to their nearest neighbors. Each pat- 
tern generating circuit is implemented with two silicon Morris-Lecar 
neurons that are connected in a reciprocally inhibitory network. We dis- 
cuss the mechanisms of oscillations in the two-cell network and explore 
system behavior based on isotropic and anisotropic coupling, and fre- 
quency gradients along the chain of oscillators. 
1 INTRODUCTION 
In recent years, neuroscientists and modelers have made great strides towards illuminat- 
ing structure and computational properties in biological motor systems. For example, 
much progress has been made toward understanding the neural networks that elicit rhyth- 
mic motor behaviors, including leech heartbeat (Calabrese and De Schutter, 1992), crus- 
tacean stomatogastric mill (Selverston, 1989) and tritonia swimming (Getting, 1989). In 
particular, segmented locomotory systems, such as those that underlie swimming in the 
lamprey (Cohen and Kiemel, 1993, Sigvardt, 1993, Grillner et al, 1991) and in the leech 
(Friesen and Pearce, 1993), are interesting from an quantitative perspective. In these sys- 
tems, it is clear that coordinated motor behaviors are a result of complex interactions 
among membrane, synaptic, circuit, and system properties. However, because of the lack 
of sufficient neural underpinnings, a complete understanding of the computational princi- 
ples in these systems is still lacking. Abstracting the biophysical complexity by modeling 
segmented systems as coupled nonlinear oscillators is one approach that has provided 
much insight into the operation of these systems (Cohen et al, 1982). More specifically, 
this type of modeling work has illuminated computational properties that give rise to 
phase constancy, a motor behavior that is characterized by intersegmental phase lags that 
are maintained at constant values independent of swimming frequency. For example, it 
has been shown that frequency gradients and asymmetrical coupling play an important 
role in establishing phase lags of correct sign and amplitude (Kopell and Ermentrout, 
1988) as well as appropriate boundary conditions (Williams and Sigvardt, 1994). 
Although theoretical modeling has provided much insight into the operation of interseg- 
720 G. N. Patel, J. H. Holleman and S. P DeWeerth 
mental systems, these models have limited capacity for incorporating biophysical proper- 
ties and complex interconnectivity. Software and/or hardware emulation provides the 
potential to add such complexity to system models. Additionally, the modularity and reg- 
ularity in the anatomical and computational structures of intersegmental systems facili- 
tate scalable representations. These factors make segmented systems particularly viable 
for modeling using neuromorphic analog very large-scale integrated (aVLSI) technology. 
In general, biological motor systems have a number of properties that make their real- 
time modeling using aVLSI circuits interesting and approachable. Like their sensory 
counterparts, they exhibit rich emergent properties that are generated by collective archi- 
tectures that are regular and modular. Additionally, the fact that motor processing is at the 
periphery of the nervous system makes the analysis of the system behavior accessible due 
to the fact that output of the system (embodied in the motor actions) is observable and 
facilitates functional analysis. 
The goals in this research are i) to study how the properties of individual neurons in a net- 
work affect the overall system behavior; (ii) to facilitate the validation of the principles 
underlying intersegmental coordination; and (iii) to develop a real-time, low power, 
motion control system. We want to exploit these principles and architectures both to 
improve our understanding of the biology and to design artificial systems that perform 
autonomously in various environments. In this paper we present an analog VLSI model of 
intersegmental coordination that addresses the role of frequency gradients and asymmet- 
rical coupling. Each segment in our system is implemented with two silicon model neu- 
rons that are connected in a reciprocally inhibitory network. A model of intersegmental 
coordination is implemented by connecting eleven such osciIlators, with nearest neigh- 
bor coupling. We present the neuron model, and we investigate the role of frequency gra- 
dients and asymmetrical coupling in the establishment of phase lags along a chain these 
neural oscillators. 
2 NEURON MODEL 
In order to produce bursting activity, a neuron must possess "slow" intrinsic time con- 
stants in addition to the "fast" time constants that are necessary for the generation of 
spikes. Hardware models of neurons with both slow and fast time constants have been 
designed based upon previously described Hodgkin-Huxley neuron models (Mahowald 
and Douglas, 1991). Although these circuits are good models of their biological counter- 
parts, they are relatively complex, with a large parameter space and transistor count, lim- 
iting their usefulness in the development of large-scale systems. It has been shown 
(Skinner, 1994), however, that pattern generation can be represented with only the slow 
time constants, creating a system that represents the envelope of the bursting oscillations 
without the individual spikes. Model neurons .with only slow time constants have been 
proposed by Morris and Lecar (1981). 
We have implemented an analog VLSI model of the Morris-Lecar Neuron (Patel and 
DeWeerth, 1997). Figure 1 shows the circuit diagram of this neuron. The model consists 
of two state variables: one corresponding to the membrane potential (V) and one corre- 
sponding to a slow variable (N). The slow variable is obtained by delaying the mere- 
VHigl 
V_- 
Ci- 
[ext 
ilsyn 
4 4 
Figure 1: Circuit diagram of silicon Morris-Lar Neuron 
ov 
A VLSIModel oflntersegmental Coordination 721 
brane potential by way of an operational transconductance amplifier (OTA) connected in 
unity gain configuration with load capacitor C 2 . The membrane potential is obtained by 
injecting two positive currents (Iex t and i. ) and two negative currents (it. and isy  ) into 
capacitor C]. Current i u raises the membrane potential towards VHig h when the mem- 
brane potential increases above V H , whereas current it. lowers the membrane potential 
towards V[o, when the delayed membrane potential increases above Vt.. The synaptic 
current, isyn, activates when the presynaptic input, Vhe, increases above Vthh. 
Assuming operation of transistors in weak inversion and synaptic coupling turned off 
( isyn = 0 ) the equations of motion for the system are. 
exp0c(V - VH)/UT) exp0c(N - VL)/UT) 
C 1 I2 = I 1 (V, N) = IextOt P + I H I + exp0c(V- VH)/UT )tP - ILl + exp0c(N - VL)/UT) rs 
C2/ - 12(V, N) = Ixtanh0c(V- N)/(2UT))(I -exp((N- Vdd )/UT))) 
The terms Ctp and GI, N, where Ctp = 1 - exp (V - VHigh)/U T and 
G/, N -' 1 -- exp(VLo, - V)/U T , correspond to the ohmic effect of transistor MI and M2 
respectively. c corresponds to the back-gate effect of a MOS transistor operated in weak 
inversion, and UT corresponds to the thermal voltage. We can understand the behavior of 
this circuit by analyzing the geometry of the curves that yield zero motion (i.e., when 
I](V,N) = 12(V,N ) = 0). These curves, referred to as nullclines, are shown in 
Figure 2 for various values of external current. 
The externally applied constant current (Ie,t), which has the effect of shifting the V 
nullcline in the positive vertical direction (see Figure 2), controls the mode of operation 
of the neuron. When the V- and N nullclines intersect between the local minimum and 
local maximum of the V nullcline (P2 in Figure 2), the resulting fixed point is unstable 
and the trajectories of the system approach a stable limit-cycle (an endogenous bursting 
mode). Fixed points to the left of the local minimum (P1 in Figure 2) or to the right of the 
local maximum (P3 in Figure 2) are stable and correspond to a silent mode and a tonic 
mode of the neuron respectively. An inhibitory synaptic current (isyn) has the effect of 
shifting the V nullcline in the negative vertical direction; depending on the state of a pre- 
synaptic cell, isy,can dynamically change the mode of operation of the neuron. 
3 TWO-CELL NETWORK 
When two cells are connected in a riprocally inhibitory network, the two cells will 
oscillate in antiphase depending on the conditions of the free and inhibited cells and the 
value of the synaptic threshold (Skinner et. al, 1994). We assume that the turn-on charac- 
teristics of the synaptic current is sharp (valid for large V.uip - V, ) such that when the 
membrane potential of a presynaptic cell reaches above Vthh, the postsynaptic cell is 
immediately inhibited by application of negative current Isy n tO its membrane potential. 
2 4 2.45 2.5 2 55 2.6 
V 
- - Ioxt=SnA 
-- Ioxt = 2.5 nA 
Ioxt = 0 nA 
  trajectories 
Figure 2: Nullcline and corresponding trajectories of silicon Morris-Lecar neuron. 
722 G. N. Patel, J. H. Holleman and S. P. DeWeerth 
If the free cell is an endogenous burster, the inhibited cell is silent, and the synaptic 
threshold is between the local maximum of the free cell and the local minimum in the 
inhibited cell, the mechanism for oscillation is due to intrinsic release. This mechanism 
can be understood by observing that the free cell undergoes rapid depolarization when its 
state approaches the local maximum thus facilitating the release of the inhibited cell. If 
the free cell is tonic and the inhibited cell is an endogenous burster (and conditions on 
synaptic threshold are the same as in the intrinsic release case), then the oscillations are 
due to an intrinsic escape mechanism. This mechanism is understood by observing that 
the inhibited cell undergoes rapid hyperpolarization, thus escaping inhibition, when its 
state approaches the local minimum. Note, in both intrinsic release and intrinsic escape 
mechanisms, the synaptic threshold has no effect on oscillator period because rapid 
changes in membrane potential occur before the effect of synaptic threshold. 
When the free cell is an endogenous burster, the inhibited cell is silent, and the synaptic 
threshold is to the right of the local maximum of the free cell, then the oscillations are 
due to a synaptic release mechanism. This mechanism can be understood by observing 
that when the membrane potential of the free cell reaches below the synaptic threshold, 
the free cell ceases to inhibit the other cell which causes the release of the inhibited cell. 
When the free cell is tonic, and the inhibited cell is an endogenous burster, and the synap- 
tic threshold is to the left of the local minimum of the inhibited cell, then the oscillations 
are due to a synaptic escape mechanism. This mechanism can be understood by observ- 
ing that when the membrane potential of the inhibited cell crosses above the synaptic 
threshold, then the membrane potential of the inhibited cell is large enough to inhibit the 
free cell. Note, increasing the synaptic threshold has. the effect of increasing oscillator fre- 
quency for the synaptic release mechanism, however, oscillator frequency under the syn- 
aptic escape mechanism will decrease with an increase in the synaptic threshold. 
By setting VHi~ h -- VLo w to a large value, the synaptic currents appear to have a sharp cut- 
off. However, because transistor currents saturate within a few thermal voltages, the 
nullclines due to the membrane potential appear less cubic-like and more square-like. 
This does not effect the qualitative behavior of the circuit, as we are able to produce anti- 
phasic oscillations due to all four mechanisms. Figure 3 illustrates the four modes of 
oscillations under various parameter regimes. Figure 3A show typical waveforms from 
two silicon neurons when they are configured in a reciprocally inhibitory network. The 
oscillations in this case are due to intrinsic release mechanism and the frequency of oscil- 
lations are insensitive to the synaptic threshold. When the synaptic threshold is increased 
above 2.5 volts, the oscillations are due to the synaptic release mechanism and the oscilla- 
tor frequency will increase as the synaptic threshold is increased, as shown in Figure 3C. 
By adjusting Iex t such that the free cell is tonic and the inhibited cell bursts endoge- 
nously, we are able to produce oscillations due to the intrinsic escape mechanism, as 
4 
1' 
0 2 4 
2 
1 
0 1 2 3 4 
Figure 3: Experimental results from two neurons connected in a reciprocally inhibitory 
network. Antiphasic oscillations due to intrinsic release mechanism (A), and intrinsic 
escape mechanism (B). Dependence of oscillator frequency on synaptic threshold for the 
synaptic release mechanism (C) and synaptic escape mechanism (D) 
A VLSIModel oflntersegmental Coordination 723 
shown in Figure 3B. As the synaptic threshold is decreased below 0.3 volts, the oscilla- 
tions are caused by the synaptic escape mechanism and oscillator frequency increases as 
the synaptic threshold is decreased. The sharp transition between intrinsic and synaptic 
mechanisms is due to nullclines that appear square-like. 
4 CHAIN OF COUPLED NEURAL OSCILLATORS 
In order to build a chain of pattern generating circuits with nearest neighbor coupling, we 
designed our silicon neurons with five synaptic connections. The connections are made 
using the synaptic spread rule proposed by Williams (1990). The rule states that a neuron 
in any given segment can only connect to neurons in other segments that are homologues 
to the neurons it connects to in the local segment. Therefore, each neuron makes two 
inhibitory, contralateral connections and two excitatory, ipsilateral connections (as well a 
single inhibitory connection in the local segment). The synaptic circuit, shown in the 
dashed box in Figure 1, is repeated for each inhibitory synapse and its complementary 
version is repeated for the excitatory synapses. In order to investigate the role of fre- 
quency gradients, each neural oscillator has an independent parameter, Iext, for setting 
the intrinsic oscillator period. A set of global parameters, I L , I H , I x , VH, VL, Vigh, 
and Vo, control the mechanism of oscillation. These parameters are set such that the 
mechanism of oscillation is intrinsic release. 
Because of inherent mismatch of devices in CMOS technology, a consequence in our 
model is that neurons with equal parameters do not necessarily behave with similar per- 
formance. Figure 4A illustrates the. intrinsic oscillator period along the length of system 
when all neurons receive the same parameters. When the oscillators are symmetrically 
coupled, the resulting phase differences along the chain are nonzero, as shown in 
Figure 4B. The phase lags are negative with respect to the head position, thus the default 
swim direction is backward. As the coupling strength is increased, indicated by the lower- 
most curves in Figure 4B, the phase lags become smaller, as expected, but do not dimin- 
ish to produce synchronous oscillations. When the oscillators are locked to one common 
frequency, f, theory predicts (Kopell and Ermentrout, 1988) that the common fre- 
quency is dependent on intrinsic oscillator frequencies, and coupling from neighboring 
oscillators. In addition, under the condition of weak coupling, the effect of coupling can 
be quantified with coupling functions that depends on the phase difference between 
neighboring oscillators: 
i i 
- = O)i + HA()i) + HD('i-1) 
where, 0 i is the intrinsic frequency of a given oscillator, H A and H a are coupling func- 
tions in the ascending and descending directions respectively, and i is the phase differ- 
ence between the (i+l)th and ith oscillator. This equation suggests that the phase lags 
must be large in order to compensate for large variations in the intrinsic oscillator fre- 
A B 
1.4 o o o 
1.3t o  300 
o.lot' 
2 4 6 8 10 2 4 6 8 10 
C 
15 
-ltl 
-151 
2 4 6 8 10 
Figure 4: Experimental data obtained from system of coupled oscillators. 
724 G. N. Patel, J. H. Holleman and S. P DeWeerth 
quencies. 
Another factor that can effect the intersegmental phase lag is the degree of anisotropic 
coupling. To investigate the effect of asymmetrical coupling, we adjusted Iex t in each 
segment so to produce uniform intrinsic oscillator periods (to within ten percent of 115 
ms) along the length of the system. Asymmetrical coupling is established by maintaining 
Vavg -- (V^s c - Vos)/2 at 0.7 volts and varying Vaet a -- V^s c - VD from 0.4 to - 0.4 
volts. VAs C and VD correspond to the bias voltage that sets the synaptic conductance 
of presynaptic inputs arriving from the ascending and descending directions respectively. 
Throughout the experiment, the average of inhibitory (contralateral) and excitatory (ipsi- 
lateral) connections from one direction are maintained at equal levels. Figure 4C shows 
the intersegmental phase lags at different levels of anisotropic coupling. Stronger ascend- 
ing weights (Vdelt a = 0.4, 0.2 volts) produced negative phase lags, corresponding to back- 
ward swimming, while stronger descending connections (Vaet a = -0.4, -0.2 volts) 
produce positive phase lags, corresponding to backward swimming. Although mathemati- 
cal models suggest that stronger ascending coupling should produce forward swimming, 
we feel that the type of coupling (inhibitory contralateral and excitatory ipsilateral con- 
nections) and the oscillatory mode (intrinsic release) of the segmental oscillators may 
account for this discrepancy. 
To study the effects of frequency gradients, we adjusted Iex t at each segment such that the 
that the oscillator period from the head to the tail (from segment 1 to segment 11) varied 
from 300 ms to 100 ms in 20 ms increments. In addition, to minimize the effect of asym- 
metrical coupling, we set Vavg = 0.8 volts and Vaet a = 0 volts. The absolute phase 
under these conditions are shown in Figure 5A. The phase lags are negative with respect 
to the head position, which corresponds to backward swimming, With a positive fre- 
quency gradient, head oscillator at 100 ms and tail oscillator at 300 ms, the resulting 
phases are in the opposite direction, as shown in Figure 5B. These results are consisfent 
with mathematical models and the trailing oscillator hypothesis as expounded by Grill- 
her et. al. (1991 ). 
5 CONCLUSIONS AND FUTURE WORK 
We have implemented and tested an analog VLSI model of intersegmental coordination 
with nearest neighbor coupling. We have explored the effects of anisotropic coupling and 
frequency gradients on system behavior. One of our results--stronger ascending connec- 
tions produced backward swimming instead of forward swimming--is contrary to the- 
ory. There are two factors that may account for this discrepancy: i) our system exhibits 
inherent spatial disorder in the parameter space due to device mismatch, and ii) the oper- 
ating point at which we performed the experiments retains high sensitivity to neuron 
parameter variations and oscillatory modes. We are continuing to explore the parameter 
space to determine if there are more robust operating points. 
We expect that the limitation of our system to only including nearest-neighbor connec- 
tions is a major factor in the large phase-lag variations that we observed. The importance 
of both short and long distance connections in the regulation of constant phase under con- 
ditions of large variability in the parameter space has been shown by Cohen and Kiemel 
(1993). To address these issues, we are currently designing a system that facilitates both 
short and long distance connections (DeWeerth et al, 1997). Additionally, to study the 
A 
o 
o 
-lO 
-20 
-8O 
-70 
-8O 
2 4 6 6 10 4 6 6 10 
Poll, on Po'ao 
Figure 5: Absolute phase with negative (A) and positive (B) frequency gradients. 
A VLSIModel oflntersegmental Coordination 725 
role of sensory feedback and to close the loop between neural control and motor behav- 
ior, we are also building a mechanical segmented system into which we will incorporate 
our aVLSI models. 
Acknowledgments 
This research is supported by NSF grant IBN-9511721. We would like to thank Avis 
Cohen for discussion on computational properties that underlie coordinated motor behav- 
ior in the lamprey swim system. We would also like to thank Mario Simoni for discus- 
sions on pattern generating circuits. We thank the Georgia Tech Analog Consortium for 
supporting students with travel funds. 
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