Bifurcation Analysis of a Silicon Neuron 
Girish N. Patel l, Gennady S. Cymbalyuk 2'3, 
Ronald L. Calabrese 2, and Stephen P. DeWeerth 1 
1School of Electrical and Computer Engineering 
Georgia Institute of Technology 
Atlanta, Ga. 30332-0250 
{ girish.patel, steve.deweerth } @ ece.gatech.edu 
2Department of Biology 
Emory University 
1510 Clifton Road, Atlanta, GA 30322 
{ gcym, rcalabre } @biology. emory.edu 
3Institute of Mathematical Problems in Biology RAS 
Pushchino, Moscow Region, Russia 142292 (on leave) 
Abstract 
We have developed a VLSI silicon neuron and a corresponding mathe- 
matical model that is a two state-variable system. We describe the cir- 
cuit implementation and compare the behaviors observed in the silicon 
neuron and the mathematical model. We also perform bifurcation analy- 
sis of the mathematical model by varying the externally applied current 
and show that the behaviors exhibited by the silicon neuron under corre- 
sponding conditions are in good agreement to those predicted by the 
bifurcation analysis. 
1 Introduction 
The use of hardware models to understand dynamical behaviors in biological systems is 
an approach that has a long and fruitful history [1][2]. The implementation in silicon of 
oscillatory neural networks that model rhythmic motor-pattern generation in animals is 
one recent addition to these modeling efforts [3][4]. The oscillatory patterns generated by 
these systems result from intrinsic membrane properties of individual neurons and their 
synaptic interactions within the network [5]. As the complexity of these oscillatory sili- 
con systems increases, effective mathematical analysis becomes increasingly more impor- 
tant to our understanding their behavior. However, the nonlinear dynamical behaviors of 
the model neurons and the large-scale interconnectivity among these neurons makes it 
very difficult to analyze theoretically the behavior of the resulting very large-scale inte- 
grated (VLSI) systems. Thus, it is important to first identify methods for modeling the 
model neurons that underlie these oscillatory systems. 
Several simplified neuronal models have been used in the mathematical simulations of 
pattern generating networks [6][7][8]. In this paper, we describe the implementation of a 
732 G. N. Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. De Weerth 
two-state-variable silicon neuron that has been used effectively to develop oscillatory net- 
works [9][10]. We then derive a mathematical model of this implementation and analyze 
the neuron and the model using nonlinear dynamical techniques including bifurcation 
analysis [11 ]. Finally, we compare the experimental data derived from the silicon neuron 
to that obtained from the mathematical model. 
2 The silicon model neuron 
The schematic for our silicon model neuron is shown in Figure 1. This silicon neuron is 
inspired by the two-state, Morris-Lecar neuron model [12][13]. Transistor M1, analo- 
gous to the voltage-gated calcium channel in the Morris-Lecar model, provides an instan- 
taneous inward current that raises the membrane potential towards VHig h when the 
membrane is depolarized. Transistor M 2 , analogous to the voltage-gated potassium chan- 
nel in the Morris-Lecar model, provides a delayed outward current that lowers the mem- 
brane potential toward VLo w when the membrane is depolarized. V H and VL are 
analogous to the half-activation voltages for the inward and outward currents, respec- 
tively. The voltages across C 1 and C 2 are the state variables representing the membrane 
potential, V, and the slow "activation" variable of the outward current, W, respectively. 
The W-nullcline represents its steady-state activation curve. Unlike the Morris-Lecar 
model, our silicon neuron model does not possess a leak current. 
Using current conservation at node V, the net current charging C 1 is given by 
C 1 $2 = Iext0[ P + iH0[p -- iL0[N 
(1) 
where iH and i are the output currents of a differential pair circuit, and c% and c N 
describe the ohmic effects of transistors M 1 and M2, respectively. The net current into C 2 
is given by 
C 2W = iXPN (2) 
where i x is the output current of the OTA, and [3p and 13 N account for ohmic effects of 
the pull-up and the pull-down transistors inside the OTA. 
VHigh" 
lpi M 1 {Xpiex t 
)-w 
VLow 
W 
\ [3p[3N i X 
:v 
Figure 1: Circuit diagram of the silicon neuron. The circuit incorporates analog building 
blocks including two differential pair circuits composed of a bias current, IBH, and 
transistors M4-M 5, and a bias current, IBL, and transistors M6-M 7, and a single follower- 
integrator circuit composed of an operational transconductance amplifier (OTA), X 1 in the 
configuration shown and a load capacitor, C 2. The response of the follower-integrator 
circuit is similar to a first-order low-pass filter. 
Bifurcation Analysis of a Silicon Neuron 733 
The output currents of the differential-pair and an OTA circuits, derived by using sub- 
threshold transistor equations [2], are a Fermi function and a hyperbolic-tangent func- 
tion, respectively [2]. Substituting these functions for i a , i n , and i x in (1) and (2) yields 
Cliff = Iext(Zp+IBH 
K(V- VH)/U T K(W- VL)/U T 
e e 
K(V_ VH)/UTCILp -- IBL K(W _ VL)/UTCILN 
1 +e 1 +e (3) 
c2W= Ixtanh(K V) p N 
where 
V - VHigh/U T VLo w - V/U T 
(Zp = 1-e (Z N = 1-e 
W - Vdd/UT -W/U T 
[3p = 1-e [3 N = 1-e 
(4) 
U T is the thermal voltage, Vdd is the supply voltage, and K is a fabrication dependent 
parameter. The terms (Zp and (Z N limit the range of V to within VHig h and VLo w, and the 
terms Op and ON limit the range of I/V to within the supply rails (Vdd and Gnd). 
In order to compare the model to the experimental results, we needed to determine val- 
ues for all of the model parameters. Vli_ h , VLo w , V H , V L , and Vdd were directly mea- 
sured in experiments. The parameters fia and IB L were measured by voltage-clamp 
experiments performed on the silicon neuron. At room temperature, U T = 0.025 volts. 
The value of K = 0.65 was estimated by measuring the slope of the steady-state activa- 
tion curve of inward current. Because W was implemented as an inaccessible node, I x 
could only be estimated. Based on the circuit design, we can assume that the bias cur- 
rents I x and IBH are of the same order of magnitude. We choose I T = 2.2 nA to fit the 
bifurcation diagram (see Figure 3). C1 and C2, which are assumed to be identical accord- 
ing to the physical design, are time scaling parameters in the model. We choose their val- 
ues (C1 = C2 = 28 pF) to fit frequency dependence on Iex t (see Figure 4). 
3 Bifurcation analysis 
The silicon neuron and the mathematical model 1 described by (3) demonstrate various 
dynamical behaviors under different parametric conditions. In particular, stable oscilla- 
tions and steady-state equilibria are observed for different values of the externally applied 
current, /ext' We focused our analysis on the influence of lext on the neuron behavior for 
two reasons: (i) it provides insight about effects of synaptic currents, and (ii) it allows 
comparison with neurophysiological experiments in which polarizing current is used as a 
primary control parameter. The main results of this work are presented as the comparison 
between the mathematical models and the experimental data represented as bifurcation 
diagrams and frequency dependencies. 
The nullclines described by (3) and for Iex t = 32 nA are shown in Figure 2A. In the 
regime that we operate the circuit, the W-nullcline is an almost-linear curve and the V- 
nullcline is an N-shaped curve. From (3), it can be seen that when IBu + Iex t > IB L the 
nullclines cross at (V, W)= (VHigh, VHigh) and the system has high voltage (about 5 
volts) steady-state equilibrium. Similarly, for Iex t close to zero, the system has one stable 
equilibrium point close to (V, W) = (VLo w, Vow). 
1The parameters used throughout the analyses of the model are VLo w = 0 V, 
VHig h = 5 V, V L = V H = 2.5 V, IB8 = 6.5 nA, /BL = 42 nA, I T = 2.2 nA, 
Vdd = 5 V, U t = 0.025 mY, and K = 0.65. 
734 G. iV. Patel, G. S. Cymbalyuk, R. L. Calabrese andS. P. DeWeerth 
A 
2.85 
2.8 
2.75 
--. 2.7 
 2.65 
2.6 
2.55 
2.5 
2.45 
I W-nullcline 
o 
o 
o 
V-nullcline 
I 
I 
I. 
I 
! 
I I I I I 
0 1 2 3 4 5 
V(volts) 
B 
3.2 
2.8 
0 2.6 
2.4 
2.2 
0 5 10 15 20 25 30 35 
time (msec) 
Figure 2: Nullclines and trajectories in the model of the silicon neuron for Iex t = 32 nA. 
The system exhibits a stable limit-cycle (filled circles), an unstable limit-cycle (untilled 
circles), and stable equilibrium point. Unstable limit-cycle separates the basins of 
attraction of the stable limit-cycle and stable equilibrium point. Thus, trajectories initiated 
within the area bounded by the unstable limit-cycle approach the stable equilibrium point 
(solid line in A's inset, and "x's" in B). Trajectories initiated outside the unstable limit- 
cycle approach the stable limit-cycle. In A, the inset shows an expansion at the 
intersection of the V- and W-nullclines. 
Bifurcation Analysis of a Silicon Neuron 735 
A 
Experimental data 
B 
5 
4 
o 
X 
o 
5 
4 
3 
>2 
o 
 x 
  
 x 
 
xxxx 
x xxx xx 
 
10 2o 3o 
le (nAmps) 
Modeling data 
40 
50 
I I I I I 
0 10 20 30 40 50 
lex t (nAmps) 
Figure 3: Bifurcation diagrams of the hardware implementation (A) and of the 
mathematical model (B) under variation of the externally applied current. In A, the steady- 
state equilibrium potential of V is denoted by "x"s. The maximum and minimum values 
of V during stable oscillations are denoted by the filled circles. In B, the stable and 
unstable equilibrium points are denoted by the solid and dashed curve, respectively, and 
the minimum and maximum values of the stable and unstable oscillations are denoted by 
the filled and untilled circles, respectively. In B, limit-cycle oscillations appear and 
disappear via sub-critical Andronov-Hopf bifurcations. The bifurcation diagram (B) was 
computed with the LOCBIF program [14]. 
736 G. N. Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. DeWeerth 
A Experimental data B Modeling data 
100  100   
 80   
. ' 
.,,'"'--"'-.,...' . 
 40  ; ''  40 
" 20 " 20 
0 0 ' ' ' 
0 10 20 30 0 10 20 30 
lex t (nAmps) lex t (nAmps) 
Figure 4: Frequency dependence of the silicon neuron (A) and the mathematical model 
(B) on the externally applied current. 
For moderate values of Iex t ([ 1 nA,34 nA]), the stable and unstable equilibrium points are 
close to (V, W)--(V m VL) (Figure 3). In experiments in which Iex t was varied, we 
observed a hard loss of the stability of the steady-state equilibrium and a transition into 
oscillations at Iex t - 7.2 nA (Iex t -- 27.5 nA ). In the mathematical model, at the criti- 
cal value of Iex t - 7.7 nA (Iex t - 27.8 nA ), an unstable limit cycle appears via a sub- 
critical Andronov-Hopf bifurcation. This unstable limit cycle merges with the stable limit 
cycle at the fold bifurcation at Iex t = 3.4 nA (Iex t '- 32.1 nA). Similarly, in the experi- 
ments, we observed hard loss of stability of oscillations at Iex t = 2.0nA 
(Iex t -32.8 nA). Thus, the system demonstrates hysteresis. For example, when 
Iex t -- 20 nA the silicon neuron has only one stable regime, namely, stable oscillations. 
Then if external current is slowly increased to Iex t - 32.8 nA, the form of oscillations 
changes. At this critical value of the current, the oscillations suddenly lose stability, and 
only steady-state equilibrium is stable. Now, when the external current is reduced, the 
steady-state equilibrium is observed at the values of the current where oscillations were 
previously exhibited. Thus, within the ranges of externally applied currents (2.0,7.2) and 
(27.5,32.8), oscillations and a steady-state equilibrium are stable regimes as shown in 
Figure 2. 
4 Discussion 
We have developed a two-state silicon neuron and a mathematical model that describes the 
behavior of this neuron. We have shown experimentally and verified mathematically that 
this silicon neuron has three regions of operation under the variation of its external current 
(one of its parameters). We also perform bifurcation analysis of the mathematical model 
by varying the externally applied current and show that the behaviors exhibited by the sili- 
con neuron under corresponding conditions are in good agreement to those predicted by 
the bifurcation analysis. 
This analysis and comparison to experiment is an important step toward our understand- 
ing of a variety of oscillatory hardware networks that we and others are developing. The 
Bifurcation Analysis of a Silicon Neuron 737 
model facilitates an understanding of the neurons that the hardware alone does not pro- 
vide. In particular for this neuron, the model allows us to determine the location of the 
unstable fixed points and the types of bifurcations that are exhibited. In higher-order sys- 
tems, we expect that the model will provide us insight about observed behaviors and 
complex bifurcations in the phase space. The good matching between the model and the 
experimental data described in this paper gives us some confidence that future analysis 
efforts will prove fruitful. 
Acknowledgments 
S. DeWeerth and G. Patel are funded by NSF grant IBN-9511721, G.S. Cymbalyuk is 
supported by Russian Foundation of Fundamental Research grant 99-04-49112, R.L. Cal- 
abrese and G.S. Cymbalyuk are supported by NIH grants NS24072 and NS34975. 
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