Homeostasis in a Silicon Integrate and Fire 
Neuron 
Shih-Chii Liu 
Institute for Neuroinformatics, ETH/UNIZ 
Winterthurstrasse 190, CH-8057 Zurich 
Switzerland 
shih@ini.phys.ethz.ch 
Bradley A. Minch 
School of Electrical and Computer Engineering 
Cornell University 
Ithaca, NY 14853-5401, U.S.A. 
minch@ ee.comell.edu 
Abstract 
In this work, we explore homeostasis in a silicon integrate-and-fire neu- 
ron. The neuron adapts its firing rate over long time periods on the order 
of seconds or minutes so that it returns to its spontaneous firing rate after 
a lasting perturbation. Homeostasis is implemented via two schemes. 
One scheme looks at the presynaptic activity and adapts the synaptic 
weight depending on the presynaptic spiking rate. The second scheme 
adapts the synaptic "threshold" depending on the neuron's activity. The 
threshold is lowered if the neuron's activity decreases over a long time 
and is increased for prolonged increase in postsynaptic activity. Both 
these mechanisms for adaptation use floating-gate technology. The re- 
sults shown here are measured from a chip fabricated in a 2-/zm CMOS 
process. 
1 Introduction 
We explored long-time constant adaptation mechanisms in a simple integrate-and-fire sili- 
con neuron. Many researchers have postulated constant adaptation mechanisms which, for 
example, preserve the firing rate of the neuron over long time invervals (Liu et al. 1998) 
or use the presynaptic spiking statistics to adapt the spiking rate of the neuron so that 
the distribution of this spiking rate is uniformly distributed (Stemmler and Koch 1999). 
Homeostasis is observed in in-vitro recordings (Desai et al. 1999) where if the K or Na 
conductances are perturbed by adding antagonists, the cell returns to its original spiking 
rate in a couple of days. 
This work differs from previous work that explore the adaptation of the firing threshold 
and the gain of the neuron through the regulation of Hodgkin-Huxley like conductances 
(Shin and Koch 1999) and regulation of the neuron to perturbation in the conductances 
(Simoni and DeWeerth 1999). Our neuron circuit is a simple integrate-and-fire neuron and 
v 
gain 
Vtu n 
Vfg 
 Vpre E 
Isyn 
M 2 
__Od 
lepsc 
V m 
Vr -IF__] Irefr 
Vo-I 
C1 
I Spike 
output, V o 
Figure 1' Schematic of neuron circuit with long time constant mechanisms for presynaptic 
adaptation. 
our adaptation mechanisms have time constants of seconds to minutes. We also describe 
adaptation of the synaptic weight to presynaptic spiking rates. This presynaptic adaptation 
models the contrast gain control curves of cortical simple cells (Ohzawa et al. 1985). 
We fabricated two different circuits in a 2-/zm CMOS process. One circuit implements 
presynaptic adaptation and the other circuit implements postsynaptic adaptation. The long 
time constant adaptation mechanisms use tunnelling and injection mechanisms to remove 
charge from and to add charge onto a floating gate (Diorio et al. 1999). We added these 
mechanisms to a simple integrate-and-fire neuron circuit (Mead 1989). This circuit (shown 
in Figure 1) takes an input current, Ievsc, which charges up the membrane, Vr. When 
the membrane exceeds a threshold, the output of the neuron, Vo, spikes. The spiking rate 
of the neuron, fo is determined by the input current, Iepsc, that is, fo = m Iepsc where 
1 
m = (C'z+C'2)Vdd is a constant. 
2 Adaptation mechanisms in silicon neuron circuit 
In order to permit continuous operation with only positive polarity bias voltages, we use 
two distinct mechanisms to modify the floating-gate charges in our neuron circuits. We use 
Fowler-Nordheim tunneling through high-quality gate oxide to remove electrons from the 
floating gates (Lenzlinger and Snow 1969). Here, we apply a large voltage across the oxide, 
which reduces the width of the Si-SiO2 energy barrier to such an extent that electrons are 
likely to tunnel through the barrier. The tunneling current is given approximately by 
It,r =Iote TM , 
where Vox = Vt,,r, - Via is the voltage across the tunneling oxide and lot and Vo are 
measurable device parameters. For the 400-)1 oxides that are typical of a 2-/zm CMOS 
process, a typical value of Vo is 1000 V and an oxide voltage of about 30 V is required to 
obtain an appreciable tunneling current. 
We use subthreshold channel hot-electron injection in an nMOS transistor (Diorio, Minch, 
and Hasler 1999) to add electrons to the floating gates. In this process, electrons in the 
channel of the nMOS transistor accelerate in the high electric field that exists in the deple- 
tion region near the drain, gaining enough energy to surmount the Si-SiO2 energy barrier 
(about 3.2 eV). To facilitate the hot-electron injection process, we locally increase the sub- 
strate doping density of the nMOS transistor using the p-base layer that is normally used 
to form the base of a vertical npn bipolar transistor. The p-base substrate implant simulta- 
neously increases the electric field at the drain end of the channel and increases the nMOS 
transistor's threshold voltage from 0.8 V to about 6 V, permitting subthreshold operation at 
gate voltages that permit the collection of the injected electrons by the floating gate. The 
hot-electron injection current is given approximately by 
Ii,,j = rllse *c/v" , 
where 18 is the source current, bac is the drain-to-channel voltage, and t/ and V/,,j are 
measurable device parameters. The value of V/,,j is a bias dependent injection parameter 
and typically ranges from 60 mV to 0.1 V. 
3 Presynaptic adaptation 
The first mechanism adapts the synaptic efficacy to the presynaptic firing rate over long 
time constants. The circuit for this adaptation mechanism is shown in Figure 1. The synap- 
tic current is generated by a series of two transistors; one is driven by the presynaptic input 
and the other by the floating-gate voltage. The floating-gate voltage stores the synaptic ef- 
ficacy of the synapse. A discrete amount of charge is integrated on a diode capacitor every 
time there is a presynaptic spike. The charge that is dumped onto the capacitor depends 
on the input frequency and the synaptic weight. The excitatory postsynaptic current to the 
membrane of the neuron depends also on the gain of the current-mirror. The tunneling 
mechanism which is controlled by Vtun is continuously on so the synaptic efficacy slowly 
decreases over time. The injection mechanism is turned on only when there is a presynaptic 
spike. This presynaptic adaptation can model the contrast gain control curves of cortical 
simple cells. 
3.1 Steady-state analysis 
In steady-state, the tunneling current, Itun, is equal to the average injection current, Iinj 
and they are as follows: 
(1) 
= - 1)nQrfi (2) 
where A is the gain of the current mirror integrator, QT = CaUT/k, Vlao is the steady-state 
floating-gate voltage, fi is the presynaptic rate and T6 is the pulse width of the presynaptic 
pulse. From Equations 1 and 2, we can solve for Vlao and thus determine the synaptic 
current, Isyn : 
k Vf g 0 1 
Isy n __-- lopb e vT -- !m/(fiT6). 
In this equation, Im is a preconstant and/ is approximately 1. The steady-state input 
current is given by Iepsc  !synTsAfi . !mA, thus it is independent of the presynaptic 
input frequency. 
3.2 Transient analysis 
With a transient change in the presynaptic frequency, fi, the initial postsynaptic frequency 
is given by: 
fo q- dfo - m  ImA(fi q- dfi) _ fo q- m * ImA(dfi/fi). (3) 
160 
140 
120 
100 
80 
40 
20 
0 
0 
Transient gain 
50 100 150 200 250 300 
Presynaptic frequency (Hz) 
350 
Figure 2: Adaptation curves of synaptic efficacy to presynaptic frequencies using long time 
constant adaptation mechanisms. 
As derived from Equation 3, we see that the transient change in the neuron's spiking rate is 
dependent on the contrast of the input spiking rate, dfi/fi. 
dfo = m   A  dfilfi = fo(dfilf) 
dfoldfi: folfi 
(4) 
Hence, the transient gain of the neuron is equal to the ratio of the postsynaptic spiking rate 
to the presynaptic input rate and it decreases with the input rate. 
3.3 Experimental results 
We measured the transient and steady-state spiking rates of the neuron around four differ- 
ent steady-state presynaptic rates of 100Hz, 150Hz, 200Hz, and 250Hz. In these measure- 
ments, the drain of the pbase injection transistor was set at 4V and the tunnelling voltage 
was set at 35.3V. For each steady-state presynaptic rate, we presented step increases and 
decreases in the presynaptic rate of 15Hz, 30Hz, 45Hz, and 60Hz. The instantaneous post- 
synaptic rate is plotted along one the four steep curves in Figure 2. After every change in 
the presynaptic rate, we returned the presynaptic rate to its steady-state value before we 
presented the next change in presynaptic rate. The transient gain of the curves decreases 
for higher input spiking rates. This is predicted by Equation 4. 
We also recorded the dynamics of the adaptation mechanisms by measuring the spiking 
rate of the neuron when the presynaptic frequency was decreased at time (t=0) from 350 
Hz to 300 Hz as shown in Figure 3. The system adapts over a time constant of minutes 
back to the initial output frequency. These data show that the synaptic efficacy adapted to a 
higher weight value over time. The time constant of adaptation can be increased by either 
increasing the tunnelling voltage or the pbase injector's drain voltage, Va. 
70 
 40 
 3o 
2o 
lO 
100 200 300 400 500 600 700 800 900 
Time (sec) 
Figure 3: Temporal adaptation of spiking rate of neuron to a decrease in the presynaptic 
frequency from 350Hz to 300Hz. The smooth line is an exponential fit to the data curve. 
4 Postsynaptic adaptation 
In the second mechanism, the neuron's spiking rate determines the synaptic "threshold". 
The schematic of this adaptation circuitry is shown in Figure 4. The floating-gate pbase 
transistor provides a quiescent input to the neuron so that the neuron fires at a quiescent rate. 
The tunneling mechanism is always turned on so the neuron's spiking rate increases in time 
if the neuron does not spike. However the injection mechanism turns on when the neuron 
spikes. The time constant of these mechanisms is in terms of seconds to minutes. The 
increase in the floating-gate voltage is equivalent to a decrease in the synaptic threshold. If 
the neuron's activity is high, the injection mechanism tums on thus decreasing the floating- 
gate voltage and the input current to the neuron. These two opposing mechanisms ensure 
that the cell will remain at a constant activity under steady-state conditions. In other words, 
the threshold of the neuron is modulated by its output spiking rate. The threshold of the 
neuron continuously decreases and each output spike increases the threshold. 
4.1 Steady-state analysis 
Similar equations as in Section 3.1 can be used to solve for Vfgo, thus leading us to the 
following expression for the steady-state input current, 
k Vf gO 
Iino = Iopt, e v, = Im/(foTs) 
where I, is a preconstant and ? is close to 1. 
4.2 Transient analysis 
When a positive step voltage is applied to Vez, the step change, AV, is coupled into the 
floating gate. The initial transient current is: 
/AV 
= o+) = 
Vo_4 
Vrefr-E 
J refr 
lin 
Vex 
Membrane 
voltage, V m 
Spike output, V o 
Adaptation ! ,,/injector 
circuitry  '-ILL)_, 
Vfg . /// 
Figure 4: Schematic of neuron circuit with long time constant mechanisms for postsynaptic 
adaptation. 
and the initial increase in the postsynaptic firing rate is 
/eAv 
fo--F dfo = fo "r 
If we assume that the step input, V/ - log(f/) (where fi is the firing rate of the presynaptic 
neuron), then the change in the floating-gate voltage is described by AV -- dfi/fi. We then 
solve for dfo, 
dfo ' k dfi 
=eV" _1 (5) 
fo UTfi 
Equation 5 shows that the transient change in the neuron's spiking rate is proportional to 
the input contrast in the firing rate. With time, the floating-gate voltage adapts back to the 
steady-state condition, so the spiking rate returns to fo. 
4.3 Experimental results 
In these experiments, we set the tunneling voltage, Vtun to 28V, and the injection voltage 
to 6.6V. We coupled a step decrease of 0.2V into the floating-gate voltage and then mea- 
sured the output frequency of the neuron over a period of 10 minutes. The output of this 
experiment is shown in Figure 5. The frequency dropped from about 19Hz to 13Hz but the 
circuit adapted after this initial perturbation and the spiking rate of the neuron returned to 
about 19Hz over 26min. A similar experiment is performed but this time a step increase 
of 0.2V was coupled into the floating gate node (shown in Figure 5). Initially, the neuron's 
rate increased from 20Hz to 28Hz but over a long period of minutes, the firing rate returned 
to 20Hz. 
5 Conclusion 
In this work, we show how long-time constant adaptation mechanisms can be added to a 
silicon integrate-and-fire neuron in a normal CMOS process. These homeostatic mecha- 
nisms can be combined with short time constant synaptic depressing synapses on the same 
neuron to provide a range of adapting mechanisms. The presynaptic adaptation mechanism 
can also account for the contrast gain curves of cortical simple cells. 
30 , 
28 
 26 
b 24 
., 22 
 2O 
200 400 600 800 
14 
12 ' ' ' 
0 1000 1200 1400 1600 
Time (see) 
Figure 5: Response of silicon neuron to an increase and a decrease of a step input of 0.2V. 
The curve shows that the adaptation time constant is in the order of about 10 min. 
Acknowledgments 
We thank Rodney Douglas for supporting this work, the MOSIS foundation for fabricating 
this circuit, and Tobias Delbrtick for proofreading this document. This work was supported 
in part by the Swiss National Foundation Research SPP grant and the U.S. Office of Naval 
Research. 
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