IEEE Circuits and Systems Magazine - Q3 2020 - 23

the same probability to be '1'. A steady hyper-state can
be reached after several runs, where the probability that
the FSM staying at each state converges [1]. By assigning
a different output ('0' or '1') to each state, exponential,
logarithmic and high-order polynomial functions can be
implemented by the FSM-based circuits. A general synthesis method is discussed in [27].
Stochastic integrators are sequential SC elements
that accumulate the difference between two input
-sequences  [1]. The stochastic integrator consists of an
RNG, an up/down counter and a comparator, as shown
in Fig. 9(a). Let the output of the comparator be S i (either
'0' or '1') at clock cycle i, the input bits be A i and B i, and
the n-bit integer stored in the counter be C i . The RNG and
the comparator work as an SNG, and the probability of
generating a '1' is c i = C i / 2 n, i.e., the n-bit counter stores
a number in a fixed-point manner and all the bits belong
to the fractional part. The counter updates its value C i at
each clock cycle by C i + 1 = C i + A i - B i, or, equivalently,
	

of the number of '1's divided by the sequence length.
The ADDIE in Fig. 10(c) is an alternative component to
estimate the probability by using a stochastic integrator
reaching the equilibrium state [1].

A
B

C
INC
DEC

2n-State Counter

>

RNG

ri

Ci +1 = Ci + Ai - Bi

A
B

c i + 1 = c i + 1/2 n (A i - B i). (1)

c k = c 0 + 1/ 2 n

k-1

/ (A i - B i), (2)

Comparator

if Ci > ri , S = 1
else
,S=0
(a)
C

+

S

-
(b)
Axon

In k clock cycles, the stochastic integrator implements
	

S

Synapse
Soma (Cell)

i=0

where c 0 is the initial (fractional) value stored in the
counter.
A stochastic integrator works in a similar manner to
a spiking neuron, as shown in Fig. 9. The stochastic integrator "fires" randomly depending on the value stored in
the counter, whereas the spiking neuron fires when the
accumulated membrane potential, V M , reaches a threshold, Vth [5]. The inputs to the neuron can be excitatory
(e i j) or inhibitory (l i j), while the input sequences, A and
B, of the stochastic integrator increase and decrease the
value stored in the counter.
The integrator has been used in a stochastic divider,
with the output stochastic sequence as the feedback signal [1], as shown in Figs. 10(a) and (b). The quotient is
obtained when the value stored in the counter reaches
an equilibrium state, i.e., the up/down counter has an
equal probability to increase or decrease. However, it
may take a long time to converge to the equilibrium state
as discussed in [6], [25]. A stochastic integrator with a
feedback loop has been used as an ADpative DIgital Element (ADDIE), shown in Fig. 10(c). The ADDIE can function as a probability follower, i.e., the probability of the
output sequence tracks the change of the probability of
the input sequence [1], [28].

Figure 9. (a) A stochastic integrator, (b) its symbol and (c) a
spiking neuron [5].

C. Probability Estimators
The computed probability of the output sequence can be
found by using a counter. It is estimated as the quotient

Figure 10. Stochastic dividers compute: (a) A/B, (b) B /( A + B ).
(c) shows an ADDIE.

THIRD QUARTER 2020 		

Dendrites

Membrane

VM (t +1) =VM (t )+ Σεij - Σιij
if VM > Vth, Reset and Fire
else , Do Not Fire
εij : Excitatory Inputs of the Neuron
ιij: Inhibitory inputs of the neuron
VM: Membrane Potential
Vth: Threshold Voltage

A
B

+

(c)

A

A /B

+

B

-

B/(A+B)

-
(b)
"

p

+

(a)
p

-
(c)

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IEEE Circuits and Systems Magazine - Q3 2020

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