348
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
properties. Function h is analytic in R. 4 We have h(0) = 0, h (q M (t)) < 0 for
any q M (t) ∈ R, h(·) is strictly increasing (h (q M (t)) > 0 for any q M (t) ∈ R),
lim q M (t)→−∞ h (q M (t)) = R off , lim q M (t)→+∞ h (q M (t)) = R on . Figure 9.4 shows
that h(·) depends upon x(−∞), i.e., the initial value of x when the memristor
is fabricated. The memristor is also flux-controlled with a characteristic q M (t) =
f (ϕ M (t)) : R → R with f = h −1 .
Note that h and f are not odd functions. To obtain an odd characteristic we
consider two HP memristors with CR q M j (t) = f (ϕ M j (t)), j = 1, 2, and suppose
they are connected in antiparallel for all t (Fig. 9.5). For the two-terminal element
obtained in this way v M (t) = v M 1 (t) = −v M 2 (t) and i M (t) = i M 1 (t) − i M 2 (t)
for any t, hence integrating between −∞ and t, q M (t) = q M 1 (t) − q M 2 (t) and
ϕ M (t) = ϕ M 1 (t) = −ϕ M 2 (t). Then
q M (t) = f (ϕ M (t)) − f (−ϕ M (t))
.
= f ap (ϕ M (t)).
This shows that the two-terminal element is equivalent to a flux-controlled memristor with an odd CR f ap : R → R. It is also charge-controlled with relation
h ap = f −1
ap .
If we consider an HP memristor with a Joglekar window and p = 1, then h ap
is odd, h ap (0) = 0, h ap is analytic in R, strictly increasing (h
ap (q M (t)) > 0)
and h
ap (q M (t)) < 0 for q M (t) > 0, h
ap (q M (t)) > 0 for q M (t) < 0. We have
lim |q M (t)|→∞ h
ap (q M (t)) = R on R off R on . As shown in Fig. 9.6, h ap depends
upon x(−∞). In particular, if x(−∞) 1, the maximum slope h
ap (0) R off /2.
9.2.1.1 An Approximating Memristor Characteristic
Next we show that under the assumption x(−∞) 1 we can perform a suitable
change of variables in order that the nonlinearity h ap (·) is brought back to a
nonlinearity closely approximating the nonlinearity s(·) of a SCNN for not too large
values of the charge. 5
Suppose to choose R on = 100 , R off = 16 k, α = 10 4 C −1 , as in the HP
memristor model in [10], and also let x(−∞) = 0.001. Consider the change of
variable q M (t) = k q q(t) and the nonlinearity ˆ
ϕ : R → R given by
ˆ
ϕ(q(t)) =
2h ap (k q q(t))
R off k q
(9.7)
where parameter k q > 0.
4 This means that h(·) has derivatives of all orders and it is the sum of its Taylor series in some
neighborhood of any q M ∈ R.
5 For simplicity we refer to a set of values for parameters R on , R off as those in [10]. However,
similar conclusions hold for other sets of parameters provided the standing assumption R on R off
is satisfied.
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
properties. Function h is analytic in R. 4 We have h(0) = 0, h (q M (t)) < 0 for
any q M (t) ∈ R, h(·) is strictly increasing (h (q M (t)) > 0 for any q M (t) ∈ R),
lim q M (t)→−∞ h (q M (t)) = R off , lim q M (t)→+∞ h (q M (t)) = R on . Figure 9.4 shows
that h(·) depends upon x(−∞), i.e., the initial value of x when the memristor
is fabricated. The memristor is also flux-controlled with a characteristic q M (t) =
f (ϕ M (t)) : R → R with f = h −1 .
Note that h and f are not odd functions. To obtain an odd characteristic we
consider two HP memristors with CR q M j (t) = f (ϕ M j (t)), j = 1, 2, and suppose
they are connected in antiparallel for all t (Fig. 9.5). For the two-terminal element
obtained in this way v M (t) = v M 1 (t) = −v M 2 (t) and i M (t) = i M 1 (t) − i M 2 (t)
for any t, hence integrating between −∞ and t, q M (t) = q M 1 (t) − q M 2 (t) and
ϕ M (t) = ϕ M 1 (t) = −ϕ M 2 (t). Then
q M (t) = f (ϕ M (t)) − f (−ϕ M (t))
.
= f ap (ϕ M (t)).
This shows that the two-terminal element is equivalent to a flux-controlled memristor with an odd CR f ap : R → R. It is also charge-controlled with relation
h ap = f −1
ap .
If we consider an HP memristor with a Joglekar window and p = 1, then h ap
is odd, h ap (0) = 0, h ap is analytic in R, strictly increasing (h
ap (q M (t)) > 0)
and h
ap (q M (t)) < 0 for q M (t) > 0, h
ap (q M (t)) > 0 for q M (t) < 0. We have
lim |q M (t)|→∞ h
ap (q M (t)) = R on R off R on . As shown in Fig. 9.6, h ap depends
upon x(−∞). In particular, if x(−∞) 1, the maximum slope h
ap (0) R off /2.
9.2.1.1 An Approximating Memristor Characteristic
Next we show that under the assumption x(−∞) 1 we can perform a suitable
change of variables in order that the nonlinearity h ap (·) is brought back to a
nonlinearity closely approximating the nonlinearity s(·) of a SCNN for not too large
values of the charge. 5
Suppose to choose R on = 100 , R off = 16 k, α = 10 4 C −1 , as in the HP
memristor model in [10], and also let x(−∞) = 0.001. Consider the change of
variable q M (t) = k q q(t) and the nonlinearity ˆ
ϕ : R → R given by
ˆ
ϕ(q(t)) =
2h ap (k q q(t))
R off k q
(9.7)
where parameter k q > 0.
4 This means that h(·) has derivatives of all orders and it is the sum of its Taylor series in some
neighborhood of any q M ∈ R.
5 For simplicity we refer to a set of values for parameters R on , R off as those in [10]. However,
similar conclusions hold for other sets of parameters provided the standing assumption R on R off
is satisfied.