9.5 Discussion
369
is to use a parallel computational approach, as that offered by a neuromorphic
architecture, together with unconventional electric devices as the memristors, that
are able to process and store information on the same physical device according
to the principle of in-memory computing. The design of the M-SCNNs here
introduced goes along this direction. In fact, the role played by the memristors in the
behavior of M-SCNNs is twofold. First, the nonlinear dynamics of memristors is
exploited during the transient analog computation; moreover, the same memristors
are also used to store the result of the processing in steady state, i.e., in a M-SCNN
processing and storing of information occur in the same physical location and are
performed by the same device.
Remark 9.8 Recall that in the (ϕ, q)-domain a memristor has an algebraic CR
(cf. (9.3)), i.e., it establishes a static nonlinear relationship between flux and charge.
The cell model here considered for a M-SCNN (cf. Fig. 9.8) differs from that of a
SCNN due to the use of memristors as the nonlinear elements in the (ϕ, q)-domain.
In a SCNN the nonlinearity is instead implemented in the (v, i)-domain by using a
saturated operational amplifier [8].
Remark 9.9 The use of memristors permits to obtain for a M-SCNN analogous
equations in the (ϕ, q)-domains as those describing SCNNs in the (v, i)-domain.
It is known however that a memristor has a CR in differential form in the (v, i)domain. In fact we have seen in Sect. 9.3.1 that a M-SCNN obeys in the (v, i)domain a system of 2n differential equations for an array of n cells. Clearly, there
is not an analogy between M-SCNNs and SCNNs when considering the (v, i)domain.
Remark 9.10 Other related articles in the literature have dealt with the design of
memristor NNs whose mathematical model in the (ϕ, q)-domain is analogous to the
Full-Range model of cellular NNs [19]. Convergence in the presence of multiple
EPs in the (ϕ, q)-domain has been studied for symmetric and also for some classes
of nonsymmetric (e.g., cooperative) cell interconnections [21]. Moreover, the issue
of global stability of the unique EP in the (ϕ, q)-domain has been addressed in
the general case of nonsymmetric interconnections [22] and also in the case where
there are delays in the interconnections [23]. In the case of convergence, or global
stability, those memristor NNs feature the same advantages as M-SCNNs in terms
of reduction of power consumption at steady state and in terms of the possibility to
implement the processing and storing phase at the same physical location.
Remark 9.11 The theoretical foundation for a class of memristor CNNs that is
conceptually similar to M-CNNs is given in [24–26]. Such NNs also adopt
a nonvolatile memristor in place of the linear resistor in the circuit of each
processing cell. A generic memristor model is considered (Chap. 2) inspired to
the mathematical description presented in [27] to capture the learning process of
a unicellular organism. The resulting cells are second order.
Two main cases are studied. In the first case, parameters are chosen such that
there are only abrupt transitions between lowest and highest resistive memristor
369
is to use a parallel computational approach, as that offered by a neuromorphic
architecture, together with unconventional electric devices as the memristors, that
are able to process and store information on the same physical device according
to the principle of in-memory computing. The design of the M-SCNNs here
introduced goes along this direction. In fact, the role played by the memristors in the
behavior of M-SCNNs is twofold. First, the nonlinear dynamics of memristors is
exploited during the transient analog computation; moreover, the same memristors
are also used to store the result of the processing in steady state, i.e., in a M-SCNN
processing and storing of information occur in the same physical location and are
performed by the same device.
Remark 9.8 Recall that in the (ϕ, q)-domain a memristor has an algebraic CR
(cf. (9.3)), i.e., it establishes a static nonlinear relationship between flux and charge.
The cell model here considered for a M-SCNN (cf. Fig. 9.8) differs from that of a
SCNN due to the use of memristors as the nonlinear elements in the (ϕ, q)-domain.
In a SCNN the nonlinearity is instead implemented in the (v, i)-domain by using a
saturated operational amplifier [8].
Remark 9.9 The use of memristors permits to obtain for a M-SCNN analogous
equations in the (ϕ, q)-domains as those describing SCNNs in the (v, i)-domain.
It is known however that a memristor has a CR in differential form in the (v, i)domain. In fact we have seen in Sect. 9.3.1 that a M-SCNN obeys in the (v, i)domain a system of 2n differential equations for an array of n cells. Clearly, there
is not an analogy between M-SCNNs and SCNNs when considering the (v, i)domain.
Remark 9.10 Other related articles in the literature have dealt with the design of
memristor NNs whose mathematical model in the (ϕ, q)-domain is analogous to the
Full-Range model of cellular NNs [19]. Convergence in the presence of multiple
EPs in the (ϕ, q)-domain has been studied for symmetric and also for some classes
of nonsymmetric (e.g., cooperative) cell interconnections [21]. Moreover, the issue
of global stability of the unique EP in the (ϕ, q)-domain has been addressed in
the general case of nonsymmetric interconnections [22] and also in the case where
there are delays in the interconnections [23]. In the case of convergence, or global
stability, those memristor NNs feature the same advantages as M-SCNNs in terms
of reduction of power consumption at steady state and in terms of the possibility to
implement the processing and storing phase at the same physical location.
Remark 9.11 The theoretical foundation for a class of memristor CNNs that is
conceptually similar to M-CNNs is given in [24–26]. Such NNs also adopt
a nonvolatile memristor in place of the linear resistor in the circuit of each
processing cell. A generic memristor model is considered (Chap. 2) inspired to
the mathematical description presented in [27] to capture the learning process of
a unicellular organism. The resulting cells are second order.
Two main cases are studied. In the first case, parameters are chosen such that
there are only abrupt transitions between lowest and highest resistive memristor