356
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
solution converges toward an EP. 6 The typical scenario in a convergent NN can
be described as follows. The NN possesses a huge number of asymptotically
stable EPs; moreover, due to convergence, the state space can be subdivided in
attraction basins of the asymptotically stable EPs, where the attraction basin of
an EP is defined as the set of initial conditions for which the corresponding
solution converges to the considered EP. The analog transient displayed by the
network during convergence to the EP can be suitably exploited for real-time
computational purposes. For example, in a content addressable memory (CAM)
the target patterns are stored in the network as asymptotically stable EPs. Given
a partial information (e.g., a pattern corrupted with noise), the NN is designed so
as to retrieve the uncorrupted pattern during the transient toward the asymptotically
stable EP. Convergence is one of the most important global dynamical properties of
NNs in view of the application to the solution of signal processing tasks in real time.
In fact, convergent NNs are potentially useful not only to implement CAMs but also
in the field of pattern formation, to solve image processing tasks and combinatorial
optimization problems [8, 9, 15, 16]. In Sect. 9.4, we will discuss some simple
applications of convergent M-SCNNs to the solution of image processing tasks
in real time.
The vector field −q + G ˆ
Φ(q) + Q 0 defining the M-SCNN (9.13) is smooth,
indeed, it is an analytic function of q. From standard results on ordinary differential
equations it follows that the M-SCNN model (9.13) enjoys the property of local
existence and uniqueness of the solution with respect to the initial conditions [17].
Moreover, it can be easily checked that the norm of the vector field increases at most
linearly with the norm of q, thus ensuring that any solution is defined on the whole
time interval [0, +∞) [17]. However, since the nonlinearity ˆ
ϕ(·) is unbounded, it is
not guaranteed a priori that solutions are bounded. The next results show that, under
a suitable constraint on the norm of G, we can guarantee boundedness of solutions
and also obtain a useful estimate of the norm of solutions starting with prescribed
initial conditions. In the statements we use the notations introduced in (9.8)–(9.11).
Property 9.1 Suppose that
G 1
.
= max
i
n
j =1
|G ij | <
R off
2R on (1 + ΔR)
(9.16)
and let
ρ min = max
⎧
⎨
⎩
¯
q M ,
G 1 + +Q 0 ∞
1 − −G 1
2R on
R off
(1 + ΔR)
⎫
⎬
⎭
(9.17)
6 Such a dynamic property is also referred to in the literature as complete stability.
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
solution converges toward an EP. 6 The typical scenario in a convergent NN can
be described as follows. The NN possesses a huge number of asymptotically
stable EPs; moreover, due to convergence, the state space can be subdivided in
attraction basins of the asymptotically stable EPs, where the attraction basin of
an EP is defined as the set of initial conditions for which the corresponding
solution converges to the considered EP. The analog transient displayed by the
network during convergence to the EP can be suitably exploited for real-time
computational purposes. For example, in a content addressable memory (CAM)
the target patterns are stored in the network as asymptotically stable EPs. Given
a partial information (e.g., a pattern corrupted with noise), the NN is designed so
as to retrieve the uncorrupted pattern during the transient toward the asymptotically
stable EP. Convergence is one of the most important global dynamical properties of
NNs in view of the application to the solution of signal processing tasks in real time.
In fact, convergent NNs are potentially useful not only to implement CAMs but also
in the field of pattern formation, to solve image processing tasks and combinatorial
optimization problems [8, 9, 15, 16]. In Sect. 9.4, we will discuss some simple
applications of convergent M-SCNNs to the solution of image processing tasks
in real time.
The vector field −q + G ˆ
Φ(q) + Q 0 defining the M-SCNN (9.13) is smooth,
indeed, it is an analytic function of q. From standard results on ordinary differential
equations it follows that the M-SCNN model (9.13) enjoys the property of local
existence and uniqueness of the solution with respect to the initial conditions [17].
Moreover, it can be easily checked that the norm of the vector field increases at most
linearly with the norm of q, thus ensuring that any solution is defined on the whole
time interval [0, +∞) [17]. However, since the nonlinearity ˆ
ϕ(·) is unbounded, it is
not guaranteed a priori that solutions are bounded. The next results show that, under
a suitable constraint on the norm of G, we can guarantee boundedness of solutions
and also obtain a useful estimate of the norm of solutions starting with prescribed
initial conditions. In the statements we use the notations introduced in (9.8)–(9.11).
Property 9.1 Suppose that
G 1
.
= max
i
n
j =1
|G ij | <
R off
2R on (1 + ΔR)
(9.16)
and let
ρ min = max
⎧
⎨
⎩
¯
q M ,
G 1 + +Q 0 ∞
1 − −G 1
2R on
R off
(1 + ΔR)
⎫
⎬
⎭
(9.17)
6 Such a dynamic property is also referred to in the literature as complete stability.