Biomedical Engineering Reference
In-Depth Information
dV
N
V
Dynamics of a Simple OBB Module
∑
C
i
=
w f
(
V
)
−
i
+
I
i
ij
j
j
i
dt
R
j
=
1
i
The SMER algorithm can be implemented sche-
matically using a network like the Hopfield model
(Hopfield & Tank, 1985) with some modifica-
tions such as nonglobal connections based on
the interconnecting topology of a locomotion
system. Like the dynamics of cellular neural
networks (Chua & Yang, 1988a,b), the input and
output voltages of each node in an OBB network
are normalised to digital low or high level while
the internal potential is continuous within the
normalised interval
[0,1]
. The SMER-based OBB
modules can be classified into two complexity
levels, namely, simple and composite. The simple
OBB modules consist of only two interconnected
nodes with prespecified reversibilities. Figure 6
shows its circuit representation. The composite
OBBs may contain an arbitrary number of cells
interconnected with any topology. Both types of
OBB modules follow
Lemma 1
for initial shared
resources allocation and configuration to avoid
possible abnormal operations, e.g., deadlock or
starvation, during oscillation.
Now consider a submultigraph of
M(N,E)
,
namely
M
ij
, having a pair of coupling nodes
n
i
and
n
j
with
r
i
and
r
j
as their reversibility, respectively.
In the SMER-based, simple OBB module, the
postsynaptic membrane potential of neuron
i
at
k
instant,
where
i=1,…,N
. This equation expresses the net
input current charging the input capacitance
C
i
of neuron
i
to a voltage potential
V
i
as the sum
of three sources: postsynaptic currents induced
in neuron
i
by presynaptic activity in neuron
j
;
leakage current due to the limit input resistance
R
i
of neuron
i
; and input currents
I
i
from other
neurons external to the circuit. The model retains
two important aspects for computation: dynam-
ics and nonlinearity (Hopfield, 1982; Hopfield &
Tank, 1986).
Hopfield classified his models into two catego-
ries, namely symmetric and asymmetric models.
Between any two coupled neurons
i
and
j
, if the
synaptic weight from neuron
i
to
j
is equal to the
weight from neuron
j
to
i
, then this is a symmetric
Hopfield network, otherwise it is named asym-
metric Hopfield network. The most prominent
property of symmetric model is its auto-associa-
tion, i.e., the system energy keeps diminishing and
finally it reaches the local (or global) minimum
energy level as the system state evolves. Unlike
the symmetric model, the asymmetric network
may exhibit oscillation and chaos when it evolves
with time. In some motor systems like CPGs, co-
ordinated oscillation is the desired computation of
the circuit. A proper combination of asymmetric
synapses can enforce chosen phase relationships
between different oscillators (Hopfield & Tank,
1986). In recognition of the properties of the net-
work dynamics, we present a novel methodology to
embed the SMER algorithm into the asymmetric
Hopfield neural networks. Specifically, we will
show that this methodology is useful in simulat-
ing all CPG patterns introduced in Golubitsky's
symmetric Hopf bifurcation theory (Golubitsky
et al., 1998, 1988).
M
i
PSP
, depends on three factors, i.e., the
potential at the last instant
(
k
)
M
i
PSP
, the impact of
( −
k
1
its coupled neuron output
, and the nega-
v
out
( −
k
1
tive feedback of neuron
i
itself
v
out
, without
considering the external impulse. The selection of
system parameters, such as the neuron thresholds
and synapse weights, are crucial for modelling. In
our model, let
r =
max(
r
i
,
r
j
) and
( −
k
1
r
=
, where
h
is a function of getting highest integer level and
multiplying it by
10
, e.g., if
r
i
= 56 and
r
j
= 381
then
'
h
(
r
)
3
rh
. We
can design the neuron
i
and
j
's thresholds θ
i
and
θ
j
and their synaptic weights as following,
(
)
=
h
(max(
56
,
381
))
=
h
(
381
)
=
10
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