Biomedical Engineering Reference
In-Depth Information
Dynamics of a Composite OBB Module
in a “whole-or-none” mechanism in terms of the
firing of its clones. It will fire if and only if all
its clones fire. Fundamentally, a composite OBB
module operates on the basis of its constituent
simple OBB modules. A schematic diagram of a
composite OBB module is shown in Figure 8.
As a generalised version, a composite OBB
module can represent a more complicated oscil-
lating neuronal network and reproduce more
rhythmic patterns than a simple one. It does not
take any fixed form in terms of the number of
constituent clones, the connection structure, etc.
Different applications specify different macroneu-
rons and their complexity. Since it is impossible to
assign a unified threshold to a macroneuron that
may have more than one independent clone, the
usual way to analyse this kind of modules is to
dissect them into the subsystems of simple mod-
ules where equations (1) - (6) are applicable. The
output of a macroneuron,
n
i
, is then determined
by all its clones. We have,
The composite OBB module is a generalisation
of the simple OBB module which can consist of
a number of simple OBB modules. A composite
OBB can have more nodes and therefore, more
complicatedly organised topologies. Suppose a
multigraph
M(N,E)
containing a set of
m
nodes
N
=
{
n
,
n
,...,
n
}
and a set of edges
E
=
0 1
ij
1
2
m
∀ ∈
, which define the connection
topology of this multigraph by using
1
ij
and
0
ij
for with and without an edge between nodes
i
and
j
, respectively. Each node
i
has its own revers-
ibility
r
i
. There are
e
ij
shared resources on the
corresponding edge
1
ij
, with their number and
configuration stipulated by
Lemma 1
.
Different from a simple OBB module in which
there are exactly two nodes coupled with each
other, a composite OBB module has more than
two nodes, and at least one node of them has con-
nection with at least two nodes. The composite
OBB module can be dissected into various simple
ones, each node of the composite OBB is split
into the corresponding copies which share the
same local clock of their maternal node. A copy
is a component of a simple module, and the total
number of copies is twice as many as the number
of edges in the composite OBB module. We then
terminologically regard a node in a composite
OBB
module as a
macroneuron
and a node's copy
as its
clone
. Their definitions are followed,
where
,
i j
N
)
n
∏
=
V
(
k
)
=
v
j
(
k
(7)
i
i
j
1
where
i
v
is the output of clone
j
of the macroneuron
i
, which couples with a cor-
responding clone of another macroneuron. The
superscript sequence
j =
1, 2, ...,
n
is the clone
number of a macroneuron
n
i
.
It is important to choose the initial postsynaptic
potential values properly for the clones which are
the simple OBB modules. Different choices will
lead to different rhythmic patterns of a composite
OBB module. To avoid system halt, no clone should
be idle if its macroneuron is designed to be firing
by equation (7). Within an appropriate parameter
range, a random selection of initial postsynaptic
potential values is allowed. After an initial dura-
tion whose length is determined by the choice of
initial postsynaptic potentials, the system will
oscillate periodically. A pseudo-code operation
of a macroneuron is given in Table 1.
,
i
≠
and
∀
i
∈
N
j
j
Definition 1.
A macroneuron is defined to be
a node i that satisfies
i
∈∀
, where N is the
set of nodes in multigraph M(N,E).
N
Definition 2.
A clone, which has independent
reversibility and represents the coupling charac-
teristics of its maternal macroneuron with one
of the neighbouring macroneurons, is a unique
component of its maternal macroneuron.
According to the principles of SMER, a mac-
roneuron of the composite OBB module operates
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