A General Rhythmic Pattern Generation Architecture for Legged Locomotion - Advancing Artificial Intelligence Through Biological Process Applications - page 208

Figure 1. A simple graph G under SER, with m=1, operation cycle p=2

Scheduling by Multiple Edge Reversal
(SMER)

It is important to know that there is always at
least one SMER solution for any target system's
topology having arbitrary prespecified reversibili-
ties at any of its nodes (Barbosa et al., 2001). In the
following sections, SER and SMER will be used
to construct the artificial CPGs by implementing
oscillatory building blocks (OBBs) as asymmetric
Hopfield-like networks, where operating sinks can
be regarded as firing neurons in purely inhibitory
neuronal networks.

SMER is a generalisation of SER where prespeci-
fied access rates to atomic resources are imposed
on processes in a distributed resource-sharing
system that is represented by a multigraph M ( N,E ).
Unlike SER, with SMER a number of oriented
edges can exist between any two nodes. Between
any two nodes i and j , i, j ∈ N , there can exist e ij
unidirected edges, e ij ≥ 0. The reversibility of
node i is r i , i.e., the number of edges that shall
be reversed by i toward each of its neighbouring
nodes, indiscriminately, at the end of the opera-
tion. Node i is an r i -sink if it has at least r i edges
directed to itself from each of its neighbours.
Each r i -sink node i operates and reverses r i edges
towards each of its neighbours , the new set of r i -
sinks will operate, and so on. Like sinks under
SER, only r i -sink nodes are allowed to operate
under SMER. It is easy to see that with SMER,
nodes are allowed to operate more than once
consecutively.
The following lemma states a basic topologic
and resource configuration constraint toward the
definition of M , where gcd is the greatest com-
mon divisor.

NATURALLy INSPIRED DISCRETE
AND ANALOG Obb MODULES

Of long-standing interest are questions about
rhythm generation in networks of nonoscilla-
tory neurons, where the driving force is not
provided by endogenous pacemaking cells. A
simple mechanism for this is based on reciprocal
inhibition between neurons, if they exhibit the
property of postinhibitory rebound (PIR) (Wang
& Rinzel, 1992). The PIR mechanism (Kuffler
& Eyzaguirre , 1955) is an intrinsic property of
many central nervous system neurons which is
referred to a period of increased neuronal excit-
ability following the cessation of inhibition. It is
often included as an element in computational
models of neural networks involving mutual
inhibition (Perkel, 1976; Roberts & Tunstall,
1990). The ensured mutual exclusion activity
between any two neighbouring nodes coupled
under SMER suggests a scheduling scheme that
resembles anti-correlated firing activity between

Lemma 1. (Barbosa, 1996; França, 1994) Let
nodes i and j be two neighbours in M . If no deadlock
arises for any initial orientation of the shared re-
sources between i and j , then

e = r + r − gcd( r , r )

ij i j i j

and

max{ r , r } ≤ e ≤ r + r − 1

.

i j ij i j