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a
b
c
1 3
1 2
1 3
12
1 4
1 1
14
1 1
15
15
16
1 0
15
10
9
10
16
9
1
8
1
8
5
2
7
2
7
1
3
6
0
1
2
3
3
6
4
5
time
t/
T
4
5
Fig. 13.7. Using the methods derived in [103, 104], networks with very dierent coupling statistics
(a), (c) can be designed to generate the same pattern of spiking activity (b). Network (a) minimizes
the L 2 -norm (
q P
P
i;j " ij ) of the coupling matrix, network (c) minimizes the L 1 -norm (
i;j j " ij j ).
(Modied from [103].)
a
b
5
t 6
6
4
t 5
t 4
1
3
t 1
2
1
10
t/
T
20
time
c
d
5
t 6
6
4
t 5
t 4
1
3
t 1
2
t/
T
1
10
time
20
Fig. 13.8. The same periodic pattern of spikes can be realized as stable invariant dynamics (b)
in one network (a) and as unstable invariant dynamics (d) in another network (c). (Reproduced
from [103].)
is predened. The equations and inequalities involve all parameters available in the
class of model systems considered, including the network topology and the coupling
strengths, the delay times and the local individual neuron dynamics [103, 104].
The high dimensionality of a typical solution space has important conceptual
consequences. For instance, the same pattern may exist in networks with very dis-
tinct topologies and coupling types, cf. Fig. 13.7. Moreover, the same pattern may
exist in networks with statistically similar topology but may be stable in one and
unstable in another network, cf. Fig. 13.8. Besides numerical analyses, for large
classes of networks even general analytical statements on the stability of such spike
 
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