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among individual oscillators is measured, and when passing a predefined threshold,
it activates a stimulation pulse.
In order to detect synchronicity among neurons, Tass proposes the calculation
of cluster variables - the center of gravity in phase space of all oscillators. Specif-
ically, if R m (
are the magnitude and phase respectively of the center
of gravity of m clusters, and
t
)
and
φ m (
t
)
Ψ j is the phase of the j th oscillator, then the cluster
variable is
N
j = 1 e im ψ j ( t ) .
1
N
e i φ m ( t ) =
Z m (
t
)=
R m (
t
)
(13.7)
Thus, if the magnitude of the cluster variable is close to 0, there is very little
synchronicity, but when it is close to unity, there is high synchronicity.
13.7.2 ALOPEX and DBS
Sanghavi [134] proposed an integrated circuit (IC) design of an adaptive DBS sys-
tem where power estimation of recorded neural activity is used as a global “error
measure” that drives the modification of stimulus pulse width, amplitude, and fre-
quency of multiple signal generators. Furthermore, the modification is accomplished
in simulation with minimal power requirements (roughly 0.8 mW) using an analog
design of the stochastic optimization algorithm ALOPEX.
Since its application to BCI [150, 62, 105, 38], the ALOPEX algorithm was ap-
plied to numerous studies involving image pattern recognition and artificial neural
networks [29]. The algorithm itself is based on the principle of Hebbian learn-
ing wherein the synaptic strength between two neurons increases in proportion
to the correlation between the activities of those neurons [140]. Similarly, given
a set of modifiable variables at iteration k , b k = {
, and a global
response estimate R k , ALOPEX recursively modifies each b j , k by using correla-
tion measures between previous changes in b j , k and changes in R k . Moreover, to
keep the algorithm from falling into an infinite loop, stochastic noise r j , k is in-
cluded. Finally, given stochastic and deterministic step sizes
b 1 , k ,
b 2 , k ,,
b N , k }
σ j , k ,are-
formulation of the algorithm in its most simplified “parity” form, as it is described
in [62], is
σ j , k and
· (
)
d j , k = (
R k 1
R k 2 )
b j , k 1
b j , k 2
| ,
(13.8)
|
R k 1
R k 2
|
|
b j , k 1
b j , k 2
b j , k =
b j , k 1
+ γ j , k ·
d j , k + σ j , k ·
r j , k .
(13.9)
Subsequently, new versions were developed including the 2T-ALOPEX algo-
rithm contributed by Sastry et al. [135] and the ALOPEX-B algorithm contributed
by Bia [18]. In particular, 2T-ALOPEX incorporates explicit probability distri-
butions into the calculation of each iteration, while ALOPEX-B is a similar but
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