Phase-selective entrainment of nonlinear oscillator ensembles
A system for entraining an oscillator ensemble is disclosed that includes a plurality of oscillators in an entrained phase pattern. The system includes an entrainment device operatively coupled to each non-linear oscillator of the oscillator ensemble, and the entrainment control device is configured to deliver a 2π-periodic control signal v(θ) to all oscillators of the plurality of oscillators to induce the entrained phase pattern.
1. A system for entraining a non-linear oscillator ensemble in an entrained phase pattern, the system comprising:
an entrainment control device operatively coupled to each of N non-linear oscillators of the non-linear oscillator ensemble, the entrainment control device is configured to deliver a 2π-periodic control signal v(θ) to each of the N non-linear oscillators, wherein each of the N non-linear oscillators endogenously oscillates at a natural frequency ω j with a corresponding natural phase ψ j , wherein j ranges from 1 to N;
an entrainment pattern generator configured to determine a 2π-periodic interaction function Λ v comprising a relationship between an entrained frequency shift Δω and an entrained phase offset φ for the N non-linear oscillators, the 2π-periodic interaction function Λ v defining the entrained phase pattern comprising an entrained phase offset φ j for each of the N non-linear oscillators, each entrained phase offset φ j consisting of a difference between the natural phase ψ j and an entrained phase θ;
an entrainment control signal generator configured to determine the 2π-periodic control signal v(θ) using the 2π-periodic interaction function Λ v and a phase response curve Z(θ), the phase response curve Z(θ) characterizing a phase shift of the N non-linear oscillators in response to a weak perturbation, wherein the 2π-periodic control signal v(θ) is configured to:
detune the natural frequency ω j by the entrained frequency shift Δω j for each of the N non-linear oscillators, the entrained frequency shift Δω j consisting of a difference between the natural frequency ω j and an entrained frequency Ω; and
shift each of the natural phases ω j ; by the entrained phase offset φ j for each of the N non-linear oscillators.
2. The system of claim 1 , wherein the N non-linear oscillators are detuned to the same entrained frequency Ω consisting of a mean of the natural frequencies ω j of the N non-linear oscillators.
3. The system of claim 1 , wherein each entrained phase offset φ j is independently selected from a range of entrained phase offsets ranging between 0 and 2π.
4. The system of claim 1 , wherein the 2π-periodic interaction function Λ v comprises a plurality of phase/detuning pairs (φ j *, Δω j ), each phase/detuning pair (φ j *, Δω j ) defining an average entrained phase offset φ j * and corresponding entrained frequency shift Δω j for each of the N non-linear oscillators in accordance with the entrained phase pattern, and wherein the 2π-periodic interaction function Λ v further comprises a negative derivative
d
d
φ
Λ
v
<
0
at each phase/detuning pair (φ j *, Δω j ).
5. The system of claim 4 , wherein the 2π-periodic interaction function Λ v comprises an interaction function Λ v *, wherein:
Λ
v
*
(
φ
)
=
r
1
+
(
r
1
-
r
N
+
1
)
σ
(
φ
-
s
1
h
N
+
1
)
+
∑
j
=
2
N
(
r
j
+
1
-
r
j
)
σ
(
φ
-
φ
j
*
h
j
)
+
∑
j
=
2
N
(
r
j
+
1
+
r
j
)
σ
(
φ
+
2
π
-
φ
j
*
h
j
)
+
(
r
1
-
r
N
+
1
)
σ
(
φ
-
s
N
+
1
h
N
+
1
)
,
r
1
=
-
Δω
1
+
1
2
(
-
Δω
1
+
Δω
2
)
,
r
j
=
1
2
(
-
Δω
1
+
Δω
j
-
1
)
for
j
=
2
,
…
,
N
,
r
N
+
1
=
-
Δω
N
+
1
2
(
-
Δω
N
-
1
+
Δω
N
)
,
s
1
=
1
2
(
φ
1
*
+
φ
N
*
-
2
π
)
,
s
j
=
1
2
(
φ
j
*
+
φ
j
-
1
*
)
for
j
=
2
,
…
,
N
,
s
N
+
1
=
1
2
(
φ
1
*
+
φ
N
*
+
2
π
)
,
h
1
=
min
{
s
2
-
φ
1
*
,
(
φ
1
*
-
s
1
)
/
2
}
,
h
j
=
min
{
s
j
+
1
-
φ
j
*
,
φ
j
*
-
s
j
}
,
for
j
=
2
,
…
,
N
,
h
N
+
1
=
min
{
1
2
(
s
N
+
1
-
φ
N
*
)
,
1
2
(
φ
1
*
-
s
1
)
}
,
σ
(
x
)
=
1
2
(
erf
(
2
x
)
+
1
)
,
erf
(
x
)
=
2
π
∫
0
x
e
-
t
2
dt
.
6. The system of claim 5 , wherein the 2π-periodic control signal v(θ) is given by:
v
(
θ
)
≈
v
r
(
θ
)
=
c
0
2
+
∑
n
=
1
r
[
c
n
cos
(
n
θ
)
+
d
n
sin
(
n
θ
)
]
,
wherein
c
0
=
2
f
0
a
0
𝒳
[
a
0
≠
0
]
,
c
n
=
2
f
n
a
n
+
b
n
g
n
a
n
2
+
b
n
2
𝒳
[
a
n
2
+
b
n
2
≠
0
]
,
d
n
=
2
f
n
b
n
+
a
n
g
n
a
n
2
+
b
n
2
𝒳
[
a
n
2
+
b
n
2
≠
0
]
,
χ[A]=1 if A is true, and χ[A]=0 otherwise,
a 0 , a n , and b n are obtained from a truncated Fourier-series expression of a phase response curve given by:
Z
(
θ
)
≈
Z
r
(
θ
)
=
a
0
2
+
∑
n
=
1
r
[
a
n
cos
(
n
θ
)
+
b
n
sin
(
n
θ
)
]
,
and f o , f n , and g n are obtained from a truncated Fourier-series expression of an interaction function Λ v r given by:
Λ
v
r
(
φ
)
=
f
0
2
+
1
2
∑
n
=
1
r
[
f
n
cos
(
n
φ
)
+
g
n
sin
(
n
φ
)
]
,
wherein
f
0
=
a
0
c
0
2
,
f
n
=
a
n
c
n
+
b
n
d
n
,
and
g
n
=
b
n
c
n
-
a
n
d
n
.
7. The system of claim 1 , wherein the phase response curve Z(θ) consists of a mean of individual phase response curves Z j (θ), each individual phase response curve Z j (θ) corresponding to the j th non-linear oscillator of the N non-linear oscillators.
8. The system of claim 7 , further comprising an oscillator characterization device operatively coupled to each of the N non-linear oscillators, the oscillator characterization device configured to determine, for each of the N non-linear oscillators, at least one of the natural frequency ω j , the natural phase ψ j , the phase response curve Z j (θ), and any combination thereof.
9. The system of claim 8 , wherein the oscillator characterization device is configured to determine a phase response curve Z(ψ(t o )) for each of the N oscillators by:
measuring a baseline phase ψ o (t o +NT) at N cycles after the absence of an applied pulse and a perturbed phase ψ 1 (t o +NT) at N cycles after the applied pulse, the applied pulse comprising an impulse with a duration Δt and a magnitude M; and
determining Z(ψ(t o ) according to:
Z
(
ψ
(
t
0
)
)
=
ψ
1
(
t
0
+
NT
)
-
ψ
0
(
t
0
+
NT
)
M
Δ
t
.
10. The system of claim 1 , wherein the system is configured to entrain the oscillator ensemble in a series of entrained phase patterns by sequentially determining a series of 2π-periodic interaction functions Δ v defining the series of entrained phase patterns, sequentially determining the series of 2π-periodic control signals v(θ) using the series of 2π-periodic interaction functions Δ v and the phase response curve Z(θ), and sequentially delivering the series of 2π-periodic control signals v(θ) to each of the N non-linear oscillators.