Rhythm processing and frequency tracking in gradient frequency nonlinear oscillator networks
View Patent ↗A method for mimicking the auditory system's response to rhythm of an input signal having a time varying structure comprising the steps of receiving a time varying input signal x(t) to a network of n nonlinear oscillators, each oscillator having a different natural frequency of oscillation and obeying a dynamical equation of the form r . = r ( α + β 1 z 2 + ε β 2 z 4 1 - ε z 2 ) + c x ( t ) cos ϕ - r ε ε r 2 - 2 ε r cos ϕ + 1 ϕ . = ω + δ 1 r 2 + ε δ 2 r 4 1 - ε r 2 - c x ( t ) sin ( ϕ ) ε r 2 - 2 ε r cos ( ϕ ) + 1 ω . = - k x ( t ) sin ϕ ε r 2 - 2 ε r cos ϕ + 1 wherein ω represents the response frequency, r is the amplitude of the oscillator and φ is the phase of the oscillator. Generating at least one frequency output from said network useful for describing said varying structure.
1. A method for mimicking the auditory system's response to rhythm of an input signal having a time varying structure comprising the steps of:
providing a network of m nonlinear oscillators, each oscillator of the m nonlinear oscillators having a different natural frequency of oscillation and obeying a dynamical equation of the form:
r
.
=
r
(
α
+
β
1
z
2
+
ε
β
2
z
4
1
-
ε
z
2
)
+
c
x
(
t
)
cos
ϕ
-
r
ε
ε
r
2
-
2
ε
r
cos
ϕ
+
1
ϕ
.
=
ω
+
δ
1
r
2
+
ε
δ
2
r
4
1
-
ε
r
2
-
c
x
(
t
)
sin
(
ϕ
)
ε
r
2
-
2
ε
r
cos
(
ϕ
)
+
1
ω
.
=
-
k
x
(
t
)
sin
ϕ
ε
r
2
-
2
ε
r
cos
ϕ
+
1
wherein ω represents the response frequency, r is the amplitude of the oscillator and φ is the phase of the oscillator;
inputting a time varying input signal x(t) to the network of m nonlinear oscillators; and
generating at least one frequency output from said network useful for describing said varying structure.
2. A method for mimicking the auditory system's response to rhythm of an input signal having a time varying structure comprising the steps of:
providing a network of m nonlinear oscillators, each oscillator having a different natural frequency of oscillation and obeying a dynamical equation of the form:
r
n
+
1
=
1
-
2
k
n
-
2
t
β
,
where
k
n
=
-
1
2
(
r
n
+
c
s
n
(
1
-
ε
r
n
)
cos
ϕ
n
-
r
n
ε
ε
r
n
2
-
2
ε
r
n
cos
ϕ
n
+
1
)
2
ϕ
n
+
1
=
ϕ
n
+
ω
T
n
+
1
-
c
s
n
(
1
-
ε
r
n
)
sin
ϕ
n
ε
r
n
2
-
2
ε
r
n
cos
ϕ
n
+
1
ω
n
+
1
=
ω
n
-
k
s
n
(
1
-
√
ε
r
n
)
sin
ϕ
n
ε
r
n
2
-
2
ε
r
n
cos
ϕ
n
+
1
wherein n indexes discrete input events, T n+1 is an inter-onset time, s n is onset strength, β is a nonlinear dampening parameter, r is the amplitude of the oscillator, c corresponds to the strength of coupling to the external stimulus and φ is the phase of the oscillator, and ω is a resonant frequency;
receiving a discrete time input signal s n at discrete times t n at the network of m nonlinear oscillators; and
generating at least one frequency output from said network useful for describing said varying structure.
3. The method of claim 2 , further comprising the steps of:
determining discrete times at which events are expected from an input stream by determining:
t
x
n
+
1
=
t
x
n
+
T
n
+
1
-
c
s
n
2
π
f
(
1
-
√
ε
r
n
)
sin
ϕ
n
ε
r
n
2
-
2
ε
r
n
cos
ϕ
n
+
1
where tx is expected event time, T n+1 is an inter-onset time, s n is onset strength, β is a nonlinear dampening parameter, r is the amplitude of the oscillator, c corresponds to the strength of coupling to the external stimulus and φ is the phase of the oscillator; and
updating expected event time, tx, according to tx=tx 1/f when t=tx, where t is real time, and outputting musical beats when t=tx.