Modeling nonlinear systems
Systems and techniques, including machine-readable instructions, for modeling of nonlinear systems. In one aspect, an apparatus includes a collection of two or more inputs configured and arranged to receive input signals, a collection of two or more outputs configured and arranged to output output signals, a processing unit configured to transform the input signals into the output signals, wherein the transformation is non-linear and treats the non-linear system as a collection of multiple input, single output non-linear systems, and a data storage that stores characteristics of the transformation.
1. An apparatus for modeling a non-linear system, comprising:
a collection of two or more inputs configured and arranged to receive input signals;
a collection of two or more outputs configured and arranged to output output signals;
a processing unit configured to transform the input signals into the output signals, wherein the transformation is non-linear; and
a data storage that stores characteristics of the non-linear transformation, including at least one of
a first characterization of an impact of a first input signal received on a first of the inputs on an effect that a second input signal received on a second of the inputs has on an output signal output by a first of the outputs, and
a second characterization of an impact of the first input signal on an effect that a third input signal on the first of the inputs has on the output signal output by a first of the outputs;
wherein the characteristics comprise values that characterize the non-linear transformation in a multiple input single output model expressed as:
w
=
u
(
k
,
x
)
+
a
(
h
,
y
)
+
n
(
σ
)
,
y
=
{
0
when
w
<
θ
1
when
w
≥
θ
,
wherein x represents input spike trains, y represents an output spike train, n represents noise with standard deviation σ, h represents a feedback kernel that describes a transformation from y to an intermediate feedback variable a, k represents feedforward kernels that describe transformation from x to an intermediate variable u, and w represents pre-threshold potential.
2. The apparatus of claim 1 , wherein the first characterization comprises a set of feedforward self-kernels and cross-kernels.
3. The apparatus of claim 1 , wherein the at least one of the first characterization and the second characterization comprises a decaying feedforward kernel.
4. The apparatus of claim 3 , wherein the decaying feedforward kernel comprises a Laguerre kernel.
5. The apparatus of claim 1 , wherein the characteristics stored by the data storage further comprise a collection of third order self-kernels.
6. A method for modeling a non-linear system, comprising:
receiving, at a data processing apparatus, machine-readable information characterizing a collection of two or more inputs and two or more outputs from a multiple input, multiple output non-linear system; and
estimating, by the data processing apparatus, a collection of parameters for modeling the non-linear system by considering the multiple input, multiple output non-linear system to be a collection of multiple input, single output systems;
wherein estimating the collection of parameters comprises estimating a collection of kernels that characterize a non-linear transformation in a multiple input single output model expressed as:
w
=
u
(
k
,
x
)
+
a
(
h
,
y
)
+
n
(
σ
)
,
y
=
{
0
when
w
<
θ
1
when
w
≥
θ
,
wherein x represents input spike trains, y represents an output spike train, n represents noise with standard deviation σ, h represents a feedback kernel that describes a transformation from y to an intermediate feedback variable a, k represents feedforward kernels that describe transformation from x to an intermediate variable u, and w represents pre-threshold potential.
7. The method of claim 6 , wherein estimating the collection of kernels comprises estimating a collection of feedforward cross kernels.
8. The method of claim 6 , wherein estimating the collection of kernels comprises estimating a collection of feedforward self kernels.
9. An apparatus for modeling a non-linear system, comprising:
a collection of two or more inputs configured and arranged to receive input signals;
a collection of two or more outputs configured and arranged to output output signals;
a processing unit configured to transform the input signals into the output signals, wherein the transformation is non-linear and treats the non-linear system as a collection of multiple input, single output non-linear systems; and
a data storage that stores characteristics of the transformation;
wherein the characteristics comprise values that characterize the non-linear transformation in a multiple input single output model expressed as:
w
=
u
(
k
,
x
)
+
a
(
h
,
y
)
+
n
(
σ
)
,
y
=
{
0
when
w
<
θ
1
when
w
≥
θ
,
wherein x represents input spike trains, y represents an output spike train, n represents noise with standard deviation σ, h represents a feedback kernel that describes a transformation from y to an intermediate feedback variable a, k represents feedforward kernels that describe transformation from x to an intermediate variable u, and w represents pre-threshold potential.
10. The apparatus of claim 9 , wherein the processing unit comprises a stochastic noise source that stochastically changes the transformation of the input signals into the output signals.
11. The apparatus of claim 9 , wherein the processing unit comprises a threshold configured to discretize the output signals.