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# nengo_spa.networks¶

Basic networks that are used by NengoSPA.

These networks do not provide any information about inputs, outputs or used vocabularies and are completely independent of SPA specifics.

 CircularConvolution(n_neurons, dimensions[, …]) Compute the circular convolution of two vectors. IdentityEnsembleArray(neurons_per_dimension, …) An ensemble array optimized for representing the identity circular convolution vector besides random unit vectors. MatrixMult(n_neurons, shape_left, …) Computes the matrix product A*B. VTB(n_neurons, dimensions[, unbind_left, …]) Compute vector-derived transformation binding (VTB). TVTB(n_neurons, dimensions[, unbind_left, …]) Compute transposed vector-derived transformation binding (TVTB).

Selection networks

 selection.IA(n_neurons, n_ensembles[, …]) Independent accumulator (IA) winner-take-all (WTA) network. selection.Thresholding(n_neurons, …[, …]) Array of thresholding ensembles. selection.WTA(n_neurons, n_ensembles[, …]) Winner-take-all (WTA) network with lateral inhibition.
class nengo_spa.networks.CircularConvolution(n_neurons, dimensions, invert_a=False, invert_b=False, input_magnitude=1.0, **kwargs)[source]

Compute the circular convolution of two vectors.

The circular convolution $$c$$ of vectors $$a$$ and $$b$$ is given by

$c[i] = \sum_j a[j] b[i - j]$

where negative indices on $$b$$ wrap around to the end of the vector.

This computation can also be done in the Fourier domain,

$c = DFT^{-1} ( DFT(a) \odot DFT(b) )$

where $$DFT$$ is the Discrete Fourier Transform operator, and $$DFT^{-1}$$ is its inverse. This network uses this method.

Parameters
n_neuronsint

Number of neurons to use in each product computation.

dimensionsint

The number of dimensions of the input and output vectors.

invert_abool, optional

Whether to reverse the order of elements in first input.

invert_bbool, optional

Whether to reverse the order of elements in the second input. Flipping exactly one input will make the network perform circular correlation instead of circular convolution which can be treated as an approximate inverse to circular convolution.

input_magnitudefloat, optional

The expected magnitude of the vectors to be convolved. This value is used to determine the radius of the ensembles computing the element-wise product.

**kwargsdict

Keyword arguments to pass through to the nengo.Network constructor.

Notes

The network maps the input vectors $$a$$ and $$b$$ of length $$N$$ into the Fourier domain and aligns them for complex multiplication. Letting $$F = DFT(a)$$ and $$G = DFT(b)$$, this is given by:

[ F[i].real ]     [ G[i].real ]     [ w[i] ]
[ F[i].imag ]  *  [ G[i].imag ]  =  [ x[i] ]
[ F[i].real ]     [ G[i].imag ]     [ y[i] ]
[ F[i].imag ]     [ G[i].real ]     [ z[i] ]


where $$i$$ only ranges over the lower half of the spectrum, since the upper half of the spectrum is the flipped complex conjugate of the lower half, and therefore redundant. The input transforms are used to perform the DFT on the inputs and align them correctly for complex multiplication.

The complex product $$H = F * G$$ is then

$H[i] = (w[i] - x[i]) + (y[i] + z[i]) I$

where $$I = \sqrt{-1}$$. We can perform this addition along with the inverse DFT $$c = DFT^{-1}(H)$$ in a single output transform, finding only the real part of $$c$$ since the imaginary part is analytically zero.

Examples

A basic example computing the circular convolution of two 10-dimensional vectors represented by ensemble arrays:

A = EnsembleArray(50, n_ensembles=10)
B = EnsembleArray(50, n_ensembles=10)
C = EnsembleArray(50, n_ensembles=10)
cconv = nengo_spa.networks.CircularConvolution(50, dimensions=10)
nengo.Connection(A.output, cconv.input_a)
nengo.Connection(B.output, cconv.input_b)
nengo.Connection(cconv.output, C.input)

Attributes
input_anengo.Node

The first vector to be convolved.

input_bnengo.Node

The second vector to be convolved.

productnengo.networks.Product

Network created to do the element-wise product of the $$DFT$$ components.

outputnengo.Node

The resulting convolved vector.

class nengo_spa.networks.IdentityEnsembleArray(neurons_per_dimension, dimensions, subdimensions, **kwargs)[source]

An ensemble array optimized for representing the identity circular convolution vector besides random unit vectors.

The ensemble array will use ensembles with subdimensions dimensions, except for the first subdimensions dimensions. These will be split into a one-dimensional ensemble for the first dimension and a subdimensions-1 dimensional ensemble.

Parameters
neurons_per_dimensionint

Neurons per dimension.

dimensionsint

Total number of dimensions. Must be a multiple of subdimensions.

subdimensionsint

Maximum number of dimensions per ensemble.

**kwargsdict

Keyword arguments to pass through to the nengo.Network constructor.

Attributes
inputnengo.Node

Input node.

outputnengo.Node

Output node.

add_neuron_input()[source]

Adds a node providing input to the neurons of all ensembles.

This node will be accessible through the neuron_input attribute.

Returns
nengo.Node

add_neuron_output()[source]

Adds a node providing neuron (non-decoded) output of all ensembles.

This node will be accessible through the neuron_output attribute.

Returns
nengo.Node

add_output(name, function, synapse=None, **conn_kwargs)[source]

This will add the attribute named name to the object.

Parameters
namestr

Name of output. Must be a valid Python attribute name.

functionfunc

Function to decode.

synapsefloat or nengo.Lowpass

Synapse to apply to the decoded connection to the returned output node.

conn_kwargsdict

Additional keywword arguments to apply to the decoded connection.

Returns
nengo.Node

Node providing the decoded output.

class nengo_spa.networks.MatrixMult(n_neurons, shape_left, shape_right, **kwargs)[source]

Computes the matrix product A*B.

Both matrices need to be two dimensional.

See the Nengo Matrix Multiplication example for a description of the network internals.

Parameters
n_neuronsint

Number of neurons used per product of two scalars.

Note

If an odd number of neurons is given, one less neuron will be used per product to obtain an even number. This is due to the implementation the Product network.

shape_lefttuple

Shape of the A input matrix.

shape_righttuple

Shape of the B input matrix.

**kwargsdict

Keyword arguments to pass through to the nengo.Network constructor.

Attributes
input_leftnengo.Node

The left matrix (A) to multiply.

input_rightnengo.Node

The left matrix (A) to multiply.

Cnengo.networks.Product

The product network doing the matrix multiplication.

outputnengo.node

The resulting matrix result.

class nengo_spa.networks.TVTB(n_neurons, dimensions, unbind_left=False, unbind_right=False, **kwargs)[source]

Compute transposed vector-derived transformation binding (TVTB).

VTB uses elementwise addition for superposition. The binding operation $$\mathcal{B}(x, y)$$ is defined as

$\begin{split}\mathcal{B}(x, y) := V_y^T x = \left[\begin{array}{ccc} V_y'^T & 0 & 0 \\ 0 & V_y'^T & 0 \\ 0 & 0 & V_y'^T \end{array}\right] x\end{split}$

with

$\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d - d' + 1} & y_{d - d' + 2} & \dots & y_d \end{array}\right]\end{split}$

and

$d'^2 = d.$

The approximate inverse $$y^+$$ for $$y$$ is permuting the elements such that $$V_{y^+} = V_y^T$$.

Note that TVTB requires the vector dimensionality to be square.

The TVTB binding operation is neither associative nor commutative. In contrast to VTB, however, TVTB has two-sided identities and inverses. Other properties are equivalent to VTB.

Parameters
n_neuronsint

Number of neurons to use in each product computation.

dimensionsint

The number of dimensions of the input and output vectors. Needs to be a square number.

unbind_leftbool

Whether to unbind the left input vector from the right input vector.

unbind_rightbool

Whether to unbind the right input vector from the left input vector.

**kwargsdict

Keyword arguments to pass through to the nengo.Network constructor.

Attributes
input_leftnengo.Node

The left operand vector to be bound.

input_rightnengo.Node

The right operand vector to be bound.

matnengo.Node

Representation of the matrix $$V_y'$$.

vecnengo.Node

Representation of the vector $$y$$.

matmulslist

Matrix multiplication networks.

outputnengo.Node

The resulting bound vector.

class nengo_spa.networks.VTB(n_neurons, dimensions, unbind_left=False, unbind_right=False, **kwargs)[source]

Compute vector-derived transformation binding (VTB).

VTB uses elementwise addition for superposition. The binding operation $$\mathcal{B}(x, y)$$ is defined as

$\begin{split}\mathcal{B}(x, y) := V_y x = \left[\begin{array}{ccc} V_y' & 0 & 0 \\ 0 & V_y' & 0 \\ 0 & 0 & V_y' \end{array}\right] x\end{split}$

with

$\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d - d' + 1} & y_{d - d' + 2} & \dots & y_d \end{array}\right]\end{split}$

and

$d'^2 = d.$

The approximate inverse $$y^+$$ for $$y$$ is permuting the elements such that $$V_{y^+} = V_y^T$$.

Note that VTB requires the vector dimensionality to be square.

The VTB binding operation is neither associative nor commutative. Furthermore, there are right inverses and identities only. By transposing the $$V_y$$ matrix, the closely related TvtbAlgebra (Transposed VTB) algebra is obtained which does have two-sided identities and inverses.

Parameters
n_neuronsint

Number of neurons to use in each product computation.

dimensionsint

The number of dimensions of the input and output vectors. Needs to be a square number.

unbind_leftbool

Whether to unbind the left input vector from the right input vector.

unbind_rightbool

Whether to unbind the right input vector from the left input vector.

**kwargsdict

Keyword arguments to pass through to the nengo.Network constructor.

Attributes
input_leftnengo.Node

The left operand vector to be bound.

input_rightnengo.Node

The right operand vector to be bound.

matnengo.Node

Representation of the matrix $$V_y'$$.

vecnengo.Node

Representation of the vector $$y$$.

matmulslist

Matrix multiplication networks.

outputnengo.Node

The resulting bound vector.

## nengo_spa.networks.selection¶

Selection networks that pick one or more options among multiple choices.

class nengo_spa.networks.selection.IA(n_neurons, n_ensembles, accum_threshold=0.8, accum_neuron_ratio=0.7, accum_timescale=0.2, feedback_timescale=0.005, accum_synapse=0.1, ff_synapse=0.005, intercept_width=0.15, radius=1.0, **kwargs)[source]

Independent accumulator (IA) winner-take-all (WTA) network.

This is a two-layered network. The first layer consists of independent accumulators (integrators), whereas the second layer does a thresholding. Once the threshold is exceeded a feedback connection will stabilize the current choice and inhibit all other choices. To switch the selection, it is necessary to provide a transient input to input_reset to reset the accumulator states.

This network is suited especially for accumulating evidence under noisy conditions and keep a stable choice selection until the processing of the choice has been finished.

Further details are to be found in [gosmann2017].

Parameters
n_neuronsint

Number of neurons for each choice.

n_ensemblesint

Number of choices.

accum_thresholdfloat, optional

Accumulation threshold that needs to be reached to produce an output.

accum_neuron_ratio: float, optional

Portion of n_neurons that will be used for a layer 1 accumulator ensemble. The remaining neurons will be used for a layer 2 thresholding ensemble.

accum_timescalefloat, optional

Evidence accumulation timescale.

feedback_timescalefloat, optional

Timescale for the feedback connection from the thresholding layer to the accumulation layer.

accum_synapseSynapse or float, optional

The synapse for connections to the accumulator ensembles.

ff_synapseSynapse or float, optional

Synapse for feed-forward connections.

intercept_widthfloat, optional

The nengo.presets.ThresholdingEnsembles intercept_width parameter.

The representational radius of the ensembles.

**kwargsdict

Keyword arguments passed on to nengo.Network.

References

gosmann2017

Jan Gosmann, Aaron R. Voelker, and Chris Eliasmith. “A spiking independent accumulator model for winner-take-all computation.” In Proceedings of the 39th Annual Conference of the Cognitive Science Society. London, UK, 2017. Cognitive Science Society.

Attributes
inputnengo.Node

The inputs to the network.

input_resetnengo.Node

Input to reset the accumulators.

outputnengo.Node

The outputs of the network.

accumulatorsnengo.Thresholding

The layer 1 accumulators.

thresholdingnengo.Thresholding

The layer 2 thresholding ensembles.

class nengo_spa.networks.selection.Thresholding(n_neurons, n_ensembles, threshold, intercept_width=0.15, function=None, radius=1.0, **kwargs)[source]

Array of thresholding ensembles.

All inputs below the threshold will produce an output of 0, whereas inputs above the threshold produce an output of equal value.

Parameters
n_neuronsint

Number of neurons for each ensemble.

n_ensemblesint

Number of ensembles.

thresholdfloat

The thresholding value.

intercept_widthfloat, optional

The nengo.presets.ThresholdingEnsembles intercept_width parameter.

functionfunction, optional

Function to apply to the thresholded values.

The representational radius of the ensembles.

**kwargsdict

Keyword arguments passed on to nengo.Network.

Attributes
inputnengo.Node

The inputs to the network.

outputnengo.Node

The outputs of the network.

thresholdednengo.Node

The raw thresholded value (before applying function or correcting for the shift produced by the thresholding).

class nengo_spa.networks.selection.WTA(n_neurons, n_ensembles, inhibit_scale=1.0, inhibit_synapse=0.005, **kwargs)[source]

Winner-take-all (WTA) network with lateral inhibition.

Parameters
n_neuronsint

Number of neurons for each ensemble.

n_ensemblesint

Number of ensembles.

inhibit_scalefloat, optional

Scaling of the lateral inhibition.

inhibit_synapseSynapse or float, optional

Synapse on the recurrent connection for lateral inhibition.

**kwargsdict

Keyword arguments passed on to Thresholding.

Attributes
inputnengo.Node

The inputs to the network.

outputnengo.Node

The outputs of the network.

thresholdednengo.Node

The raw thresholded value (before applying function or correcting for the shift produced by the thresholding).