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_neurons (int) – Number of neurons to use in each product computation.
dimensions (int) – The number of dimensions of the input and output vectors.
invert_a (bool, optional) – Whether to reverse the order of elements in first input.
invert_b (bool, 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_magnitude (float, 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.
**kwargs (dict) – Keyword arguments to pass through to the nengo.Network constructor.

input_a¶

The first vector to be convolved.

Type: nengo.Node

input_b¶

The second vector to be convolved.

Type: nengo.Node

product¶

Network created to do the element-wise product of the \(DFT\) components.

Type: nengo.networks.Product

output¶

The resulting convolved vector.

Type: nengo.Node

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)

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.

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_dimension (int) – Neurons per dimension.
dimensions (int) – Total number of dimensions. Must be a multiple of subdimensions.
subdimensions (int) – Maximum number of dimensions per ensemble.
**kwargs (dict) – Keyword arguments to pass through to the nengo.Network constructor.

input¶

Input node.

Type: nengo.Node

output¶

Output node.

Type: nengo.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: The added node.
Return type: 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: The added node.
Return type: nengo.Node

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

Adds a new decoded output.

This will add the attribute named name to the object.

Parameters

name (str) – Name of output. Must be a valid Python attribute name.
function (func) – Function to decode.
synapse (float or nengo.Lowpass) – Synapse to apply to the decoded connection to the returned output node.
conn_kwargs (dict) – Additional keywword arguments to apply to the decoded connection.

Returns

Node providing the decoded output.

Return type

nengo.Node

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_neurons (int) –
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_left (tuple) – Shape of the A input matrix.
shape_right (tuple) – Shape of the B input matrix.
**kwargs (dict) – Keyword arguments to pass through to the nengo.Network constructor.

input_left¶

The left matrix (A) to multiply.

Type: nengo.Node

input_right¶

The left matrix (A) to multiply.

Type: nengo.Node

C¶

The product network doing the matrix multiplication.

Type: nengo.networks.Product

output¶

The resulting matrix result.

Type: nengo.node

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.

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_neurons (int) – Number of neurons for each choice.
n_ensembles (int) – Number of choices.
accum_threshold (float, 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_timescale (float, optional) – Evidence accumulation timescale.
feedback_timescale (float, optional) – Timescale for the feedback connection from the thresholding layer to the accumulation layer.
accum_synapse (Synapse or float, optional) – The synapse for connections to the accumulator ensembles.
ff_synapse (Synapse or float, optional) – Synapse for feed-forward connections.
intercept_width (float, optional) – The nengo.presets.ThresholdingEnsembles intercept_width parameter.
radius (float, optional) – The representational radius of the ensembles.
**kwargs (dict) – Keyword arguments passed on to nengo.Network.

input¶

The inputs to the network.

Type: nengo.Node

input_reset¶

Input to reset the accumulators.

Type: nengo.Node

output¶

The outputs of the network.

Type: nengo.Node

accumulators¶

The layer 1 accumulators.

Type: nengo.Thresholding

thresholding¶

The layer 2 thresholding ensembles.

Type: nengo.Thresholding

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.

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_neurons (int) – Number of neurons for each ensemble.
n_ensembles (int) – Number of ensembles.
threshold (float) – The thresholding value.
intercept_width (float, optional) – The nengo.presets.ThresholdingEnsembles intercept_width parameter.
function (function, optional) – Function to apply to the thresholded values.
radius (float, optional) – The representational radius of the ensembles.
**kwargs (dict) – Keyword arguments passed on to nengo.Network.

input¶

The inputs to the network.

Type: nengo.Node

output¶

The outputs of the network.

Type: nengo.Node

thresholded¶

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

Type: nengo.Node

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_neurons (int) – Number of neurons for each ensemble.
n_ensembles (int) – Number of ensembles.
inhibit_scale (float, optional) – Scaling of the lateral inhibition.
inhibit_synapse (Synapse or float, optional) – Synapse on the recurrent connection for lateral inhibition.
**kwargs (dict) – Keyword arguments passed on to Thresholding.

input¶

The inputs to the network.

Type: nengo.Node

output¶

The outputs of the network.

Type: nengo.Node

thresholded¶

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

Type: nengo.Node