How interconnected are neurons in ensembles?

[HowardC]

If one connects an ensemble back to itself with a decoded connection

e.g. nengo.Connection(ens_A, ens_A) with ens_A having 50 neurons does this mean each neuron only connects to itself and not to any other neurons in the ensemble

if I do this directly:
for i in range (0,50):
nengo.Connections(ens_A.neurons[i], ens_A.neurons[i]) it gives possibly similar results to the decoded connection.

if I connect everything to everything; i.e.
for i in range (0,50):
for j in range (0,50):
nengo.Connection(ens_A.neurons[i], \ ens_A.NEURONS[J])

[HowardC]

this message was sent by accident. My intuitions have problems with an ensemble of neurons that have no interconnections (which seems to be the default. What am I missing? Thanks

[xchoo]

Hi @HowardC, and welcome back!

@HowardC

this message was sent by accident.

For future reference, the Nengo forums allows you to edit your posts!
You can edit posts by clicking on the “…” button right next to “Reply”, and then clicking on the pencil button. Deleting a post is done similarly, by pressing the “…” button, then the garbage can button.

Now, let’s get to answering your questions!

@HowardC

If one connects an ensemble back to itself with a decoded connection

e.g. nengo.Connection(ens_A, ens_A) with ens_A having 50 neurons does this mean each neuron only connects to itself and not to any other neurons in the ensemble

This is a fantastic conceptual question to ask, and the quick answer is no. If you have a recurrent (decoded) connection, every neuron in ens_A connects to every neuron in ens_A, including itself. To see why this is the case, let us consider the more straight-forward 2-ensemble connection. Consider the following Nengo code:

ens_A = nengo.Ensemble(5, 1)
ens_B = nengo.Ensemble(5, 1)
nengo.Connection(ens_A, ens_B)

Conceptually, what this amounts to is something like this (please excuse my crude handdrawn diagrams), where $\vec{d}$ are the decoders for ens_A and $\vec{e}$ are the encoders for ens_B.

Mathematically, the encoders and decoders can be combined to form a fully connected weight matrix, so the diagram below is the mathematical equivalent of the network:

Now, to understand what is happening in the case of a recurrent connection, we can consider a recurrent connection as a rolled up version of a 2-ensemble connection, where the neurons in ens_A and ens_B are the same. The decoded version of a recurrent connection the looks like this:

Converting this into the fully connected weight matrix version yields this (I’ve only drawn the connections for the two neurons, it gets messy if I include all of the connections. I’ve coloured the connections differently to differentiate the source neuron):

[HowardC]

thanks so much. I will read this over carefully.