Generating Coherent Patterns of Activity from Chaotic Neural Networks

Sussillo, D., Abbott, L.F. (2009). Generating Coherent Patterns of Activity from Chaotic Neural Networks. Neuron 63 544-557.

I wanted to get into the Larry Abbott stuff as it seemed to pertain to the local bending model I was envisioning. So this paper covers these ideas of using chaotic random networks as the basis for learning arbitrary patterns. Essentially because of the high dimensional space the pattern is easy to read out with some simple linear combination. They expand upon previous work and make the model more complex and incorporate "FORCE" learning.

The idea of FORCE learning is that you start with a fast learn (pretty easy because its just a linear thing), but you don't immediately try to get the error all the way to zero. You keep a small amount of error that then decreases slowly. The reason for this is that the networks tend to destabilize and ruin the output. Part of the key of keeping the network in line is that the output feedsback into the network, and the chaos gets overcome by the pattern. Essentially with no stimuli the network tends to be chaotic, but when given a periodic stimuli to learn, the network loses its chaos.

From the examples its quite good at learning arbitrary patterns. They do an interesting PCA analysis of the network after its learned a particular pattern. The network is initiated with the same random connectivity, and then taught to learn the same pattern. The readout weights are initiated at different random strengths. Once the pattern is learned, the major components (the first few PCs) converge to the same point, regardless of where the readout weights were initialized. The later components end up being at a different place.

The learning performs best right on the verge of chaos. By tuning a network parameter (essentially the average synaptic strength), they can vary the amount of chaos in the network. It performs poorly at learning tasks when the network is not in the chaotic regime, but it also cannot work when the network is too chaotic. The learned pattern has to be able to overcome the intrinsic network chaos, and the network needs some chaos so that it can learn the pattern.

Finally they show a network that can do interesting input-output transformations. They add some control inputs, which essentially connect to the network randomly and train the outputs when the appropriate control inputs come on.

(A) Network with control inputs used to produce multiple output patterns (synapses and readout weights that are modifiable in red).
(B) Five outputs (one cycle of each periodic function made from three sinusoids is shown) generated by a single network and selected by static control inputs.
(C) A network with four outputs and eight inputs used to produce a 4-bit memory (modifiable synapses and readout weights in red).
(D) Red traces are the four outputs, with green traces showing their target values. Purple traces show the eight inputs, divided into ON and OFF pairs associated with the output trace above
them. The upper input in each pair turns the corresponding output on (sets it to +1). The lower input of each pair turns the output off (sets it to  1). After learning, the network has implemented a 4-bit memory, with each output responding only to its two inputs while ignoring the other inputs.


The 4-bit memory is a pretty cool example. For each bit there is an on and off input pulse, which flips the state of the corresponding readout unit. So in the network there are 16 fixed points corresponding to all possible bit combinations.

Then they have a much more complex example of the network implementing running and walking like behaviors. They got some motion capture data and trained the network to mimic the joint angles for 95 different joints. And the network was able to learn both walking and running.

The network synaptic structure is linear. Firing rate model that goes through a tanh output non-linearity