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figure 4 shows example unsupervised classification results as implemented using a three layer fully connected (fc) neural network. the maps produced by the model closely resemble the brain regions identified by spatial ica and spatial general linear models, and importantly provide additional information on the types of representations in the different networks. while the performance of all neural networks in the framework was similar (fig. 5a), deep learning models performed consistently better (further information on network parameters can be found in the supplementary information and the work that will be explored in future studies). to test generalization to different levels of the hierarchy, the model was trained to classify signals from various positions of the nodes in the model. the performance of the model at each level of the hierarchy is shown in figure 5b. training the model to perform classification at the level of the network nodes yielded consistently high performance in all subjects, suggesting that the activations in the networks are meaningful. this model could be further refined, such as by introducing back-propagation through time to learn about earlier network nodes. to address modeling the fine-grained nature of information carried in task-related bold signals, the model could also be extended to classify task-related fmri time series, perhaps by utilizing a temporal convolutional neural network (see introduction).
figure 2: a topological analysis of a representative vortex knot. the knot k6-2 is depicted in blue, with white lines outlining the six contour crossings that constitute the knot (each represented by a circle). we observe that the knot (a) unties via a sequence of vortex reconnections that result in k6-2(b), the unknotted vortex. the knot can be built from the trivial unknotted unknot at right (k-0) by adding local complex features to each black dot on the figure, much as a two-dimensional knotted polygon can be built from a series of elementary knots.
after each vortex topology transition we begin to see evidence of the system achieving a plateau in both energy and entanglement entropy, and at this point the system remains in one specific vortex configuration for a period of time. we define a plateau in the energy or entropy at time t t as a period in which the system evolves forward, either towards a lower or higher energy state, but has no net change in energy or entropy, i.e. ∑ dt [e(t) − e(t-1)] = ∑ dt [e(t) − e(t+1)] = 0. after some time, the system begins to transition again: one of the previous configurations is lost, and a new topology is formed. to quantify the timescale of energy and entropy plateaus, we monitor the fraction of time ∑ dt [e(t) − e(t-1)] = ∑ dt [e(t) − e(t+1)] = 0 and ∑ dt [s(t) − s(t-1)] = ∑ dt [s(t) − s(t+1)] = 0.
we used a recurrent neural network model with an architecture consisting of 2 layers of hidden grus (having 64 latent variables) and 2 output layers, inspired by descriptions of long short-term memory (lstm) networks in the literature [33]. each of the hidden gru layers were preceded by a max-pooling layer to combine information across nearby neurons, yielding a compact representation of the input history that is common across the layers. each of the gru and pooling layers were followed by a normalization layer: (1) the gru layer was normalized by z-scoring (i.e. centered to zero mean and unit standard deviation), and (2) the pooling layer was normalized to a unit standard deviation of 1 to help remove differences in total brain activity across different runs/participants. after pooling, the representation of the input was common across the hidden gru layers and the pooling layer. we chose a temporal input of up to 30 seconds, which translated to a nx = 2892 dimensionality. this was informed by the need for maximum pooling, while retaining temporal continuity. this representation included the parcellation-specific information learned by the grus. the final representations of the inputs consisted of nx = 2892 dimensions and nx = 1 dimensions (representation summary). the final representation is common across the two hidden gru layers.
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