1. Symmetry Breaking in Space-Time Hierarchies Shapes Brain Dynamics and Behavior 
Ajay S. Pillai, Viktor K. Jirsa 
Neuron 
Volume 94, Issue 5, Pages (June 2017)
DOI: /j.neuron
Copyright © 2017 Elsevier Inc. Terms and Conditions

2. Figure 1
Behavior Is the Set of Coordinated Actions in a Task-Specific Context
(A–C) Top: behavior evolves in a context determined by internal and external variables, receiving sensory information from the outside world and acting upon it with its effectors. It emerges from high-dimensional brain networks with complex connectivity. These brain networks interact among themselves and also with the environment in a dynamic fashion with characteristic timescales. Bottom: behavior evolves over time.
(A) For different initial conditions, but all other constraints being identical, behavioral state variables may still evolve differentially in time. Models of behavior are typically captured by a set of ordinary differential equations, which are of lower dimension than the state space of network space.
(B) A geometric representation of a generative model represented by its flow in state space. The trajectories are plotted corresponding to the time series shown in (A). The oscillating variables from (A) form a limit cycle, and the variables that go downward are part of the fixed-point dynamics. Depending on where you start in the state space you can end in either state. A line that separates these states (separatrix) is shown.
(C) The same flow is illustrated graphically in a higher-dimensional state space. Here the trajectories collapse onto a lower dimensional manifold (here a hemispherical surface, which is 2D manifold in a 3D space) and remain on this manifold during the functionally relevant dynamics.
Movies S1 and S2 show simulations corresponding to (B) and (C). Movie S1 shows a limit cycle oscillator on a planar manifold. Movie S2 shows a spherical manifold with bistable noisy dynamics where a stimulus causes a transition of the system dynamics from one state to another.
Neuron  , DOI: ( /j.neuron )
Copyright © 2017 Elsevier Inc. Terms and Conditions

3. Figure 2 Emergence of Functional Dynamics from Network Interactions
The interactions are constrained by connectivity composed of symmetric and symmetry-breaking components. In the context of SFM theory, the symmetric connectivity will give rise to the low-dimensional space or manifold (i.e., constrain the space). Mathematically, the symmetric connectivity corresponds to the scaling parameter μ being zero.
(A) Network with symmetric connectivity.
(B) For symmetric network connectivity shown in (A), the dynamics collapses on to the manifold. Once the trajectories reach the low-dimensional manifold (shown here as an undulating plane in 3D), there will be no further time evolution and the flow along the manifold is zero.
(C) Network with asymmetric connectivity.
(D) When μ is not zero, the symmetry of the connectivity is broken as shown in (C) and a structured flow emerges on the manifold. Parametric changes in the network connectivity produce different structured flows.
Movie S3 shows simulations corresponding to the periodic flow shown in (D).
Neuron  , DOI: ( /j.neuron )
Copyright © 2017 Elsevier Inc. Terms and Conditions

4. Figure 3 Fitts’ Task Dynamics Embedded in a Neuronal Network Dynamics
(A) The Hooke’s portrait (vertical axis is the acceleration and the horizontal axis the position) shows the projections of the network variables upon the functional subspace and illustrates the changes in the control mechanism. The blue trajectory is for a high-difficulty (ID > 5.5) pointing task between two targets. The orange trajectory is for low-difficulty task (ID < 5.5).
(B) The projections of the network variables upon the functional subspace (ξ-system) are plotted for a single trial and one task difficulty (shown are only the first 30 variables plotted as time series over time). Movie S4 presents corresponding simulations in ξ space.
(C) All 100 time series of the neural network variables (q-system) are shown (same trial as in Figure 3B).
Neuron  , DOI: ( /j.neuron )
Copyright © 2017 Elsevier Inc. Terms and Conditions

5. Figure 4 Brain Function Is Distributed Across the Neuronal Network
For conceptual clarity, we distinguish between the dynamics in the function space (spanned by the state variables ξi) and dynamics in the network space (spanned by the state variables q).
(A) Dynamics in the function coordinate space ξi. Specifically, it shows a single trajectory’s evolution in a 5D space where the manifold is 3D-spherical (inset). As shown, the trajectory quickly collapses onto the spherical manifold and then oscillates around the equator of the sphere (ξ1,2). The other dimensions (ξ3-5) converge to zero.
(B) The same dynamics in the distributed network coordinates qi. As can be seen from the time series in the q space, all 5 dimensions are oscillating.
(C) Dynamics of the system used to create redundant networks, i.e., different networks with different connectivities exhibiting the same dynamics. For clarity, only the first two dimensions from the 5D system are shown.
(D) Two different mappings, Pa in blue and Pb in red (see main text), result in the same dynamics and provide two networks that exhibit the same behavior. The trajectories show the evolution of the dynamics of the two networks (qa in blue, and qb in red). As highlighted by the dashed oval, the difference of the two systems is in the initial transients to the manifold. Once on the manifold, the dynamics of the two networks are the same.
Neuron  , DOI: ( /j.neuron )
Copyright © 2017 Elsevier Inc. Terms and Conditions

6. Figure 5
Distributed Functional Networks Exhibit Robustness to Perturbation
(A) The similarity of the structured flows on the manifolds between any two networks as measured by the deviation angle (DA), defined as the angle between the vector fields of the full system and the lesioned system. Network sizes ranged from 50 nodes to 10,000 nodes. The horizontal axis shows the percentage of nodes lesioned, and the vertical axis shows DA. Standard deviations are plotted as the error bars. The results show that as the size increases the networks become more robust as evidenced by lines flattening out with increasing network size.
(B) The mean deviation angle across lesions for a network of given size (error bars show the standard deviation). As the network size doubles, the robustness increases almost exponentially.
Neuron  , DOI: ( /j.neuron )
Copyright © 2017 Elsevier Inc. Terms and Conditions