Composed solutions of synchronized patterns in multiplex networks of Kuramoto oscillators.

Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to investigate these networks, obtaining mathematical descriptions for the dynamics of multilayer and multiplex systems is still an open problem. Here, we combine ideas and concepts from linear algebra and graph theory with nonlinear dynamics to offer a novel approach to study multiplex networks of Kuramoto oscillators. Our approach allows us to study the dynamics of a large, multiplex network by decomposing it into two smaller systems: one representing the connection scheme within layers (intra-layer), and the other representing the connections between layers (inter-layer). Particularly, we use this approach to compose solutions for multiplex networks of Kuramoto oscillators. These solutions are given by a combination of solutions for the smaller systems given by the intra- and inter-layer systems, and in addition, our approach allows us to study the linear stability of these solutions.

Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks.

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.

Neuronal activation sequences in lateral prefrontal cortex encode visuospatial working memory during virtual navigation.

Working memory (WM) is the ability to maintain and manipulate information 'in mind'. The neural codes underlying WM have been a matter of debate. We simultaneously recorded the activity of hundreds of neurons in the lateral prefrontal cortex of male macaque monkeys during a visuospatial WM task that required navigation in a virtual 3D environment. Here, we demonstrate distinct neuronal activation sequences (NASs) that encode remembered target locations in the virtual environment. This NAS code outperformed the persistent firing code for remembered locations during the virtual reality task, but not during a classical WM task using stationary stimuli and constraining eye movements. Finally, blocking NMDA receptors using low doses of ketamine deteriorated the NAS code and behavioral performance selectively during the WM task. These results reveal the versatility and adaptability of neural codes supporting working memory function in the primate lateral prefrontal cortex.

Algebraic approach to spike-time neural codes in the hippocampus.

Although temporal coding through spike-time patterns has long been of interest in neuroscience, the specific structures that could be useful for spike-time codes remain highly unclear. Here, we introduce an analytical approach, using techniques from discrete mathematics, to study spike-time codes. As an initial example, we focus on the phenomenon of "phase precession" in the rodent hippocampus. During navigation and learning on a physical track, specific cells in a rodent's brain form a highly structured pattern relative to the oscillation of population activity in this region. Studies of phase precession largely focus on its role in precisely ordering spike times for synaptic plasticity, as the role of phase precession in memory formation is well established. Comparatively less attention has been paid to the fact that phase precession represents one of the best candidates for a spike-time neural code. The precise nature of this code remains an open question. Here, we derive an analytical expression for a function mapping points in physical space to complex-valued spikes by representing individual spike times as complex numbers. The properties of this function make explicit a specific relationship between past and future in spike patterns of the hippocampus. Importantly, this mathematical approach generalizes beyond the specific phenomenon studied here, providing a technique to study the neural codes within precise spike-time sequences found during sensory coding and motor behavior. We then introduce a spike-based decoding algorithm, based on this function, that successfully decodes a simulated animal's trajectory using only the animal's initial position and a pattern of spike times. This decoder is robust to noise in spike times and works on a timescale almost an order of magnitude shorter than typically used with decoders that work on average firing rate. These results illustrate the utility of a discrete approach, based on the structure and symmetries in spike patterns across finite sets of cells, to provide insight into the structure and function of neural systems.

Algebraic approach to the Kuramoto model.

We study the Kuramoto model with attractive sine coupling. We introduce a complex-valued matrix formulation whose argument coincides with the original Kuramoto dynamics. We derive an exact solution for the complex-valued model, which permits analytical insight into individual realizations of the Kuramoto model. The existence of a complex-valued form of the Kuramoto model provides a key demonstration that, in some cases, reformulations of nonlinear dynamics in higher-order number fields may provide tractable analytical approaches.