Experimental studies have begun revealing essential properties of the structural connectivity and the spatiotemporal activity dynamics of cortical circuits. To integrate these properties from anatomy and physiology, and to elucidate the links between them, we develop a novel cortical circuit model that captures a range of realistic features of synaptic connectivity. We show that the model accounts for the emergence of higher-order connectivity structures, including highly connected hub neurons that form an interconnected rich-club. The circuit model exhibits a rich repertoire of dynamical activity states, ranging from asynchronous to localized and global propagating wave states. We find that around the transition between asynchronous and localized propagating wave states, our model quantitatively reproduces a variety of major empirical findings regarding neural spatiotemporal dynamics, which otherwise remain disjointed in existing studies. These dynamics include diverse coupling (correlation) between spiking activity of individual neurons and the population, dynamical wave patterns with variable speeds and precise temporal structures of neural spikes. We further illustrate how these neural dynamics are related to the connectivity properties by analysing structural contributions to variable spiking dynamics and by showing that the rich-club structure is related to the diverse population coupling. These findings establish an integrated account of structural connectivity and activity dynamics of local cortical circuits, and provide new insights into understanding their working mechanisms.

The formation of dynamic patterns such as localized propagating waves is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended systems including neural systems, in which they might play important functional roles. Here we derive a type of two-dimensional neural-field model with refractoriness to study the formation mechanism of localized waves. After comparing this model with existing neural-field models, we show that it is able to generate a variety of localized patterns, including stationary bumps, localized waves rotating along a circular path, and localized waves with longer-range propagation. We construct explicit bump solutions for the two-dimensional neural field and conduct a linear stability analysis on how a stationary bump transitions to a propagating wave under different spatial eigenmode perturbations. The neural-field model is then partially solved in a comoving frame to obtain localized wave solutions, whose spatial profiles are in good agreement with those obtained from simulations. We demonstrate that when there are multiple such propagating waves, they exhibit rich propagation dynamics, including propagation along periodically oscillating and irregular trajectories; these propagation dynamics are quantitatively characterized. In addition, we show that these waves can have repulsive or merging collisions, depending on their collision angles and the refractoriness parameter. Due to its analytical tractability, the two-dimensional neural-field model provides a modeling framework for studying localized propagating waves and their interactions.

Responding to various stimuli, some neurons either remain resting or can fire several distinct patterns of action potentials, such as spiking, bursting, subthreshold oscillations, and chaotic firing. In particular, Wilson's conductance-based neocortical neuron model, derived from the Hodgkin-Huxley model, is explored to understand underlying mechanisms of the firing patterns. Phase diagrams describing boundaries between the domains of different firing patterns are obtained via extensive numerical computations. The boundaries are further studied by standard instability analyses, which demonstrates that the chaotic neural firing could develop via period-doubling and/or period- adding cascades. Sequences of the firing patterns often observed in many neural experiments are also discussed in the phase diagram framework developed. Our results lay the groundwork for wider use of the model, especially for incorporating it into neural field modeling of the brain.

A range of perceptual and cognitive processes have been characterized from the perspective of probabilistic representations and inference. To understand the neural circuit mechanism underlying these probabilistic computations, we develop a theory based on complex spatiotemporal dynamics of neural population activity. We first implement and explore this theory in a biophysically realistic, spiking neural circuit. Population activity patterns emerging from the circuit capture realistic variability or fluctuations of neural dynamics both in time and in space. These activity patterns implement a type of probabilistic computations that we name fractional neural sampling (FNS). We further develop a mathematical model to reveal the algorithmic nature of FNS and its computational advantages for representing multimodal distributions, a major challenge faced by existing theories. We demonstrate that FNS provides a unified account of a diversity of experimental observations of neural spatiotemporal dynamics and perceptual processes such as visual perception inference, and that FNS makes experimentally testable predictions.