Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article


Doctor of Philosophy


Applied Mathematics


Muller, Lyle


Thanks to recent advances in neurotechnology, waves of activity sweeping across entire cortical regions are now routinely observed. Moreover, these waves have been found to impact neural responses as well as perception, and the responses themselves are found to be structured as traveling waves. How exactly do these waves arise? Do they confer any computational advantages? These traveling waves represent an opportunity for an expanded theory of neural computation, in which their dynamic local network activity may complement the moment-to-moment variability of our sensory experience.

This thesis aims to help uncover the origin and role of traveling waves in the visual cortex through three Works. In Work 1, by simulating a network of conductance-based spiking neurons with realistically large network size and synaptic density, distance-dependent horizontal axonal time delays were found to be important for the widespread emergence of spontaneous traveling waves consistent with those in vivo. Furthermore, these waves were found to be a dynamic mechanism of gain modulation that may explain the in-vivo result of their modulation of perception. In Work 2, the Kuramoto oscillator model was formulated in the complex domain to study a network possessing distance-dependent time delays. Like in Work 1, these delays produced traveling waves, and the eigenspectrum of the complex-valued delayed matrix, containing a delay operator, provided an analytical explanation of them. In Work 3, the model from Work 2 was adapted into a recurrent neural network for the task of forecasting the frames of videos, with the question of how such a biologically constrained model may be useful in visual computation. We found that the wave activity emergent in this network was helpful, as they were tightly linked with high forecast performance, and shuffle controls revealed simultaneous abolishment of both the waves and performance.

All together, these works shed light on the possible origins and uses of traveling waves in the visual cortex. In particular, time delays profoundly shape the spatiotemporal dynamics into traveling waves. This was confirmed numerically (Work 1) and analytically (Work 2). In Work 3, these waves were found to aid in the dynamic computation of visual forecasting.

Summary for Lay Audience

The brain is organized into distinct regions of neurons. Within a single such region, the neurons connect to one another intricately, forming a web-like network called a recurrent network. Through our senses, such as vision, these recurrent networks receive stimulation from the outside world, and use this stimulation to help the organism execute meaningful actions, such as movement. Exactly how the neurons in a recurrent network orchestrate, however, is unclear. Often, the electrical neural activity is observed to show waves traveling across the recurrent network, reminiscent of turbulent water waves that wash up on shore. It is unknown exactly how these traveling waves arise and how useful they are.

In Work 1, large computer simulations of a recurrent network were performed. The results showed that a crucial component to include in the simulation is the communication transmission delays between neuron pairs. With these delays included, the simulated electrical neural activity in the network exhibited traveling waves that agreed with real biological recordings, thereby shedding light on the potential origins of such waves in the brain.

In Work 2, a mathematical description of these waves was sought, since mathematical descriptions permit the deepest understanding of physical phenomena. For this purpose, the Kuramoto model was used. The Kuramoto model is a simple yet powerful mathematical model capable of describing traveling waves. This model included time delays similar to those in Work 1. Thus, a mathematical understanding of traveling waves that emerge from delayed recurrent networks was gained.

In Work 3, we asked how traveling waves in the visual part of brain could help in making predictions. To answer this, we used networks similar to the ones in the prior two Works, which are known to cause traveling waves. We then stimulated the network with frames of a movie, and tasked the network with forecasting what the future movie frames might look like. The network was able to learn how to predict these movies. Successful predictions were tightly linked with traveling waves, thereby supporting that these waves are useful for visual prediction in the brain.

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License