Zhuo-Cheng Xiao

How do human brains function? This question is vital for us to understand ourselves and the intelligence emerge from the complex interactions between cells. Carrying out experiments on human and animals, and measuring their brain activities and behaviors, can provide us valuable insights. However, animal experiments are costly, ethically complex, and occasionally hindered by technological limitations. Here is where mathematics lends a helping hand! Through computational models, we can create digital versions of brain parts. Generally, models can summarize what we know and infer what we don’t know. They can explain physiological mechanism behind crucial brain functions and predict the activities of the brain, and eventually, help us better understand the brain.

I primarily work on the following two research directions (more technical):

1. Multiscale models for cortex: Efficiency vs. Biological realism:

Modeling the human cortex is extremely challenging due to its structural and dynamic complexity. Biologically detailed models can incorporate many details of real cortical circuits but are computationally costly and difficult to scale up. This constrains the modeling to small patches of cortex, thus limiting the range of phenomena that can be studied. Alternatively, reduced and phenomenological models are much easier to build and run. But there is a trade-off: The more a model is simplified, the more difficult it is to compare directly to experimental data. To strike a balance between biological realism and computational efficiency, we aim for mathematically reduced models retaining many essential features of the real cortex at a small fraction of the cost.

We propose a multiscale modeling framework for cortex by combining coarse-graining and local dynamic equilibrium. We are currently working on using the modeling framework to explore crucial parameters and detailed network architecture of the primary visual cortex (V1), aiming for a comprehensive, multi-layer model in the next few years.

 

2. Mathematical theories for spiking networks: a focus on neural oscillations

Spiking Neural Networks (SNNs) are at the forefront of modeling the brain's complex dynamics, aiming to mimic the processing and transmission of information through electrical pulses. Achieving an ideal, comprehensive mathematical theory to map the architecture, parameters, and stimuli of SNNs to their dynamics faces significant challenges, including dimensionality, dynamical heterogeneity, and spike singularity.

We introduce a Markovian framework for SNNs, focusing on neural oscillations across various timescales and the metastability of dynamics' multiple attractors. Our goal is to develop a mathematical theory that accommodates a finite number of neurons while capturing both firing rates and synchronicities. Unlike traditional differential equation-based theories for SNNs, our Markov framework simplifies analysis by treating spike production and other network activities uniformly as transitions between states, offering a significant analytical advantage.