Linear-Threshold Network Models for Describing and Analyzing Brain Dynamics

TL;DR

This article illustrates stability and stabilization properties of LTR dynamics and how they are related to goal-driven selective attention, multistability and its relationship with declarative memory, and bifurcations and oscillations and their role in modeling seizure dynamics in epilepsy.

Abstract

Over the past two decades, an increasing array of control-theoretic methods have been used to study the brain as a complex dynamical system and better understand its structure-function relationship. This article provides an overview on one such family of methods, based on the linear-threshold rate (LTR) dynamics, which arises when modeling the spiking activity of neuronal populations and their impact on each other. LTR dynamics exhibit a wide range of behaviors based on network topologies and inputs, including mono- and multi-stability, limit cycles, and chaos, allowing it to be used to model many complex brain processes involving fast and slow inhibition, multiple time and spatial scales, different types of neural behavior, and higher-order interactions. Here we investigate how the versatility of LTR dynamics paired with concepts and tools from systems and control can provide a computational theory for explaining the dynamical mechanisms enabling different brain processes. Specifically, we illustrate stability and stabilization properties of LTR dynamics and how they are related to goal-driven selective attention, multistability and its relationship with declarative memory, and bifurcations and oscillations and their role in modeling seizure dynamics in epilepsy. We conclude with a discussion on additional properties of LTR dynamics and an outlook on other brain processess that for which they might be play a similar role.

Figures (14)

• Figure 1: Graph-theoretic model of a brain network. Excitatory neurons and connections are shown in red, while inhibitory neurons and populations are shown in blue.
• Figure 2: An intracellular recording showing a spike train as is used for communication between neurons (top) and the corresponding firing rate (bottom), estimated by binning spikes in $100^{\mathrm{ms}}$ bins and smoothing using a Gaussian window with $500^{\mathrm{ms}}$ standard deviation EN-JC:21-tacIDAH-ZB-JC-MAM-KDH-GB:00DAH-KDH-ZB-JC-AM-HH-AS-GB:09-crcns.
• Figure 3: The sigmoidal (left) and linear-threshold (right) activation functions are commonly used for defining firing rate models.
• Figure 4: Feedback and feedforward mechanisms of control within brain networks. The left panel shows an inhibitory feedback loop, where an excitatory signal (red) from the main neuronal population (grey) stimulates an inhibitory interneuron (blue), which in turn inhibits the main population. The right panel shows the feedforward inhibition mechanism, where an excitatory signal from a separate neuronal population stimulates an inhibitory interneuron as well as the main population. The interneuron then inhibits the main population, typically resulting in its net inhibition.
• Figure 5: Brain region divided into task-irrelevant (grey) and task-relevant (red and blue) neuron populations. The task-irrelevant nodes make up $\mathbf{x}^0$ in the partition of the state, while the task-relevant nodes form $\mathbf{x}^1$. The control input $\mathbf{u}(t)$ is used to selectively inhibit the task-irrelevant populations, while the input $\tilde{\mathbf{d}}^1$ recruits the task-relevant populations to an equilibrium.

Theorems & Definitions (19)

• Definition 1
• Theorem 2
• Theorem 3
• Remark 4
• Definition 5
• Definition 6
• Theorem 7
• Definition 8
• Definition 9
• Definition 10