[Paper Review] Mean-field approximations of networks of spiking neurons with short-term synaptic plasticity

Richard Gast|arXiv (Cornell University)|Jan 15, 2021

Neural dynamics and brain functionNeuroscience52 references27 citations

TL;DR

This paper develops two novel mean-field models for networks of quadratic integrate-and-fire (QIF) neurons with pre-synaptic short-term plasticity (STP), overcoming limitations of prior stochastic spike-timing approximations. Using the Ott-Antonsen ansatz, it derives exact low-dimensional equations that capture bursting and bi-stable dynamics, revealing how STP shapes macroscopic network behavior.

ABSTRACT

In Parkinson's disease (PD), large parts of the brain transition into states of enhanced neural synchronization. These phase transitions have been associated with the death of dopaminergic neurons as well as with impaired motor function. In this thesis, we address the much-debated question of how parkinsonian synchronization depends on dopamine depletion in the basal ganglia (BG). To this end, we develop spiking neural network (SNN) models of BG circuits and study them via bifurcation analysis. First, we derive mean-field models that allow to account for various forms of short-term plasticity in SNNs. We show that such short-term plasticity mechanisms can lead to highly synchronous, periodic bursting dynamics and discuss the relevance of this bursting regime for PD. Second, we find that the external pallidum, an important part of the BG, cannot cause parkinsonian oscillations autonomously. However, our results suggest that the external pallidum may contribute to the emergence of cross-frequency coupling that has been reported for parkinsonian oscillations. Finally, we describe an open-source Python toolbox that we developed to implement and analyze mean-field models of neural dynamics. Together, this thesis provides insight into BG synchronization processes as well as the mathematical basis and software for future studies of neural synchronization.:1 Introduction 1.1 A complex systems perspective of the brain 1.2 Brain function and the phase transition to synchronized neural activity 1.3 Low-dimensional manifolds of synchronized neural activity 1.4 Phase transitions to synchronized neural activity in Parkinson’s disease 1.5 Thesis overview 2 Mathematical Models and Methods 2.1 A non-linear oscillator model of neural activity 2.2 Dynamical systems methods for the study of neural network models 2.3 Dynamics of a single QIF neuron 3 Low-Dimensional Dynamics in Spiking Neural Networks 3.1 Mean-field approaches in neuroscience 3.2 Dynamics of QIF networks with post-synaptic STP 3.3 Dynamics of QIF networks with spike-frequency adaptation 3.4 Mean-field dynamics of QIF networks with pre-synaptic STP 3.5 Discussion 4 Phase Transitions and Neural Synchronization in the External Pallidum 4.1 A new perspective on GPe structure and function 4.2 GPe model definition and analysis 4.3 Phase transitions in the GPe under static and periodic input 4.4 Discussion 5. Modeling of Neural Mean-Field Dynamics Via PyRates 5.1 Computational modeling in neuroscience 5.2 The Framework 5.3 Pre-implemented methods for neural modeling workflows 5.4 Results 5.5 Discussion 6. Conclusion and Outlook

Motivation & Objective

• To extend mean-field theory to spiking neural networks with pre-synaptic short-term plasticity, which previous methods fail to capture.
• To address the limitation of stochastic spike-timing approximations in deterministic QIF networks with distributed parameters.
• To derive mathematically rigorous mean-field equations that preserve the dynamics of networks with synapse-specific plasticity.
• To investigate how pre-synaptic STP influences macroscopic network behavior, such as bursting and multistability.
• To provide a foundation for future meso- and macroscopic modeling of neural circuits with dynamic synapses.

Proposed method

• Applies the Ott-Antonsen ansatz to derive mean-field equations for all-to-all coupled QIF neurons with pre-synaptic STP.
• Introduces two distinct approaches to model pre-synaptic STP: one based on synaptic resource dynamics and another on a modified adaptation formalism.
• Uses bifurcation analysis to explore the stability and transitions between dynamic regimes in the derived mean-field systems.
• Compares the proposed models against a recent stochastic spike-timing approximation, demonstrating its inaccuracy in deterministic settings.
• Employs the standard Tsodyks-Markram model for short-term plasticity, incorporating both depression and facilitation mechanisms.
• Derives closed-form mean-field equations by assuming slow adaptation of synaptic variables relative to membrane potential dynamics.

Experimental results

Research questions

• RQ1 Can exact mean-field equations be derived for QIF networks with pre-synaptic short-term plasticity using the Ott-Antonsen method?
• RQ2 How do pre-synaptic STP mechanisms like vesicle depletion affect macroscopic network dynamics compared to post-synaptic models?
• RQ3 Does the stochastic spike-timing approximation accurately reproduce the dynamics of deterministic QIF networks with distributed parameters?
• RQ4 What dynamic regimes—such as bursting or bi-stability—emerge in QIF networks with pre-synaptic STP?
• RQ5 How do the derived mean-field models compare in accuracy and predictive power to existing approximations?

Key findings

• The proposed mean-field models accurately reproduce macroscopic activity in deterministic QIF networks with distributed parameters, unlike the stochastic spike-timing approximation which fails.
• The models capture complex network dynamics, including periodic bursting and bi-stable regimes, arising from pre-synaptic short-term plasticity.
• Bifurcation analysis reveals that pre-synaptic STP enables the emergence of sustained bursting and multistable states in QIF networks.
• The derived mean-field equations are mathematically more involved than prior approximations but provide a more accurate and exact description of network dynamics.
• The study demonstrates that pre-synaptic STP cannot be treated as a global macroscopic variable unless all synapses of a neuron are affected simultaneously, which is not the case in vesicle depletion models.
• The results validate the use of the Ott-Antonsen ansatz for networks with synapse-specific dynamic variables, extending its applicability beyond neuron-specific adaptation.