Low-dimensional model for adaptive networks of spiking neurons

We investigate a large ensemble of Quadratic Integrate-and-Fire (QIF) neurons with heterogeneous input currents and adaptation variables. Our analysis reveals that for a specific class of adaptation, termed quadratic spike-frequency adaptation (QSFA), the high-dimensional system can be exactly reduced to a low-dimensional system of ordinary differential equations, which describes the dynamics of three mean-field variables: the population’s firing rate, the mean membrane potential, and a mean adaptation variable. The resulting low-dimensional firing rate equations (FRE) uncover a key generic feature of heterogeneous networks with spike frequency adaptation: Both the center and the width of the distribution of the neurons’ firing frequencies are reduced, and this largely promotes the emergence of collective synchronization in the network. Our findings are further supported by the bifurcation analysis of the FRE, which accurately captures the collective dynamics of the spiking neuron network, including phenomena such as collective oscillations, bursting, and macroscopic chaos.

💡 Research Summary

This paper presents a significant breakthrough in the mean-field reduction of spiking neural networks with additional dynamics, specifically focusing on spike-frequency adaptation (SFA). The authors study a large population of heterogeneous Quadratic Integrate-and-Fire (QIF) neurons subject to SFA. The key innovation is the introduction of a specific adaptation rule termed “Quadratic Spike-Frequency Adaptation” (QSFA), where the drive to the adaptation variable is proportional to the square of a neuron’s instantaneous firing rate.

The primary and remarkable result is that under QSFA, the high-dimensional system of coupled spiking neurons can be exactly reduced to a low-dimensional system of ordinary differential equations. This reduced model describes the dynamics of just three mean-field variables: the population firing rate (R), the mean membrane potential (V), and a mean adaptation variable. The derivation leverages the fact that with QSFA and Lorentzian-distributed heterogeneous inputs, the distribution of adaptation variables itself remains Lorentzian at all times. This property allows for the application of the exact mean-field reduction technique originally developed for simple QIF networks, extending its power to a more biologically realistic model.

The analysis of the resulting firing rate equations (FRE) reveals a fundamental, generic principle of SFA in heterogeneous networks: it not only reduces the mean firing rate of the population (a well-known effect) but also crucially reduces the heterogeneity (the width) of the firing rate distribution across neurons. The authors demonstrate that neurons with higher intrinsic firing rates experience stronger adaptation. This homogenizing effect actively counteracts the intrinsic diversity in the network, which is typically a desynchronizing force. Consequently, SFA promotes the emergence of collective synchronization through a dual mechanism: providing slow negative feedback and reducing network heterogeneity.

The bifurcation analysis of the low-dimensional FRE shows that it accurately captures the full repertoire of collective dynamics observed in direct simulations of the original spiking network. This includes transitions between asynchronous states, collective oscillations, bursting, and even macroscopic chaos. The phase diagrams illustrate how increasing the adaptation strength expands the parameter region supporting synchronized states, directly linking the theoretical homogenization effect to enhanced synchronizability.

In summary, this work provides a novel, exact low-dimensional model for adaptive spiking networks. It offers not only a powerful analytical tool for studying networks with adaptation but also delivers a profound insight: spike-frequency adaptation is a potent mechanism for controlling collective dynamics by modulating the level of heterogeneity within a neural population.