Exact macroscopic equations for globally coupled heterogeneous neurons


Many phenomena in the natural and biological sciences reflect the collective behavior of large numbers of interacting dynamical units. Insight into such phenomena can therefore by gleaned by studying idealized systems in which the dynamics of the individual units are taken to be as simple as possible, while still retaining the most salient features for the physical system of interest. A well-known example is the Kuramoto model, a canonical example of collective synchronization. In the field of neuroscience, the individual unit is the neuron and the collective behavior of interest is often the response of the neuronal network to external inputs. Here, as opposed to the Kuramoto model, the parameter order of interest is not necessarily the degree of synchronization, but rather the average number of electrical impulses or “spikes” generated by the network per unit of time, known as the “firing rate”.

Here I will discuss the derivation of an exact mean-field model for a large network of coupled, heterogeneous model neurons [Montbrio15]. The neural model, that is the individual element, is the “quadratic integrate-and-fire” neuron, a canonical model which approximates how spikes are emitted in a large class of more realistic neuronal models. We find that the relevant macroscopic variables are the mean firing rate as well as the mean voltage drop across the cellular membrane of neurons in the network; the collective dynamics are described by a system of two coupled ODEs. Interestingly, we can show that there is a deep connection between this neuronal description of collective behavior, and the more traditional Kuramoto order parameter which measures collective synchronization. Specifically, the conformal transformations which map between the unit circle and the positive half plane allow us to switch between these two descriptions and tie our work to the so-called Ott-Antonsen ansatz [Ott08], which provides an exact solution to the Kuramoto model.

[Montbrio15] E. Montbrió, D. Pazó, and A. Roxin. Macroscopic description for networks of spiking neurons. , 5, 2015.

[Ott08] E. Ott and T. M. Antonsen. Low-dimensional behavior of large systems of globally coupled oscillators. , 18:037113, 2008.