A multiple timescales approach to bridging spiking- and population-level dynamics

Abstract

A rigorous bridge between spiking-level and macroscopic quantities is an on-going and well-developed story for asynchronously firing neurons, but focus has shifted to include neural populations exhibiting varying synchronous dynamics. Recent literature has used the Ott-Antonsen ansatz (2008) to great effect, allowing a rigorous derivation of an order parameter for large oscillator populations. The ansatz has been successfully applied using several models including networks of Kuramoto oscillators, theta models, and integrate-and-fire neurons, along with many types of network topologies. In the present study, we take a converse approach: given the mean field dynamics of slow synapses, we predict the synchronization properties of finite neural populations. The slow synapse assumption is amenable to averaging theory and the method of multiple timescales. Our proposed theory applies to two heterogeneous populations of excitatory-dimensional and inhibitory-dimensional oscillators with homogeneous synaptic weights. We then demonstrate our theory using two examples. In the first example, we take a network of excitatory and inhibitory theta neurons and consider the case with and without heterogeneous inputs. In the second example, we use Traub models with calcium for the excitatory neurons and Wang-Buzsáki models for the inhibitory neurons. We accurately predict phase drift and phase locking in each example even when the slow synapses exhibit non-trivial mean-field dynamics.

Interplay of different synchronization modes and synaptic plasticity in a system of class I neurons

“…For QIF neurons, the boundaries of ATs were obtained analytically both at fixed synaptic weights and in the presence of STDP. Near resonances T 2 /T 1 ≈ n , where n is a natural number, we generalized these analytical results for an arbitrary class I neuron model and confirmed their correctness on WB 16,17 and ML 18,19 biophysically plausible class I neuron models.…”

“…The QIF neuron is considered as the main model in our study, as it allows obtaining analytical results. In addition, we demonstrate the generality of our results using Wang–Buzsáki (WB) 16 , 17 and Morris–Lecar (ML) 18 , 19 biophysically plausible class I neuron models. Class I neurons generally have a purely positive PRC (also called type I PRC), indicating that perturbations always produce an advance (and not a delay) of their phase 20 .…”

“…The dynamics of neuron’s membrane potentials and ions activation/inactivation variables are governed by the equations 16 , 17 with the following functions We used the parameters: , mV and . At direct current SNIC bifurcation occurs in an isolated ( ) neuron.…”

Bumps and oscillons in networks of spiking neurons

“…The equation above can be cast as a 4D, first-order spatial dynamical system in the vector (u, u , u , u ), which we omit here for brevity. To construct localized solutions to (3) we proceed in the same spirit as [27][28][29][30][31][32]34,35 : to each homogeneous steady state of (3) corresponds one value r j in (8), and hence one value u j in (17), and one constant solution (u j , 0, 0, 0) to (20); in addition, there exists a region in parameter space where r 1 and r 3 coexist and are stable (see also Figure 2a). A localized steady state of (3) is identified with a bounded, sufficiently regular function u : R → R which satisfies (20) with boundary conditions…”

“…A challenge faced in the derivation of neural field models is to establish an accurate mean-field description of the spiking dynamics of the underlying microscopic neural network. Classical neural field models recover the microscopic dynamics only in the limit of slow synapses 9 , and the derivation of neural mass or neural field description from network models of spiking neurons is still an active area of research [10][11][12][13][14][15][16][17] . In addition, in neural fields the network firing rate is not an emergent quantity, but rather the result of a modelling choice.…”